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Deformation behaviour of a model aluminum-brass two phase material Bathla, Rajat 2003

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DEFORMATION BEHAVIOUR OF A MODEL ALUMINUM-BRASS TWO PHASE MATERIAL by R A JA T B A T H L A B.E., University Of Roorkee, Roorkee, India, 1998  A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR THE D E G R E E OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (DEPARTMENT OF M E T A L S A N D M A T E R I A L S ENGINEERING)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A N O V E M B E R 2003  © Raj at Bathla, 2003  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 The University of British Columbia Vancouver, Canada  DE-6 (2/88)  11  ABSTRACT The present work is primarily concerned with investigating the relationship between the mechanical properties and the volume fraction and the geometrical distribution of a second phase a two-phase model material where plasticity occurs in both phases. The deformation behaviour of two-phase materials with elastic reinforcements has been widely studied in the literature. However, relatively few studies have examined two-phase materials where both phases have elastic-plastic properties and deform plastically under an applied far field strain. The mechanical response for such a two-phase material has been studied experimentally and with a finite element model for an aluminum/brass model composite. In this model composite, AA6061 age hardening alloy is used as a matrix phase and C3600brass fibres are used as reinforcement phase. Samples were made by extrusion at room temperature followed by a variety of heat treatments. The heat treatments were carefully controlled to vary the relative properties of the fibres and matrix. Fibre volume fractions of 20 % with square or triangular arrangements and 35 % with a square arrangement were tested using plane strain deformation experiments where the fibre axis was transverse to the loading axis. The results indicate that strain partitioning between the two phases is very sensitive to the yield stress and work hardening characteristics of the different phases. The geometric arrangement of the second phase had little influence on the macroscopic stressstrain response but significantly affected the pattern of local deformation in the matrix.  iii  TABLE OF CONTENTS ABSTRACT  ii  T A B L E OF C O N T E N T S  iii  LIST OF FIGURES  iv  LIST OF T A B L E S  x  ACKNOWLEDGEMENT  xi  CHAPTER: 1 INTRODUCTION  1  CHAPTER: 2 L I T E R A T U R E R E V I E W  4  2.1 Introduction: 2.1.1 Characterization of two-phase materials 2.1.2 Two-phase material with reinforcement spacing >10 um 2.1.2.1 Low deformation regime 2.1.2.1 High deformation regime (s>5)  4 4 6 8 13  2.2 Deformation theories 16 2.2.1 Upper and lower bound analysis 16 2.2.2 Eshelby's models 17 2.2.3 Shear lag model for short fibres 19 2.2.4 Extension of Eshelby model to co-deforming materials (Weng's Secant method). 19 2.2.5 Finite Element Method (FEM) calculations 21 CHAPTER: 3 SCOPE A N D OBJECTIVES OF PRESENT W O R K  23  CHAPTERS EXPERIMENTAL METHODOLOGY  24  4.1 Sample fabrication  24  4.2 Constituent properties  29  4.3 Fiducial grid preparation  29  4.5 Strain calculation from the grid deformation  30  4.6 Mechanical testing (Channel die compression test)  32  4.7 Finite element model 4.7.1 Material properties 4.7.1 Boundary conditions  ;  34 36 36  CHAPTER: 5 RESULTS - P L A N E STRAIN COMPRESSION  38  5.1 Constituent stress-strain response  38  5.1.1 Curve extrapolation  40  5.2 Channel die compression tests  41  5.2.1 20% triangular samples: 5.2.1.1 Stress-strain response and macroscopic shape change of samples: 5.2.1.2 Localized fibre strain and F E M strain distribution in the sample: 5.2.1.3 Stresses parallel to the loading direction (a 2). 5.2.1.4 Stresses perpendicular to the loading direction (an) 5.2.2 20% square samples: 5.2.2.1 Stress-strain response and macroscopic shape change of samples: 5.2.2.2 Localized fibre strain and F E M strain distribution in the sample: 5.2.2.3 Stresses along the loading direction (022) 5.2.2.4 Stresses perpendicular to the loading direction (an) 2  41 41 44 •• 48 49 52 52 55 60 62  5.2.3 35% square samples: 5.2.3.1 Stress-strain response and macroscopic shape change of samples: 5.2.3.2 Stresses along (0*22) and perpendicular (an) to the loading direction  65 65 66  CHAPTER: 6 DISCUSSION OF RESULTS  68  6.1 Mechanical response 6.1.1 Effect of geometry 6.1.2 Strain distribution in the matrix 6.1.2.1 Triangular geometry 6.1.2.2 Square geometry 6.1.3 Strain partitioning in the fibres 6.1.3.1 Matrix having lower yield strength than the fibres 6.1.3.2 Matrix having higher yield strength than the fibres 6.1.4 Effect of volume fraction on the stress-strain response (FEM calculations)  68 68 70 70 71 71 71 73 74  6.2 Stresses inside the fibres 6.2.1 Stresses perpendicular to loading direction (an) 6.2.2 Effect of volume fraction on the stresses (G\1) inside the fibres  76 77 79  6.3 F E M model results on virtual composite with variable fibre/matrix properties 6.3.1 Sample set 1: Matrix and fibres have same yield strength  81 82  6.3.2 Sample set 2: Fibres have 1.25 times higher yield strength than matrix  84  CHAPTER: 7 CONCLUSIONS A N D FUTURE W O R K  86  7.1 Conclusions  86  7.2 Future work  87  REFERENCES.  89  vi  LIST OF FIGURES Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7  Figure 2.8  Effect of geometrical arrangement of fibres on the strain distribution inside the matrix. Results of Poole et al [9] for non-deforming fibres. Percentage reduction of embedded reinforcements as a percentage reduction of sample.Unckel, H. [15]. Strain hardening curve of inclusion(A) and surrounding matrix (B). Unckel [15]. 10 Influence of volume fractions of the phases on the recrystallization temperature of (a) a phase (b) P phase. Clarebrough and Perger [17] 11 The strain distribution between phases in two different dual-phase steels Balliger et.al. [18] 12 Role of geometric arrangement of fibres on the overall stress-strain response of the composite. Poole et al. [9] 13 F E M modeling results showing localized shear in a pearlite with ferrite/cementite plates orientated (a) parallel and (b) perpendicular to the wire axis, and (c) atomic force micrograph illustrating localized shear in a pearlitic steel. Zelin [22] 15 Eshelby's cutting and welding exercise and how the constrained strain (e ) is related to transformation strain (s ) by Eshelby tensor (S). Clyne et al. [6] 18 Secant modulus, E ' , when phase was deforming plastically and the modulus (E) when the phase was elastic. 21 Cross-section of Drill geometry on 24mm diameter AA6061 bar (a) 20% square fibre distribution where x=1.58mm, y=3.15mm and z=1.58mm, (b) 20% triangular fibre distribution where x=1.58mm, y=3.15mm and z=1.58rnm and (c) 35% square distribution of fibres x=1.58mm, y=2.4mm and z=0.8mm. 26 c  Figure 2.9 Figure 4.1  Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5  T  Sample fabrication steps  27  Extruded sample and die  27  Extracted Brass fibres from the extruded Sample  28  Sample geometry (a) 20% square fibre distribution, (b) 20% triangular 28  Figure 4.6  fibre distribution and (c) 35% square distribution of fibres.  Figure 4.7  Back scatter S E M image of gold grid on the sample surface.  30  Figure 4.8  Deformation of square into a parallelogram.  31  Schematic of channel die compression test  33  F E M mesh for composite with 20% square fibre arrangement  35  F E M mesh for composite with 20% triangular fibre arrangement  35  F E M mesh for composite with 35% square fibre arrangement  36  Figure 4.8(a) Figure 4.8(b) Figure 4.8(c)  vii Figure 5.1  Figure 5.2 Figure 5.3  Stress strain response of the constituent phases (a) Fibres have higher yield stress than the matrix (b) Matrix has higher yield stress than the fibres Stress- strain response of 20% fibre volume fraction composite with triangular fibre arrangement Sample picture of 20% triangular distribution and matrix having lower rj (a) Before deformation (b) after Stepl, s=10.8% (c) after Step2, 8=20.4% (d) after Step3, e=29.3% Sample picture of 20% triangular distribution and matrix having higher a (a) Before deformation (b) after Step3, E=24% 20 triangular composite and matrix having lower a (a) Experimental vs. F E M calculated fibre strain (b) von Mises contour plots of strain distribution in the sample after imposed true strain of 0.25. 20% triangular composite and matrix having higher a (a) von Mises contour plots of strain distribution in the sample after imposed true strain of 0.25 (b) Experimental vs. F E M calculated fibre strain. Stress distribution along the loading direction (022) within the composite sample (matrix having lower o- ) containing 20% fibre in triangular distribution after the imposed deformation of 0.15 Stress distribution along the loading direction (022) within the composite sample (matrix having higher a ) containing 20% fibre in triangular distribution after the imposed deformation of 0.15. Stress distribution perpendicular to loading direction (an) within the composite sample (Matrix having lower a ) containing 20% fibre in triangular distribution after the imposed deformation of (a) 0.15 and b) 0.25  39 42  ys  Figure 5.4  y s  F i g u r e 5.5  F i g u r e 5.6  Figure 5.7  43  44  y s  46  y s  46  ys  Figure 5.8  48  ys  Figure 5.9  49  ys  Figure 5.10  Stress distribution perpendicular to loading direction (an) within the composite sample (Matrix having higher o ) containing 20% fibre in triangular distribution after the imposed deformation of (a) 0.15 and (b) 0.25 5.11 Stress- strain response of 20% fibre volume fraction composite with square fibre arrangement compared with F E M calculated curves. 5.12 Sample picture of 20% square distribution and matrix having lower a (a) Before deformation (b) after Stepl, s=14.5% (c) after Step2, 8=26.3% 5.13 Sample picture of 20% square distribution and matrix having higher CT (a) Before deformation (b) after Step3, 8=24% 5.14(a) 20% square composite and matric having lower a , Experimental vs. F E M calculated fibre strain  50  ys  Figure Figure  Figure  53  y s  ys  Figure  51  54 54  ys  Figure 5.14(b) 20% square composite and matric having lower a  (b) von Mises contour plots of strain distribution in the sample after imposed true and 5.14(c) strain of 0.25 (c) deformation pattern from contour plot. Figure 5.15(a) 20% square composite and matrix having higher a , Experimental vs. F E M calculated fibre strain  56  y s  57  ys  58  viii Figure 5.15(b) 20% square composite and matrix having higher r j (b) von Mises contour plots of strain distribution in the sample after imposed true and 5.15 (c) strain of 0.25 (c) deformation pattern from contour plot. Figure 5.16 Stress distribution along the loading direction (rj 2) within the composite sample (matrix having lower a ) containing 20% fibre in square distribution after the imposed deformation of 0.15 Figure 5.17 Stress distribution along the loading direction (022) within the composite sample (matrix having higher a ) containing 20% fibre in square distribution after the imposed deformation of 0.15 Figure 5.18 Stress distribution perpendicular to the loading direction (an) within the composite sample (Matrix lowera ) containing 20% fibre in square distribution after the imposed deformation of (a) 0.15 and b) 0.25 Figure 5.19 Stress distribution perpendicular to the loading direction ( a n ) within the composite sample (Matrix higher a ) containing 20% fibre in square distribution after the imposed deformation of (a) 0.15 and b) 0.25 Figure 5.20 Stress- strain response of 35% fibre volume fraction composite, fibres in square arrangement. Figure 5.21 Sample picture of 35% square distribution and matrix having lower rj (a) Before deformation (b) after Stepl, s=8.5% (c) after Step3, e=30.4% Figure 5.22 Stress distribution along the loading direction (0*22) within the composite sample (matrix having lower a ) containing 35% fibre in square distribution after the imposed deformation of 0.15 Figure 5.23 Stress distribution perpendicular to the loading direction ( a n ) within y s  59  2  ys  61  ys  61  ys  63  y s  64 65  ys  66  y s  the composite sample (matrix having lower r j ) containing 35% fibre in square distribution after the imposed deformation of 0.15 Effect of geometrical arrangement of fibres on the mechanical response of the composite, Matrix has yield strength than the fibres Effect of geometrical arrangement of fibres on the mechanical response of the composite, Matrix has higher yield strength than the fibres. Effect of geometrical arrangement of fibres on the accumulated strain in the fibres for the composite having matrix of lower a . Effect of geometrical arrangement of fibres on the accumulated strain in the fibres for the composite having matrix of higher a Effect of volume fraction of fibres on the mechanical response of the composite (a) matrix having lower yield strength than the fibres and (b) matrix having higher yield strength than the fibres. Percentage of debonded fibres with imposed deformation, in 20% volume fraction composite samples.  