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Fiber reinforced concrete : characterization of flexural toughness & some studies on fiber-matrix bond-slip… Dube, Āśīsha 1999

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FIBER REINFORCED  CONCRETE:  CHARACTERIZATION OF F L E X U R A L TOUGHNESS S O M E STUDIES O N F I B E R - M A T R I X BOND-SLIP INTERACTION by  Ashish Dubey B.Eng., Devi Ahilya University, 1988 M.Eng., Rani Durgavati University, 1991 M . A . S c , The University of British Columbia, 1993  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES (The Department of Civil Engineering)  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A April, 1999 © Ashish Dubey, 1999  &  in  presenting  degree freely  at  this  the  available  copying  of  department publication  of  in  partial  fulfilment  University  of  British  Columbia,  for  this or  thesis  reference  thesis by  this  for  his thesis  and  study.  scholarly  or for  her  purposes  gain  of  Civil  The University of British Vancouver, Canada  DE-6 (2788)  fc  y\chT> ef_9u r v  Columbia  requirements  shall  that  agree  may  representatives.  financial  the  I agree  I further  permission.  Department  of  It not  be is  that  the  Library  permission  granted  by  understood be  for  allowed  an  advanced  shall for  the that  without  make  it  extensive  head  of  my  copying  or  my  written  Abstract  One major problem associated with the testing of fiber reinforced concrete specimens under flexural loading is that the measured post-cracking response is severely affected by the stiffness of the testing machine. As a consequence, misleading results are obtained when such a flexural response is used for the characterization of composite toughness. Unfortunately, many existing standards allow the use of such a flexural response for toughness characterization. As a part of this research program, assessment of a new toughness characterization technique termed the Residual Strength Test Method (RSTM) has been made. In this technique, a stable narrow crack is first created in the specimen by applying a flexural load in parallel with a steel plate under controlled conditions. The plate is then removed, and the specimen is tested in a routine manner in flexure to obtain the post-crack load versus displacement response. Flexural response for a variety of fiber reinforced cementitious composites obtained using the Residual Strength Test Method has been found to correlate very well with those obtained with relatively stiffer test configurations such as closed-loop test machines.  A good agreement between the flexural  response obtained from the aforementioned methods seems to validate the Residual Strength Test Method. This method is simple, and can be carried out easily in any commercial laboratory equipped with a test machine with low stiffness. The Residual Strength Test Method is found to be effective in differentiating between different fiber types, fiber lengths, fiber configurations, fiber volume fractions, fiber geometries and fiber moduli. In particular, the technique has been found to be extremely useful for testing cement-based composites containing fibers ait very low dosages (< 0.5% by volume).  As another major objective of this research program, an analytical model based on shear lag theory is introduced to study the problem of fiber pullout in fiber reinforced composites. The proposed model eliminates limitations of many earlier models and captures essential features of pullout process, including progressive interfacial debonding, Poisson's effect, and variation in interfacial properties during the fiber pullout process. Interfacial debonding is modeled using an interfacial shear strength criterion.  Influence of normal contact stress at the fiber-matrix  interface is considered using shrink-fit theory, and the interfacial frictional shear stress over the ii  debonded interface is modeled using Coulomb's Law. Stresses required to cause initial, partial and complete debonding of the fiber-matrix interface are analyzed, and closed form solutions are derived to predict the complete fiber pullout response. Analysis shows that the initial debonding stress strongly depends upon fiber length and fiber elastic modulus. The process of interfacial debonding turns catastrophic at the instant the fiber pullout stress begins to drop with an increase in debond length. This condition is satisfied when the difference between the change in the frictional component and the adhesional component of pullout stress occurring due to increase in debond length equals to zero. The magnitude of interfacial frictional shear stress along the embedded fiber length is found to vary as a result of the Poisson's contraction of the fiber. Moreover, Poisson's effect manifests itself in the form of a non-linear relationship between the peak pullout stress versus embedded fiber length plot.  Based on energy considerations, an  analytical solution is derived to compute the interfacial coefficient of friction.  This solution  depicts the dependence of the interfacial coefficient of friction on fiber pullout distance.  For  both steel and polypropylene fibers, interfacial coefficient of friction is found to decrease exponentially with increase in pullout distance. Matrix wear resulting from fiber pullout appears to be responsible for the aforementioned physical phenomena. Parametric studies are carried out to investigate the influence of fiber-matrix interfacial properties (adhesional bond shear strength, normal contact stress and coefficient of friction) and elastic modulus of the fiber.  Results  suggest that for a given set of interfacial properties, initial debonding stress, maximum pullout stress, stability of debonding process, catastrophic debond length, interfacial shear stress distribution, and overall pullout response significantly depend upon the elastic modulus of the fiber. Given the fiber elastic modulus, recommendations are made as to how efficiency of fiber in pullout may be improved by modifying different interfacial properties..  iii  Table of Contents  Abstract  ii  Table of Contents  iv  List of Tables  ix  List of Figures  xi  Notations  xx  Acknowledgement  xxiii  1  Introduction  1  1.0 Objectives and Scope  2  1.1 Thesis Organization  3  Literature Review  5  2.0 Introduction  5  2  2.1 Issues of Significance Concerning Characterization of Toughness of Fiber Reinforced Concretes  6  2.2 Micromechanical Models for Investigating the Fiber-Matrix Interfacial Behavior  3  10  2.3 Experimental Studies on the Fiber Pullout Behavior  26  2.4 Mechanical Behavior of Fiber Reinforced Cementitious Composites  30  Macromechanical Behavior of Fiber Reinforced Concrete (FRC) & Measurement of Flexural Response of Low Toughness F R C  37  3.0 Introduction  37 iv  3.1 Material Variables Influencing Macromechanical Behavior and Toughness of F R C  38  3.1.1  Mineral Admixtures  40  3.1.2  Fiber Volume Content  44  3.1.3  Fiber Aspect Ratio  46  3.1.4  Surface Characteristics of Fiber  47  3.1.5  Fiber Geometry  50  3.1.6  Shrinkage Properties of Matrix  51  3.2 Characterization of Flexural Toughness of Fiber Reinforced Concretes 3.2.1  3.2.2  4  53  Residual Strength Test Method (RSTM) - Assessment and Calibration  55  3.2.1.1 Experimental Procedure  57  3.2.1.2 Results  60  3.2.1.3 A Round-Robin Test Program to Validate R S T M  65  3.2.1.4 Discussion  67  Residual Strength Test Method (RSTM) Performance of Various Composites  69  3.2.2.1 Materials, Mixes and Testing  69  3.2.2.2 Results  71  3.2.2.3 Discussion  79  3.3 Conclusions  80  Bond-Slip Performance of Fibers Embedded in Cementitious Matrices  82  4.0 Introduction  82  4.1 Bond-Slip Performance of Fibers - Influence of Pullout Parameters  82  4.1.1  4.1.2  Straight-smooth, Stainless Steel Fibers Embedded in Normal Strength Matrix  84  4.1.1.1 Influence of Fiber Length  85  Straight, Stainless Steel Fibers with Rough Surface V  4.1.3  4.1.4  4.1.5  4.1.6  5  Embedded in Normal Strength Matrix  86  4.1.2.1 Influence of Surface Roughness  87  4.1.2.2 Influence of Fiber Length  88  Straight-smooth, Stainless Steel Fibers Embedded in Non-Shrink Grout Matrix  88  4.1.3.1 Influence of Matrix Shrinkage Behavior  90  4.1.3.2 Influence of Fiber Length  90  Straight-smooth, Stainless Steel Fibers Embedded in High-Strength Matrix  91  4.1.4.1 Influence of Water/Cement Ratio  92  Straight-smooth, Stainless Steel Fibers Embedded in Silica-fume Modified High-Strength Matrix  93  4.1.5.1 Influence of Silica-fume Modification  94  Straight, Smooth Polypropylene Fibers Embedded in Normal Strength Matrix  95  4.1.6.1 Influence of Fiber Length  96  4.2 Conclusions  97  Progressive Debonding Model for Fiber Pullout  99  5.0 Introduction  99  5.1 The Proposed Progressive Debonding Model  103  5.1.1  Stage 1: Fiber Completely Bonded Along Its Entire Embedded Length  104  5.1.1.1 Fiber Axial Stress Distribution, 07  109  5.1.1.2 Interfacial Shear Stress Distribution, x  109  5.1.1.3 Fiber Displacement, Ub  110  a  5.1.1.4 Debonding Criterion and Initial Debonding Stress, Gd  111  vi  5.1.2  Stage 2: Fiber Partially Debonded Along Its Embedded Length  112  5.1.2.1 Fiber Axial Stress Distribution, 07  115  5.1.2.2 Interfacial Frictional Shear Stress Distribution, tf  115  5.1.2.3 Pullout Stress versus Debond Length Relationship, a vs. Id  116  a  5.1.2.4 Fiber Displacement versus Debond Length Relationship, U dvs.ld  116  p  5.1.2.5 Bond and Frictional Components of Pullout Stress, o ,bond and 0  5.1.3  117  <7  0ifric  5.1.2.6 Catastrophic Debonding  118  Stage 3: Fiber Completely Debonded and Pulling Out  119  5.1.3.1 Fiber Axial Stress Distribution, 07  120  5.1.3.2 Interfacial Frictional Shear Stress Distribution, Xj  121  5.1.3.3 Fiber Pullout Stress, a  121  0  5.1.3.4 Pullout Displacement, U  122  pj  5.1.4  6  Calibration of Interfacial Parameters - o , /n and T c  122  s  5.2 Conclusions  125  Progressive Debonding Model for Fiber Pullout: Validation  127  6.0 Introduction  127  6.1 Calibration of Interfacial Properties and Validation of Model  127  6.1.1  Source Code for Progressive Debonding Model  6.1.2  Straight-smooth, Stainless Steel Fiber Embedded in  127  Normal Strength Matrix 6.1.3  128  Straight, Stainless Steel Fibers with Rough Surface Embedded in Normal Strength Matrix  6.1.4  '  130  Straight-smooth, Stainless Steel Fibers Embedded in Non-Shrink Grout Matrix vii  132  6.1.5  6.1.6  7  Straight, Smooth Polypropylene Fibers Embedded in Normal Strength Matrix  135  Pullout Data Found in the Literature  136  6.2 Conclusions  139  Progressive Debonding Model For Fiber Pullout: Parametric Studies  141  7.0 Introduction  141  7.1 Parametric Studies  141  7.1.1  Influence of Adhesional Bond Strength, r  141  7.1.1.1 Fiber Elastic Modulus, E = 210 GPa  141  7.1.1.2 Fiber Elastic Modulus, E = 3.5 GPa  146  Influence of Interfacial Contact Stress, a  151  7.1.2.1 Fiber Elastic Modulus, E = 210 GPa  151  7.1.2.2 Elastic Modulus, E = 3.5 GPa  154  Influence of Interfacial Coefficient of Friction, ji  157  7.1.3.1 Fiber Elastic Modulus, E = 210 GPa  157  7.1.3.2 Elastic Modulus, E = 3.5 GPa  160  s  f  f  7.1.2  c  f  f  7.1.3  f  f  8  7.2 Conclusions  163  Conclusions  166  Bibliography  173  Appendix A  188  Appendix B  193  Appendix C  209 viii  List o f Tables  Table 3.1.1  Mix Proportions  38  Table 3.1.2  Description of fibers investigated  38  Table 3.1.3  Properties of silica fume and high-reactivity metakaoline investigated  Table 3.1.4  41  JSCE Absolute Toughness and Flexural Toughness Factor Influence of Pozzolan  Table 3.1.5  42  JSCE Absolute Toughness and Flexural Toughness Factor Influence of Pozzolan Type  Table 3.1.6  44  JSCE Absolute Toughness and Flexural Toughness Factor Influence of fiber volume  Table 3.1.7  46  JSCE Absolute Toughness and Flexural Toughness Factor Influence of fiber aspect ratio  Table 3.1.8  47  JSCE Absolute Toughness and Flexural Toughness Factor Influence of fiber surface characteristics  Table 3.1.9  JSCE Absolute Toughness and Flexural Toughness Factor Influence of fiber geometry  Table 3.1.10  49 51  JSCE Absolute Toughness and Flexural Toughness Factor Influence of matrix shrinkage  52  Table 3.2.1  Mix Proportions  57  Table 3.2.2  Fresh Properties of Concrete Mixes  58  Table 3.2.3  Residual Strengths, RS for various mixes  63  Table 3.2.4  Test planning  65  Table 3.2.5  Residual Strengths, RS - Canadian Round-Robin Test Program [221]  66  Table 3.2.6  Maximum concrete stress at peak load with and without steel plate [219]  68  Table 3.2.7  Mix Proportions  69  Table 3.2.8  Fibers Investigated in Set 1 and Details of Test Program [220]  70  Table 3.2.9  Fibers investigated in Set 2 [220]  70  Table 3.2.10  Various Mixes Investigated in Set 2 and Details of Test Program [220]  71  Table 3.2.11  Fresh properties of Mixes in Set 2 [220]  76  Table 3.2.12  Compressive strengths (28-Day) for Mixes in Set 1 [220]  76  Table 3.2.13  Compressive strengths (28-Day) for Mixes in Set 2 [220]  76  ix  Table 3.2.14  Detailed results for fibers in Set 1  77  Table 3.2.15  Detailed results for fibers in Set 2  78  Table 4.1.1  Experimental pullout test results for straight-smooth, stainless steel fibers embedded in normal strength matrix  Table 4.1.2  Experimental pullout test results for straight, stainless steel fibers with rough surface embedded in normal strength matrix  Table 4.1.3  95  Various parameters taken into consideration in the theoretical models of fiber pullout  Table 6.1.1  93  Experimental pullout test results for straight, smooth polypropylene fibers embedded in normal strength matrix  Table 5.0.1  91  Experimental pullout test results for straight-smooth, stainless steel fibers embedded in silica-fume modified normal strength matrix  Table 4.1.6  89  Experimental pullout test results for straight-smooth, stainless steel fibers embedded in high strength matrix  Table 4.1.5  86  Experimental pullout test results for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  Table 4.1.4  85  100  Theoretical and experimental peak pullout load and displacement corresponding to peak pullout load for a straight-smooth, stainless steel fiber embedded in normal strength matrix  Table 6.1.2  130  Theoretical and experimental peak pullout load and the displacement corresponding to peak pullout load for a steel fiber with rough surface embedded in normal strength matrix  Table 6.1.3  132  Theoretical and experimental peak pullout load and the displacement corresponding to peak pullout load for a straight-smooth, stainless steel fiber embedded in non-shrink grout matrix  Table 6.1.4  134  Theoretical and experimental peak pullout load and the displacement corresponding to peak pullout load for a straight, smooth polypropylene fiber embedded in normal strength matrix  Table 6.1.5  136  Comparison of peak pullout loads as predicted by (i) the proposed progressive debonding model, and (ii) finite element analysis, Mallikarjuna, et al. [225]  138  List of Figures  Figure 2.2.1  Variation of maximum fiber pullout load with embedded fiber length factor for various friction conditions [55]  Figure 2.2.2  17  Variation of pullout load with fiber displacement for various interfacial frictions conditions [109]  17  Figure 2.2.4  Interfacial bond stress vs. slip relationship [61]  24  Figure 2.4.1  Stress-strain response for a fiber reinforced cementitious composite showing strain hardening response  31  Figure 3.1.1  Test setup for flexural toughness test according to A S T M C 1018 [4]  39  Figure 3.1.2  Comparison of flexural load versus deflection response for fiber reinforced concrete mixes with and without silica fume and containing straight-smooth, stainless steel fiber at a dosage of 0.76%  Figure 3.1.3  42  Comparison of flexural load versus deflection response for fiber reinforced concrete mixes different types of pozzolans  43  Figure 3.1.4a Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber volume, Vf = 0.76% and 5.0% of straight-smooth, stainless steel fiber (STL-STR1)  45  Figure 3.1.4b Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber volume, Vf = 1.0% and 5.0% of straight, smooth polypropylene fibers (PP-STR1)  45  Figure 3.1.4c Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber volume, Vf = 1.0% and 5.0% of straight polyvinyl alcohol fibers (PVA-1) Figure 3.1.5  46  Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber aspect ratio, Vf = 0.50% of straight, smooth polypropylene fibers (PP-STR1 and PP-STR2)  47  Figure 3.1.6a Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber surface characteristics, Steel fiber - STL-STR @ V = 0.76% f  Figure 3.1.6b Comparison of flexural load versus deflection response for fiber  XI  48  reinforced concrete mixes - influence of fiber surface characteristics, Polyvinyl alcohol fiber - PVA-1 @ V = 1.00% f  Figure 3.1.7  Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber geometry  Figure 3.1.8  49  50  Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of shrinkage properties of matrix  Figure 3.2.1  52  Comparison of Closed-loop Test Method with Open-Loop Test Method for Flexural Tests on Fiber Reinforced Concrete. Notice the large load instability that occurs during at the peak load during an open-loop test, and the related damage [6]  Figure 3.2.2a Schematic of Residual Strength Test Method experimental setup  54 56  Figure 3.2.2b Test on a Fiber Reinforced Concrete Beam using the Residual Strength Test Method. Note the steel plate under the beam  58  Figure 3.2.3a Test on a Fiber Reinforced Concrete Beam using the Closed-Loop Test Method  59  Figure 3.2.3b Schematic of controls in a flexural Closed-loop Test Method Figure 3.2.4  Comparison of Closed-loop and Open-loop curves for concrete with 0.1% of Fibrillated Polypropylene Fiber (PP1)  Figure 3.2.5  60  Comparison of Closed-Loop and Open-loop curves for concrete with 0.3% of Monofilament Polypropylene Fiber (PP2)  Figure 3.2.8  60  Comparison of Closed-loop and Open-loop curves for concrete with 0.1 % of Monofilament Polypropylene Fiber (PP2)  Figure 3.2.7  60  Comparison of Closed-Loop and Open-loop curves for concrete with 0.3% of Fibrillated Polypropylene Fiber (PP1)  Figure 3.2.6  60  60  Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.1 % of Fibrillated Polypropylene Fiber (PP1)  Figure 3.2.9  61  Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.3% of Fibrillated Polypropylene Fiber (PP1)  61  Figure 3.2.10 Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.5% of Fibrillated Polypropylene xu  Fiber (PP1) Figure 3.2.11 Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.1% of Monofilament Polypropylene Fiber (PP2) Figure 3.2.12 Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.3% of Monofilament Polypropylene Fiber (PP2) Figure 3.2.13 Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.5% of Monofilament Polypropylene Fiber (PP2) Figure 3.2.14 Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.5% of Monofilament Nylon Fiber (NL1) Figure 3.2.15a Residual Strength for Various Mixes using the Closed-Loop Test Method Figure 3.2.15b Residual Strength for Various Mixes using the Residual Strength Test Method Figure 3.2.16 Percentage Difference (6) in RS Values between the Closed-Loop Test Method and the Residual Strength Test Method Figure 3.2.17 Comparison of Closed-Loop and Open-Loop Curves for Concrete with 0.3% of Fibrillated Polypropylene Fiber (PP1) Figure 3.2.18 Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.3% of Fibrillated Polypropylene Fiber (PP1) Figure 3.2.19a Load-Deflection Curves for FRC with PPF-1 Fiber at Dosage Rate of 0.2% Figure 3.2.19bLoad-Deflection Curves for F R C with PPF-1 Fiber at Dosage Rate of 0.4% Figure 3.2.19c Load-Deflection Curves for F R C with PPF-1 Fiber at Dosage Rate of 0.5% Figure 3.2.19d Load-Deflection Curves for F R C with PPF-1 Fiber at Dosage Rate of 0.6% Figure 3.2.19e Load-Deflection Curves for F R C with PPF-1 Fiber at Dosage Rate of 0.8% xiii  Figure 3.2.20 Load-Deflection Curves for F R C with NLM-1 Fiber at Dosage Rate of 0.4%  73  Figure 3.2.21 Residual Strength Values Plotted as a Function of Fiber Volume for Fibers in Set 1  74  Figure 3.2.22 Load-Deflection Curves for F R C with PP-STR Fiber at a Dosage Rate of 1.6%  75  Figure 3.2.23 Load-Deflection Curves for FRC with a Hybrid Combination of PP-STR Fiber @ 1.6% and Steel Fiber (ST-HKD) @ 0.25%  75  Figure 3.2.24 Load-Deflection Curves for F R C with PVA-1 Fiber at a Dosage Rate of 1.6%  75  Figure 3.2.25 Load-Deflection Curves for F R C with Steel Fiber S T - H K D at a Dosage Rate of 0.76%  79  Figure 3.2.26 Load-Deflection Curves for F R C with Steel Fiber ST-CR1 at a Dosage Rate of 0.76%  79  Figure 4.1.1a Schematic of a standard pullout test specimen Figure 4.1.2  Pullout response of straight-smooth, stainless steel fibers embedded in normal strength matrix  Figure 4.1.3  86  Pullout response of straight, stainless steel fibers with rough surface embedded in normal strength matrix  Figure 4.1.5  85  Embedded fiber length vs. maximum pullout load curve for straight-smooth, stainless steel fibers  Figure 4.1.4  83  87  Comparison of the pullout response of 30 mm long straight-smooth, stainless steel fibers and 30 mm long straight, stainless fibers with rough surface (normal strength matrix)  Figure 4.1.6  Embedded fiber length vs. maximum pullout load curve for straight, stainless steel fibers with rough surface  Figure 4.1.7  89  Pullout response of 30 mm long fibers embedded in normal strength matrix (CSA Type 10 cement) and in non-shrink grout matrix  Figure 4.1.9  88  Pullout response of straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  Figure 4.1.8  87  90  Embedded fiber length vs. maximum pullout load curve for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  91 XIV  Figure 4.1.10 Pullout response of straight-smooth, stainless steel fibers embedded in high strength matrix  92  Figure 4.1.11 Pullout response of straight-smooth, stainless steel fibers embedded in normal strength matrix and high strength matrix  93  Figure 4.1.12 Pullout response of straight-smooth, stainless steel fibers embedded in silica fume modified high strength matrix  94  Figure 4.1.13 Pullout response of straight-smooth, stainless steel fibers embedded in high strength matrix and silica fume modified high strength matrix  94  Figure 4.1.14 Pullout response of straight, smooth polypropylene fibers embedded in normal strength matrix  96  Figure 4.1.15 Embedded fiber length vs. maximum pullout load curve for straight, smooth polypropylene fibers Figure 5.0.1  Single fiber embedded in concrete matrix. Figure depicts one side of a two-sided pullout test specimen  Figure 5.0.2  97  99  A comparison between the experimental pullout response for a steel fiber with the predicted responses using the existing pullout models. The experimental pullout response is by Naaman and Shah [142]  Figure 5.1.1  102  The principle of a single fiber pullout test - The three stages of pullout process (Stang and Shah [106])  104  Figure 5.1.2  Fiber completely bonded over its embedded length  105  Figure 5.1.3  Fiber partially debonded along its embedded length  112  Figure 5.1.4  Free body diagram for a fiber element of length dz  112  Figure 5.1.5  Fiber completely debonded over its length and pulling out  120  Figure 6.1.1  Coefficient of friction versus pullout displacement curves for straight-smooth, stainless steel fibers embedded in normal strength matrix  Figure 6.1.2  129  Comparison of experimental and theoretical pullout response straight-smooth, stainless steel fiber fiber (embedded length - 30 mm) embedded in normal strength matrix  Figure 6.1.3  Coefficient of friction versus pullout displacement curves for straight, stainless steel fibers with rough surface embedded in XV  130  normal strength matrix Figure 6.1.4  131  Comparison of experimental and theoretical pullout response straight, stainless steel fiber with rough surface (embedded length 30 mm) embedded in normal strength matrix  Figure 6.1.5  132  Coefficient of friction versus pullout displacement curves for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  Figure 6.1.6  133  Comparison of experimental and theoretical pullout response straight-smooth, stainless steel fiber (embedded length - 30 mm) embedded in non-shrink grout matrix  Figure 6.1.7  134  Coefficient of friction versus pullout displacement curves for straight, smooth polypropylene fibers embedded in normal strength matrix  Figure 6.1.8  13 5  Comparison of experimental and theoretical pullout response straight, smooth polypropylene fiber (embedded length - 30 mm) embedded in normal strength matrix  Figure 6.1.9  136  Comparison of experimental and theoretical pullout response for a straight, smooth steel fiber. Experimental curve is by Naaman and Shah [ 142]  137  Figure 6.1.10 Comparison of experimental and theoretical pullout response for a straight, smooth polypropylene fiber. Experimental curve is by Wang, L i and Backer [69]  139  Figure 7.1.1a Influence of adhesional bond strength on pullout response (E/=210 GPa)  142  Figure 7.1.1b Influence of adhesional bond strength on the pullout load at initial debonding and the peak pullout load (Zs/=210 GPa)  144  Figure 7.1.1c Influence of adhesional bond strength on displacement at the peak pullout load (Ej=210 GPa)  144  Figure 7.1.1 d Variation in pullout load and pullout load components as a function of debond length (Ef=210 GPa)  144  Figure 7.1.1 e Variation in pullout load and its components as a function of debond length at different values of adhesional bond strength, T (£y=210GPa)  145  S  XVI  Figure 7.1 .If Axial load distribution in fiber at initial debonding (£/=210 GPa)  145  Figure 7.1.1g Interfacial shear stress distribution at initial debonding (£)=210 GPa)  145  Figure 7.1.1 h Axial load distribution at completion of debonding (Ef=210 GPa)  146  Figure 7.1. Ii Interfacial shear stress distribution at completion of debonding (£p210GPa)  146  Figure 7.1.2a Influence of adhesional bond strength on pullout response (£/=3.5 GPa)  147  Figure 7.1.2b Influence of adhesional bond strength on pullout load at initial debonding and maximum pullout load (£)=3.5 GPa)  147  Figure 7.1.2c Influence of adhesional bond strength on pullout displacement corresponding to peak pullout load (£/=3.5 GPa)  147  Figure 7.1.2d Variation in pullout load and pullout load components as a function of debond length (Ej=3.5 GPa)  149  Figure 7.1.2e Variation in pullout load and its components as a function of debond length at different values of adhesional bond strength, x  s  (£y=3.5 GPa)  149  Figure 7.1.2f Fiber Axial load distribution in fiber at initial debonding (£,=3.5 GPa)  150  Figure 7.1.2g Interfacial shear stress distribution at initial debonding (£,=3.5 GPa)  150  Figure 7.1.2h Fiber axial load distribution at completion of debonding, (£)=3.5 GPa)  150  Figure 7.1.2i Interfacial shear stress distribution at completion of debonding, (£/=3.5GPa)  150  Figure 7.1.3a Influence of interfacial contact stress on pullout response (£,=210 GPa)  151  Figure 7.1.3b Influence of interfacial contact stress on pullout response (£,=210 GPa)  151  Figure 7.1.3c Influence of interfacial contact stress on pullout load at initial debonding and peak pullout load (£/=210 GPa)  152  Figure 7.1.3d Influence of interfacial contact stress on pullout displacement at the peak pullout load (£p210 GPa) Figure 7.1.3e Variation in pullout load and its components as a function of XVII  152  debond length (£/=210 GPa) Figure 7.1.3f Fiber axial load distribution at completion of debonding (£,=210 GPa) Figure 7.1.3g Interfacial shear stress distribution at completion of debonding (£,=210 GPa) Figure 7.1.4a Influence of interfacial contact stress on prepeak pullout response (£,=3.5 GPa) Figure 7.1.4b Influence of interfacial contact stress on pullout response (£/=3.5 GPa) Figure 7.1.4c Influence of interfacial contact stress on pullout load at initial debonding and peak pullout load (£/=3.5 GPa) Figure 7.1.4d Influence of interfacial contact stress on pullout displacement at the peak pullout load (£/=3.5 GPa) Figure 7.1.4e Variation in pullout load and its components as a function of debond length (Ej=3.5 GPa) Figure 7.1.4f Fiber axial load distribution at completion of debonding (Ej=3.5 GPa) Figure 7.1.4g Interfacial shear stress distribution at completion of debonding (£/=3.5 GPa) Figure 7.1.5a Influence of interfacial coefficient of friction on pullout response (£/=210 GPa) Figure 7.1.5b Influence of interfacial coefficient of friction on pullout response (£,=210 GPa) Figure 7.1.5c Influence of interfacial coefficient of friction on pullout load at initial debonding and peak pullout load (£/=210 GPa) Figure 7.1.5d Influence of interfacial coefficient of friction on pullout displacement at the peak pullout load (£/=210 GPa) Figure 7.1.5e Variation in pullout load and its components as a function of debond length (£,=210 GPa) Figure 7.1.5f Fiber axial load distribution at completion of debonding (£,=210 GPa) Figure 7.1.5g Interfacial shear stress distribution at completion of debonding (£,=210 GPa) xviii  Figure 7.1.6a Influence of interfacial coefficient of friction on prepeak pullout response (E/=3.5 GPa) Figure 7.1.6b Influence of interfacial coefficient of friction on pullout response (Ef=3.5 GPa) Figure 7.1.6c Influence of interfacial coefficient of friction on pullout load at initial debonding and peak pullout load (Ef=3.5 GPa) Figure 7.1.6d Influence of interfacial coefficient of friction on pullout displacement at the peak pullout load (E/=3.5 GPa) Figure 7.1.6e Variation in pullout load and its components as a function of debond length (£>=210 GPa) Figure 7.1.6f Fiber axial load distribution at completion of debonding  (Ej=3.5 GPa) Figure 7.1.6g Interfacial shear stress distribution at completion of debonding (£/=3.5 GPa)  xix  Notations  n  -  modulus ratio between concrete and steel  b  -  width of concrete beam  d  -  depth of concrete beam  L  -  length of concrete beam  5  -  span of concrete beam  8150  -  beam deflection of magnitude S/150  Tja  -  JSCE Absolute Toughness  Gb  -  JSCE Flexural Toughness Factor  dp  -  depth of the steel plate  y  -  depth of neutral axis  M  -  bending moment at cracking  o  -  stress carried by the beam at cracking  s  -  stress carried by steel plate at the instant beam cracks  P  cr  -  cracking load of beam  8  cr  -  deflection at cracking load of beam  a  -  fiber radius  b  -  outer radius of the matrix coaxial cylinder in a pullout test geometry  r  -  radial direction in a pullout specimen  z  -  axial fiber direction in a pullout specimen  L  -  embedded fiber length in a pullout specimen  -  matrix shear stress  r  a  -  interfacial shear stress  Ts  -  fiber-matrix interfacial shear strength  Tf  -  interfacial shear stress over the debonded interface  w  -  matrix axial displacement  w  a  -  matrix displacement at the interface (i.e., r=d)  M>b  -  matrix displacement at the surface of coaxial cylinder (i.e., at r=b)  <y  -  fiber pullout stress  Of  -  fiber axial stress  c  c  o  T  m  m  0  XX  G  -  matrix axial stress  G  -  matrix axial stress at the interface (i.e., r=a)  m  a  Go.bond -  bond component of pullout stress  Oo.fric -  frictional component of pullout stress  Go.peak -  fiber peak pullout stress  Gb  -  matrix axial stress at the surface of coaxial cylinder (i.e., at r=b)  G  -  contact stress at the fiber-matrix interface  G  -  contact stress at fiber-matrix interface after Poisson's contraction of fiber  Gd  -  fiber debonding stress  8  -  fiber-matrix misfit  \i  -  interfacial coefficient of friction  jl  -  initial value of the interfacial coefficient of friction  ji  -  steady-state value of the interfacial coefficient of friction  c  -  rate at which the interfacial coefficient of friction decays with increase in p  A  -  pullout displacement  Apeak  -  pullout displacement corresponding to peak pullout load  Aa  -  decrease in fiber radius due to Poisson's contraction of fiber  Id  -  interfacial debond length  ld,cat  -  catastrophic debond length of interface  Pd  -  rigid body displacement of fiber in a pullout test  Pdi  -  rigid body displacement of the fiber in a pullout test  Pd2  -  rigid body displacement of the fiber in a pullout test  Ppdi  -  pullout load corresponding to the pullout distance, pdi  P d2  -  pullout load corresponding to the pullout distance, pd2  Ub  -  fiber displacement when fiber is completely bonded  Ud  -  fiber displacement during partial interfacial debonding  c  cp  r  t  ss  p  p  Updpeak -  d  fiber displacement corresponding to peak pullout load fiber displacement beyond completion of interfacial debonding  W  p  -  Work of fiber pullout when the rigid body displacement of the fiber increases from pdi to p 2 d  v  m  -  Poisson ratio of matrix material  xxi  Vf  -  Poisson's ratio of fiber material  E  m  -  matrix elastic modulus  Ef  -  fiber elastic modulus  MOR -  Modulus of Rupture  RS  -  Residual Strength  RSI  -  Residual Strength Index  xxii  Acknowledgement  I extend my deepest gratitude to my research supervisor, Dr. N . Banthia for providing the opportunity to work toward this research program, and for his continued advice, support and encouragement throughout the program. In particular, the freedom he allowed me during the course of this research greatly helped me to nurture research and independent thinking skills. I remain indebted for his direction, pragmatism and trust.  I am also profoundly grateful to Mr. Rick Hingson of Synthetic Industries, Chatanooga, USA, for his personal initiative and invaluable guidance.  His involvement undoubtedly enabled  blossoming of this research. Especial thanks are also due to Mr. Cheng Yan for his dynamism and help during my research. Thanks are also extended to Dr. Sydney Mindess, Dr. Reza Vaziri, and Dr. Tom Troczynski for the guidance they provided from time to time. I am also grateful to Mr. Dennis Teo for performing a part of the experimental work, and to all personnel of the Department of Civil Engineering, including, Mr. Doug Smith, Mr. John Wong, Mr. Max Nazar and Mr. Dick Postgate, who in one way or another contributed to the success of this study. Furthermore, I am grateful to all my colleagues at U B C who made the atmosphere quite enriching. Thanks are also due to researchers around the globe from whom I have benefited immeasurably either through their writings or through personal contacts.  I extend my deepest appreciation to Anima and Nandu for being there during the toughest moments and for sharing with me the joyous moments. Without them, my work would not have been as encouraging and fulfilling as it has been. Last but not least, I remain obligated to every being on this planet and to the mother-nature.  x x i i  Chapter 1  Introduction  Plain concrete is a brittle material characterized by its low tensile strength, low tensile strain capacity and low fracture toughness. To render concrete a usable structural material, it has to be reinforced.  Historically, concrete reinforcements have been in the form of continuous steel  reinforcing bars and stirrups, which are placed strategically in the structure to withstand the imposed tensile and shear stresses. In the last couple of decades, another type of concrete reinforcement that has gained immense popularity is discontinuous short fibers [1-3]. Continuous bars and discontinuous short fibers perform different roles in concrete.  While  conventional reinforcing bars are used to increase the tensile and shear capacities of concrete, the primary reason for inclusion of discontinuous short fibers in the concrete matrix is to improve its post-cracking response (i.e., to augment apparent ductility, energy absorption capacity, impact resistance and cracking resistance of concrete).  Because of its superior energy absorption  capability, fiber reinforced concrete has been widely used in a variety of civil engineering applications including thin sheet products, large precast products, shotcrete, offshore structures, seismic structures, pavements, overlays, crash barriers, machine foundations, hydraulic structures and thin and thick repairs.  While the influence of fibers on the pre-cracking behavior is only marginal, the potency of fiber addition becomes apparent after the brittle concrete matrix has cracked. In the post-cracking stage, fibers that bridge a crack apply closing pressure at the crack front.  Closing pressure  suppresses the tendency of crack growth by reducing the stress intensity at the crack tip. In this context, efficiency of fibers in suppressing crack growth is directly dependent upon the magnitude and the nature of the closing pressure field generated due to the interaction between fiber and matrix in the wake region of the crack tip. With the opening of the crack, fibers pull out of the matrix and absorb energy in the process.  This energy that is absorbed at the  micromechanical level in the fiber pullout process appears at the macromechanical level as toughness of the fiber reinforced composite.  In this context, the significance of interfacial 1  micromechanics becomes apparent, given the dependence of both the the fiber-matrix interaction and macromechanical composite properties on the micromechanical properties of the fibermatrix interface. Despite numerous theoretical and experimental studies on the micromechanics of interfaces that have been carried out, it is not clear as to how micromechanical properties of an interface must be tailored to improve fiber pullout efficiency. It is also of utmost importance to understand the combined influence of fiber elastic modulus and interfacial properties on fiber pullout response, since fibers with a wide variety of elastic moduli are commonly employed to produce fiber reinforced cementitious composites. The aforementioned aspects are explored as part of this research program. Ductility or the energy absorption capability of fiber reinforced concrete is commonly measured by loading a specimen in flexure and measuring the corresponding load versus displacement response. The area under the load versus displacement response represents the flexural toughness of the composite. Some testing standards for characterizing flexural toughness that fall under this category include ASTM C1018 [4] and JSCE-SF4 [5]. Although not widely recognized, one major problem that is associated with the testing of a specimen under flexural loading (for instance, as per ASTM C1018 or JSCE SF-4) is that the measured post-cracking response is severely affected by the stiffness of the testing machine [6]. For a given testing system, the severity of this effect is more pronounced at low fiber dosage rates and when fibers with a low modulus are used. To accurately characterize the flexural behavior of fiber reinforced cementitious composites, a new test method called the Residual Strength Test Method (RSTM) has recently been adopted as a testing standard by the American Society of Testing and Materials [7]. Another major objective this research program is to investigate the validity of the Residual Strength Test Method. 1.1 Objectives and Scope  The present work on fiber reinforced concrete is categorized under the following two categories: macromechanical studies, and micromechanical studies  2  The objectives that fall under the category of macromechanical studies are: to examine the influence of machine stiffness on the measured flexural load versus deflection response of fiber reinforced concretes (in particular, fiber reinforced concretes with low toughness); •  to assess the validity of the newly developed Residual Strength Test Method (RSTM) for the 1  measurement of the post cracking response of fiber reinforced concrete; to investigate the influence of various material variables on the flexural toughness of fiber reinforced concrete.  The objectives that fall under the category of micromechanical studies are: to examine the influence of various pullout parameters on the fiber pullout responsei; •  to develop a fiber pullout model in order to understand the mechanics of interaction between the fiber and the cement based matrices, and to understand the progressive debonding behavior of the fiber-matrix interfaces;  •  to conduct parametric studies using the developed fiber pullout model from the viewpoint of understanding the influence of the fiber-matrix interfacial properties and the mechanical properties of the constituents on the fiber pullout response.  1.2 Thesis Organization The pertinent literature is reviewed in Chapter 2. The influence of various material variables on the flexural toughness of F R C is described in Chapter 3. In addition, this chapter explores the influence of machine stiffness on the flexural load versus deflection response of FRC. Two testing schemes are used for this purpose - an open-loop test scheme and a closed-loop test scheme. The stiffness of the former testing machine is lower than that of the latter. Also, in the latter test scheme, a feedback signal from the beam deflection or the crack opening displacement is utilized in order to crack the beam at a controlled rate.  Furthermore, in Chapter 3, an  assessment of the Residual Strength Test Method is made.  Apart from the intensive  investigations carried out in this context at the University of British Columbia, a Round Robin Test Program was accomplished between the Ministry of Transportation of Ontario, the Technical University of Nova Scotia and the University of British Columbia. Results from this test program are also reported in this chapter.  Residual Strength Test Method (RSTM) is now an American Society of Testing & Materials ( A S T M ) test standard - A S T M 1399-98 [7]. 1  3  The influence of various pullout parameters on the fiber pullout response is described in Chapter 4.  Pullout parameters investigated include water/binder ratio, matrix shrinkage properties,  matrix modification by pozzolans, fiber length and fiber surface roughness. Pullout tests were carried out on fibers with different elastic moduli. To understand the mechanics of interaction between the fiber and the matrix, an analytical model for the problem of fiber pullout is developed and presented in Chapter 5. Validation of the proposed model is carried out in Chapter 6. Further, parametric studies are carried out using the proposed pullout model to study the influence of the fiber-matrix interfacial properties and the mechanical properties of the constituents on the fiber pullout response. Results form the aforementioned parametric studies are presented in Chapter 7. Finally, Chapter 8 summarizes the important conclusion drawn from this work.  4  Chapter 2  Literature Review  2.0 Introduction The fundamental reason behind the addition of discontinuous fibers to cementitious materials is to suppress the inherent tendency of the brittle matrix to crack at low strains.  While  conventional reinforcing bars increase the tensile and shear capacities of concrete, discontinuous fibers, when added to a concrete matrix, enhance the material's ability to deform and absorb energy. Thus, toughness of fiber reinforced concrete is a measure of the energy absorption capability (i.e., deformability) of the composite and its ability to resist fracture when subjected to applied loads. There are several approaches that may be adopted to evaluate the toughness of fiber reinforced concrete.  The primary difference between these approaches stems from the  characteristics of the applied load on the structure. These characteristics include: i.  mode of loading: tension, compression, or flexure  ii.  rate of loading: static, dynamic, or impact  Flexural toughness under static loading is most commonly used to characterize the toughness of fiber reinforced cementitious composites. Some flexural toughness testing standards that fall under this category include A S T M C1018 [4] and JSCE SF-4 [5]. A brief literature review on the characterization of flexural toughness of fiber reinforced concretes is presented in this chapter.  While the influence of fibers on the pre-cracking behavior is only marginal, the potency of fiber addition becomes apparent only after concrete has cracked. In the post-cracking stage, fibers that bridge a crack apply a closing pressure at the crack front. This tends to close the crack and suppresses the tendency of crack growth by reducing the stress intensity at the crack tip. In this context, efficiency of fibers in suppressing crack growth is directly dependent upon the magnitude and the nature of the closing pressure field resulting from the interaction between fiber and matrix. Given the dependence of composite response on interfacial micromechanical properties, it becomes critical to tailor the interfacial micromechanical properties for achieving 5  the desired composite properties. For this reason, experimental and theoretical investigations on the micromechanics of interfaces actively continue. Several experimental test methods have been developed for the characterization of the fiber-matrix interfacial properties, such as the single fiber pullout test, microdebond test, single fiber fragment test, and fiber pushout test (also known as indentation test) [8]. Among these, the single fiber pullout test is most commonly employed in the field of fiber reinforced concrete. Yet another objective of this chapter is to present a literature review on the topics of micromechanical and macromechanical properties of fiber reinforced cementitious composites. First, the relevant micromechanical models for the problem of fiber pullout are reviewed. This is followed by a review of important experimental fiber pullout studies carried out to this date.  Finally, a brief review of important  micromechanical models describing the tensile behavior of fiber reinforced cementitious composites is presented.  2.1  Issues O f Significance Concerning Characterization of Toughness of Fiber Reinforced Concretes  Several definitions of toughness, all in the form of energy absorption, have been developed to quantify the toughness of fiber reinforced cementitious composites under static loading. For instance, flexural toughness, compressive toughness and tensile toughness are some definitions, which correspond to toughness obtained through flexural, compressive and tensile tests, respectively. In the following, flexural toughness is discussed in detail, as it is one of the most common loading states for fiber reinforced concrete applications.  Extensive work [4,5,9-51] has been done towards the characterization of flexural toughness of fiber reinforced concretes, and many different test methods have been developed for this purpose.  Most of these test methods use a static third-point bending configuration, and the  energy absorbed by the specimen is computed from the area under the load-deflection curve. In addition, several different toughness definitions and descriptions exist in various test standards and in the literature. Kasperkiewicz and Skarendahl [9] have classified the various toughness characterization methods into two categories: i.  Absolute value descriptions  ii.  Relative value descriptions  6  Methods based on an 'absolute value description' measure the total energy required either to fracture a specimen of arbitrary size, or to load the specimen to a particular end-point deflection. Thus, an absolute description of toughness does not use any ideal material as a reference. Methods that fall under this category include: Japanese Society of Civil Engineers, JSCE - SF4 [5], Dutch CUR-Recommendation N° 10 [10], Belgian Standard N B N B15-238 [11], Spanish A E N O R U N E 83-510-89 [12], German D B V Recommendations [13-15], RJEEM Energy Absorption [16].  As an example of the absolute approach, the Japanese Society of Civil  Engineers in its standard "Method of Test for Flexural Strength and Flexural Toughness of Fiber Reinforced Concrete - JSCE-SF4 [5]" defines a toughness parameter called the Absolute Toughness,  TJCI  as the area under the load-deflection curve to a deflection of loading span/150.  Methods based on a 'relative value description' normalize the energy absorbed up to a specified deflection by the energy absorbed up to approximately the elastic limit of the material, or by the energy absorbed in fracturing the unreinforced matrix. Methods falling under this category include: A S T M C1018 [4], A C I Committee 544 Recommendations [17], Barr Indices [18-22], French Standard PI8-409 [23], Wang and Backer Index [24], Ward and L i Indices [25], and Shah et al. [26]. A n example of the relative approach is A S T M C1018 standard titled "Standard test method for flexural toughness and first-crack strength of fiber reinforced concrete (using beam with third-point loading)." Toughness parameters of the A S T M C1018 [4] technique are based on determining the amount of energy required first to deflect and crack a fiber reinforced concrete beam loaded at its third-points and then to selected multiples of the first crack deflection. Toughness Indices, I5, I , 1 2 0 . I30 and LJO are then calculated by taking the ratios of 10  the energy absorbed to a certain multiple of first crack deflection and the energy consumed up to the occurrence of the first crack itself.  