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Modelling the effect of grain size distribution on the mechanical response of metals Raeisinia, Babak 2008

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MODELLING THE EFFECT OF GRAIN SIZE DISTRIBUTION ON THE MECHANICAL RESPONSE OF METALS  by  BABAK RAEISINIA B.Sc., Tehran University, Iran, 2000 M.A.Sc., The University of British Columbia, 2003  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADAUTE STUDIES (MATERIALS ENGINEERING)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  November 2008  © Babak Raeisinia, 2008  Abstract Recent experimental studies have pointed to the potential for producing metals with improved mechanical properties based on manipulation of the grain size distribution. It is, however, unclear how these improvements are brought about and whether grain size distribution manipulation can effectively be used to tailor the mechanical response of metals. In this work, these issues are examined using two novel grain size dependent self-consistent models, an elastoplastic and a viscoplastic, where plasticity is assumed to occur by dislocation slip.  For this purpose, monotonic deformation of a number of model f.c.c.  polycrystals with moderate stacking fault energy (such as copper) is examined. Polycrystals with lognormal distributions, having average grain sizes ranging from 100 nm to 50 p.m, and bimodal distributions are considered. It is found that increasing the width of the lognormal grain size distribution, while keeping the average grain size constant, decreases the yield strength of the polycrystal and increases its work hardening rate. This behaviour is attributed to the increasing volume fraction of grains larger than the average, as the distribution is widened, which have lower threshold stresses and higher work hardening rates than the average. The simulation results are summarized in the form of new property maps where the range of ultimate tensile strength-uniform elongation combinations which can be achieved through grain size distribution manipulation are shown. These maps also demonstrate that bimodal polycrystals demonstrate better overall properties as compared to lognormal polycrystals. The observed grain size distribution effects are, however, found to be dependent on the nature of the constitutive relationship assumed for the grains. The developed maps provide a first guide  11  for materials engineers interested in the modification of the mechanical properties of polycrystals through grain size distribution manipulation.  111  Table of Contents Abstract  .  Table of Contents List of Tables List of Figures List of Symbols Acknowledgements Chapter 1  -  ii iv  vii Viii XV  xix  Introduction  1  Chapter 2 Literature Review -  5  2.1. Introduction  5  2.2. Grain Size Effects in Polycrystals  6  2.2.1. Mechanisms of Plastic Flow  6  2.2.2. Yield Strength  7  2.2.3. Work Hardening 2.3. Grain Size Distribution Effects in Polycrystals  11 18  2.3.1. Grain Size Distribution in Polycrystalline Metals 2.3.2. Experimental Studies  20  2.3.3. Modelling Studies  25  2.4. Polycrystal Deformation Models Based on Homogenization Theori es 2.4.1. Local Constitutive Behaviour  18  28 29  2.4.2. Homogenization Theories  31  2.4.3. Comparison with Other Modelling Approaches  35  2.5. Summary: Challenges and Opportunities Chapter 3  -  Scope and Objectives  Chapter 4 Model Development -  4.1. Introduction 4.2. Model Polycrystals 4.2.1. Grain Size Distribution 4.2.2. Texture 4.3. Self-Consistent Formulations  36 38 40 40 40 41 43 46  iv  4.3.1. The Elastoplastic Self-Consistent (EPSC) Model 4.3.2. The Viscoplastic Self-Consistent (VPSC) Model  47  48 4.3.3. Incorporating Grain Size Dependence 50 Chapter 5 Simulation of Polycrystal Deformation Using the Elastoplastic Self-Consistent Model 58 5.1. Introduction 58 5.2. Methodology 58 5.3. Results 59 5.4. Discussion 62 5.4.1. Grain Size Distribution Effect 62 -  5.4.2. Modelling the Evolution of Volume Fraction of Yielde d Grains 5.4.3. Characterization of Yielding 5.4.4. Impact of the Local Hardening Law 5.5. Summary Chapter 6  -  65 71 78  82 Simulation of Polycrystal Deformation Using the Viscoplastic Self-Consistent  Model  84  6.1. Introduction 6.2. Methodology 6.3. Results 6.3.1. Unimodal Polycrystals 6.3.2. Bimodal Polycrystals 6.4. Discussion 6.4.1. Grain Size Distribution Effect 6.4.2. Comparison of Unimodal and Bimodal Grain Size Distrib utions 6.4.3. Model Assumptions and Variables  84 85 86 86 90 92 92 95 98  6.4.3.1. Yield Strength Definition  99  6.4.3.2. Spatial Distribution of Grains  99  6.4.3.3. Grain-Matrix Interaction  102  6.4.3.4. Local Hardening Relationship  104  6.4.4. Model Validation 6.5. Summary  111 116 V  Chapter 7 Concluding Remarks  118  7.1. Summary of Observations  118  7.2. Future Work  121  -  References Appendix Al Appendix A2 Appendix A3 Appendix A4 Appendix A5  123 —  —  —  —  —  Kocks-Mecking Strain Hardening Model  133  Lognormal Grain Size Distributions in Metals  137  EPSC and VPSC Polycrystal Models  140  Volume Fraction Probability Density Function  149  Afanesev Theory of Microdeformation in Polycrystals  151  vi  List of Tables Table 2.1. Different relationships predicted for the Hall-Petch slope  10  Table 2.2. Commonly used models in studies of polycrystal deformation  36  Table 4.1. Values of parameters in Equation (4.16) used in the analysis  55  Table A2. 1. Examples of experimentally determined lognormal grain size distributions.... 