F I B E R R E I N F O R C E D C O N C R E T E : C H A R A C T E R I Z A T I O N O F F L E X U R A L T O U G H N E S S & S O M E S T U D I E S O N F I B E R - M A T R I X B O N D - S L I P I N T E R A C T I O N 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 SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (The Department of Civi l Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A April, 1999 © Ashish Dubey, 1999 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil fc y\chT> ef_9u rv The University of British Columbia Vancouver, Canada DE-6 (2788) 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 i i 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 i i 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 2 Literature Review 5 2.0 Introduction 5 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 10 2.3 Experimental Studies on the Fiber Pullout Behavior 26 2.4 Mechanical Behavior of Fiber Reinforced Cementitious Composites 30 3 Macromechanical Behavior of Fiber Reinforced Concrete (FRC) & Measurement of Flexural Response of Low Toughness FRC 37 3.0 Introduction 37 iv 3.1 Material Variables Influencing Macromechanical Behavior and Toughness of FRC 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 5 0 3.1.6 Shrinkage Properties of Matrix 51 3.2 Characterization of Flexural Toughness of Fiber Reinforced Concretes 53 3.2.1 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 3.2.2 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 4 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 Straight-smooth, Stainless Steel Fibers Embedded in Normal Strength Matrix 84 4.1.1.1 Influence of Fiber Length 8 5 4.1.2 Straight, Stainless Steel Fibers with Rough Surface V Embedded in Normal Strength Matrix 86 4.1.2.1 Influence of Surface Roughness 87 4.1.2.2 Influence of Fiber Length 88 4.1.3 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 4.1.4 Straight-smooth, Stainless Steel Fibers Embedded in High-Strength Matrix 91 4.1.4.1 Influence of Water/Cement Ratio 92 4.1.5 Straight-smooth, Stainless Steel Fibers Embedded in Silica-fume Modified High-Strength Matrix 93 4.1.5.1 Influence of Silica-fume Modification 94 4.1.6 Straight, Smooth Polypropylene Fibers Embedded in Normal Strength Matrix 95 4.1.6.1 Influence of Fiber Length 96 4.2 Conclusions 97 5 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, xa 109 5.1.1.3 Fiber Displacement, Ub 110 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, aa vs. Id 116 5.1.2.4 Fiber Displacement versus Debond Length Relationship, Updvs.ld 116 5.1.2.5 Bond and Frictional Components of Pullout Stress, o0,bond and <70ifric 117 5.1.2.6 Catastrophic Debonding 118 5.1.3 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, a0 121 5.1.3.4 Pullout Displacement, Upj 122 5.1.4 Calibration of Interfacial Parameters - oc, /n and Ts 122 5.2 Conclusions 125 6 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 127 6.1.2 Straight-smooth, Stainless Steel Fiber Embedded in Normal Strength Matrix 128 6.1.3 Straight, Stainless Steel Fibers with Rough Surface Embedded in Normal Strength Matrix ' 130 6.1.4 Straight-smooth, Stainless Steel Fibers Embedded in Non-Shrink Grout Matrix 132 vii 6.1.5 Straight, Smooth Polypropylene Fibers Embedded in Normal Strength Matrix 135 6.1.6 Pullout Data Found in the Literature 136 6.2 Conclusions 139 7 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, rs 141 7.1.1.1 Fiber Elastic Modulus, Ef= 210 GPa 141 7.1.1.2 Fiber Elastic Modulus, Ef= 3.5 GPa 146 7.1.2 Influence of Interfacial Contact Stress, ac 151 7.1.2.1 Fiber Elastic Modulus, Ef= 210 GPa 151 7.1.2.2 Elastic Modulus, Ef= 3.5 GPa 154 7.1.3 Influence of Interfacial Coefficient of Friction, ji 157 7.1.3.1 Fiber Elastic Modulus, Ef= 210 GPa 157 7.1.3.2 Elastic Modulus, Ef= 3.5 GPa 160 7.2 Conclusions 163 8 Conclusions 166 Bibliography 173 Appendix A 188 Appendix B 193 Appendix C 209 viii List o f Tables Table 3.1.1 Mix Proportions 3 8 Table 3.1.2 Description of fibers investigated 38 Table 3.1.3 Properties of silica fume and high-reactivity metakaoline investigated 41 Table 3.1.4 JSCE Absolute Toughness and Flexural Toughness Factor -Influence of Pozzolan 42 Table 3.1.5 JSCE Absolute Toughness and Flexural Toughness Factor -Influence of Pozzolan Type 44 Table 3.1.6 JSCE Absolute Toughness and Flexural Toughness Factor -Influence of fiber volume 46 Table 3.1.7 JSCE Absolute Toughness and Flexural Toughness Factor -Influence of fiber aspect ratio 47 Table 3.1.8 JSCE Absolute Toughness and Flexural Toughness Factor -Influence of fiber surface characteristics 49 Table 3.1.9 JSCE Absolute Toughness and Flexural Toughness Factor -Influence of fiber geometry 51 Table 3.1.10 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 Table 3.2.15 Table 4.1.1 Table 4.1.2 Table 4.1.3 Table 4.1.4 Table 4.1.5 Table 4.1.6 Table 5.0.1 Table 6.1.1 Table 6.1.2 Table 6.1.3 Table 6.1.4 Table 6.1.5 Detailed results for fibers in Set 1 77 Detailed results for fibers in Set 2 78 Experimental pullout test results for straight-smooth, stainless steel fibers embedded in normal strength matrix 85 Experimental pullout test results for straight, stainless steel fibers with rough surface embedded in normal strength matrix 86 Experimental pullout test results for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix 89 Experimental pullout test results for straight-smooth, stainless steel fibers embedded in high strength matrix 91 Experimental pullout test results for straight-smooth, stainless steel fibers embedded in silica-fume modified normal strength matrix 93 