In addition to the above definitions there exist several other toughness characterization schemes that do not fall in the aforementioned categories. For example, Mobasher and Shah [27] define the fracture toughness, Gf as the area under the load-deflection response (extrapolated to the value of zero load), divided by the net cross-sectional area of the specimen. A few other proposed methods describe the toughness in the form of equivalent flexural strength, for instance, Banthia and Trottier [28] quantify toughness by a parameter called 'post crack strength',  which is evaluated at various deflections.  Similarly, Norwegian Concrete  Association's guidelines for shotcrete, N B P No 7 [29], describe the toughness parameters in the 7  form of residual flexural strength at specified deflections. According to the Norwegian NBP No 7 guidelines, toughness properties of shotcrete are determined according to four different classes - the two highest classes (Class 2 and Class 3) involve determination of: •  residual flexural strength from the load-deflection curve (residual flexural stress at fixed deflections of 1 and 3 mm, respectively)  •  flexural strength  •  minimum fiber content as a function of fiber length and material.  The E F N A R C [30] recommendation also uses a toughness classification similar to that of the Norwegian N B P No 7.  As can be noted, there are several definitions of flexural toughness and several test standards to characterize the flexural toughness of fiber reinforced concretes. As a result of the multitude of definitions and standards, there is considerable controversy amongst the users of FRC about the best way to characterize flexural toughness. Both approaches (absolute and relative) provide important information about the mechanical properties of fiber reinforced concretes, but neither approach is sufficient. For instance, it is possible for two materials to have different mechanical performances but still have the same value of absolute toughness. Similarly, in context of the relative approaches, it is possible for two materials that are mechanically different to have the same A S T M C1018 toughness indices.  Several investigators [24,41,42,44,45,47,49,51] have expressed their doubts about the definition and determinations of first crack according to A S T M C1018. Doubts have also been cast about the measurement of net beam deflection and instability after the peak load for materials with low toughness, which could lead to substantial errors in the evaluation of toughness parameters. Gopalaratnam et al. [42] and Trottier and Banthia [45] pointed out that the toughness indices were not very sensitive in distinguishing amongst different fiber contents. Johnston [32-34] on the contrary, found the reverse to be true. Balaguru et al. [49] reported that A S T M indices I5 and I10 were not good indicators of the variation present in the load-deflection response, and recommended that indices such as I100 occurring at much higher deflections, rather than I5 and I10, should be used for the toughness evaluation of FRC.  Some issues of significance pertinent to toughness testing and toughness characterization are summarized below:  i.  Deflection measuring procedure  Large inaccuracies in toughness computation may result if beam deflections are not. measured accurately. Large extraneous deformations result due to local crushing of the test beam at the supports, the elastic and inelastic deformations of the loading fixtures, and deformations in the support system. Note that, up to first crack, extraneous deformations contribute most of the measured deflections, and after the first crack, extraneous deformation constitute only a small fraction of the total beam deflection. For characterizing toughness accurately, these extraneous deflections should be excluded from the deflection measurements. Further details on this issue can be found in Reference 42.  ii.  Specimen Geometry and Size Effect  It is well established that the strength of concrete structures is dependent on the size of the structure.  Both strength (also, first-crack strength) and post-peak ductility decrease with an  increase in structural size. Consequently, toughness parameters are also affected by the size of the specimen and hence are size dependent.  The extremely small ratio of shear-span to beam depth ratio as specified by most Standards causes high shear stresses similar to those in deep beams and walls. pronounced influence of shear on the test results.  This could lead to a  Furthermore, this state of stress is  unrepresentative of the most common applications of fiber reinforced concrete, for example, tunnel linings and slabs-on-grade [41].  iii.  Stiffness of testing machine  During testing, the loading system deforms and stores energy. At first crack or at the peak load, part of this energy is suddenly released and absorbed by the specimen. For concretes with low fiber content, this normally causes a sudden instability in the load-deformation response, which consequently affects the toughness evaluation. The degree of this instability depends on the relative stiffness of the loading system, with softer loading systems releasing larger amounts of energy. Banthia and Trottier [44] have pointed out the effects of instability on the measurement of A S T M Toughness Indices.  9  iv.  Identification of first crack  In the ASTM C1018 method, the indices depend upon the energy absorbed by the specimen up to the occurrence of the first crack; therefore, it is very important to locate the first crack exactly. ASTM C1018 identifies the first crack on the load-deflection curve as "the point of the loaddeflection curve at which the form of the curve first becomes nonlinear". Note that, according to this definition, objectively locating the point of first crack in the ascending part of the loaddeflection response is not possible and is always subject to human-judgement error. This becomes an even bigger problem for beams that exhibit post-cracking strengthening. Thus the need is to establish an objective definition of first crack for fiber reinforced concretes with different volume fractions of fibers.  The following recommendations were made by Gopalaratnam and Gettu [41] with regard to the characterization of flexural toughness of FRC using the four-point bending test: •  Avoid the use of first-crack deflection to define toughness. If first crack is needed it should be objectively defined.  •  Use shallow beams (1/d ratios > 5) to minimize structural effects and large shear spans to minimize the effect of shear stresses.  •  Ensure stability of the test at all times through the use of a stiff machine and/or a servocontrolled machine.  •  Use absolute energy and associated equivalent flexural strength at prescribed deflection limits for a standard beam as sensitive measures of toughness.  •  Use deflection limits that are related to specimen size.  In addition, Gopalaratnam and Gettu [41] recommended the use of the equivalent post-cracking approach for incorporating energy absorption of FRC in structural design. Further, they suggest that this approach should be coupled with serviceability-related and application specific limits on deflection or crack width.  2.2  M i c r o m e c h a n i c a l Models for Investigating the F i b e r - M a t r i x Interfacial Behavior  Depending on the choice of criterion that is used for fiber-matrix interfacial debonding, the theoretical analysis of the problem of fiber pullout can be classified into two distinct approaches: strength based approach [52-78] and fracture mechanics based approach [78-99]. Theoretical 10  models based on the former approach use maximum interfacial shear stress as the interfacial debonding criterion such that when the interfacial shear stress exceeds the interfacial bond strength, debonding is supposed to occur. On the other hand, if theoretical models based on the concepts of fracture mechanics, the debonded zone is considered as an interfacial crack, and the extension of the crack is dependent on the energy criterion being satisfied. Cox [52] developed the first strength-based analytical model to describe the transfer of stress between fiber and matrix.  This model assumes that the tensile stresses in the matrix are  negligible relative to those in the fiber and the shear stresses in the fiber are small compared to those in the matrix. Assuming compatibility of fiber and matrix displacements at the interface, 1  i.e., no slip, Cox [52] derived analytical expressions for the axial stress distribution in the fiber and the shear stress distribution at the interface. The analytical model by Greszczuk [53] was also based on similar assumptions. Greszczuk [53] further postulated that at the instant when the shear strength of the interface was attained, catastrophic debonding would occur over the entire embedded length of fiber. However, in reality, debonding may be limited to the zone in which the elastic shear stress exceeds the adhesional shear bond strength, and in that scenario, the process of load transfer will be comprised of frictional shear transfer at the debonded zone and elastic shear transfer over the remaining length of fiber. Greszczuk's model [53] did not include the possibility of the existence of frictional bond, which constituted a major limitation of the model. The combined stress transfer mechanism was first treated analytically by Lawrence [55]. In this model, interfacial frictional shear stresses over the entire debonded zone were assumed to remain constant.  Models developed by Gopalratnam and Shah [55], Nammur.et al. [61],  Gopalaratnam and Cheng [65], Stang et al. [78] also took into account the combined stress transfer mechanisms.  It is apparent that the shear stresses (both elastic and frictional) that  develop parallel to the fiber-matrix interface are of utmost importance in controlling the fibermatrix stress transfer mechanism. However, stresses and strains may also develop normal to the fiber-matrix interface as a result of the Poisson effect, volume changes, and multiaxial loading. They may induce considerable variations in the resistance to frictional slip, which is sensitive to normal stress.  A comprehensive approach to the stress-transfer problem therefore requires  simultaneous treatment of all the above-mentioned effects, namely, elastic shear transfer, frictional slip, debonding and normal stresses and strains.  Analytical models developed by  Takaku and Arridge [54] and Hsueh [66-68] are more comprehensive than the previously cited  1  Models based on this assumption are now known as Shear-lag Models. 11  since these models take into consideration influence of Poisson's contraction of the fiber on the pullout response. The fracture mechanics approach is characterized by the assumption that the propagation of the debonding zone requires a certain amount of energy, and that this energy is characteristic of the bond between the fiber and the matrix [106]. Thus, the debonding energy is proportional to the magnitude of the increase of the debonded zone. Based on the concepts of fracture mechanics, the total energy release rate, G, can be written as [106]: dW G  =  dW  dW  f  ^lex__^21e_ da  da  L  2.1  da  where, a is the increase in debonded area due to extension of the crack, W is the work done by ex  external load, W is the elastic strain energy stored in the system, and Wf is the energy dissipated e  in inelastic parts of the structure, e.g., work done by friction on the debonded interface. In fiber pullout problems it is often assumed that the critical energy release rate corresponds to the mode II toughness of the interface, G" ", i.e., the contribution of mode I to the energy release rate is r  insignificant. Thus, with this assumption, the criterion for interfacial debonding is given by equating the energy release rate with the critical energy release rate G-,"' of the interface: G™' =G  2.2  Bowling and Groves [87], Stang and Shah [64, 78], Gao et al. [92], Morrison et al. [85], Zhou et al. [89], and Kim et al. [88] have proposed fiber pullout models based on the concepts of fracture mechanics. A review of some important fiber pullout models follows. Greszczuk [53] Greszczuk [53] assumed that the fiber load was transferred to the matrix through the elastic shear stresses only. Further, it was assumed that, when the magnitude of this elastic shear stress exceeded the matrix shear strength at the location where fibers entered the matrix, catastrophic debonding took place along the fiber length. He proposed a method to determine the shear strength and shear modulus of the fiber. This method is based on the pullout test on fibers that 12  are embedded in the matrix to various lengths. Based on the shear lag theory, the final solution for the interfacial shear stress at a point located at a distance x from where the fiber enters the matrix in terms of the average shear stress T  T(  a v  = P llmrl  was derived as:  X)  = a/{sinh(rxc) - coth(a/) cosh(cec)}  2.3  -11/2  where,  a=  2G,  2.4  b,rE,  and, G, = shear modulus of the matrix bi = effective thickness of the interface r = fiber radius Ef = fiber modulus From the above equation, the maximum shear stress would occur at x = 0, hence:  ^  = cd{coth(od)}  2.5  According to the above equation, as od—>0, the limit x /x -4 max  av  1, i.e., x x —> t . Greszczuk ma  av  proposed that this condition can be used to determine the shear strength of the interface. Since, in the above equation, a is a constant, therefore,  T  m a x  /T  a v  would be a function of embedded fiber  length only. Thus, by conducting pullout tests on fibers that are embedded in the matrix to various lengths, the shear strength, x versus /, and extrapolating i av  a v  T  m a x  ,  of the interface can be estimated by plotting a curve of  at / = 0.  As mentioned above, for a given fiber and matrix type, the value of interfacial shear strength is established. Now, for a fiber with a given embedded length, the average shear stress T obtained from the experimental pullout test result, and thereafter the ratio  i /T a v  m a  x  a v  T  m a x  can be  be evaluated.  Using Equation 2.5, a value of cd that corresponds to the calculated ratio x /x av  max  can be  computed. With length of the fiber / and cd known, the value of a can be computed. Finally, using Equation 2.4, the shear modulus G, of the interface can be calculated. Here, Equation 2.4 assumes that the thickness of the interface £>, is known. 13  Lawrence [55] The model developed by Lawrence [55] includes the effects of both the interfacial elastic shear stresses and the frictional shear stresses, and it recognizes the conditions for either a gradual, or an instantaneous debonding of the interface. He has shown that the form of the distribution of the shear stress and the load along the fiber length depends upon the elastic properties of constituents and the fiber embedded length. Lawrence [55] extended the theory developed by Greszczuk [53] by taking into account the process of progressive debonding of the fiber-matrix interface. He suggested that the maximum fiber pullout load would occur at the instant when debonding of that part of fiber length where the elastic bond is still intact takes place in a catastrophic manner. Lawrence [55] derived the following equation for the distribution of load P(x) along the fiber length:  „. . „ sinh-y/a x  P(x)=P  f  2.6  ?i=—  sinhVtt l  e  where, Pf is the fiber pullout load, x is the distance of the point under consideration measured from the embedded fiber end, l is the embedded fiber length, and a is an elastic constant given e  by:  a = CK\  1  1  AE  A E  f  j  f  j  2.7  m m  where, C is the fiber perimeter, Af and A are the areas of the fiber and the matrix, respectively, m  Ef and E are the elastic modulus of the fiber and the matrix, respectively. K is a constant that m  relates the interfacial elastic shear stress Twith the virtual displacement (u-v) of the interface by the equation r=K (u-v), where u is the virtual displacement in the direction of the fiber at a point on the fiber located at a distance x from the embedded fiber end, and v is the virtual displacement of the matrix at the same point, if the fiber was replaced by the matrix. Based on the Equation 2.6, Lawrence pointed out that the hyperbolic sine function is approximately linear for small values and exponential at large values.  As a result, the  distribution of the relative load P(x)/Pf is affected by the embedded fiber length l . Accordingly, e  14  for fibers with small lengths, the buildup of load along the fiber length is approximately linear, however, this is not the case for the fibers with longer lengths. Further, Lawrence derived the following equation for the distribution of shear stress T(X) along the embedded fiber length:  T(X) =  KP cosh ~J~a x  1  f  1  2.8  •Jot sinhVa l A E f  e  f  Following Equation 2.8, the maximum shear stress occurs at the point where the fiber enters the matrix, and is given by:  EL  cothVa I  1 AE f  f  1  2.9  AE m  m  The above equation also depicts that for a given value of a, the distribution of the relative shear stress along the embedded fiber length is influenced by the fiber length. Lawrence assumed that it was possible for the fiber-matrix interface to partially debond, rather than the catastrophic debonding assumed by Greszczuk [53]. Further, in his analysis he took into consideration the existence of frictional shear stresses T, over the debonded fiber length. According to Equation 2.9, when the fiber load Pf is such that the interfacial shear stress equals x (the interfacial elastic shear strength), the fiber would debond from the matrix at the s  point where it enters the matrix. Whether the fiber would continue to debond at a constant load Pf or whether an increase in pullout load would be necessary would depend on the embedded length of the fiber l , and the ratio of the elastic shear strength Ts and the frictional shear strength e  T, of the fiber-matrix interface. The maximum value of fiber pullout load P ™* would occur f  when the distance, x, of the debonded point from the embedded fiber end equals: 1 -"•max  —  I—  cosh"  1  p-  2.10  From the above equation, it is clear that the stage at which the debonding becomes catastrophic is dependent upon the ratio T/T,. When T / T ; >cosh V a / (i.e., when x = x^, 2  S  e  debonding process becomes catastrophic as soon as it commences. 15  = l ), the e  On the contrary, if  T / Ti < cosh -fa l , a further increase in fiber pullout load, Pf, is necessary for debonding to 2  s  e  continue. Accordingly, the maximum value of the fiber pullout load P  max /  required to achieve  complete debonding and initiate the fiber pullout is given by the following equations, •A,E  ,-y/a K  f  A EA E f  m m } }  f  K  m  ^  f  J  S  A E AE m  m  f  Ip  e  ''AE-A E m  tanhVa l  ^  2.11  Xn  f  tanh 4a x  max  + T,C(Z - x )  l >  max  e  Xn  2.12  f  From the above equations it is clear that the maximum fiber pullout load that a fiber can achieve is not only a function of the ratio T/T, (i.e.  but it is also a function of the embedded fiber  length l . In addition, Equation 2.10 implies that, for any fiber length that is less than or equal to e  Xmax,  the process of debonding would be catastrophic.  At the instant when the entire fiber length is debonded, the resistance to fiber pullout would drop to a value of T, C l , and subsequently it would decrease as the fiber pulls out from the matrix. e  When the interfacial frictional shear strength T, is equal to zero, then according to Equations 2.10 and 2.11 catastrophic debonding of the interface would take place; i.e., the fiber pullout load would drop from maximum to zero as soon as elastic shear stress at the location where the fiber enters the matrix equals the interfacial shear strength T . S  For the determination of the shear strength pullout load P  m a x f  of the interface experimentally, the maximum  for fibers with different embedded length l is plotted against the function e  l . The shape of the curve so obtained is a function of the ratio T/T, as shown in Figure e  2.2.1. The shape of the experimental fiber pullout load versus displacement curve obtained is also dependent on this ratio.  Figure 2.2.2 depicts the schematic fiber pullout load versus  displacement curves for the various ratios of T/T,.  16  Figure 2.2.1: Variation of maximum fiber pullout load with embedded fiber length factor for various friction conditions [55]  8  (a)  Tyr,- = 1  5  (b)  1 < T / T ; < oo  (C)  T/T/=oo  Figure 2.2.2: Variation of pullout load with fiber displacement for various interfacial frictions conditions [109]  17  When T/T,- =1, the fiber pullout load versus slip response is as shown in Figure 2.2.2a, and the P  m a x /  versus  l curve is a straight line as shown in Figure 2.2.1. The shear strength of the e  interface can be computed from the equation,  2.13  In this situation, debonding is not catastrophic, and the fiber pullout load decreases linearly from maximum to zero as shown in the Figure 2.2.2a. When 1 < T/T, < oo, the fiber pullout load versus displacement curve is as shown in Figure 2.2.2b, and the P  m a x /  versus -fa l curve is as shown in Figure 2.2.1. In the P e  m a x /  versus 4oc l  e  curve a point of discontinuity occurs and the slope of the curve, A, becomes constant. The point of discontinuity corresponds to the fiber length l that equals x^ e  as given by Equation 2.10.  Lawrence points out that the slope of the curve A is related to the frictional resistance to fiber pullout after the fiber-matrix interface has completely debonded, and it can be set equal to T, C/a. Thus,  %• =  2.14 C  1  Also, from the definition of  (Equation 2.10), the magnitude of the interfacial shear strength  can be determined as,  2.15  When the P  m a x /  versus -Ja l curve has no obvious discontinuity and becomes linear at long e  embedment lengths, this implies that the frictional shear resistance is very small, and for this case the fiber pullout load versus slip curve is as shown in the Figure 2.2.2c. Accordingly, the ratio T/T,= O O , and all embedded fiber lengths are less than x  niax  as given by the Equation 2.10.  The interfacial shear strength Ts can be determined from the asymptotic value of P P  m a x /  = P °°) from the P f  m a x /  max /  (i.e.,  versus -fa l curve. Rearranging Equation 2.10, one can obtain, e  18  2.16  Gopalaratnam and Shah [56] With regard to interfacial shear stresses, Gopalaratnam and Shah [56] made similar assumptions to those made by Lawrence [55] to obtain the solution to the fiber pullout problem. Thus, this model takes into consideration the following: the existence of interfacial elastic shear stresses prior to the inception of fiber-matrix interfacial debonding, the existence of both the interfacial elastic shear stresses and the interfacial frictional shear stresses when the fiber-matrix interface is partially debonded, and the existence of interfacial frictional shear stresses after the fiber-matrix interface has completely debonded and is pulling out. In the model developed by Lawrence [55], the parameter K that relates the interfacial elastic shear stresses, T, with the virtual displacement of the interface, u-v, was inadequately defined. Gopalaratnam and Shah [56] assumed a square packing geometry of the fibers and related the interfacial elastic shear stress,T, with the shear displacement of the matrix at the interface u-v in the following way,  T=  . Ju-v) r l n ^ / r / V , /2) .  2.17  Gm  where r is the fiber radius, G is the shear modulus of the matrix, V) is the volume fraction of the m  fibers, v is the fiber displacement in the axial direction at the interface and u is the matrix displacement at the surface of the coaxial cylinder. When the fiber is partially debonded, the axial stress distribution in the fiber, F(x) and the interfacial shear stress distribution, r (x), over the bonded length of fiber were derived as e  follows: F(x) = C, cosh fix + C sinh fix + C 2  3  2.18 T (x) = — — (C, sinh fix + C cosh fix) e  2  19  and, the axial stress distribution in the fiber, F(x), and the interfacial shear stress distribution, r (x) over the debonded length of fiber were derived as follows: e  F(x) = P-2Ttrx  —  i  X  2  2.19 X (x) = -x e  i  where P is the pullout load, Xj is the interfacial frictional shear stress at the interface, and constants fi, Cu C2 and C3 are described in reference [56].  Takaku and Arridge [54] Takaku and Arridge [54] developed Greszczuk [53] model by taking into consideration the existence of frictional shear stresses at the debonded interface when the fiber was completely debonded.  However, the possibility of progressive interfacial debonding was not considered.  Thus, according to this model, catastrophic debonding of the interface takes place at the instant when the interfacial elastic shear stress at the location where the fiber enters the matrix exceeds the elastic bond strength of the interface. The relationship between the embedded fiber length, /, and the shear strength,  ^  was assumed to be the same as that derived by Greszczuk, i.e.,  Xmwc  = cd{coth(od)}  2.20  However, they defined the elastic parameter a in a different manner, as given by the following equation, -11/2  a=  2.21 f  r  E  i  H m f) r  /r  where G is the shear modulus of the matrix, ry is the fiber radius, r is the radius of matrix m  m  surrounding the fiber, and E is the fiber modulus. To evaluate the values of shear strength of the f  interface, T ^ , and the elastic parameter a, they proposed a graphical method based on the results obtained from the experimental pullout tests.  20  Unlike Greszczuk [53], however Takaku and Arridge [54], considered the frictional resistance to fiber pullout after complete debonding had occurred, and derived the following relationship for the tensile stress a(x) in the fiber at any distance x from the embedded fiber end, a (x) = A[l - exp(-fljc)]  2.22  s  Putting x=x (embedded fiber length), the relationship between the initial fiber pullout stress o; e  and the embedded fiber length x was written as: e  CT. = A[l-exp(-Bx )]  2.23  e  where A is a function of the normal compressive stress, o~, exerted by the matrix on the fiber 0  across the interface, and the elastic properties of the fiber and the matrix. Its value can be obtained from the following equation:  A=°° < E  ( 1 +  V  J  where v and vy are the Poisson's ratio for the matrix and the fiber, respectively; B is a function m  of the coefficient of friction between the fiber and matrix at the interface, fl, and the elastic properties of the fiber and the matrix; Its value can be obtained from the following equation:  B=  - ' " E, r , ( l + v j 2  £  2.25  V  CT and ji can be determined using Equation 2.23 and the experimental relationship between o~, 0  andx .. e  Pinchin and Tabor [57] Pinchin and Tabor [57] hypothesized that a 'misfit' occurs between the fiber and the matrix since the matrix shrinks during the process of setting, hardening, and curing. Defining the fiber-matrix misfit, 6, as the difference between the radius of the fiber and the radius of the hole in the matrix in the absence of the fiber, they provided a theoretical elastic analysis of fiber pullout stress, a(x), in terms of the fiber-matrix misfit, 8, the coefficient of friction fi at the fiber-matrix interface, and the elastic constants of the materials. 21  Their analysis demonstrated that the  frictional resistance to pullout, and hence the fiber pullout force, was very sensitive to the fibermatrix misfit. Due to fiber-matrix misfit, strain, e (=8/rf, where, r, is fiber radius) between the fiber and the 0  matrix is produced normal to the interface.  And, as a consequence, an interfacial contact  pressure, P, develops normal to the interface at every point along the length of the fiber, the magnitude of which is given by the equation:  (l +  2.26  v )/E +(l-v )E m  m  f  f  where v and v, are the Poisson's ratio of matrix and fiber, respectively, and E and Ef are the m  m  elastic modulus of matrix and fiber, respectively. When the fiber is loaded along its length by a stress <7f(x), it undergoes a Poisson contraction (ef=VfOf(x)/Ef), and this results in a reduction in the fiber-matrix misfit. Following Equation 2.26, a reduction in misfit strains by an amount e (e f  causes a reduction in the interfacial contact pressure. Taking all the above mentioned  = e -£f) 0  factors into consideration, Pinchin and Tabor derived an expression to calculate the stress in the fiber o(x) at any distance x from the embedded fiber end:  8 E  <J (x) = r v  t  f  f  When E » f  f  1-exp  -2v /ix f  {E r {(l f  f  +  vJ/E +(l-v )/E } m  f  2.27  f  E , Equation 2.27 reduces to Equation 23 derived by Takaku and Arridge [54]. A n m  estimate of the magnitude of fiber-matrix misfit can be obtained from the experimental relationship between the fiber pullout stress and the embedded fiber length, and using Equation 2.27. Based on experimental pullout test results obtained for steel fibers, they concluded that during the process of pullout the frictional resistance markedly decreased. They hypothesized that the observed decrease in pullout load was not due to the wear of the matrix but it was due to the compaction and densification of the cement particles near the fiber surface which caused a nonreversible reduction in the fiber-matrix misfit.  22  Hsueh [67] Hsueh [67] modeled progressive debonding of the fiber-matrix interface using shear lag theory. Stresses required to debond the fiber-matrix interface and to pull out a fiber were analyzed as a function of embedded length of fiber. With an assumption that interfacial debonding initiates when the maximum interfacial shear stress exceeds the interfacial shear strength, T„ the initial debonding stress, Od (i.e., the applied pullout stress Cy required to initiate debonding), was derived as: ( ,~ ,  X,  .  s f. . „ 1 ,  (2/a)[(l+vj{l (b la 2  +  9  , ^ 2 xlf, 9 ,  ,, . 2 x  ,,9  92 , ,„TP/2 2 2 A  (b /a -\)(EJE )]p \n(b/a)-(b -a )/2f 2  f  2.28  \)(E IE )coth(ar) + 2/{(exp(ar) - exp(-ar)}  2  m  f  where, a is fiber radius, b is the outer radius of the coaxial cylinder of the pullout specimen, t is fiber length, v is matrix Poisson's ratio, £ / a n d E are the elastic modulus of fiber and matrix, m  m  respectively, and a is defined by:  a E +(b -a )E a =— a E (\+v )^ ln(b/a)-(b -a )/2\ 2  1  2  2  f  2  f  m  2  2.29  2  m  From the above equation, it can be seen that the initial debonding stress, Cy, depends upon the length of fiber, t. Also, for a partially debonded fiber with a debond length equal to h, the debonding stress, Cy, can be obtained by replacing t by t-h in the Equation 2.28. It was shown that at the debonded interface pullout of the fiber was resisted by the frictional shear stress arising due to the existence of the residual clamping stress, a at the interface. c  Shrinkage of the matrix or the thermomechanical mismatch between fiber and matrix that take place during fabrication were identified as the phenomena responsible for the presence of a residual clamping stress at the interface. As assumed in other models [54,57], Hsueh [67] also assumed a constant value of residual clamping stress, o , along the entire fiber length. Further, it c  was demonstrated that the radial compressive stress at the interface decreases due to Poisson's contraction of fiber, and as a result, the interfacial frictional shear stress distribution along the debonded length of fiber becomes nonlinear.  With these assumptions, the axial stress  distribution, 07, in a partially debonded fiber was derived as:  23  — { l - exp(m z)}+ 5{exp(m, z) - exp(m z)}+ a exp(m z) 2  2  d  2.30  2  2  l  where z=0 at the embedded end and is equal to L (fiber length) at the location where the fiber enters the matrix; specimen geometry and material property dependent coefficients A , A3, B, mi, 2  and m2 are defined in the reference [67].  Nammur et al. [61] and Naaman et al. [62,63] The fiber pullout model introduced by Nammur, Naaman and Clark [61] is a cohesive interface type model [106]. A cohesive interface type model assumes that only relative displacements between the fiber and the matrix can activate stress transfer at the interface. Also, in these types of models the interfacial traction is described as a function of the displacement discontinuity, and since there exists a unique relationship between interface traction and interface displacement discontinuity, it is not required to distinguish between the debonded and the bonded interface. In this model, the relationship between the interfacial bond stress versus relative slip between fiber and matrix was assumed to be of the type shown in Figure 2.2.3.  slip Figure 2.2.3: Interfacial bond stress vs. slip relationship [61]  As can be noticed in the above figure, the bond-slip curve is assumed to be linear elastic up to the point where the bond strength  Xmax  of the interface is attained. In the elastic region, the slope  of the bond stress vs. slip curve represents the bond modulus, K. Beyond the slip corresponding to  Xmax,  a constant frictional shear stress equal to T, is assumed to exist at the interface.  Furthermore, it was assumed that T, cannot exceed  Xmax-  Since the interfacial bond due to  chemical adhesion is not slip induced, the application of the bond stress vs. slip relationship assumed in this model is limited to the cases where chemical adhesion is negligible. The other 24  major limitation of this model is that it assumes a constant value of interfacial shear stress at the debonded front. Applying the bond stress vs. slip constitutive relationship to a cylindrical fiber-matrix coaxial pullout model, relationships were derived for interfacial shear stress distribution, axial shear stress distribution, and fiber displacement at the various stages of pullout loading. In the interest of brevity these relationships are not included in this review. Applying shrink-fit theory to the problem of pullout and hypothesizing that the radial misfit between fiber and matrix decreases as a fiber is pulled out of the matrix, Naaman et al. [62,63] modified the previous model developed by Nammur et al. [61]. It was shown that as the fiber pulled out from the matrix, the interfacial frictional shear stress at the debonded  interface  decreased as a result of the decrease in radial misfit.  Wang, L i and Backer [69] In this work [69], the authors emphasize the triviality of fiber-matrix interfacial elastic shear bond on the overall pullout response of both high as well as low modulus fibers. They compared the theoretical pullout response for steel fibers embedded in two different types of matrix using the model developed by Gopalaratnam and Shah [56].  With the first kind of matrix, the  interfacial fiber matrix bond was considered to be purely frictional, and with the second kind of matrix, both elastic and frictional bonds were assumed to exist. They demonstrated that, for both cases, the theoretical pullout responses predicted by the Gopalaratnam and Shah model [56] were very similar. Commenting on the influence of fiber modulus, they pointed out that for fibers with lower E / E ratios, the effect of elastic shear stresses on pullout response would be even f  m  less. They concluded that for most cases, the pullout analysis could be simplified by neglecting the elastic stress field existing at the fiber-matrix interface. Further, they developed a model to predict the pullout load versus displacement response of polymeric fibers. For modeling purposes, they assumed a linear elastic constitutive behavior for both polymeric materials studied (nylon and polypropylene).  Also, the effect of Poisson's  contraction of the fiber, and the effect of the elastic shear stress field were ignored.  In the  pullout curves reported by the authors, the fiber pullout load increased with the pullout distance. They attributed this increase to an increase in interfacial frictional shear stresses while the fiber pulled out from the matrix. They pointed out that this increase in frictional shear stresses was 25  due the wear of the fiber surface during the fiber pullout process. Based on this observation they developed a model to predict the pullout response, which took into account the variation of the interfacial frictional bond strength as a function of fiber pullout distance.  They assumed the  frictional bond strength t(s), to be a quadratic function of the slippage distance, s, r(s) = a +a s + a s  2.31  2  0  ]  2  where a , au and a are constants, determined empirically such that the theoretical pullout curves 0  2  match the experimental curves. However, they also pointed out that an alternative expression based on the theory for fiber-cement friction and wear would be more desirable to represent the %-s relationship.  2.3 Experimental Studies on the Fiber Pullout Behavior Interfacial bond defined as the shearing stress at the interface between the fiber and the surrounding matrix allows transfer of stress between the two materials.  As discussed in the  previous section, the fiber-matrix interface has been modeled and studied extensively because of its important role in composite materials. It is well accepted that the contribution of fiber in increasing the toughness of a composite is primarily governed by the energy dissipation mechanisms associated with the pullout of fibers from the matrix. Because of the importance of the interfacial properties on the overall composite response, various researchers [110-153] have put considerable effort into the investigation of interfacial properties via single fiber pullout tests. Fiber pullout tests are also routinely carried out in other fiber reinforced composites, such as fiber reinforced ceramics [169-177] and fiber reinforced metals [8,178,179].  In addition,  because of the dependence of micromechanical properties on the interfacial microstructure, considerable effort has also been put toward the microstructural investigation of cementitious interfaces [160-168].  A brief review of some important experimental fiber pullout studies  follows. Gopalaratnam and Abu-Mathkour [125] investigated the influence of fiber embedded length, fiber diameter and matrix quality on the pullout behavior of steel fibers. They concluded that the average bond strength was inversely related to the embedded length and that the average bond strength increased with an increase in fiber diameter.  Additionally, they found that the peak  pullout load did not increase with an increase in the matrix compressive strength. Based on this 26  result they concluded that the frictional bond strength might be unrelated to the matrix compressive strength. Similarly, Gray and Johnston [107] also found no correlation to exist between the matrix compressive strength and the average interfacial bond strength for the case of straight fibers. Investigating the pullout response of deformed and straight fibers, Naaman and Najm [123] found that for deformed fibers the load increased with matrix strength even beyond the peak pullout load, i.e., in the descending branch, while for smooth fibers the increase was mostly noticeable in the ascending branch. Thus, they concluded that as the matrix strength increases, the bond between the fiber and matrix also increases; however, the debonding rate following the peak load is greater for high strength matrices. They also investigated the influence of various mineral admixtures: micro-silica, fly ash, and latex.  They found that micro-silica did not  improve the interfacial bond strength. On the other hand, small improvements were obtained with the addition of fly ash. Similarly, addition of latex led to a significant increase in the peak pullout load but had no effect on the post-peak response, suggesting that addition of latex increased the adhesional bond strength but had no effect on the frictional bond strength. Larson and Bayasi [139] also reported that bond strength drastically increased with the addition of latex. Banthia [131] found that addition of silica fume improved the adhesional bond strength but at the same time made the matrix more brittle. Also, pullout resistance was found to be better at subzero temperatures (-55°C) in comparison to that at normal temperature (22°C). It was also found that the mortar matrices offered a greater resistance to fiber pullout in comparison to cement paste.  It was suggested that the sand particles possibly act as tiny crack arresters  (blunting and deflecting crack) in the vicinity of the fiber and help arrest debonding of the interfacial crack. Influence of curing temperature on the pullout response was also studied in this investigation. Pinchin and Tabor [141,146-147] reported that compaction of the concrete surrounding a fiber enhanced the frictional bond, and that the pullout load increased almost linearly with increase in confining pressure. They also reported that pullout load was proportional to the fiber-matrix misfit, defining misfit as the difference between the radius of the wire and that of the hole in the matrix in the absence of the wire. Similar conclusions have been made recently by Geng and Leung [120].  In an earlier study, Geng and Leung [119] found that the post-peak pullout  response displayed different trends for fibers with different materials. 27  For instance, for steel  fibers the post-peak load decreased rapidly, for polypropylene fibers the post-peak load decreased slowly and for nylon fibers the post-peak pullout load increased with the sliding distance. Wang et al. [115,116] observed that synthetic fibers like polypropylene and nylon are easily abraded by cement particles during the process of pullout.  The severity of surface  abrasion of fiber was found to increase with increase in fiber slip and it was accompanied with an increase in the apparent fiber/matrix bond strength. Gokoz and Naaman [145] reported that the peak pullout load, which is indicative of the elastic bond on the interface, was not sensitive to strain-rate. Banthia and Trottier [127] and Banthia et al. [134] also investigated strain rate sensitivity of bond for straight and deformed fibers. They found that pullout performance for deformed fibers was sensitive to the rate of loading. On the other hand, for straight fibers no definite conclusion could be made with regard to the strain rate sensitivity of the interfacial bond. In particular, for deformed fibers they found that both the peak pullout load and the pullout energies increased with an increase in the rate of loading. They concluded that since steel, cementitious matrices and the fiber-matrix interface all have properties dependent upon the rate of loading, the capability of the fiber to resist pullout largely depends upon the rate at which fiber pullout occurs. In addition, in these studies, cement paste matrices were found to be less sensitive to loading rate in comparison to the mortar matrices. Pacios et al. [138] also investigated the influence of rate of loading on interfacial response between fibers and cementitious matrices. They found that both pullout load and slip increased with increase in the rate of loading. Naaman and Shah [142] found that the efficiency of bond in the case of pullout of an inclined steel fiber (with respect to loading axis) was at least as good as that of fibers aligned parallel to the loading direction. Ouyang el al. [86] also found that pullout resistance of inclined fibers was generally greater than that of aligned fibers. The maximum inclination of fiber with respect of loading axis investigated in this study was 37°. On the other hand, Banthia and Trottier [127] investigated a wider range of fiber inclination (0° - 60°) and concluded otherwise, stating that the bond efficiency of aligned fibers was greatest, and increasing inclinations of fiber with respect to the loading axis reduced the efficiency of interfacial bond. Maage [140,144] also investigated the influence of fiber inclination and reported that the bond efficiency increased to a maximum and then decreased as the angle between the fiber and the loading direction increased.  28  Gray and Johnston [107] found that the bond strength depended on the direction of casting, concluding that vertically cast specimens developed a higher interfacial bond strength than horizontally cast specimens. Additionally they found that an increase in the sand-cement ratio in the mortar matrix led to a decrease in interfacial bond for the vertically cast specimens and an increase in bond for the horizontally cast specimens. Burakiewicz [143], on the other hand, reported otherwise, stating that the bond strength was virtually insensitive to vibration and fiber orientation during setting and hardening of the matrix. Shao et al. [154] studied the processes of matrix cracking and fiber-matrix interfacial debonding using high-sensitivity Moire interferometry techniques.  The initiation and propagation of  cracking and interfacial debonding were observed during uniaxial tension tests, and fiber stress, interfacial slip, interfacial shear stress, and the matrix strain distribution were calculated from the experimental information. Based on the experimental study, they questioned the validity of the fiber-matrix interfacial shear stress versus slip constitutive relationships assumed in several fiber pullout models [60-63]. Kawamura and Igarashi [149] carried out a fluorescence microscopic study of the interfacial zone between steel fibers and cementitious matrices subjected to pullout load. They reported that progressive debonding of the interface originated from the point where the fiber entered the matrix, as predicted by shear-lag theory based analytical pullout models. A similar conclusion was made by L i et al. [81,82], based on the fluorescence optical microscopic investigation carried out on uniaxial tensile specimens. Kawamura and Igarashi [149] also found silica fume modified cementitious matrices to be less tough in comparison to the matrices without silica fume.  Further, they reported that during the propagation of interfacial cracks, local failures  around a fiber in the silica fume matrices were limited to smaller regions around the interface because of the presence of denser and more homogeneous bulk phases in the matrix; on the other hand, the toughening mechanisms of non-silica fume matrices were responsible for deflection and branching of cracks due to unhydrated cement particles and Ca(OFf)2 crystals. In an another study, Kawamura and Igarashi [151] reported that the maximum pullout load increased with increasing silica fume content (20% replacement) but the drop in load immediately after the maximum load was considerably small. Also, the total work of pullout was considerably greater with the silica fume matrix and authors attributed this to the increase in frictional resistance along the debonded surfaces due to the addition of silica fume. In this study authors also found that the values of microhardness (measured using the ultramicrohardness tester with Vickers 29  indenter - minimum load=0.1gm) in the bulk cement paste matrix as well as in the interfacial transition zone were increased by the addition of silica fume.  2.4 Mechanical Behavior of Fiber Reinforced Cementitious Composites Especially since the pioneering work of Romualdi and Batson [180] on steel fiber reinforced concrete, extensive research into the mechanical properties of fiber reinforced cementitious composites has taken place [1-3, 181-184]. Moreover, in recent years, several advancements have been made with regard to matrix, fibers, interfaces and the composite production process. Some examples include [182]: i.) the commercial introduction of a new generation of additives such as superplasticizers that allow high matrix strengths to be readily achieved with little loss in workability; ii.) the increasing use of active or inactive micro-fillers such as silica fume and fly ash and a better understanding of their effect on matrix porosity, strength, and durability; iii.) the increasing availability for use in concrete of fibers of different types and properties which can add significantly to the strength, ductility, and toughness of the resulting composite; iv.) the use of polymer addition or impregnation of concrete which adds to its strength and durability but also enhances the bond between fibers and matrix thus increasing the efficiency of fiber reinforcement; and, v.) innovations in production processes to achieve uniform mixing of high volume fiber fractions with reduced effects on the matrix porosity.  