137 Table A2.2. Parameters relating lognormal distributions of grain weights (m) and grain volumes (V) to lognormal distribution of grain diameters (D) 138  vii  List of Figures Figure 2.1. Formation of dislocation forests near grain boundaries as hypothesized in the models of Li [48] and Meyers and Ashworth [52] 9 Figure 2.2. Room temperature work hardening rate plotted against flow stress for copper of different grain sizes. Curves replotted using data in [8 1,82] 15 Figure 2.3. Compressive stress-strain curve of copper with approximately 200 rim grain size. Curve plotted using data in [87] 17 Figure 2.4. Schematic representation of grain diameter lognormal distribution on (a) linear and (b) logarithmic scales. The dashed line delineates the mean grain size of the distribution. Note the Gaussian (normal) distribution of grain sizes in (b). P(D)dD denotes the fraction of grains which fall between D and D + dD 19 Figure 2.5. (a) Room temperature tensile stress-strain curves for Cu with different grain structures: fine-grained (curve A), coarse-grained (curve B), and bimodal (curve C). Grain size distributions corresponding to stress-strain curves A and C are shown, respectively, in (b) and (c). Figures replotted using data provided in [91,110] 21 Figure 2.6. Electron microscope micrographs of a bimodal aluminum specimen in the (a) asprocessed and (b) as-deformed states. The ultrafine-grained (ufg) matrix and the embedded microcrystalline (mc) grains are evident in (a). The extensive slip activity in the coarse grain of the deformed sample is clear in (b). Micrographs reproduced with permission from [109]. 23 Figure 2.7. Electron microscope micrograph of the surface of a bimodal aluminum sample, same sample shown in Figure 2.6, showing a crack stopped by a micron size grain. Image reproduced with permission from [109] 24 Figure 2.8. Calculated Hall-Petch relationship for copper polycrystals with varying lognormal grain size distributions. The numbers on each line are indicative of the width of the size distribution with larger numbers corresponding to wider distributions. The decrease in the slope of the lines (i.e. the Hall-Petch slope) with the width of the grain size distribution is evident. Lines replotted using data provided in [98,116] 26 Figure 2.9. Polycrystal response, P, described in terms of constitutive relationships of its grains, g , using homogenization theory H. Note that ci. and 1 are the grain stress and strain while o and are the polycrystal stress and strain values 29 Figure 2.10. Schematic representation of the (1-site) self-consistent homogenization scheme. The grains are first approximated by their equivalent ellipsoidal inclusions. Each inclusion is then assumed to be embedded in the Homogeneous Equivalent Medium (HEM) whose properties are those of the polycrystal itself 33 viii  Figure 4.1. Generated lognormal grain size distributions. The grain size values, D, are normalized by the mean grain size of the distribution, p. The o-/p ratio noted on each plot is a measure of the width of the distribution 42 Figure 4.2. Three Euler angles angles 4)1, and 4)2 relating the sample reference frame, XYZ, to crystal reference frame, xTz’. 4)1 is the angle between the X axis and the projection of z’ axis onto the XI plane, 1 is simply the angle between the Z and Z’ axes, and 4)2 is the angle between the Y’ axis and the zone axis of the two planes whose normals are the Zand Z axes 44 ,  Figure 4.3. Equal-area projections of [100], [110], and [111] poles of 2000 discrete orientations generated by the aforementioned approach. Each point on the images represents a single orientation or grain. X and Y denote the sample axes 45 Figure 4.4. Density distribution of generated crystal orientations. The peaks on distribution profiles with in excess of 1000 orientations occurs at multiples of average density (MAD) ranging from 0 to 1.5 45 Figure 4.5. Schematic representation of(a) the evolution of threshold stress with accumulated strain in the grain for a number of different grain sizes and (b) the hardening rate (8 = dr /dr) as a function of threshold stress for a given grain size D as described by Equation (4.16) 53 Figure 4.6. Experimental tensile stress-strain curves from corresponding VPSC fit (a). For the VPSC fits, Equation hardening law and polycrystals having grain size distributions to those reported for the experiments and o-/i ratios equal to the corresponding experimental work hardening rates  [81,82] compared with their (4.16) is used as grain-level with average grain sizes equal 0.4 are used. The inset shows 54  Figure 4.7. Work hardening rate plotted against flow stress for IF steel of different grain sizes tested at room temperature. Rates calculated from stress-strain data in [163] 56 Figure 5.1. Calculated von Mises equivalent stress-strain and work hardening rate, e, curves for polycrystals with 300 nm, (a) and (c), and 20 jim, (b) and (d), mean grain sizes and varying widths of distribution. E denotes Young’s modulus 60 Figure 5.2. The evolution of volume fraction of grains yielded as a function of polycrystal von Mises equivalent strain for two polycrystals with (a) 300 nm and (b) 20 tm mean grain sizes and varying widths of distribution 61 Figure 5.3. Generated lognormal grain size distributions shown in terms of number fraction and volume fraction of grains. The volume fraction distribution is plotted using Equation (5.3) 63  ix  Figure 5.4. Predicted yield strength as a function of square root of mean grain size for a variety of unimodal polycrystals with varying distribution widths. Yield strength defined as the 0.2 % offset stress. Slope of the lines decreases from 0.24 MNm 312 at go/p 0 to 0.14 2 at /t 0.8 ” 3 MNm 64 =  Figure 5.5. The effect of (a) mean grain size and (b) o-/p ratio of lognormal grain size distribution on the evolution of volume fraction of yielded grains as predicted from Equation (5.9) 67 Figure 5.6. Fit of Equation (5.9) (dashed lines) to EPSC predicted volume fraction of yielded grains for polycrystals with varying widths of grain size distribution and (a) 300 nm and (b) 20 J.tm mean grain size 68 Figure 5.7. Comparison of (a) mean grain size and (b) standard deviation determined from fit to model predicted volume fraction of yielded grains (p and and values used as input in the model (‘input and O_jnput) Dotted lines represent cases where the input and fitted values are equal  69  Figure 5.8. The evolution of volume