Experimental pullout test results for straight, smooth polypropylene fibers embedded in normal strength matrix 95 Various parameters taken into consideration in the theoretical models of fiber pullout 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 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 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 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 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] 17 Figure 2.2.2 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% 42 Figure 3.1.3 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) 46 Figure 3.1.5 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 f = 0.76% 48 Figure 3.1.6b Comparison of flexural load versus deflection response for fiber XI reinforced concrete mixes - influence of fiber surface characteristics, Polyvinyl alcohol fiber - PVA-1 @ V f = 1.00% 49 Figure 3.1.7 Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of fiber geometry 50 Figure 3.1.8 Comparison of flexural load versus deflection response for fiber reinforced concrete mixes - Influence of shrinkage properties of matrix 52 Figure 3.2.1 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] 54 Figure 3.2.2a Schematic of Residual Strength Test Method experimental setup 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 60 Figure 3.2.4 Comparison of Closed-loop and Open-loop curves for concrete with 0.1% of Fibrillated Polypropylene Fiber (PP1) 60 Figure 3.2.5 Comparison of Closed-Loop and Open-loop curves for concrete with 0.3% of Fibrillated Polypropylene Fiber (PP1) 60 Figure 3.2.6 Comparison of Closed-loop and Open-loop curves for concrete with 0.1 % of Monofilament Polypropylene Fiber (PP2) 60 Figure 3.2.7 Comparison of Closed-Loop and Open-loop curves for concrete with 0.3% of Monofilament Polypropylene Fiber (PP2) 60 Figure 3.2.8 Comparison of Closed-Loop and Residual Strength Test Method Curves for Concrete with 0.1 % of Fibrillated Polypropylene Fiber (PP1) 61 Figure 3.2.9 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 FRC with PPF-1 Fiber at Dosage Rate of 0.4% Figure 3.2.19c Load-Deflection Curves for FRC with PPF-1 Fiber at Dosage Rate of 0.5% Figure 3.2.19d Load-Deflection Curves for FRC with PPF-1 Fiber at Dosage Rate of 0.6% Figure 3.2.19e Load-Deflection Curves for FRC with PPF-1 Fiber at Dosage Rate of 0.8% xiii Figure 3.2.20 Load-Deflection Curves for FRC 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 FRC 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 FRC with PVA-1 Fiber at a Dosage Rate of 1.6% 75 Figure 3.2.25 Load-Deflection Curves for FRC with Steel Fiber ST-HKD at a Dosage Rate of 0.76% 79 Figure 3.2.26 Load-Deflection Curves for FRC with Steel Fiber ST-CR1 at a Dosage Rate of 0.76% 79 Figure 4.1.1a Schematic of a standard pullout test specimen 83 Figure 4.1.2 Pullout response of straight-smooth, stainless steel fibers embedded in normal strength matrix 85 Figure 4.1.3 Embedded fiber length vs. maximum pullout load curve for straight-smooth, stainless steel fibers 86 Figure 4.1.4 Pullout response of straight, stainless steel fibers with rough surface embedded in normal strength matrix 87 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 Figure 4.1.6 Embedded fiber length vs. maximum pullout load curve for straight, stainless steel fibers with rough surface 88 Figure 4.1.7 Pullout response of straight-smooth, stainless steel fibers embedded in non-shrink grout matrix 89 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 90 Figure 4.1.9 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 97 Figure 5.0.1 Single fiber embedded in concrete matrix. Figure depicts one side of a two-sided pullout test specimen 99 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] 102 Figure 5.1.1 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 129 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 130 Figure 6.1.3 Coefficient of friction versus pullout displacement curves for straight, stainless steel fibers with rough surface embedded in XV normal strength matrix 131 Figure 6.1.4 Comparison of experimental and theoretical pullout response -straight, stainless steel fiber with rough surface (embedded length -30 mm) embedded in normal strength matrix 132 Figure 6.1.5 Coefficient of friction versus pullout displacement curves for straight-smooth, stainless steel fibers embedded in non-shrink grout matrix 133 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 134 Figure 6.1.7 Coefficient of friction versus pullout displacement curves for straight, smooth polypropylene fibers embedded in normal strength matrix 13 5 Figure 6.1.8 Comparison of experimental and theoretical pullout response -straight, smooth polypropylene fiber (embedded length - 30 mm) embedded in normal strength matrix 136 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 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, TS (£y=210GPa) 145 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, xs (£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) 152 Figure 7.1.3e Variation in pullout load and its components as a function of XVII 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) x v i i i 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 Mc - bending moment at cracking oc - stress carried by the beam at cracking os - stress carried by steel plate at the instant beam cracks Pcr - cracking load of beam 8cr - 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 T m - matrix shear stress ra - interfacial shear stress Ts - fiber-matrix interfacial shear