In general, the parameters governing the behavior of fiber reinforced cementitious composites can be classified under three categories. The first category consists of the parameters related to the fibers, such as elastic modulus, tensile strength, ductility, length, diameter, and volume fraction. The second category consists of the parameters related to the matrix such as fracture toughness, elastic modulus, ultimate tensile strength, ultimate tensile strain, and initial flaw size. Finally, the third category consists of the micromechanical parameters related to fiber-matrix interface, such as interfacial bond strength. Traditionally, research on fiber reinforced concrete has focused on studying the dependence of composite behavior on one or two of the aforementioned properties at a time. Typically, these properties have been fiber volume fraction and fiber aspect ratio. Thus, it is important to recognize that fiber volume fraction, for instance, is only one of an array of parameters under our control when designing fiber reinforced cementitious composites.  For this reason it is not sufficient to understand the individual  influence of each parameter on composite properties. 30  Composite optimization requires  establishing the combined influence of all relevant parameters on composite properties. Through composite optimization, composites with tailored properties can be produced with only moderate volume fraction of fibers.  In this regard, development of high-performance cementitious  composites, (i.e., composites with an optimized combination of properties, such as quasi-strain hardening behavior, multiple cracking, strength, toughness, energy absorption, durability,  and  corrosion  resistance)  require  an  understanding  of  the  stiffness,  fundamental  micromechanisms controlling the composite behavior.  Strain AB - multiple cracking  BC - softening response  Strain  Figure 2.4.1: Stress-strain response for afiberreinforced cementitious composite showing strain hardening response  Figure 2.4.1 shows the stress-strain response in tension for a fiber reinforced cementitious composite exhibiting multiple cracking. The overall tensile response can be divided into three regions. First, the initial linear region up to the Bend-Over-Point (BOP) represents the elastic loading of the composite (i.e., the region OA). Beyond this region, multiple cracking of the matrix takes place, and the stress-strain curve rises in a non-linear fashion showing a quasi-strain hardening response (i.e., the region A B ) . Finally, at the end of multiple cracking, strain localization takes place and the stress-strain curve shows a softening response (i.e, region BC). The extent of these stages depends on the properties of the fiber, matrix and interface. So far, considerable research efforts have been put into modeling the tensile behavior of fiber reinforced composites [185-214].  However, as Shah and Ouyang [105] have pointed out, despite the  extensive modeling efforts, the phenomenon of crack propagation in fiber reinforced cement 31  based matrices still remains poorly understood.  A brief review of literature describing the  conditions leading to quasi strain hardening and multiple cracking behavior is presented below. These solutions generally identify a number of controlling parameters, such as fiber reinforcing parameters, the interfacial fiber-matrix bond, critical fiber volume that lead to quasi strain hardening and multiple cracking.  Aveston et al. [185] proposed an analytical model to predict the conditions required to obtain multiple matrix cracking and crack spacing in the event of multiple cracking. This model is commonly known as the A C K model. The authors suggested that if one of the two constituents (fiber or matrix) of a fibrous composite breaks at a much lower elongation than the other, then provided the non-broken constituent is able to bear the load, a tensile specimen will show continued or multiple fracture of the more brittle phase until the specimen finally breaks when the ultimate strength of the stronger phase is reached. The above statement suggests that if the breaking strain of the matrix is less than that of the fiber then the matrix will be successively fractured into shorter lengths by a process of multiple fracture, until the fiber attains its failure strain. Likewise, if the breaking strain of the fiber is less than that of the matrix, then the fiber will be successively fractured into shorter lengths by a process of multiple fracture, until the matrix attains its failure strain. The condition for multiple cracking of matrix was derived as: 2.32 where Of and a u  mu  are fiber and matrix strength, respectively, a'f is the fiber stress required to  produce a strain equal to the breaking matrix strain, and Vy and V are volume fractions of fibers m  and matrix, respectively. The right hand side of the above equation represents the composite stress at the instant matrix attains failure strain. Thus, the above equation suggests that the fibers are able to sustain the additional load thrown upon them when the matrix fails. The minimum fiber volume fraction required to achieve multiple cracking (i.e., critical fiber volume fraction) can be obtained by using the equality expressed in the above equation. Furthermore, assuming a linear distribution of shear stress in the debonded zone, authors calculated the crack spacing x' as:  2.33  x =  32  where a  is the matrix fracture stress, r is the fiber radius, and r i s the average fiber-matrix  mu  interfacial bond strength. By assuming that the crack spacing at the beginning of multiple cracking is 2x', (i.e., sliding occurs along the entire fiber-matrix interface), authors derived the solution for failure strain of matrix, e from the energy balance as: mM  -il/3  ,  \2vt E V) m  ^ mu  s  2.34  E E rV 2  c  m  m  where y is the fracture surface work in forming a crack in matrix, E is the elastic modulus of m  c  composite, E is the elastic modulus of the matrix. m  Budiansky et al. [186] generalized the A C K model to include the three types of fiber interface conditions: perfectly bonded fibers, debonded sliding fibers and unbonded, sliding fibers. This model is commonly known as the B H E model. Authors obtained distributions of stress and strain in the fiber and matrix based on shear lag analysis. The strain energy release rate of composites was calculated from the obtained stress distributions, and the multiple cracking response was predicted from the energy balance, which equated the strain energy release rate of composites to the sum of energies resulting from debonding and sliding, and the matrix surface For a perfectly bonded fiber, the B H E model predicts the composite stress, o f, for  energy.  C  multiple cracking as: CT.  lV E G p f  f  mc  2.35  rE E  y  m  c  where 1 2K, 6E P=7T V | _ - 6 1 o g ( y ) - 3 V . ( 3 - V ) E (l + v ) 3  /  m  1  /  f  ll/2  m  where v is Poisson's ratio of the matrix, Gmc is the fracture resistance of matrix, Ef, E and E m  m  c  are the elastic modulus of fiber, matrix and composite, respectively, and Vf and V are the m  volume fraction of fiber and matrix, respectively.  Naaman [187] proposed a solution based on the mechanics of composite materials for predicting conditions required for achieving multiple cracking of matrix. Assuming a three dimensional fiber distribution, he derived equations for predicting the first cracking stress, o , and the cc  33  maximum post-cracking stress (or bridging stress), a . pc  The composite tensile stress at first  cracking of matrix was derived as: o" = c (l-V ) cc  mu  + a a W Ud  f  l  2  2.36  /  where Vf =  volume fraction of fibers  L  =  fiber length  d  =  fiber diameter  c  =  matrix tensile strength  x  =  average bond strength at the fiber matrix interface  mu  a,i =  coefficient representing the fraction of bond mobilized at first matrix cracking  a  efficiency factor of fiber orientation in the uncracked state of the composite  2  =  The maximum post cracking stress, a , was derived assuming that a critical crack exists across pc  the entire section of the tensile member, the crack is normal to the tensile stress field, and the contribution of the matrix is negligible. Thus, the maximum post cracking stress, o  pc  was  derived as: G =\X X xV Lld pc  2  i  2.37  f  where Xi = expected pullout length ratio (equal to 1/4 from probability considerations) X2 = efficiency factor of orientation in the cracked state X3 = group efficiency factor associated with the number of fibers pulling out per unit area  For multiple cracking to Occur, the maximum post-cracking stress must be larger than the first cracking stress, i.e.:  Combining Equations 2.36-38, the minimum fiber volume required to guarantee multiple cracking (i.e., critical fiber volume, Vf it) was obtained as: Cr  V f,cril  V  =  T  2 39  L  1+——(hKK-tt&i)  34  The above equation depicts that the fiber aspect ratio and the ratio of fiber-matrix interfacial bond strength and matrix tensile strength are influential parameters in controlling the fiber volume fraction required for multiple cracking to occur. Parametric studies conducted using the above equation depict that a higher fiber aspect ratio and a lower interfacial bond strength to matrix tensile strength ratio lead to reduction in critical fiber volume.  L i and Wu [188] assumed a random fiber orientation and used the fracture mechanics based Jintegral approach (originally proposed by Marshall and Cox [202]) to derive the condition leading to multiple cracking and quasi-strain hardening. They derived the critical fiber volume fractions as: V  f  •=  gr(L /d f  2 40  )8  f  a  where J is the composite crack tip toughness, which may be related to the matrix fracture c  toughness, K , as suggested in Reference [195]. m  Lf and df are fiber length and diameter,  The snubbing factor, g, and interfacial frictional bond strength, T, are the  respectively.  parameters describing the interaction between fiber and matrix. The snubbing factor accounts for the variation in bridging force across a matrix crack when a fiber is pulled out at an inclined angle. And, finally, 8 in the above equation represents the crack opening at which the fiber Q  bridging stress reaches a maximum, a , and is given by: 0  * ~ ~°  2.41  f  E d (l+ri) f  f  where n = (VfE )/(V E ), and V and Ef are the fiber volume fraction and fiber elastic modulus, f  m  m  respectively, and V and E  f  m  m  are the matrix volume fraction and matrix elastic modulus,  respectively.  For a pseudo strain hardening material, the ultimate strength of the composite, a coincides with cu  the maximum bridging stress, o~, given by: 0  (  " cu  2  6  /  T  ZL  ^  2.42  From the viewpoint of composite design, Equation 2.40 is significant, since it provides guidelines to tailor the micromechanical parameters for strain hardening materials such that Vf  iCrit  35  is minimized. In an another article, L i [196] has recommended the following with regard to the tailoring of micromechancial parameters:  Matrix Toughness Tailoring: With a decrease in matrix toughness, J , smaller amount of c  fibers is needed to make the transition from the quasi-brittle failure mode to the strainhardening failure mode. From the viewpoint of matrix design, in order to tailor the matrix toughness, adjustments can be made to the water/cement ratio, size and type of aggregates, and microfillers such as silica fume.  Interfacial Bond Tailoring:  Enhancing the interfacial frictional bond, T, while still  ensuring that fiber fracture does not occur, would assist in lowering the amount of fibers required to achieve multiple cracking.  The ductility of fiber reinforced cementitious  composites is associated with the inelastic strain generated as a consequence of multiple cracking.  Moreover, the composite inelastic strain is a function of the multiple crack  density and crack opening. The multiple crack density can be expected to increase with the interfacial bond strength, T, which controls the stress transfer from the bridging fiber into the matrix material. To improve the interfacial bond strength, several methods can be employed, such as fiber surface modification, fiber deformation, and transition zone modification.  Fiber Length Tailoring: Equation 2.40 depicts that the critical fiber volume fraction, Vf ,, Cri  is inversely proportional to L , (since S is proportional to Lf), suggesting that fibers with 3  0  long lengths are preferable. However, difficulty in processing places a limitation on the choice of fiber length.  In addition to the models discussed above, several other micromechanical models have been proposed for predicting cracking behavior in fiber reinforced cementitious composites (including the ones by Leung and L i [190,191], Tjiptobroto and Hansen [200,201], Cox and Evans [203,204], Nemat-Nasser and Hori [205], Marshall, Mai [206], Mori and Mura [207], Yang et al. [208], Hillerborg, Modeer and Peterson [209], Foote, Mai and Cotterell [212], Jenq and Shah [213] and Ouyang, Mobasher and Shah [214]).  36  Chapter 3  Macromechanical Behavior of Fiber Reinforced Concrete (FRC) & Measurement of Flexural Response of Low Toughness FRC  3.0 Introduction Cementitious materials have low strain capacity and ductility. The tensile strain capacity and energy absorption capability (i.e., ductility) of these materials can be dramatically improved by inclusion of fibers in the matrix. The extent of improvement obtained due to the addition of fibers depends on the mechanical properties of the fiber and the matrix, and on the nature of interaction between the two. Optimal design of fiber reinforced concretes can be accomplished only when the influence of various material variables on the macromechanical behavior of fiber reinforced concrete composite is properly understood. Therefore, one objective of this research is to experimentally investigate influence of various important material variables on the postcracking response in flexure and the ensuing toughness of fiber reinforced concretes.  Measured flexural response of fiber reinforced concretes is also influenced by the various test variables [41,42]. One such test variable is stiffness of testing machine. The present A S T M standard (ASTM C1018 [4]) and the JSCE standard (JSCE-SF4 [5]) for evaluating flexural response and toughness of fiber reinforced concretes are appropriate only for fiber reinforced concretes with high toughness [6]. For low toughness fiber reinforced concretes, the measured flexural response based on the aforementioned standards has been found to grossly depend upon the relative stiffness of testing machine [6]. Thus, another major objective of this research is to address the issue of rationally measuring the true flexural response of fiber reinforced concretes with low toughness. A S T M has recently accepted a new test method [7] as a test standard for measurement of flexural performance of fiber reinforced concretes. This method is referred as the Residual Strength Test Method in this chapter. Validity of this method is explored as a part of this research program and the results are reported in this chapter.  37  3.1 Material Variables Influencing Macromechanical Behavior and Toughness of FRC This section investigates the influence of various material variables on the macromechanical behavior and toughness of FRC. Important material variables investigated include properties of matrix, volume content of fibers, fiber geometry, fiber aspect ratio, surface properties of fibers, and shrinkage properties of matrix.  Materials and Mixes The materials and the mixes used in this investigation are given below in Table 3.1.1. C S A Type 10 normal Portland cement was used. Clean river sand with a fineness modulus of 2.3 was used as the fine aggregate, and pea gravel with a maximum size of 10 mm was used as the coarse aggregate. A l l mixes contained air-entraining agent at a dosage of 0.1 ml/kg of cement. For a few other mixes, different mix proportions were used and those will be indicated at appropriate locations in this section. Geometry and mechanical properties of fibers used in this investigation are given in the Table 3.1.2. Table 3.1.1: Mix Proportions Cement  Water  Sand  (kg)  (kg)  400  200  (kg)  Aggregate (10 mm) (kg)  Air Entraining Agent (ml/kg of cement)  800  1000  0.1  Table 3.1.2: Description of fibers investigated Fiber  Material  Geometry  Cross-  Length  Diameter  sectional shape  (mm)  (mm)  (MPa)  (GPa)  STL-STR  Stainless Steel  Straight  Circular  40  0.76  1100  210  PP-STR1  Polypropylene  Straight  Oval  40  0.63  450  3.5  PP-STR2  Polypropylene  Straight  Oval  38  0.19  450  3.5  PVA-1  PVA*  Straight  Oval  29  0.42  1100  23  ST-HKD  Steel  Hooked-end  Circular  35  0.55  1115  210  ST-HKD2  Steel  Hooked-end  Circular  30  0.50  1115  210  ST-CR1  Steel  Crimped*  Crescent  38  2.0 x 0.35  1037  210  * PVA - Polyvinyl Alcohol; * Crimp amplitude=lmm and Wave length=7.5mm; - Tensile strength of fiber; E - Elastic Modulus of fiber f  Specimens and Test Methods Cylinders for compressive strength test: Cylindrical specimen 200 mm long and 100 mm in diameter were used to test the compressive strength of F R C and plain concrete mixes. Cylinders  38  were tested in compression in a 1000 kN compression testing machine as per A S T M C 39 at 7 and 28 days. Beams for flexural toughness test: Beam specimens conforming to the dimensions recommended in A S T M C1018 - (100 mm x 100 mm x 350 mm) were used for flexural testing of FRC and plain concrete mixes. A pan type concrete mixer (capacity: 0.18 m ) was used for mixing the 3  concrete and the procedure outlined in A S T M C192 "Method of Making and Curing Concrete Specimens in the Laboratory" [215] was followed. A l l coarse aggregate and some of the mixing water were first mixed, followed by the addition of fine aggregate, cement, water, pozzolanic admixture in slurry form (when included), air entraining agent and fibers.  Figure 3.1.1: Test setup for flexural toughness test according to ASTM C 1018 [4]  For flexural tests, a 150-kN capacity floor mounted, Instron electro-mechanical testing machine was used. A l l operations were controlled by a microprocessor based central processing unit. Two LVDTs mounted on a Japanese Yoke, shown in Figure 3.1.1, recorded the net deflection of the beam at the mid-span. With the use of a Japanese Yoke, extraneous deflections resulting from settlement of supports, crushing at load points, and load-fixture deformations are eliminated and only the true deflections are recorded [42,44]. The load was applied at a constant cross-head displacement rate of O.lmm/min. A PC-based data acquisition system was used to digitally record loads and displacements at a frequency of 10 Hz. The output signals from the testing machine and LVDTs were acquired by a computer-based high-speed data acquisition system. The data acquisition system comprised a PC-486 computer, a 16 channel A / D conversion board, 39  and a 16 1/0 channel A C and D C conditioner unit in which each channel was equipped with amplifiers with gains of up to 1000. The maximum sampling rate of this data acquisition system is 10 million points per second, i.e., 1 M H z . The software package used for acquiring data was "Quicklog" by Strawberry Inc.  The resulting load versus mid-span deflection plots were  analyzed for JSCE flexural toughness parameters [5]. The test procedure described above is also referred as the Open-loop Test Method (OLTM) in this thesis.  3.1.1 Mineral Admixtures Constituents of the binder have a significant influence on the chemistry and microstructure of the matrix in the interfacial regions. Both chemistry and microstructure of matrix are modified to a considerable extent when a portion of cement is replaced by a mineral admixture such as pozzolan.  For example, a silica fume modified matrix gives rise to a very densified  microstructure at the fiber-matrix interface [105]. It can be expected that the post-cracking response of F R C would be affected as a consequence of the modified interfacial properties brought about by the inclusion of pozzolans. The objective of the following studies was to investigate the influence of matrix modification by pozzolans on the post-cracking response of FRC.  Influence of Matrix Modification by Silica Fume Pozzolan The influence of pozzolanic modification of the matrix on the post-cracking response of fiber reinforced concrete was investigated.  Pozzolan-silica fume was chosen for this purpose;  properties of the silica fume used are given in Table 3.1.3. The two mixes investigated contained straight-smooth, stainless steel fibers at a volume fraction of 0.76%. The mix proportion used was - water : binder : sand : aggregate = 0.35 : 1.00 : 1.50 : 1.75. The control mix did not contain silica fume and the silica fume mix contained silica fume at a dosage of 10% by weight of binder as shown in Table 3.1.4. The twenty-eight day compressive strength for the control mix was approximately about 69 MPa and that for the silica fume mix was about 71 MPa.  40  Table 3.1.3: Properties of Pozzolans Property Average Particle size, fl m Reactive Component (minimum percent by mass) Specific Gravity  Silica fume  High-Reactivity Metakaoline  0.1  1.5  Si0 (97%)  SiOi+AlzOj+FejOj (95%)  2.2  2.5  2  Five beams were tested for each mix. The average flexural responses for these mixes are compared in Figure 3.1.2. It can be noticed that the post-peak flexural response in the case of mix containing silica fume was relatively inferior in comparison to the mix without silica fume. JSCE toughness parameters [5], namely, "Absolute Toughness (7j )" and "Flexural Toughness CI  Factor (Ob)" were calculated from the average flexural response. The Absolute Toughness, T  JCI  is  defined as the area under the load-deflection curve to a deflection of span/150. The Flexural Toughness Factor, o , (a measure of the equivalent flexural strength) is defined as: b  where S is beam span, w is beam width, d is beam depth, and 8  150  is 5/150. The calculated JSCE  toughness parameters are given in Table 3.1.4. Apparently, the observed disparity in the flexural performance in the two cases is related to the micromechanics of fiber-matrix interface. The influence of matrix modification by mineral admixtures on fiber pullout response is a subject of investigation in Chapter 4.  41  30  STL-STR @ V,=0.76% 20 with silica fume without silica fume 10  0 0  1  2  3  Deflection (mm)  Figure 3.1.2: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes with and without silica fume and containing straight-smooth, stainless steel fiber at a dosage of 0.76%  Table 3.1.4: JSCE Absolute Toughness and Flexural Toughness Factor - Influence of Pozzolan Fiber Id.  STL-STR  SRL-STR  Fiber Description  Binder Type  Fiber Volume  (% weight)  (%)  Straight-smooth, stainless steel  100% CSA Type 10NPC*  0.76  Straight-smooth, stainless steel  90% CSA Type 10 NPC + 10% Silica fume  0.76  NPC - Normal Portland Cement;  MOR (MPa)  Absolute Toughness, TJCI  Flexural Toughness Factor, Oi,  (N-m)  (MPa)  14.9  2.25  (0.51)  (1.28)  (0.19)  5.46  12.5  1.88  (0.54)  (0.90)  (0.14)  5.51 @  - Figures in brackets indicate standard deviations  Influence of Pozzolan Type While the addition of pozzolans densifies the matrix microstructure, it also increases the brittleness of matrix. Increased matrix brittleness is of particular concern when deformed fibers are involved. Increased matrix brittleness can cause crushing and splitting of the matrix and in turn, curtail the ability of fibers to transfer stresses during pullout, thus reducing the overall toughness. In addition to the water/binder ratio, one may anticipate that the type, chemical reactivity, and particle size of the pozzolan used will have a decisive effect on the brittleness of the matrix and on the fiber-matrix bond. The objective of this investigation was to study the influence of various matrix modifications on the flexural toughness performance of steel fiber reinforced concrete. Two types of pozzolanic admixtures were investigated, namely, i.  silica fume, and  ii.  high-reactivity metakaolin (HRM) 42  High-reactivity metakaolin (HRM) is produced by calcining purified kaoline clay in the temperature range of 700 to 800°C.  H R M is a poorly crystallized white powder with a high  pozzolanic reactivity, which earns it the name super-pozzolan. It comprises nearly 95% of (Si0 +Fe 03+Al20 ) and has elongated particles about l/Jrn in size on average. 2  2  3  Physical  characteristics and chemical properties of silica fume and H R M investigated are given in Table 3.1.3.  Mix C F (control mix) contained only CSA Type 10 ( A S T M Type I) cement as the cementing material.  The mix proportion (by weight) for the control mix was:' water : cement : fine  aggregate : coarse aggregate = 0.35 : 1.0 : 1.5 : 2.25. In M i x SF, on the other hand, 10% of cement was replaced by an equal quantity of silica fume, and in M i x M F , 10% of cement was replaced by an equal quantity of high-reactivity metakaolin. Finally in Mix SMF, a combination of 5% silica fume and 5% high-reactivity metakaolin replaced 10% of cement. In all the mixes, hooked-end steel fibers, 30 mm long and 0.5 mm in diameter were added at 1% by volume. 35  -r  30 -  —-X  with 10% HRM  25 -  ^C'^^-CT^'^^ ^-^ 51  20 -J  100% Cement  15 10 -  with 10% Silica fume  1  1  5 0 0.0  1  —1  0.5  . 1  with 5%Silica fume + 5% HRM  0  1  1  1.5  2.0  2.5  Deflection (mm)  Figure 3.1.3: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes different types of pozzolans  Representative load versus mid-span deflection plots for the various steel fiber reinforced concrete mixes are given in Figure 3.1.3. It can be observed that the post-peak performance of fiber reinforced concrete with H R M was superior to other composites.  Silica fume concrete  (Mix SF) exhibited a relatively brittle behavior with a steeper drop in load beyond peak. The load-deflection performance of the control, non-pozzolan concrete (Mix CF) and the hybridpozzolan concrete (Mix SMF) was between that of concretes with H R M (Mix MF) and silica fume (Mix SF). As expected, at a high fiber volume fraction of 1% used in this investigation, 43  some pseudo-strain hardening occurred in the post-peak region for all composites.  JSCE  toughness parameters, shown in Table 3.1.5, also depict the superior performance of concrete containing H R M . Lower JSCE numbers for silica fume concrete (mix SF) are directly related to the relatively steeper drop in the load that occurred after peak, indicating a greater rate of softening and damage in these composites. Visually, one could notice that the silica fume matrix had cracked more extensively, and fibers as a result could not develop adequate mechanical anchorage with the matrix. These fibers then pulled out with minimal straightening of the hooks and absorbed much less energy. For fibers in concrete with high-reactivity metakaolin, on the other hand, the ends had been straightened for a greater number of fibers signifying an increased absorption of energy. Table 3.1.5: JSCE Absolute Toughness and Flexural Toughness Factor - Influence of Pozzolan Type Mix Id,  Fiber Description  Binder Type  Fiber Volume  (% weight)  MOR  #  (MPa)  Absolute Toughness,  (%)  TJCI  Flexural . Toughness Factor, Ob  (N-m)  (MPa)  CF  Hooked end steel ST-HKD  100% Cement  1.0  7.19 (0.53)  39.20 (3.8)  5.88 (0.44)  SF  Hooked end steel ST-HKD  90% Cement + 10% Silica fume  1.0  8.37 (0.30)  32.0 (3.36)  4.80 (0.50)  MF  Hooked end steel ST-HKD  90% Cement + 10% HRM  1.0  8.39 (0.82)  41.4 (3.24)  6.40 (0.49)  SMF  Hooked end steel ST-HKD  90% Cement + 5% Silica fume + 5% HRM  1.0  8.39 (0.32)  36.8 (4.30)  5.81 (0,45)  * - MOR - Modulus of Rupture;  3.1.2  @  @  - Figures in brackets indicate standard deviation  Fiber Volume Content  To investigate the influence of fiber volume content, three types of fibers were chosen. Fiber types and the corresponding fiber volumes investigated were, i.  Straight-smooth, stainless steel fiber, STL-STR - V = 0.76% and 5.0%  ii.  straight, smooth polypropylene fiber, PP-STR1 - V = 1.0% and 5.0%  iii.  straight polyvinyl alcohol fiber, PVA-1 - V = 1.0% and 5.0%  f  f  f  44  30 k  strain hardening due to multiple cracking of matrix  ""•""••""-^^STL-STR @ V,=5.00%  STL-STR @ V =0.76% f  1  0  1  2  3  4  5  Deflection (mm)  Figure 3.1.4a: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes Influence of fiber volume, V = 0.76% and 5.0% of straight-smooth, stainless steelfiber(STL-STR1) f  Five beams were tested for each mix. The average flexural responses for the various mixes are shown in Figures 3.1.4a, b and c. JSCE Flexural Toughness and Flexural Toughness Factor for the various mixes are reported in Table 3.1.6.  It can be noticed that the flexural toughness  increases with increase in fiber volume. Also, both the prepeak and the postpeak response are a function of properties of fibers incorporated in the matrix. In specimens reinforced with 5% steel and polyvinyl alcohol fibers, considerable multiple cracking of the matrix took place and this gave rise to strain hardening response in the prepeak region. On the other hand, for specimens reinforced with 5% polypropylene fibers, no multiple cracking of the matrix took place. Moreover, the strain hardening response observed in the postpeak region was most likely related to the constitutive behavior of polypropylene fibers. 30  1  ,  20  1  PP-STR1 @ V,=5.00%  10 PP-STR1 @ V =1.00% f  0 0  1  2  3  4  5  Deflection (mm)  Figure 3.1.4b: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes Influence of fiber volume, V = 1.0% and 5.0% of straight, smooth polypropylene fibers (PP-STR1) f  45  In the specimens reinforced with steel and polypropylene fibers it was also observed that the majority of crack bridging fibers pulled out during crack opening.  On the other hand, for  specimen reinforced with polyvinyl alcohol fibers, the majority of crack bridging fibers fractured during pullout, which led to relatively steeper drop in post-crack load.  High interfacial  adhesional bond strength and low transverse shear strength of polyvinyl alcohol fibers could be the possible reasons responsible for fracture of fibers.  Table 3.1.6: JSCE Absolute Toughness and Flexural Toughness Factor - Influence of fiber volume Fiber Id.  Fiber Description  Fiber Volume  MOR  Absolute Toughness, T  ja  (%)  (N-m)  (MPa)  Flexural Toughness Factor, oj, (MPa)  . STL-STR  Straight-smooth, stainless steel  0.76  5.51 (0.51)  14.9(1.28)  2.25 (0.19)  5.0  8.59 (0.73)®  42.8(2.87)  6.43 (0.43)  PP-STR1  Straight, smooth polypropylene  1.0  5.14(0.55)  7.41 (0.63)  1.11 (0.09)  5.0  5.19(0.65)  32.8 (3.25)  4.91 (0.10)  PVA-1  Straight polyvinyl alcohol  1.0  6.21 (0.60)  21.5(1.29)  3.23(0.19)  5.0  6.81 (0.86)  39.0(1.74)  5.85 (0.26)  - Figures in brackets indicate standard deviations 30  0 -I 0  I  h  1  2  3  4  5  Deflection (mm)  Figure 3.1.4c: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes Influence of fiber volume, V = 1.0% and 5.0% of straight polyvinyl alcohol fibers (PVA-1) f  3.1.3 Fiber Aspect Ratio Fiber aspect ratio is defined as the ratio of fiber length to fiber diameter. The influence of fiber aspect ratio was investigated by comparing the flexural toughness response of concretes reinforced with the following polypropylene fibers, namely: 46  i.  straight, smooth polypropylene fiber, PP-STR 1 (L=38 mm, d=0.63 mm, L/d=6Q)  ii.  straight, smooth polypropylene fiber, PP-STR2 (L=38 mm, d=0.19 mm, Z/*=200)  Both fibers were virgin polypropylene with an elastic modulus of approximately 3.5 GPa. Five beams were tested for each mix. The average flexural responses for F R C reinforced with these fibers are shown in Figure 3.1.5, and the corresponding JSCE flexural toughness parameters are reported in Table 3.1.7. Two features are apparent in the load versus deflection plots: A higher fiber aspect ratio led to an increase in total energy absorption. Instability was more pronounced in the case of fibers with lower aspect ratio.  30  T  PP-STR2 @ V=0.50%  1  0  2  3  4  5  Deflection (mm)  Figure 3.1.5: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber aspect ratio, V = 0.50% of straight, smooth polypropylene fibers (PP-STR 1 and PP-STR2) f  Table 3.1.7: JSCE Absolute Toughness and Flexural Toughness Factor - Influence of fiber aspect ratio Fiber Id.  Fiber Description  Aspect ratio 1/d  Fiber Volume  MOR  Absolute Toughness,  (%)  (MPa)  TJCI  Flexural Toughness Factor, oj,  (N-m)  (MPa)  PP-STR 1  Straight, smooth polypropylene  68  0.50  5.86 (0.83)  3.5 (0.34)  0.52 (0.05)  PP-STR2  Straight, smooth polypropylene  200  0.50  5.49 (0.35)  6.88 (0.61)  1.03 (0.09)  @  - Figures in brackets indicate standard deviations  3.1.4 Surface Characteristics of Fiber Influence of surface characteristics of fiber on toughness of F R C was.investigated by testing concretes reinforced with the following fibers: 47  i.  straight-smooth, stainless steel fiber, STL-STR; V = 0.76%  ii.  grease coated, straight-smooth, stainless steel fiber, STL-STR; V = 0.76%  iii.  straight polyvinyl alcohol fiber, P V A - 1 ; Vf = 1.0%  iv.  grease coated, straight polyvinyl alcohol fiber, P V A - 1 ; Vf= 1.0%  f  f  Five beams were tested for the each mix mentioned above. Figure 3.1.6a compares the average flexural response for virgin and grease coated steel fibers (STL-STR). The corresponding JSCE toughness parameters are reported in the Table 3.1.8. In the figure, it can be noticed that the prepeak response remained practically unaltered as a result of grease coating of steel fibers. On the other hand, considerable variation in the postpeak flexural response was seen as a result of the grease coating - the postpeak flexural performance of concrete with grease coated steel fibers was inferior in comparison to that of concrete with virgin steel fibers.  This variation in  performance is possibly related to the reduction in fiber-matrix interfacial coefficient of friction (and hence, reduction in interfacial frictional bond) brought about by the coating of grease. 30  ,  n  0 4  1  1  1  1  1  0  1  2  3  4  5  Deflection (ram)  Figure 3.1.6a: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes Influence of fiber surface characteristics, Steel fiber - STL-STR @ V = 0.76% f  48  Table 3.1.8: JSCE Absolute Toughness and Flexural Toughness Factor - Influence of fiber surface characteristics Fiber Id.  Fiber Description  Fiber Volume  MOR  (%)  Absolute Toughness, T  Flexural Toughness Factor, oj,  (MPa)  (N-m)  (MPa)  ]a  STL-STR  Straight, smooth steel  0.76  5.51 (0.51)®  14.9(1.28)  2.25 (0.19)  STL-STR  Grease coated, straight steel  0.76  5.31 (0.43)  10.72 (0.23)  1.61 (0.03)  PVA-1  Straight, smooth PVA  1.00  6.21 (0.60)  21.5 (1.29)  3.23 (0.19)  PVA-1  Greased coated, straight PVA  1.00  5.76 (0.66)  30.31 (1.32)  4.55 (0.20)  - Figures in brackets indicate standard deviations  Figure 3.1.6b compares the average flexural response for virgin and grease coated, polyvinyl alcohol fibers (PVA-1), and the corresponding JSCE toughness parameters are reported in Table 3.1.8. In the figure it can be noted that the total energy absorption was superior in the case of grease coated polyvinyl alcohol fibers. Visually, it was noticed that in the concrete reinforced with virgin polyvinyl alcohol fibers, most fibers had fractured during crack opening. On the other hand, in the concrete reinforced with grease coated polyvinyl alcohol fibers, a greater number of fibers had pulled out during crack opening. It appears that a high adhesional bond between fiber and matrix leads to fracture of virgin fibers, and as a result, energy absorption remains low. The coating of grease possibly reduces the adhesional bond between fiber and matrix. Consequently, a greater number of fibers are able to pull out, and in this process they absorb a greater amount of energy relative to the energy absorbed by the concrete specimen reinforced with virgin fibers.  0 A  1  1  1  1  i  0  1  2  3  4  5  Deflection (mm)  Figure 3.1.6b: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes Influence of fiber surface characteristics, Polyvinyl alcohol fiber - PVA-1 @ V = 1.00% f  49  From the above experimental investigation it can be concluded that the flexural toughness response of fiber reinforced concretes is strongly dependent upon the characteristics of the fiber surface. Dependence of micromechanical properties of interface on characteristics of the fiber surface will be a subject of investigation in Chapters 4-7.  3.1.5 Fiber Geometry Influence of fiber geometry was investigated by comparing the flexural toughness response of concretes reinforced with three different steel fiber types, namely, i.  Straight-smooth, stainless steel fiber, STL-STL, V = 0.76%  ii.  hooked-end steel fiber, S T L - H K D , V = 0.76%  iii.  crimped steel fiber, STL-CR1, V = 0.76%  f  f  f  STL-STR |  /  STL-CR1 STL-HKD  0  1  1  1  2  3  4  5  Deflection (mm)  Figure 3.1.7: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence offibergeometry  Fiber beam specimens were tested for the each case mentioned above.  Flexural load versus  deflection responses for fiber reinforced concretes reinforced these fibers are shown in Figure 3.1.7, and the corresponding JSCE flexural toughness parameters are reported in Table 3.1.9. Comparing the performance of the three fibers, it is clear that for stiff fibers like steel, the geometry of the fiber has a considerable influence on flexural toughness response.  Concrete  specimens reinforced with deformed steel fibers absorbed more energy in comparison to those reinforced with straight-smooth steel fibers. Also, different fiber geometrical configuration gave different degrees of energy absorption. It appears that the main energy absorption mechanism is derived from the cold working of steel that takes place when deformed portions of fibers are 50  pulled out. For the crimped fiber, it was noticed that considerable matrix damage took place during the process of fiber pullout, and this had an unfavorable influence on the overall energy absorption. Table 3.1.9: JSCE Absolute Toughness and Flexural Toughness Factor - Influence of fiber geometry Fiber Id.  Fiber Description  Fiber Volume  MOR  (%)  Absolute Toughness, T  Flexural Toughness Factor, Oi,  (MPa)  (N-m)  (MPa)  ja  STL-STR  Straight, smooth steel  0.76  5.51 (0.51)®  14.9(1.28)  2.25 (0.19)  STL-HKD  Hooked-end steel  0.76  6.21 (0.56)  34.1 (1.98)  5.11 (0.30)  STL-CR1  Crimped steel  0.76  5.87 (0.64)  22.7(1.74)  3.41 (0.26)  - Figures in brackets indicate standard deviations  3.1.6 Shrinkage Properties of Matrix Shrinkage of cementitious matrix largely depends upon the setting, hardening and curing history of concrete and upon the properties of the matrix. For a given history of setting, hardening and curing, it can be expected that a cementitious matrix containing shrinkage compensating admixtures will experience reduced shrinkage. A type of cementitious matrix that falls under this category is available commercially under the name 'non-shrink grout'. Below, comparison is made for the flexural toughness performance for fiber reinforced concretes with two different kinds of matrix, namely: i.  C S A Type 10 normal Portland cement matrix, and  ii.  Non-shrink grout matrix  51  30  V^O.76%  20  4  STL-STR (CSA Type 10 Cement Matrix)  •a 3  STL-STR 10  (Non-shrink Grout Matrix)  0 -f  1  1  1  1  1  0  1  2  3  4  5  Deflection (mm)  Figure 3.1.8: Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of shrinkage properties of matrix  Flexural toughness performance of composites reinforced with straight-smooth, stainless steel fibers at Vf=0.76% was investigated. Five beams were tested for each combination of matrix and fiber. Figure 3.1.8 compares the average flexural responses of fiber reinforced concretes for the two types of matrices investigated. The corresponding results are tabulated in the Table 3.1.10. It can be seen that the fiber reinforced concrete specimens with normal Portland cement (CSA Type 10) matrix absorbed more energy in comparison to the non-shrink grout matrix. This behavior is possibly related to the to the higher fiber pullout resistance offered by the normal Portland cement matrix. This aspect is explored from the micromechanical perspective in Chapters 4.  Table 3.1.10: JSCE Absolute Toughness and Flexural Toughness Factor - Influence of matrix shrinkage Fiber Id.  STL-STR  Fiber Description  Straight, smooth steel  Matrix  Fiber Volume  MOR  (%) CSA Type 10 Normal Portland Cement Non-shrink Grout  Absolute Toughness, T  Flexural Toughness Factor, Oj,  (MPa)  (N-m)  (MPa)  0.76  5.51 (0.51)®  14.9(1.28)  2.25(0.19)  0.76  5.22 (0.60)  9.1 (0.76)  1.36(0.11)  - Figures in brackets indicate standard deviations  52  ja  3.2 Characterization of Flexural Toughness of Fiber Reinforced Concretes Toughness is the most important property of any fiber reinforced concrete. One variable that significantly  influences the measured  toughness  of F R C is the  stiffness  of testing  machine/loading system on which the test is performed. When a loading system applies a load on a test specimen, there is a reactive force on the loading system itself, and due to this reactive force, the loading system deforms and stores energy. The extent of energy built up in the loading system depends upon its stiffness - the softer the testing system, the greater the energy built up in it. At the instant when the beam cracks, a part of the stored energy is suddenly released into the specimen. The specimen absorbs this incoming energy, and as a consequence, the amount of energy absorbed by the specimen from the external load is reduced. If an F R C specimen is inadequately reinforced, the sudden release of energy causes unstable propagation of cracks, thus influencing the post-cracking flexural response. Unfortunately, the currently available methods of quantifying toughness using a flexural specimen are riddled with ambiguities and considered unacceptable.  A S T M C1018 test  procedure [4] to characterize flexural toughness of concrete suffers from the above-mentioned problem. Thus, the test procedure is particularly worrisome when concrete reinforced with a low volume fraction of synthetic or other fiber is tested using an open-loop test machine with low stiffness under load control. Under such circumstances, the brittle nature of concrete manifests itself in the post-peak region of the load-displacement curve and two specific features become apparent (Figure 3.2.1):  •  Immediately following the peak load, the load drops suddenly in an uncontrolled and unstable manner depending upon the stiffness of the machine. The data-logger even when running at a very high frequency does not usually record any points in this region.  •  After the unstable part, the curve attains a stable softening level, during which loads are again functions of the test machine characteristics. If the machine stiffness is too low, the release of energy during the unstable part will be high and the specimen may suffer a greater damage due to the higher incoming energy. If so, the loads during softening will be lower and the post peak load carrying response will be inferior as compared to close-loop testing (high stiffness) where this sudden release of energy and related damage are not allowed to occur (Figure 3.2.1).  53  20 Open-loop Test Method 16  Close-loop Test Method INSTABILITY  1  4  Damage due to Instability  0 0.0  0.5  1.0  1.5  2.0  Deflection (mm) Figure 3.2.1: Comparison of Close-loop Test Method with Open-Loop Test Method for Flexural Tests on Fiber Reinforced Concrete. Notice the large load instability that occurs during at the peak load during an open-loop test, and the related damage [6]  For most fiber reinforced concretes, rather than the pre-peak response, which is not expected to be very different from plain concrete, it is the post-peak response that is of a primary interest. With the above mentioned artifacts in the post-peak load displacement part of the curve, any analysis carried out using the standardized A S T M C1018 technique (or any other technique for that matter) is going to be of limited use and even misleading. One possible solution, naturally, would be to always use a close-loop machine, especially when pronounced instability in the postpeak part of the curve is expected. Unfortunately, close-loop tests are difficult to run and time consuming.  These tests are also expensive and necessitate sophisticated and complex  instrumentation and data acquisition devices. Since most testing laboratories are equipped only with open-loop test machines, any new methodology developed should be designed such machines and still provide a valid post-peak response.  A new test method, called the Residual Strength Test Method [6,7], has recently been developed which characterizes the softening part of the curve by first loading the specimen in conjunction with a steel plate. Steel plate absorbs the energy released by the machine at the occurrence of matrix cracking, and in this manner, damage to the specimen resulting from the sudden release of energy is minimized. After first cracking, the specimen is unloaded, the steel plate is removed, and the specimen is reloaded. The post-peak response thus obtained on reloading is expected to give the true stress transfer capability (and the toughening capability) of fibers across a crack. 54  Until this date, no systematic and exhaustive study has been carried out to assess the validity of the Residual Strength Test Method. This aspect has been explored in the following investigation. The major objectives of this investigation are: •  To assess the validity of the Residual Strength Test Method. This is achieved by examining the data obtained from the proposed test method and comparing it with the results obtained from an open-loop test machine with low stiffness and a close-loop test with high stiffness.  •  To assess the capability of the Residual Strength Test Method to identify the influence of various fiber characteristics such as type, length, configuration, volume fraction, geometry, and the elastic modulus.  3.2.1  Residua] Strength Test Method (RSTM) - Assessment and Calibration  In the Residual Strength Test Method, fiber-reinforced beams are pre-cracked under third-point flexural loading in parallel with a 12 mm thick steel plate as shown in Figure 3.2.2a and b. The steel plate provides the support and absorbs the energy that is released from the machine at the occurrence of the peak load when compliance of specimen changes suddenly. At a net deflection between 0.25 mm and 0.50 mm, the cracked beam is unloaded, residual beam deflection on complete unloading is recorded (relaxation of beam during unloading is monitored by LVDT's), and this is followed by the removal of the steel plate. The cracked beam is then reloaded without the steel plate in four-point bending to obtain the residual load-deflection curve. The loads supported by this beam at 0.5, 0.75, 1.0 and 1.25 mm of beam deflection are then averaged and normalized to obtain Residual Strength, RS values by using an elastic analysis. In other words,  4 where P .s, 0  P0.75,  P1.0,  P1.25  J  •  "  correspond to the load values at 0.5, 0.75, 1.0 and 1.25 mm beam  deflection, respectively, L is the test span, b is width of the beam and d is the depth of the beam. Notice that the Residual Strength, RS, which has the units of stress, is not the true stress in the specimen (which has already cracked and the application of the elastic equation is strictly not valid), but a measure of an approximate "engineering" stress.  55  Loading machi crosshead  Concrete beam specimen ' 100 x 100 x350 mm  «JBt Japanese Yoke  Stainless steel plate 12 x 100 x 350 mm  Further, the Residual Strength Index, RSI can be calculated as:  RSI{%) =  MOR  x 100  3.3  where RS is the Residual Strength as defined in Equation 3.2, and MOR is the modulus of rupture of the material calculated by testing beams (without the steel plate) in accordance with the 56  A S T M Test Method C 78 [218]. This test procedure, an assessment of which was the principal objective of this investigation, is abbreviated as R S T M .  3.2.1.1 Experimental Procedure Materials and Mixes To assess Residual Strength Test Method, three types of polymeric fibers were investigated: i.  Fiber PP1: Fiber PP1 was 100% polypropylene with a special blend of fibers with lengths in the range of 11-20 mm. This fiber was fibrillated.  ii.  Fiber PP2: Fiber PP2 was also 100% polypropylene with a length of 19 mm. This was a monofilament fiber.  iii.  