fraction of yielded grains with macroscopic strain for when all the grains have the same crystallographic orientation (dashed lines) and when the grains are randomly orientated (solid lines). The mean grain size of the polycrystals is 300 nm 71 Figure 5.9. Hall-Petch plots for polycrystals with varying grain size distributions (uo/p ratios noted on each plot). For each case, three different definitions of yield strength are used. True yield strength refers to stress at which 99 % of the volume of the polycrystal is plastic/yielded 73 Figure 5.10. Comparison of the model predicted volume fraction of yielded grains (solid lines) and that determined using Equation (5.13) (dashed lines) for polycrystals with (a) 300 nm and (b) 20 m mean grain size 75 Figure 5.11. Comparison of the Hall-Petch plots for polycrystals with different grain size distributions (u /p ratios noted on each plot) when the true yield strength of the polycrystals 0 and the yield strength determined through the Saada approach is used 77 Figure 5.12. Variation of the Hall-Petch slope with c/p when different definitions of yield strength are used 78 Figure 5.13. (a) Calculated von Mises equivalent stress-strain curves for polycrystals with 300 nm mean grain size and varying width of distribution. (b) Predicted yield strength as a function of square root of mean grain size for a variety of polycrystals with varying distribution widths. Yield strength defined as the stress at which the volume fraction of  x  yielded grains reaches 0.99. Local hardening law where only the onset of plasticity is grain size dependent in each grain is used 79 Figure 5.14. Comparison of the model predicted volume fraction of yielded grains (solid lines) and that determined using the Saada approach (dashed lines) for polycrystals with (a) 300 rim and (b) 20 jim mean grain size. Local hardening law where only the yielding of the grains is size dependent is used 80 Figure 5.15. Comparison of the Hall-Petch plots for polycrystals with different grain size distributions ( /,u ratios noted on each plot) when the true yield strength of the polycrystals 0 and the yield strength determined through the Saada approach are used 81 Figure 6.1. Computed von Mises equivalent stress-strain curves for a number of polycrystals with varying mean grain size. The polycrystals do not have a size distribution (i.e. Jo//I 0). =  86 Figure 6.2. Predicted yield strength as a function of square root of mean grain size for a variety of unimodal polycrystals with varying distribution widths. Yield strength defined as the von Mises equivalent stress calculated after the first deformation step. The slope of the lines decreases from 0.23 MNm 312 at o-/p 0 to 0.13 MNm 312 at cro/p 0.8 87 =  =  Figure 6.3. Calculated von Mises equivalent stress-strain curves and work hardening plots of polycrystals with 700 rim, (a) and (c), and 10 jim, (b) and (d), mean grain sizes and varying widths of distribution. In (c) and (d) the ® lines correspond to the Considère criterion. =  88 Figure 6.4. Predicted relative von Mises equivalent stresses and strains as a function of grain size for three model polycrystals at three different stages of deformation corresponding to macro von Mises equivalent strains of 0.01, 0.10, and 0.50. The mean grain size of the polycrystals and the width of their grain size distribution (o-/p) is indicated in the images. The top row corresponds to the relative strain values and the bottom row to the relative stress values. Local refers to grain-level values 89 Figure 6.5. The effect of adding different volume fractions of coarse 3 jim grains to the mix of fine 200 nm grains as predicted by the model. Percent fractions are noted on the figure. The 0 % and the 100 % indicate polycrystals with unimodal grain sizes of 200 rim and 3 jim, respectively 91 Figure 6.6. The variation of (a) ultimate tensile strength and (b) uniform elongation as a function of the volume fraction of coarse constituents for a number of bimodal polycrystals having 200 nm fine and 1 or 10 jim coarse constituents 95 Figure 6.7. Ultimate tensile strength versus uniform elongation predicted for different unimodal and bimodal polycrystals. The unimodal polycrystals are of different average grain sizes (p) and grain size distribution widths (cro). The bimodal polycrystals, delineated by the xi  dashed lines, all have 200 nm grains as their fine constituents and either 1 or 10 m coarse constituents. Arrows on the dashed lines show the direction of increase in the volume fraction of the coarse constituents of the bimodal polycrystals (see Figure 6.6 for details). Note that in this plot p values ranging from 100 nm to 50 pm are used and for each case, u 0 is determined by varying the cr /p ratio from 0 to 0.8 at 0.2 increments. Polycrystals with no 0 uniform elongation are not depicted in this plot 96 Figure 6.8. Plot of ultimate tensile strength versus uniform elongation for different bimodal polycrystals having fine constituents of 100 or 300 rim with coarse constituents of 1 or 10 tm. Open symbols/solid lines correspond to polycrystals with 10 im coarse grains and solid symbols/dashed lines refer to polycrystals with 1 im coarse grains. The arrows delineate the direction of increase in volume fraction of coarse grains 97 Figure 6.9. Variation of the Hall-Petch slope with uc/p as predicted from the elastoplastic and viscoplastic simulations. The 0.2 % offset definition is used for the elastoplastic simulations 100 Figure 6.10. Predicted von Mises equivalent stress-strain curves for two bimodal polycrystals consisting of 70 volume percent 3 trn grains arid 30 volume percent 200 nm grains with fine and coarse-grained elements of the polycrystal arranged in equal-strain and random configurations. The stress-strain curves are plotted to the end of uniform elongation as calculated with the Considére criterion. The inset schematically shows the equal-strain configuration. The white and gray colors in the inset represent