67  ys  Figure 6.1 Figure 6.2  Figure 6.5 Figure 6.6 Figure 6.7  Figure 6.8  y s  y s  67 69  69 72 73  75 76  IX  Figure 6.9  F E M calculated stresses inside the fibres perpendicular to the loading direction (an ) for 20% volume fraction composite having (a)matrix of lower a (b) matrix of highera . Effect of volume fraction on the stresses perpendicular to direction (an) inside the fibres (a) composite having matrix of lower yield strength than the fibres and (b) composite having matrix of higher yield strength than the fibres. Fibre/matrix property set where fibres and matrix have the same yield strength. Keeping the fibre work hardening (0 ) the same and varying the work hardening of matrix from 0.150 to 0.650. (a) Fibre strain vs. farfield strain of four samples (b) strain ratio (fibre/farfield) vs. work hardening ratio( matrix/fibre) Fibre/matrix property set where fibres 1.25 times yield strength of the matrix. Keeping the fibre work hardening (0 ) the same and varying the work hardening of matrix from 0.150 to 0.650. (a) Fibre strain vs. farfield strain of four samples (b) Farfield strain required for fibres to start deforming vs. work hardening ratio. y s  Figure 6.10  Figure 6.11  Figure 6.12 Figure 6.13  Figure 6.14  ?8  ys  80  82 83  84 85  X  LIST OF TABLES Table 4.1  Chemical composition for Brass C-3600  24  Table 4.2  Chemical composition for AA6061  24  Table 4.3  Composition used for removing oxide from brass fibre  25  Table 4.4  Aluminum dissolution recipe  29  xi  ACKNOWLEDGEMENT  First of all, I would like to thank my supervisor Dr. Warren J. Poole for his constant guidance and support throughout this work. I am grateful to him for being available whenever I needed his help. I would also like to thank N S E R C Canada for fully funding this project. M y sincere thanks also goes to Dr. Partha Ganguly for helping me in the initial stages of this project and making me literate to use A B A Q U S . I also wish to express my gratitude to Mr. Mohammad Mazinani for the fruitful technical discussions related to my work. I am thankful to Ms. Mary Mager for her help in the experimental work involving S E M . Special thanks goes to the machine shop, especially Mr. Carl N g and Mr. Ross Macleod for fabricating the samples, required in this work. Finally, this work would not have been complete without the help of my family and close friends.  1  CHAPTER: 1 INTRODUCTION M a n y technologically important alloys are two-phase materials where both phases are capable o f substantial plastic deformation. Examples o f engineering materials where plastic deformation occurs i n two ductile phases include a/fi brasses, Duplex steels and C u N b alloys. W h e n these materials which are comprised o f two phases with  different  mechanical properties are subjected to strain, the phases react differently to the applied strain. This results i n inhomogeneous stress and strain distributions in the two-phase material and affects the overall strength and recrystallization behaviour o f the two-phase material. It is very important to develop a sound understanding o f the deformation behaviour o f these types o f alloy systems, which may help to improve the properties o f existing materials or develop materials with improved properties. Very little experimental work is available i n the literature on the deformation behaviour o f two-phase material where the stress-strain curves o f individual phases are accurately known. The majority o f work has been focussed on finding the mechanical response o f individual phases using analytical equations and measurements o f strain partitioning in the reinforcing phase using indirect methods e.g. hardness measurements, recrystallization behaviour, residual stress measurements, metallography etc. Thus, the major objective o f this work is to develop new fundamental knowledge on deformation o f twophase material where plasticity occurs i n both phases. In the present work, stress-strain  curves o f individual phases were  obtained  experimentally and measurements o f strain partitioning were done b y a direct method i.e. the change in shape o f a reference grid on the composite sample after deformation. The local  2 distribution of strain in a model composite was obtained experimentally as well as by Finite Element method calculations as a function of the geometric arrangement, volume fraction and mechanical behaviour of the constituent phases. The information obtained from experiments is utilized to understand the stress and strain partitioning in the phases and may later be extended to study the recrystallization behaviour and fracture behaviour of twophase materials. The approach taken in this study is as follows: •  Choosing the appropriate material for constituent phases in sample fabrication and standardising a processing route to make the composite materials.  •  Conducting channel die compression tests on fabricated samples with 20% and 35% volume fraction of fibres in different geometrical arrangements (i.e. square and triangular).  •  Experimentally measuring of strain partitioning in reinforcing phase by measuring the change in shape of the reference grid before and after deformation of the sample.  •  Simultaneously, conducting finite element method simulations for geometrically similar samples to those used in the experiments. This thesis is organized in the following manner. Chapter 2 reviews some of the  available literature on the deformation of two-phase materials where plasticity occurs in both phases. In addition, the analytical and computational method available for the determination of mechanical response of these composite systems is reviewed. Chapter 3 describes the scope and objectives for this work. Chapter 4 describes in detail the experimental and numerical techniques used in this work. Chapter 5 presents results from experiments and F E M simulations and Chapter 6 is a discussion on the obtained results. Finally, Chapter 7 is  3  a summary of the important conclusions that can be drawn from the present work and suggestions for future work.  4  CHAPTER: 2 LITERATURE REVIEW This chapter is organized in the following manner. Section 2.1 gives a brief introduction and reviews the work on two-phase alloy systems. Deformation theories for two-phase materials are reviewed in Section 2.2.  2.11ntroduction Two-phase materials can be defined as those materials having two constituents that can be physically or visibly distinguished. The major volume fraction constituent in these materials is generally called the matrix and the minor volume fraction constituent is called the reinforcements. These reinforcements are also referred to as inhomogeneities because their properties such as stiffness and coefficient of thermal expansion differ from that of the matrix.  2.1.1  Characterization of two-phase materials It is important to classify the scale of two-phase materials so that suitable  strengthening mechanisms that can be used to describe the stress-strain behaviour of the alloy. Microstructurally, these materials can be characterized by the spacing of reinforcements into three broad categories: i)  Reinforcement spacing of 10-100 nm  ii)  Reinforcement spacing of 100-500 nm  iii)  Reinforcement spacing >5 um  5 Case (i) encompasses most of the precipitation hardening alloys. The radius of these reinforcements is in the range of 1-10 nm and volume fraction varies from 1-5% [1]. These reinforcements are uniformly distributed in the matrix and give a very high contribution to strength because they provide obstacles to the motion of dislocations. As such, strengthening in these type of alloys can be best described by dislocation strengthening models [2]. The maximum strengthening due to precipitates is found for the optimal balance between strength of individual precipitates and spacing. Alloys of the second category are often known as dispersion hardening alloys. In these alloys, insoluble oxide reinforcements are dispersed in the matrix during the alloy manufacturing process and the volume fraction of reinforcement in these alloys ranges from 1-10% [2]. Dislocation strengthening models can explain the strengthening effect from these second phase reinforcements. Ebling and Ashby [3] studied the strengthening effect of SiC»2 dispersoids produced by internal oxidation in a Copper matrix. These dispersoids can not be sheared by dislocations and thus significantly change the work hardening rate of the matrix. The mechanism of strengthening was first explained by Orowan [4] and later modified by Ashby to produce a more refined estimate of strengthening [5]. Case (iii) encompasses materials such as metal matrix composites [6] and dual phase steels. Here the second phase volume fraction is 10-40% and the reinforcement size is of the order of 5-40 um. Strengthening mechanisms at these length scales can be successfully explained by continuum models. For example, an excellent review of continuum approaches to the study of M M C ' s has been presented by Clyne and Withers [6].  6  It is important to note here that at intermediate length scales i.e. 0.1-10 um both dislocation and continuum approaches overlap. A preliminary approach to account for both theories has been attempted by Nan and Clarke [7]. This work addresses materials where the length scale of the microstructure is typical of Case (iii). Therefore this particular case is further described in detail in the subsequent sections. 2.1.2 Two-phase material with reinforcement spacing >10 pm Cho and Gurland [8] have further classified two-phase alloys of this length scale into two subgroups. Alloys that belong to the first group are composed of a soft matrix and a small amount of high strength second phase such as found in commercial metal matrix composites or dual phase steels. Here the matrix deforms plastically while the second phase or the reinforcement deforms elastically. Material behaviour of this group has been studied extensively, ranging from deformation behaviour by Poole et al. [9] to the debonding and reinforcement failure studies by Kiser et al [10] and Ganguly et al. [11]. Finite element models have evolved over a period of time due to the flexibility of this approach to study geometric effects. Further, good agreement has been observed between these models and experiments. For example, Poole et al. [9] have extensively studied the non-homogeneous deformation of the matrix in a Cu-W model composite system using experiments and F E M models. These workers have shown that effectively there are only two regions in the matrix. One region is the region surrounding the fibres extending 1.1 times the diameter of fibres where the matrix deformation is controlled by the interaction of the matrix and the fibres. The second region can be considered to consist of the remainder of the matrix where the nature of the deformation depends upon the geometric arrangement of fibres. Figure 2.1  shows the different patterns of deformation in the matrix by changing the geometrical arrangement of fibres from square to triangular.  ODO  •• •  Compression  0 7.  Axis  a)  25 X  0 X  Compression  Axis  25  X  Figure 2.1 Effect ofgeometrical arrangement offibres on the strain distribution inside the matrix. Results of Poole et al [9] for non-deforming fibres.  Alloys of the second group, which will be the focus of this study, consist of plastically deformable matrix and a substantial fraction of a plastically deformable second phase. Examples of the second group of alloys that were studied over the years by researchers are oc/p brasses [12], a/p T i alloys [13], Cu-Nb nano composites [14] and a/y duplex stainless steels [8]. As a group these alloys offer great combination of strength and ductility, which  8  are favorable properties for forming processes. The deformation behaviour of these alloys can be further subdivided into two categories based on imposed deformation in the sample: 2.1.2.1 Low deformation regime In general, this regime can be defined by deformations in the range of 5-30%. For this low deformation regime, the issues of strain and stress partitioning are of critical importance and significantly affect the overall behaviour of the material [9]. Also the volume fraction of the reinforcing phase and its distribution in the matrix affects the stress/strain partitioning and in turn may affect the stress-strain response of the composite. When the composite alloy is deformed, the reinforcing phase may deform more than or less than the matrix or may not even deform at all depending upon its mechanical properties (yield strength, work hardening etc) and the factors discussed above. A n early experimental study of this deformation behaviour for different combinations of matrices and second phases was done by Unckel [15]. He chose several alloy systems where the relative hardness of the constituent phases was varied. In some cases the second phase was softer (e.g. Alloy 1leaded brass) while in other cases it was harder (Alloy 2- a/(3 brass, Alloy3-Complex brass, Alloy 4-Copper with iron and Alloy 5- Tin-Bronze). The deformation of the reinforcing phase was measured as a function of the sample deformation (see Figure 2.2). It was clearly observed that when the reinforcements are softer than the matrix, they deform more than the total sample deformation (Alloy 1) and when the inclusions are harder their deformation is lower than the sample deformation (Alloys 2,3,4 and 5). Another interesting observation in these experiments was that all the curves have a tendency to approach a slope of 45° at some level of sample deformation indicating that at higher deformations, the tendency for codeformation exists in these materials. While these observations were of great interest, several  important factors such as the effect of exact relative work hardening of each phase their geometrical arrangement and volume fraction were not accounted for. Figure 2.3 can be used to help understand the strain partitioning in the two phases. As shown in the Figure, the inclusion (curve A ) will start to deform only when matrix (curve B) has suffered a reduction Ri, i.e. when the matrix work hardens to the yield stress of the inclusion.  