Fiber N L 1 : Fiber NL1 was 100% nylon with a length of 19 mm. Like the fiber PP2, this was a monofilament fiber without fibrillation. Table 3.2.1: Mix Proportions Cement (kg)  Silica-fume (kg)  Water (kg)  Sand (kg)  460  51  179  767  Aggregate (10 mm) Superplasticizer (ml/100 kg of Cement) (kg) 893  600  The mix proportions of the concrete matrix are given in Table 3.2.1. The design compressive strength was approximately 70 MPa. A particularly high design compressive strength was chosen to exaggerate the problem of instability. Cement, aggregate and part of the silica fume were first dry mixed in a pan type mixer. The rest of the silica-fume and the superplasticizer were added to water to form a slurry. This slurry was then added to the dry-mix to obtain a workable wet-mix. Lastly, fibers were added gradually to avoid any fiber balling. Fiber addition rates were 0.1, 0.3 and 0.5 percent by volume of the PP1 and PP2 fibers and 0.5 percent by volume of the NL1 fibers.  Six compression cylinders of size 100 mm x 200 mm and 15 beams of size 100 mm x 100 mm x 350 mm were cast for each mix. External vibration was used for consolidating the concrete into the moulds. Specimens were demolded 24 hours after casting and then cured in a tank with lime saturated water for 28 days before testing.  57  Typical values of the slump, air content and unit weight for the control and 0.5% fiber volume concretes are presented in Table 3.2.2. Notice that there was some drop in the slump values due to fiber reinforcement, but generally the mixes were well workable and in the practical range.  Table 3.2.2: Fresh Properties of Concrete Mixes Mix Control Plain FRC - PP1 Fiber* FRC - PP2 Fiber* FRC - NL1 Fiber*  Slump (mm) Air Content (%) Unit Weight (kg/m ) 3  140 120 110 80  1.6 1.5 2.0 1.4  2405 2402 2382 2413  Compressive S trength (MPa) 7 days 28 days 54.5 71.4 52.9 71.5 54.5 71.8 55.2 72.8  *Fiber volume fraction of 0.5%  Test Procedure Close-loop testing was conducted in a 150 k N MTS machine with a function generator. Figure 3.2.3a shows the experimental test setup and Figure 3.2.3b shows the schematic of the controls.  Figure 3.2.3a: Test on a Fiber Reinforced Concrete Beam using the Close-Loop Test Method.  As seen in the Figure 3.2.3a, a crack mouth opening displacement (CMOD) transducer monitored the crack opening displacement and supplied the feed-back signal to the servocontroller. A Japanese Yoke [5] was mounted around the specimen to record the net deflections so that the extraneous deflections resulting due to the settlement of supports, crushing at the load points, and the loading fixture deformations were eliminated and only the net deflections were 58  recorded. Two LVDT were used to record the net deflection of the beam at mid-span and data from these were then averaged. At the start of the program, specimens were tested under control signals first from the CMOD transducer and then from the two LVDTs determine whether the source of feedback signal made any difference to the load-displacement curves. No particular difference was noticeable. The rest of the close-loop tests were then performed under CMOD control, as illustrated in Figure 3.2.3b. In a typical test, the CMOD, the applied load, and the vertical net displacement values were recorded using a digital data acquisition system based on a personal computer. The close-loop Test Method is abbreviated as CLTM. Note that the Open-loop Test Method (OLTM) has already been described in Section 3.1.  SIGNAL GENERATOR  PRIOR TO FIRST CRACK STRAIN RATE = 2X1C)jsec  CONTROL SIGNAL  EXTERNAL PROGRAM  AFTER FIRST CRACK CMOD RATE = 0.0005 mm/sec  t  SERVO CONTROLLER  EXTERNAL FEEDBACK FEEDBACK SELECTOR  A.  SERVO VALVE  t INTERNAL TRANSDUCER OUTPUT (not used)  OUTPUT SIGNAL (EXTERNAL)  HYDRAULIC ACTUATOR  a  SPECIMEN  CMOD TRANSDUCER (SENSITIVITY = 600 wV/mm)  ERROR DETECTOR  UMIT DETECTOR  TO SYSTEM FAILSAFE INTERLOCK  Figure 3.2.3b: Schematic of controls in a flexural Close-loop Test Method  59  3.2.1.2 Results The differences between the curves from the close-loop and the open-loop tests were discussed previously and are illustrated in Figure 3.2.1. Typical close-loop and open-loop curves for concrete with PP1 fiber are compared in Figures 3.2.4 and 3.2.5 for 0.1 and 0.3% fiber volume fractions, respectively. The same for F R C with PP2 fibers are compared in Figure 3.2.6 and 3.2.7.  In general, for both fiber volume fractions, a large instability region was noticeable  immediately after the peak load when the displacement jumped from -0.04 mm at the peak load, to -1.0 mm (25 fold) at the end of the instability. This sudden release of energy from the machine in all likelihood created some damage in the specimen, with cracks undergoing a large uncontrolled opening, and the load carrying capacity beyond the instability was affected.  25  25 Fiber PP1 @ V, = 0.1S  20 L ^ \  Close-loop Test Method  15  1 10  Open-loop .^Test Method  — I  0.0  1  0.5  '  Fiber PP1 @ V = 0.3% f  Close-loop Test Method  5*.\ >*.\  Open-loop Test Method  ^*=  a  1  1.0  1.5  — i  2.0  0.0  0.5  Deflection (mm)  1-  '  1.0  1.5  ::  2.0  Deflection (mm)  Figure 3.2.4: Comparison of Close-loop and Open-loop  Figure 3.2.5: Comparison of Close-Loop and Open-loop  curves for concrete with 0.1% of Fibrillated Polypropylene  curves for concrete with 0.3% of Fibrillated Polypropylene  Fiber (PP1)  Fiber (PP1)  Close-loop Test Method  2  Open-loop Test Method  0.5  1.0  1.5  2.0  0.5  Deflection (mm)  1.0  1.5  Deflection (mm)  Figure 3.2.6: Comparison of Close-loop and Open-loop  Figure 3.2.7: Comparison of Close-Loop and Open-loop  curves  curves for concrete with 0.3% of Monofilament  for  concrete  with  0.1%  of  Monofilament  Polypropylene Fiber (PP2)  Polypropylene Fiber (PP2)  60  In Figures 3.2.8-10, the close-loop load-deflection plots for concrete with PP1 fiber are compared with those from the Residual Strength Test Method (RSTM) for fiber volume fractions of 0.1, 0.3 and 0.5%, respectively. The same for FRC with PP2 fibers are compared in Figures 3.2.11-13. The corresponding curves for FRC with NL1 fibers at a fiber volume fraction of 0.5% are compared in Figure 3.2.14. Notice that the curves from the C L T M and R S T M match quite well in the post-peak region. The R S T M curves start at a certain finite displacement, which is the same as the residual displacement in the specimen after the first loading with the steel plate. Notice also that the R S T M curves start at varying initial displacements,, indicating that varying levels of damage were induced in these beams during the first loading with the steel plate. These minor variations in the induced damage during the first loading, however, do not appear to alter or influence the load carrying capacity during the second loading.  25 Fiber PP1 @ V = 0.1<i  20  Fiber PP1 @ V = 0.3%  f  15 -  /  10 -  Close-loop Test Method  I  Close-loop Test Method Residual Strength Test Method  Residual Strength ^ Test Method  5 0 — 0.0  f  y " i  MMSMMUMMHBESE  1  0.5  1.0  1.5  2.0  0.5  Deflection (mm)  • 1.0  1.5  2.0  Deflection (mm)  Figure 3.2.8: Comparison of Close-Loop and Residual  Figure 3.2.9: Comparison of Close-Loop and Residual  Strength Test Method Curves for Concrete with 0.1% of  Strength Test Method Curves for Concrete with 0.3% of  Fibrillated Polypropylene Fiber (PP1)  Fibrillated Polypropylene Fiber (PP1)  61  25 20  Fiber PP1 @ V, = 0.5% I  Qose-loop  '  Test Method  Fiber PP2 @ V=0.1< Close-loop Test Method  Residual Strength Test Method  2 Residual Strength Test Method  •+-  0.5  1.0  1.5  2.0  0.5  Deflection (mm)  1.0  1.5  2.0  Deflection (mm)  Figure 3.2.10: Comparison of Close-Loop and Residual  Figure 3.2.11: Comparison of Close-Loop and Residual  Strength Test Method Curves for Concrete with 0.5% of  Strength Test Method Curves for Concrete with 0.1% of  Fibrillated Polypropylene Fiber (PP1)  Monofilament Polypropylene Fiber (PP2)  25 Fiber PP2 @ V, = 0.3%  Fiber PP2 @ V, = 0.5%  Close-loop  Close-loop  Test Method  Test Method  Residual Strength  Residual Strength  Test Method  0.5  Test Method  1.0  1.5  2.0  0.5  Deflection (mm)  1.0  1.5  2.0  Deflection (mm)  Figure 3.2.12: Comparison of Close-Loop and Residual  Figure 3.2.13: Comparison of Close-Loop and Residual  Strength Test Method Curves for Concrete with 0.3% of  Strength Test Method Curves for Concrete with 0.5% of  Monofilament Polypropylene Fiber (PP2)  Monofilament Polypropylene Fiber (PP2)  25  Deflection (mm) Figure 3.2.14: Comparison of Close-Loop and Residual Strength Test Method Curves for Concrete with 0.5% of Monofilament Nylon Fiber (NL1)  62  In Table 3.2.3, the Residual Strength (RS) values (Equation 3.2), for the various fibers, are reported. Generally speaking, the RS values obtained using the Open-Loop Test Method were the lowest and those obtained using the close-loop method were the highest. The values from the proposed R S T M were in-between these two extremes, but were closer to values from the closeloop method. This is to be expected, given that substantial damage is induced in the specimen during the open-loop testing. In the R S T M , on the other hand, the presence of the steel plate substantially reduces the damage and the technique more or less imitates close-loop testing. The good agreement between the C L T M and the R S T M data indicates that the proposed method is capable of accurately measuring the capacity of fiber reinforced concrete to carry loads beyond cracking.  Table 3.2.3: Residual Strengths, RS for various mixes Residual Strength, RS (MPa) Mix/Fiber  FRC- PP1 Fiber Fibrillated Polypropylene  FRC-PP2 Fiber Monofilament Polypropylene  FRC-NL1 Fiber Monofilament Nylon  V (%)  Open-Loop Test  Close-Loop Test  Method (RSOLTM)®  Method (RSCLTM)  0.1  0.61 (0.11)*  0.67 (0.08)  0.63 (0.12)  0.3  0.78 (0.35)  1.27 (0.02)  1.19(0.13)  0.5  —  1.88 (0.21)  1.75 (0.17)  0.1  0.33 (0.06)  0.32 (0.04)  0.31 (0.02)  0.3  0.4(0.12)  0.74 (0.09)  0.66 (0.08)  0.5  —  0.87 (0.24)  0.69 (0.22)  0.1  ~  -  -  0.3  _  0.5  —  1.16(0.19)  1.02 (0.15)  f  Residual Strength Test Method  (RS ) RSTM  * Figures in brackets indicate standard deviation; ® For the Open-loop Test Method where P  A  5  and P JS were in the unstable region, the Residual Strength (RS) was 0  calculated by ignoring the load values (in Equation 3.2) falling under the unstable region.  The C L T M results for the various fiber types are further compared with the R S T M results in Figure 3.2.15. Note again a very good match between these techniques at all fiber volume fractions. Notice also that at a certain fiber volume fraction, the C L T M predicts the PP1 fiber to be more effective than the PP2 fiber, and an identical conclusion can be drawn from the R S T M data. Further, the nylon fiber (NL1) is less effective than the PP1 fiber but more effective than the PP2 fiber as per the close-loop data, and an identical conclusion can be drawn from the Residual Strength Test Method. These data indicate very clearly that the R S T M is entirely 63  capable of distinguishing between not only the various fiber volume fractions of the same fiber type but also between the various fiber types at a given volume fraction. _  2 0.9!<  cu CO  OS  1 0.8  ant  Residual Strengh Test Method  1  1  o.sy  !  m-gg  n Fibrillated Monofilament Monofilament polypropylene polypropylene nylon fibers fibers (PP1) fibers (PP2) (NL1)  — i  Fibrillated Monofilament Monofilament polypropylene polypropylene nylon fibers fibers (PP1) fibers (PP2) (NL1)  Figure 3.2.15a: Residual Strength for Various Mixes using  Figure 3.2.15b: Residual Strength for Various Mixes using  the Close-Loop Test Method  the Residual Strength Test Method  One way to assess the effectiveness of the R S T M as compared to the C L T M , is to calculate the difference, S, as: <5 =  RS CLTM  xlOO  3.4  25 -,  PP2 fibers  t£  20 o c  I  15 -  5  10 -  NL1 fiberd  <&  u c  5 0 -  —i 0.1  r  0.2  0.3  0.4  0.5  0.6  0.7  Fiber volume (%) Figure 3.2.16: Percentage Difference (5) in RS Values between the Close-Loop Test Method and the Residual Strength Test Method  These values of the difference, 8, are plotted in Figure 3.2.16 for the various fiber types. Notice that 8 is always positive (i.e., the R S T M always underestimates the capacity) and is in the range of 5-25 percent depending on the fiber type.  It also appears that the difference 8 is directly  related to fiber efficiency; a fiber with lower efficiency indicates a higher 8. This is logical since 64  a less effective fiber with a poorer pull-out resistance may lead to more damage during the first loading cycle and thus sustain lower loads in the second loading cycle.  3.2.1.3 A Round-Robin Test Program to Validate RSTM To assess the validity and inter-laboratory reproducibility of the proposed Residual Strength Test Method, a round-robin test program was carried out between the University of British Columbia (UBC), The Ministry of Transportation of Ontario (MTO), and the Technical University of Nova Scotia (TUNS). This section describes the results obtained from this test program. Twenty-five fiber reinforced concrete beams (100 mm x 10 mm x 350 mm) were cast at UBC with mix details given in Table 3.2.1. Only one fiber volume addition rate of 0.3% was investigated. Fifteen beams were retained at UBC, five beams were sent to MTO and the remaining five to TUNS. The division of tasks for the three laboratories is outlined in Table 3.2.4. All of the tests were performed at the specimen age of 14 days. Table 3.2.4: Test planning Institute UBC  A S T M C1018  Residual Strength Test  A S T M C1018  (Close-loop)  Method  (Open-loop)  V  V  V  MTO  V  TUNS  V  Results of the open-loop and close-loop tests performed on identical beams at UBC are given in Figure 3.2.17. Notice that although the two test configurations result essentially in identical peak loads, the post-peak load-displacement behaviors are markedly different for the reasons stated earlier. Figure 3.2.18 compares the load-displacement curves recorded in close-loop conditions with those recorded during the second stage loading of the proposed RSTM technique by the three institutes. Notice that for a proper comparison, the starting displacements for the RSTM curves are shifted by the amount of residual displacement in the specimen (considering relaxation) after the first loading with the steel plate. The residual displacement at the end of first loading ranged from 0.25 mm to 0.75 mm and was noticeably different for each of the three laboratories. One would expect that the residual displacement, apart being a function of machine stiffness, is also a function of operator experience. UBC, having had more experience with the test procedure, managed to halt the loading sufficiently rapidly after the creation of the crack in the specimen, and thus had lower residual displacements at the end of the first stage. 65  Closed-loop versus  Close-loop  Residual Strength Test Method  versus  Open-loop  Open-loop - Close-loop  Closed-loop  1 .3  •s o J  TUNS -UBC MTO  RSTM  M i i *1PP 0.00  0.50  1.00  1.50  0.00  2.00  0.50  Deflection (mm)  for  Concrete  with  0.3%  1.50  2.00  Figure 3.2.18: Comparison of Close-Loop and Residual  Figure 3.2.17: Comparison of Close-Loop and Open-Loop Curves  1.00 Deflection (mm)  of  Strength Test Method Curves for Concrete with 0.3% of  Fibrillated  Fibrillated Polypropylene Fiber (PP1)  Polypropylene Fiber (PPl)  In spite of the variation in the residual displacement from one laboratory to the other, interestingly, the residual load carrying capacity remained relatively unaffected as seen in Figure 3.2.18. As stipulated in the proposed technique, the load-displacement curves were analyzed for Residual Strength, RS, which are reported in Table 3.2.5. Notice that the Residual Strength reported by the three laboratories are comparable, and the within batch standard deviations (0.17, 0.08 and 0.18) are not very different from the inter-laboratory standard deviation (0.14). A variation in the residual displacement (and hence, in the extent of damage in the beam) at the onset of the Residual Strength test does not appear to affect the residual strength capacity in any way; the specimens tested at TUNS suffered most damage during the first stage of loading, but also recorded the highest Residual Strength.  Table 3.2.5: Residual Strengths, RS - Canadian Round-Robin Test Program [221] Residual Strength, RS (MPa) Open-loop  Close-loop  UBC  UBC  0.78  0.96  Residual Strength Test Method UBC  MTO  TUNS  0.97 (0.17)*  0.81 (0.08)  1.03 (0.18)  Average=0.94 (0.14)  * Figures in brackets indicate standard deviation  Thus, from the results obtained from the Canadian Round-Robin Test Program it was concluded that the Residual Strength Test Method is capable of characterizing the contribution of fibers in the post-matrix cracking stage with acceptably low within-batch and inter-laboratory variation. 66  3.2.1.4 Discussion In the following, the stress state in the composite (FRC beam + steel plate) at the instant matrix cracking occurs is analyzed (i.e., at the instant beam is unloaded). The response of a concrete beam to an applied load while supported by a plate is expected to be significantly different from that of the beam loaded without the plate. Assuming that the concrete beam and the steel plate act monolithically (i.e., there is no slip between the two and strain compatibility is maintained), one can show that for the composite beam, the depth of the neutral axis prior to cracking is: nbd. d+^-  +bd  2 Vbd + nbd )  y=-  (d)  3.5 r  where, n = modulus ratio between concrete and steel (~6), d and b = depth and width of concrete beam, and d = depth of the steel plate. p  The maximum tensile stress carried by the beam at the instant of cracking is given by [219]: CT  M (d-y)  3.6  c  =  V B  '  D  12  +  B  J  -  D  ~2  y  d f + nb- - + nbd d n  12  "  V  d  3  E  +  -L-y  and the corresponding stress in steel plate is given by [219]: CT  nM (d + c  =  bd  3  12  _ d + bd\  + nb- L  12  d-y) + nbd d +  '  n  3.7  \  f  2  •y  where, M is the moment at cracking. Assuming full compatibility between steel plate and c  concrete beam (i.e., both steel plate and concrete beam flex about a combined neutral axis) at the instant beam cracks, the maximum tensile stress in concrete, c calculated using Equation 3.6 is c  given in Table 3.2.6 for the various mixes. Clearly, the maximum stress in concrete attained with the steel plate underneath remains significantly lower than the corresponding modulus of rupture value for the various mixes. This indicates that when the concrete beam is loaded in parallel with the steel plate, at failure it carries peak stresses far below its flexural capacity and 67  that its compliance is not significantly altered. Moreover, it can also be shown that stresses in concrete and the corresponding damage to the concrete beam will reduce further as the thickness of steel plate is increased. Table 3.2.6: Maximum concrete stress at peak load with and without steel plate C oncrete beam + Steel Plate Modulus of Rupture MOR (MPa)  Cracking  Deflection at  Load  Cracking Load  (kN)  (mm)  (MPa)  5.64(0.41)*  28.3 (5.83)  0.037 (0.0096)  1.60  5.46 (0.64)  26.9 (5.15)  0.036 (0.0092)  1,52  4.83 (0.52)  34.1 (9.21)  0.038 (0.0086)  1.93  $  Mix/Fiber  FRC- PP1 Fiber Fibrillated Polypropylene FRC-PP2 Fiber MonofilamentPolypropylene FRC-NL1 Fiber Monofilament Nylon  $  Theoretical $  Per  Concrete Stress® a  c  - Data for beams with different V (Table 3.2.3) combined for calculating average values and standard deviations reported in the table ® - Maximum concrete stress calculated assuming full compatibility between concrete beam and steel plate * - Figures in brackets indicate standard deviation f  Note that full compatibility between steel plate and concrete beam is assumed in the above calculations. In reality, however, some slip will occur between the two, which will increase the stress in concrete, and the actual value of maximum concrete stress, cr , may be expected to lie c  somewhere between the theoretical value of o (Table 3.2.6) and the corresponding Modulus of c  Rupture values.  The exact distribution of the applied load between the concrete beam and the steel plate will depend upon the extent of slip possible between the two. If the two remain monolithic slip, the steel plate will carry a much larger share of the load and the stresses in the concrete as described above will remain low. As the possibility of slip between the two increases, so do the stress in concrete and the portion of the load carried by concrete beam. At the limit, if concrete beam and steel plate act totally independently of each other (i.e., both steel plate and concrete beam flex about their own centroidal axes), the concrete beam will carry nearly all of the applied load with only a very minor portion (< 2%) carried by the plate.  68  3.2.2  Residual Strength Test Method - Performance of Various Composites  After having established the validity of the test technique, the technique was applied to fiber reinforced concrete carrying various volume fractions of polymeric and steel fibers. Based on the extensive Residual Strength data reported below, it is clear that the technique is highly suitable for characterizing the mechanical response of fiber reinforced composites.  3.2.2.1 Materials, Mixes and Testing Two sets of beams were cast as described below. In Set 1, the objective was to investigate fibrillated and monofilament polymeric fibers at low to medium dosage rates in concrete. In Set 2, concretes reinforced with high volume fractions of steel, P V A and deformed polypropylene fibers were investigated. The concrete mix proportions are given in Table 3.2.7. Cement, sand and aggregate were first mixed dry in an Omni-mixer, followed by the addition of water. Lastly, fibers were added gradually to avoid any fiber balling. For each mix, eight beams of size 100 mm x 100 mm x 350 mm were cast using external vibration. In addition, six compression cylinders of size 100 mm x 200 mm were cast for compressive strength determination.  Table 3.2.7: Mix Proportions Cement (kg)  Water (kg)  Sand (kg)  Aggregate (10 mm) (kg)  Air Entraining Agent (ml/kg of cement)  400  200  1000  800  0.1  Set 1: Polymeric Fibers at Low to Medium Dosages In the first set, a total of seven different polymeric fibers—five micro-denier and two macrodenier—were investigated. The details of the test program are given in Table 3.2.8. As seen, both the Open-Loop Test Method (OLTM) and the Residual Strength Test Method (RSTM) were employed to characterize the composites (see Sections 3.1 and 3.2 for details). Of the seven fibers investigated, six were of polypropylene and one was of nylon (Table 3.2.8). Further, of the six polypropylene fibers, four were fibrillated (PPF-1 to PPF-4) while the remaining two were monofilament type (PPM-1 and PPM-2).  The nylon fiber (NLM-1) was also a  monofilament type. A l l four fibrillated fibers had similar geometrical configurations and were procured from different manufacturers. The two monofilament polypropylene fibers investigated were also procured from different manufacturers. Monofilament PPM-1 fiber was 38 mm long  69  and had a diameter of 0.19 mm. Monofilament PPM-2 fiber was also 38 mm long but had a diameter of 0.63 mm.  Table 3.2.8: Fibers Investigated in Set 1 and Details of Test Program [220] Fiber and Mix Id.  Fiber Information  Length  Test Method  v = 0.2%  (mm)  f  OL"  RS  v = 0.4% f  OL"  +  RS  +  Fibrillated 38 X X X X Polypropylene Fibrillated PPF-2 38 X X X X Polypropylene Fibrillated PPF-3 38 Polypropylene Fibrillated PPF-4 38 Polypropylene Monofilament PPM-1 38 Polypropylene Monofilament PPM-2 38 X X X X Polypropylene Monofilament NLM-1 38 X X X X Nylon OL - Open-Loop Test Method; RS - Residual Strength Test Method PPF-1  v = 0.5% t  OL~  RS  X  v = 0.6% f  OL~  RS  X  X  X  X  X  v = 0.8% f  OL"  RS  X  X  X  X  X  X  X  X  -  -  -  -  X  X  -  -  -  -  X  X  -  •, -  -  X  X  X  X  X  X  X  X  X  X  X  X  +  :  +  +  +  Set 2: Large Diameter Steel and Polymeric Fibers at High Dosages The four commercially available macro fibers (large diameter) investigated in Set 2 are described in Table 3.2.9. The mixes and the test program are described in Table 3.2.10. Of the four fibers investigated, one was polypropylene fiber, another was polyvinyl alcohol fiber and the remaining two were deformed steel fibers.  One hybrid mix was also investigated, where both  polypropylene and steel fibers were combined in the same mix. As in the case of the Set 1 specimens, both Open-Loop and Residual Strength Test Methods were employed.  Table 3.2.9: Fibers investigated in Set 2 Fiber Material Geometry  Crosssectional shape  Length Diameter  Size Amplitude  (mm)  (mm)  E  J f  (mm)  Wave length (mm)  (MPa)  (GPa)  PP-STR  Polypropylene  Straight  Oval  50  0.63  -  -  450  3.5  PVA-1  PVA*  Flattenedends  Oval  29  0.42  -  -  1100  23  ST-HKD  Steel  Hooked-end  Circular  35  0.55  -  1115  210  ST-CR1  Steel  Crimped  Crescent  38  2.0 x 0.35  7.5  1037  210  * PVA - Polyvinyl Alcohol;  @ f - tensile strength; t  •  $ E - Fiber Elastic Modulus f  70  -  1.0  •  Table 3.2.10: Various Mixes Investigated in Set 2 and Details of Test Program Fiber and Mix Id.  Fiber Material and Type  Test Method V = 0.76% f  PP-STR PVA-1 ST-HKD ST-CR1 HBRD-1  Straight Polypropylene Straight Polyvinyl Alcohol Hooked-end Steel Crimped Steel Hybrid 1.6% PP-STR +0.25% ST-HKD  OL~  RS  -  v = 1.0%  v = 1.6%  f  OL~  RS  -  -  -  -  X  f  OL"  RS  -  X  X  X  X  -  x  X  -  -  +  v = 3.0% f  Vf=PP @ 1.6% + Steel @ 0.25% OL~ RS  OL"  RS  X  X  X  -  -  X  X  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  X  X  +  +  +  +  Typical values of slump, V - B time, air content, and unit weights for the mixes in Set 2 are given in Table 3.2.11. Notice that all mixes were adequately workable and the V - B time increased with an increase in the fiber dosage rate. Testing procedures are similar to those described in the Sections 3.1 and 3.2.  3.2.2.2 Results Compressive Strengths Twenty-eight day compressive strengths for mixes belonging to Set 1 are given in Table 3.2.12. Clearly, influence of fiber type and fiber volume on compressive strength was not seen. The twenty-eight day compressive strength for mixes belonging to Set 2 are given in Table 3.2.13. Again, no influence of fiber type and fiber volume fraction on the compressive strength was noticeable.  Analysis of Flexural Toughness: Specimens from Set 1 In Set 1 as described, the performance of four fibrillated polypropylene (PPF-1 to PPF-4), two monofilament polypropylene (PPM-1 and PPM-2), and one monofilament nylon (NLM-1) fibers was investigated at fiber volume fractions of up to 0.8%. To illustrate the general performance, the curves for fiber PPF-1 are shown in Figures 3.2.19a-e for an increasing fiber volume fraction from 0.2% to 0.8%. Curves for the other fibers depicted similar qualitative trends and hence are not reproduced in the interest of brevity. In Figures 3.2.19a-e, notice that for concrete mixes with very low fiber volume fractions (Vr=0.2%), considerable instability occurred in the openloop tests. Even in such cases, the Residual Strength Test Method was able to capture the  71  toughening due to fiber reinforcement. The instability problem worsened for mixes containing nylon fibers (NLM-1) where even for fiber volume fractions as high as 0.4%, no reading could be recorded in the post-peak region (Figure 3.2.20). Once again the Residual Strength Test Method was able to capture fiber toughening in the case of nylon fiber at all fiber volumes including 0.4%.  72  Fiber NLM-1 @Vf=0.4%  Fiber PPF-1 @Vf=0.8% 20  o 0  2  3  4  0  Deflection (mm)  Figure 3.2.19e: Load-Deflection Curves  1  m  i 1  i  i  i  i  2  3  4  5  for  F R C with  Deflection (mm) Figure  for F R C with  3.2.20: Load-Deflection  Curves  N L M - 1 Fiber at Dosage Rate of 0.4%  PPF-1 Fiber at Dosage Rate of 0.8%  As the fiber dosage is increased (Figures 3.2.19a-e), one can notice a decrease in the load instability in the O L T M . One can also notice an increase in the post-peak load carrying capacity as measured by the R S T M and hence a related increase in the Residual Strength (RS) values (Equation 3.2). The R S T M was thus capable of predicting the post-peak toughness performance for composites with a wide range of various fiber volume fractions—both low and high.  The results obtained for all the fibers tested in Set 1 are given in Table 3.2.14, where the modulus of rupture (MOR) from the open-loop test, and the Residual Strength (RS, Equation 3.2) and Residual Strength Index (RSI, Equation 3.3) values from the Residual Strength Test Method are reported. The RS values are plotted in Figure 3.2.21 as a function of fiber volume fraction for the various fibers. reproduced.  The RSI values follow similar trends and hence are not  As seen clearly in Figure 3.2.21, the Residual Strength, RS, increases almost  linearly with an increase in the fiber volume fraction. For the fiber volumes tested, interestingly, Residual Strength, RS did not reach an asymptotic limit as is often expected. The performance of fibrillated fibers PPF-1 and PPF-2 was very similar at various fiber volume fractions. Also, the performance of the other two fibrillated fibers (PPF-3 and PPF-4), which were tested only at a fiber dosage rate of 0.5%, compared well with that of PPF-1 and PPF-2 fibers at the given volume fractions. Monofilament polymeric fibers had inferior performance in comparison to their fibrillated counterparts. Among the two monofilament fibers investigated, the performance of the small diameter fiber PPM-1 was superior to that of the large diameter fiber PPM-2. A small diameter resulted in an increased fiber surface area for a given volume fraction and hence a  73  greater total frictional resistance during the process of fiber pullout during a test. In general, for the fibrillated fibers, Residual Strength increased by nearly 7 to 8 times when the fiber volume increased from 0.2% to 0.8%; the corresponding increase in the case of the monofilament fibers was only of the order of 5 to 6 times.  Fiber Volume, Vf (%)  Figure 3.2.21: Residual Strength Values Plotted as a Function of Fiber Volume for Fibers in Set 1  Analysis of Flexural Toughness: Specimens from Set 2 In Set 2, performance of eight commercially available large diameter fibers was investigated at high volume fractions equal to or exceeding 0.76%. Given that high volume fractions of fibers were investigated in Set 2, the damage resulting due to instability even in an open-loop test was marginal.  The results are compiled in Table 3.2.15.  Some illustrative flexural load versus  deflection curves for F R C with various fibers are shown in Figures 3.2.22 to 26. In Figure 3.2.22 data for the straight polypropylene fiber (PP-STR) are given at 1.6% by volume. Notice that the Residual Strength values for concrete reinforced with 1.6% of straight polypropylene fiber (PPSTR) were inferior than those for concrete reinforced with even a lower volume fraction of the fibrillated fiber (PPF-1 and PPF-2) (Tables 3.2.14 and 3.2.15). This is most likely due to the large diameter of the PP-STR fiber; a similar trend was observed previously when comparing the PPM-1 fiber with the PPM-2 fiber and the fibrillated fibers (Figure 3.2.21). In Figure 3.2.23, the results for the hybrid mix (HRBD-1) with a combination of the straight polypropylene fiber (PPSTR) at 1.6% by volume and hooked-end steel fiber (ST-HKD) at 0.25% by volume are presented. When Figures 3.2.22 and 3.2.23 are compared, a significant improvement due to the 74  addition of steel fibers is clearly noticeable. This is also clear from the RS values reported in Table 3.2.15.  25 Mix HBRD-1 20 15  |  OLTM  Fiber PP-STR @Vf=l.6%  | /  + Fiber ST-HKD @ Vf= 0.25%  10 5  2  3  V  "RSTM  "  0 M  1  1  1  1  0  1  2  3  4  Deflection (mm)  5  Deflection (mm)  Figure 3.2.22: Load-Deflection Curves for FRC with PP-  Figure 3.2.23: Load-Deflection Curves for FRC with a  STR Fiber at a Dosage Rate of 1.6%.  Hybrid Combination of PP-STR Fiber @ 1.6% and Steel Fiber (ST-HKD) @ 0.25%.  Finally, in Figures 3.2.24-26, the results for the P V A fiber (PVA-1), the hooked-end steel fiber (ST-HKD) and the crimped steel fiber (ST-CR1), respectively, are presented.  Notice that the  high modulus P V A fiber performed better than the low modulus polypropylene fibers (Table 3.2.15) and that the instability was minimized due to the higher modulus. Steel fibers with the hooked ends (Figure 3.2.25) gave the highest RS values of all composites tested even at a modest volume fraction of 0.76%. The crimped steel fiber, on the other hand, was not as effective as the hooked-end steel fiber (Figures 3.2.25 and 3.2.26) and this clearly demonstrates the ability of the R S T M to distinguish between steel fibers of various geometries. Fiber PVA-1 @ Vf=1.6%  OLTM  Deflection (mm) Figure 3.2.24: Load-Deflection Curves for FRC with PVA-1 Fiber at a Dosage Rate of 1.6%.  75  E8  •J  III  1  1  1  >o  1  1  1  e P •  II l £  I  CU  6 o U  o  r?0>  1  73  O  o\  = 0.76% of Steel 1.0% <>fPVA V-B ACT (sec) (%)  1  i  1  r-  CS oo CO cs  I  uo  o  CO  1  Cv  >/-]  1  r>  t >  1  x  cs  o  £  o  00  1/0  ">> c c •> CD 73  ii o U  o  as  1,1  CD  CU  JS "o  •a > 00  cu  I  cu CU  1  >  8  1 < >  •2 O CD  E 71  'C  o a u 3C GO  ac fi CO  i GO  •8 u  GO  T3 'C £  cu vs  cs  t >  Q  00  ac  u  CD CD  e= >  cs  I  03 DC — + >I +  2  Oi ca  ac  GO  a. E 55 • E. 53  I  i  1  1  •  1  1  •  •  i  •  a  ca•  •a  L  ^ •* d  Q  CD  ?  & E o  u <  c  GO  3  02 CQ  fi  VO  CD  CO cs  +  Fiber Material and Type  1  VO ro cs  1  *BB GO  Fiber and Mix Id.  1  CU  o  u  cs ro u  •  NLM-1  o  GO  S  •  u  H  39.6 (5.1) 37.5 (2.4)  cs  ON 00  vi  JS  1  cu •*  | Vf=0.5%  Tj  >  CU Cu  00 03  E ? 55 J ,  *  1  Q  05 GO  37.1 (2.1) 40.1 (2.2)  II >•  •  cs  o I •o • C w> cs o •c u o 2  o o DC  39.7 (0.4) 40.4 (1.1)  oo  ^  .5  a  39.4 (3.5) 38.9 (2.9)  y s  00  PPM-2  <  & fi  + X)  ssive Strengt  vo  '3  C/3  Fiber Information  ^  O0  •a Ui  § 2 S3 *  1  i  •  CJ X>  GO  CD  "2  PPM-1  ro cs  JS  38.9 (2.8) 39.1 (2.7) 42.1 (4.3) 41.9 (3.5)  ON 00 CO cs  vo  n.  1  •  ^ oo  o  a, 6 o U  Ui  £ ^ .a  •o c  o  Cu  PPF-4  0-  1  o  1  i  Fiber & Mix Id.  >perties  Sim (mm)  cs  1  • ? 8 a, B « Q  PPF-3  1  r~  3  2  tPo lyp  ro  o  pyle  H •o  1  i  o  JS O  CD C  PPF-2  1  o  42.1 (3.1) 40.6 (2.2)  1  3^,  •  40.44 (4.4) 39.8 (4.2)  i  •  GO  Maten;  1  oo  5  38.9 (1.6) 40.3 (4.9) 37.9  i  CS  ro CN  ©,  ^ 05 <  41.1 (3.3) 39.2 (2.7)  1  •  vo  ro  i  #  1  !>  £ "E b "56  VO  >5  37.8 (2.1) 41.3 (3.6)  BB  II t£  Fibrillated Polypropylene Fibrillated Polypropylene Fibrillated Polypropylene Fibrillated Polypropylene Monofilament Polypropylene Monofilament Polypropylene Monofilament Nylon  }  Cu.  PPF-1  £.  V-B (sec)  <  6%(PP)  O +  ca  ON CO CS  i  3  3  E  Q  CN  cn  0.85 (0.11)  0.71 (0.05)  6.03 (0.15)  (  1  m ^ \o cn i n g.  1  •  1  '  1  5.61 (0.14)  1 1  • •  • •  •  •  • •  • 1  •  •  • •  1 1 <  • • • • • •  . 1  1  • •  1  •  12.7  ••  0.41  •  oo  (0.08)  1  5.10 (0.58)  0.88 (0.14)  17.3  5.56 (0.32)  1.15 (0.07)  1.21 (0.08)  1  5.9 (0.14)  1  17.7  •  1.04 (0.18)  •  . 5.86 (0.27)  •  5.54 (0.57)  14.1  •  0.45 (0.09)  •  5.79 (0.20)  •  0.04 (0.02)  •  1  23.1  •  5.04 (0.44)  1.26 (0.35)  5.46 (0.25)  1  29.5  •  NLM-1  1.69 (0.11)  5.73 (0.23)  2.62 (0.11)  5.77 (0.22) 34.3  1.85 (0.16)  5.39 (0.10)  •  0.23 (0.12)  1.46 (0.02)  5.98 (0.53)  24.4  27.3  1.54 (0.20)  5.65 (0.30)  21.2  2.58 (0.07)  5.54 (0.51)  34.6  1.86 (0.10)  5.37 (0.07)  •  6.92 (0.52)  1.18 (0.12)  5.57 (0.36)  27.7  1.5 (0.23)  5.41 (0.48)  21.7  RS (MPa)  MOR (MPa)  RS (MPa)  Vf = 0.8%  MOR (MPa)  Vf = 0.6%  1  45.4  46.6  20.7  20.5  •  PPM-2  1.27 (0.04)  5.85 (0.26)  RS (MPa)  MOR (MPa)  Vf = 0.5%  1  •  PPM-1  RS (MPa)  MOR (MPa)  Vf = 0.4%  Fiber Volume, Vf  %i  PPF-4  »n oo  PPF-3  0.44 (0.03)  '  5.63 (0.50)  00  PPF-2  1 0.33 (0.07)  MOR* (MPa)  Vf = 0.2%  3  PPF-1  Mix Id.  Fiber and  E  IS  t-  _o c  <0  •? CJ  C3  cn  -a  o  •a c  1  co  o  0 -  «1  M  <<  +  Vi  5  oo  01 %  x o  £ a  oi  fc,  *  i  CO 1  1 1  • • • •  • •  1 1  1  1 1 1  1  •  •  '  • • •  59.7  •  *  3.34  l  (0.55)  1  5.59  1 1  (0.63)  •  • •  • •  1 1  • •  1 1  • •  • • • • • •  i  1 1 l  1 1  i  1 1 1 1 1  1  03  t>  x  3  u Q  <£  earn  *  1 •  rage of  1 57.2  •  3.27  MOR* (MPa) •  (0.20)  1  5.72  71.2  RS** (MPa) 1  (0.34)  1  91.0  1  5.74  3.88 (0.48)  5.45  •  3.19 (0.14)  •  (0.65)  44.3  1 1  Aver age of  1 •  5.96  2.47 (0.08)  5.58 (0.24)  1  HBRD-1  1 1 •  (0.43)  1 53.6  • '  (0.19)  31.2  MOR* (MPa)  Vf = 1.6%(PP) + 0.25%(Steel)  1  ST-CRl  a~ •  6.31  a~ 1.83  * * ^ (0.19)  a~  5.87  RS** (MPa)  a~ at  (0.21)****  MOR* (MPa)  Vf = 3.0%  . S  (0.22)  RS** (MPa) 1 1  MOR* (MPa)  Vf = 1.6% I—1 ^  ST-HKD  RS** (MPa)  Vf=1.0%  Fiber Volume, Vf  *  PVA-1  '3  a~  PP-STR  OS x>  MOR* (MPa)  E  Vf = 0.76%  X  Mix Id.  Fiber and  oo c  *  #  '£ •?  c o  1  u -a •o •o c  C3  o  oi O  Xl 0) CJ  X!  00  <  <u >  oS  X 0 0  OO  II  ja  X  OS  a  E  Deflection (mm)  Deflection (mm)  Figure 3.2.25: Load-Deflection Curves for F R C with Steel  Figure 3.2.26: Load-Deflection Curves for F R C with Steel  Fiber S T - H K D at a Dosage Rate of 0.76%.  Fiber S T - C R l at a Dosage Rate of 0.76%.  3.2.2.3 Discussion The purpose of the research reported in this section was to make an assessment of the validity of the newly developed Residual Strength Test Method for toughness characterization of fiber reinforced concrete.  The Residual Strength Test Method was seen to be highly effective in  differentiating between different fiber types, fiber lengths, fiber configurations, fiber volume fractions, fiber geometries and fiber moduli. With respect to the specific composites tested, the following conclusions could be drawn: •  Based on the results from specimens in Set 1, it is clear that at a given fiber volume fraction, fibrillated polypropylene fibers provide better toughening than monofilament polypropylene fibers or the monofilament nylon fibers. In the range of fiber volume fractions investigated (0.2%-0.8%), both the Residual Strengths and the Residual Strength Indices increase almost linearly with the fiber volume fraction. Other parameters remaining the same, performance of F R C with small diameter fibers is better.  .  Data from specimens tested in Set 2 indicate that the Residual Strength Test Method equally well predicts the toughening capabilities of steel and other macro-fibers at large volume fractions. The post-peak instability in these composites, however, was minimal. Steel fibers provided very high Residual Strengths, but among the steel fibers the hooked-end steel fiber demonstrated  better toughening capability than the crimped steel fiber.  Combining  polypropylene and steel fibers in the same mix had a synergistic effect and composites with very high Residual Strengths were obtained. 79  3.3 Conclusions •  Material factors that were found to influence toughness of fiber reinforced concrete include fiber aspect ratio, fiber surface characteristics, fiber geometry, fiber volume content, shrinkage properties of matrix, and mineral admixtures (pozzolans).  •  The influence of test machine stiffness on the measured flexural load versus deflection response of fiber reinforced concrete composites has been evaluated. It has been found that the measured flexural response of fiber reinforced cementitious composites, particularly of the ones containing low fiber volume fractions, is greatly influenced by the machine configuration. For machines with open-loop test configuration with low stiffness, the load drops suddenly in an uncontrolled and unstable manner immediately following the peak load, the extent of this instability being dependent upon machine stiffness and loading rate. In addition, after the unstable part, the curve attains a stable softening level, during v/hich loads are functions of the test machine characteristics. For the aforementioned reasons the use of such load versus deflection curves to quantify toughness often translates into meaningless toughness parameters. Unfortunately, the existing standards to characterize toughness make use of such flexural load versus deflection curves, and therefore toughness measures using these standards are highly suspect.  •  A n evaluation of the Residual Strength Test Method (RSTM) for the measurement of postcracking performance and flexural toughness of fiber reinforced concrete composites has been carried out. In the test method, a stable narrow crack is first created in the specimen by applying flexural load in parallel with a steel plate under controlled conditions. The plate is then removed, and the specimen is tested in a routine manner in flexure to obtain the postcrack load versus displacement response.  In the experimental test program, several fiber  reinforced concrete mixes containing a variety of fibers at different volume fractions were evaluated.  Results from the test program suggest that the R S T M holds promise in the  measurement of post-cracking performance of fiber reinforced concrete composites. •  The Residual Strength Test Method is seen to be highly effective in differentiating between different fiber types, fiber lengths, fiber configurations, fiber volume fractions, fiber geometries and fiber moduli. In particular, the technique is very useful for testing cementbased composites containing fibers at very low dosages.  80  It is recommended that proper care must be exercised while removing the steel plate before reloading the cracked specimen. Specimen mishandling may increase the damage and the consequent results. The post-peak load-deformation curves obtained from the Residual Strength Test Method correlated well with those obtained from the Close-Loop Test Method for all three types of fibers (PP1, PP2 and NL1 fibers) investigated at various fiber volume fractions up to 0.5%. Slightly lower Residual Strength (RS) values are obtained from the Residual Strength Test Method as compared to the close-loop method. The error is magnified to a small extent for low efficiency fibers. When various fiber types are compared using the close-loop method, fibrillated polypropylene (PP1) is found to be the most effective followed by the nylon (NL1) fiber and the monofilament polypropylene (PP2) fiber. The Residual Strength Test Method was also capable of accurately predicting these important trends. At a given fiber volume fraction, fibrillated polypropylene fibers provide better toughening than monofilament polypropylene fibers or the monofilament nylon fibers. In the range of fiber volume fractions investigated (0.2%-0.8%) of these fibers, both the Residual Strengths and the Residual Strength Indices increase almost linearly with increase in fiber volume fraction. Other parameters remaining the same, F R C with smaller diameter fibers have better performance. The Residual Strength Test Method equally well predicts the toughening capabilities of steel and other macro-fibers at large volume fractions. composites, however, is minimal.  The post-peak instability in these  Steel fibers provide very high Residual Strengths, but  among the steel fibers, the hooked-end steel fiber demonstrates better toughening capability than the crimped steel fiber. Combining polypropylene and steel fibers in the same mix has a synergistic effect on the toughness of FRC.  81  Chapter 4  Bond-Slip Performance of Fibers Embedded in Cementitious Matrices  4.0 Introduction In the previous chapter, it was observed that the energy absorption capability of fiber reinforced concrete was dependent upon the properties of its constituents, i.e., properties of fibers and properties of matrix. It was also noted that the main energy absorption mechanism in fiber reinforced concrete composite is associated with pullout of fibers from the matrix.  More  specifically, the micromechanical properties of fiber-matrix interface dictate the fiber pullout response and the consequent composite toughening or strengthening.  If optimization of  composite toughening or composite strengthening is sought, it becomes critical to identify different variables that influence micromechanical properties of the fiber-matrix interface and the consequent fiber pullout response. Therefore, the objective of the present investigation is to shed some light on this important aspect through experimental fiber pullout studies.  4.1 Bond-Slip Performance of Fibers - Influence of Pullout Parameters Despite numerous micromechanical fiber pullout studies done in the past, the extent of dependence of micromechanical properties (and the consequent fiber pullout response) on the various pullout parameters is not properly understood. The objective of the present investigation is to critically explore this important aspect. In this regard, dependence of experimental fiber pullout response on the following pullout parameters has been explored: •  Fiber elastic modulus  •  Fiber length  •  Fiber surface roughness  •  Matrix shrinkage properties  •  Matrix water/cement ratio  •  Matrix modification by pozzolan - silica fume  82  Single fiber pullout tests were performed using the specimen shown in Figure 4.1.1. The specimen preparation is as follows - the specimens are cast in two parts separated by a plastic separator film. The lower part (labeled T in Figure 4.1.1) is cast first with the fiber embedded in it, and is allowed to cure for 24 hours. Once the lower half has hardened, the other half (labeled 'IF in Figure 4.1.1) of the specimen is cast. The assembly is further cured for a period of 28 days. For preparing specimens with fiber lengths longer than the standard specimen size shown in Figure 4.1.1, an attachment is added to the standard mould while casting. Pullout tests were performed in a 150-kN floor mounted testing machine. Load was applied at a cross-arm travel rate of 0.1 mm/min through a 5-kN load cell. LVDTs (Linear Variable Differential Transformers).  Displacements were measured by two Loads and displacements were digitally  recorded using a 16-bit data acquisition system operating at a frequency of 10 Hz.  65 mm concrete matrix  <b -  A  Pullout  T  load  steel anchorage ring —. 5  mm  simulated crack surface (plastic separator) 130 mm LVDT LVDT holder  Figure 4.1.1: Schematic of a standard pullout test specimen  83  Three fiber types were investigated, namely, i.  straight-smooth, stainless steel fiber (fiber diameter - 0.44 mm)  ii.  straight, stainless steel fiber with rough surface (fiber diameter - 0.44 mm)  iii.  straight, smooth polypropylene fiber (fiber diameter - 0.95 mm)  1  Matrix types investigated were: i.  normal strength matrix (Concrete matrix, C S A Type 10 Portland Cement) •  Compressive strength - 36 MPa  ii. high strength matrix (Concrete matrix, C S A Type 10 Portland Cement) •  Compressive strength - 69 MPa  iii. non-shrink grout matrix (Mortar matrix) •  Compressive strength - 18 MPa  iv. silica fume modified high strength matrix (Concrete matrix, C S A Type 10 Portland Cement) •  Compressive strength - 71 M P a  The experimental fiber pullout test results are reported below.  4.1.1  Straight-smooth, Stainless Steel Fibers Embedded in Normal Strength Matrix  The experimental pullout responses of straight-smooth, stainless steel fibers embedded in a normal strength matrix were investigated, and the corresponding results are reported here. The mix proportion used was - water : cement: sand : aggregate = 0.50 : 1.0 : 2.5 : 2.0. Cement used was of C S A Type 10 specification.  The above mix proportion gave an average 28-day  compressive strength of approximately 36 MPa, as determined by testing five 200 mm x 100 mm <p cylinders. Fibers with embedded lengths equal to 15 mm, 30 mm, 60 mm and 90 mm on both sides of the crack were chosen, and for each fiber length, five fibers were tested for pullout. Figure 4.1.2 shows the pullout response of fibers tested. The average pullout curves have also been plotted in the same figure (thick lines). Results from these pullout tests are tabulated in Table 4.1.1. The total pullout energy, E u was measured by numerical integration of pullout ma  curve. 1  Straight-smooth stainless steel fibers were sand blasted for about 25 minutes to produce the straight stainless steel fibers with rough surface 84  180  0  10  20  30  40  50  60  70  80  90  Displacement (mm) Figure 4.1.2: Pullout response of straight-smooth, stainless steel fibers embedded in normal strength matrix  Table 4.1.1: Experimental pullout test results for straight-smooth, stainless steel fibers embedded in normal strength matrix  Fiber  Matrix  Straight-smooth, stainless steel  Pullout Response  Embedded Fiber Length (mm)  Normal strength  Ppeak  8peak  (N)  (mm)  (N-mm)  15  51.2(6.6)*  0.034 (0.011)  226 (32)  30  95.1 (9.0)  0.120(0.038)  769 (166)  60  100.5 (8.3)  0.387 (0.099)  1940 (288)  90  134.5 (21.3)  0.484 (0.054)  3818 (374)  Peak load; 8 - Displacement at peak pullout load; - Figures in brackets indicate standard deviation  Ppeak -  peak  E  m  a  l  - Total pullout energy  4.1.1.1 Influence of Fiber Length In the Table 4.1.1, it can be noticed that increase in fiber length led to an increase in peak pullout load. However, the rate of increase in peak pullout load diminished with increase in fiber length. The peak pullout load reached an asymptote at very long fiber lengths as seen in Figure 4.1.3.  