the coarse and fine-grained constituents of the bimodal polycrystal, respectively 100 Figure 6.11. The effect of grain-matrix interaction on predicted von Mises equivalent stressstrain response of a polycrystal with mean grain size of 700 nm and cr /p ratio of 1. Note 0 that in this analysis a value of 20 is used for n instead of 65 to keep the numerical convergence of the Second Order interaction manageable 103 Figure 6.12. The effect of grain-matrix interaction on partitioning of strain among grains of a polycrystal with mean grain size of 700 nm and cr /p ratio of 1. The plot corresponds to 0 macroscopic von Mises equivalent strain of 0.01 103 Figure 6.13. The influence of the local hardening law on the ultimate tensile strength-uniform elongation relationship for a number of unimodal polycrystals. The black and white symbols correspond to o-/i ratios equal to 0 and 0.8, respectively 105 Figure 6.14. Predicted von Mises equivalent stress-strain curves for a number of unimodal polycrystals having Equation (6.2) as their local constitutive behaviour. The corresponding work hardening rate for the curves, eeq, is also depicted 106 Figure 6.15. Ultimate tensile strength versus uniform elongation for different unimodal ( /p 0 ratio varying from 0 to 0.8) and bimodal (mixture of 2 tm coarse and 100 nm fine grains) xii  polycrystals. Percent values indicate the percent volume fraction of coarse 2 jim grains. The 0 % and the 100 % correspond to polycrystals with unimodal grain sizes of 100 nm and 2 jim, respectively. The arrow points to unimodal polycrystal with mean grain size of 240 nm and co/p 0.8 whose ultimate tensile strength is approximately the same as 70 % bimodal polycrystal 107 =  Figure 6.16. Comparison of the von Mises equivalent stress-strain curves and work hardening rates, ecq, for unimodal polycrystal with mean grain size of 240 nm and o/p 0.8 and 0 bimodal polycrystal with 70 volume percent 2 jim and 30 volume percent 100 nm grains. 108 =  Figure 6.17. Predicted relative von Mises equivalent strains as a function of grain size for unimodal polycrystal with mean grain size of 240 nm and o/p 0.8 at three different stages of deformation corresponding to macro von Mises equivalent strains of (a) 0.01, (b) 0.10, and (c) 0.5. The average strain in the coarse (2 tim) and fine (100 nm) grains of 70 % bimodal polycrystal having approximately the same ultimate tensile strength is also indicated on the plots as dashed lines 109 =  Figure 6.18. The effect of local Hall-Petch parameter, k, on ultimate tensile strength-uniform elongation relationship for a number of unimodal polycrystals. The black and white symbols correspond to k = 0.094 MNm 2 and k = 0.188 MNm ” 3 , respectively. Equation (4.16) is 312 used as local hardening law, i.e. both the yield strength and work hardening are grain size dependent 110 Figure 6.19. Experimental tensile stress-strain curve of copper with an average grain size of 400 nm from the work of Shen et al. [174] compared with its corresponding VPSC simulation (‘a). The grain size distribution used for the simulation has an average of 400 nm and cr /p ratio of 0.55. The inset shows the corresponding work hardening rate for the 400 0 nm copper of Shen et al. plotted along with the work hardening rates of copper reported in [8 1,82]. Young’s modulus of copper is delineated by the dashed line. The point where the work hardening rate of the 400 nm sample collapses on top of the other work hardening curves is demarcated by the arrows on both the stress-strain and work hardening plots 112 Figure 6.20. Comparison of experimental results of Wang et al. [91,110] with VPSC model predictions. Data points labeled A correspond to copper sample with unimodal grain size distribution and those labeled B to copper sample with bimodal grain size distribution. The inset shows the experimental grain size distributions for the unimodal and bimodal copper samples. Note that the original grain size distribution corresponding to sample A was provided as number fraction and here, by assuming spherical grains, the number fraction is converted to volume fraction 114 Figure A1.1. Schematic representation of(a) Equation (Al.4) and (b) Equation (A1.5). The insets show plots of the strain hardening rate of each equation as a function of stress 135 Figure AS.1. Schematic illustration of the probability density function describing volume fraction distribution of grains with yield strengths c-.. For this plot, the polycrystal grain size xiii  distribution is assumed to be lognormal and the grain size and grain yield strength are assumed to be related through the Hall-Petch relationship 151  xiv  List of Symbols localization tensor b  magnitude of Burgers vector Burgers vector of system S  CJk/  elastic stiffness tensor  D  grain size critical grain size  E  Young’s modulus  EPSC  elastoplastic self-consistent model  F (D)  probability density function of grain volume fractions  f  volume fraction of grains  1oarye  volume fraction of coarse grains  fe  volume fraction of elastic grains volume fraction of fine grains  ffine  volume fraction of yielded grains G  shear modulus  G (o)  probability density function of volume fraction of grains with yield strengths 1 cT  1 g  constitutive relationship of grain i  H  homogenization theory  HEM  homogeneous equivalent medium self-hardening rate on system S  hss  hardening moduli  h” (S  5’)  latent-hardening rate of system S caused by slip on system S’  ‘jk1  identity tensor  K k  Hall-Petch slope  ,  1 k  ,  ,  3 K 1c 3 ,  work hardening model constants xv  geometric mean free path Luk,  grain elastoplastic stiffness tensor  L]kl  polycrystal elastoplastic stiffness tensor  Lkl  elastoplastic interaction tensor  MD  mean of the logarithnis of grain sizes  M/kt  grain viscoplastic compliance tensor mean of the logarithms of grain weights mean of the logarithms of grain volumes  MAD  multiples of average density polycrystal viscoplastic compliance tensor viscoplastic interaction tensor  in  total length of dislocations emitted per unit area of grain boundary; weight  mS  Schmid factor associated with system S  rn.  