strength Tf - interfacial shear stress over the debonded interface wm - matrix axial displacement wa - matrix displacement at the interface (i.e., r=d) M>b - matrix displacement at the surface of coaxial cylinder (i.e., at r=b) 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 servo-controlled 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 Micromechanical Models for Investigating the Fiber-Matrix 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.1 Assuming compatibility of fiber and matrix displacements at the interface, 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 fiber-matrix 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 dW dWf G = ^lex__^21e_ L 2.1 da da da where, a is the increase in debonded area due to extension of the crack, Wex is the work done by external load, We is the elastic strain energy stored in the system, and Wf is the energy dissipated 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"r", i.e., the contribution of mode I to the energy release rate is 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 a v = P llmrl was derived as: T( X) = a/{sinh(rxc) - coth(a/) cosh(cec)} 2.3 where, a = -11/2 2G, b,rE, 2.4 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 xmax/xav-4 1, i.e., xmax —> t a v. Greszczuk 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, T m a x , of the interface can be estimated by plotting a curve of x a v versus /, and extrapolating i a v at / = 0. As mentioned above, for a given fiber and matrix type, the value of interfacial shear strength T m a x is established. Now, for a fiber with a given embedded length, the average shear stress T a v can be obtained from the experimental pullout test result, and thereafter the ratio i a v /T m a x be evaluated. Using Equation 2.5, a value of cd that corresponds to the calculated ratio xav/xmax 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)=Pf ?i=— 2.6 sinhVtt le where, Pf is the fiber pullout load, x is the distance of the point under consideration measured from the embedded fiber end, le is the embedded fiber length, and a is an elastic constant given by: a = CK\ 1 1 AfEf A E j j m m 2.7 where, C is the fiber perimeter, Af and Am are the areas of the fiber and the matrix, respectively, Ef and Em are the elastic modulus of the fiber and the matrix, respectively. K is a constant that 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 le. Accordingly, 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 ) = KPf cosh ~J~a x •Jot sinhVa le 1 1 AfEf 2.8 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 1 AfEf AmEm 2.9 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 xs (the interfacial elastic shear strength), the fiber would debond from the matrix at the 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 le, and the ratio of the elastic shear strength Ts and the frictional shear strength T, of the fiber-matrix interface. The maximum value of fiber pullout load Pf™* would occur 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 S / T ; >cosh2 V a /e(i.e., when x = x^, = le), the debonding process becomes catastrophic as soon as it commences. On the contrary, if 15 T s / T i < cosh2 -fa le, a further increase in fiber pullout load, Pf, is necessary for debonding to continue. Accordingly, the maximum value of the fiber pullout load P / m a x required to achieve complete debonding and initiate the fiber pullout is given by the following equations, ,-y/a K •A,Ef A E A f E f m m } } tanhVa le Ip ^ Xn 2.11 K ''AE-AfEf ^ m m J S AmEmAfEf tanh 4a xmax + T,C(Z - xmax) le > Xn 2.12 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 le. In addition, Equation 2.10 implies that, for any fiber length that is less than or equal to 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 le, and subsequently it would decrease as the fiber pulls out from the matrix. 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 TS. For the determination of the shear strength of the interface experimentally, the maximum pullout load Pf m a x for fibers with different embedded length le is plotted against the function le. The shape of the curve so obtained is a function of the ratio T/T, as shown in Figure 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 5 (a) Tyr,- =1 (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 le curve is a straight line as shown in Figure 2.2.1. The shear strength of the 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 le curve is as shown in Figure 2.2.1. In the P / m a x versus 4oc le curve a point of discontinuity occurs and the slope of the curve, A, becomes constant. The point of discontinuity corresponds to the fiber length le that equals x^ 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 1 C 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 le curve has no obvious discontinuity and becomes linear at long 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 xniax as given by the Equation 2.10. The interfacial shear strength Ts can be determined from the asymptotic value of P / m a x ( i .e . , P / m a x = Pf°°) from the P / m a x versus -fa le curve. Rearranging Equation 2.10, one can obtain, 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 = . .Gm Ju-v) 2.17 r l n ^ / r / V , /2 ) where r is the fiber radius, Gm is the shear modulus of the matrix, V) is the volume fraction of the 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, re(x), over the bonded length of fiber were derived as follows: F(x) = C, cosh fix + C2 sinh fix + C 3 2.18 Te (x) = — — (C, sinh fix + C2 cosh fix) 19 and, the axial