85  160  Fiber embedded length (mm)  Figure 4.1.3: Embeddedfiberlength vs. peak pullout load curve for straight-smooth, stainless steel fibers embedded in normal strength matrix  4.1.2 Straight, Stainless Steel Fibers with Rough Surface Embedded in Normal Strength Matrix The experimental pullout response of straight, stainless steel fibers with a rough surface embedded in a normal strength matrix was investigated and the corresponding results are reported here. The mix proportion used and the concrete compressive strength were same as mentioned in the previous sub-section. Fibers with embedded lengths equal to 15 mm, 30 mm, 60 mm and 90 mm on both sides of the crack were chosen, and for each fiber length five fibers were tested for pullout. Figure 4.1.4 shows the pullout response of the fibers tested. The average pullout curves have also been plotted in the same figure (thick lines). Results from these pullout tests are tabulated in Table 4.1.2.  Table 4.1.2: Experimental pullout test results for straight, stainless steel fibers with rough surface embedded in normal strength matrix  Fiber  Straight, fiber with rough surface  Matrix  Normal strength  Embedded Fiber Length (mm)  Pullout Response Ppeak  (N)  (mm)  (N-mm)  15  105.9(14.3)*  0.052 (0.019)  450(145)  30  116.1 (9.6)  0.123 (0.038)  1160 (228)  60  127.9 (14.0)  0.546 (0.131)  3201 (557)  90  135.9 (10.39)  0.591 (0.109)  5916(1045)  Peak load; 8peak- Displacement at peak pullout load; - Figures in brackets indicate standard deviation  Ppeak #  86  E i lota  -  Total pullout energy  Displacement (mm) Figure 4.1.4: Pullout response of straight, stainless steelfiberswith rough surface embedded in normal strength matrix  4.1.2.1 Influence of Surface Roughness To study the influence of fiber surface roughness on the fiber pullout response, the pullout curves for 30 mm long smooth and rough fibers are compared in Figure 4.1.5. It can be seen that peak pullout loads were higher for fibers with rough surface. Also, the peak pullout displacement and the total pullout energy were higher for fibers with rough surface (compare Table 4.1.1 and Table 4.1.2). With increase in fiber length, disparity between the peak pullout loads for smooth fiber and rough fiber diminished.  rough steel fiber  •smooth steel fiber  10  15  20  25  30  Displacement (mm)  Figure 4.1.5: Comparison of the pullout response of 30 mm long straight-smooth, stainless steel fibers and 30 mm long straight, stainless fibers with rough surface (normal strength matrix)  87  4.1.2.2 Influence of Fiber Length For fiber with rough surface, it can be seen that an increase in fiber length led to an increase in the peak pullout load (Figure 4.1.6).  However, the rate of increase in peak pullout load  diminished with increase in fiber length as seen in Table 4.1.2 and Figure 4.1.6, and reached an asymptotic value. The asymptotic value of pullout load for straight-rough, stainless steel fibers was very close to that obtained for the straight-smooth, stainless steel fibers (compare Figure 4.1.5 and Figure 4.1.6). 160 z  w  120  / 0  X  x  x  Experimental -Trendline  20  40  60  80  100  Fiber embedded length (mm)  Figure 4.1.6: Embeddedfiberlength vs. peak pullout load curve for straight, stainless steelfiberswith rough surface  4.1.3  Straight-smooth, Stainless Steel Fibers Embedded in Non-Shrink Grout Matrix  The frictional resistance to fiber pullout is dependent upon the normal contact stresses (acting perpendicular to the longitudinal fiber axis) that develop at the fiber-matrix interface.  The  normal contact stress at the fiber-matrix interface results from matrix shrinkage taking place during the curing, setting and hardening of the matrix. The greater the matrix shrinkage, the higher the magnitude of interfacial normal contact stress and the corresponding frictional resistance. Thus, it can be inferred that a cementitious matrix that experiences less shrinkage will offer lower resistance to fiber pullout.  A type of cementitious matrix that undergoes  relatively lesser shrinkage is available commercially under the name 'non-shrink grout' (mortar matrix). Fiber pullout response of straight-smooth, stainless steel fibers embedded in non-shrink grout matrix was investigated. The 28-day compressive strength of the non-shrink grout matrix was approximately 18 MPa, as determined by testing five 200 mm x 100 mm 0 cylinders. Fibers with embedded lengths equal to 15 mm, 30 mm, 60 mm and 90 mm on both sides of the crack  88  were chosen, and for each fiber length, five fibers were tested for pullout. Figure 4.1.7 shows the pullout response of fibers tested. The average pullout curves have also been plotted in the same figure (thick lines). Results from these pullout tests are tabulated in Table 4.1.3. 70 Straight-smooth, stainless steel fiber embedded in non-shrink grout matrix  60 4  -o  2 60 mm  a  90 mm  0  10  20  30  40  50  60  70  80  90  Displacement (mm) Figure 4.1.7: Pullout response of straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  Table 4.1.3: Experimental pullout test results for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  Fiber  Straight-smooth, stainless steel  Matrix  Non-shrink grout  Embedded Fiber Length (mm)  Pullout Response Ppeak  (mm)  (N-mm)  15  26.1 (3.9)  0.013 (0.003)  83 (12)  30  36.6 (4.6)  0.044 (0.009)  269 (67)  60  39.5 (5.8)  0.148(0.049)  754 (145)  90  42.5 (5.9)  0.160(0.057)  1560 (240)  #  Peak load; S - Displacement at peak pullout load; - Figures in brackets indicate standard deviation  Ppeak -  Etotal  (N)  p e a k  89  E  m  a  i  - Total pullout energy  4.1.3.1 Influence of Matrix Shrinkage Behavior To study the influence of matrix shrinkage on the fiber pullout response, the pullout curves for 30 mm long straight, smooth fibers are compared in the Figure 4.1.8 for the following two types of cementitious matrices: i.  C S A Type 10 normal strength matrix  ii.  Non-shrink grout matrix  0  5  10  15  20  25  30  Displacement (mm)  Figure 4.1.8: Pullout response of 30 mm long fibers embedded in normal strength matrix (CSA Type 10 cement) and in non-shrink grout matrix  In the figure, it can be noticed that the peak pullout load attained in the case of non-shrink grout matrix was much lower in magnitude in comparison to that obtained in the case of C S A Type 10 normal strength matrix. Also, the displacement at peak pullout load and the total pullout energy was lower in the case of non-shrink grout matrix.  4.1.3.2 Influence of Fiber Length It can be seen that increase in fiber length led to an increase in peak pullout load. However, the rate of increase in peak pullout load diminished with increase in fiber length. The peak pullout load reached an asymptote at very long fiber lengths as seen in Figure 4.1.9. The asymptotic value of pullout load obtained in this case (= 43 N) was much lower than that obtained in the case of C S A Type 10 normal strength concrete matrix (= 135 N).  90  Fiber embedded length (mm)  Figure 4.1.9: Embedded fiber length vs. peak pullout load curve for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  4.1.4  Straight-smooth, Stainless Steel Fibers Embedded in High-Strength Matrix  The experimental fiber pullout response of straight-smooth, stainless steel fibers embedded in a high strength matrix was investigated, and the corresponding results are reported here. The mix proportion used was - water : cement: sand : aggregate = 0.35 : 1.00 : 1.50 : 1.75. Cement used was of C S A Type 10 specification.  The above mix proportion gave an average 28-day  compressive strength of approximately 69 MPa, as determined by testing five 200 mm x 100 mm (j) cylinders. Fibers with one-side embedded lengths equal to 30 mm were tested for pullout. Figure 4.1.10 shows the pullout response of the fibers tested. The average pullout curve has also been plotted in the same figure (thick line). Results from these pullout tests are tabulated in Table 4.1.4.  Table 4.1.4: Experimental pullout test results for straight-smooth, stainless steel fibers embedded in high strength matrix Pullout Response  Embedded Fiber  Straight-smooth, stainless steel  Matrix  High strength  Fiber Length  Ppeak  (mm)  (N)  (mm)  (N-mm)  30  102.80 (9.95)  0.134 (0.038)  1026 (271)  &peak  Peak load; 8peak- Displacement at peak pullout load; £,„,„/- Total pullout energy - Figures in brackets indicate standard deviation  Ppeak -  91  0  5  10  15  20  25  30  Displacement (mm)  Figure 4.1.10: Pullout response of straight-smooth, stainless steel fibers embedded in high strength matrix  4.1.4.1 Influence of Water/Cement Ratio To study the influence of water/cement ratio on fiber pullout response, the pullout curves for 30 mm long straight, smooth fibers are compared in Figure 4.1.11 at the following two water/cement ratios: i.  w/c = 0.50  ii.  w/c = 0.35  In Figure 4.1.11 it can be seen that the pullout loads in the descending branch of the pullout curve at a w/c ratio of 0.35 are greater in comparison to those at a w/c ratio of 0.50. The postpeak pullout loads are a function of the frictional shear resistance that mobilizes at the fibermatrix interface. A superior post-peak descending pullout curve obtained at lower w/c ratio (= 0.35) implies that the interfacial frictional resistance was greater at this w/c ratio. Autogeneous shrinkage of matrix associated with the low w/c ratio matrices is one possible reason that may influence the frictional resistance. It must be noted that an enhanced matrix shrinkage leads to an increase in fiber-matrix interfacial contact stress and the consequent interfacial frictional bond. The physical characteristics of the sliding surfaces between the fiber and the matrix is yet another factor that to some extent may influence the interfacial frictional bond.  92  Displacement (mm)  Figure 4.1.11: Pullout response of straight-smooth, stainless steel fibers embedded in normal strength matrix and high strength matrix  4.1.5  Straight-smooth, Stainless Steel Fibers Embedded in Silica-fume Modified HighStrength Matrix  The experimental fiber pullout response of straight-smooth, stainless steel fibers embedded in a silica-fume modified normal strength matrix was investigated and the corresponding results are reported here. The mix proportion used was - water : binder : sand : aggregate = 0.35 : 1.00 : 1.50 : 1.75. The binder contained a mixture of cement (90% by weight) and silica fume (10% by weight). Cement used was of C S A Type 10 specification. The above mix proportion gave an average 28-day compressive strength of about 71 MPa, as determined by testing five 200 mm x 100 mm 0 cylinders. Fibers with one-side embedded lengths equal to 30 mm were tested for pullout. Figure 4.1.12 shows the pullout response of the fibers tested. The average pullout curve has also been plotted in the same figure (thick line).  Results from these pullout tests are  tabulated in Table 4.1.5. Table 4.1.5: Experimental pullout test results for straight-smooth, stainless steel fibers embedded in silica-fume modified high strength matrix Pullout Response  Embedded Fiber  Straight-smooth, stainless steel  Matrix  Silica fume Modified High strength  Fiber Length  Ppeak  (mm)  (N)  30  101.9 (4.60)  #  Peak load; 8 - Displacement at peak pullout load; * - Figures in brackets indicate standard deviation Ppeak -  8peak  peak  93  E  m  a  l  Etolal  (mm)  (N-mm)  0.125 (0.032)  1295 (209)  - Total pullout energy  120  Displacement (mm) Figure 4.1.12: Pullout response of straight-smooth, stainless steelfibersembedded in silica fume modified high strength matrix  4.1.5.1 Influence of Silica-fume Modification To study the influence of matrix modification by silica-fume on the fiber pullout response, pullout curves for 30 mm long straight-smooth, stainless steel fibers pulled out from two different types of matrices are compared in Figure 4.1.13. These matrices are: i.  High strength matrix  ii.  Silica fume modified high strength matrix  Displacement (mm)  Figure 4.1.13: Pullout response of straight-smooth, stainless steelfibersembedded in high strength matrix and silica fume modified high strength matrix 94  In Figure 4.1.13, it can be seen that the fiber pullout response was superior in the case of the silica fume modified high strength matrix. Disparity in the fiber pullout response with the above two matrix systems could be the result of one or more of the following factors: chemical nature of the interface, shrinkage behavior of the matrix, and the physical characteristics of the sliding surfaces during fiber pullout.  4.1.6 Straight, Smooth Polypropylene Fibers Embedded in Normal Strength Matrix The experimental pullout response of straight, smooth polypropylene fibers embedded in a normal strength matrix was investigated and the corresponding results are reported here. The mix proportion used was - water : cement: sand : aggregate = 0.5 : 1.0 : 2.5 : 2.0. Cement used was of C S A Type 10 specification.  The above mix proportion gave an average 28-day  compressive strength of about 36 MPa, as determined by testing five 200 mm x 100 mm 0 cylinders. Fibers with embedded lengths equal to 5 mm, 15 mm, 30 mm, and 60 rnm on both sides of the crack were chosen, and for each fiber length, five to six fibers were tested for pullout. Figure 4.1.14 shows the pullout response of fibers tested. The average pullout curves have also been plotted in the same figure (thick lines). Results from these pullout tests are tabulated in Table 4.1.6.  Table 4.1.6: Experimental pullout test results for straight, smooth polypropylene fibers embedded in normal strength matrix Pullout Response  Embedded Fiber  Matrix  Straight, smooth  Normal strength  polypropylene  Fiber Length  D r  peak  (mm)  (N)  (mm)  (N-mm)  5  12.4 (2.00)*  0.047.(0.02)  28 (3)  15  14.9(1.25)  0.300 (0.24)  139 (59)  30  17.6(1.00)  1.251 (0.89)  306 (46)  60  22.3 (2.70)  2.983(1.46)  894 (87)  Peak load; S - Displacement at peak pullout load; - Figures in brackets indicate standard deviation  Ppeak -  p e a k  95  E  t o t a l  - Total pullout energy  Displacement (mm)  Figure 4.1.14: Pullout response of straight, smooth polypropylene fibers embedded in normal strength matrix  4.1.6.1 Influence of Fiber Length It can be seen that increase in fiber length led to an increase in peak pullout loads. However, the rate of increase in peak pullout load diminished with increase in fiber length. The peak pullout load reached an asymptote at longer fiber lengths as shown in the Figure 4.1.15. Also, it can be noticed that for any given fiber length, the peak pullout load attained for polypropylene fiber was much lower in comparison to that for steel fiber (compare Figure 4.1.3 and Figure 4.1.15). Furthermore, for polypropylene fibers, the asymptotic value of pullout stress was reached at a much smaller fiber length in comparison to that for steel fibers.  96  ox 0  20  40  60  80  Fiber embedded length (mm)  Figure 4.1.15: Embeddedfiberlength vs. peak pullout load curve for straight, smooth polypropylene fibers  4.2 Conclusions •  Increase in fiber length led to an increase in peak pullout load. However, at long fiber lengths, the peak pullout load reached an asymptote.  •  For a given fiber length, an increase in fiber surface roughness increased the peak pullout load. However, disparity between the peak pullout load for a smooth steel fiber and a rough steel fiber decreased with increase in fiber length. The asymptotic value of peak pullout load attained at long fiber lengths in the case of rough steel fiber was very similar to that obtained for smooth steel fiber. The displacement corresponding to peak pullout load and the total energy absorption were also found to increase with increase in surface roughness.  •  For a given fiber length, peak pullout loads decreased substantially when steel fibers were pulled out from non-shrink grout matrix. The observed decrease in peak pullout load is likely related to the shrinkage behavior of the non-shrink grout matrix.  •  The post-peak pullout response for steel fibers embedded in high-strength matrix was superior in comparison to that for steel fibers embedded in normal strength matrix. This observation implied that frictional resistance in the former was greater.  97  For any given fiber length, the peak pullout load attained for polypropylene fiber was much lower in comparison to that for steel fiber. Also, for polypropylene fibers, the asymptotic value of pullout stress was reached at a much smaller fiber length in comparison to that for steel fibers.  98  Chapter 5  Progressive Debonding Model for Fiber Pullout  5.0 Introduction Both one-sided and two-sided pullout specimens are commonly used to investigate the fibermatrix interfacial properties [108].  In a two-sided pullout test specimen, a matrix crack is  simulated at the center of the specimen. A single fiber embedded on both sides of the simulated crack provides resistance to the crack opening. Figure 5.0.1 depicts one side of a two-sided pullout test specimen. A fiber of radius a and length 2L is embedded at the center of the matrix coaxial cylinder of radius b. The gripping fixtures of the testing machine hold the matrix coaxial cylinders on each side of the simulated crack, and the specimen is subjected to tensile stress in the axial direction (normally at a constant rate of displacement).  The pullout load versus  displacement response is recorded to yield the pullout curve. Simulated surface  Figure 5.0.1: Singlefiberembedded in concrete matrix. Figure depicts one side of a two-sided pullout test specimen  Interpretation of pullout curves is commonly carried out using the analytical models based on the shear-lag analysis [52-78]. The shear-lag analysis assumes that the extensional stresses in the matrix are small relative to those in the fiber and that the shear stresses in the fiber are small  99  compared to those in the matrix [109]. Some important shear-lag models [53-56,61,67,68] that are widely used to study the interfacial debonding phenomena in the problem of fiber pullout were reviewed in Chapter 2. Table 5.0.1 shows the important features captured by these models. A brief discussion on these models follows.  Adhesional shear bond Progressive debonding  X  Dependence of debond stress on fiber length  X  V  V  V  V  X  V  X  X  X  (67,68)  Hsueh  Gopalaratnam .&Shah (56)  (61)  Nammur et al.  Takaku &Arridge (54)  (55)  Lawrence  (53)  Parameter  Greszczuk  Table 5.0.1: Various parameters taken into consideration in the theoretical fiber pullout models  V  V  Poisson's effect (Non-constant interfacial frictional shear bond) Evolution of interfacial properties during pullout x - not considered  X  X  V  X  X  X  X  X  X  x-  X  V - considered  Models by Greszczuk [53], Takaku and Arridge [54], Lawrence [55], Gopalaratnam and Shah [56], Naamur et al. [61] and Hsueh [67,68] acknowledge the existence of the adhesional shear bond at the bonded interface. Furthermore, these models use a strength based criterion to govern interfacial debonding (i.e., interfacial debonding is assumed to occur when the interfacial shear stress equals the adhsesional shear bond strength of the interface). Greszczuk [53] was the first to derive an interfacial debonding criterion using the shear-lag theory. Progressive debonding was not considered in his analysis, since he postulated that catastrophic debonding would occur with the initiation of debonding. Thus, his solution did not consider the stabilization of the debonding process that may take place due to the existence of frictional shear bond at the debonded interface.  For the first time, existence of frictional shear bond at the debonded  interface was included in the analysis by Lawrence [55]. Assuming existence of adhesional  100  bonded interface and a constant frictional shear bond at the debonded interface, Lawrence [55] derived a solution for progressive debonding. Later, Gopalaratnam and Shah [56] and Nammur et al. [61] also developed solutions for progressive debonding with an assumption of a constant frictional shear bond at the debonded interface. Note that the magnitude of the frictional shear bond at the debonded interface is actually dependent upon the interfacial contact stress and the coefficient of friction of the sliding surfaces. Since the interfacial contact stress varies with the fiber axial force and coefficient of friction may vary with the sliding distance, it is not necessary that the frictional shear bond remain constant over the debonded region.  Therefore, the  assumption of a constant frictional shear bond constitutes a major limitation of the abovementioned progressive debonding models. It is also important to include the influence of Poisson's contraction of the fiber in the analysis. This is because Poisson's contraction of the fiber reduces the net contact stress at the interface and, in turn, the frictional shear bond. Models by Lawrence [55], Gopalaratnam and Shah [56], and Naamur et al. [61] do not include the influence of Poisson's effect in the analysis. Neither do these models explicitly include interfacial properties such as coefficient of friction and interfacial contact stress in the analysis.  These constitute major limitations of the above  mentioned models. It must be noted that the dependence of interfacial frictional bond on the interfacial contact stress and the coefficient of friction warrants explicit inclusions of these parameters into the analysis. Only, the models by Takaku and Arridge [54] and Hsueh [67,68] include both parameters in the analysis. The model by Takaku and Arridge [54] was the first to consider the influence of Poisson's contraction of fiber in the analysis. However, these aspects were considered in the analysis only after the occurrence of complete interfacial debonding (i.e., in the fiber pullout case as explained later). Thus, the influence of Poisson's contraction during progressive debonding remained unaccounted, which constituted a major limitation of this model.  The model by Hsueh [54] considers Poisson's effect during progressive debonding,  however the analysis and the closed-form solutions presented are complex to use. During the process of fiber pullout, considerable matrix wear close to the interface is expected to occur. In such a scenario, the interfacial coefficient of friction would diminish with increase in pullout distance. Therefore, to objectively characterize the fiber-matrix interface, variation in the interfacial coefficient of friction with pullout distance must be taken into consideration.  101  Surprisingly, the existing theoretical fiber pullout models have not addressed this important aspect. In light of this fact, it may be possible to explain why the predictions obtained from the existing pullout models deviate considerably from the experimental results. To appreciate this fact profoundly, in what follows, predictions from the pullout models by Takaku and Arridge [54], Gopalaratnam and Shah [56] and Hsueh [67,68] are compared with the experimental results.  Figure 5.0.2 compares the experimental pullout curve for a steel fiber reported by  Gopalaratnam and Shah [56] with the predicted responses using the models mentioned above. Note that the fiber diameter was 0.25 mm and the fiber length was 12.5 mm. For making predictions using the models by Takaku and Arridge [54] and Hsueh [67,68] , the fiber length 1  2  was assumed to be 12 mm, since the source codes were written to accept only integer values for fiber length. In the figure it can be seen that all three theoretical predictions deviate considerably from the experimental curve. The reason for such wide disparity could possibly be attributed to the fact that these models disregard the evolution of interfacial properties during the pullout process.  0  3  6  9  12  15  Displacemert (rrrrj  Figure 5.0.2: A comparison between the experimental pullout response for a steel fiber with the predicted responses using the existing pullout models. The experimental pullout response is by Naaman and Shah [142]  1  2  The source code for Takaku and Arridge model [54] is attached in Appendix A. The source code for Hsueh model [67,68] is attached in Appendix B.  102  The above discussion brings to attention the inadequacy of the existing fiber pullout models, and also the fact that a need exists for a model that realistically captures the physical phenomenon occurring during the process of fiber pullout. A new fiber pullout model is proposed below that considers the evolution of the interfacial coefficient of friction during the process of fiber pullout. Additionally, the proposed model takes into account the following aspects that are either considered or ignored in the earlier models: •  Dependence of the initial debonding stress on the embedded fiber length.  •  Radial dependence of the axial stress in the matrix.  •  Explicit inclusion of the interfacial properties such as the contact stress and the coefficient of friction.  •  Poisson's effect (in the event of a debonded fiber).  5.1 The Proposed Progressive Debonding Model Consider a fiber of radius a and length L embedded at the center of the matrix coaxial cylinder with inner radius a and outer radius b, as shown in Figure 5.0.1. A cylindrical coordinate system is selected so that the z-axis corresponds to the fiber axial direction and the r-axis corresponds to the radial direction. The embedded end of the fiber is located at z=0, and the other end where the fiber exits the matrix is located at z=L. The exit-end of the fiber (i.e., at z=L) is subjected to the tensile stress, a , as shown in Figure 5.0.1. Both fiber and matrix are assumed to be elastic. 0  Transfer of stress between the fiber and the matrix is via interfacial shear stresses. The entire pullout process can be divided into three stages [106], as shown in Figure 5.1.1: i.  Stage 1 - Fiber completely bonded along the length of thefiber:During stage 1, fiber and matrix displacements at the interface remain compatible, and the resistance to fiber pullout is derived from the adhesional shear stresses at the interface. At the end of stage 1, debonding of the interface is initiated at the location where the fiber enters the matrix.  ii.  Stage 2 - Fiber partially bonded along its embedded length: During stage 2, progressive debonding of the interface is initiated at the location where the fiber exits the matrix. The  103  adhesional shear stresses at the bonded interface and the frictional shear stresses at the debonded interface resist the fiber pullout. At the end of stage 2, the fiber is completely debonded along its embedded length.  iii.  Stage 3 - Fiber completely debonded over its embedded length and pulling out: At the end of stage 2, pullout of the fiber is initiated, and thereafter, the interfacial frictional shear stresses resist the pullout of fiber from the matrix.  Analysis for these three stages of the pullout process is presented below.  Debonding  s///s/////////ss Perfect bond  Perfect bond (ii) Debonding  (i) Perfect bond  mm  M„>j  ^  Debonded interface  sVr'j (iii) Pull-out  Figure 5.1.1: The principle of a singlefiberpullout test - The three stages of pullout process (Stang and Shah [106])  5.1.1 Stage 1: Fiber Completely Bonded Along Its Entire Embedded Length Consider the case where the fiber is completely bonded to the matrix (i.e., fiber and matrix displacements remain compatible at the interface) over its entire embedded length as shown in Figure 5.1.2.  G represents the fiber pullout stress applied at the emerging fiber end. The 0  essence of the formulation presented below is similar to the one presented earlier by Hsueh [67,68] in light of the fact that the analysis takes into consideration the radial dependence of the axial stresses in the matrix. The models by Greszczuk [53], Lawrence [55], Takaku and Arridge [54], Nammur et al. [61] and Gopalaratnam and Shah [56] do not consider the radial dependence of the axial stress in the matrix.  104  Assuming that the shear force in the matrix decreases linearly in the radial direction with a maximum at the fiber-matrix interface (i.e., at r=a) and zero at the free surface of the coaxial cylinder (i.e., at r=b), the matrix shear stress, x , at any radial distance, r, can be expressed as: m  b-r  —T •a r  for r > b  a  5.1  where x represents the shear stress at the interface, i.e., at r=a. The above equation satisfies the a  equilibrium in the axial direction, approximately, when the fiber diameter, a, is small in comparison to the radial dimension, b and the axial stress gradient in the matrix is small. In the models by Gopalaratnam and Shah [56] and Hsueh [57], the formulation of the stress transfer problem is based on the assumption that the shear stress in the matrix varies inversely with the radial distance (with a maximum at the interface). In the model by Greszczuk [53], the problem formulation is based on the shear deformation of a thin layer of interface. Based on the earlier work by Cox [52], the models by Lawrence [55] and Takaku and Arridge [54] use a rather awakard scheme to define the shear deformation of the matrix material.  Wb  matrix displacement profile T =0 m  a  0  Z=0  z=L  Figure 5.1.2: Fiber completely bonded over its embedded length  105  ->  z  If w represents the displacement of matrix in the axial direction, the corresponding shear stress m  in the matrix (ignoring the radial displacements) is given by:  f  =  m  E  dw  -  -  5.2  2 ( 1 + 0 dr  Combining Equations 5.1 and 5.2, and integrating: b  a IF-a)  C(b-r) , -dr =  T  a  -a)}  w  b  E  t dw  m  s  5.3 n  2(l + v m )J  r  where w and Wb are the axial displacement in the matrix at r=a and r=b, respectively. Solving a  the above equation, the interfacial shear stress, T , can be written as: a  E  T„ = -  m (  2(l + v )a m  W  b  ~  (b-a)  W  a )  5.4  log  -1 \  a  J  Combining 5.1 and 5.4, the shear stress in the matrix, z , at any radial distance, r, is given by: m  E (w -w )(b-r) m  b  a  2(l + um)r b\og  5.5  -(b-a)  Combining Equations 5.2 and 5.5 and integrating, the axial displacement in the matrix, vv , can m  be written as: w -w w = w +(b\ m a Mog -(b-a) b  \  a  a  felog \  5.6  + a-r a  J  j  1 0 6  Ignoring the Poisson's effect (i.e., treating the problem as one-dimensional), the axial stress in the matrix, a , (in the z direction) is given by: m  a m =£,m  dw„ dz  5.7  Combining Equations 5.6 and 5.7, the axial stress in the matrix at any radial distance, r, is given by:  b log  "=i7 '  CT  CT  +  Mog  5.8  + a-r  -(b-a)  The above equation describes the radial dependence of the axial stress in the matrix. As mentioned earlier, the models presented in the References 53, 55, 54, 61 56 do not consider the radial dependence of the axial stress in the matrix.  The condition for mechanical equilibrium between the external applied stress, o" , and the 0  internal stress distribution at any section can be written as: b  ° f  +  —  \  <  J  m  r  d  r  =  5.9  °0  Combining Equations 5.8 and 5.9, the axial stress in the matrix, a b is obtained as: m=  y + a-yqa m  5.10  where,  107  a=  a  ^logf^  1 a Mog  \  -(b-a)  2  a  ^-a )  J  a(b -a ) 2  2  +  2  (b -a ) 3  3  Now, considering the equilibrium between the axial stress in the fiber and the interfacial shear stress: f dz  2T a  d<J  =  5.11  A  Combining Equations 5.4 and 5.11: do  f  ~dz  = -6  E (w -w ) m  b  5.12  a  where -ii - i  d = \a (l + v ) 2  m  (b-a)  log  Differentiating the above equation: da 2  dz  T~--0  dw  dw  dz  dz  h  E  m  n  5.13  108  From which we obtain: da 2  = SE„  L  dz  5.14  7  Substituting for Oi, from Equation 5.10 in Equation 5.14: da  Q  1  ]  e_  — ( 7  ~de  +  5.15  a - yqa) + ad <? - —<y f  n  ri  m  The above equation is a second order linear differential equation and it dictates the stress transfer between the fiber and the matrix for a bonded interface.  5.1.1.1 Fiber Axial Stress Distribution, 07 The solution to the above differential equation represents the axial stress distribution over the embedded fiber length (z ^L) prior to debonding and at initiation of debonding. With boundary conditions, oy= o at z=L and 07= 0 at z=0, the solution of the differential equation becomes: Q  7  +  a - yna + an  where, fl -  6_ yn  (y +  (a-yna  ,sinh(/fe)  + an) ,  )  H  sinh[j3(L-z)]  J - y — r )  sinh( pL)  ^  T  sinh(pL)  u  +y  5.16  1/2  a-yqa)+ad  For any fiber pullout stress, <7 , that is less than or equal to the initial fiber debonding stress, a , 0  d  Equation 5.16 represents the fiber axial stress distribution, 07.  5.1.1.2 Interfacial Shear Stress Distribution, r  a  The interfacial shear stress, z distribution can be obtained by combining Equations 5.11 and a  5.16:  109  -apo  .cosh(pz)  0  2 ( 7 + a - yncc + an)  cosh[J8(L-z)]'  sinh(pX)  5.17  sinh(pjL)  For any fiber pullout stress, rj , that is less than or equal to the initial fiber debonding stress, Od, c  Equation 5.17 represents the interfacial shear stress distribution, x along the embedded fiber a  length.  5.1.1.3 Fiber Displacement, Ub Fiber displacements, U , when the fiber is completely bonded over its embedded length is b  obtained by integrating the strains along the fiber length: L  E (y + a- yna + afj) f  , (a-yna  JI  .sinh(jfe) sinh[p\L-z)] + ar!) , , „ [ - y — r + 7 dz sinh(pL) sinh(/JL) H  r  t  t  n  T  X  J  5.18  Solving: . cosh( BL) -1 , (a - yna + an • •r) ^-—t— + %, E (y + a- yna + an) /Jsinh(p7_) '  5.19  f  For a two-sided pullout test, fiber displacement is given by:  U„  2o\  EAy + a-yna  + an)  . (a-yna  + an-y)  , c o s h ( p X ) - l ., . V* ' +yL p sinh( pL)  5.20  where, the total fiber length is equal to 2L and embedded fiber length on each side of the pullout specimen is equal to L.  110  5.1.1.4 Debonding Criterion and Initial Debonding Stress, Od Using the maximum shear stress as the debonding criterion, the applied pullout stress, Od, required to initiate debonding can be obtained from Equation 5.17 by putting T =r at z=L a  s  (Ts=shear strength of the interface). Thus, the debonding criterion can be written as: -1-1  2r (y + cc- yncc + an) (a - yncc + an) cosh(/?L) + y s  5.21  sinh(/?L)  Thus, Od, in the above equation represents the fiber pullout stress, o , at which interfacial 0  debonding initiates. The above equation also depicts dependence of the initial debonding stress on fiber length. On plotting Od as a function of L, it can be seen that with increase in the fiber length, the initial debonding stress increases and attains an asymptotic value at long fiber lengths. Some earlier works on the problem of fiber pullout [54,55,68,93,176] also demonstrate the dependence of the initial debonding stress on the embedded fiber length.  Ill  5.1.2 Stage 2: Fiber Partially Debonded Along Its Embedded Length Takaku and Arridge [54] assume fiber and matrix to be held together in a shrink-fit configuration [224] to derive a differential equation of stress transfer for a debonded interface. However, they reserve the application of this equation to the case of a completely debonded fiber (i.e., stage 3). In the following, the application of their analysis is extended to the case of a partially debonded fiber. After the inception of interfacial debonding, a debonded zone of length l is created as shown in d  Figure 5.1.3. During partial debonding, fiber pullout is resisted by the adhesional shear stresses acting at the bonded interface of length L-l , and the frictional shear stresses acting at the d  debonded interface of length l . The interfacial frictional shear stress at the debonded zone is d  mobilized because of the existence of the interfacial contact pressure resulting due to matrix shrinkage [1,106]. The fiber axial stress, 07, at the junction of the bonded and the debonded zone is termed here as the progressive debonding stress, <y , and as a first approximation, o for any d  d  debond length, l , can be obtained by replacing L by L-l in Equation 5.21. d  d  'o  'o  Id z=0  z=L-l  d  z=L  Figure 5.1.3: Fiber partially debonded along its embedded length  07+ do/  Figure 5.1.4: Free body diagram for afiberelement of length dz  112  Over the debonded interface, transfer of stress between fiber and encapsulating matrix takes place through interfacial frictional shear stresses, Xf. Equilibrium between the fiber axial stress, 07, and the interfacial frictional shear stress, X/, is given by (Figure 5.1.4):'  da f dz  f  5.22  a  From Coulomb's Law, we obtain:  da f dz  f  2^o- cp  a  5.23  a  where fi is the interfacial coefficient of friction, and a  cp  is the contact pressure between fiber and  matrix at the interface after the fiber has undergone Poisson's contraction. In a shrink-fit configuration [224], the radial fiber-matrix misfit mobilizes radial contact pressure at the fiber-matrix interface. The fiber-matrix misfit is the difference between the fiber radius and the matrix hole radius after shrinkage in the absence of fiber. The fiber-matrix misfit is also equal to the sum of the increase in the inner radius of the matrix cylinder and the decrease in the radius of the fiber in the presence of radial contact pressure.  For an unstressed fiber, the  relationship between the interfacial contact pressure, a , and the radial fiber-matrix misfit, 8 , can c  r  be written as [224]: 5.24  Due to the Poisson's effect, the radial fiber-matrix misfit and the interfacial contact pressure decreases when fiber is stressed in the axial direction. For an axially stressed fiber, reduction in fiber-matrix misfit is given by:  5.25  113  where 07 is the fiber axial stress. The resultant contact pressure, <j , at the interface can be cp  obtained by substituting S -Aa for 8 and a-Aa (=a) for a in the Equation 5.24: r  r  (S -Aa) r  cp  a  a +b b -a 2  5.26  2  2  + t>„  7  E  1  f i  f  Substituting for Aa in the above equation:  •+ -  a a +b E~ b —a 2  2  +  ^[l-v ) E f  f  / J  ?L E„  va f  a +b 2  5.27  f  2  +  Thus, when tensile stress in the fiber is 07, the resultant contact pressure, o , at the interface is cp  given by: a  5.28  =<7 + (0(7 ,  cp  c  /  where co =  a +b 0 —a 2  2  Since Takaku and Arridge [54] developed their model for stiff fibers, the constant co was defined differently in their model.  Substituting for o  cp  da  f  dz  _  2/f  a  in the equation of stress transfer (Equation 5.23):  [a +(oo ] c  5.29  f  114  Rearranging:  1  dz  +—COG,  =•  2H  5.30  a  The above equation is a first order linear differential equation, and it dictates stress transfer between fiber and matrix over a debonded interface. This equation was first derived by Takaku and Arridge [54], and was applied to the case of a completely debonded fiber. In the following, the application of this equation is extended to the case of a partially debonded fiber.  5.1.2.1 Fiber Axial Stress Distribution, 07 The solution of the above differential equation represents the axial stress, 07 over the debonded fiber length (i.e., L > z > L-ld). With the boundary condition, Of=<7d at z-L-U, the solution is obtained as: 2a>fi(L-l )  -lajiz  d  '  co  CO  e  .  a  e  5.31  a  The fiber axial stress, 07, along the bonded region (z <L-U) can be obtained by replacing L by Lld and <y by  in Equation 5.16.  0  5.1.2.2 Interfacial Frictional Shear Stress Distribution, if From Coulomb's law, the distribution of interfacial frictional shear stress, T/over the debonded fiber length can be obtained as: r =H<y =n((j +CQCJ ) f  cp  c  5.32  f  The above equation depicts that the dependence of interfacial frictional shear stress, Tf, on fiber axial stress, 07. Substituting for 07, the interfacial frictional shear stress, 17 over the debonded region (L >z> L-ld) is obtained as:  115  "I  T,  = aco o +— e d  2<QLl(L-l )  -2CQLtZ  d  a  e  CO  5.33  a  x , along the bonded region (z <L-ld) can be obtained by replacing L  The interfacial shear stress,  a  by L-ld and o by o in Equation 5.17. 0  d  5.1.2.3 Pullout Stress versus Debond Length Relationship, a vs. U 0  The relationship between the fiber pullout stress, a , and the debond length, l , is obtained by 0  d  using Equation 5.31 at z=L: a.  a  5.34  co  Note that the fiber pullout stress, a , is equal to the initial debonding stress, G , when the debond 0  d  length is equal to zero, i.e., 0"=Cv when ld=0. 0  5.1.2.4 Fiber Displacement versus Debond Length Relationship, U d vs. U P  The total fiber displacement, U d, can be obtained by integrating the fiber strains along the p  embedded fiber length:  -7  E (7 + a - yqcc + an)  sinh[/?(L-/,-z)]  smh[fi(L-l )]  + y\dz  d  f  -|  2a>n(L-l ) d  -2cofiz  dz  116  5.35  Solving:  u  Pd  =  cosh[/3(L-/,)l-l ^ Bsmh[j3(L-l )]  E (y + a- yna + an) (a-yna + an-y) f  n  + r  (L-/.) d )  d  -2/i<u/  rf  a 2E \ico  E co f  °c_  5.36  CO  f  and, for a two-sided pullout problem, the total fiber displacement, U d, assuming debonding p  occurs symmetrically, can be derived as:  2a  A  E (y + a- yna + an)  (a-yna + an-y)  cosh[fi(L-l )]-l , ^ _ ' , Bsmh[j3(L-l )] d  n  , ' +y(L-l ) t T  J  r  x 1  d  d  f  2a  E co f  °c_  -1  co  2E /icol f  5.37  where the total fiber length is equal to 2L, and the embedded length on each side of the pullout specimen is equal to L.  5.1.2.5 Bond and Frictional Components of Pullout Stress, a ,bond and o jric 0  0  At any debond length Id, the fiber pullout stress, o , is a summation of two components - the one 0  arising due to the adhesional shear bond (o bond) and the other arising due to the frictional shear 0i  bond ((Jofric)a =a o  u  That is:  J  5.38  +cj .  o,bond  f  o,fnc  From the above solution, the two components of pullout stress are obtained as:  ® o.bond ~ ° d >  -1  o.fric  117  c ° d  +  CO  5.39  5.1.2.6 Catastrophic Debonding The interfacial debonding process becomes catastrophic when the fiber pullout stress, a begins 0  to drop with an increase in debond length, l . Thus, the condition for catastrophic debonding can d  be written as:  dl  d  The above condition will be satisfied when the difference between the increase in the frictional component of the pullout stress (i.e., o" / - ) and the decrease in the adhesional component of the 0 n  pullout stress (i.e.,  o,fric  da  dl  d  <  o ,bond) 0  c  with increase in l is equal to or less than zero. That is: d  o,bond  dG  5.41  dl  d  Let the debond length at which the debonding process turns catastrophic be represented by l  .  dxat  With the mechanical properties of constituent materials, interfacial properties, fiber length and fiber radius all known, the length of debond zone, h, at which the debonding process will turn catastrophic can be obtained by satisfying the equality expressed by the Equation 5.40. Substituting for o in Equation 5.40, we obtain the slope of <y vs. l curve as: 0  0p  d  2/IGXC,  0  /3bc  4 +  p\jc  3  1\ia  c  d  5.42  dh  118  where  ap x =(a-yna + 2  x =(a3  ari)cosh[P(L-l )]+Y d  yna + ct77)sinh[/J(L - l )] d  ap  At the instant that catastrophic debonding occurs, the slope of the a vs. Id curve will be zero. 0  Therefore, the debond length, ld, t, at which debonding process becomes catastrophic can be ca  obtained by equating the right hand side of the Equation 5.42 to zero. The peak pullout load and the fiber displacement corresponding to peak pullout load can be obtained by substituting the calculated value of catastrophic debond length, l , t, into Equation 5.34 and Equation 5.37, d ca  respectively.  5.1.3 Stage 3: Fiber Completely Debonded and Pulling Out Consider the case when the fiber is completely debonded along its embedded length, and let p  d  be the rigid body displacement of the fiber as shown in Figure 5.1.5. In this case, transfer of stress between the fiber and the encapsulating matrix takes place entirely through the interfacial frictional shear bond. The governing differential equation of stress transfer for the pullout case has been derived by Takaku and Arridge [54]: da 2// 2n —— H—— coo =——o f  dz  a  f  a  5.43  c  Solving this equation, Takaku and Arridge [54] derived a solution for the fiber axial stress distribution, 07. However, their solution implicitly assumed that the coefficient of friction remain constant during the fiber pullout process (i.e., \i is independent of pd). This limitation is  119  relaxed in the analysis proposed below. Based on the energy considerations, a procedure to calibrate interfacial properties is proposed in the next section; this procedure effectively describes the dependence of the coefficient of friction, p on the fiber pullout distance, pd.  The work of Takaku and Arridge [54] is further extended here and closed-form solutions are derived for the interfacial frictional shear stress distribution, T/, the fiber pullout stress, o~, and 0  the fiber displacement, U  .  z=L-p  d  Pullout distance  Figure 5.1.5: Fiber completely debonded over its length and pulling out  5.1.3.1 Fiber Axial Stress Distribution, 07 The solution of the above differential equation (Equation 5.43) represents the axial stress along the embedded length of a completely debonded fiber. With the boundary condition, 07 = 0 at z=0, the solution is obtained as:  where 0 < z ^L-pj. In the above equation note that ji is a function of pd.  120  5.1.3.2 Interfacial Frictional Shear Stress Distribution, Xf Following Coulomb's law, the interfacial frictional shear stress, Xf, over the embedded length can be obtained as follows: x  f  =iio =n{o +coa ) cp  c  5.45  f  Substituting for 07, the interfacial frictional shear stress,  H<J e  Xfis  obtained as:  5.46  a  c  where 0 < z ^L-pd-  In the above equation note that fi is a function of pd- This equation depicts that interfacial frictional shear stress varies along the embedded length of fiber, with a minimum at the exit fiber end and a maximum at the embedded fiber end. It can also be noted that during stage 2, when ld=L, Equation 5.33 reduces to Equation 5.46.  5.1.3.3 Fiber Pullout Stress, a  0  The relationship between fiber pullout stress, a , and pullout distance, pd, can be obtained from 0  Equation 5.44: -2a)ii(L-p ) d  \-e  5.47  a  The fiber pullout stress, o , at pd=0, termed here as the initial frictional pullout stress, 0  <7 , if  can be  written as: -2cofiL  co  1-e  5.48  a  121  In the above equations note that \i is a function of pd.  5.1.3.4 Fiber Displacement, U  The total fiber displacement, U , is obtained by integrating the fiber strains along the total fiber Pd  length: L -ICOfiZ  =P«-  U  l-e  E co  Pd  -2con(L-p ) d  dz--^- I I l - e E co  a  dz  5.49  f  f  L-Pd  Solving: -2/ico(L-p )  -2n<a(.L-p )  d  U  P d  =P ~ d  E co f  (  i  -  p  '  )  +  ^ r  '  '  d  _  \  1  +  p  ' r  5.50  e  For a two-sided pullout problem, the fiber displacement, U , is given by:  U  D  =P ~ d  f  2c,  -2tm(L-p ) d  1  -2tio)(L-p )"  f  d  5.51  E co f  where total fiber length is 2L and embedded length of fiber on each side of the pullout specimen is L. In the above equations note that ji is a function of pd-  5.1.4 Calibration of Interfacial Parameters - c , \i and t c  s  Interfacial contact stress, c : When the fiber is axially loaded, the resultant contact stress, c , c  cp  is given by Equation 5.28:  c =c +coc cp  c  f  122  where 07 is the axial fiber stress, a is the interfacial contact stress for an unloaded fiber, and co is c  a constant. This equation depicts that with an increase in axial fiber stress the interfacial contact stress decreases due to Poisson's effect (note that a is negative). For fibers with long lengths, a c  critical value of fiber axial stress will be reached such that the contact between fiber and matrix will disappear and the resultant contact stress, <j , will become zero. When this happens, the cp  interfacial debonding will continue without requiring any increase in the applied pullout stress. From the asymptotic value of fiber pullout stress, a = 0  cr ,asymptotic, 0  obtained from the peak pullout  stress verses embedded fiber length plot, the interfacial contact stress, o~, can be calculated as: c  Interfacial coefficient of friction, fi: During the process of fiber pullout, gradual abrasion and wear of the interface layer takes place. As a consequence, the interfacial coefficient of friction is expected to decrease with increase in the pullout displacement.  