Schmid tensor associated with system S  N  number of grains  N 0 1  total number of grains  n  rate-sensitivity exponent  n  parameter controlling strength of interaction of grain with HEM nonnal vector to slip plane S  P  constitutive relationship of polycrystal  P (D)  probability density function of grain sizes  q’  latent-hardening matrix  S  standard deviation of the logarithms of grain sizes Eshelby tensor standard deviation of the logarithms of grain weights  S  standard deviation of the logarithms of grain volumes  xvi  V  volume volume of coarse grains  Vfi,ze  volume of fine grains  VPSC  viscoplastic self-consistent model  XYZ  sample reference frame  X’Y7’  crystal reference frame strain  6 e q  von Mises equivalent strain strain of grain i grain strain tensor  ç  characteristic strain applied strain rate average plastic strain rate polycrystal strain polycrystal strain tensor grain strain rate tensor applied strain rate tensor polycrystal strain rate tensor  F  cumulative strain in grain  0 F  grain size dependent characteristic strain  y  shear strain shear strain on system S shear strain rate on system S reference shear strain rate on system S  9  polycrystal work hardening rate  9  local work hardening rate initial work hardening rate (local)  xvii  01  asymptotic work hardening rate (local)  OS  work hardening rate on system S  p  arithmetic mean grain size  v  Poisson’s ratio  p  dislocation density; density stress standard deviation von Mises equivalent stress  eq  lattice friction stress, stress of grain i grain stress tensor yield stress grain deviatoric stress tensor grain stress rate tensor polycrystal stress polycrystal stress tensor polycrystal deviatoric stress tensor polycrystal stress rate tensor shear stress Peierls-Nabarro lattice friction back-extrapolated threshold stress  +  mechanical threshold stress on system S resolved shear stress on system S saturation stress critical stress for propagation of slip across a grain boundary mechanical threshold stress rate on system S l ,1 42 ,  Euler angles  xviii  Acknowledgements During the course of this work, not only have I grown immensely as a researcher, but more importantly, as a person. I thank God foremost for providing me with this opportunity. I would also like to express my utmost gratitude to the many people who made this thesis possible. I am forever indebted to my supervisors Dr. Warren J. Poole and Dr. Chad W. Sinclair for their continuous trust, support, and encouragement. I especially want to thank them for always (even at 9 p.m. on a Friday night) being there to hear my seemingly endless problems and questions. My sincere thanks goes to Dr. Carlos Tome at Los Alamos National Laboratory for kindly providing me with the EPSC and VPSC packages.  His  constructive advice and expertise were instrumental in getting me started on this project and are extremely appreciated. I also wish to thank Dr. Paul R. Dawson from Cornell University for the stimulating and fruitful discussions through which I learned to view materials problems from the continuum perspective. I would like to express my gratitude to all my friends for their helpfulness throughout this work and also for providing the friendly environment in our group.  My special thanks to Fateh Fazeli, Henry Proudhon, André  Phillion, Reza Roumina, Hamid Azizi-Alizamini, Leo Colley, and (by his own request) Sujay Sarkar for all the help and fruitful discussions. Last but not least, I would like to thank my family, in particular my mom and dad, for their continuous support, encouragement, understanding, and sacrifice during the period of this work. Thank you, thank you, thank you. This thesis is dedicated to you.  xix  CHAPTER 1 Introduction There are different ways through which metals can be strengt hened. One of the most rudimentary (i.e. does not require any alloying additions) and well studied of these methods is grain size refinement. The strengthening effect of grain bound aries has long been realized and the Hall-Petch relationship, which relates the yield strengt h of a metal to its average grain size, is one of the most widely recognized relationships in materials science. This basic approach to strengthening metals has benefited from a resurgence of interest in recent years. This has primarily been spurred by technological advancements in processing of metals with average grain sizes below 1 jim. These ‘new’ metals possess interes ting properties including high strength. The introduction of these fine grained metals as a new generation of highstrength engineering material has, however, proven to be a challen ging task. In addition to a number of processing and manufacturing challenges, these materi als have been shown to exhibit limited uniform elongation, usually less than 5 % in tension , making them unfit for conventional forming operations or for use in load bearing components.  The low work  hardening capability in these fine grained metals, or their limited ability to sustain work hardening beyond small strains, has been identified as the cause of their reduced uniform elongation. In polycrystals with grain sizes larger than 1 jim, the yield strengt h and work hardening response are generally insensitive to the character of the grain size distribution of the polycrystal. This is evident from the success of the Hall-Petch relatio nship, which only depends on the average of the grain size distribution. However as the average grain size is  1  refined, it has been found experimentally that the mechanical response of the polycrystal demonstrates an increased sensitivity to the character of its grain size distribution. Thus, grain size distribution can potentially be used as an additional variable in controlling the mechanical response of polycrystals. Accordingly, the notion of tailoring the grain structure of fine grained metals has been put forth as a means to increase their low work hardening capability. Tailoring the grain structure essentially means that a fine grained polycrystal be treated as a composite material with hard fine grains and soft coarse grains. The mechanical response, in particular the uniform elongation of the polycrystal, can be engineered in a similar manner to conventional composite materials by varying the relative fraction and distribution of these hard and soft ‘phases’.  