z  u 2 O  , V..T.  'k^,  tl  >°V>  U  z;  '  G Z Q »U  /  /  /  /  y  /  >: /  •'  "  « /• ><»-  /  .>•  Q  TO'  20  JO  40  50  60  7,0  80  90  100  REDtteroON;OF WHOtE TEST-PIECE; PER CENT. Figure :2.2 Percentage reduction of embedded reinforcements as a percentage reduction of sample. Unckel, H. [15].  Another interesting study on the strain partitioning between the phases of a laminate was done by Oztiirk and co-workers [16]. In his study, the relative hardness of the reinforcing phase of the laminate with respect to the matrix was varied from 2.5 to 13. Oztiirk concluded that for higher relative hardness of the reinforcing phase, there would be lower strain partitioning to the reinforcing phase. These results are almost in parallel to the results by Unckel [15] except that they were better quantified in terms of relative hardness of  10 the phases. However, the nature of dependence of this behaviour based on the geometric arrangement of the second phase was still unknown.  I REDUCTION.  PER CENT.  Figure 2.3 Strain hardening curve of inclusion(A) and surrounding matrix (B). Unckel [15]. Clarebrough and Perger [17] observed the relative deformation of a and p phases in an a/p brass and the effect of this on recrystallization behaviour. The results from this work are based on the start of recrystallization for the a and P phases at certain imposed deformation. If the recrystallization start temperature changes at any imposed deformation it will indicate the non-homogeneous deformation behavior of that phase. It can be observed from Figure 2.4 that for any volume fraction of p more than 0.35, the deformation of the phases is uniform throughout the sample i.e. both a and P phases co-deform. While for low volume fraction brass (volume fraction of P-phase is less than 0.35) the largest cc-phase deformation is achieved for the p-phase volume fraction of 0.20.  Figure 2.4 Influence of volume fractions of the phases on the recrystallization temperature of (a) a phase (b) (5phase. Clarebrough and Perger [17]  12 Another example with significant industrial implications is the strain partitioning in dual phase (DP) steels which was characterized by Balliger and Gladman [18]. DP steels, with a micro structure consisting of martensite islands dispersed in a ferrite matrix, have received a great deal of attention due to their useful combination of high strength, high work hardening rate and ductility. Figure 2.5 shows the strain distribution between phases for a ferrite/pearlite and a ferrite/martensite steel, containing the same volume fraction (14%) of second phase [18]. As can be seen, for the pearlitic dual-phase steel, there is no strain partitioning between ferrite and pearlite phases. However, strain partitioning occurs in the martensitic dual-phase steel in such a way that martensite phase only plastically deforms in the regions close to the fracture surface of the specimen (i.e. the regions of largest imposed strain).  i  i  i  I  I  1-2 10 08 0-6 8  i—r  i  (a)J  specimen strain J  constituent strain Ferrite-14% pearlite  0-4  0-2  01  i  •  •  » •  08 V 06-  I <W1 Ferrite-14% martensite  020_ Uniform deformation Necked region  Undeformed tensile head  DISTANCE FROM FRACTURE SURFACE , mm Figure 2.5 The strain distribution between phases in two different dual-phase steels Balliger etal. [18].  13 As has been observed from the examples above, the strain partitioning and load transfer to the reinforcing phase affect strengthening from the reinforcing phase. However, the geometric arrangement of second phase may also be an important factor in the abovementioned behaviour. Until now the results of geometric arrangement were only available for elastic-plastic matrix reinforced with completely elastic reinforcement e.g. Poole et al. [9]. These authors have found that the geometric arrangement of fibres does not affect the overall stress strain response at low fibre volume fractions (20%) but does not have a significant effect at higher volume fractions (30%) (see Figure 2.6). 300.00  OH  200.00  ,73  100.00  20 — — 20 30 30 0.00  volume volume volume volume  •ri-rtn-i-t-i-n-i-r-r-ri-riTH-'-i-i-T-t-i-i-rt-j-T-r  0.00  0.10  0.20 .  0.30  Strain  fraction (Tri. f r a c t i o n (Sqr) fraction (Tri) fraction (Sqr)  0.40  ' .0.50  Figure 2.6 Role ofgeometric arrangement offibres on the overall stress-strain response of the composite. Poole et al. [9]  2.1.2.1 High deformation regime (e>5) Ductile two-phase materials deformed into this regime are mainly used for making high strength wires. The strength in these composites derives from the high amount of work  14 hardening due to the large strains that are applied (~10 or more) without annealing. Bevk et al. [19] have studied the high strain deformation behaviour of these alloy systems. Detailed literature is available to understand the strengthening and deformation response of these composites at high strains [14,20,21]. A few examples of this material class include Cu-Nb nano composites and pearlitic steels. Cu-Nb Analysis of longitudinal sections of Cu-Nb wires by Spitzig [14] provides insight into how the microstructure develops during the deformation process. Nb reinforcements are transformed from an initial spherical shape into fine filaments. These filaments become finer and closer together with increasing levels of imposed deformation. The spatial distribution pattern of the filaments becomes more homogeneous as the deformation process proceeds to very large strains (i.e. s > 10). It has been observed that at these high strains these two phases co-deform with equal strain in each phase. Pearlitic steels Similar observations can be found for pearlitic steels. For example, with increasing levels of imposed deformation, the pearlite colonies become elongated along the deformation axis i.e. the pearlite becomes thinner and the filament spacing decreases. Interestingly, Zelin [22] has found by experiments and by F E M calculations that there is a inhomogeniety of strain in the sample during the deformation due to the formation of shear bands. These shear bands run at 30-45° along the deformation axis and form a diamond like cell structure. The inhomogeniety of plastic flow during the deformation can cause a substantial decrease in total elongation of these wires (see Figure 2.7).  15  0  1.00  2.00  3.00 UM  Figure 2.7 FEM modeling results showing localized shear in a pearlite with ferrite/cementite plates orientated (a) parallel and (b) perpendicular to the wire axis, and (c) atomic force micrograph illustrating localized shear in a pearlitic steel. Zelin [22]  As seen in this section the deformation of ductile two-phase alloy systems, at high deformation strains occurs by co-deformation of the constituent phases where the strain in both phases is equal to the applied strain. However, when the deformation strain in similar composites (pearlitic steel, Cu-Nb etc) is low, load transfer and strain partitioning between the phases becomes an important parameter from the point of view of shape change of pearlite/Nb phase, inhomogeneous deformation and texture development of the material. Thus one important aspect of this thesis will be to understand the deformation of ductile twophase alloy systems at lower deformation strains.  16  2.2 Deformation theories For plastically deforming ductile two-phase alloys where the matrix is softer than the reinforcing phase, the reinforcements have two possible modes of deformation: i)  the reinforcements may remain elastic (e.g. in M M C ' s ) .  ii)  i f the stress transfer process from matrix to the fibre is efficient enough then plastic deformation in the reinforcements may also occur.  Plastic deformation in these alloys is characterized in terms of large differences in stresses and strains between the two phases. The overall mechanical properties of these alloys reflect a synergistic interaction of the properties of the constituent phases. In addition, the plastic incompatibilities between the phases causes a strong contribution to strain hardening which causes high local stress concentrations that may lead to premature fracture by void initiation and propagation. Therefore to improve the properties of these alloys it is important to understand the localized stress and strain partitioning in these alloys. Several deformation theories have been proposed in the past to describe the flow behaviour of multiphase materials. Since the scale of the reinforcements in the present work is much higher than for dislocation based theories, this section will focus on continuum based models only.  2.2.1 Upper and lower bound analysis The simplest approach is based on either equal stress or equal strain partitioning between matrix and the reinforcement under an applied load. The strain equality (Voigt bound) and stress equality (Reuss bound) represent upper and lower bound solutions for two phase deformation. In general these two solutions are quite far apart and are not sufficient accurate to predict the deformation behaviour when the geometry of the second phase lies  17 between the two bounds (e.g. particles, whiskers or short fibers). Hence there are some advanced analytical and numerical solutions available to understand, the composite behaviour containing particulate or whiskers as a second phase. Most important of them is Eshelby's equivalent inclusion method [23]. There are various forms in which Eshelby's technique can be applied to determine the composite response, an excellent review of which has been presented by Clyne and Withers [6]. The original Eshelby's solution was further developed into models such as the self-consistent approach [24-28], which allows for more accurate predictions of stresses for high volume fraction composites. 2.2.2 Eshelby's models The original Eshelby solution was for a single elastic reinforcement embedded in an infinitely large elastic matrix. The model has been extended to plastically deformable alloys by Brown and Stobbs [29,30], Corbin et al. [27,28] and Weng et. al [31,32]. The original Eshelby technique is based on replacing the actual inclusion by one made out of matrix material, which has an appropriate misfit strain, such that the stress field is the same for actual inclusion. Eshelby's approach can be best understood by visualizing a series of cutting and welding exercises as shown in Figure 2.8. The following steps can explain the exercise: 1) Cut the inclusion from the matrix and assume it to undergoes a stress free shape change or transformation strain (s ). T  2) Apply surface tractions to fit this inclusion back in the matrix. 3) After Step 2 the surface tractions are removed. Now the matrix will constrain (s ) the c  inclusion so that it cannot attain its transformed shape. The resulting stresses on the reinforcement can be represented using Hook's law.  18  v ~  ^  =  ii -  C  ii  {  M  e  -  c  e  T  (2.1)  )  Transformation Strain  C £  o T =  £>£  Figure :2.8 Eshelby's cutting and welding exercise and how the constrained strain (£) is related to transformation strain (J) by Eshelby tensor (S). Clyne et al. [6]  Where: (7y = is the stress on the reinforcement, cr? = Constrained stress on the reinforcement c r j j =  e  T  Transformed stress,  e  c  =  Constrained strain on reinforcement  =Transformed strain on reinforcement, and  C  M  = Stiffness tensor of matrix  The next step in this analysis is correlating £ with e ,c  S, known as Eshelby tensor to correlate e e  with e  c  c  =  S  e  T  T  T  . Eshelby developed a tensor,  . (2.2)  Eshelby tensor S is a correlation between inclusion aspect ratio and the Poisson's ratio of its material. Brown and Clarke [33] have calculated values of S for various inclusion  19 geometries ranging from long fibers to plates. Eshelby's model predicts that the stresses inside the reinforcements are uniform, and are higher than the applied farfield stress for the reinforcements stiffer than the matrix.  2.2.3 Shear lag model for short fibres The shear-lag model due to Cox [34] is widely used to calculate the deformation properties of elastic fibrous materials such as short fiber composites and random fiber networks in a perfectly elastic matrix. Chad and Robert [35] extended the original shear lag model to a plastically deformable matrix. The assumption in the original elastic/elastic model is that the tensile stresses in the matrix are transferred to the fibres by generation of shear stresses at the fibre matrix interface. The model is useful to calculate the stress in the composite having fibres of high aspect ratio especially for the cases that may be difficult to deal with Eshelby's equivalent inclusion method.  