Based on the energy  considerations of the problem of fiber pullout, a method is proposed below to determine variation in coefficient of friction as a function of fiber pullout displacement. pullout, W , when fiber pullout displacement increases from p p  dl  to p  d2  Work of fiber  can be written as:  5.53  where o is the fiber pullout stress given by Equation 5.47. Substituting for o in Equation 5.53 a  0  we obtain: Pdi  5.54 Pdi  123  Within a small displacement domain (p  dl  to p ), the coefficient of friction, ji, can be assumed to d2  remain constant, and the above equation can be solved as: 7Z» cr  -2a>liL  2  CO  ,  Pdl ~ Pdl  a  _  e  a  5.55  2copi  To obtain W from the experimental pullout curve, the relationship between the total pullout p  displacement and the rigid body displacement of the fiber must be known.  As a first  approximation, the rigid body displacement of the fiber, p is taken equal to A-A ,  where, A is  d  peak  the total pullout displacement, and A k is the total pullout displacement corresponding to the pea  peak pullout load. With W and a known, the coefficient of friction, fi, can be calculated from p  c  the above non-linear equation. This equation depicts the dependence of coefficient of friction, fi, on the pullout displacement, p . d  Adhesional bond strength, x : For very small fiber lengths, the peak pullout stress is governed by s  the Equation 5.21. Differentiating Equation 5.21 with respect to fiber embedded length we obtain: da  d  _-2T (y s  + a-yncc + ari) \[(a-yna  dL  af3  \  + ar])cosh(RL) + y]pcosh(i3L)  [(a-yr/a + arj)cosh(/?L)-i-y]  2  5.56 P(a - yqa + ar]) sinh (/3L) 2  [(a - yna + ar]) cosh( f5L) + y ]  2  At L = 0 the above equation simplifies to: 2T„  dL  5.57  a  JL-O  Rearranging the above equation we obtain:  a do  5.58  d  dL  JL=0  124  Thus, substituting the initial slope of the experimental maximum pullout stress versus embedded fiber length plot in the above equation, the interfacial adhesional shear strength, x , can be s  calculated.  5.2 Conclusions •  An analytical model for the problem of fiber pullout is proposed in this chapter.  The  proposed model is unique because of its ability to take into consideration the evolution of the interfacial properties during the pullout process. The proposed model captures the essential features of the pullout process, including the progressive interfacial debonding and the Poisson's effect in the event of a debonded fiber. Analysis is divided into three stages, and for the each stage, closed-form solutions are derived for the fiber axial load distribution, the interfacial shear stress distribution and the fiber displacement.  Complete pullout response  can be predicted using the proposed progressive debonding model. •  During stage 1, fiber and matrix displacements at the interface remain compatible, and the resistance to fiber pullout is derived from the adhesional shear stresses mobilized at the interface.  Closed-form solutions are derived for the fiber axial stress distribution and the  interfacial shear stress distribution along the fiber length, the fiber displacement, and the initial debonding stress (i.e., the fiber pullout stress required to initiate interfacial debonding). The closed-form solution for the initial debonding stress is derived based on a maximum shear stress criterion. This solution indicates that, among other factors, the initial debonding stress depends upon fiber length and fiber elastic properties. At the end of stage 1, interfacial debonding initiates at the location where the fiber enters the matrix. •  During stage 2, the fiber is partially bonded along its embedded length. The adhesional shear stresses at the bonded interface and the frictional shear stresses at the debonded interface resist the fiber pullout.  The influence of Poisson's contraction of fiber is taken into  consideration in the analysis. It is shown that for any debond length, the fiber pullout stress is a summation of two components - the one arising due to the adhesional shear bond and the other arising due to the frictional shear bond. Closed-form solutions are derived for fiber axial stress distribution over the bonded and the debonded interfaces, interfacial adhesional shear stress distribution over the bonded interface, interfacial frictional shear stress  125  distribution over the debonded interface, fiber pullout stress versus debond length relationship, and fiber displacement versus debond length relationship. It is demonstrated that debonding process becomes catastrophic at the instant when the fiber pullout stress begins to drop with increase in debond length. This condition is satisfied when the difference between change in the frictional component of pullout stress and the adhesional component of pullout stress resulting due to change in debond length becomes equal to zero. A closed-form solution is derived to calculate the catastrophic debond length, given the mechanical properties of constituent materials, the interfacial properties and the geometry of the pullout specimen. Closed-form solutions are also derived to calculate the peak pullout stress and the displacement corresponding to the peak pullout stress. During stage 3, the fiber is completely debonded along its embedded length and fiber pullout is initiated. Frictional shear stresses existing over the debonded interface resist pullout of fiber from the matrix. Closed-form solutions are derived for fiber axial stress distribution, interfacial frictional shear stress distribution, fiber pullout stress, and fiber displacement at different stages of pullout process. A procedure to calibrate interfacial properties is described in this chapter. It is shown that interfacial contact stress can be calculated using the asymptotic value of pullout stress on the peak pullout stress versus embedded length plot. It is recognized that the coefficient of friction may decrease with increase in pullout displacement.  Based on the energy  considerations, a method is proposed to calculate the coefficient of friction as a function of pullout displacement. The adhesional bond strength can be calculated from the initial slope of the peak pullout stress versus embedded fiber length plot.  126  Chapter 6  Progressive Debonding Model for Fiber Pullout: Validation  6.0 Introduction In the previous chapter, an analytical model to study progressive debonding of fiber-matrix interfaces and fiber pullout response was proposed. Validation of this model is achieved in the present chapter.  For this purpose, experimental fiber pullout data from Chapter 4 are used.  Theoretical predictions are also made for the pullout data found in the literature.  6.1 Calibration of Interfacial Properties and Validation of Model In order to use the proposed model, calibration of three interfacial properties, namely, o , ji and c  t , is required. A procedure for calibrating interfacial properties was described in the previous s  chapter. Clearly, these parameters will depend upon the properties of the constituents involved. For instance, it can be expected that matrices with different shrinkage behavior will yield different values of <T , and also that fibers having different surface roughness characteristics will C  yield different values of \i. In this regard, the experimental pullout curves presented in Chapter 4 are meaningful, since fibers and matrices with different properties led to different pullout curves suggesting dependence of pullout response on interfacial properties. The pullout data of Chapter 4 are used here to calibrate the interfacial properties and to validate the proposed Progressive Debonding Model. Validity of the proposed model is also confirmed by comparing the model predictions with the pullout data found in the literature.  6.1.1 Source Code for Progressive Debonding Model To theoretically predict fiber pullout response using the proposed Progressive Debonding Model, a source code is written in C programming language. This source code is attached in Appendix C.  127  6.1.2 Straight-smooth, Stainless Steel Fiber Embedded in Normal Strength Matrix In Section 4.1.1, experimental pullout curves for straight-smooth, stainless steel fibers embedded in a normal strength matrix were presented. Here, those data are used to calibrate the interfacial properties.  The elastic modulus of fiber and matrix are taken as 210 GPa and 30 GPa,  respectively. The fiber radius, a and the outer radius of matrix cylinder, b are taken as 0.22 mm and 22 mm, respectively.  Substituting the asymptotic value of pullout stress from the pullout stress versus embedded length plot in Equation 5.52, the magnitude of the interfacial contact stress, o~, is obtained as c  -29.3 MPa.  The coefficient of friction versus pullout distance relationship was calculated from Equation 5.55. In this equation, the interval between p  dl  and p  d2  was chosen as 0.5 mm, and the area under  the pullout curve, W lying between p i and p was calculated as: p  (p  = "  d2  +P )  *{p ,- )  Kp  Wp  d  6.1  PdJ 2  d  Pdl  where P i and P  are the pullout load values corresponding to the pullout distances p i and p ,  pd2  pd  d  d2  respectively. Equation 5.55 is a non-linear equation, and it was solved for fi using the Mathcad PLUS 6.0 Professional Edition software.  The coefficient of friction versus pullout distance  curves obtained using Equation 5.55 for fibers of different lengths are plotted in Figure 6.1.1. In the figure, it can be noted the coefficient of friction, fl, decreases exponentially with increase in pullout distance. The evolution law for the coefficient of friction can be described by the following equation: fi = (fi..fij -  cpd  e  +fi  6.2  ss  where, fit - initial coefficient of friction fi  - steady state value of coefficient of friction attained at large pullout distances  c  - a constant that governs the rate at which coefficient of friction decays with increase in  ss  pullout distance The following values of fit, fi  ss  and c gave an appropriate data fit describing variation in  coefficient of friction with increase in pullout distance: fit = 0.12, fi - 0.035 and c = 0.7. ss  128  0.14 -fr 15 mm 30 mm 60 mm 90 mm Trendline  0.12 4 +0.035  0.02  +  4  o  o  5  10  15  Displacement, p  20 d  25  30  (mm)  Figure 6.1.1: Coefficient of friction versus pullout displacement curves for straight-smooth, stainless steel fibers embedded in normal strength matrix  Substituting these values in Equation 6.2, the evolution law for coefficient of friction, /! can thus be written as ji = 0.085 e~ ' +0.035. Note that the observed decrease in coefficient of friction 01p  1  is attributable to matrix wear and consequent smoothening of interface layer as the fiber pulls out of the matrix. It can be readily appreciated that the evolution law for coefficient of friction, ji, may take different forms depending upon the age of matrix, matrix porosity/air content, size of voids, and loading rate.  On substituting the initial slope of maximum pullout stress versus fiber embedded length curve in Equation 5.58, the magnitude of adhesional bond strength, x , is obtained as -2.4 MPa. It must s  be noted that a more precise prediction of adhesional bond strength can be obtained by testing fibers of even smaller embedded length.  Using the above values of interfacial properties, the theoretical pullout response is predicted for a straight-smooth, stainless steel fiber with 30 mm embedded length (Section 4.1.1). Figure 6.1.2 compares the experimental pullout curves with the theoretical. A good correspondence between the theoretical prediction and the experimental curves is noticeable in the figure.  129  140  120 4  Displacement (mm)  Figure 6.1.2: Comparison of experimental and theoretical pullout response - straight-smooth, stainless steel fiber fiber (embedded length - 30 mm) embedded in normal strength matrix  Theoretically predicted peak pullout load and the displacement corresponding to peak pullout load are compared with experimental values in the Table 6.1.1. Model predicted results agree well with the experimental ones.  Table 6.1.1: Theoretical and experimental peak pullout load and displacement corresponding to peak pullout load for a straight-smooth, stainless steelfiberembedded in normal strength matrix  Fiber  Straightsmooth, stainless steel Ppeak  6.1.3  Embedded Fiber Length (mm)  Fiber Diameter  30  0.44  - Peak pullout load;  PpeakW)  Matrix  8  (mm) Normal strength  Speak  (mm)  Experimental  Model  Experimental  Model  95.10  89.16  0.120  0.099  - Displacement at peak pullout load  peak  Straight, Stainless Steel Fibers with Rough Surface Embedded in Normal Strength Matrix  In Section 4.1.2, experimental pullout curves for straight, stainless steel fibers with a rough surface embedded in a normal strength matrix were presented.  Here, those data are used to  calibrate the interfacial properties. The elastic modulus of fiber and matrix are taken as 210 GPa and 30 GPa, respectively. The fiber radius, a and the outer radius of matrix cylinder, b are taken as 0.22 mm and 22 mm, respectively. Substituting the asymptotic value of pullout stress from the pullout stress versus embedded length plot in Equation 5.52, the magnitude of interfacial 130  contact stress, cr , is obtained as -29.5 MPa. Coefficient of friction versus pullout distance curves c  obtained using Equation 5.55 are plotted in Figure 6.1.3 for fibers of different lengths. The magnitudes of p and p t  are obtained as 0.26 and 0.068, respectively, thus indicating that even  ss  for rough steel fiber the coefficient of friction, fi, decreases with increase in pullout distance. However, for rough steel fiber, both the initial coefficient of friction, jii and the steady-state coefficient of friction, p , are greater than those for straight-smooth, stainless steel fibers. The ss  decay rate for coefficient of friction, characterized by the constant c, is found to be independent of a fiber's surface roughness. Substituting these values in Equation 6.2, the evolution law for coefficient of friction, fj,, can thus be written as p, = 0.192 e~  01pd  +0.068.  Note that this  evolution law is different than that obtained for smooth steel fiber, and this suggests the dependence  of interfacial coefficient of friction on fiber surface roughness.  Superior  interlocking between fiber and matrix probably is responsible for greater values of the coefficient of friction in the case of rough steel fibers.  • c o +  15 mm 30 mm 60 mm 90 mm Trendline  o U  5  10  15  Displacement, p  20 d  25  30  (mm)  Figure 6.1.3: Coefficient of friction versus pullout displacement curves for straight, stainless steel fibers with rough surface embedded in normal strength matrix  Using these values of interfacial properties, the theoretical pullout response is predicted for a fiber with 30 mm embedded length (Section 4.1.2). Figure 6.1.4 compares the model prediction with the experimental pullout curves. A good correspondence between the model prediction and the experimental curves is distinctly noticeable in the figure.  131  Displacement (mm)  Figure 6.1.4: Comparison of experimental and theoretical pullout response - straight, stainless steelfiberwith rough surface (embedded length - 30 mm) embedded in normal strength matrix  Model predictions for the peak pullout load and the displacement corresponding to peak pullout load are compared with the experimental values in Table 6.1.2. Model predicted values agree well with the experimental ones.  Table 6.1.2: Theoretical and experimental peak pullout load and the displacement corresponding to peak pullout load for a steelfiberwith rough surface embedded in normal strength matrix  Fiber  Straight, stainless steel fiber with rough surface Ppeak  Embedded Fiber Length (mm)  Fiber Diameter  30  0.44  - Peak pullout load;  Matrix  8  (mm) Normal strength  V ( N )  Speak  (mm)  Experimental  Model  Experimental  Model  116.1  122.04  0.120  0.156  - Displacement at peak pullout load  peak  6.1.4 Straight-smooth, Stainless Steel Fibers Embedded in Non-Shrink Grout Matrix In Section 4.1.3, experimental pullout curves for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix were presented. Here, those data are used to calibrate the interfacial properties.  The elastic modulus of fiber and matrix are taken as 210 GPa and 30 GPa,  respectively. The fiber radius, a and the outer radius of matrix cylinder, b were taken as 0.22 mm and 22 mm, respectively. Substituting the asymptotic value of pullout stress from the pullout stress versus embedded length plot in the Equation 5.52, the magnitude of interfacial  132  contact stress, o~, is obtained as -11.7 MPa. Clearly, the magnitude of interfacial contact stress c  obtained with non-shrink grout matrix is much smaller than that obtained with normal strength matrix.  This decrease is directly attributable to the reduced shrinkage of non-shrink grout  matrix. Elastic modulus of fiber and matrix are taken as 210 GPa and 30 GPa, respectively. The fiber radius, a and the outer radius of matrix cylinder, b were taken as 0.22mm and 22 mm, respectively. Coefficient of friction versus pullout displacement curves obtained using Equation 6.55 are plotted in the Figure 6.1.5 for fibers of different lengths. It can be noted that coefficient of friction, fi, decreases with increase in pullout distance, and that the evolution law for coefficient of friction can be described by the equation fi = 0.085 e~  onpd  +0.035. This evolution  law for coefficient of friction, fi, is same as that obtained for a straight-smooth, stainless steel fiber embedded in a normal strength matrix.  On substituting the initial slope of maximum  pullout stress versus fiber embedded length curve in Equation 5.58, the magnitude of adhesional bond strength, t , is obtained as -1.3 MPa. Using these values of interfacial properties, s  theoretical pullout response is predicted for a fiber with 30 mm embedded length (Section 4.1.3). Figure 6.1.6 compares the theoretical pullout curve with the experimental ones.  A good  correspondence between the theoretical prediction and the experimental curves is distinctly noticeable in the figure. 0.14 i • •  0.12 -  tf o  1  0.1 -  u=0.085e^"" +0.035  +  0.08  15 m m 30mm 60mm 90mm Trendine  o  I 8  0.06 • 0.040.02 0 10  15  Displacement, p  20 d  25  30  (mm)  Figure 6.1.5: Coefficient of friction versus pullout displacement curves for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix  Model predictions for the peak pullout load and the displacement corresponding to peak pullout load are compared with the experimental values in Table 6.1.3. compared well with the experimental ones.  133  Model predicted values  50  Displacement (mm)  Figure 6.1.6: Comparison of experimental and theoretical pullout response - straight-smooth, stainless steel fiber (embedded length - 30 mm) embedded in non-shrink grout matrix  Table 6.1.3: Theoretical and experimental peak pullout load and the displacement corresponding to peak pullout load for a straight-smooth, stainless steel fiber embedded in non-shrink grout matrix  Fiber  Straightsmooth, stainless steel Ppeak  Embedded Fiber Length (mm)  Fiber Diameter  30  0.44  Matrix  (mm)  - Peak pullout load; S peak  S  Ppeak (N)  Nonshrink Grout  p e a k  (mm)  Experimental  Model  Experimental  Model  36.6  35.6  0.044  0.039  Displacement at peak pullout load  134  6.1.5 Straight, Smooth Polypropylene Fibers Embedded in Normal Strength Matrix 0.14 0.12 A o •a  E  15 mm  0.1 0.08 \  A  30 mm  +  60 mm Trendline  o  °  + ++  0.02 \ 0 0  5  10  15  20  25  30  Displacement, p (mm) d  Figure 6.1.7: Coefficient of friction versus pullout displacement curves for straight, smooth polypropylene fibers embedded in normal strength matrix  In Section 4.1.6, experimental pullout curves for straight, smooth polypropylene fibers embedded in normal strength matrix were presented. Here, those experimental pullout data are used to calibrate the interfacial properties. The elastic modulus of fiber and matrix are taken as 3.5 GPa and 30 GPa, respectively. The fiber radius, a and the outer radius of matrix cylinder, b are taken as 0.475 mm and 47.5 mm, respectively. Substituting the asymptotic value of pullout stress from the pullout stress versus embedded length plot in the Equation 5.52, the magnitude of interfacial contact stress, <7 , is obtained as -11.0 MPa. This value of interfacial contact stress, C  <7 , is smaller than that obtained for stainless steel fibers embedded in a similar type of matrix. C  Since interfacial frictional bond and interfacial contact stress, o , are related by Equation 5.46, it c  can be concluded that magnitude of interfacial frictional bond will also be lower in the case of polypropylene fibers. The coefficient of friction versus pullout distance curves obtained using Equation 5.55 for fibers of different lengths are plotted in the Figure 6.1.7. It can be noted that the coefficient of friction, pt, decreases with increase in pullout distance, and that the evolution law for coefficient of friction can be described by the equation pi = 0.02 e~°  iPd  + 0.03 . Note that  this evolution law is different than that obtained for steel fibers embedded in a similar type of matrix. On substituting the initial slope of the maximum pullout stress versus fiber embedded length curve in Equation 5.58, the magnitude of adhesional bond strength, T is obtained as -0.61 s  MPa. Using these values of interfacial properties, theoretical pullout response is predicted for a fiber with 15 mm embedded length.  Figure 6.1.8 compares the model prediction with the 135  experimental pullout curves, and it can be seen that model prediction agrees well with the experimental ones.  Model predictions for the peak pullout load and the displacement corresponding to peak pullout load are compared with the experimental values in Table 6.1.4. The predicted peak pullout load agrees well with the experimental value, however, the experimental displacement at peak pullout load is much greater than the predicted value.  Displacement (mm)  Figure 6.1.8: Comparison of experimental and theoretical pullout response - straight, smooth polypropylene fiber (embedded length - 15 mm) embedded in normal strength matrix Table 6.1.4: Theoretical and experimental peak pullout load and the displacement corresponding to peak pullout load for a straight, smooth polypropylene fiber embedded in normal strength matrix  Fiber  Straightsmooth, polypropylene Ppeak  Embedded Fiber Length (mm)  Fiber Diameter  15  0.95  - Peak pullout load;  Ppeak (N)  Matrix  8  (mm) Normal strength  Speak (mm)  Experimental  Model  Experimental  Model  16.84  14.90  0.300  0.106  - Displacement at peak pullout load  peak  6.1.6 Pullout Data Found in the Literature Naaman and Shah [ 142] Figure 6.1.9 compares the experimental fiber pullout response reported by Naaman and Shah [142] for a steel fiber and the theoretical prediction based on the proposed Progressive 136  Debonding Model. Length of fiber was 12.7 mm and diameter of fiber was 0.25 mm. The elastic modulus of fiber and matrix are taken as 210 GPa and 30 GPa, respectively. The outer radius of matrix cylinder, b is taken as 12.5 mm. Following interfacial properties are assumed to obtain the theoretical prediction: T, = -2.4  MPa  Cv= -29.3  MPa  fl = 0.085  e~°  lpd  + 0.035  (/i, = 0.12  and ^  s  = 0.035)  It can be seen that very good agreement between the theoretical prediction from the progressive debonding model and the experimental response is obtained. Note that, interfacial properties assumed here to make theoretical prediction are same as the ones used previously to predict pullout response of straight, smooth steel fibers embedded in normal strength matrix (Section 6.1.2).  For comparison, the predictions obtained from the other theoretical models are also  shown in the same figure.  30  Disriacemert (rrrrj Figure 6.1.9: Comparison of experimental and theoretical pullout response for a straight, smooth steel fiber. Experimental curve is by Naaman and Shah [142]  137  Mallikarjuna, Fafard and Banihia [225] Table 6.1.5 compares the peak pullout loads predicted by the proposed Progressive Debonding Model and the Finite Element Analysis results obtained by Mallikarjuna et al.  [225].  Theoretical  predictions are obtained for fibers with three different diameters - 0.50, 0.75 and 1.00 mm. In all cases the embedded fiber length is 18 mm. The elastic modulus of fiber and matrix are taken as 210 GPa and 30 GPa, respectively. The following interfacial properties are assumed to obtain theoretical predictions: T, = -2.4 MPa <Jc= -29.3 M P a ji = 0.085 e^ " 7p  +0.035  (rx, = 0.12 and /4s = 0.035)  Table 6.1.5: Comparison of peak pullout loads as predicted by (i) the proposed progressive debonding model, and (ii) finite element analysis, Mallikarjuna, et al. (225) Embedded Fiber Length (mm)  Fiber  Straightsmooth, steel  Fiber Diameter  18  Peak pullout load, P  peak  (N)  (mm)  Finite Element Analysis  Proposed Progressive Debonding Model  0.50  77.7  75.8  0.75  119.5  124.0  1.00  159.6  172.9  Very good agreement is obtained between the two results.  Agreement is better for smaller  diameter fibers, and disparity between the two result decreases with decrease in fiber diameter.  Wang, Li and Backer [69] Figure 6.1.10 compares the experimental fiber pullout response reported by Wang et al. [69] for a straight, smooth polypropylene fiber and the theoretical prediction based on the proposed Progressive Debonding Model.  Length and diameter of fiber are 50 mm and 0.508 mm,  respectively. The elastic modulus of fiber and matrix are taken as 3.5 GPa and 30 GPa, respectively. The outer radius of matrix cylinder, b is taken as 25 mm. The following interfacial properties are assumed to obtain theoretical prediction: t = -0.61 M P a s  cr =-14.0 MPa c  ji = 0.02 e~°  3pd  + 0.03 (fr = 0.05 and ^ = 0.03) 138  In the figure, a good agreement between the theoretical and the experimental response is noticeable.  20  30  40  60  Emplacement (mm)  Figure 6.1.10: Comparison of experimental and theoretical pullout response for a straight, smooth polypropylene fiber. Experimental curve is by Wang, Li and Backer [69]  6.2 Conclusions •  The main objective of this chapter was to validate the proposed Progressive Debonding Model.  Theoretical predictions from the proposed model compare well with the  experimental pullout data presented in Chapter 4 and the pullout data found in literature. Good agreement between the theoretical and the experimental results thus validates the proposed model. •  The evolution law for the coefficient of friction indicates that coefficient of friction decays exponentially with increase in fiber pullout distance. This observation is found to be valid for both steel as well as polypropylene fibers. Smoothening of the interface (due to matrix wear) with increase in pullout distance appears to be responsible for this behavior.  •  A comparison between the evolution laws for the coefficient of friction for rough steel fiber and a smooth steel fiber indicates that the coefficient of friction values for the former are greater than for the latter. A superior interlocking mechanism between fiber and matrix in the case of rough steel fibers appears to be the reason for higher values of coefficient of friction in the case of rough steel fibers.  This observation depicts the dependence of  interfacial frictional bond on fiber's surface roughness.  139  A comparison between the evolution laws for the coefficient of friction for the smooth polypropylene fibers and the smooth steel fibers indicates that the coefficient of friction values for the former are smaller than for the latter. This may be attributable to the following two reasons: the smoother surface of the polypropylene fibers and the hydrophobic nature of polypropylene material. The hydrophobic property of polypropylene will prevent penetration of the matrix into the fine cavities on the fiber surface.  As a result, a good interlocking  mechanism between fiber and matrix will not be established, and the coefficient of friction will remain low. A comparison between the evolution laws for the coefficient of friction for the smooth polypropylene fibers and the smooth steel fibers indicates that the coefficient of friction decays from the initial value to a steady-state value at a faster rate for the steel fibers. The magnitude of the interfacial contact stress obtained with a Portland cement matrix (CSA Type 10) is found to be greater than that obtained with a non-shrink grout matrix. This observation depicts the dependence of interfacial contact stress and interfacial frictional bond on shrinkage properties of the matrix. For polypropylene fibers embedded in a normal strength matrix, the interfacial contact "stress is found to be smaller than that for steel fibers embedded in a similar type of matrix. This observation depicts the dependence of interfacial contact stress and interfacial frictional bond on the elastic properties of constituents.  140  Chapter 7  Progressive Debonding Model for Fiber Pullout: Parametric Studies  7.0 Introduction In the Chapter 4, it was experimentally observed that fiber pullout response depends upon various factors, such as fiber-matrix interfacial properties, mechanical properties of constituents and physical dimensions of the fiber. From the viewpoint of optimization of toughness/strength, it is critical to identify the relative importance of these factors, given the dependence of composite properties on fiber pullout response.  Thus, the objective of this chapter is to  systematically investigate influence of various factors on the progressive debonding behavior and fiber pullout response. This objective has been achieved by conducting parametric studies using the proposed Progressive Debonding Model.  7.1 Parametric Studies The influence of the following factors on progressive debonding and pullout response has been investigated: •  Adhesional bond strength, T  •  Interfacial contact stress, a  •  Interfacial coefficient of friction, ix  s  c  Important results obtained from parametric studies are reported below.  7.1.1  Influence of Adhesional Bond Strength, x  s  The influence of adhesional bond strength, r , on progressive debonding and fiber pullout s  response is investigated in this section. For this purpose, parametric studies are carried out on two fibers with different elastic moduli. The first fiber has an elastic modulus of 210 GPa and the same for the second fiber is 3.5 GPa.  7.1.1.1 Fiber Elastic Modulus, E = 210 GPa f  Parametric studies are carried out for three different values of adhesional bond strength, T : -1 s  MPa, -5 M P a and -10 MPa. Assumed values of the other interfacial properties are: o = -15.0 c  141  MPa and /z = 0.065 e~°  +0.035 (/i =0.1 and fi =0.035). Mechanical properties of fiber are  JPd  t  ss  assumed as: £}=210 GPa (elastic modulus) and, V/=0.30 (Poisson's ratio), and the same for matrix are assumed as 30 GPa and 0.20, respectively. Total fiber length, L , is taken as 50 mm (i.e., one side embedded length=25 mm), and fiber diameter is taken as d=l.O mm. The assumed value of b (the outer radius of matrix cylinder) is 50 mm. Note that an elastic modulus of 210 GPa corresponds to that of steel. 160 120 Z  J 80 a S.  40  0 0  0.01  0.02  0.03  0.04  0.05  0.06  Displacement (mm) Figure 7.1.1a: Influence of adhesional bond strength on pullout response (£^=210 GPa)  Figure 7.1.1a compares the fiber pullout response for three different values of adhesional bond strength, x . The initial linear part of the pullout curves seen in the figure depicts elastic loading s  of fibers. In the linear region of the pullout curve, the maximum interfacial shear stress remains below the interfacial adhesional bond strength, T , therefore, the interface along the entire s  embedded length remains fully bonded. On further loading, the pullout load increases in a nonlinear fashion until the peak pullout load is attained. In this region interfacial debonding initiates and continues in a progressive fashion, i.e., pullout load increases with increase in debond length. Also, over the debonded interface, interfacial frictional shear stresses are mobilized. The pullout load during progressive debonding increases because the rate at which the frictional component of pullout load increases with change in debond length is greater than the corresponding rate at which adhesional component of the pullout load decreases, i.e., do  o  fric  Idl > do d  obond  Idl  d  (Equation 5.41). At the peak pullout load, the rate of increase in the  frictional component of pullout load becomes equal to the rate of decrease in adhesional component of the pullout load.  142  On further loading, the remaining intact portion of the interface debonds catastrophically, i.e., no increase in pullout load is required to further debond the interface. Thus, pullout load drops in this region, which is accompanied by decrease in pullout displacement. The pullout load in this region drops because the rate at which frictional component of pullout load increases due to change in debond length is smaller than the corresponding rate at which the adhesional component of the pullout load decreases.  The theoretically predicted decrease in pullout  displacement is not observed in the experiments since pullout tests are normally carried out at a constant rate of pullout displacement. In Figure 7.1.1a it can also be noticed that no catastrophic debonding occurs when T = -1 MPa. S  In Figure 7.1.1b, the pullout load at initial debonding and the peak pullout load are plotted as a function of adhesional bond strength, T . Both the pullout load at initial debonding and the peak S  pullout load increase with increase in adhesional bond strength. However, the rate of increase of the former is greater than that of the latter. In Figure 7.1.1c, pullout displacement at the peak pullout load is plotted as a function of adhesional bond strength, r . In the figure it can be seen s  that pullout displacement at the peak pullout load increases with increase in adhesional bond strength, x . s  After initial debonding, further debonding requires the applied pullout load to overcome the interfacial frictional shear stresses at the debonded interface and adhesional shear stresses at the bonded interface. As a result, the pullout load required to further debond the interface depends upon the extent of prior debonding. Figure 7.1.1d shows variation in pullout load as a function of debond length for the case when x = -5 MPa. In the same figure, components of pullout load, s  (i.e., the adhesional and the frictional components) are also plotted. The following points can be noted in the figure: •  Interfacial debonding initiates at the surface where the fiber enters the matrix. At initiation of debonding the fiber pullout load (i.e., the initial debonding load) is equal to the adhesional component of pullout load, since the frictional component of pullout load is equal to zero.  •  With increase in debond length, fiber pullout load continues to increase until debond length corresponding to the peak pullout load is attained. Upon further debonding, fiber pullout load begins to decrease;  With increase in debond length, the adhesional component of  pullout load decreases, on the other hand, the frictional component of pullout load increases.  143  The peak pullout load on the pullout load vs. debond length curve corresponds to the point when slope of the curve becomes zero.  This condition is satisfied when slope of the  adhesional component of pullout load vs. debond length curve is equal and opposite in sign to that of the frictional component of pullout load vs. debond length curve. The debond length corresponding to this point is termed the catastrophic debond length,  ld, t ca  (Section  5.1.2.6), since the debonding process turns catastrophic upon further debonding.  Peak pullout load  X  120  z  i •2 80  | 1 Pullout load at initial debonding  3  JD  ~5 a.  \  V-'"  40  0  2  X  0.02 X  . -'  * * *'  3  £ =210GPa /  £7=210 GPa cr =-15MPa  ^ o.oi  c  o\=-15 MPa fl,  4  6  =0.1  8  10  12  2  4  6  8  10  Adhesional bond strength, T , (MPa)  Adhesional bond strength, T , (MPa)  Figure 7.1.1b: Influence of adhesional bond strength on the  Figure 7.1.1c: Influence of adhesional bond strength on  pullout load at initial debonding and the  displacement at the peak pullout load  peak pullout load (£}=210 GPa)  (£,=210 GPa)  Figure 7. L i e shows pullout load and its components as a function of debond length at different values of adhesional bond strength, t . s  From this figure, the following important observations  can be made: •  The pullout load corresponding to any given debond length increases with increase in adhesional bond strength, t . s  However, at complete debonding, the magnitude of pullout  load is independent of adhesional bond strength, T . s  •  Prior to complete debonding, the adhesional component of pullout load increases with increase in adhesional bond strength; on the other hand, the frictional component of pullout load decreases with increase in adhesional bond strength.  •  Catastrophic debonding takes place at t = -5.0 MPa and -10 MPa; on the other hand, atx = s  s  1.0 MPa, the debonding process is completely stable, i.e., pullout load continues to increase until the fiber is completely debonded. •  For a given fiber length, catastrophic debond length, l  diCaU  adhesional bond strength, T^.  144  decreases with increase in the  160  160  £ , = 2 1 0 GPa 0% =-15  120  T,=-5 MPa  \  i  120 • /J.=O.I  Total^—-  £ , = 2 1 0 GPa  T,=-10MPa  MPa  \  \  -  v  t,=-5MPa -  O 3  Frictional c ornporffint^--*^'^  <r =-15MPa  80  c  / i , = 0 1 ^ ^  1  1 Adhesional component 0  5  10  15  . . - ' ' r,=-l MPa  80  3  -TV?-V  o  40  \ \  20  5  25  Debond length /,,/Axlal position £-z (nm)  10  15  20  25  Debond length, /j^\xial position £-z (mm)  Figure 7.1. Id: Variation in pullout load and pullout load  Figure 7 . L i e : Variation in pullout load and its components  components as a function of debond length  as a function of debond length at different  (Ej=210 GPa)  values of adhesional bond strength, r  s  ( £ ) = 2 1 0 GPa)  Figure 7.1.If shows the variation in axial load distribution in the fiber at initiation of debonding for different values of adhesional bond strength, x . In the figure, it can be seen that fiber axial s  load increases with increase in adhesional bond strength, T5. It can also be noticed that fiber axial load is a maximum at the loaded end of the fiber. Axial load decreases hyperbolically towards the embedded end of fiber and it vanishes near the center of the embedded fiber length.  A  similar trend can be seen for the interfacial shear stress distribution along the embedded fiber length at initiation of debonding, Figure 7.1.1 g. 100 £ , = 2 1 0 GPa er =-15MPa  80  r,=-10MPa  c  M,=0.1  /  CL.  5  r,=-l MPa  \  \  60 r,=-5 MPa  /  T,=-5MPa " ^ ^ ^ ^  40 r,=-l MPa  20  /  E, =210  /  GPa r,=-10MPa  cr,=-15 MPa  a  M,=0.1 1  0  5  10  15  20  0  25  10  15  20  25  Axial position, z (mm)  Axial position, z (mm)  Figure 7.1.If: Axial  5  load distribution in fiber at initial  Figure 7.1.lg: Interfacial shear stress distribution at initial  debonding ( £ / = 2 1 0 GPa)  debonding (£^=210 GPa)  Figure 7.1.1h shows the variation in axial load distribution at completion of debonding for different values of adhesional bond strength, %. In the figure it can be seen that the axial load distribution along the fiber length is independent of adhesional bond strength, T , at completion S  of interfacial debonding. It can also be noticed that fiber axial load is maximum at the loaded fiber end and it decreases almost linearly to a value of zero at the embedded fiber end. Figure 145  7.1.1i shows the interfacial shear stress distribution at completion of interfacial debonding. In this figure it can be seen that interfacial shear stress is maximum at the embedded fiber end, and it gradually decreases towards the exit fiber end. Poisson's contraction of the fiber is responsible for the observed interfacial shear stress distribution along the embedded length of fiber. Comparison of Figures 7.1.1g and 7.1.1i clearly depicts that the axial load distributions at initiation of debonding and at completion of debonding are very different.  Axial position, z (mm)  Axial position, z (mm)  Figure 7.1.1h: Axial load distribution at completion of  Figure 7.1.Ii: Interfacial shear stress distribution at  debonding (^=210 GPa)  completion of debonding (£/==210GPa)  7.1.1.2 Fiber Elastic Modulus, E/= 3.5 GPa Parametric studies are carried out for three different values of adhesional bond strength, %: -1 MPa, -5 M P a and -10 MPa. Assumed values of other interfacial properties are: c = -15.0 M P a c  and ji = 0.02 e - "' +0.03 Qi =0.05 and fi =0.03). Mechanical properties of fiber are assumed -0  3  t  ss  as: Ef=3.5 GPa (elastic modulus) and, ty=0.35 (Poisson's ratio), and the same for matrix are assumed as 30 GPa and 0.20, respectively. Total fiber length, L, is taken as 50 mm (i.e., one side embedded length=25 mm), and fiber diameter is taken as d=l.O mm. The assumed value of b (the outer radius of matrix cylinder) is 50 mm.  Note that an elastic modulus of 3.5 GPa  corresponds to that of polypropylene.  Figure 7.1.2a compares the pullout response of fibers with Ej- 3.5 GPa at three different values of adhesional bond strength, x . In this figure, it can be seen that the difference between the s  pullout responses at three different values of adhesional strength, x , is relatively insignificant. s  At Tj =-1 M P a the debonding process is completely stable. On the other hand, the debonding process turns catastrophic at T^=-5 M P a and -10 MPa; however, the load drop during 146  catastrophic debonding is very small and not noticeable in the figure.  X j =-1U M r a  T,=-5 MPa  //*~~~~~  £ , = 3 . 5 GPa  j[  CT =-15 MPa  T,=-1 MPa  0  0.1  C  H,=0.05 0.2  0.3  0.4  0.5  Displacement (mm)  Figure 7.1.2a: Influence of adhesional bond strength on pullout response (Ey=3.5 GPa)  0.5  50 -i £ , = 3 . 5 GPa ad(h  40 -  illout  _o  o\=-15 MPa  Peak pullout load  ^,=0.05 \  30 ...  3K  3KW  ^  20 •  a.  II  0 •  .-A  A"  \  X X 0.3 1  Pullout toad at initial debonding 10 -  X  0.4  £ , = 3 . 5 GPa cr =-15MPa  8.  -  e  li =0.05 l  0.2 2  4  6  8  10  2  Adhesional bond strength, r, (MPa)  4  6  8  10  12  Adhesional bond strength, T , (MPa)  Figure 7.1.2b: Influence of adhesional bond strength on  Figure 7.1.2c: Influence of adhesional bond strength on  pullout load at initial debonding and  pullout displacement corresponding to peak  maximum pullout load (E/=3.5 GPa)  pullout load (Ef=3.5 GPa)  In Figure 7.1.2b, the pullout load at initial debonding and the peak pullout load are plotted as a function of adhesional bond strength, T,. Both the pullout load at initial debonding and the peak pullout load increase with increase in adhesional bond strength; however, the rate of increase of the latter is smaller than that of the former. Comparing Figures 7.1.1b and 7.1.2b it can also be noted that for any given adhesional bond strength, the initial debonding load for low modulus fibers is much smaller than that for high modulus fibers. Moreover, disparity between the two increases with increase in adhesional bond strength. It can also be noted that when adhesional bond strength increases from -1 MPa to -10 MPa, peak pullout load increases by about 49% for high modulus fiber, on the other hand, peak pullout load increases only by about 7% for low modulus fibers. 147  In Figure 7.1.2c, displacement at the peak pullout load is plotted as a function of adhesional bond strength, T,. In the figure, it can be seen that the displacement at peak pullout load increase with increase in adhesional bond strength, r . Comparing Figures 7.1.1c and 7.1.2c it can be noted s  that at any given adhesional bond strength, the displacement at peak pullout load for low modulus fibers is much greater than for high modulus fibers.  Figure 7.1.2d shows variation in pullout load as a function of debond length for the case when % = -5 MPa. In the same figure, components of pullout load (i.e., the adhesional component and the frictional component) are also plotted. It can be noted that variation in the pullout load and its components with increase in debond length is similar to that for high modulus fibers.  Figure 7.1.2e shows pullout load and its components as a function of debond length at different values of adhesional bond strength, T . From this figure, the following important observations S  can be made: •  Pullout load corresponding to any given debond length increases with increase in adhesional bond strength, x . s  However, at complete debonding, magnitude of pullout load is  independent of adhesional bond strength, x . s  •  Prior to complete debonding, the adhesional component of pullout load increases with increase in adhesional bond strength. On the other hand, the frictional component: of pullout load decreases with increase in adhesional bond strength.  •  In Figures 7.1.1e and 7.1.2e it can be seen that when adhesional bond strength is increased from -1 M P a to -10 MPa, the increase in the peak pullout load for high modulus fibers is about 49%. On the other hand, the same for low modulus fibers is only about 7%. The reason for this disparity is that for low modulus fibers, much of the increase in the adhesional component of pullout load obtained with increase in adhesional bond strength is compensated by the corresponding decrease in the frictional component of pullout load. From the viewpoint of optimization of interfacial properties, this observation is significant, since it demonstrates that efficiency of low modulus fibers cannot be improved substantially by solely increasing adhesional bond strength.  148  •  Catastrophic debonding takes place at Ts = -5.0 MPa and -10 MPa. On the other hand, at Ts = -1.0 MPa the debonding process is completely stable, i.e., pullout load continues to increase until the fiber is completely debonded.  •  Other parameters remaining same, catastrophic debond length, ld  iCat  decreases with increase  in the adhesional bond strength, T . Also, the catastrophic debond length is greater for low s  modulus fibers in comparison to its high modulus counterparts.  Debond length, / lAxial position, L-z (mm)  Debond length, / /Axial position, L-z (mm)  d  d  Figure 7.1.2d: Variation in pullout load and pullout load  •  Figure 7.1.2e: Variation in pullout load and its components  components as a function of debond length  as a function of debond length at different  CE/=3.5 GPa)  values of adhesional bond strength, % (£/=3.5 GPa)  Figure 7.1.2f shows the axial load distribution in the fiber at the initiation of debonding for different adhesional bond strengths, x . In the figure, it can be seen that the fiber axial load s  increases with increase in adhesional bond strength, T^. It can also be noticed that axial load in the fiber is maximum at the loaded fiber end and it decreases very rapidly along the embedded fiber length. Thus, for low modulus fibers only a very small length of the fiber is mobilized in the stress transfer process prior to initiation of interfacial debonding. Comparing Figures 7.1.lf and 7.1.2f it can be seen that prior to initiation of interfacial debonding, the rate of decrease in axial load along the embedded fiber length for low modulus fibers is very rapid in comparison to that for high modulus fibers. This means that prior to initiation of interfacial debonding, the length of fiber mobilized in the stress transfer process is longer for high modulus fibers in comparison to that for low modulus fibers. Observations similar to those above can be made for the interfacial shear stress distribution along the embedded fiber length of low modulus fibers (Figure 7.1.2g).  149  16  £  -2H  12  0\=-15 MPa  E  -6H  /t,=0.05  •a  1  4  a  0  5  10  15  20  25  5  10 Axial position, z  Axial position, z (mm)  Figure 7.1.2f: Fiber Axial load distribution in fiber at initial  15  20  25  (mm)  Figure 7.1.2g: Interfacial shear stress distribution at initial  debonding (Ey=3.5 GPa)  debonding (£}=3.5 GPa)  Figure 7.1.2h shows the variation in axial load distribution at completion of debonding for different values of adhesional bond strength, T . In the figure, it can be seen that axial load s  distribution along the fiber length is independent of adhesional bond strength, x (at completion s  of interfacial debonding). It can also be noticed that fiber axial load is a maximum at the loaded fiber end and it decreases to a value of zero at the embedded fiber end. Comparison of Figures 7.1.2f and 7.1.2h clearly depicts that the axial load distributions at initiation of debonding and at completion of debonding are very different. In the former case only a part of the fiber is axially loaded, whereas in the latter case, the whole fiber is axially loaded. Figure 7.1.2i shows the interfacial shear stress distribution at completion of interfacial debonding. It can be seen that interfacial shear stress is a maximum at the embedded fiber end, and it decreases towards the exit fiber end. Poisson's contraction of the fiber is responsible for the observed variation in shear stress distribution along the embedded length of fiber. Comparing Figures 7.1.1i and 7.1.2i it  150  can also be seen that for low modulus fibers, the rate at which interfacial shear stress decays is more rapid in comparison to that for high modulus fibers.  