stress distribution in the fiber, F(x), and the interfacial shear stress distribution, re(x) over the debonded length of fiber were derived as follows: F(x) = P-2Ttrxi Xe(x) = -xi — X 2 2.19 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, Xmwc was assumed to be the same as that derived by Greszczuk, i.e., ^ = cd{coth(od)} 2.20 However, they defined the elastic parameter a in a different manner, as given by the following equation, a = -11/2 rf E i H rm/rf) 2.21 where Gm is the shear modulus of the matrix, ry is the fiber radius, rm is the radius of matrix surrounding the fiber, and Ef is the fiber modulus. To evaluate the values of shear strength of the 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 s (x) = A[l - exp(-fljc)] 2.22 Putting x=xe (embedded fiber length), the relationship between the initial fiber pullout stress o; and the embedded fiber length xe was written as: CT. = A[l-exp(-Bxe)] 2.23 where A is a function of the normal compressive stress, o~0, exerted by the matrix on the fiber 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 m and vy are the Poisson's ratio for the matrix and the fiber, respectively; B is a function 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= 2 £ - V ' " 2.25 E, r , ( l + v j C T 0 and ji can be determined using Equation 2.23 and the experimental relationship between o~, andxe.. 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. Their analysis demonstrated that the 21 frictional resistance to pullout, and hence the fiber pullout force, was very sensitive to the fiber-matrix misfit. Due to fiber-matrix misfit, strain, e0 (=8/rf, where, r, is fiber radius) between the fiber and the 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 + vm)/Em+(l-vf)Ef 2.26 where vm and v, are the Poisson's ratio of matrix and fiber, respectively, and Em and Ef are the 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 ef (e = e0-£f) causes a reduction in the interfacial contact pressure. Taking all the above mentioned 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: *.\ Open-loop Test Method a^*=:: — i 1- ' 0.0 0.5 1.0 1.5 Deflection (mm) 2.0 Figure 3.2.5: Comparison of Close-Loop and Open-loop curves for concrete with 0.3% of Fibrillated Polypropylene Fiber (PP1) 2 Close-loop Test Method Open-loop Test Method 0.5 1.0 1.5 Deflection (mm) 2.0 Figure 3.2.6: Comparison of Close-loop and Open-loop curves for concrete with 0.1% of Monofilament Polypropylene Fiber (PP2) 0.5 1.0 1.5 Deflection (mm) Figure 3.2.7: Comparison of Close-Loop and Open-loop curves for concrete with 0.3% of Monofilament 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 20 15 -10 -5 0 Fiber PP1 @ Vf = 0.1o O < £ . + 1 1 1 • 6%(PP) V-B (sec) 1 I i } BB • 1 i o £ "E b "56 CS vo r o CN 1 • i 1 ro 1 1 i 1 oo r~ c s 1 1 i 1 >perties Sim (mm) o 1 • 1 0 - ^ oo vo r o c s ON 00 CO c s • i 1 ^ y s vo < ^ oo c s • 1 II > • 03 Tj c s ON 00 1 • 1 E ? 55 J , o O 1 i 1 vi 1 o\ VO r o c s r-r> CO c s CS oo CO c s I of Steel >fPVA ACT (%) uo o CO 1 = 0.76% 1.0% < V-B+ (sec) 1 Cv c s >/-] 1 * *BB GO £ o 1/0 o 00 o as Fiber Material and Type CD c CD 1,1 •a > 00 cu ">> c •> 73 CU JS "o 1 8 •a c CD •2 O CD o a 3C GO •8 E 71 ' C u u GO Q i GO T3 cu e= ' C vs £ c s DC — + Fiber and Mix Id. I cu CU 1 < > ac fi CO 2 u 00 Q Oi ca ac CD o U u < ca • CD CD > I 03 I > + a. E 55 • E . 53 a, 6 o U x i i a o u c s r o u ca Cu. e P CU 6 o U • J E 8 3 II t£ II l£ > 5 ^ 05 < r?0> ! > GO • o CD H C •o pyle 73 2 en; lyp Mat tPo U i JS CJ O0 X> •a U i C/3 § 2 S3 * cu •* VO r o © , GO CU Cu 5 3^, o JS O o 3 o Cu '3 > CU •o c CD "2 o o DC H GO GO I •c u 05 u fi GO • ? Q 8 a , B « £ ^ .a & fi . 5 0 0 a •o o • C w> c s o o 2 + X ) Q 02 CQ ac 3 00 t > 42.1 (3.1) 40.6 (2.2) I i 1 39.6 (5.1) 37.5 (2.4) 1 ? L 40.44 (4.4) 39.8 (4.2) 1 • 1 37.1 (2.1) 40.1 (2.2) ssive Strengt | Vf=0.5% 38.9 (1.6) 40.3 (4.9) 37.9 38.9 (2.8) 39.1 (2.7) 42.1 (4.3) 41.9 (3.5) & ^ E •* o d > 41.1 (3.3) 39.2 (2.7) 1 • • 39.7 (0.4) 40.4 (1.1) cs t > 37.8 (2.1)# 41.3 (3.6) • i • 39.4 (3.5) 38.9 (2.9) Fiber Information Fibrillated Polypropylene Fibrillated Polypropylene Fibrillated Polypropylene Fibrillated Polypropylene Monofilament Polypropylene Monofilament Polypropylene Monofilament Nylon Fiber & Mix Id. PPF-1 PPF-2 PPF-3 PPF-4 PPM-1 PPM-2 NLM-1 3 E VO <0 00 o E Q CN cn Vf = 0.8% IS 46.6 45.4 • • 20.5 20.7 Vf = 0.8% RS (MPa) 2.58 (0.07) 2.62 (0.11) • • • • • • 1.21 (0.08) 1.15 (0.07) Vf = 0.8% MOR (MPa) 5.54 (0.51) 5.77 (0.22) • • • • 1 1 5.9 (0.14) 5.56 (0.32) Vf = 0.6% %i 34.6 34.3 1 1 • •• • • 17.7 17.3 Vf = 0.6% RS (MPa) 1.86 (0.10) 1.85 (0.16) 1 1 1 < 1.04 (0.18) 0.88 (0.14) Vf = 0.6% MOR (MPa) 5.37 (0.07) 5.39 (0.10) • • • • • • . 5.86 (0.27) 5.10 (0.58) Fiber Volume, Vf Vf = 0.5% 27.7 27.3 24.4 29.5 23.1 14.1 12.7 Fiber Volume, Vf Vf = 0.5% RS (MPa) 1.5 (0.23) 1.54 (0.20) 1.46 (0.02) 1.69 (0.11) 1.26 (0.35) 0.85 (0.11) 0.71 (0.05) Fiber Volume, Vf Vf = 0.5% MOR (MPa) 5.41 (0.48) 5.65 (0.30) 5.98 (0.53) 5.73 (0.23) 5.46 (0.25) 6.03 (0.15) 5.61 (0.14) Vf = 0.4% 21.7 21.2 1 1 1 1 . 