2.2.4 Extension of Eshelby model to co-deforming materials (Weng's Secant method) The Weng secant method [31,32] predicts the overall stress-strain response of ductile two-phase alloy system using the properties of individual constituents and an Eshelby type model. During the deformation of a ductile two-phase alloy system, internal stresses are generated due to the strain difference of the two phases. It is very important for the prediction of the stress-strain curves of these alloys to take internal stress relaxation into account and Weng's secant method is suitable since it gives stress or strain partitioning after the plastic relaxation. In terms of internal stress relaxation, plastic deformation of the matrix is accounted for by treating the plastic matrix as a material with a decreasing stiffness calculated  20 from plastic stress-strain curve. Stress or strain partitioning parameter B connecting stress S  Q  and strain of the composite to that of individual constituents is described as:  B  s  /?o(G| - < J Q ) + GQ  _  °  [/ + 0 - / ) ^ ] ( G , - G  s 0  (  )+G  2  3  v  s 0  Where, f is the volume fraction of the inclusion and /? is the Eshelby's tensor for the S  0  spherical inclusions, /? =2(4-5I/O)/15(1-VQ). G and v refer to shear modulus and S  s  0  Poisson's ratio where the subscript 0,1 and s refer to matrix, inclusion and secant moduli respectively. The parameter B  s 0  changes with the deformation, which makes the predictions  of this model more realistic. Composite stress can be calculated using this model: . , . Flow stress of the constituent Stress in the composite= load transfer parameter There can be three possible stages during the deformation of a composite: Stage 1: Matrix and reinforcements are in elastic regime Stage 2(a): Matrix deforms plastically and reinforcement remains elastic or Stage 2(b): Matrix remains elastic and reinforcement deforms plastically Stage 3: Matrix and reinforcement deform plastically A set of equations can be generated for these stages using the variables discussed above. It is important to note here that when the plasticity in any phase occurs the secant modulus is used to characterize the stiffness of the phase and its value is less than the modulus when the material was elastic. Figure 2.9 illustrates how secant modulus changes with the amount of plastic deformation.  21  Si  s  Figure :2.9 Secant modulus, E [ , when phase was deforming plastically and the modulus (E) when the phase was elastic.  2.2.5 Finite Element Method (FEM) calculations Numerical models using approaches such as the Finite Element Method have become increasingly utilized to study deformation in two-phase materials due to the fact that complex microstructures can be modeled enabling a better understanding of the effect of geometrical arrangement of reinforcements [36]. In the F E M approach, a two-phase material having finite geometry is discretised into finite elements and then displacement or forces can be added as boundary conditions [37]. The discretisation process involves dividing the geometry into finite number of nodes and elements, forming a mesh. The elements of the mesh can be assigned to contain the material properties that define how the material of that element will behave under the applied load. After application of the appropriate boundary conditions, a computer program is used to solve for the internal forces and displacements such that the total energy of the system is minimized. Use of F E M models has helped researchers understand the effect of reinforcement geometrical distribution, aspect ratio, etc. on the stress-strain distribution in the composite materials. However, most of the work in the literature has focused on the ductile matrices reinforced with perfectly elastic reinforcement  22  [9,36]. However, a recent attempt has been made by Al-Abbasi and Nemes [38] to model the effect of martensite reinforcement size distribution in dual phase steel on the overall stressstrain response. Al-Abbasi and Nemes have concluded using F E M models, that the reinforcement size distribution affects the stress-strain curve of the DP steel only at high volume fractions. To summarize, detailed literature is available on high strain deformation behaviour of ductile two-phase alloy systems. However, on the other hand, little understanding is available for the cases where the deformation strains are low and especially in relation to the stress/strain partitioning which occurs in these systems. It is very important to develop a sound understanding of the deformation behaviour for these types of alloy systems as it may help to improve the properties of existing materials or develop materials with improved properties. To the author's best knowledge no literature is available so far where the stressstrain response of individual constituents are determined experimentally and a ductile two phase material is made out of these constituents. In addition to this, there is a gap in the literature on the effects of constituent properties, volume fraction and geometrical distribution on the overall stress-strain response of composite studies at low deformation strains for ductile two-phase alloy systems. The importance of current work lies herein.  23  CHAPTER: 3 SCOPE AND OBJECTIVES OF PRESENT WORK  The present work aims at examining the effect of volume fraction and geometric arrangement of the reinforcing phase on the deformation behaviour of ductile two phase materials. The scope of this work includes: •  Fabrication of a model composite system, having square and triangular fibre arrangement, with volume fraction between 20 and 35% where the length scale of reinforcements is such that continuum approaches can be used.  •  Constructing finite element models for the above mentioned experimental samples and comparing the obtained results with the experimental ones. Validation of F E M models with the experimental results will give confidence in building models for more complex geometries.  The objective of this work is to understand the effects of: •  Spatial arrangement of reinforcements to the overall stress-strain response of the sample and localized deformation behaviour of the constituent phases.  •  Constituent relative flow stress and relative work hardening rate on the strain partitioning in the reinforcing phase.  24  CHAPTER: 4 EXPERIMENTAL METHODOLOGY The principal aim of the experimental work was to fabricate two-phase material with 20% and 35% volume fraction of second phase having a suitable set of constituent properties. These samples were then tested under channel die compression and the strain partitioning between the reinforcing phase and the overall sample was studied. In parallel finite element models for these samples were also developed and the F E M results were compared with experimental results.  4.1 Sample fabrication Model composites were fabricated with the aluminum alloys AA6061 as the matrix phase and brass C-3600 as the second phase. The chemical composition of these alloys is given in Table 4.1 and 4.2. Table 4.1 Chemical composition for Brass C-3600: Brass C3600  Cu  Fe  Pb  Zn  Other  6063%  Max 0.35%  2.53.7%  35.5%  Max 0.5%  Table 4.2 Chemical composition for AA6061: A16061  Al  Cr  Cu  Fe  Mg  Mn  98%  0.04 0.35%  0.15 0.4%  Max 0.7%  0.81.2%  Max 0.15%  Si 0.40.8%  Ti  Zn  Max 0.15%  Max 0.25%  Samples were fabricated by co-extrusion of brass and aluminum. Brass fibres, 1.58 mm in diameter and 27 mm in length, were used to fabricate the model composite. The initial step in processing was the heat treatment of fibres at 560°C for 10 min to soften the material for subsequent extrusion. After the heat treatment, the fibres were chemically cleaned to remove  25 any oxidized layer on its surface using the acid mixture shown in Table 4.3 at temperature of 50°C for 20s: Table 4.3 Composition usedfor removing oxide from brass fibres:  Chemical Volume %  Phosphoric acid  Acetic acid  Nitric acid  HC1  55  25  20  0.5  Extruded AA6061 bar stock of 24 mm diameter and 35 mm in length was used to form the matrix. Holes 1.58 mm in diameter were drilled with the spatial arrangements shown in Figure 4.1. After drilling, the aluminum block was cleaned with acetone. The brass fibres were then inserted in predrilled aluminum block prior to extrusion. A schematic of the extrusion process is shown in the Figure 4.2. Extrusion was carried out in a closed die on a MTS machine of maximum load capacity 220 kN. The AA6061 block with the brass fibres inserted was then extruded from an initial diameter of 24 mm to a final diameter of 18 mm. The extruded sample was mechanically fixed in the die. To remove the sample from the die, the die plus sample was heat-treated at 350°C for 5 minutes and quenched in water. After this treatment the sample could be removed from the die. Figure 4.3 shows the extruded sample and die after the sample has been taken out of the die. The initial 15 mm of the extruded sample was found to have undergone non-homogeneous deformation as can be seen from Figure 4.4, therefore this section of the extrusion was removed and discarded. The remainder of the extrusion was cut into 5 mm thick discs. These discs were further machined into 5 mm thick square or rectangular shape samples. Subsequently a solution treatment of 10 minutes at 560°C plus aging heat treatments at room temperature for 6 hours or 210°C for 30 minutes were given to the final samples to control the stress-strain curves of individual constituents. Figure 4.5 shows the final sample geometry used in this work.  26  X I  1  0 o0  (!)  o o OO o o OO o QP O • z  (<0 Figure 4.1 Cross-section of Drill geometry on 24 mm diameter A A6061 bar (a) 20% square fibre distribution where x=1.58 mm, y=3.15 mm and z=1.58 mm, (b) 20% triangular fibre distribution where x=1.58 mm, y=3.15 mm and z=1.58 mm and (c) 35% square distribution of fibres x=1.58 mm, y=2.4 mm and z=0.8 mm.  AI-6061  Brass Extrusion  Composite for h/t  5mm Discs  Extruded Sample  Figure 4.2 Sample fabrication steps  Figure 4.3 Extruded sample and die  28  Non- Uniform Deformation A  Figure 4.4 Extracted Brassfibresfrom the extruded Sample  I  i  9.9 mm  o o o o  o o o o o o o o o o o o  o  o o o o o o o o o o o o o o o o o  12.5 mm  1r  9.75 mm  (a) 7.6 mm  12.6 mm  o o o o  o o o o  o o o o  o o o o  (b)  7.6 mm Figure 4.5 Sample geometry (a) 20% squarefibredistribution, (b) 20% triangularfibredistribution and (c) 35% square distribution of fibres.  (c)  29  4.2 Constituent properties The stress-strain response of constituent phase was obtained by passing the individual constituents through the same deformation history as the model composite. The following procedure was used for each phase: i) Aluminum: Compression samples were deformed 57% using the Gleeble 3500. Samples were then subsequently heat-treated and compression tested. ii) Brass: Brass fibres were extracted from the extruded sample by dissolving the aluminum matrix using the chemical solution shown in Table 4.4. The cleaning solution was used at 85-90°C. Figure 4.4 shows a macrograph of extracted brass fibres. Brass fibres so obtained were heat-treated and then tensile tested. Table 4.4 Aluminum dissolution recipe Chemical g/1  NaOH  Na S  Triethanolamine(TEA)  185  15  30  2  4.3 Fiducial grid preparation A rectangular array of gold squares was fabricated on the surface of channel die compression samples. This array of gold squares allowed for the measurement of local strain in the deformed samples based on the change in the shape of the grid. The grids were prepared by evaporating gold trough a commercially available 1000 mesh-size grid, which was kept in close contact with the sample. The following procedure was used: 1. Sample faces containing fibres were metallographically prepared down to 1pm diamond paste. 2. Commercially available 1000-mesh grid was placed on one of the faces. 3. The whole setup was then placed in a high vacuum gold evaporator.  30 4. Under a high vacuum of 10" torr or less, gold is evaporated in the chamber which passes 5  through the commercial grid and leaves square shape equidistant gold points on the sample. Figure 4.6 is a S E M picture of the grid on sample surface before deformation.  Figure 4.6 Back scatter SEM image ofgold grid on the sample  surface.  4.5 Strain calculation from the grid deformation The approach to strain calculation from a deformed grid adopted was similar to previous researchers (e.g. Sowerby [39] who measured strain in metal stamping and Poole et al. [40] who experimentally measured localized strains in Cu-W composite). The first step in the process was to calculate the deformation gradient tensor, F , from the centroids of four gold points which form an element as shown in Figure 4.6, before and after deformation. Figure 4.7 is an exaggerated picture of that element before and after deformation where the deformed shape is shown by dotted lines.  31  X  Figure 4.7 Deformation of square into a parallelogram.  