7.1.2  Influence of Interfacial Contact Stress, a  c  The influence of interfacial contact stress, o" , on progressive debonding and fiber pullout c  response is investigated below. For this purpose, parametric studies are carried out on two fibers with different elastic moduli. The first fiber has an elastic modulus of 210 GPa and the same for the second fiber is 3.5 GPa.  7.1.2.1 Fiber Elastic Modulus, E = 210 GPa f  Parametric studies are carried out for three different values of interfacial contact stress, c : -5 c  MPa, -15 M P a and -30 MPa. Assumed values of other interfacial properties are: T = -1.0 M P a s  and ix = 0.065 e~  0JPd  +0.035  (Lit  =0.1 and LI =0.035).  Mechanical properties of fiber are  SS  assumed as: £/=210 GPa (elastic modulus) and, v/=0.30 (Poisson's ratio), and the same for matrix are assumed as 30 GPa and 0.20, respectively. Total fiber length, L , is taken as 50 mm (i.e., one side embedded length=25 mm), and fiber diameter is taken as <i=1.0 mm. The assumed value of b (the outer radius of matrix cylinder) is 50 mm. Note that an elastic modulus of 210 GPa corresponds to that of steel. 240 200 Z  160  •2  120  240  £7=210 GPa  H  T,=-l  o-=-30 MPa c  £ , = 2 1 0 GPa  200  MPa  o =-30 MPa  T,=-l  c  n, =0.1  Z  160  M  i  MPa  ,=0.1  o.=-15 MPa  o =-15 MPa c  1° 8  1  H  40  j/  s  80  V /  o =-5 MPa c  o =-5 MPa c  40  0  0  0.01  0.02  0.03  0.04  0.05  0  0.06  0  5  10  15  20  25  Displacement (mm)  Displacement (mm)  Figure 7.1.3a: Influence of interfacial contact stress on  Figure 7.1.3b: Influence of interfacial contact stress on  pullout response (£)=210 GPa)  pullout response (£}=210 GPa)  The prepeak part of pullout curves and the complete pullout curves for different values of interfacial contact stress, a are shown in Figures 7.1.3a and 7.1.3b, respectively. In the former c  figure it can be noticed that the prepeak pullout curves become nonlinear at very small values of pullout loads. It can also be noticed that both the peak pullout load and the displacement at peak 151  pullout load increase with increase in interfacial contact stress, o . In the latter figure it can be c  noticed that the postpeak pullout response varies greatly at different values of interfacial contact stress, o , and in general, postpeak pullout loads increase with increase in interfacial contact c  stress, o . In other words, energy absorbed during the process of fiber pullout increases with an c  increase in interfacial contact stress, o . c  240  E  f  200  S  =210 GPa  0.04  Peak pullout load  f ,=-1 MPa /J =0.1  3  t  160  o  « o  120  •-  _o  S.  X 0.03  I!  n Z  3.  8-  Pullout load at initial debonding  80  o.02  X  Cu  £ , = 2 1 0 GPa 0.01  T,=-l  X  MPa  ij,=0.1  A  40 0  5  10  15  20  25  30  35  40  0  0 Normal contact stress, - o" (MPa)  5  10  15  20  25  Normal contact stress, -a  30  c  e  Figure 7.1.3c: Influence of interfacial contact stress on  35  40  (MPa)  Figure 7.1.3d: Influence of interfacial contact stress on  pullout load at initial debonding and peak  pullout displacement at the peak pullout load  pullout load (£/=210 GPa)  (£/=210 GPa)  In Figure 7.1.3c, the pullout load at initial debonding and the peak pullout load are plotted as a function of interfacial contact stress, o . In this figure it can be seen that the peak pullout load c  increases and the pullout load at initial debonding remains unchanged with increase in interfacial contact stress, o . In Figure 7.1.3d, displacement at the peak pullout load is plotted as a function c  of interfacial contact stress, o . In the figure, it can be seen that pullout displacement at peak c  pullout load increases with increase in interfacial contact stress, o . c  Given the dependence of pullout performance on interfacial contact stress, o , two approaches c  can be used for improving fiber efficiency: •  Using a matrix that shrinks more during curing, setting and hardening, so that a higher value of contact stress, o , is generated at the interface. c  •  Intelligently designing the fiber such that interfacial contact stress, o , increases during the c  process of fiber pullout.  152  240 £ , = 2 1 0 GPa  200  g  c  Adhesional  n=o.\  160  Pullout  component  load  CS  \  Frictional component  3  I  o- =-30MPa  T,=-l  o- =-15MPa_^ c  80  J\' o- =-5 MPa r  40  0  0  5  10  Debond length, l  15  20  25  /Axial position, L-z (mm)  d  Figure 7.1.3e: Variation in pullout load and its components as a function of debond length (Zy=210 GPa)  Figure 7.1.3e shows pullout load and its components as a function of debond length at different values of interfacial contact stress, o . From this figure, the following important observations c  can be made: •  Pullout load corresponding to any given debond length increases with increase in interfacial contact stress, o . c  •  Prior to complete debonding, the frictional component of pullout load increases with increase in interfacial contact stress, o . On the other hand, the adhesional component of pullout load c  is not affected by change in interfacial contact stress, o . c  a.  a  5  =-5 MPa  =-15 MPa  1  1 1 . 10  15  20  —  -3  =-30 MPa  c  £/=210GPa T,=-1 MPa  .  1 — —  5  25  —  m  — i -  10  1—  15  u,=0.1  20  25  Axial position, z (mrp)  Axial position, z (mm)  Figure 7.1.3f: Fiber axial load distribution at completion of  Figure 7.1.3g: Interfacial shear stress distribution at  debonding (£y=210GPa)  completion of debonding (Ef=210 GPa)  Figure 7.1.3f shows the variation in axial load distribution at completion of debonding for different values of interfacial contact stress, o . In the figure, it can be seen that axial load along c  the embedded fiber length increases with increase in interfacial contact stress, o . It can also be c  153  noticed that the fiber axial load is maximum at the loaded fiber end and it decreases almost linearly to a value of zero at the embedded fiber end. Figure 7.1.3g shows the interfacial shear stress distribution at completion of interfacial debonding. From this figure two observations can be made. First, interfacial shear stress increases with increase in interfacial contact stress, o~. c  Secondly, for any given value of interfacial contact stress, a , the interfacial shear stress is c  maximum at the embedded fiber end and it gradually decreases towards the exit fiber end. Poisson's contraction of the fiber is responsible for the observed interfacial shear stress distribution along the embedded length of fiber.  7.1.2.2 Elastic Modulus, E = 3.5 GPa f  Parametric studies are carried out for three different values of interfacial contact stress, <7 : -5 C  MPa, -15 M P a and -30 MPa. Assumed values of other interfacial properties are: T = -1.0 MPa s  and  Ll =  0.02 e~ " + 0.03 (/i =0.05 and a3p  t  fi  ss  =0.03). Mechanical properties of fiber are assumed  as: Ef=3.5 GPa (elastic modulus) and v/=0.35 (Poisson's ratio), and the same for matrix are assumed as 30 GPa and 0.30, respectively. Total fiber length, L, is taken as 50 mm (one side embedded length=25 mm), and fiber diameter is taken as cf=1.0 mm. The assumed value of b (the outer radius of matrix cylinder) is 50 mm.  Note that an elastic modulus of 3.5 GPa  corresponds to that of polypropylene. 80  80 £ , = 3 . 5 GPa  £ 3 . 5 GPa  T =-1 MPa  T =-l MPa  / =  s  5  o =-30 MPa  ^,=0.05  \i, =0.05  c  Z  i  3  40  '".,^=-30 MPa  1  ^ = - 1 5 MPa  3  o =-15 MPa  I  c  if  / 40  n,=-5 MPa  -  q.=-5 MPa  o+ 0  0.2  0.4  0.6  0.8  0  Displacement (mm)  5  10  15  20  25  Displacement (mm) •  Figure 7.1.4a: Influence of interfacial contact stress on  Figure 7.1.4b: Influence of interfacial contact stress on  prepeak pullout response (E/=3.5 GPa)  pullout response (£}=3.5 GPa)  The prepeak part of pullout curves and the complete pullout curves for different values of interfacial contact stress, cr , are shown in Figures 7.1.4a and 7.1.4b, respectively. In the former c  figure, it can be noticed that the prepeak pullout curves become nonlinear at very small values of pullout loads. It can also be noticed that both the peak pullout load and the displacement at peak 154  pullout load increase with increase in interfacial contact stress, o~. In the latter figure it can be c  noticed that the postpeak pullout response varies greatly at different values of interfacial contact stress, a , and in general, postpeak pullout loads increase with increase in interfacial contact c  stress, a . In other words, energy absorbed during the process of fiber pullout increases with c  increase in interfacial contact stress, G . c  Given the dependence of pullout performance on interfacial contact stress, a , two approaches c  can be used for improving fiber efficiency as outlined in the previous sub-section (Section 7.1.2.1). 80 Peak pullout load  ~  0.8  X  a 5 „  M "8  40  ti  1  Pullout load at initial debonding  i.........  0  5  10  15  20  25  30  35  40  0.6  0 4  0.2  X  £,=3.5 GPa r,=-l MPa /i,=0.05 .  X  0 10  15  20  25  Normal contact stress, -a  Normal contact stress, - rj (MPa)  c  c  Figure 7.1.4c: Influence of interfacial contact stress on  30 (MPa)  Figure 7.1.4d: Influence of interfacial contact stress on  pullout load at initial debonding and peak  pullout displacement at the peak pullout load  pullout load (£y=3.5 GPa)  (£,=3.5 GPa)  In Figure 7.1.4c, the pullout load at initial debonding and the peak pullout load are plotted as a function of interfacial contact stress, cr . In this figure it can be seen that the peak pullout load c  increases and the pullout load at initial debonding remains unchanged with increase in interfacial contact stress, o . In Figure 7.1.4d, displacement at the peak pullout load is plotted as a function c  of interfacial contact stress, o . In this figure, it can be seen that the displacement at the peak c  pullout load increases with increase in interfacial contact stress, a . Comparing Figures 7.1.3d c  and 7.1.4d it can be noticed that at any given interfacial contact stress, a , the displacement at the c  peak pullout load for low modulus fibers is much greater than that for their high modulus counterparts.  155  0  5  10  15  20  25  Debond lergth, / /ficAA portion, L-z (rrm) d  Figure 7.1.4e: Variation in pullout load and its components as a function of debond.lengfh (£/=3.5 GPa)  Figure 7.1.4e shows pullout load and its components as a function of debond length at different values of interfacial contact stress, a . c  In the figure, it can be seen that the pullout load  corresponding to any given debond length increases with increase in interfacial contact stress, o . c  It can be also seen that the frictional component of pullout load increases with increase in interfacial contact stress, a . On the other hand, the adhesional component of pullout load is not c  affected by change in interfacial contact stress, <7 . C  Axial position, z (mm)  Axial position, z (mm)  Figure 7.1.4f: Fiber axial load distribution at completion of  Figure 7.1.4g: Interfacial shear stress distribution at  debonding (£/=3.5 GPa)  completion of debonding (£/=3.5 GPa)  Figure 7.1.4f shows the variation in axial load distribution at completion of debonding for different values of interfacial contact stress, o . In the figure, it can be seen that axial load along c  the embedded fiber length increases with increase in interfacial contact stress, o . It can also be c  noticed that the fiber axial load is maximum at the loaded fiber end and it decreases to a value of 156  zero at the embedded fiber end. Figure 7.1.4g shows the interfacial shear stress distribution at completion of interfacial debonding. From this figure, two observations can be made. First, interfacial shear stress increases with increase in interfacial contact stress, <7 , and secondly, for C  any given value of interfacial contact stress, o~, the interfacial shear stress is maximum at the c  embedded fiber end and it decreases towards the exit fiber end. Poisson's contraction of the fiber is responsible for the observed interfacial shear stress distribution along the embedded fiber length. Comparison of Figures 7.1.3g and 7.1.4g also indicates that for low modulus fibers the rate of decay of interfacial shear stress along the embedded length is more rapid in comparison to that for their high modulus counterparts.  7.1.3  Influence of Interfacial Coefficient of Friction, fi  The influence of the interfacial coefficient of friction, ji, on progressive debonding and fiber pullout response is investigated below. For this purpose parametric studies are carried out on two fibers with different elastic moduli. The first fiber has an elastic modulus of 210 GPa and the same for the second fiber is 3.5 GPa.  7.1.3.1 Fiber Elastic Modulus, E/= 210 GPa Parametric studies are carried out by varying the initial coefficient of friction, chosen values of /x, are: 0.1, 0.25 and 0.50,  and the corresponding evolution laws for the  coefficients of friction selected are: /i = 0.065 e~°  7Pd  jl = 0.465 e~°  7pd  The three  +0.035, /I = 0.215 e~°  npd  +0.035, and  +0.035, respectively. Assumed values of the other interfacial properties were:  Ts = -1.0 M P a and o~ = -15.0 MPa. Mechanical properties of the fiber are assumed as: £/=210 c  GPa (elastic modulus), and, v/=0.30 (Poisson's ratio), and the same for the matrix are assumed as 30 GPa and 0.20, respectively. Total fiber length, L, is taken as 50 mm (i.e., one side embedded length=25 mm), and fiber diameter is taken as <i=1.0 mm. The assumed value of b (the outer radius of matrix cylinder) is 50 mm. Note that an elastic modulus of 210 GPa corresponds to that of steel.  157  300  300 fii=0.50  250 Z  200  i  z  g i  150  I  3 O  / /  |  £ =210GPa  250 /ii=0.25  r  \  /J,=0.50  T,=-l MPa o- =-15 MPa  200 150  c  \ \  ^,=0.25  3  c.-=o.i  \ \  100  /i,=0.1  £y=210 GPa 50  T,=-l MPa  50  o- =-15MPa e  0 0  0.05  0.1  0 0.15  0  5  10  Displacement (mm)  15  20  25  Displacement (mm)  Figure 7.1.5a: Influence of interfacial coefficient of friction  Figure 7.1.5b: Influence of interfacial coefficient of friction  on pullout response ( £ / = 2 1 0 GPa)  on pullout response ( £ } = 2 1 0 GPa)  The prepeak part of the pullout curves and the complete pullout curves for different values of interfacial coefficient of friction, /z„ are shown in Figures 7.1.5a and 7.1.5b, respectively. In the former figure, it can be noticed that the prepeak pullout curves become nonlinear at very small values of pullout loads, and also that both the peak pullout load and the displacement at peak pullout load increase with increase in interfacial coefficient of friction, / i , . In the latter figure, it can be noticed that the postpeak pullout response varies greatly at different values of interfacial coefficient of friction, fx and in general, postpeak pullout loads increase with increase in h  interfacial coefficient of friction. In other words, energy absorbed during the process of fiber pullout increases with increase in interfacial coefficient of friction. 300 £,=210GPa 250  S  1 Z  T,=-l MPa  £ =210GPa  o- =-15MPa  T,=-l MPa  /  c  200  o\=-15 MPa  11  Peak pullout load  150  3  S.  ioo X Pullout load at initial debonding  50 A  0 0  --A  X X  --• A y  0.25  0.5  0.75  0.25  Coefficient of friction, /i  0.5  0.75  Coefficient of friction, ft,  Figure 7.1.5c: Influence of interfacial coefficient of friction  Figure 7.1.5d: Influence of interfacial coefficient of friction  on pullout load at initial debonding and peak  on pullout displacement at the peak pullout  pullout load ( £ / = 2 1 0 GPa)  load (£>=210 GPa)  In Figure 7.1.5c, the pullout load at initial debonding and the peak pullout load are plotted as a function of interfacial coefficient of friction, ju . In this figure, it can be seen that the peak (  pullout load increases with increase in interfacial coefficient of friction, jU,. From the viewpoint 158  of optimization of interfacial properties, this observation is important, since it demonstrates that efficiency of high modulus fibers can be significantly improved by increasing the coefficient of friction.  In the Figure 7.1.5d, the displacement at the peak pullout load is plotted as a function of interfacial coefficient of friction, zz,. In this figure it can be seen that the displacement at the peak pullout load increases with increase in the interfacial coefficient of friction, /z,. £ , = 2 1 0 GPa T,=-l MPa (T =-15MPa c  Pullout load  Jii =0.25  Frictional component^,/.'' V \  Adhesional component 0  5  10  15  20  25  Debond length,I /Axial position, L-z (mm) d  Figure 7.1.5e: Variation in pullout load and its components as a function of debond length (£y=210 GPa)  Figure 7.1.5e shows pullout load and its components as a function of debond length at different values of interfacial coefficient »of friction, rx,.  From this figure, the following important  observations can be made: •  Pullout load corresponding to any given debond length increases with increase in interfacial coefficient of friction, zz,.  •  Prior to complete debonding, the frictional component of pullout load increases with increase in interfacial coefficient of friction, /z,. On the other hand, the adhesional component of pullout load is not affected by a change in interfacial coefficient of friction, zz,.  Figure 7.1.5f shows the variation in axial load distribution at completion of debonding for different values of interfacial coefficient of friction, zz,. In the figure, it can be seen that the axial load along the embedded fiber length increases with increase in interfacial coefficient of friction, zz,-. It can also be noticed that the fiber axial load is maximum at the loaded fiber end and it decreases to a value of zero at the embedded fiber end. Figure 7.1.5g shows the interfacial shear 159  stress distribution at completion of interfacial debonding.  In the figure it can be noticed that  interfacial shear stress increases with increase in interfacial coefficient of friction, fx . Also, for t  any given value of interfacial coefficient of friction, fit the interfacial shear stress is maximum at the embedded fiber end and it decreases towards the exit fiber end. It can also be noticed that the rate of decay of interfacial shear stress along the embedded fiber length increases with increase in interfacial coefficient of friction, Lij. Poisson's contraction of fiber is responsible for the observed interfacial shear stress distribution along the embedded length of fiber.  Axial position, z (mm)  Axial position, z (mm)  Figure 7.1.5f: Fiber axial load distribution at completion of  Figure 7.1.5g: Interfacial shear stress distribution at  debonding (£/=210GPa)  completion of debonding  (£ =210GPa) y  7.1.3.2 Fiber Elastic Modulus, E = 3.5 GPa f  Parametric studies are carried out by varying the initial coefficient of friction, Li . The three t  chosen values of Lij are: 0.05, 0.25 and 0.50, and the corresponding evolution laws for coefficient Ll -  0.47  e~°  of friction 3pd  selected  are:  LL  = 0.02e~°  3pd  +0.03,  LL  = 0.22e** * +0.03, and v  + 0.03, respectively. Assumed values of the other interfacial properties were:  x  s  = -1.0 M P a and o~ = -15.0 MPa. Mechanical properties of the fiber are assumed as: E/=3.5 GPa c  (elastic modulus), and, V/=0.35 (Poisson's ratio), and the same for matrix are assumed as 30 GPa and 0.30, respectively.  Total fiber length, L , is taken as 50 mm (i.e., one side embedded  length=25 mm), and fiber diameter is taken as d=l.O mm. The assumed value of b (the outer radius of matrix cylinder) is 50 mm. Note that an elastic modulus of 3.5 GPa corresponds to that of polypropylene.  160  ^i/0.25  JJ,=0.50  £ , = 3 . 5 GF'a / / / I ,-=0.25  11  T,=-l  '  z  f \ .  N  MPa  o%=-15MPa  . \lii=0.50  //J,=0.05 JJ,=0.05  [  £ , = 3 . 5 GPa T,=-l  MPa  <r =-15MPa c  0 0  0.1  0.2  0.3  0.4  0.5  0.6  0  5  10  Displacement (mm)  15  20  25  Displacement (mm)  Figure 7.1.6a: Influence of interfacial coefficient of friction  Figure 7.1.6b: Influence of interfacial coefficient of friction  on prepeak pullout response (£/=3.5 GPa)  on pullout response (£/=3.5 GPa)  The prepeak part of pullout curves and the complete pullout curves for different values of interfacial coefficient of friction, (ii, are shown in Figures 7.1.6a and 7.1.6b, respectively. In the former figure, it can be noticed that the prepeak pullout curves become nonlinear at very small values of pullout loads. Also, increase in interfacial coefficient of friction, /i„ beyond a certain value does not produce any increase in peak pullout load. From Figure 7.1.6b, it can be observed that the total pullout energy does not increase significantly with increase in coefficient of friction. 40  Z  Peak pullout load  S -g  £ , = 3 . 5 GPa  20  T,=-l  0.6  X  X  MPa  o",=-15 MPa  X  •a  Pullout load at initial debonding  8.  Ef =3.5 GPa '  0.2  T,=-l  MPa  o- =-15 MPa c  0  0.25  0.5  0  0.75  Coefficient of friction, ll;  0.25  0.5  0.75  Coefficient of friction, ti;  Figure 7.1.6c: Influence of interfacial coefficient of friction  Figure 7.1.6d: Influence of interfacial coefficient of friction  on pullout load at initial debonding and peak  on pullout displacement at the peak pullout  pullout load (£/=3.5 GPa)  load (E/=3.5 GPa)  In Figure 7.1.6c, the pullout load at initial debonding and the peak pullout load are plotted as a function of interfacial coefficient of friction, /i,-.  In this figure, it can be seen that the peak  pullout load initially increases, and then it becomes constant with increase in interfacial coefficient of friction, //,. From the viewpoint of optimization of interfacial properties, this observation is important, since it demonstrates that the efficiency of low modulus fibers cannot 161  be significantly improved by solely increasing the coefficient of friction.  In Figure 7.1.6d,  displacement at the peak pullout load is plotted as a function of interfacial coefficient of friction, zz„ and it can be seen that the displacement at the peak pullout load increases at a decaying rate with increase in interfacial coefficient of friction, /z,. 40  Pullout load  M  30 Z  z  /j,=0.05  20  j f  3  2  fli =0.25 £ , = 3 . 5 GPa  Frictional component  / **  10  T,=-l MPa o- =-15 MPa c  Adhesional comDonent 5  1  1  10  15  —  20  25  Debond length,/,, /Axial position, L-z (mm)  Figure 7.1.6e: Variation in pullout load and its components as a function of debond length (£^=210 GPa)  Figure 7.1.6e shows pullout load and its components as a function of debond length at different values of interfacial coefficient of friction, zz*. In this figure, it can be seen that in the lower range of coefficient of friction, zz, (= 0.05 to 0.25), pullout load corresponding to any given debond length increases with increase in interfacial coefficient of friction, /Z;. On the other hand, in the higher range of coefficient of friction, zz, (= 0.25 to 0.50), pullout loads at smaller debond lengths {Id < 10 mm) increase with increase in coefficient of friction, zz„ while the difference between pullout loads due to increase in coefficient of friction vanishes at longer debond lengths.  /*i=0.50  /  •a 20  //J,=0.25  It i =0.05^  //  0  /i,=0.50  £ 3 . 5 GPa / =  If '  '  T,=-l  £ / = 3 . 5 GPa  MPa  T,=-l MPa  0\=-15 MPa 1  L_  5  10  o- =-15 MPa  -10 15  20  25  c  0  Axial position, z (mm)  5  10  15  20  25  Axial position, z (mm)  Figure 7.1.6f: Fiber axial load distribution at completion of  Figure 7.1.6g: Interfacial shear stress distribution at  debonding (£>=3.5 GPa)  completion of debonding (£y=3.5 GPa)  162  Figure 7.1.6f shows variation in axial load distribution at completion of debonding for different values of interfacial coefficient of friction,  It can be noticed that the fiber axial load is  maximum at the loaded fiber end and it decreases to a value of zero at the embedded fiber end. In the figure, it can also be seen that at large values of interfacial coefficient of friction, Li , axial t  load remains constant along the major portion of embedded fiber length. Figure 7.1.6g shows the interfacial shear stress distribution at completion of interfacial debonding for different values of interfacial coefficient of friction, /j,-. In this figure, it can be noticed that the peak value of interfacial shear stress increases with increase in interfacial coefficient of friction,  Also, for  any given value of interfacial coefficient of friction, Li , the interfacial shear stress is maximum at t  the embedded fiber end and it decreases towards the exit fiber end.  Moreover, the rate of  decrease in shear stress increases with increase in the interfacial coefficient of friction, LL  V  Another very interesting feature that can be seen in this figure is that when /ij=0.5, the interfacial shear stress distribution along a major part of embedded fiber length is smaller than the same at jli = 0.05 and 0.25. Poisson's contraction of fiber is responsible for the observed interfacial shear stress distribution along the embedded fiber length. Comparison of Figures 7.1.5g and 7.1.6g also indicates that for low modulus fibers, the interfacial shear stress along the embedded length decays more rapidly in comparison to their high modulus counterparts.  7.2 Conclusions •  Prior to initial debonding pullout curve is linear and the stress transfer between fiber and matrix is purely elastic via adhesional shear stresses.  •  After initial debonding, further debonding (i.e., progressive debonding) requires the applied pullout load to overcome the interfacial frictional shear stresses at the debonded interface and adhesional shear stresses at the bonded interface.  As a result, the pullout load required to  further debond the interface depends upon the extent of prior debonding. During progressive interfacial debonding, the pullout curve becomes nonlinear. Pullout load during progressive debonding increases because the rate of increase in the frictional component of pullout load with increase in debond length is greater than the corresponding rate at which the adhesional component of the pullout load decreases. At the peak pullout load, the rate of increase in the frictional component of pullout load becomes equal to the rate of decrease in the bond component of the pullout load. Beyond the peak pullout load, the remaining bonded portion of interface debonds catastrophically. 163  Both the pullout load at initial debonding and the peak pullout load increase with increase in adhesional bond strength.  For low modulus fibers (3.5 GPa) these increases are much  smaller relative to high modulus fibers. This is because, for low modulus fibers much of the increase in the bond component of pullout load obtained with increase in adhesional bond strength is compensated by the corresponding decrease in the frictional component of pullout load.  From the viewpoint of optimization of interfacial properties, this observation is  important, since it demonstrates that efficiency of low modulus fibers cannot be improved significantly by solely increasing adhesional bond strength. Prior to complete debonding, the bond component of pullout load increases with increase in adhesional bond strength.  On the other hand, the frictional component of pullout load  decreases with increase in adhesional bond strength. Other parameters remaining the same, debond length at catastrophic debonding decreases with increase in adhesional bond strength. The interfacial debonding process is completely stable at very low values of adhesional bond strength for both the high modulus and the low modulus fibers. Prior to initiation of debonding, the rates of decrease in axial load and interfacial shear stress along the embedded fiber length for low modulus fibers are much more rapid in comparison to those for high modulus fibers. As a result, for low modulus fibers, only a very small embedded length of the fiber is mobilized in the stress transfer process prior to initiation of debonding. On the other hand, for high modulus fibers, a relatively longer fiber length is mobilized in the stress transfer process prior to initiation of debonding. For a given set of interfacial properties, catastrophic debond length decreases with increase in fiber elastic modulus, i.e., the higher the fiber modulus, the smaller is the catastrophic debond length. At completion of debonding and during fiber pullout interfacial shear stress is maximum at the embedded fiber end and it decreases towards the exit fiber end. Poisson's contraction of the fiber in the radial direction reduces the resultant contact stress at the interface and, hence, the interfacial shear stress.  Because low modulus fibers undergo higher Poisson's  contraction, interfacial shear stresses along the fiber length decrease at a faster rate in comparison to those for high modulus fibers. The peak pullout load increases with increase in interfacial contact stress, and this is because the frictional component of pullout load increases with increase in interfacial contact stress. Also, pullout loads on the post peak descending branch of the pullout curve increase with 164  increase in interfacial contact stress. Thus, the energy absorbed during the process of fiber pullout also increases with increase in interfacial contact stress. The above observations are valid for fibers of different modulus. The magnitude of interfacial shear stresses mobilized over the debonded interface increases with increase in interfacial contact stress. Also, for any given value of interfacial contact stress, the interfacial shear stress is maximum at the embedded fiber end and it decreases towards the exit fiber end. These observations are true for both high modulus as well as low modulus fibers. For a given set of interfacial properties, the rate of decrease of interfacial shear stress along the embedded length is greater for low modulus fiber. Given the dependence of pullout performance on interfacial contact stress, two approaches can be applied to improve fiber efficiency - i). using a matrix that shrinks more during curing, setting and hardening such that a higher value of interfacial contact stress is generated at the interface, and ii). intelligently designing fiber such that interfacial contact stress increases during the process of fiber pullout. The peak pullout load for high modulus fibers increases with increase in interfacial coefficient of friction.  From the viewpoint of optimization of interfacial properties, this  observation is important, since it demonstrates that efficiency of high modulus fibers can be significantly improved by increasing the coefficient of friction. On the other hand, for low modulus fibers, the peak pullout load initially increases and then it becomes constant with increase in interfacial coefficient of friction. Again, from the viewpoint of optimization of interfacial properties, this observation is important, since it demonstrates that efficiency of low modulus fibers cannot be significantly improved by solely increasing the coefficient of friction. The magnitude of interfacial shear stresses over the debonded interface increases with increase in interfacial coefficient of friction. For any given value of interfacial coefficient of friction, the interfacial shear stress is maximum at the embedded fiber end and it decreases towards the exit fiber end. The interfacial shear stress along the fiber length decreases more rapidly with increases in the interfacial coefficient of friction. These observations are true for both high modulus as well as low modulus fibers.  165  Chapter 8  Conclusions  Principal conclusions drawn from this research can be grouped into the following categories:  Macromechanical Behavior of FRC & Toughness Characterization •  Important material factors that influence post-cracking response (strengthening/toughness) of fiber reinforced concrete include fiber aspect ratio, fiber surface characteristics, fiber geometry, fiber volume content, shrinkage properties of matrix, and properties of mineral admixtures such as pozzolans.  •  The measured flexural load versus deflection response of fiber reinforced cement-based composites, particularly of the ones containing low fiber volume fractions of steel or synthetic fibers is greatly affected by the machine configuration. For machines with an openloop test configuration and low stiffness, the applied flexural load drops suddenly in an uncontrolled and unstable manner immediately following the peak load - the extent of this instability is a function of the machine stiffness and the rate of loading. Beyond the zone of instability, the flexural load versus deflection plot attains a stable softening level, during which loads are functions of the test machine characteristics.  For the aforementioned  reasons, the use of such load versus deflection curves to quantify toughness translates into meaningless toughness parameters.  Unfortunately, the existing standards to characterize  toughness (for example, A S T M C1018 [4] and JSCE SF4 [5]) allow the use of such flexural load versus deflection curves. •  Assessment of a new toughness characterization technique termed the Residual Strength Test Method (RSTM) has been made. In this technique, a stable narrow crack is first created in the specimen by applying flexural load in parallel with a steel plate under controlled conditions. The plate is then removed, and the specimen is tested in a routine manner in flexure to obtain the post-crack load versus displacement response.  Post-peak flexural  response obtained using this technique correlates very well with those obtained with relatively stiffer test configurations such as closed-loop test machines. A good agreement between the flexural response obtained from the aforementioned methods seems to validate 166  the Residual Strength Test Method. The Residual Strength Test Method is simple, and can be carried out easily in any commercial laboratory equipped with a test machine with low stiffness. •  The Residual Strength Test Method is seen to be effective in differentiating between different fiber types, fiber lengths, fiber configurations, fiber volume fractions, fiber geometries and fiber moduli. In particular, the technique is extremely useful for testing cement-based composites containing fibers at very low dosages (< 0.5% by volume).  Bond-Slip Performance of Fibers Embedded in Cementitious Matrices •  The maximum pullout load is influenced by the embedded fiber length. With increases in fiber length, the maximum pullout load increases, attaining an asymptote at long fiber lengths. The aforementioned observation is found to be valid for fibers with different elastic moduli.  •  For a given fiber length and diameter, the maximum pullout load increases with increase in fiber surface roughness. Additionally, for a fiber of a given length, the displacement corresponding to the maximum pullout load and the total energy absorption increase with increase in surface roughness.  The disparity between the maximum pullout loads for a  smooth steel fiber and a rough steel fiber decreases with increase in fiber length.  In  particular, the asymptotic value of maximum pullout load attained at long fiber lengths in the case of rough steel fibers is approximately similar to that for smooth steel fibers. •  The maximum pullout load is also influenced by the shrinkage properties of the matrix - for a fiber of given length and diameter, the maximum pullout load with non-shrink grout matrix is substantially lower than that obtained with normal portland cement matrix.  This  observation is valid for fibers of different length, and therefore, the asymptotic value of pullout stress attained in the case of non-shrink grout matrix is much lower in comparison to that with normal portland cement matrix. •  For a given type of matrix and fiber length, steel fibers attained a greater peak pullout load in comparison to polypropylene fibers. In addition, for polypropylene fibers, the asymptotic value of pullout stress is attained at much smaller fiber length in comparison to that with steel fibers.  167  Progressive Debonding Model for Fiber Pullout •  To understand the mechanics of interaction between fibers and matrices, a shear-lag model is proposed to study the problem of fiber pullout. The proposed model is unique because of its ability to take into consideration the evolution of the interfacial properties during the pullout process. The analysis of the problem can be divided into three stages: During stage 1, the fiber remains completely bonded along its embedded length, and the displacements at the fiber-matrix interface are compatible.  Analysis is based on the  shear-lag theory with the maximum shear stress as the criterion for fiber-matrix interfacial debonding. During this stage, resistance to fiber pullout is derived through the interfacial adhesional shear stress. Closed-form solutions are derived for the fiber axial stress distribution, the interfacial shear stress distribution, the fiber displacement, and the initial debonding stress. The closed-form solution for the initial debonding stress depicts that the initial debonding stress primarily depends upon the elastic properties of the fiber and the fiber length. During stage 2, the fiber is partially bonded along its embedded length. The adhesional shear stresses at the bonded interface and the frictional shear stresses at the debonded interface resist the fiber pullout. Interfacial friction is modeled using the Coulomb's law and the Poisson's effect along the debonded interface is modeled by considering the fiber and the matrix to be held together in a shrink-fit configuration. It has been shown that for any interfacial debond length, the fiber pullout stress is a summation of the two components - the one arising due to the adhesional shear bond and the other arising due to the frictional shear bond. Closed-form solutions are derived for the fiber axial stress distribution, the interfacial shear stress distribution, the fiber pullout stress versus debond length relationship, and the fiber displacement versus debond length relationship. .  The interfacial debonding process becomes catastrophic at the instant when the fiber pullout stress begins to drop with increase in debond length. This condition is satisfied when the difference between the change in the frictional component of pullout stress and the change in adhesional component of pullout stress occurring due to increase in debond length minimizes. Closed-form solutions are derived for the catastrophic debond length, the peak pullout stress and the displacement corresponding to peak pullout stress.  .  During stage 3, the fiber is completely debonded along its embedded length and fiber pullout is initiated. Frictional shear stresses existing over the debonded interface resist 168  pullout of the fiber from the matrix. Closed-form solutions are derived for the fiber axial stress distribution, the interfacial frictional shear stress distribution, the pullout stress and the fiber displacement. Using the proposed progressive debonding model it is possible to predict the complete pullout response. Theoretical predictions from the proposed model compare well with the experimental pullout data. A procedure to calibrate interfacial properties from experimental pullout data is established. It is shown that interfacial contact stress can be calculated using the asymptotic value of pullout stress on the peak pullout stress versus embedded length plot. The adhesional bond strength can be calculated from the initial slope of the peak pullout stress versus embedded length plot.  Based on energy considerations, a method is proposed to calculate the  coefficient of friction as a function of pullout distance. The evolution law for coefficient of friction depicts that interfacial coefficient of friction decays exponentially with increase in fiber pullout distance. This observation is found to be valid for fibers of different elastic modulus. Smoothening of the interface as a result of the matrix wear during fiber pullout appears to be the reason for this behavior. Prior to the initial debonding, the stress transfer between fiber and matrix is purely elastic (i.e., via adhesional shear stresses) and the corresponding pullout curve is linear. After the initial debonding, further debonding (i.e., progressive debonding) requires the applied pullout load to overcome the interfacial frictional shear stresses at the debonded interface and adhesional shear stresses at the bonded interface. As a result, the pullout load required to further debond the interface depends upon the extent of prior debonding. During progressive interfacial debonding, the pullout curve becomes nonlinear. Pullout load during progressive debonding increases because the rate of increase in the frictional component of the pullout load with increase in debond length is greater than the corresponding rate at which the adhesional component of the pullout load decreases. At the peak pullout load, the rate of increase in the frictional component of pullout load becomes equal to the rate of decrease in the bond component of the pullout load. Consequently, beyond the peak pullout load, the remaining bonded portion of interface debonds catastrophically. Both the pullout load at initial debonding and the peak pullout load increase with increase in adhesional bond strength. For the low modulus fibers (=3.5 GPa) these increases are not as significant relative to those obtained in the case of high modulus fibers. This is because for 169  low modulus fibers much of the increase in the bond component of the pullout load obtained with increase in the adhesional bond strength is compensated by the corresponding decrease in the frictional component of the pullout load. From the viewpoint of optimization of the interfacial properties, this observation is important, since it demonstrates that efficiency of low modulus fibers cannot be improved significantly solely by increasing the adhesional bond strength. For a given debond length, the bond component of pullout load increases and the frictional component of pullout load decreases with increase in adhesional bond strength. The other parameters remaining constant, the debond length at the occurrence of catastrophic debonding decreases with increase in the adhesional bond strength. Prior to initiation of debonding, the rates of decrease in axial load and interfacial shear stress along the embedded fiber length are greater for low modulus fibers in comparison to high modulus fibers. Consequently, for low modulus fibers, only a very small embedded fiber length is mobilized in the stress transfer process. On the other hand, for high modulus fibers, a relatively longer fiber length is mobilized in the stress transfer process. For a completely debonded interface, interfacial frictional shear stress is maximum at the embedded fiber end and it decreases towards the exit fiber end. Poisson's contraction of fiber in the radial direction reduces the resultant contact stress at the interface and the consequent interfacial frictional shear stress. Because the low modulus fibers undergo a higher Poisson's contraction, the interfacial frictional shear stresses along the fiber length decrease at a greater rate in comparison to those in the case of high modulus fibers. The peak pullout load increases with increase in interfacial contact stress. This is because the frictional component of pullout load increases with increase in interfacial contact stress. Also, pullout loads on the post-peak descending branch of the pullout curve increase with increase in interfacial contact stress. Energy absorbed during the process of fiber pullout also increases with increase in interfacial contact stress. The above observations are valid for fibers of different moduli. The magnitude of interfacial shear stresses mobilized over the debonded interface increases with an increase in interfacial contact stress. Also, for any given value of interfacial contact stress, the interfacial shear stress is maximum at the embedded fiber end and it decreases towards the exit fiber end. These observations are true for both high modulus as well as low 170  modulus fibers. For a given set of interfacial properties, the rate of decrease in interfacial shear stress along the embedded length is greater for low modulus fiber. •  Given the dependence of pullout performance on interfacial contact stress, two approaches can be used to improve fiber efficiency - i). using a matrix that shrinks more during curing, setting and hardening such that a higher value of interfacial contact stress is generated at the interface, and ii). intelligently designing fiber such that interfacial contact stress increases during the process of fiber pullout.  •  The peak pullout load for high modulus fibers increases with increase in interfacial coefficient of friction. From the viewpoint of optimization of the interfacial properties this observation is important, since it demonstrates that efficiency of high modulus fibers can be significantly improved by increasing the coefficient of friction. However, the above is not the case with low modulus fibers.  Recommendations for Future Studies •  In the current research program, fiber pullout response under static loading was investigated. However, the micromechanical properties of the interface and the mechanical properties of the constituents, (i.e., fiber and matrix) may be expected to depend upon the rate of loading. In such a scenario, fiber pullout response will be a function of the rate of loading. It is recommended that studies be carried out to investigate the influence of loading rate on fiber pullout response. In this context, it will be worthwhile to examine: •  The influence of matrix modification by mineral admixtures, such as silca-fume, fly-ash, high reactivity metakaoline, etc.  •  The influence of matrix modification by chemical admixtures, such as air entraining agents, high-range water-reducing admixtures, etc.  • •  The influence of polymer viscoelasticity in the case of polymeric fibers.  Fiber-matrix interfacial properties are expected to change as a result of the treatment of fiber surface with chemical coatings such as Organo-functional silanes. It is recommended that studies be carried out to investigate the extent of dependence of fiber pullout response on different chemical coatings.  171  Fibers with extremely good interfacial bond with matrix may fracture during the pullout process. This appears to be the case with polyvinyl alcohol fibers (fiber PVA-1). Thus, for polyvinyl alcohol fibers, improved pullout efficiency may be expected with reduction in interfacial bond. This reduction can possibly be achieved either through the application of appropriate chemical coating on fiber surface or through a change in the chemical properties of the base polymer. Studies are recommended in this context. It is recommended that studies be carried out to understand the influence of fiber inclination and ductility of fiber material on the pullout response. Also there is need to model the pullout response for these scenarios. In the current research, sensitivity of fiber pullout response on the interfacial contact stress was observed.  It was seen that the interfacial contact stress was dependent: upon the  shrinkage properties of the matrix. However, the interfacial contact stress may also depend upon any external confining stress applied to the fiber reinforced cementitious composite. It is recommended that further studies be carried out to investigate and characterize the influence of external confining stresses on fiber pullout response. In addition to the initial water/cement ratio, the shrinkage behavior of cementitious matrix is highly dependent upon the environmental conditions existing during setting, hardening and curing of matrix. Since the fiber-matrix interfacial properties and the pullout response are strongly dependent upon the shrinkage behavior of cementitious matrix (as seen in this study), it becomes important to diligently control the environmental parameters while producing fiber reinforced cementitious environmental  conditions  composites.  remains to be  recommended to explore this aspect.  172  investigated.  Now, what constitute optimal Detailed investigations  are  Bibliography  [I] .  Bentur, A . and Mindess, S., "Fiber Reinforced Cementitious Composites," Elsevier Applied Science, London, 1990.  [2].  