1 1 oo t-Vf = 0.4% RS (MPa) 1.27 (0.04) 1.18 (0.12) • • 0.45 (0.09) 0.41 (0.08) Vf = 0.4% MOR (MPa) 5.85 (0.26) 5.57 (0.36) • • • • • • 5.79 (0.20) 5.54 (0.57) Vf = 0.2% »n oo ( 1 1 • • 1 1 cn Vf = 0.2% 3 1 0.33 (0.07) 0.44 (0.03) ' ' • • • • 0.23 (0.12) 0.04 (0.02) Vf = 0.2% MOR* (MPa) m ^ \o cn i n g. 5.63 (0.50) 1 1 • • 6.92 (0.52) 5.04 (0.44) Fiber and Mix Id. PPF-1 PPF-2 PPF-3 PPF-4 PPM-1 PPM-2 NLM-1 < < c _o C3 • ? CJ -a 1 •a c co o « 1 0 -5 M Vi oo 01 % x £ o a oi fc, + * i oo c X E OS x> '3 u Q CO Vf = 1.6%(PP) + 0.25%(Steel) * a~ 1 1 1 1 • • 59.7 Vf = 1.6%(PP) + 0.25%(Steel) RS** (MPa) 1 1 1 1 1 l 3.34 (0.55) Vf = 1.6%(PP) + 0.25%(Steel) MOR* (MPa) • • 1 1 1 1 • ' 5.59 (0.63) Vf = 3.0% * * . S I—1 ^ a~ 44.3 • • • • • • Vf = 3.0% at 2.47 (0.08) • • 1 1 • • Vf = 3.0% MOR* (MPa) 5.58 (0.24) 1 1 Fiber Volume, Vf Vf = 1.6% * a~ 31.2 71.2 • • • • • • Fiber Volume, Vf Vf = 1.6% RS** (MPa) 1.83 (0.19) 3.88 (0.48) 1 1 Fiber Volume, Vf Vf = 1.6% MOR* (MPa) 5.87 (0.21)**** 5.45 (0.14) ' • • • • • Vf=1.0% * * ^ a~ • • 53.6 • • • • • • Vf=1.0% RS** (MPa) 1 1 3.19 (0.65) i 1 1 l 1 1 Vf=1.0% MOR* (MPa) 1 1 5.96 (0.19) i 1 1 1 1 1 Vf = 0.76% # a~ 1 1 1 1 91.0 57.2 1 1 Vf = 0.76% RS** (MPa) 1 1 1 1 5.74 (0.43) 3.27 (0.20) Vf = 0.76% MOR* (MPa) 1 1 • • 6.31 (0.22) 5.72 (0.34) Fiber and Mix Id. PP-STR PVA-1 ST-HKD ST-CRl HBRD-1 earn oi 0 3 earn O t> X l x 0) 00 CJ 3 X ! oS <£ X of rage of 0 0 age rage of II Aver OO Aver < OS c o ' £ •? u -a •o •o c C3 o ja X 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 FRC 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. • An evaluation of the Residual Strength Test Method (RSTM) for the measurement of post-cracking 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 post-crack 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 cement-based 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, FRC 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. The post-peak instability in these composites, however, is minimal. 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. Displacements were measured by two LVDTs (Linear Variable Differential Transformers). Loads and displacements were digitally recorded using a 16-bit data acquisition system operating at a frequency of 10 Hz. 65 mm j Debonded interface sVr'j (iii) Pull-out Figure 5.1.1: The principle of a single fiber pullout 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. G0 represents the fiber pullout stress applied at the emerging fiber end. The 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, xm, at any radial distance, r, can be expressed as: —T a for r > b 5.1 r b-r •a where xa represents the shear stress at the interface, i.e., at r=a. The above equation satisfies the 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. a0 matrix displacement Wb profile Tm=0 Z=0 z=L -> z Figure 5.1.2: Fiber completely bonded over its embedded length 105 If wm represents the displacement of matrix in the axial direction, the corresponding shear stress in the matrix (ignoring the radial displacements) is given by: E dw f = - - 5.2 m 2 (1+0 dr Combining Equations 5.1 and 5.2, and integrating: b wb a C(b-r) , Em t a- -dr = s dwn -a)} r 2(l + vm)J T IF-a) 5.3 where wa and Wb are the axial displacement in the matrix at r=a and r=b, respectively. Solving the above equation, the interfacial shear stress, Ta, can be written as: T„ = - E m ( W b ~ W a ) 2(l + vm)a (b-a) log \ a J -1 5.4 Combining 5.1 and 5.4, the shear stress in the matrix, zm, at any radial distance, r, is given by: Em(wb-wa)(b-r) 2(l + um)r b\og -(b-a) 5.5 Combining Equations 5.2 and 5.5 and integrating, the axial displacement in the matrix, vvm, can be written as: w = w +-m a wb-wa Mog (b\ \ a j -(b-a) felog \ a J + a-r 5.6 106 Ignoring the Poisson's effect (i.e., treating the problem as one-dimensional), the axial stress in the matrix, am, (in the z direction) is given by: a =£, dw„ m m dz 5.7 Combining Equations 5.6 and 5.7, the axial stress in the matrix at any radial distance, r, is given by: CT"=i7CT'+ Mog -(b-a) b log + a-r 5.8 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"0, and the internal stress distribution at any section can be written as: b ° f + — \ < J m r d r = °0 5.9 Combining Equations 5.8 and 5.9, the axial stress in the matrix, am=b is obtained as: y + a-yqa m 5.10 where, 107 a = a 1 a2 Mog -(b-a) ^ l o g f ^ ^ - a 2 ) \ a J + a(b2-a2) (b3-a3) Now, considering the equilibrium between the axial stress in the fiber and the interfacial shear stress: d„ 5.26 Ef 1 f i Substituting for Aa in the above equation: a E~ a2+b2 b —a • + -vfaf + ^[l-vf) ?L Ef / J E„ a2+b2 5.27 + Thus, when tensile stress in the fiber is 07, the resultant contact pressure, ocp, at the interface is given by: a =<7 + (0(7 , cp c / where co = 5.28 a2+b2 0 —a Since Takaku and Arridge [54] developed their model for stiff fibers, the constant co was defined differently in their model. Substituting for ocp in the equation of stress transfer (Equation 5.23): da f _ 2/f dz a [ac+(oof] 5.29 114 Rearranging: 1 +—COG, =• dz 2H a 5.30 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: ' co CO 2a>fi(L-ld) -lajiz e a. e a 5.31 The fiber axial stress, 07, along the bonded region (z z> L-ld) is obtained as: 115 T , = aco od +— e "I 2n(L-ld) -2cofiz dz 5.35 116 Solving: uPd = Ef(y + a- yna + an) cosh[/3(L-/,)l-l (a-yna + an-y) ^ + r ( L - / . ) n Bsmh[j3(L-ld)] d ) a Efco 2Ef\ico °c_ CO -2/i o.fric -1 ° d + c CO 5.39 117 5.1.2.6 Catastrophic Debonding The interfacial debonding process becomes catastrophic when the fiber pullout stress, a0 begins to drop with an increase in debond length, ld. Thus, the condition for catastrophic debonding can be written as: dld The above condition will be satisfied when the difference between the increase in the frictional component of the pullout stress (i.e., o"0/n-c) and the decrease in the adhesional component of the pullout stress (i.e., o0,bond) with increase in ld is equal to or less than zero. That is: dao,fric < dGo,bond dld dld 5.41 Let the debond length at which the debonding process turns catastrophic be represented by ldxat. 