Although there is experimental evidence in  support of this proposition, details of how precisely and to what extent the character of the grain size distribution influences the behaviour of the polycrystal are not clear. Furthermore, it is increasingly becoming possible, both at laboratory- and industrial-scale, to exert control over the grain structure of polycrystals. There is, therefore, the need to be able to identify grain size distributions which produce desirable properties so such microstructures can be designed and then manufactured. To address these issues, this work examines the role of grain size distribution on the deformation of polycrystals.  For this purpose, a modelling approach is adopted.  This  approach allows access to trends and information not accessible experimentally, making it possible to develop a better understanding of the associated phenomena. Two novel grain size dependent polycrystalline models are used in this work, an elastoplastic self-consistent model and a viscoplastic self-consistent model. Assuming plastic deformation of the grains is accommodated by (1 iO){1 1 i} dislocation slip, the models incorporate grain size  2  dependent constitutive laws to allow for inclusion of grain size distribution as an additional model variable.  Using the models, the full deformation spectrum from the elastoplastic  transient behaviour to the fiuiiy plastic behaviour is examined. For simplicity, loading of the polycrystals is constrained to monotonic loading only. In this process, the general case of polycrystals with varying lognormal grain size distributions (with average grain sizes in the 100 nrn to 50 .im range) is examined, although the special case of bimodal grain size distributions is also considered for large strain deformations. This thesis is organized as follows.  First, a review of the pertinent literature is  provided in Chapter 2. In Chapter 3, the scope and objectives of the work are outlined in detail.  Chapter 4 delineates the process through which the models are constructed.  Specifically, the generation of the model polycrystals and the specifics of the grain-level constitutive laws are described.  In Chapter 5, the elastoplastic self-consistent simulation  results are presented and discussed. The impact of grain size distribution on the yielding behaviour of polycrystals is examined in this chapter.  In particular, how varying the  character of the grain size distribution, i.e. its mean and standard deviation, can influence the percolation of plasticity through the polycrystal.  For this purpose, the evolution of the  volume fraction of yielded grains as a function of macroscopic strain is correlated to the grain size distribution of the polycrystal. This information is also used to examine a new methodology for characterizing the yielding behaviour from uniaxial tensile or compression tests.  In Chapter 6, the viscoplastic self-consistent simulation results are presented and  discussed.  Here, the grain size distribution effect on the fully plastic response of the  polycrystals is investigated. The impact of varying the width of the grain size distribution is evaluated for a variety of polycrystals, specifically how the work hardening response of the  3  polycrystals is manipulated. These results are evaluated by compa ring the overall stressstrain curves of the different polycrystals with each other. The partitio ning of stress and strain among the grains of the polycrystals is also used in this analys is.  To explore the  effectiveness of grain size distribution manipulation as a means to tailor uniform elongation in polycrystals, maps of the ultimate tensile strength versus uniform elongation are constructed for the examined polycrystals. This analysis is extended to the special case of bimodal grain size distributions to develop a sense of the extent to which grain size distribution manipulation can control the behaviour of polycrystals.  4  CHAPTER 2 Literature Review 2.1. Introduction Over the years, notable progress has been mad e in understanding the complex problem of deformation in polycrystals.  There is, accordingly, an extensive body of  literature dedicated to this topic. See for example the reviews by Gil Sevillano et al. [1] and Gil Sevillano [2]. The aim of this chapter is not to review all the literature associated with polycrystalline deformation but instead, to provide a brief and general insight into current understanding of plastic deformation in polycryst als with emphasis on grain size and grain size distribution effects. The literature review is organized as follows. First, Section 2.2 provides a brief overview of grain size effects in polycrystals. In this section, grain size effects as they relate to mechanisms of plastic flow , yield strength, and work hardening of polycrystals are examined. The current state of research on influence of grain size distribution on deformation of polycrystals is exam ined in Section 2.3 where both empirical and theoretical investigations are discussed.  In Section 2.4, the topic of simulation of  deformation in polycrystalline systems is treated. Fina lly, in Section 2.5, a summary of the chapter is provided; highlighting the challenges and opportunities at hand with regard to deformation in metal polycrystals. The focus of the current work is on grain sizes which exhibit ‘classical’ behaviour  —  i.e. deformation dominated by slip. Therefore, discu ssion of  nanocrystalline materials with grain sizes  <  1 OOnm will be limited in this chapter.  