Balaguru, P. N . and Shah, S. P., "Fiber-Reinforced Cement Composites," McGraw-Hill, Inc., New York, 1992.  [3].  Hannant, D. J., "Fiber Cements and Fiber Concretes," John Wiley & Sons, Chichester, 1978.  [4].  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Mallikarjuna, M . , Fafard, M . , and Banthia, N . , " A New Three-Dimensional Interface (Contact) Element for Fiber Pull-out Behavior in Composites," Computers and Structures, V . 44, No. 4, 1992, pp. 753-764.  187  Appendix A  188  /* /*  SOURCE CODE FOR THE FIBER PULLOUT MODEL BY TAKAKU AND ARRIDGE [REFERENCE 54]  ttinclude #include #include #include #include  <stdio.h> <float.h> <math.h> <stdlib.h> <memory.h>  */ */  \  #define TT 200 #define N 3 main { FILE FILE FILE FILE FILE FILE FILE FILE FILE FILE FILE  () *input_file; *STRS_DISTR; *STRS_DISTR2; *PO_DISP; *P0_DISP2; *DEBLEN_DEBSTRSS; *DEBLEN_DEBSTRSS2; *PO_STRESSvsDL; *P0_STRESSvsDL2; *TEST; *TESTIN;  int L, d l , zz, parts,  pdist;  double a, b, Poss_m, Poss_f, Em, Ef, ABS, mu, contact_stress, Initial_debonding_stress, row, alpha, epsl, eps2, eps3, eps4, epsilon, thetal, theta2, theta, betal, beta, c n s t n t l , cnstnt2, mega, deb_len, deb_pos, Fiber_area, z, zzz, debond_length, omegal, LL, PDS, FD1, FD2, FAS1, FAS2, FAS3, pd, pullout_distance, FDP1, FDP2, FDP3, FASP1, FASP2, zero_stress, l_cata, l _ c a t a l , l_cata3, xxxx, yyyy, zzzz, wwww, alphaa_ta, alpha_ta, sigma_o_peak, U_d; float Pullout_Bond_component[TT] [N], Pullout_Friction_component[TT] [N], Fiber_disp_2sides[TT] [N], Pullout_Load[TT] [N], Fiber_axial_stress[TT] [N], Interfacial_shear_stress[TT] [N], Fiber_axial_load[TT] [N], Fiber_axl_load_bond_compo[TT] [N], • Fiber_axl_load_frictional_compo[TT] [N], ISS[TT], Fiber_displacement[TT], Fiber_displacement2[TT], Progressive_dbnd_stress[TT], Progressive_POstress[TT], P0_LD_Bond[TT], P0_LD_Fric[TT], Progressive_P01oad[TT]; input_file = fopen("inp","r"); STRS_DISTR = fopen("stress","w"); / * stress d i s t r i b u t i o n i n fiber * / STRS_DISTR2 = fopen("stress2", "w"); PO_DISP = fopen("disp","w"); / * pullout stress and fiber displacement */ P0_DISP2 = fopen("disp2", "w"); TESTIN = fopen("in", "w");  189  * read the input f i l e * / / * ABS - Adhesional Bond Strength */ fscanf(input_file, "%lf %lf %lf %lf %lf %lf %d %lf %lf %lf ", &a, &b, &Poss_f, &Poss_m, &Em, &Ef, &L, &ABS, &mu, &contact_stress); /* /*  _ Fiber completely bonded along i t s length  */ */ _  /*  fprintf(PO_DISP, "Deb Len Dbnd Strs Bnd PO Ld F r i c PO Ld\n"); fprintf(P0_DISP2, "Deb Len Dbnd Strs Bnd PO Ld F r i c PO Ld\n"); fprintf(STRS_DISTR "Dbnd Strs Deb Lt Strs Axl Lod Bnd Ld F r i c Ld \n"); (  PO Strs PO Strs Deb Pos  */  Fib Disp  Progr PO Ld  Fib Disp2 Axl Pos  Progr PO Ld  Axl Strs  Shr  omega=Poss_f*Em/(Ef*(l+Poss_m)); Fiber_area = 3.14*a*a; alphaa_ta=(2*3.14159*Em)/(a*Ef*(l+Poss_m)*log(b/a)); alpha_ta=sqrt(alphaa_ta); deb_len=0.0; deb_pos=0.0; dl=deb_len; zero_stress=0.0; Initial_debonding_stress=-2*ABS*tanh(alpha_ta*L)/(a*alpha_ta); Progressive_dbnd_stress[dl]=Initial_debonding_stress; Progressive_dbnd_stress[dl]=Initial_debonding_stress; Progressive_POstress[dl] = zero_stress; Progressive_P01oad[dl]=0.0; P0_LD_Bond[dl]=0.0; P0_LD_Fric[dl]=0.0; Fiber_displacement[dl]=zero_stress/(Ef*a*cosh(alpha_ta*L)); Fiber_displacement2[dl] = 2*Fiber_displacement[dl]; fprintf(P0_DISP, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement[dl], Progressive_P01oad[dl], P0_LD_Bond[dl], P0_LD_Fric[dl]); fprintf(P0_DISP2, "%7.21f %7.21f %7.21f %7.51f %7.21f . %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement2[dl], Progressive_POload[dl], P0_LD_Bond[dl], P0_LD_Fric[dl]); deb_len=0.0; deb_pos=0.0; dl=deb_len; Initial_debonding_stress=-2*ABS*tanh(alpha_ta*L)/(a*alpha_ta); Progressive_dbnd_stress[dl]=Initial_debonding_stress; Progressive_POstress[dl] = Initial_debonding_stress; Progressive_P01oad[dl]=Progressive_POstress[dl]*Fiber_area; P0_LD_Bond[dl]=Progressive_dbnd_stress[dl]*Fiber_area; P0_LD_Fric[dl]=Progressive_P01oad[dl]-P0_LD_Bond[dl];  190  Fiber_displacement[dl]=Initial_debonding_stress/(Ef*(row+alpharow*epsilon*alpha+alpha*epsilon))*((alpha-row*epsilon*alpha+alpha*epsilon row)*(cosh(beta*L)-1)/(beta*sinh(beta*L))+row*L); Fiber_displacement2[dl] = 2*Fiber_displacement[dl]; fprintf(PO_DISP, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement[dl], Progressive_P01oad[dl], PO_LD_Bond[ d l ] , PO_LD_Fric[dl]); fprintf(PO_DISP2, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement2[dl], Progressive_P01oad[dl] PO_LD_Bond[dl], PO_LD_Fric[dl]);  /*  /* /*  */  Fiber completely debonded along i t s length _  */ •__ * /  parts=4; for(pdist=0; pdist<=parts*L; ++pdist) { pullout_distance=pdist; pd=pullout_distance/parts; / * L_new=L-pout_dist; */ Progressive_dbnd_stress[pdist]= 0.0; / * calculate progressive pullout stress and fiber displacement * / Progressive_POstress[pdist]=-contact_stress/omega*(1-exp(-2 *mu*omega*(Lpd)/a)); Progressive_P01oad[pdist]=Progressive_POstress[pdist]*Fiber_area; PO_LD_Fric[pdist]=Progressive_P01oad[pdist]; PO_LD_Bond[pdist]= 0.0; FDPl=contact_stress/(Ef*omega); FDP2=a/(2*mu*omega)*(exp(-2*mu*omega*(L-pd)/a)-1); FDP3=pd*(l-exp(-2*mu*omega*(L-pd))); Fiber_displacement[pdist]=pd-FDPl*((L-pd)+FDP2+FDP3); Fiber_displacement2[pdist] = pd-FDPl*(2*(L-pd)+2*FDP2+FDP3); fprintf(PO_DISP, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", pd, Progressive_dbnd_stress[pdist], Progressive_POstress[pdist], Fiber_displacement[pdist], Progressive_P01oad[pdist], PO_LD_Bond[pdist], PO_LD_Fric[pdist]); fprintf(PO_DISP2, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", pd, Progressive_dbnd_stress[pdist], Progressive_POstress[pdist], Fiber_displacement2[pdist], Progressive_P01oad[pdist], PO_LD_Bond[pdist], PO_LD_Fric[pdist]); / * calculate fiber a x i a l stress d i s t r i b u t i o n & i n t e r f a c i a l shear stress d i s t r i b u t i o n over the debonded zone of the p a r t i a l l y debonded fiber */ for(zz=0; zz<=parts*(L-pd); ++zz) { zzz=zz; z=zzz/parts;  191  FASP1=contact_stress/omega; FASP2=exp(-2*omega*mu*z/a); Fiber_axial_stress[zz] [3]=-FASPl*(1-FASP2); Interfacial_shear_stress[zz] [3]=mu*contact_stress*FASP2; Fiber_axial_load[zz] [3] = Fiber_axial_stress[zz] [3] * Fiber_area Fiber_axl_load_bond_compo[zz] [3] = Progressive_dbnd_stress[pdist] Fiber_area; Fiber_axl_load_frictional_compo[zz] [3] = Fiber_axial_load[zz] [3] Fiber_axl_load_bond_compo[zz] [2]; fprintf(STRS_DISTR, "%7.21f %7.21f %7.21f %7.21f %7.21f %7.21 %7.21f %7.21f %7.21f \ n " , Progressive_dbnd_stress[pdist], pd, z, Fiber_axial_stress[zz] [3], Interfac ial_shear_s tress[zz] [3], Fiber_axial_load[zz] [3], Fiber_axl_load_bond_compo[zz] [3], Fiber_axl_load_frictional_compo[zz] [3]);  192  Appendix B  193  /*  SOURCE CODE FOR THE PULLOUT MODEL BY HSUEH [REFERENCES 67 & 68]  #include #include #include #include #include ttinclude  */  <stdio.h> <float.h> <math.h> <stdlib.h> <memory.h> <malloc.h>  #define TT 200 main { FILE FILE FILE FILE FILE FILE FILE FILE  () *input_file; *STRS_DISTR; *PO_DISP; *DEBLEN_DEBSTRSS; *PO_STRESSvsDL; *TEST; *TESTIN; *AREA;  int Debnd_stress, rad, bb;  L, Counter, zz, d l , ITER,  i n c r , steps, zx, zy, zzyy,  zxpo,  double a, b, Poss_m, Poss_f, Em, Ef, ISS, Alpha, z, LZ, DS_Num, DS_Numerator, DS_Denominator, Initial_debonding_stress, p u l l o u t _ s t r s , Pullout_stress[200], Fiber_axial_stress[200], FAS1, FAS2, FAS3, Interfacial_shear_stress[200], IS3, ISS1, ISS2, ISS3, FD1, FD2, FD3, Difference, pullout_stress, Fiber_disp[2 00], alph, ppp, x, y, log_ba, increment, DST_Num, DST_Numerator, DST_Denominator, debond_stress_current [200], deb_len, b l , D, denom, A l , A2, A3[200], mu, stress_pout[200], stress_shrnk, A1A2, ml, m2, B[200], SP1, SP2a, SP2b, SP2c, SP2, SP3a, SP3b, SP3c, SP3 , xx, xy, xz, xxz, xxzz, mid, m2d, mlm2, mlm2d, u b l , ub2,ub2a, ub2b, ub2c, ub[200], LDEB1, LDEB2, US1, US2, US3, US4, US[200], USSS, A3a, A3b, mldexp, m2dexp, po_len, emb_len, debond_stress_current3, stress_pout3[200], USP1, USP2, USP3, USP4, USP[200], UP3[200], mlz, m2z, ASI, AS2, AS3, mmzl, mmz2, ISSS1, ISSS2, ISSS3, deb_pos, steer, zzz, debond_stress_current01, pullout_strs01, ISSbl, ISSb2, ISSb3, ISb3, EMBD_LEN, mlm2z, POS, embd_len_po, separation, deb_component, fric_component, Total_bond_load, shrnk_strss_net[200], mm2dexp, row, modulus_ratio, epsl, eps2, eps3, eps4, epsilon, t h e t a l , theta2, theta, betal, beta, DSC1, DSC2, DSC3, DSC[200], matrix_stress_b, matrix_stress, r a d l , k; float Pullout_Bond_component[TT], Pullout_Friction_component[TT], Fiber_disp_2sides[TT], Fiber_area, Pullout_Load_Bond_component[200], Pullout_Load_Friction_component[200], Pullout_Load[TT], Fiber_axial_load[TT], Fiber_axl_load_bond_compo[TT], Fiber_axl_load_frictional_compo[TT], areal[TT]; i n p u t _ f i l e = fopen("inptl","r"); STRS_DISTR = fopen("stress","w");  / * stress d i s t r i b u t i o n ' i n fiber * /  194  PO_DISP = fopen("disp","w"); / * pullout stress and fiber displacement * / DEBLEN_DEBSTRSS = fopen("dbsdbl","w"); / * debond length versus debond stress */ PO_STRESSvsDL = fopen("posdl","w"); / * pullout stess vs debond length */ TEST = fopen("testout", "w"); TESTIN = fopen("in", "w"); AREA- = fopen("area", "w"); / * read the input f i l e * / fscanf(input_file, "%lf %lf %lf %lf %lf %lf %d %lf %lf %lf ", &a, &b, &Poss_f, &Poss_m, &Em, &Ef, &L, &ISS, &mu, &stress_shrnk) ; fprintf(TESTIN, "%lf %lf %lf %lf %lf %lf %d %lf \ n " , a, b, Poss_f, Poss_m, Em, Ef, L , ISS, mu); /* /*  fprintf(DEBLEN_DEBSTRSS, "Debond Length fprintf(AREA, "pdist pd area \n"); * /  Debond Stress \n");  */  row=a*a/(b*b-a*a); modulus_ratio=Em/Ef; epsl=2/(a*a* (b*log(b/a) - (b-a) ) )'; eps2=(0.5*b*b*b*log(b/a)-b*(b*b-a*a)/4); eps3=0.5*a*(b*b-a*a); eps4=(b*b-a*a)/3; epsilon=epsl*(eps2+eps3-eps4); thetal=a*a*(l+Poss_m); theta2=(b/(b-a))*log(b/a)-1; theta=l/(thetal*theta2); betal= (theta*(row+modulus_ratiorow*epsilon*modulus_ratio)/(row*epsilon))+modulus_ratio*theta; beta=pow(betal,0.5); Fiber_area = 3.14*a*a; / * calculate the i n i t i a l debonding stress (which i s same as- the fiber p u l l out test) * / log_ba = log(b/a),• alph=(a*a*Ef+(b*b-a*a)*Em)/(Ef*(l+Poss_m)*(b*b*log_ba-(b*b-a*a))); Alpha = pow(alph,0.5)/a; /* fprintf(TEST, "%7.21f %7.21f \n",log_ba, alph); * / /* fprintf(TEST, "%7.21f %7.21f %7.21f \n",log_ba, alph, Alpha); */ DS_Num = ((l+Poss_m)*(1+(b*b/(a*a)-1)*(Em/Ef))*(b*b*log_ba-(b*b-a*a)12) ); DS_Numerator = (2/a) * pow(DS_Num,0.5); DS_Denominator = (((b*b/(a*a)-1)*(Em/Ef))/tanh(Alpha*L))+ 2/(exp(Alpha*L)exp(-Alpha*L)); Initial_debonding_stress = -ISS * DS_Numerator/DS_Denominator;  195  /* */  /* */ /*  FIBER COMPLETELY BONDED ALONG ITS ENTIRE LENGTH  */  / * Calculate pullout stress vs displacement relationship when fiber i s e l a s t i c a l l y bonded to the matrix throughout the embedded fiber length * / / * NOTE: I n i t i a l debonding stress w i l l be same as the fiber pullout test */  / * Also calculate the a x i a l stress d i s t r i b u t i o n i n fiber and the i n t e r f a c i a l shear stress d i s t r i b u t i o n for pullout load less than or equal to debonding load * / fprintf(STRS_DISTR, "CTR PO Strs Deb Lt Deb Pos Axl Pos Axl Strs Shr Strs Axl Ld Bnd Ld F r i c Ld\n"); Counter=0; pullout_strs=0; incr=100; steps=(Initial_debonding_stress/100); for(ITER=l; ITER <= steps+2; ++ITER) { dl=0; Counter=Counter+l; Pullout_stress[ITER] = pullout_strs; Pullout_Friction_component[dl]= 0.0; Pullout_Bond_component[dl]=Pullout_stress[ITER]; Pullout_Load[ITER] = Pullout_stress[ITER] * Fiber_area; Pullout_Load_Friction_component[dl] = 0.0; Pullout_Load_Bond_component[dl] = Pullout_Bond_component[dl]  * Fiber_area;  for(zz=0; zz<=2*L; ++zz) { deb_len=0.0; deb_pos=0.0; z = 0 .5*zz ; LZ = L - z ; debond_stress_current[dl] = Initial_debonding_stress; / * calculate a x i a l stress d i s t r i b u t i o n i n fiber */ FAS1 = (a*a*Ef*pullout_strs)/(a*a*Ef+(b*b-a*a)*Em); FAS2 = ((b*b/(a*a))-1) * (Em/Ef) * (exp(Alpha*z)-exp(Alpha*z))/(exp(Alpha*L)-exp(-Alpha*L)); FAS3 = (exp(-Alpha*(L-z)) - exp(Alpha*(L-z)))/(exp(Alpha*L) - exp(Alpha*L)); Fiber_axial_stress[zz] = FAS1*(1+FAS2+FAS3); Fiber_axial_load[zz] = Fiber_axial_stress[zz] * Fiber^area; Fiber_axl_load_bond_compo[zz]= Fiber_axial_load[zz]; Fiber_axl_load_frictional_compo[zz]=0.0; / * calculate i n t e r f a c i a l shear stress d i s t r i b u t i o n */ 1551 = ((b*b/(a*a))-1) * (Em/Ef) * (exp(Alpha*z) + exp(Alpha*z))/(exp(Alpha*L)-exp(-Alpha*L)); 1552 = (exp(-Alpha*(L-z)) + exp (Alpha*(L-z)))/(exp(Alpha*L) - exp(Alpha*L));  196  IS3 = ((l+Poss_m)*(1+(b*b/(a*a)-1)*(Em/Ef))*(b*b*log_ba-(b*b-a*a)12)) ; ISS3 = (2/a)* pow(IS3, 0.5); Interfacial_shear_s'tress [zz] = -pullout_strs*((ISS1+ISS2)/ISS3); /* fprintf(TEST, "ISS1=%7.llf ISS2=%7.21f ISS3=%7.21f ISS[z=%7.21f]=%7.21f\n", ISS1, ISS2, ISS3, z, Interfacial_shear_stress[zz][dl]); */ fprintf(STRS_DISTR, "%d %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Counter, p u l l o u t _ s t r s , deb_len deb_pos, z, Fiber_axial_stress[zz], Interfacial_shear_stress[zz], Fiber_axial_load[zz] Fiber_axl_load_bond_compo[zz], Fiber_axl_load_frictional_compo[zz]); matrix_stress_b = p u l l o u t _ s t r s / e p s i l o n - (row+modulus_ratiorow*epsilon*modulus_ratio)*Fiber_axial_stress[zz]/(row*epsilon); bb=b; for(rad=l; rad<=bb; ++rad) { k=rad; radl=log(k/a); matrix_stress=modulus_ratio*Fiber_axial_stress[zz] + matrix_stress_bmodulus_ratio*Fiber_axial_stress[zz]*(b*radl+a-rad)/(b*log(b/a)-(b-a)); fprintf(DEBLEN_DEBSTRSS, "PO STRS=%7.21f zz=%d rad=%d matrix_stress_b=%7.21f matrix_stress=%7.21f \ n " , p u l l o u t _ s t r s , zz, rad, matrix_stress_b, matrix_stress); } }  / * calculate fiber displacement * / FD1 = (a*a*pullout_strs)/(a*a*Ef+(b*b-a*a)*Em); FD2 = (((b*b/(a*a))-1)*(Em/Ef)-1)/Alpha; FD3 = (exp(Alpha*L)+exp(-Alpha*L)-2)/(exp(Alpha*L)-exp(-Alpha*L)); Fiber_disp[ITER] = FD1 * (L + FD2*FD3); Fiber_disp_2sides[ITER] = 2*Fiber_disp[ITER]; /* fprintf(P0_DISP, "%7.21f %7.21f %7.21f %7.21f \ n " , pullout_strs, FD1, FD2, FD3); */ /* fprintf(P0_DISP, "FDl=%7.21f FD2=%7.21f FD3=%7.21f\n", FD1, FD2, FD3); * / fprintf(P0_DISP, "%d %7.21f %7.21f %9.61f %9.61f %7.21 %7.21f %7.21f . %7.21f %7.21f %7.21f\n", ITER, deb_len, debond_stress_current[dl], Fiber_disp[ITER], Fiber_disp_2sides[ITER], Pullout_stress[ITER], Pullout_Bond_component[dl], Pullout_Friction_component[dl], Pullout_Load[ITER], Pullout_Load_Bond_component[dl], Pullout_Load_Friction_component[dl]); if(ITER<steps+l) pullout_strs = pullout_strs + i n c r ; else pullout_strs = Initial_debonding_stress; /* fprintf(P0_DISP,"ITR=%d \n", ITER); }  197  */  /* */  /* */  FIBER PARTIALLY DEBONDED ALONG ITS LENGTH  /* */  /*  Calcualte pullout stress vs displacement r e l a t i o n s h i p when fiber-matrix are p a r t i a l l y bonded  / * Calcualte debonding stress (debond_stress_current length * / for(dl=0; dl<=2*L; ++dl)  [dl]  */  for each debond  {  if(dl  < 2*L)  { deb_len = 0.5*dl; b l = L - deb_len; / * Counter = Counter+1; * / DST_Num = ((l+Poss_m)*(1+(b*b/(a*a)-1)*(Em/Ef))*(b*b*log_ba-(b*b• a*a)/2)); DST_Numerator = (2/a) * pow(DST_Num,0.5); DST_Denominator = (((b*b/(a*a)-1)*(Em/Ef))/tanh(Alpha*(L-deb_len)))+ 2/ (exp(Alpha* (L-deb_len) ) -exp(-Alpha* (L-deb_len) ) ) ,debond_stress_current  [dl]  = -ISS * DST_Numerator/DST_Denominator;  DSC[dl]=0.0; DSC1=2*(row+modulus_ratiorow*epsilon*modulus_ratio+modulus_ratio*epsilon)/(a*beta); DSC2=(modulus_ratiorow*epsilon*modulus_ratio+modulus_ratio*epsilon)/tanh(beta*(Ldeb_len)); DSC3= 2*row/(exp(beta*(L-deb_len))-exp(-beta*(L-deb_len))),DSC[dl]=-ISS*DSCl/(DSC2+DSC3); fprintf(DEBLEN_DEBSTRSS, "%7.21f %7.21f %7.21f Alpha=%7.21f row=%7.21f beta=%7.21f epsilon=%7.21f \ n " , deb_len, debond_stress_current [dl], DSC[dl], Alpha, row, beta, epsilon); */  /*  }  else { deb_len = 0.5*dl; b l = L - deb_len; debond_stress_current [dl] = 0.0; fprintf(DEBLEN_DEBSTRSS, "%7.21f debond_stress_current [dl]);  %7.21f  \ n " , deb_len,  } }  D = ( (b*b+a*a) / (b*b-a'*a) ) + Poss_m + (Em*(l-Poss_f)/Ef); denom = (l+Poss_m)* (b*b*log_ba - (b*b-a*a)12); A l = a*(l-b*b/(a*a))*D/(2*mu*Poss_m*denom); A2 = ((l-b*b/(a*a))*(Em*Poss_f/(Ef*Poss_m))-1)/denom;A1A2 = (A1*A1-4*A2); ml = (-A1 + pow(AlA2,0.5))12; m2 = (-A1 - pow(AlA2,0.5))/2;  198  fprintf(TEST, "D=%7.21f Al=%7.21f A2=%7.21f ml=%7.21f m2=%7.21f \n",D, A l , A2, ml, m2); fprintf(TEST, "deb_len ml m2 (ml+m2)*deb_len debond stress current SP1 SP2 SP3 Pullout Stress\n"); */  / * Caluclate pullout stress ( p a r t i a l debond pullout stress) debond lengths */  for various  fprintf(PO_STRESSvsDL, "Debond length Debond stress Pullout stress \n"); /* fprintf(STP, "SP1, SP2a, SP2b SP2c, SP2, SP3a, SP3c, SP3 Pullout stress \n"); */ ITER=ITER-1; for(dl=0; dl<=2*L; ++dl) { Counter=Counter+l; deb_len = 0.5*dl; mlm2 = ml+m2; mid = ml*deb_len; m2d = m2*deb_len; mlm2d = (ml+m2)* deb_len; SP1 = debond_stress_current[dl]*(ml-m2)*exp(mlm2d); / * fprintf(TEST, "%7.21f %7.21f %7.21f %7.21f %7.21f %50.21f", deb_len, ml, m2, mlm2d, debond_stress_current [dl ] , SP1) ,*/ SP2a=(a*a-b*b)*D*Ef*(m2*exp(m2d)-ml*exp(mld)+(ml-m2)*exp(mlm2d)); SP2b=(a*a-b*b)*Em*Poss_f-a*a*Ef*Poss_m; SP2c=(2*mu/a)*(exp(mid)-exp(m2d)); SP2 = ((SP2a/SP2b)-SP2c)* stress_shrnk; SP3a= (a*a-b*b) *Em*Poss_f * (ml*exp (mid) -m2*exp (m2d) ) -a*a*Ef*Poss_m* (mlm2) *exp(mlm2d) ; SP3b=(a*a-b*b)*Em*Poss_f-a*a*Ef*Poss_m; SP3c=2*mu*Em*Poss_f*(exp(mid)-exp(m2d))/(a*D*Ef); SP3=((SP3a/SP3b)+SP3c); stress_pout[dl]= (SP1+SP2)/(SP3); / * stress_pout[dl] i s p a r t i a l debond pullout stress */ / * For the calculated fiber pullout stress, check i f separation of matrix-fiber interface has taken place i n the debonded region * / if(stress_pout[dl] < debond_stress_current[dl]) { / * means, separation of fiber and matrix has taken place i n the debonded region & hence f r i c t i o n a l stress i s zero i n the debonded region */ stress_pout[dl] = debond_stress_current[dl]; separation=l; } else { separation=0; }  Pullout_stress[ITER]=stress_pout[dl]; Pullout_Bond_component[dl] = debond_stress_current[dl];  199  Pullout_Friction_component[dl] = Pullout_stress[ITER] Pullout_Bond_component[dl] ; Pullout_Load[ITER] = Pullout_stress[ITER] * Fiber_area; Pullout_Load_Friction_component[dl] = Pullout_Friction_component[dl] * Fiber_area; Pullout_Load_Bond_component[dl] = Pullout_Bond_component[dl] * Fiber_area; fprintf(PO_STRESSvsDL, "%7.21f %7.21f %7.21f \n", deb_len, debond_stress_current[dl], stress_pout[dl]); /* fprintf(STP, "%7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \n" , SP1, SP2a, SP2b, SP2c, SP2, SP3a, SP3c, SP3, Pullout_stress[ITER]) ; */ A3a=stress_pout[dl] ; A3b= (stress_shrnk/Poss_m)*(l-b*b/(a*a))*D; A3[dl] = -(A3a+A3b)/denom; if(dl>0) B[dl] = (stress_pout[dl]-debond_stress_current[dl]*exp(m2d)(A3[dl]/A2)*(1-exp(m2d)))/(exp(mid)-exp(m2d)); else B[dl]= 0; fprintf(PO_STRESSvsDL, "%7.21f %7.21f \n", deb_len, debond_stress_current[dl], stress_pout[dl]); / * fprintf(TEST, stress_pout[dl]); / * fprintf(TEST,  %7.21f  "%50.21f %50.21f %10.21f \ n " , SP2, SP3, */ "A3[dl]=%20.21f B[dl]=%20.21f \ n " , A3[dl], B [ d l ] ) ;  */  /*  Calculate fiber displacement for each p a r t i a l debond pullout stress if(dl<2*L) { / * calculate fiber displacement for the bonded part * / LDEBl=Alpha*(L-deb_len); LDEB2 = - Alpha *(L-deb_len); mldexp = exp(mid); m2dexp = exp(m2d); ubl = (a*a*debond_stress_current[dl])/(a*a*Ef+(b*b-a*a)*Em); ub2a = (((b*b/(a*a))-1)*(Em/Ef)-1)/Alpha; ub2b = exp(LDEBl)+exp(LDEB2)-2; ub2c = exp(LDEB1)-exp(LDEB2); ub2 = ub2a*ub2b/ub2c; ub[dl] = ubl*((L-deb_len)+ ub2); / * Caculate fiber displacement for the debonded portion of the embedded length */ if(separation==0) / * That i s i n t e r f a c i a l f r i c t i o n a l exists on the interface * / {  US1=A3[dl]*deb_len/(A2*Ef); /* fprintf(TEST," A3[%7.21f]=%7.21f A2=%7.21f m2=%7.21f m2dexp=%27.llf \ n " , deb_len, A3[dl], A2, m2, m2dexp); */  200  TJS2=A3 [dl] * (l-m2dexp) / (A2*m2*Ef) ; US3=B[dl]*(((mldexp-1)/ml) - ((m2dexp-l)/m2))/Ef; US4=(m2dexp-l)*debond_stress_current[dl]/(m2*Ef); USSS=US4+US3+US2; US[dl] = (US1+US2+US3+US4); Fiber_disp[ITER] = ub[dl] + US[dl]; Fiber_disp_2sides[ITER] = 2*Fiber_disp[ITER]; fprintf(PO_DISP, "%d %7.21f %7.21f %9.61f ' %9.61f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \n", ITER, deb_len, debond_stress_current[dl], Fiber_disp[ITER], Fiber_disp_2sides[ITER], Pullout_stress[ITER], Pullout_Bond_component[dl], Pullout_Friction_component[dl], Pullout_Load[ITER], Pullout_Load_Bond_component[dl], Pullout_Load_Friction_component[dl]); / * fprintf(TEST, "A3[%d]=%7.21f B[%d]=%7.21f DSC[%d]=%7.21f USl=%37.11f US2=%37.11f US3=%37.11f US4=%37.11f USSS=%37.11f\n", d l , A3[dl], d l , B [ d l ] , d l , debond_stress_current[dl] US1, US2, -US3, US4, USSS); */ }  else / * That i s when interface is not i n contact i n the debonded region, separation==0 * / {  US[dl]= debond_stress_current[dl]*deb_len/Ef; / * that i s , fiber pullout stress equal to current debond stress * / Fiber_disp[ITER] = ub[dl] + US[dl]; Fiber_disp_2sides[ITER] = 2*Fiber_disp[ITER]; fprintf(PO_DISP, "%d %7.21f %7.21f %9.61f %9.61f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \n", ITER, deb_len, debond_stress_current[dl], Fiber_disp[ITER], Fiber_disp_2sides[ITER], Pullout_stress[ITER], Pullout_Bond_component[dl], Pullout_Friction_component[dl], Pullout_Load[ITER], Pullout_Load_Bond_component[dl], Pullout_Load_Friction_component[dl])  )  )  else  / * that is when dl=2*L (ie,  debond length equal to fiber length  */  {  if(separation==0) region * /  / * that i s , when interface is i n contact i n the debonded  {  mldexp=exp(mid); m2dexp=exp(m2d); ub[dl]=0.0; US1=A3[dl]*deb_len/(A2*Ef); US2=A3[dl]*(l-m2dexp)/(A2*m2*Ef); US3=B[dl]*(((mldexp-1)/ml) - ((m2dexp-l)/m2))/Ef; US4=(m2dexp-l)*debond_stress_current[dl]/(m2*Ef); USSS=US4+US3+US2+US1; US[dl] = (US1+US2+US3+US4); Fiber_disp[ITER] = ub[dl] + US[dl]; Fiber_disp_2sides[ITER] = 2*Fiber_disp[ITER];  201  f p r i n t f (PO_DISP, "%d %7.21f %7.21f %9.61f %9..61f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , ITER, deb_len, debond_stress_current[dl], Fiber_disp[ITER], Fiber_disp_2sides[ITER], Pullout_stress[ITER], Pullout_Bond_component[dl], Pullout_Friction_component[dl], Pullout_Load[ITER], Pullout_Load_Bond_component[dl], Pullout_Load_Friction_component[dl]); / * fprintf(TEST, "%7.61f %7.61f %7.61f %7.61f \ n " , US1,US2,US3,US4); */ / * fprintf(TEST, "A3[%d]=%7.21f B[%d]=%7.21f A2=%7.31f m2=%8.51f m2dexp=%8.51f,DSC[%d]=%7.21f USl=%37.11f US2=%37.11f US3=%37.11f US4=%37.11f USSS=%37.llf\n",dl,A3[dl],dl,B[dl],A2,m2,m2dexp,dl,debond_stress_current[dl], US1,US2, US3, US4, USSS); */ /* fprintf(PO_DISP, "%d %7.21f %7.31f %7.21f %7.11f USP[dl]=%7.31f UP3=%7.31f\n", ITER, emb_len, Fiber_disp[ITER], Pullout_stress[ITER], debond_stress_current3, USPfdl], UP3[dl]); */ } else / * that is when interface i s not i n contact in. the debonded region * / { ub[dl]=0.0; US[dl]= debond_stress_current[dl]*deb_len/Ef; / * that i s , fiber pullout stress equal to current debond stress * / Fiber_disp[ITER] = ub[dl] + US[dl]; Fiber_disp_2sides[ITER] = 2*Fiber_disp[ITER]; fprintf(PO_DISP, "%d %7.21f %7.21f %9.61f %9.61f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \n", ITER, deb_len, debond_stress_current[dl], Fiber_disp[ITER], Fiber_disp_2sides[ITER], Pullout_stress[ITER], Pullout_Bond_component[dl], Pullout_Friction_component[dl], Pullout_Load[ITER], Pullout_Load_Bond_component[dl], Pullout_Load_Friction_component[dl]); /* fprintf(PO_DISP, "%d %7.21f %7.31f %7.21f %7.11f USP[dl]=%7.31f UP3[dl]=%7.31f\n", ITER, emb_len, Fiber_disp[ITER], Pullout_stress[ITER], debond_stress_current3, USP[dl], UP3[dl]); */ }  / * Calculate stress d i s t r i b u t i o n i n fiber and interface when fiber p a r t i a l l y debonded */ / * calculate stress d i s t r i b u t i o n i n the bonded region if(dl<2*L) {  for(zzyy=2*L-dl; zzyy>=0; --zzyy) { zy=2 *L-dl-zzyy; deb_pos=0.0; z = 0 . 5 * zy; LZ = L-deb_len-z; pullout_strs = debond_stress_current[dl]; / * calculate a x i a l stress d i s t r i b u t i o n i n fiber * /  202  */  FAS1 = (a*a*Ef*pullout_strs)/(a*a*Ef+(b*b-a*a)*Em); FAS2 = ((b*b/(a*a))-1) * (Em/Ef) * (exp(Alpha*z)-exp(Alpha*z))/(exp(Alpha*(L-deb_len))-exp(-Alpha*(L-deb_len))); FAS3 = (exp(-Alpha*(L-deb_len-z)) - exp(Alpha*(L-deb_lenz)))/(exp(Alpha*(L-deb_len)) - exp(-Alpha*(L-deb_len))); Fiber_axial_stress[zy] = FAS1*(1+FAS2+FAS3); Fiber_axial_load[zy] = Fiber_axial_stress[zy] * Fiber_area; Fiber_axl_load_bond_compo[zy]= Fiber_axial_load[zy]; Fiber_axl_load_frictional_compo[zy]= 0.0; Total_bond_load = Fiber_axl_load_bond_compo[zy]; / * calculate i n t e r f a c i a l shear stress d i s t r i b u t i o n */ ISSbl = ((b*b/(a*a))-1) * (Em/Ef) * (exp(Alpha*z) + exp(Alpha*z))/(exp(Alpha*(L-deb_len))-exp(-Alpha*(L-deb_len))); ISSb2 = (exp(-Alpha*(L-deb_len-z)) + exp (Alpha*(L-deb_lenz)))/(exp(Alpha*(L-deb_len)) - exp(-Alpha*(L-deb_len))); ISb3 = ((l+Poss_m)*(l+(b*b/(a*a)-1)*(Em/Ef))*(b*b*log_ba-(b*ba*a)12)) ;  ISSb3 = (2/a)* pow(ISb3, 0.5); Interfacial_shear_stress[zy] = -pullout_strs*((ISSbl+ISSb2)/ISSb3) /* fprintf(TEST, "ISSl=%7.11f ISS2=%7.21f ISS3=%7.21f ISS[z=%7.21f]=%7.21f\n", ISS1, ISS2, ISS3, z, Interfacial_shear_stress[zz][dl]); */ fprintf(STRS_DISTR, "%d %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Counter, stress_pout[dl], deb_len, deb_pos, z, F i b e r _ a x i a l _ s t r e s s [ z y ] , Interfacial_shear_stress[zy], Fiber_axial_load[zy], Fiber_axl_load_bond_compo[zy], Fiber_axl_load_frictional_compo[zy]; } }  else { /* fprintf(STRS_DISTR, "I am here \n"); */ deb_pos=0.0; zy=0; z=0.05*zy; zzz=0.01; LZ = L-deb_len-z; DST_Num = ((l+Poss_m)*(1+(b*b/(a*a)-1)*(Em/Ef))*(b*b*log_ba-(b*bA  a*a)12));  DST_Numerator = (2/a) * pow(DST_Num,0.5); DST_Denominator = (((b*b/(a*a)-1)*(Em/Ef))/tanh(Alpha*(0.01)))+ 21(exp(Alpha*(0.01))-exp(-Alpha*(0.01))); debond_stress_current01 = -ISS * DST_Numerator/DST_Denominator; pullout_strs01 = debond_stress_current01; pullout_strs = debond_stress_current[dl]; / * calculate a x i a l stress d i s t r i b u t i o n i n fiber * / FAS1 = (a*a*Ef*pullout_strs)/(a*a*Ef+(b*b-a*a)*Em); • /* fprintf(STRS_DISTR, "but i moved s l i g h t l y l \ n " ) ; * / FAS2 = ((b*b/(a*a))-1) * (Em/Ef) * (exp(Alpha*z)-exp(Alpha*z))/(exp(Alpha*(L-deb_len+0.00000001))-exp(-Alpha*(Ldeb_len+0.00000001))) ; /* fprintf(STRS_DISTR, "but i moved s l i g h t l y 2\n"); */  203  FAS3 = (exp(-Alpha*(L-deb_len-z)) - exp(Alpha*(L-deb_lenz)))/(exp(Alpha*(L-deb_len+0.00000001)) - exp(-Alpha*(Ldeb_len+0.00000001))); /* fprintf(STRS_DISTR, "but i moved s l i g h t l y \n"); */ Fiber_axial_stress[zy] = FAS1*(1+FAS2+FAS3); Fiber_axial_load[zy] = Fiber_axial_stress[zy] * Fiber_area; Fiber_axl_load_bond_compo[zy]= Fiber_axial_load[zy]; Fiber_axl_load_frictional_compo[zy]= 0.0; Total_bond_load = Fiber_axl_load_bond_compo[zy]; /* fprintf(STRS_DISTR, "but i moved s l i g h t l y 3\n"); */ / * calculate i n t e r f a c i a l shear stress d i s t r i b u t i o n */ ISSbl = ((b*b/(a*a))-1) * (Em/Ef) * (exp(Alpha*z) + exp(Alpha*z))/(exp(Alpha*(L-deb_len+0.01))-exp(-Alpha*(L-deb_len+0. 01))) ; ISSb2 = (exp(-Alpha*(L-deb_len-z)) + exp (Alpha*(L-deb_lenz)))/(exp(Alpha*(L-deb_len+0.01)) - exp(-Alpha*(L-deb_len+0.01))); ISb3 = ((l+Poss_m)*(1+(b*b/(a*a)-1)*(Em/Ef))*(b*b*log_ba-(b*b-a*a)12)) ; ISSb3 = (2/a)* pow(ISb3, 0.5); Interfacial_shear_stress[zy] = -pullout_strs01*((ISSbl+ISSb2)/ISSb3); fprintf(STRS_DISTR, "%d %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Counter, stress_pout[dl], deb_len, deb_pos, z, Fiber_axial_stress[zy], Interfacial_shear_stress[zy], Fiber_axial_load[zy], Fiber_axl_load_bond_compo[zy], Fiber_axl_load_frictional_compo[zy]); } fprintf(STRS_DISTR,  "debonded! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !  \n") ;  / * Calculate stress d i s t r i b u t i o n i n the debonded region * / fprintf(TEST, "separation=%7.21f \ n " , separation); for(zx=0; zx<=dl; ++zx) { if(separation==0) / * that is when the interface i s i n contact i n the debonded region */ {  z=0.5*zx; deb_pos=L-deb_len+z; mlz=ml*z; m2z=m2*z; / * a x i a l stress d i s t r i b u t i o n i n fiber when fiber p a r t i a l l y debonded */ Fiber_axial_stress[zx] = (A3[dl]/A2) * (1-exp(m2z)) + B[dl]*(exp(mlz) exp(m2z)) + debond_stress_current[dl]*exp(m2z); Fiber_axial_load[zx] = Fiber_axial_stress[zx] * Fiber_area; Fiber_axl_load_bond_compo[zx] = Total_bond_load; Fiber_axl_load_frictional_compo[zx]= Fiber_axial_load[zx] Total_bond_load; / * deb_component= debond_stress_current[dl]*exp(m2z); */ /* fric_component= (A3[dl]/A2) * (l-exp(m2z)) + B[dl]*(exp(mlz) exp(m2 z)); */  204  -  /* \n",  fprintf(STRS_DISTR, " d e b comp = % 7 . 2 1 f deb_component, f r i c _ c o m p o n e n t ) ; */  fric  corapo =  %7.21f  / * i n t e r f a c i a l s h e a r s t r e s s d i s t r i b u t i o n when f i b e r p a r t i a l l y d e b o n d e d */ Interfacial_shear_stress[zx]= -(a/2)*((-A3[dl]/A2)*m2*exp(m2z) + B [ d l ] * ( m l * e x p ( m l z ) - m2*exp(m2z)) + debond_stress_current[dl]*m2*exp(m2z)); / • c a l c u l a t e r e d u c t i o n i n r a d i a l c o m p r e s s i v e s t r e s s due t o p o i s s o n ' s c o n t r a c t i o n of f i b e r */ shrnk_strss_net[zx]=stress_shrnk-(Interfacial_shear_stress[zx]/mu stress_shrnk); fprintf(STRS_DISTR, "%d %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Counter, s t r e s s _ p o u t [ d l ] , deb_len, deb_pos, z, Fiber_axial_stress[zx], Interfacial_shear_stress[zx], Fiber_axial_load[zx], Fiber_axl_load_bond_compo[zx], Fiber_axl_load_frictional_compo[zx], shrnk_strss_reduction[zx]); } else / * t h a t i s when t h e { z=0.5*zx; deb_pos=L-deb_len+z; mlz=ml*z; m2z=m2*z;  interface  separated  i n the debonded r e g i o n  Fiber_axial_stress[zx]=debond_stress_current[dl]; Fiber_axial_load[zx]=Fiber_axial_stress[zx]*Fiber_area; Fiber_axl_load_bond_compo[zx] = Total_bond_load; Fiber_axl_load_frictional_compo[zx]= 0.0; Interfacial_shear_stress[zx]=0; shrnk_strss_net[zx]=stress_shrnkstress_shrnk);  (Interfacial_shear_stress[zx]/mu  -  } fprintf(STRS_DISTR, "%d %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Counter, s t r e s s _ p o u t [ d l ] , deb_len, deb_pos, z, Fiber_axial_stress[zx], Interfacial_shear_stress[zx], Fiber_axial_load[zx], Fiber_axl_load_bond_compo[zx], Fiber_axl_load_frictional_compo[zx], shrnk_strss_.net[zx]);  /* fprintf(STRS_DISTR, "%d %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , C o u n t e r , s t r e s s _ p o u t [ d l ] , deb_len,.deb_pos, z, Fiber_axial_stress[zx], Interfacial_shear_stress[zx], Fiber_axial_load[zx], Fiber_axl_load_bond_compo[zx],  %7.21f  Fiber_axl_load_frictional_compo[zx]);  } ITER-ITER+1; }  205  */  */  /*  /* /*  */  FIBER COMPLETELY DEBONDED ALONG ITS LENGTH & IS BEING PULLED OUT  */ */  / * Calculate pullout stress (pullout case) for various pullout lengths */  ITER=ITER-1; for(dl=0; dl<=2*L; ++dl) { po_len = 0.5*dl; emb_len=L-po_len; EMBD_LEN= 2 * emb_len; mlm2 = ml+m2; mid = ml*emb_len; m2d = m2*emb_len; mlm2d = (ml+m2)* emb_len; SP1 = 0 . 0 ; SP2a=(a*a-b*b)*D*Ef*(m2*exp(m2d)-ml*exp(mid) + (ml-m2)*exp(mlm2d) ); SP2b=(a*a-b*b)*Em*Poss_f-a*a*Ef*Poss_m; SP2c=(2*mu/a)*(exp(mid)-exp(m2d)); SP2 = ((SP2a/SP2b)-SP2c)* stress_shrnk; SP3a=(a*a-b*b)*Em*Poss_f*(ml*exp(mid)-m2*exp(m2d))-a*a*Ef*Poss_ra*(mlm2)*exp(mlm2d); SP3b=(a*a-b*b)*Em*Poss_f-a*a*Ef*Poss_m; SP3c=2*mu*Em*Poss_f*(exp(mid)-exp(m2d))/(a*D*Ef); SP3=(SP3a/SP3b)+SP3c; stress_pout3[dl]= (SP1+SP2)/SP3; / * stress_pout[dl] i s p a r t i a l debond pullout stress */ Pullout_stress[ITER]=stress_pout3[dl]; debond_stress_current3=0.0; debond_stress_current[dl]=0.0; Pullout_Bond_component[dl] = 0.0; Pullout_Friction_component[dl] = Pullout_stress[ITER]; Pullout_Load[ITER] = Pullout_stress[ITER] * Fiber_area; Pullout_Load_Friction_component[dl] = Pullout_Friction_component[dl] Fiber_area; Pullout_Load_Bond_component[dl] = 0.0; fprintf(PO_STRESSvsDL, "%7.21f emb_len, debond_stress_current3,  %7.21f stress_pout3[dl]);  %7.21f  \n"  / * calculate A3[dl] and B[dl] for various pullout lengths * / A3a=stress_pout3[dl]; A3b= (stress_shrnk/Poss_m)*(l-b*b/(a*a))*D; A3[dl] = -(A3a+A3b)/denom; / * Infact, A3[dl] remains same for a l l pullout lengths * / if(dl<2*L) B[dl] = (stress_pout3[dl]-(A3[dl]/A2)*(l-exp(m2d)))/(exp(mld)exp(m2d)); /*B[dl] is zero for a l l pullout lengths * / else B[dl]= 0; / * calculate fiber displacement for the pullout case * / / * Caculate fiber displacement for the debonded portion of embedded length */ USP1=A3[dl]*emb_len/(A2*Ef);  206  USP2=A3[dl]*(l-exp(m2d))/(A2*m2*Ef); USP3=B[dl]*(((exp(mld)-1)/ml) - ((exp(m2d)-1)/m2))/Ef; USP4=(exp(m2d)-1)*debond_stress_current3/(m2*Ef); USP[dl] = (USP1+USP2+TJSP3+USP4) ; mm2dexp=exp(m2d); / * fprintf(TEST, "USPl=%7.61f USP2=%7.61f USP3=%7.61f A2 = %7.21f m2=%8.51f mm2dexp=%8 . 51f \n" , USP1, USP2^, USP3 , A2 , m2 , mm2dexp); */ / * fprintf(TEST, "A3[2*L]=%7.21f B[2*L]=%100.981f mldexp=%20.151f m2dexp=%20.151f ml=%20.151f m2=%20.151f Ef=%7.21f USP3=%7.21f\n", A3[2*L], B[2*L], mldexp, m2dexp, ml, m2, Ef, USP3); */ / * calculate fiber displacement for the pulled out portion of the fiber */ UP3[dl]=0.5*dl*(l+stress_pout3[dl]/Ef); / * Add * / Fiber_disp[ITER] = USP[dl]+UP3[dl]; Fiber_disp_2sides[ITER] = 2*USP[dl] + 0.5*dl*(1+2 * stress_pout3[dl]/Ef); fprintf(PO_DISP, "%d %7.21f %7.21f %9.61f %9.61f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \n", ITER, po_len, debond_stress_current[dl], Fiber_disp[ITER], Fiber_disp_2sides[ITER], Pullout_stress[ITER], Pullout_Bond_component[dl], Pullout_Friction_component[dl], Pullout_Load[ITER], Pullout_Load_Bond_component[dl], Pullout_Load_Friction_component[dl]); / * find area under the pullout curve pd_previous=parts; pd_last=pd-l/pd_previous; area[pdist]=(PO_LD_Fric[pdist]+PO_LD_Fric[pdist-l])/(2*parts); areal[ITER]=(Pullout_stress[ITER]+Pullout_stress[ITER-1]) *(Fiber_disp[ITER]-Fiber_disp[ITER-1])12; f p r i n t f (AREA, "%d %7.21f \ n " , d l , areal [ ITER].) ;  */  / * Calculate stress d i s t r i b u t i o n i n fiber and interface when fiber being pulled out * / for(zxpo=0; zxpo<=EMBD_LEN; ++zxpo) { z=0.5*zxpo; mlz =ml* z; m2 z =m2 * z; mlm2z=(ml+m2)*z; / * a x i a l stress d i s t r i b u t i o n i n fiber when fiber p a r t i a l l y debonded */ / * note: A3[dl] same for a l l pullout lengths */ / * note: B[dl] same for a l l pullout lengths - infact equal to zero */  Fiber_axial_stress[zxpo] = (A3[dl]/A2) * (l-exp(m2z)) + B[dl]*(exp(mlz) - exp(m2z)); Fiber_axial_load[zxpo] = Fiber_axial_stress[zxpo] * Fiber_area; Fiber_axl_load_bond_compo[zxpo]= 0.0; Fiber_axl_load_frictional_compo[zxpo]= Fiber_axial_load[zxpo];  207  /*  interfacial  shear  s t r e s s d i s t r i b u t i o n when  fiber  partially  debonded  Interfacial_shear_stress[zxpo]= -(a/2)*((-A3[dl]/A2)*m2*exp(m2z) B[dl]*(ml*exp(mlz) - m2*exp(m2z)));  +  f p r i n t f ( S T R S _ D I S T R , "%d %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Counter, s t r e s s _ p o u t 3 [ d l ] , p o _ l e n , emb_len, z, F i b e r _ a x i a l _ s t r e s s [ z x p o ] , Interfacial_shear_stress[zxpo], Fiber_axial_load[zxpo], Fiber_axl_load_bond_compo[zxpo], Fiber_axl_load_frictional_compo[zxpo])  } ITER=ITER+1;  208  Appendix C  209  / * Computer Program to Predict Pullout Load versus Displacement Response Using the Progressive Debonding Model */ #include <stdio.h> ttinclude <float.h> #include <math.h> ttinclude <stdlib.h> #include <memory.h> #include <malloc.h> #define TT 200 #define N 3 main () { FILE * i n p u t _ f i l e ; FILE *STRS_DISTR; FILE *STRS_DISTR2; FILE *PO_DISP; FILE *PO_DISP2; FILE *DEBLEN_DEBSTRSS; FILE *DEBLEN_DEBSTRSS2; FILE *PO_STRESSvsDL; FILE *PO_STRESSvsDL2; FILE *TEST; FILE *TESTIN; FILE *AREA; int L , d l , zz, parts, pdist; double a, b, Poss_m, Poss_f, Em, Ef, ABS, mu, contact_stress, Initial_debonding_stress, row, alpha, epsl, eps2, eps3, eps4, epsilon, thetal, theta2, theta, betal, beta, c n s t n t l , cnstnt2, omega, deb_len, deb_pos, Fiber_area, z, zzz, debond_length, omegal, LL, PDS, FD1, FD2, FAS1, FAS2, FAS3, pd, pullout_distance, FDP1, FDP2, FDP3, FASP1, FASP2, zero_stress, m u i n i t i a l , mufinal, C, pd_previous', pd_last; float Pullout_Bond_component[TT] [N], Pullout_Friction_component[TT] [N], Fiber_disp_2sides[TT] [N], Pullout_Load[TT] [N], Fiber_axial_stress[TT] [N],_Interfacial_shear_stress[TT] [N], Fiber_axial_load[TT] [N], Fiber_axl_load_bond_compo[TT] [N], Fiber_axl_load_.frictional_compo[TT] [N], ISS[TT], Fiber_displacement[TT], Fiber_displacement2[TT], Progressive_dbnd_stress[TT], Progressive_POstress[TT], P0_LD_Bond[TT], P0_LD_Fric[TT], Progressive_POload[TT], area[TT], areal[TT]; i n p u t _ f i l e = fopen("inpmu","r"); STRS_DISTR = fopen("stress","w"); / * stress d i s t r i b u t i o n i n fiber */ STRS_DISTR2 = fopen("stress2", "w"); P0_DISP = fopen("disp","w"); / * pullout stress and fiber displacement */ P0_DISP2 = fopen("disp2", "w"); DEBLEN_DEBSTRSS = fopen("dbsdbl","w"); /*debond length vs. debond stress */ DEBLEN_DEBSTRSS2 = fopen("dbsdbl2", "w"); PO_STRESSvsDL = fopen("posdl","w"); / * pullout stress vs. debond length * /  210  DEBLEN_DEBSTRSS2 = fopen("dbsdbl2", "w"); PO_STRESSvsDL = fopen("posdl","w"); / * pullout stress vs debond length P0_STRESSvsDL2 = fopen("posdl2", "w") ; TEST = fopen("testout", "w"); TESTIN = fopen("in", "w" ) ; AREA = fopen("area", "w") ; / * read the input f i l e */ fscanf(input_file "%lf %lf %lf %lf %lf %lf %d %lf %lf %lf %lf %lf", &a, &b, &Poss_f, &Poss_m, &Em, &Ef, &L, &ABS, &mu, &contact_stress, &mufinal, &C); (  fprintf(TESTIN, "%lf %lf %lf %lf %lf Poss_f, Poss_m, Em, Ef, L , ABS, mu); fprintf(DEBLEN_DEBSTRSS, /*  "Debond Length  %lf  %d  %lf %lf \ n " , a, b,  Debond Stress \n");  ABS - Adhesional Bond Strength */ row=a*a/(b*b-a*a); alpha=Em/Ef; epsl=2/(a*a*(b*log(b/a)-(b-a))); eps2=(0.5*b*b*b*log(b/a)-b*(b*b-a*a)/4) ; eps3 = 0.5*a*(b*b-a*a) ; eps4=(b*b-a*a)/3 ; epsilon=epsl*(eps2+eps3-eps4); thetal=a*a*(l+Poss_m); theta2=(b/(b-a))*log(b/a)-1; theta=l/(thetal*theta2); betal= (theta*(row+alpha-row*epsilon*alpha)/(row*epsilon))+alpha*theta; beta=pow(betal,0.5); cnstntl=row+alpha-row*epsilon*alpha+alpha*epsilon; cnstnt2=alpha-row*epsilon*alpha+alpha*epsilon; omegal=(Ef/Em)*((a*a+b*b)/(b*b-a*a)+Poss_m)+(l-Poss_f); omega=Poss_f/omegal; Fiber_area = 3.14*a*a;  fprintf(PO_DISP, "Deb Len Dbnd Strs PO Strs Fib Disp Progr PO Ld Bnd PO Ld F r i c PO Ld\n"); fprintf(P0_DISP2, "Deb Len Dbnd Strs PO Strs Fib Disp2 Progr PO I Bnd PO Ld F r i c PO Ld\n"); fprintf(STRS_DISTR, "Dbnd Strs Deb Lt Deb Pos Axl Pos Axl Strs Shr Strs Axl Lod Bnd Ld F r i c Ld \n"); fprintf(PO_STRESSvsDL, "PO Strss Deb Len Prog Dbnd Strss \n"); fprintf(AREA, "pdist pd_previous pd area \n"); fprintf(TESTIN, "alpha=%10.51f beta=%10.51f row=%10.51f epsilon=%10.51f omega=%7.51f \ n " , alpha, beta, row, epsilon, omega);  211  /*  /* /*  _  */  Fiber completely bonded along the entire embedded fiber length _ _  */ */  .  deb_len=0.0; deb_pos=0.0; dl=deb_len; zero_stress=0.0; Initial_debonding_stress=-2*ABS*(row+alpharow*epsilon*alpha+alpha*epsilon)*sinh(beta*L)/((a*beta)*((alpharow*epsilon*alpha+alpha*epsilon)*cosh(beta*L)+row)); Progressive_dbnd_stress[dl]=Initial_debonding_stress; Progressive_POstress[dl] = zero_stress; Progressive_P01oad[dl]= 0.0; PO_LD_Bond[dl]= 0.0; PO_LD_Fric [dl ] = 0 . 0 ; Fiber_displacement[dl]=zero_stress/(Ef*(row+alpharow*epsilon*alpha+alpha*epsilon))*((alpha-row*epsilon*alpha+alpha*epsilonrow)*(cosh(beta*L)-1)/(beta*sinh(beta*L))+row*L); Fiber_displacement2[dl] = 2*Fiber_displacement[dl]; f p r i n t f (PO_DISP, "%7.21f %7.21f %7.21f %7.5'lf %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement[dl], Progressive_P01oad[dl], PO_LD_Bond[dl], PO_LD_Fr i c [ d l ] ) ; fprintf(PO_DISP2, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement2[dl], Progressive_P01oad[dl], PO_LD_Bond[dl], PO_LD_Fric[dl]); deb_len=0.0; deb_pos=0.0; dl=deb_len; Initial_debonding_stress=-2*ABS*(row+alpharow*epsilon*alpha+alpha*epsilon)*sinh(beta*L)/((a*beta)*((alpharow*epsilon*alpha+alpha*epsilon)*cosh(beta*L)+row)); Progressive_dbnd_stress[dl]=Initial_debonding_stress; Progressive_POstress[dl] = Initial_debonding_stress; Progressive_P01oad[dl]=Progressive_POstress[dl]*Fiber_area; PO_LD_Bond[dl]=Progressive_dbnd_stress[dl]*Fiber_area; PO_LD_Fric[dl]=Progressive_P01oad[dl]-PO_LD_Bond[dl]; Fiber_displacement[dl]=Initial_debonding_stress/(Ef*(row+alpharow*epsilon*alpha+alpha*epsilon))*((alpha-row*epsilon*alpha+alpha*epsilonrow)*(cosh(beta*L)-1)/(beta*sinh(beta*L))+row*L); Fiber_displacement2[dl] = 2*Fiber_displacement[dl]; fprintf(PO_DISP, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement[dl], Progressive_P01oad[dl], PO_LD_Bond[dl], PO_LD_Fric[dl]); fprintf(PO_DISP2, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement2[dl], Progressive_P01oad[dl], PO_LD_Bond[dl], PO_LD_Fric[dl]);  212  /*  fprintf(PO_DISP, "%7.21f %7.21f %7.21f %7.51f \ n " , deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement[dl]); */ / * fprintf(P0_DISP2, "%7.21f %7.21f %7.21f %7.51f \n" , deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement2[dl]) ; */ fprintf(PO_STRESSvsDL, "%7.21f %7.21f %7.21f \ n " , Progressive_POstress[dl] , Fiber_displacement[dl], Progressive_dbnd_stress[dl]); parts=2; for(zz=0; zz<=parts*L; ++zz) { deb_len=0.0; deb_pos=0.0; zzz=zz; z=zzz/parts; Fiber_axial_stress[zz] [1] (Initial_debonding_stress/cnstntl)*(cnstnt2*sinh(beta*z)/sinh(beta*L)row*sinh(beta*(L-z))/sinh(beta*L)+row); Interfacial_shear_stress[zz] [1] = (a*beta*Initial_debonding_stress/(2*cnstntl))*(cnstnt2*cosh(beta*z)/sinh(b eta*L)+row*cosh(beta*(L-z))/sinh(beta*L)); ISS[zz] = (a*beta*Initial_debonding_stress/(2*cnstntl))*(cnstnt2*cosh(beta*z)/sinh(b eta*L)+row*cosh(beta*(L-z))/sinh(beta*L)); Fiber_axial_load[zz] [1] = Fiber_axial_stress[zz] [1] * Fiber_area; Fiber_axl_load_bond_compo[zz] [1] = Fiber_axial_load[zz] [1]; Fiber_axl_load_frictional_compo[zz] [1] = 0.0; fprintf(TESTIN, "%lf %lf\n", Initial_debonding_stress, Fiber_displacement); fprintf(STRS_DISTR, "%7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Initial_debonding_stress, deb_len, deb_pos, z, Fiber_axial_stress[zz] [1], Interfacial_shear_stress[zz] [1], Fiber_axial_load[zz] [1], Fiber_axl_load_bond_compo[zz] [1], Fiber_axl_load_frictional_compo[zz] [1]);  /*  /* /*  */  Fiber p a r t i a l l y bonded along i t s embedded length  */ v  for(dl=0; dl<=parts*L; ++dl) { debond_length=dl; deb_len=debond_length/parts ; deb_pos=L-deb_len; LL=parts*L; if(dl<LL) / * calculate progressive debonding stress * / Progressive_dbnd_stress[dl]=-2*ABS*(row+alpharow*epsilon*alpha+alpha*epsilon)*sinh(beta*(L-deb_len))/((a*beta)*((alpharow*epsilon*alpha+alpha*epsilon)*cosh(beta*(L-deb_len))+row)); else  213  Progressive_dbnd_stress[dl]=0.0; / * calculate progressive pullout stress and fiber displacement * / Progressive_POstress[dl]=contact_stress/omega+(Progressive_dbnd_stress[dl]+contact_stress/omega)*exp (-2*mu*omega*deb_len/a); if(Progressive_POstress[dl]<Progressive_dbnd_stress[dl]) Progressive_POstress[dl]=Progressive_dbnd_stress[dl]; Progressive_P01oad[dl]=Progressive_POstress[dl]*Fiber_area; PO_LD_Bond[dl]=Progressive_dbnd_stress[dl]*Fiber_area; PO_LD_Fric[dl]=Progressive_P01oad[dl]-PO_LD_Bond[dl]; FDl=Progressive_dbnd_stress[dl]/(Ef*(row+alpharow*epsilon*alpha+alpha*epsilon))*((alpharow*epsilon*alpha+alpha*epsilon-row)*(cosh(beta*(L-deb_len))1)/(beta*sinh(beta*(L-deb_len+0.000000001)))+row*(L-deb_len)); FD2=-contact_stress*deb_len/(Ef*omega)a/(2*Ef*mu*omega)*(Progressive_dbnd_stress[dl]+contact_s tress/omega)*(exp(2*mu*omega*deb_len/a)-1); Fiber_displacement[dl]=FD1+FD2; Fiber_displacement2[dl] = 2*Fiber_displacement[dl]; fprintf(PO_DISP, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement[dl], Progressive_P01oad[dl], PO_LD_Bond[dl], PO_LD_Fric[dl]); fprintf(PO_DISP2, "%7.21f %7.21f %7.21f %7.51f . %7.21f %7.21f %7.21f\n", deb_len, Progressive_dbnd_stress[dl], Progressive_POstress[dl], Fiber_displacement2[dl], Progressive_P01oad[dl], PO_LD_Bond[dl], PO_LD_Fric[dl]); fprintf(PO_STRESSvsDL, "%7.21f %7.21f %7.21f \ n " , Progressive_POstress[dl], Fiber_displacement[dl], Progressive_dbnd_stress[dl]); / * calculate fiber a x i a l stress d i s t r i b u t i o n & i n t e r f a c i a l shear stress distribution */ / * over the bonded zone of the p a r t i a l l y debonded f i b e r * / for(zz=0; zz<=parts*L-dl; ++zz) { zzz=zz; z=zzz/parts; Fiber_axial_stress[zz] [2] = (Progressive_dbnd_stress[dl]/cnstntl)*(cnstnt2*sinh(beta*z)/sinh(beta*( L-deb_len))-row*sinh(beta*(L-deb_len-z))/sinh(beta*(L-deb_len))+row); Interfacial_shear_stress[zz] [2] = (a*beta*Progressive_dbnd_stress[dl]/(2*cnstntl))*(cnstnt2*cosh(beta*z)/sin h(beta*(L-deb_len+0.000001))+row*cosh(beta*(L-deb_len-z))/sinh(beta*(Ldeb_len+0.000001))); Fiber_axial_load[zz] [2] = Fiber_axial_stress[zz] [2] * Fiber_area; Fiber_axl_load_bond_compo[zz] [2] = Fiber_axial_load[zz] [2]; Fiber_axl_load_frictional_compo[zz] [2] = 0.0; / * fprintf(TESTIN, "%lf %lf\n", Initial_debonding_stress, Fiber_displacement); * / fprintf(STRS_DISTR, "%7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Progressive_dbnd_stress[dl], deb_len, deb_pos, z, Fiber_axial_stress[zz] [2], Interfacial_shear_stress[zz] [2],  214  Fiber_axial_load[zz] [2], Fiber_axl_load_bond_compo[zz] Fiber_axl_load_frictional_compo[zz] [2]);  [2],  }  / * calculate fiber a x i a l stress d i s t r i b u t i o n & i n t e r f a c i a l shear stress distribution */ / * over the debonded zone of the p a r t i a l l y debonded fiber * / for(zz=parts*L-dl; zz<=parts*L; ++zz) { zzz=zz ; z=zzz/parts; FASl=contact_stress/omega; FAS2=Progressive_dbnd_stress[dl]+contact_stress/omega; FAS3=exp(2*omega*mu*(L-deb_len)/a)*exp(-2*omega*mu*z/a); Fiber_axial_stress[zz] [2]=-FASl+FAS2*FAS3; Interfacial_shear_stress[zz] [2]=mu*omega*FAS2*FAS3; Fiber_axial_load[zz] [2] = Fiber_axial_stress[zz] [2] * Fiber_area; Fiber_axl_load_bond_compo[zz] [2] = Progressive_dbnd_str'ess [dl] * Fiber_area; Fiber_axl_load_frictional_compo[zz] [2] = Fiber_axial_load[zz] [2] Fiber_axl_load_bond_compo[zz] [2]; / * fprintf(TESTIN, "%lf %lf\n", Initial_debonding_stress, Fiber_displacement); * / fprintf(STRS_DISTR, "%7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Progressive_dbnd_stress[dl], deb_len, deb_pos z, Fiber_axial_stress[zz] [2], Interfacial_shear_stress[zz] [2], Fiber_axial_load[zz] [2], Fiber_axl_load_bond_compo[zz] [2], Fiber_axl_load_frictional_compo[zz] [2]); } }  /* /* /*  Fiber completely debonded along i t s length  _  */ */ */  muinitial=mu; for(pdist=0; pdist<=parts*L; ++pdist) { pullout_distance=pdist; pd=pullout_distance/parts; / * L_new=L-pout_dist; * / Progressive_dbnd_stress[pdist]=0.0; / * calculate mu * / mu= (muinitial-muf inal) *exp (-C*pd) +muf i n a l ; / * calculate progressive pullout stress and fiber displacement */ Progressive_POstress[pdist]=-contact_stress/omega*(l-exp(-2*mu*omegci*(Lpd)/a)); Progressive_P01oad[pdist]=Progressive_POstress[pdist]*Fiber_area; PO_LD_Fric[pdist]=Progressive_P01oad[pdist]; P0_LD_Bond[pdist]=0.0; FDPl=contact_stress/(Ef*omega); FDP2=a/(2*mu*omega)*(exp(-2*mu*omega*(L-pd)/a)-1); FDP3=pd*(l-exp(-2*mu*omega*(L-pd)));  215  Fiber_displacement[pdist]=pd-FDPl*((L-pd)+FDP2+FDP3); Fiber_displacement2[pdist] = pd-FDPl*(2*(L-pd)+2*FDP2+FDP3); fprintf(PO_DISP, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", pd, Progressive_dbnd_stress[pdist], Progressive_POstress[pdist], Fiber_displacement[pdist], Progressive_P01oad[pdist], PO_LD_Bond[pdist], PO_LD_Fric[pdist]); fprintf(P0_DISP2, "%7.21f %7.21f %7.21f %7.51f %7.21f %7.21f %7.21f\n", pd, Progressive_dbnd_stress[pdist], Progressive_POstress[pdist], Fiber_displacement2[pdist], Progressive_P01oad[pdist], PO_LD_Bond[pdist], PO_LD_Fric[pdist]); fprintf(PO_STRESSvsDL, "%7.21f %7.21f %7.21f \ n " , Progressive_POstress[pdist], Fiber_displacement[pdist], Progressive_dbnd_stress[pdist]); / * find area under the pullout curve * / pd_previous=parts; pd_last=pd-l/pd_previous; area[pdist]=(PO_LD_Fric[pdist]+PO_LD_Fric[pdist-1])/(2*parts); areal[pdist]=(PO_LD_Fric[pdist]+PO_LD_Fric[pdist1])*(Fiber_displacement[pdist]-Fiber_displacement[pdist-1])12; fprintf(AREA, "%d %7.21f %7.21f %7.21f %7.21f\n", p d i s t , pd_last, pd, area[pdist], a r e a l [ p d i s t ] ) ; / * calculate fiber a x i a l stress d i s t r i b u t i o n & i n t e r f a c i a l shear stress distribution */ / * over the debonded zone of the p a r t i a l l y debonded fiber * / for(zz=0; zz<=parts*(L-pd); ++zz) { zzz=zz; z=zzz/parts; FASPl=contact_stress/omega; FASP2=exp(-2*omega*mu*z/a) ; Fiber_axial_stress[zz] [3]=-FASPl*(1-FASP2); . Interfacial_shear_stress[zz] [3]=mu*contact_stress*FASP2; Fiber_axial_load[zz] [3] = Fiber_axial_stress[zz] [3] * Fiber_area; Fiber_axl_load_bond_compo[zz] [3] = Progressive_dbnd_stress[pdist] * Fiber_area; Fiber_axl_load_frictional_compo[zz] [3] = Fiber_axial_load[zz] [3] Fiber_axl_load_bond_compo[zz] [2]; fprintf(STRS_DISTR, "%7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f %7.21f \ n " , Progressive_dbnd_stress[pdist], pd, z, z, Fiber_axial_stress[zz] [3], Interfacial_shear_stress[zz] [3], Fiber_axial_load[zz] [3], Fiber_axl_load_bond_compo[zz] [3], Fiber_axl_load_frictional_compo[zz] [3] ) ; }  } }  216  

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