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 o0 in Equation 5.40, we obtain the slope of liL Pdl ~ Pdl 2copi , a _ e a 5.55 To obtain Wp from the experimental pullout curve, the relationship between the total pullout displacement and the rigid body displacement of the fiber must be known. As a first approximation, the rigid body displacement of the fiber, pd is taken equal to A-Apeak, where, A is the total pullout displacement, and Apeak is the total pullout displacement corresponding to the peak pullout load. With Wp and ac known, the coefficient of friction, fi, can be calculated from the above non-linear equation. This equation depicts the dependence of coefficient of friction, fi, on the pullout displacement, pd. Adhesional bond strength, xs: For very small fiber lengths, the peak pullout stress is governed by the Equation 5.21. Differentiating Equation 5.21 with respect to fiber embedded length we obtain: dad _-2Ts(y + a-yncc + ari) \[(a-yna + ar])cosh(RL) + y]pcosh(i3L) dL af3 \ [(a-yr/a + arj)cosh(/?L)-i-y]2 5.56 P(a - yqa + ar]) sinh 2 (/3L) [(a - yna + ar]) cosh( f5L) + y] 2 At L = 0 the above equation simplifies to: 2T„ dL JL-O a 5.57 Rearranging the above equation we obtain: a dod dL 5.58 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, xs, can be 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, oc, ji and ts, is required. A procedure for calibrating interfacial properties was described in the previous 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 doobond Idld (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 TS = -1 MPa. 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, TS. 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 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, rs. In the figure it can be seen that pullout displacement at the peak pullout load increases with increase in adhesional bond strength, xs. 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 xs = -5 MPa. In the same figure, components of pullout load, (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,cat (Section 5.1.2.6), since the debonding process turns catastrophic upon further debonding. 120 z i •2 80 3 JD ~5 a. 40 Peak pullout load Pullout load at initial debonding * * *' £ / = 2 1 0 G P a \ . -' V-'" o\=-15 MPa fl, =0.1 10 12 0 2 4 6 8 Adhesional bond strength, T , (MPa) Figure 7.1.1b: Influence of adhesional bond strength on the pullout load at initial debonding and the peak pullout load (£}=210 GPa) | 1 0.02 3 ^ o.oi X X X £7=210 GPa crc=-15MPa 2 4 6 8 10 Adhesional bond strength, T , (MPa) Figure 7.1.1c: Influence of adhesional bond strength on displacement at the peak pullout load (£,=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, ts. 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, ts. However, at complete debonding, the magnitude of pullout load is independent of adhesional bond strength, Ts. • 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 ts = -5.0 MPa and -10 MPa; on the other hand, atxs = -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, ldiCaU decreases with increase in the adhesional bond strength, T .^ 144 160 120 - 80 3 O 1 £ , = 2 1 0 GPa t,=-5MPa =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 c=-15MPa Pullout load Jii =0.25 Frictional component^,/.'' V \ Adhesional component 0 5 10 15 20 25 Debond length,I d /Axial position, L-z (mm) 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, fxt. Also, for 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 (£ y=210GPa) 7.1.3.2 Fiber Elastic Modulus, Ef = 3.5 GPa Parametric studies are carried out by varying the initial coefficient of friction, Lit. The three chosen values of Lij are: 0.05, 0.25 and 0.50, and the corresponding evolution laws for coefficient of friction selected are: LL = 0.02e~° 3 p d +0.03, LL = 0.22e**v* +0.03, and Ll - 0.47 e~°3pd + 0.03, respectively. Assumed values of the other interfacial properties were: xs = -1.0 MPa and o~c = -15.0 MPa. Mechanical properties of the fiber are assumed as: E/=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 (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 JJ,=0.50 / / / I ,-=0.25 ' 11 //J,=0.05 [ £ ,=3 .5 GPa T , = - l MPa =3.5 GPa) 20 25 0 5 10 15 Axial position, z (mm) Figure 7.1.6g: Interfacial shear stress distribution at 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, Lit, axial 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, Lit, the interfacial shear stress is maximum at 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, LLV 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 open-loop 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 composites. Now, what constitute optimal environmental conditions remains to be investigated. Detailed investigations are recommended to explore this aspect. 