F. =  d X i  dX,  iJ  Since in practice Ax' and Ax" will be different due to non-uniform deformation, an average value at the center of the element will be calculated. Consider an example for the calculation of F from the Figure 4.7. For this case: (Ax'+Ax") 2AX  Next Green's deformation tensor is calculated from the deformation gradient tensor, i.e. C  =F + F 2  ^xx  r  Cxy  C  2  l l ^  \ 2  r  Cyx ~  ^12^22  =F +F YY  1  21  T  1  22  Green's deformation tensor is a symmetric matrix containing terms that are only functions of Fjj. From this matrix square of elongation ratios X-^ will be calculated, i.e the Eigenvalues. }2 12 hi ' 2 2 / t  (C„+C ),1 w  32 Finally, the principal true strains are calculated by. e  u  s  2 2  =ln(A ) n  =ln(A ) 22  The experimental procedure for strain measurement can be summarised as: 1. S E M picture of grid before deformation was taken at a magnification of 350x and then imported into the U T H S C S A [41] image analysis program for the acquisition of centroids of each gold point on the sample. 2. Sample was deformed and S E M picture of deformed grid was taken at the same magnification as undeformed grid and the centroids of each gold point were remeasured using the same procedure as in Stepl. 3. A computer program based on the equations described above for strain measurement was used to calculate the value of principal strain in each grid. The calculations in this program were based on two assumptions: (i) constant volume after deformation and (ii) zero strain in thickness direction ( S 3 3 ) . 4. A n average was determined based on the analysis of at least 20 elements.  4.6 Mechanical testing (Channel die compression test) A l l tests were conducted on a MTS/Instron 8500 servo-hydraulic machine. The setup of the channel die and the sample is shown in Figure 4.8. Low friction conditions were maintained during the test by careful machining the sample according to channel die dimensions and appropriate lubricant condition between sample and die walls. To reduce the effect of friction teflon tape was placed between sample and die walls. One side of the teflon tape was coated with moly-disulphide spray and was placed in contact with the channel die walls. Deformation of the sample was carried out in three steps of approximately 10%  33  Loading Direction  42  Constraint Direction  Flow Direction  Channel Die  Composite Sample  Figure 4.8 Schematic of channel die compression test  strain. After each strain increment, the sample dimensions (height, width and thickness) were measured followed by observation under the S E M to record the distortion in the gold grid. During mechanical tests, data (time, load and displacements) were acquired directly on a PC using the Instron Wavemaker software. The stress strain curves from channel die were generated from the load displacement data using following set of equations: _ Load &  channel-die  -i £  channel-die  ~  Ah  ~j  ~  (1  n m  U  +  ~7  +  —  \  K  )  ,  /  where: Ao= Initial contact area with the punch, Ah= change in displacement and h = initial 0  height of the sample. The stress strain response from the channel die was converted to true von Mises equivalent stress and strain by the following formulae:  34  ^"von Mises  £  A ^  2 —  von Mises  ^"channel-die  F  ^J~^ channel-die  4.7 Finite element model F E M models with similar geometries to those used in the experiments were constructed using the A B A Q U S 6.3 software package. The model development involved the following steps: 1. Square and triangular geometries were generated using A B A Q U S - C A E  software  package. Sample was represented in 2-D with the fibre axis perpendicular to the plane of paper. 2. While no mesh convergence studies were conducted in this work but similar work from other researchers (Poole [40] and Ganguly [42]) indicated that more than 3000 element would sufficient enough to capture localized strain and overall stress-strain response of the model composite. 3. 2-D quadrilateral plane strain elements with reduced integration were used to mesh these geometries. Fibres were meshed with a higher mesh density than the matrix. 4. Interface between fibre and matrix was assumed to have perfect contact and not allowed to debond at any imposed strain. 5. The contact between top and bottom face of the sample with the rigid platens was modeled as non-penetrating frictionless contact. Figure 4.8(a),(b) and (c) shows the F E M mesh for 20% square, 20% triangular and 35% square models, respectively.  Figure 4.8(b) FEM mesh for composite with 20% triangularfibrearrangement  36  2  '  •  1  Figure 4.8(c) FEM mesh for composite with 35% square fibre arrangement  4 . 7 A Material properties The matrix and the second phase were both treated as elastic-plastic materials. Elastic properties were obtained from the literature and plastic properties were taken from the experimentally determined stress-strain curves of the matrix and the second phase. 4.7.2 Boundary conditions Since the sample was deformed by the movement of rigid surfaces that are in contact with the sample therefore the displacement/rotation boundary conditions were put on top and bottom reference nodes of top and bottom rigid surfaces •  Boundary conditions on the top rigid surface: ui=0 ; U2=amount of displacement required to achieve the required strain in the sample and W3 (rotational displacement )=0.  •  Boundary conditions on the bottom rigid surface:  37 ui=0 ; u =0 and Rotational displacement(ur3)=0. 2  In addition to these boundary conditions on the sample, there were additional stability conditions on the sample, namely: >  Free ends of the sample (edges not in contact with rigid body) were free to move i.e. the stresses were zero there.  > Nodes of the sample in contact with the rigid body would always remain in contact with the rigid body at any deformation of the sample.  38  C H A P T E R :  R E S U L T S  - P L A N E  STRAIN  5  C O M P R E S S I O N  This chapter describes the results of plane strain compression tests and compares the results with the F E M models. The channel die compression tests were conducted on five different combinations of matrix and second phase properties. The results are described in terms of the macroscopic stress-strain response, the shape change of sample, the plastic strain in the fibres and stress distribution perpendicular and along the loading direction.  5.1 Constituent stress-strain response For each of the geometries shown in Figure 4.5 two samples were prepared. With these samples, different heat treatments were conducted to ensure that, i) the fibres had higher yield stress than the matrix, and ii) the matrix had higher yield stress than the fibres.  Case: 1 Fibres have higher yield stress'than the matrix: A solution treatment at 560°C for 10 minutes was followed by room temperature aging for 6 hours. Figure 5.1(a) shows the individual stress-strain response of the fibres and matrix after this heat treatment, i.e. from extracted fibres and monotonically defined matrix material as described in Section 4.2. The yield stress for the fibre and matrix were 142 M P a and 101 MPa, respectively. The work hardening rate of the fibres exceeds that of the matrix for all strains.  39  500  Extrapolated curve brass 400 H  300 co  CO CD  55  2  0  0  Aluminum matrix Brass fibres  100  0.00  0.05  0.10  0.15  0.20  0.25  0.30  Strain (a) 500  400  £  300  co  CO  £  200  Aluminum matrix Brass fibres  100 -\  0.00  0.05  0.10  0.15  0.20  0.25  0.30  Strain  (b) Figure 5.1 Stress strain response of the constituent phases (a) Fibres have higher yield stress than the matrix (b) Matrix has higher yield stress than the fibres  40 Case: 2 Matrix has higher yield stress than the fibres: A solution treatment 560°C for 10 minutes was followed by aging at 210°C for 30 minutes. Figure 5.1(b) shows the individual stress-strain response of the constituent phases after this heat treatment. In this case the matrix and fibres had yield stresses of 270 MPa and 142 MPa, respectively. The work hardening rate of the matrix was considerably lower than for the fibres and at a strain of approximately 18% the two curves intersect. It is worth noting here that the stress-strain curve of brass fibres is the same in both Figures 5.1(a) and 5.1(b). The reason for this behaviour is because the initial treatment of the composite samples at 560°C makes the brass fibres unaffected by subsequent treatments at temperatures lower than 560°C (aging at 210°C and at room temperature). For both Case.l and Case:2, only the matrix properties are changing. Therefore, for the sake of simplicity in the following sections we will call Case:l property set as "Matrix lower a " and Case:2 ys  property set as "Matrix higher o" ". ys  As explained in Section 4.2, the brass fibres after extraction from the extruded sample were tensile tested. It was found that the tensile tests on brass fibres could only yield the stress-strain curve up to 15% of strain, i.e. after that the fibres fail. The composite samples on the other hand were tested in compression for up to a strain of 25%. In order to compare the stress-strain curves of experimental samples with the F E M predictions and individual constituents the brass fibres curve has been extrapolated to a strain of 25%. 5.1.1 Curve extrapolation True stress-strain curve was first converted to true stress vs. true plastic strain curve and the available data points were fitted to Ludwik equation, i.e.cr = Ke". This yielded the value of  K(=645) and n(=0.35) for that data set. From these available K and n  41 values of true stress for any plastic true strain can be calculated. In general, a large extrapolation of the Holloman equation is not reliable but for small extrapolations used in this work it was considered to be reasonable (i.e. from strains of 0.15 to 0.25)  5.2 Channel die compression tests Tests on three geometries, i.e. 20% triangular, 20% square and 35% square, are described in terms of stress-strain response, macroscopic shape changes, strain in the fibres and stress distribution in the sample in the following subsections.  5.2.1 20% triangular samples: 5.2.1.1 Stress-strain response and macroscopic shape change of samples: Figure 5.2 shows the von Mises equivalent stress-strain response for the 20% triangular arrangement composite with the aluminum matrix of higher and lower yield stress viz. the fibres. The discontinuities in the experimental curve (i.e. line with no symbols) arise when the compression test is stopped and the lubrication replaced. The following observations can be drawn from these curves: i)  For both combinations of fibre and matrix properties, good agreement is observed between the experimental and F E M curves for the first step of deformation.  ii)  The experimental curve for the high a  y s  matrix shows a substantial deviation  from the F E M results at strains greater than 10%.  42  500 20% triangular arrangement  400 4  Matrix lowerCT^(Exp) — Matirx lower  (FEM)  Matrix higher cr^ (Exp)  -A—  0 4r 0.0  0.1  0.2  Matix higher a (FEM)  0.3  0.4  Strain Figure 5.2 Stress- strain response of 20% fibre volume fraction composite with triangular fibre arrangement  The stepwise macroscopic shape changes of these samples are shown in Figure 5.3 and Figure 5.4 for the different combinations of matrix/fibre properties. It can be observed from Figures 5.3 and 5.4 that the fibres take up the shape of an ellipse when the sample is deformed but the aspect ratio of the ellipse is higher in Figure 5.4 than in Figure 5.3. For the lower yield strength matrix, the fibres start to debond after Step-2 (however from Figure 5.3(b) debonding is not clearly visible due to low magnification) and a significant level of debonding is present after Step-3 (debonding is clearly visible in Figure 5.3(c)). The composite having higher yield strength matrix in Figure 5.4 shows that the level of debonding is almost the same as for the composite with lower yield strength matrix i.e. thirteen fibres out of eighteen have debonded.  