5  2.2. Grain Size Effects in Polycrystals All other parameters of the microstructure being equal, grain size exerts a strong influence on a polycrystal’s properties [3]. In particular, through the interaction of this size parameter with characteristic dislocation length scales in the polycrystal (such as the radius of curvature for a dislocation at a given stress) the nature of plastic flow in a polycrystal may be modified. This modification ultimately can lead to changes in mechanisms of plastic flow and grain size dependences such as those of yield strength and work hardening.  These  effects are examined in the following sections.  2.2.1. Mechanisms of Plastic Flow Plastic flow in polycrystals can occur by motion of dislocations, diffusive flow of atoms, or displacement of individual grains with respect to each other through grain boundary sliding.  The established view of plasticity in polycrystals, at low homologous  temperatures and high stress levels, is that it primarily occurs through dislocation mediated processes [4]. This is referred to as dislocation-based plasticity. The premise of dislocationbased plasticity is that dislocations, i.e. the carriers of deformation, can be generated and propagated in the material [5]. This means that any factor that might hinder the generation and/or propagation of dislocations can potentially limit, modify, or even halt this type of plasticity and ultimately cause alternative modes of deformation to be activated. The grain size has been identified as one of such factors and it is now believed that as the grain size of a polycrystal is reduced, its deformation mechanism evolves from being dislocation-based to one mediated by grain boundaries (e.g. by grain boundary sliding) [6-9]. This latest view of plasticity in polycrystals is founded on numerous experimental observations [10-20] and theoretical studies (namely molecular dynamics) [21-29] which report on deviations from  6  ‘traditional’ deformation behaviour in polycrystals with average grain sizes smaller than 1 jim. It has been difficult to assert a unified view of plastic flow in polycrystals based on the noted observations since there are a number of outstanding issues which still need to be resolved. For example, i) there is a material dependence of the behaviour which is currently not well understood (e.g. how the stacking fault energy of the material affects its deformation behaviour [24,28]), ii) other microstructural features of the material may also have an influence on its behaviour (e.g. the character of its grain boundaries, whether it is high or low angle boundary [30]), and iii) the type of loading (e.g. role of hydrostatic pressure) also affects the mechanism of flow [17]. Furthermore, the notion of crossover from dislocation driven to grain boundary mediated deformation itself is also still debated. A number of recent in-situ TEM deformation studies [31,32] have shown that dislocation mediated plasticity still plays a dominant role in the deformation of polycrystals with grain sizes as small as 10 nm. Thus, despite the extensive body of work, the evolution of the mechanisms of deformation with grain size is still to be resolved.  2.2.2. Yield Strength The yield strength of polycrystals has been found to have a close correlation with their grain size. Hall [33] and Petch [34] empirically showed that the lower yield stress of low carbon steels is dependent on the grain size D by an equation of the form =u  (2.1)  where o is the lattice friction stress and k is the Hall-Petch slope. It is now established that the application of this equation is not limited to steels and the yield strengths of a wide range of materials, single phase and poly-phase, also obey such a relationship [35-37]. Note that 7  although more than 50 years have passed since the introduction of Equation (2.1), the exponent of—l/2 in this equation has yet to be definitively verified experimentally. Baldwin [38] has demonstrated that due to scatter in experimental data, a reasonable fit to the data could also be obtained using exponents of—i or —1/3. The original theoretical considerations for Equation (2.1) were based on grain boundaries acting as barriers to dislocation motion [39,40]. Dislocations emanating from a source inside the grain are held up against the grain boundary and fon-n a pileup. Once the stress concentration from this pileup reaches a critical value, the dislocation flow will propagate across the boundary by penetration or by operation of a source in the boundary or in the neighbouring grain. There is extensive experimental evidence that grain boundaries can act as both sources and barriers for dislocations [41].  Etch pit [42-44] and transmission electron  microscopy [45-47] studies point to the fact that in the early stages of plastic flow grain boundaries act primarily as sources for dislocations. Realization of the importance of grain boundaries as sources of dislocation rather than barriers, as argued in the early models, has lead to the development of alternative theories.  These theories avoid the description of  stresses at grain boundaries and instead concentrate on the influence of grain size/grain boundaries on dislocation density [2,48-50]. Li [48] assumed that at the onset of yielding, grain boundaries act as sources of dislocations and ‘pump’ dislocations into the grains. As a result, as shown in Figure 2.1, forests of dislocations are formed in regions close to the boundary.  The yield stress is accordingly described as the stress required to move  dislocations through these forests. Using the Taylor formulation [51], i.e. r  cc  the yield  stress is calculated. A Hall-Petch grain size dependence is obtained by postulating that for a  8  * Figure 2.1. Formation of dislocation forests near grain boundaries as hypothesized in the models of Li [48] and Meyers and Ashworth [52].  