172 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 \ #define TT 200 #define N 3 main () { FILE *input_file; FILE *STRS_DISTR; FILE *STRS_DISTR2; FILE *PO_DISP; FILE *P0_DISP2; FILE *DEBLEN_DEBSTRSS; FILE *DEBLEN_DEBSTRSS2; FILE *PO_STRESSvsDL; FILE *P0_STRESSvsDL2; FILE *TEST; FILE *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, cnstntl , 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_catal , 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"); STRS_DISTR2 = fopen("stress2", "w"); PO_DISP = fopen("disp","w"); */ P0_DISP2 = fopen("disp2", "w"); TESTIN = fopen("in", "w"); /* stress distr ibution in fiber */ / * pullout stress and fiber displacement 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 PO Strs Fib Disp Progr PO Ld Bnd PO Ld Fric PO Ld\n"); fprintf(P0_DISP2, "Deb Len Dbnd Strs PO Strs Fib Disp2 Progr PO Ld Bnd PO Ld Fric PO Ld\n"); fprintf(STRS_DISTR ( "Dbnd Strs Deb Lt Deb Pos Axl Pos Axl Strs Shr Strs Axl Lod Bnd Ld Fric Ld \n"); 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+alpha-row*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[ 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]); /* */ / * 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*(L-pd)/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 axial stress distribution & interfac ia l shear stress distribution over the debonded zone of the par t ia 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 #define TT 200 main () { FILE *input_file; FILE *STRS_DISTR; FILE *PO_DISP; FILE *DEBLEN_DEBSTRSS; FILE *PO_STRESSvsDL; FILE *TEST; FILE *TESTIN; FILE *AREA; int Debnd_stress, L, Counter, zz, d l , ITER, incr, steps, zx, zy, zzyy, zxpo, rad, bb; double a, b, Poss_m, Poss_f, Em, Ef, ISS, Alpha, z, LZ, DS_Num, DS_Numerator, DS_Denominator, Initial_debonding_stress, pullout_strs, 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, ubl, 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, thetal, theta2, theta, betal, beta, DSC1, DSC2, DSC3, DSC[200], matrix_stress_b, matrix_stress, radl , 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]; input_file = fopen("inptl","r"); STRS_DISTR = fopen("stress","w"); / * stress d i s tr ibut ion' in 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 Debond Stress \n"); */ / * fprintf(AREA, "pdist pd area \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_ratio-row*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 is 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 is e las t ica l ly 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 axial stress distribution in fiber and the interfac ia l shear stress distribution 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 Fric 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 * z z ; LZ = L-z ; debond_stress_current[dl] = Initial_debonding_stress; /* calculate axial stress distribution in 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 interfac ia l shear stress distribution */ 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, pullout_strs, 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 = pullout_strs/epsilon - (row+modulus_ratio-row*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_b-modulus_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", pullout_strs, 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(ITER0) 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 %7.21f \n", deb_len, debond_stress_current[dl], stress_pout[dl]); / * fprintf(TEST, "%50.21f %50.21f %10.21f \n", SP2, SP3, stress_pout[dl]); */ /* fprintf(TEST, "A3[dl]=%20.21f B[dl]=%20.21f \n", A3[dl], B[dl]); */ / * Calculate fiber displacement for each part ia 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 is interfac ia l f r i c t iona 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[dl], d l , debond_stress_current[dl] US1, US2, -US3, US4, USSS); */ } else /* That is when interface is not in contact in 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) /* that i s , when interface is in contact in the debonded region */ { 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 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, "%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 is not in 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 distribution in fiber and interface when fiber par t ia l ly debonded */ /* calculate stress distribution in 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 axial stress distribution in 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_len-z)))/(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 interfac ia l shear stress distribution */ 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_len-z)))/(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*b-a*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, Fiber_axial_stress[zy], 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"); */ A 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*b-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 axial stress distribution in fiber */ FAS1 = (a*a*Ef*pullout_strs)/(a*a*Ef+(b*b-a*a)*Em); • / * fprintf(STRS_DISTR, "but i moved s l ight ly 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*(L-deb_len+0.00000001))) ; / * fprintf(STRS_DISTR, "but i moved s l ight ly 2\n"); */ 203 FAS3 = (exp(-Alpha*(L-deb_len-z)) - exp(Alpha*(L-deb_len-z)))/(exp(Alpha*(L-deb_len+0.00000001)) - exp(-Alpha*(L-deb_len+0.00000001))); /* fprintf(STRS_DISTR, "but i moved s l ight ly \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 ight ly 3\n"); */ / * calculate interfac ia l shear stress distribution */ 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_len-z)))/(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 distribution in the debonded region */ fprintf(TEST, "separation=%7.21f \n", separation); for(zx=0; zx<=dl; ++zx) { if(separation==0) / * that is when the interface is in contact in the debonded region */ { z=0.5*zx; deb_pos=L-deb_len+z; mlz=ml*z; m2z=m2*z; /* axial stress distribution in 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 / * f p r i n t f ( S T R S _ D I S T R , "deb comp = %7.21f f r i c corapo = %7.21f \ n " , deb_component , f r i c _ c o m p o n e n t ) ; * / / * 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 debonded * / I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x ] = - ( a / 2 ) * ( ( - A 3 [ d l ] / A 2 ) * m 2 * e x p ( m 2 z ) + B [ d l ] * ( m l * e x p ( m l z ) - m2*exp(m2z)) + d e b o n d _ s t r e s s _ c u r r e n t [ d l ] * m 2 * e x p ( m 2 z ) ) ; / • 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 o f f i b e r * / s h r n k _ s t r s s _ n e t [ z x ] = s t r e s s _ s h r n k - ( I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x ] / m u -s t r e s s _ s h r n k ) ; 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 %7.21f \ n " , C o u n t e r , s t r e s s _ p o u t [ d l ] , d e b _ l e n , d e b _ p o s , z , F i b e r _ a x i a l _ s t r e s s [ z x ] , I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x ] , F i b e r _ a x i a l _ l o a d [ z x ] , F i b e r _ a x l _ l o a d _ b o n d _ c o m p o [ z x ] , F i b e r _ a x l _ l o a d _ f r i c t i o n a l _ c o m p o [ z x ] , s h r n k _ s t r s s _ r e d u c t i o n [ z x ] ) ; } e l s e / * t h a t i s when t h e i n t e r f a c e s e p a r a t e d i n t h e debonded r e g i o n * / { z = 0 . 5 * z x ; d e b _ p o s = L - d e b _ l e n + z ; m l z = m l * z ; m2z=m2*z; F i b e r _ a x i a l _ s t r e s s [ z x ] = d e b o n d _ s t r e s s _ c u r r e n t [ d l ] ; F i b e r _ a x i a l _ l o a d [ z x ] = F i b e r _ a x i a l _ s t r e s s [ z x ] * F i b e r _ a r e a ; F i b e r _ a x l _ l o a d _ b o n d _ c o m p o [ z x ] = T o t a l _ b o n d _ l o a d ; F i b e r _ a x l _ l o a d _ f r i c t i o n a l _ c o m p o [ z x ] = 0 . 0 ; I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x ] = 0 ; s h r n k _ s t r s s _ n e t [ z x ] = s t r e s s _ s h r n k - ( I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x ] / m u -s t r e s s _ s h r n k ) ; } 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 %7.21f \ n " , C o u n t e r , s t r e s s _ p o u t [ d l ] , d e b _ l e n , d e b _ p o s , z , F i b e r _ a x i a l _ s t r e s s [ z x ] , I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x ] , F i b e r _ a x i a l _ l o a d [ z x ] , F i b e r _ a x l _ l o a d _ b o n d _ c o m p o [ z x ] , F i b e r _ a x l _ l o a d _ f r i c t i o n a l _ c o m p o [ z x ] , s h r n k _ s t r s s _ . n e t [ z x ] ) ; / * 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 " , C o u n t e r , s t r e s s _ p o u t [ d l ] , d e b _ l e n , . d e b _ p o s , z , F i b e r _ a x i a l _ s t r e s s [ z x ] , I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x ] , F i b e r _ a x i a l _ l o a d [ z x ] , F i b e r _ a x l _ l o a d _ b o n d _ c o m p o [ z x ] , F i b e r _ a x l _ l o a d _ f r i c t i o n a l _ c o m p o [ z x ] ) ; * / } 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*(ml-m2)*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] is part ia 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 %7.21f %7.21f \n" emb_len, debond_stress_current3, stress_pout3[dl]); / * 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; fprintf (AREA, "%d %7.21f \n", d l , areal [ ITER].) ; /* Calculate stress distribution in 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; /* axial stress distribution in 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 / * 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 debonded I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x p o ] = - ( a / 2 ) * ( ( - A 3 [ d l ] / A 2 ) * m 2 * e x p ( m 2 z ) + B [ d l ] * ( m l * e x p ( m l z ) - m 2 * e x p ( m 2 z ) ) ) ; 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 " , C o u n t e r , s t r e s s _ p o u t 3 [ d l ] , p o _ l e n , e m b _ l e n , z , F i b e r _ a x i a l _ s t r e s s [ z x p o ] , I n t e r f a c i a l _ s h e a r _ s t r e s s [ z x p o ] , F i b e r _ a x i a l _ l o a d [ z x p o ] , F i b e r _ a x l _ l o a d _ b o n d _ c o m p o [ z x p o ] , F i b e r _ a x l _ l o a d _ f r i c t i o n a l _ c o m p o [ z x p o ] ) } ITER=ITER+1; 208 Appendix C 209 / * Computer Program to Predict Pullout Load versus Displacement Response Using the Progressive Debonding Model */ #include ttinclude #include ttinclude #include #include #define TT 200 #define N 3 main () { FILE *input_file; 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, cnstntl , 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, muinit ial , 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]; input_file = fopen("inpmu","r"); STRS_DISTR = fopen("stress","w"); /* stress distr ibution in 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 %lf %d %lf %lf \n", a, b, Poss_f, Poss_m, Em, Ef, L, ABS, mu); fprintf(DEBLEN_DEBSTRSS, "Debond Length 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 Fric PO Ld\n"); fprintf(P0_DISP2, "Deb Len Dbnd Strs PO Strs Fib Disp2 Progr PO I Bnd PO Ld Fric PO Ld\n"); fprintf(STRS_DISTR, "Dbnd Strs Deb Lt Deb Pos Axl Pos Axl Strs Shr Strs Axl Lod Bnd Ld Fric 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+alpha-row*epsilon*alpha+alpha*epsilon)*sinh(beta*L)/((a*beta)*((alpha-row*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+alpha-row*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 %7.5'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[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]); deb_len=0.0; deb_pos=0.0; dl=deb_len; Initial_debonding_stress=-2*ABS*(row+alpha-row*epsilon*alpha+alpha*epsilon)*sinh(beta*L)/((a*beta)*((alpha-row*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+alpha-row*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[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 par t ia 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