43  -jinaii  j^ i  ijntrii'li  -iimi  H 4b 4b  •  (fe i J  • • •  (d)  F/gare 5.J Sam/>/e picture of 20% triangular distribution and matrix having lower a (a) Before deformation (b) after Stepl, £=10.8% (c) after Step2, e=20.4% (d) after Step3, £=29.3% ys  44  (a)  (b)  Figure 5.4 Sample picture of 20% triangular distribution and matrix having higher <j (a) Before deformation (b) after Step3, e=24% ys  5.2.1.2 Localized fibre strain and F E M strain distribution in the sample: Strain in the fibre was calculated by analyzing the deformation of the gold grid on the surface of the sample. Fibre strain has been calculated in terms of the average strain for at least 20 elements on three innermost fibres after deformation. The elements were chosen from the inner most fibres to avoid sample edge effects. Further, F E M calculations showed that the strain was not uniform within a fibre, but due to the limitation of the strain measurement technique used in the present work it was not possible to determine the strain distribution within the fibres. F E M simulated von Mises strains are compared with experimental values in this section. Results from the sample where the matrix has a lower initial strength are shown in Figure 5.5. Figure 5.5(a) compares the average value of experimentally measured von Mises  45 strain in fibres to that of F E M calculated values. A curve with unit slope is also shown in this graph for comparison i.e. fibres and matrix have the same level of strain. It can be observed that the experimental points follow the same trend as the F E M curve and the gap between the two is significant only at 29% of strain. It can also be observed from Figure 5.5(b) that since the fibres have higher strength than the matrix they have accumulated lower strain than the matrix. In Figure 5.5(b), it can be observed that the deformation of the overall composite is very uniform, except for the regions horizontally in between the two fibres which experience almost 1.2 times the farfield strain resulting in some horizontal banding in the material. The area of the matrix, in between the three fibres which form a unit cell, experiences a similar amount of strain as experienced by the fibres. Results for the higher yield stress matrix are shown in Figure 5.6. It can be observed that the experimental points follow the same trend as the F E M curve and they are in very close agreement with the predicted values. Again, it can also be observed in Figure 5.6(b) that since the fibres are softer than the matrix they accumulate strain faster than the matrix. The deformation in the matrix is not as uniform as that in the fibres but still the overall range of localized deformation inside the composite is very low. Areas of the matrix horizontally in between the two fibre experience lower strains than the farfield strain. The area of the matrix at the centre of three fibres experiences the same amount of strain as experienced by the fibres.  46  Figure 5.5 20 triangular composite and matrix having lower a (a) Experimental vs. FEM calculated fibre strain (b) von Mises contour plots of strain distribution in the sample after imposed true strain of 0.25. ys  47  (b)  Figure 5.6 20% triangular composite and matrix having higher a (a) von Mises contour plots of strain distribution in the sample after imposed true strain of 0.25 (b) Experimental vs. FEM calculatedfibrestr ys  48 5.2.1.3 Stresses parallel to the loading direction (022) F E M calculated contour plots of stress parallel to the loading direction are shown in Figure 5.7 for a composite having a lower yield strength matrix and Figure 5.8 for the higher yield strength matrix. From these plots following conclusions can be drawn: i)  Stresses (o ) are compressive in nature at all the places in the sample.  ii)  For lower yield strength matrix, the fibres experience higher compressive  22  stress than the matrix (see Figure 5.7). Fibre stress to the matrix stress is . -290 MPa. approximately ( ). - 2 7 0 MPa iii)  Fibres experience lower compressive stresses than the matrix when they are softer than the matrix (see Figure 5.8). The ratio of fibre stress to the matrix . ,-360MPa, , stress is approximately ( ) the stress distribution is less uniform - 400 MPa than the previous case, i.e. Figure 5.7.  Figure 5.7 Stress distribution along the loading direction (a ) within the composite sample (matrix having 22  lower a ) containing 20% fibre in triangular distribution after the imposed deformation of 0.15 ys  49  Figure 5.8 Stress distribution along the loading direction (a d within the composite sample (matrix having 2  higher a ) containing 20% fibre in triangular distribution after the imposed deformation of 0.15. vs  5.2.1.4 Stresses perpendicular to the loading direction (an) Casel: Matrix lower yield stress The nature of the stresses perpendicular to the loading direction (o"n) is tensile in the fibres and as the imposed deformation increases, the tensile stresses in the fibres also increase. Comparing Figures 5.9(a) and 5.9(b) (i.e. imposed strains of 0.15 and 0.25) it can be concluded that the pattern of stress distribution remains the same with increasing the imposed deformation; only the intensity changes.  50  (b) Figure 5.9 Stress distribution perpendicular to loading direction (o~u) within the composite sample (Matrix having lower a ) containing 20% fibre in triangular distribution after the imposed deformation of (a) 0.15 and b) 0.25 vs  Case2: Matrix higher yield stress The contour plot of stress distribution perpendicular to loading direction ( a n ) in the sample is shown in Figure 5.10 for an imposed deformation of 0.15 and 0.25 respectively. Since the fibres are softer than the matrix, they yield when the composite is initially  51  (b) Figure 5.10 Stress distribution perpendicular to loading direction (a,,) within the composite sample (Matrix having higher 0}J containing 20% fibre in triangular distribution after the imposed deformation of (a) 0.15 and (b) 0.25  52 and as a result compressive stresses develop in the fibres (see Figure 5.10(a)). This behaviour is opposite to Case: 1 where the fibres experience tensile stresses. As the material is deformed further (higher than 20% of imposed strain), the fibres work harden to a strength higher than the matrix and then the nature of stresses in the fibres takes the form of tensile stresses (see Figure 5.10(b)).  5.2.2 20% square samples:  5.2.2.1 Stress-strain response and macroscopic shape change of samples: Figure 5.11 shows the von Mises equivalent stress-strain response for the 20% square arrangement composite. The following observations can be drawn from these curves: i)  Lower c  y s  matrix: good agreement between experimental and F E M curves  along the entire curve. ii)  Higher rj matrix: deviation between experimental and F E M curves from the ys  very beginning. Upon closer examination of the dimensional data for this sample, it was observed that the sample showed considerable amount of out of plane deformation. The reason for this observation is not clear but it would be consistent with the results (i.e. a deviation from plane strain conditions.).  53  500  20% square arrangement 400 A  Matrix lower  (Exp)  —#— Matrix low er (FEM) Matrix higherCT^(Exp) — • — Matrix higher a (FEM)  0 40.0  0.1  0.2  0.3  0.4  Strain Figure 5.11 Stress- strain response of 20% fibre volume fraction composite with square fibre arrangement compared with FEM calculated curves.  The macroscopic shape change of these two samples are shown in Figure 5.12 for 20% square distribution (lower a (higher a  y s  y s  matrix) and in Figure 5.13 for 20% square distribution  matrix). It can be observed from Figures 5.12 and 5.13 that the fibre shape  becomes elliptical when the sample is deformed but the aspect ratio of the ellipse is higher in Figure 5.13 than in Figure 5.12. This observation is similar to the results for the 20% triangular results discussed in Section 5.2.1. Further, it can be observed from Figure 5.12 and Figure 5.13 that for the same level of imposed deformation, fibre debonding is more pronounced for the lower yield strength matrix (see Figure 5.12c) compared to the higher yield strength matrix case (Figure 5.13b).  54  :  0  . % ft d S  •  •  •  p •  i  1  •  •  • •  (b)  1  (c)  Figure 5.12 Sample picture of 20% square distribution and matrix having lower a (a) Before ys  deformation (b) after Step], e=14.5% (c) after Step2, e=26.3%  (a)  (b)  Figure 5.13 Sample picture of 20% square distribution and matrix having higher a (a) Before ys  deformation (b) after Step3, £=24%  55 5.2.2.2 Localized fibre strain and F E M strain distribution in the sample: Results for the low yield stress matrix are shown in Figure 5.14 and for the high yield stress matrix in Figure 5.15. Figure 5.14(a) compares the average value of experimentally measured von Mises strain in fibres to that of F E M calculated values. It can be observed that the experimental points follow the same trend as the F E M curve, however, the experimental results are higher than the model prediction. As can be seen from Figure 5.14(b), a result of fibres being stronger than the matrix is that the fibres have accumulated lower strain than the matrix. The deformation of matrix is not as uniform as that of the fibres. Areas of the matrix vertically and horizontally in between the two fibres experience almost 1.5 times the farfield strain and the area of matrix surrounded by four fibres (center of a unit cell) experiences strain equal to fibre strain (see Figure 5.14(c)). The contour plot reveals that the overall strain variation within the sample is much greater in square geometry sample compared to the samples with triangular fibre arrangement (compare Figure 5.5(b) and Figure 5.14(b)). Further, it can also be observed from Figure 5.14(b) that the area of the matrix at the centre of the four fibres experiences the same amount of strain as experienced by the fibres.  Figure 5.14 (a) 20% square composite and matric having lower o}.,, Experimental vs. FEM calculatedfibre strain  57  PEEQ e. C r i t . : ' + 4 . 0 00 e-01 + 3 . 6 6 7 e - 01 +3 . 33 3e - 0 1 + 3 . 0 0 0 e - 01 +2.66?e-01 + 2 . 3 3 3 e-01 + 2 . 0 0 0e-01 + 1.66 / e- 01 + 1.33 3e-01 + 1.0 0 0 e- 01 + 6.6 6 I e - 0 2 + 0 . 0 0 0e+0 C  Low strain regions in the matrix (strain almost equal to fibre strain)  Large strain regions (40% strain for imposed strain of 25%)  (c)  Figure 5.14 20% square composite and matric having lower Oy (b) von Mises contour plots of strain distribution in the sample after imposed true strain of 0.25 (c) deformation patternfromcontour plot. S  Results of the high yield stress matrix are shown in Figure 5.15. It is important to note here that the fibres have lower yield stress than the matrix but work harden more quickly than the matrix (see Figure 5.1(b)) and for strain levels more than 18% there is a crossover in the two stress-strain curves. Figure 5.15(a) compares the average value of  58 experimentally measured von Mises strain in the fibres to that of the F E M calculated values. It can be observed that the experimental points follow the same trend as the F E M curve and are in a very close agreement with the predicted values. From Figure 5.15(a) it can be observed that for strains greater than 0.15, the strain in the fibres is lower than the farfield strain. Upon comparing with the case discussed previously, i.e. lower a  y s  matrix (see Figure  5.14(b)), the deformation of the matrix for higher rj matrix is also not uniform (see Figure ys  5.15(b)). Areas of the matrix vertically and horizontally in between the two fibres experience almost 1.2 times the farfield strain. The area of the matrix surrounded by four fibres forming a unit cell experience the same amount of strain as experienced by the fibres. For this case, the contour plot reveals that again the overall strain variation within the sample is much greater in square geometry sample as compared to the samples of triangular fibre arrangement (compare Figure 5.6(b) and Figure 5.15(b)). 0.4  20% square composite, Matrix higher ay s 0.3  A  c JO  •  0.2  FEM calculated Slope =1 Experimental data  0.3  0.4  Farfield strain (a) Figure 5.15(a) 20% square composite strain  and matrix having higher a  ys  Experimental  vs. FEM calculated  fibre  Figure 5.15 20% square composite and matrix having higher a (b) von Mises contour plots of strain d in the sample after imposed true strain of 0.25 (c) deformation patternfromcontour plot. vs  60 5.2.2.3 Stresses along the loading direction (022) F E M calculated contour plots of the stress along the loading direction are shown in Figure 5.16 for the lower yield strength matrix and in Figure 5.17 for the higher yield strength matrix. From these plots the following conclusions can be drawn: i)  Stresses (cr ) are compressive in nature at all the places in the sample.  ii)  A very distinct vertical banding is present in both the Figures 5.16 and Figure  22  5.17, which was not the case for triangular geometry samples. iii)  For the lower yield strength matrix, the fibres experience higher compressive stress than the matrix (see Figure 5.16). The ratio of fibre stress to the matrix stress is approximately (—-^10 MPa ^ - 260 MPa  ^ ^  a r e a s  j  n m  e  composite  except the areas of matrix right vertically in between two fibres where the stress ratio is approximately (—^K^MPa  .^ ^  close to 1.  - 3 0 0 MPa iv)  Fibres experience lower compressive stresses than the matrix when they are softer than the matrix (see Figure 5.17). The ratio of fibre stress to the matrix stress is approximately (—^OfJMPa -410MPa  ^  a s  a  m  o  r  e  un  i f o r m stress  distribution than the previous case, i.e. the lower yield strength matrix (see Figure 5.16).  61  Figure 5.17 Stress distribution along the loading direction (a ) within the composite sample (matrix hav higher <r J containing 20%fibrein square distribution after the imposed deformation of 0.15 22  y  62 5.2.2.4 Stresses perpendicular to the loading direction ( a n ) Casel: Matrix lower yield stress The contour plot of the stress distribution perpendicular to loading direction (an) in the sample is shown in Figure 5.18 for the imposed deformation of 0.15 and 0.25 respectively. Stresses within the fibres are tensile in nature and with increasing level of imposed deformation the tensile stresses within the fibre increases. The horizontal banded pattern of stresses is very similar for both Figures 5.18(a) and 5.18(b). Casel: Matrix higher yield stress The contour plot of the stress distribution perpendicular to loading direction (an) in the sample is shown in Figure 5.19 for the imposed deformation of 0.15 and 0.25 respectively. Since the fibres are softer than the matrix, they yield first when the composite is deformed and exhibit compressive stresses in the matrix perpendicular to the loading direction (see Figure 5.19(a)). This behaviour is the opposite to Case:l where the fibres experience tensile stresses. As this material is deformed further (higher than 20% of imposed strain) and the fibres work harden the nature of stresses in the fibres reverses and becomes tensile (see Figure 5.19(b)). These results are similar to the results for triangular arrangements discussed in Section 5.2.1.4. However, the magnitude of the stresses is dependent on the geometric arrangement with the tensile stresses being higher for the triangular arrangement. Further, a clear horizontal banding of the stress can be observed in Figure 5.19(a) which then breaks off as the deformation is increased to 25% in Figure 5.19(b).  Figure 5.18 Stress distribution perpendicular to the loading direction (a,,) within the composite sample (Matrix lowero~ J containing 20% fibre in square distribution after the imposed deformation of (a) 0.15 and b) 0.25 y  Figure 5.19 Stress distribution perpendicular to the loading direction (On) within  the composite sample (Matrix  higher o~ ) containing 20% fibre in square distribution after the imposed deformation of (a) 0.15 and b) 0.25 ys  65  5.2.3 35% square samples: 5.2.3.1 Stress-strain response and macroscopic shape change of samples: The stress-strain response of this material is shown in Figure 5.20 and the macroscopic deformation of sample is shown in Figure 5.21. The following observations can be drawn from these figures: i)  There is good initial agreement between experimental and F E M and then there is a large deviation.  ii)  As the sample deformation increases severe debonding and matrix failure is observed leading to flattening of stress-strain curve (see Figure 5.21(c)).  500  3 5 % square arrangement  CD  D.  400  H  300  H  Matrix lower Matrix lower  0 *0.0  0.1  0.2  0.3  (Exp) (FEM)  0.4  Strain Figure 5.20 Stress- strain response of 35%fibrevolume fraction composite,fibresin  square arrangement.  66  (b)  «  Figure 5.21 Sample picture of 35% square distribution and matrix having lower <j (a) Before deformation (b) after Stepl, e=8.5% (c) after Step3, e=30.4% ys  5.2.3.2 Stresses  along (022) and perpendicular  (an) to the loading  direction  Increasing the fibre volume fraction in the composite affects the intensity of stresses in the various regions but the pattern of stresses remains similar to that of the 20% volume fraction composite. The pattern of stresses inside the 35% volume fraction sample is similar to 20% square composite sample, i.e. compare Figure 5.17 and Figure 5.22 forCT22and Figure 5.18 and Figure 5.23 for a . However, the stresses in the fibres are larger for the u  higher volume fraction sample than for the lower volume fraction at the same level of deformation.  67  Stress distribution perpendicular to the loading direction (an) within the composite sampl (matrix having lower OjJ containing 35%fibrein square distribution after the imposed deformation of 0.1 Figure 5.23  68  CHAPTER: 6 DISCUSSION OF RESULTS This chapter will present the analysis of the results obtained and has been organized in three sections. In the Section 6.1, the mechanical response of the experimental samples will be analyzed in a detailed and comparative manner. In Section 6.2 F E M stresspartitioning results of the samples will be examined. Section 6.3 presents F E M results on virtual composite systems.  6.1 Mechanical response 6.1.1 Effect of geometry Figure 6.1 shows that according to the F E M calculations there is no significant effect of the geometric arrangement of the reinforcements on the overall stress-strain response of the composite. This observation is also consistent with the experimental results on the 20% fibre volume fraction composite containing a matrix of lower yield strength than the fibres (see Figure 6.1). For the 20% triangular arrangement samples having matrix of higher rj , the F E M ys  results again show very little effect due to the geometric arrangement of fibres. However, in this case, the deviation of the experimental results is larger. As pointed out in section 5.2.2.1, the results for square arrangement can probably be explained due to unusually large out of plane deformation. For the triangular arrangement there is initially a good agreement between the model and the experiments but a larger discrepancy is observed at higher strains (s>0.1), i.e. after the lubricant was replaced. These results from the current study on the effect of geometry  600  0.00  0.05  0.10  0.20  0.15  0.25  0.30  True Strain Figure 6.1 Effect of geometrical arrangement offibres composite, Matrix has yield strength than the fibres  on the mechanical response of the  600 20% composite, Matrix higher a,ys  -•—  0.00  0.05  0.10  Triangular arrangement (Exp) Square arrangement (Exp) Triangular (FEM) Square (FEM)  0.15  0.20  0.25  0.30  True Strain Figure 6.2 Effect of geometrical arrangement offibres on the mechanical response of the composite, Matrix has higher yield strength than the fibres.  70  are consistent with the results that Poole et.al. [9] obtained for composite material with a non-deforming reinforcing phase. However, geometry plays a very significant role in the localized deformation behaviour inside the sample. Fibres and matrix deform very differently for different sample geometries (compare Figure 5.5(b) and Figure 5.14(b)). The detailed behaviour is explained in the following subsections. 6.1.2 Strain distribution in the matrix The strain distribution in the matrix is non-uniform for both square and triangular fibre arrangements. However, for comparison purposes the results from the present work on a composite with lower rj matrix are compared with the the results in the literature on the ys  composites having nondeformable fibres (see Section 2.1.2). Suitable literature results are not available to compare with the results of composite having higher a  y s  matrix.  6.1.2.1 Triangular geometry Case 1: Matrix lower Comparing the results of Poole et al. [9] in Figure 2.1 with Figure 5.5(b) it can be concluded that a similar matrix deformation pattern are obtained in case of deforming fibres. The high strain regions exist horizontally in between the two fibres (see Figure 5.5(b)). Case 2: Matrix higher The deformation pattern is almost reverse than the previous case since high strain regions exist inside the fibres now and in certain pockets in the matrix (see Figure 5.6(b)).  71  6.1.2.2 Square geometry Case 1: Matrix lower g . y  It can be observed from Figure 5.14(b) that the high deformation regions in the matrix are located horizontally and vertically in between the two fibres. The center region of the unit cell experiences low strain due to the preferential flow of matrix around fibres and the formation of dead zone in the center. Upon closer examination of these strains, it is observed that the strain at the centre of a unit cell is almost the same as experienced by the fibres. Case 2: Matrix higher o  E  The matrix deformation behaviour for this geometry at 25% of composite strain is the same as that for lower a  y s  matrix (compare Figure 5.14(b) and Figure 5.15(b)). High  deformation regions in the matrix are located horizontally and vertically in between two fibres. 6.1.3 Strain partitioning in the fibres It was observed in section 6.1.2 that the matrix deformation was non-uniform in the various regions of the composite samples. On the other hand, the strain distribution in the fibres is almost uniform during the deformation of a composite sample. Therefore this section deals with the effect of geometrical arrangement on the deformation behaviour of these fibres. Two matrix/fibre property set will be analyzed in the following section:  6.1.3.1 Matrix having lower yield strength than the fibres B y examining the stress-strain response of this property set in Figure 5.1(a), it can be observed that in a composite of this type the fibres have a tendency to deform less than the  72 matrix. As the matrix work hardens, the fibres start to deform and their shape becomes elliptical. Figure 6.5 shows the effect of the geometrical arrangement of the fibres on the strain accumulated in the fibres during the deformation of the composite. As can be seen from this figure, the fibres tend to deform less when they are in square arrangement compared to the triangular arrangement. It is not immediately obvious why this is the case but it must be related to how the flow of the matrix around the fibres differs in these two cases. From Figure 6.5 it can be observed that the slope of the curve decreases with the increased strain in the samples i.e. lesser strain is partitioned to the fibres. It could be related to the decrease in work hardening rate ratio of the constituent phases with imposed deformation. This effect of work hardening ratio on the strain partitioning in the fibres will be examined in more detail in Section 6.3.  Farfield strain  Effect ofgeometrical arrangement offibres on the accumulated strain in the fibres for the composite having matrix of lower a . Figure 6.5  ys  73  6.1.3.2 Matrix having higher yield strength than the fibres This is a very interesting case since the fibres are initially softer than the matrix but they become stronger after a certain deformation of the composite (see Figure 5.1(b)). As a composite of this constituent property set is deformed, initially the fibers deform more than the matrix but after a certain sample deformation a stage will come when the fibre will work harden more than the matrix which will result in a lower strain accumulation in the fibres. Figure 6.6 shows the effect of the geometrical arrangement of the fibres on the strain accumulated in the fibres during the deformation of the composite. For this case it is also observed that the fibres have a lower tendency to deform when they are in square arrangement. 0.4  20% composite, Matrix higher CT ys 0.3  'ro  2  -±s-  0.2  20% Triangular Slope =1 20% Square  0.3  0.4  Farfield strain Figure 6.6 Effect of geometrical arrangement offibres on the accumulated strain in thefibresfor the composite having matrix of higher a . ys  74 6.1.4 Effect of volume fraction on the stress-strain response (FEM calculations) While the experimental results on high volume fraction samples are incomplete, it is worthwhile to compare the F E M calculated results of low volume fraction samples (20% fibres) with the high volume fraction samples (35% fibres). In general, increasing the fibre content in the matrix shifts the stress-strain response of the resulting composite system closer towards fibre stress-strain response. This observation is further confirmed by F E M results in this section. The important observations in the two composite systems are summarized as follows: i)  Composites having lower yield strength matrix: the stress-strain for the higher volume fraction sample shifts up because of the net increase in the volume fraction of the stronger phase (see Figure 6.7(a)). However, the difference between the curves of lower and higher volume fraction samples is very small.  ii)  Composites having higher yield strength matrix: Increase of volume fraction shifts the stress-strain curve of the sample down because of the net increase in the volume fraction of the softer phase. The difference in the stress-strain curves of lower and higher volume fraction composite is larger at the beginning (see Figure 6.7(b)). This difference tends to decrease with the increase in imposed strain and at approximately 20% strain both the stress-strain curves of lower and higher volume fraction fall on top of each other. This is due to the fact that there is a crossover in the fibre and matrix stress-strain curves at about 20% strain. To summarize, there is a very small effect of volume fraction for the fibre/matrix  property set studied in this work. The effect of volume fraction in this work is much less than observed in the work by Poole et al. [9] on composites having non-deforming fibres.  75 600  20% vs 35% composite, Matrix lower oys  #-  20% Square (FEM) 35% Square (FEM) Brass fibres Aluminum matrix  0.30  0.25  600  20% vs 35% composite, Matrix higher a  y s  20% Square (FEM) - • - 35% Square (FEM) Brass fibres •• •• Aluminum matrix  0.00  0.05  0.10  0.15  0.20  0.25  0.30  True Strain (b)  Figure 6.7 Effect of volume fraction offibreson the mechanical response of the composite (a) matrix having lower yield strength than thefibresand (b) matrix having higher yield strength than the fibres.  76 6.2 Stresses inside the fibres A n advantage of the F E M calculations is the ability to examine the stress distribution in the composite. This is useful to understand the debonding behaviour of the samples and the strain response of fibres. If one knows the variation in rji\ with the imposed strain it is possible to predict the debonding behaviour of fibres in the composite sample. Figure 6.8 summarizes the percentage of debonded fibres in the sample with the imposed deformation. A l l the samples have shown good bonding characteristics during the first stage of deformation but as the deformation proceeds fibres start to debond. 100  80  « \  .a - 60 CD •a c o  I—  XI  •g 40  i —  20% s q u a r e , Matrix higher <r, ys 20% s q u a r e , Matrix lower o, ys 20% triangular, Matrix higher o, ys 20% triangular, Matrix lower a , ys  20  10  20  —i—  30  40  Deformation strain in the sample (%) Figure 6.8 Percentage of debonded fibres with imposed deformation, in 20% volume fraction composite samples.  It can be observed from Figure 6.8 that the debonding progresses more quickly in samples having a lower yield strength matrix and for the sample in triangular arrangement. The reason for this behaviour is directly related to the tensile stress generation at the fibre matrix interface during the deformation of the composite. Debonding does not occur i f the  77  stresses at the interface are compressive in nature. Debonding will occur only when these interfacial stresses are tensile and are higher than the bond strength of the interface. If experimental results for channel die compression and macroscopic pictures are analyzed together with the F E M results on the evolution of an, an estimate can be made for the bond strength of the interface.  6.2.1 Stresses perpendicular to loading direction ( a n ) Figure 6.9 presents the effect of constituent properties on the O n values inside the fibres for 20% square and 20% triangular fibre arrangement composite. Qualitatively the evolution of a,, is the same for one type of property set in both Figures 6.9(a) and 6.9(b) irrespective of geometrical arrangement of fibres. Casel: Matrix lower a : a n value is always positive (i.e. tensile stress) and ys  increased with imposed deformation for this composite. The rate of increase of <J\\ is significantly larger for 20% triangular samples than for 20% square sample. The reason why these stresses are positive is due to the fact that lower deformation of fibres results in load transfer to the fibres. The observation, that the magnitude of G\\ in triangular sample increases more quickly for the square sample, suggests that debonding should occur at lower strains in triangular sample, however this observation was not experimentally confirmed. Examination of Figure 6.8 shows that for these samples the debonding occurs for an imposed strain higher than 10%. Based on the F E M results shown Figure 6.9(a), it is then possible to estimate the interfacial strength to be in the range of 15-20 MPa or even less.  78  50  79 Case2: Matrix higher cr  ys:  In this case cs in the fibres is initially compressive in n  nature since the fibres are softer than the matrix. As the deformation of the sample proceeds and the fibre work hardens, the nature of stresses inside the fibres gradually shift towards tension. Figure 6.9(b) shows the G stress becomes positive at lower U  strains in triangular sample than for square sample and as a result one would expect more debonding in triangular sample as observed in the experimental results shown in Figure 6.8. A n important point to note here is that the number of debonded fibres also includes those debonded fibres that are on the edge of the sample. The stress state on the fibres on the edge of the sample is very different than ones in the centre of the sample that makes these fibres more prone to debonding. This may explain why some debonding is observed for the square arrangement even though the F E M results show very low interfacial stress for this case.  6.2.2 Effect of volume fraction on the stresses ( a n ) inside the fibres  Further, the effect of volume fraction on the stress perpendicular to loading direction is shown in Figure 6.10. Again considering the two cases: 1.  Sample having matrix of lower a : In Figure 6.10(a) both the curves start off together ys  and as the deformation proceeds the higher volume fraction composite takes up higher stresses. One important thing to note here is that a n is always of tensile nature in the fibres for this combination of fibre/matrix properties. This leads to the conclusion that debonding in 35% square sample should occur earlier than the 20% square sample. The experimental results confirm this effect, although this appears to be a rather small effect in the F E M calculations.  80  60  Composite sample have matrix lower cr,ys "i? 0L  50  40  30  20  20% square 35% square  10  0.00  0.05  0.10  0.15  0.20  0.25  0.30  Strain (a) 60  Composite samples have matirx of higher a,ys 40 cn £  20  .=  -20  20% square 35% square  -40  -60 0.00  0.05  0.10  0.15  0.20  0.25  0.30  Strain (b)  Figure 6.10 Effect of volume fraction on the stresses perpendicular to direction (cr,,) inside the fibres (a) composite having matrix of lower yield strength than the fibres and (b) composite having matrix of hig yield strength than thefibres.  81  2. Sample having matrix of higher  cr : ys  Figure 6.10 (b) illustrates that with increase in  volume fraction, 0"n is less compressive in the beginning and as the deformation proceeds the difference tend to decrease. A stage comes during the deformation when both low and high volume fraction curves overlap and the nature of the stress approaches tension. This leads to the conclusion that the debonding behaviour for 35% square sample should be very close to the 20% square sample. However, there are no experimental results in the current work to confirm this prediction.  6.3 FEM model results on virtual composite properties  with variable fibre/matrix  Having validated the F E M models against the experimental results, it is of interest to consider some virtual two-phase materials and execute these models on these materials in an attempt to develop more generalized knowledge. In addition to the understanding developed from Unckel's experimental work, described in Section 2.1.2.1, which shows the dependence of relative work hardening rate on the strain partitioning to the second phase it will be interesting to analyze this effect in more detail using the F E M model. Therefore two sets of idealized materials were considered where the yield strength ratio between the fibre and the matrix was varied between 1 and 1.25. The work hardening behaviour of the fibre and the matrix was assumed to be linear and the effect of different relative work hardening rates was examined by keeping the same stress-strain response of the fibres and only varying the hardening rate of the matrix in one sample set. These F E M simulations were conducted on 20% square fibre arrangement samples.  82 6.3.1 Sample set 1: Matrix and fibres have same yield strength Fibre/matrix property sets for four samples are shown in Figure 6.11 and the strain in the fibres vs. imposed deformation has been analyzed in Figure 6.12. From Figure 6.12(a) it may be concluded that as the strain hardening rate of the matrix approaches the strain rate of the fibres, the strain accumulated in the fibres increases. Figure 6.12(b) is a refined representation of Figure 6.12(a) i.e. as the work hardening ratio between the phases approaches to unity, equal strain partitioning in between the phases is observed as would be expected from a material where the constituent phases are the same. 1000  800 H CD Q.  600  CO  £  CO CD  3  400  200  0.4  0.6  0.8  True Strain (plastic) Figure 6.11 Fibre/matrix property set wherefibresand matrix have the same yield strength. Keeping thefibrework hardening (6) the same and varying the work hardening of matrix from 0.15 6 to 0.656.  Figure 6.12 (a) Fibre strain  vs. farfield strain offour samples (b) strain ratio (fibre/farfield) vs. wor hardening ratiof matrix/fibre)  84 6.3.2 Sample set 2: Fibres have 1.25 times higher yield strength than matrix Fibre/matrix property sets for four samples are shown in Figure 6.13. From Figure 6.14 it can be concluded that the fibres will not start deforming as the composite is deformed because the yield strength of two constituent phases are different. The delay in plastic deformation of the fibres depends upon the relative work hardening ratio of matrix/fibre. The higher the work hardening ratio the earlier the fibres will start to deform (see Figure 6.14b). Once the fibres start deforming they tend to follow a similar pattern (slopes) as shown in the previous case (compare Figure 6.14(a) and Figure 6.14(a)).  1000  Matrixl ,0.150 Matrix2 , 0.279 Matrix3,0.459 Matrix4 ,0.659  Fibre work hardening = 9 0 0.0  0.2  0.4  0.6  0.8  1.0  1.2  True Strain (plastic)  Figure 6.13 Fibre/matrix property set where fibres 1.25 times yield strength of the matrix. Keeping the fibre work hardening (0) the same and varying the work hardening of matrix from 0.156 to 0.656.  0.25  Matrixl , 0.159  -•-  Matrix2 , 0.278 Matrix3 , 0.450 Matrix4 , 0.659  0.00  0.05  0.10  0.15  0.20  0.25  Farfield strain (a)  0.09 -i  0.02 H 0.1  1  1  1  1  1  0.2  0.3  0.4  0.5  0.6  Work hardening ratio (e  matrix  /9  fibre  1 0.7  )  (b) Figure 6.14 (a) Fibre strain vs. farfield strain offour samples (b) Farfield strain required for fibres to start deforming vs. work hardening ratio.  86  CHAPTER: 7 CONCLUSIONS AND FUTURE WORK 7.1 Conclusions The primary objective of this work was to develop an understanding of the effect of geometrical arrangement and constituent properties on the deformation behaviour of ductile two-phase model composite systems. Model composites were fabricated by co-extrusion where AA6061 age hardenable aluminum alloy was used as the matrix phase and C3600brass fibres were used as the reinforcement phase. The composite was deformed in plane strain compression with the fibres perpendicular to the loading axis. An important conclusion can be drawn regarding the effect of geometrical arrangement of fibres to the overall stress-strain response. It was confirmed experimentally as well as with F E M results that the geometrical arrangement does not play significant role to the overall stress-strain response of the composite. Strain partitioning to the fibres was studied experimentally and with F E M models. Experimental results on strain partitioning match very well or follows a similar trend as F E M results. It was also concluded with the F E M results that the strain partitioning in the fibres is a function of the geometrical arrangement of the fibres in a composite. Fibres in the square arrangement tend to deform less than the triangular arrangement during composite deformation. With F E M results it was concluded that increasing the fibre volume fraction from 20% to 35% in the composite does not significantly change the stress-strain response. In other words for the fibre/matrix property set studied in this work, the volume fraction of  87  fibres has a rather a small effect on the overall stress-strain response. However, the local stress distribution in the higher volume fraction composite has stresses of higher magnitude. Stresses perpendicular to the loading direction at the fibre/matrix the interface were calculated from F E M results. From the evolution of these stresses and the debonding behaviour of the samples the bond strength of the interface can be roughly estimated. From these calculations the interface bond strength in these composite lies in the range of 15-25 MPa. Finally, F E M simulations on samples with idealized constituent work hardening behaviour were conducted. It was observed from these simulations that the strain partitioning to the fibres is a function of relative yield strength and relative work hardening rate of the constituent phases.  7.2 Future work Based on this study the following are some of the topics which may be considered for future work: 1. Due to time constraints of this project, it was not always possible to repeat every test. It would be worthwhile to conduct a series of additional tests, especially on the samples where the matrix had higher yield stress than the fibres. 2. Due to the non-homogeneous deformation, particularly in the matrix phase it would be of interest to conduct the texture development  studies and also to examine the  recrystallization behaviour in these composite samples. 3. Having validated the F E M model for perfectly bonded fibre/matrix interface, more complex models could be generated which take into account the debonding phenomenon  88 at the interface. The resulting knowledge would be helpful in failure studies of these composite systems. Having validated the F E M model for simpler geometries such as square and triangular arrangement, more complex models could be generated which take into account the effect of clustering and size distribution of the reinforcing phase.  89  REFERENCES 1) S. Esmaeili, D.J. Lloyd and W.J. Poole, Acta Mater., 2003, 5 1 , pp. 2243-3357. 2) M . Nembach, Particle strengthening of metals and alloys, John Wiley and sons: New York, 1996. 3) R. Ebling and M . F . Ashby, Phil. Mag. A, 1966, 13 , pp. 805-834. 4) E. Orowan, Symposium on internal stresses in metals and alloys, Institute of metals: London U K , 1948, pp. 451-453. 5) M . F . Ashby,  Proc.  Second Bolton  Landing  conference on  Oxide  Dispersion  Strengthening, Gordon and Breach: New York N Y , 1966, pp. 143-212. 6) T.W. Clyne and P.J. Withers, An introduction to metal matrix composites, Cambridge University Press: Cambridge, 1993. 7) C. 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