given strain, the dislocation density is an inverse function of grain size, i.e. p cx l/D. Another approach was proposed by Conrad [53]. He assumed that the average free slip distance traveled by a dislocation is proportional to the grain size and since the Orowan relationship [54] states that this average distance is inversely related to the dislocation density, the dislocation density is inversely related to the grain size. This means that fine grained specimens are assumed to have higher dislocation densities as compared to coarse grained specimens at a given value of plastic strain. The Taylor equation is once more used and given the above relations between dislocation density and grain size, a Hall-Petch equation is derived. Finally, Meyers and Ashworth [52] have also proposed another model. However, unlike the two previous theories, the strengthening in this model is not rationalized in terms of the Taylor equation. Similar to Li [48], they consider grain boundaries as sources of dislocations and that the generated dislocations remain in the vicinity of the boundaries, 9  forming a work hardened layer encompassing each grain. The polycrystal is then treated as a composite comprised of a connected network of hard material with soft cores, Figure 2.1. The fraction of the hard network varies with grain size and this ultimately results in a relationship for general yielding of the composite. The above discussed models all provide a grain size dependence of D” 2 but they lead to different formulations of the Hall-Petch slope.  Table 2.1 summarizes the  relationships noted for the Hall-Petch slope for the different models. Note that the tabulated slopes should be differentiated from that reported in Equation (2.1) since these slopes relate the grain size to the shear stress on a slip plane and not the macroscopic normal stress acting on the polycrystal as a whole. Aside from the shear modulus (G), the magnitude of the Burgers vector (b), and the Poisson’s ratio (v), these relationships contain parameters and constants whose values and dependences to strain, temperature, or chemical composition are not well defined.  It is therefore difficult to quantitatively define the slopes using these  relationships. Nonetheless, normalized values (i.e. k/(G/)) in the range of 0.06 to 0.18 are generally reported [2].  Table 2.1. Different relationships predicted for the Hall-Petch slope  model  Eshelby  k /Gbr*(2 —v) 1 I a \I 2,r(1—v)  Conrad  a,GJby//3  Li  a,Gbf  ,a,,a 1 a , 3 8: constants; G: shear modulus; b: magnitude of the Burgers vector; U: Poisson’s ratio; y shear strain; i: critical stress for propagation of slip; in: total length of dislocations emitted per unit area of grain boundary  10  In the end, it is worth noting that an inverse Hall-Petch effect, i.e. grain size softening, has been reported for nanocrystalline materials with grain sizes smaller than 50 nm, see e.g. [55-57]. Arguments based on the reconsideration of the dislocation line tension [3,58] or activation of grain boundary based mechanisms [57,59,60] have been put forth to  explain this behaviour.  Existence of processing flaws in samples, such as porosity, has  however been noted as one source of these observed deviations [61].  2.2.3. Work Hardening A general understanding of the dislocation processes responsible for work hardening in metals is currently available [62]. This is particularly true for the case of single crystals (see the seminal review article by Nabarro, Basinski, and Holt [63]). It is now generally accepted that short range interaction of gliding dislocations with dislocations threading their slip planes (i.e. forest dislocations [64]) is responsible for work hardening. As far as these interactions are concerned, there is no difference between single crystals and polycrystals. In polycrystals, however, additional considerations are to be taken into account. One of such issues is with regard to grain size. There have been a number of attempts to describe work hardening as a function of grain size, but a general consensus regarding this matter is currently not available. This is despite the extensive body of literature dedicated to the study of grain size and its effect on work hardening.  The lack of consensus on this issue can in part be traced back to  uncertainties associated with the evolution of the mechanisms of deformation with grain size, as discussed in Section 2.2.1. Ambiguities regarding the influence of the processing route on the material behaviour also contribute to this uncertainty. For example, it is reported that polycrystals processed by equal channel angular pressing (ECAP) have a high level of  11  internal stress and that their grain boundaries contain very high dislocation densities (i.e. non-equilibrium grain boundaries) [65). These materials thus show very low work hardening rate during tension and compression which has been attributed to enhanced recovery processes at the grain boundaries [65]. Therefore, care should be taken when comparing the work hardening behaviour of a coarse grained material processed by conventional deformation and recrystallization methods with fined grained material processed by ECAP. In the following, a number of models will be examined which demonstrate the current understanding of grain size effect on work hardening.  Discussion is limited to work  hardening models based on the one-parameter framework of Kocks and Mecking [62,66] described in Appendix Al. Note that alternative, yet more complex, frameworks where more than one microstructural parameter are used to describe work hardening are available (for example see [67,6 8]). A popular approach to treating the grain size dependence of work hardening in models has been through the use of the geometrically-necessary dislocation concept [69,70]. Geometrically-necessary dislocations are dislocations stored in the polycrystal as a result of inhomogeneous plastic deformation. They accommodate the deformation gradients in the polycrystal and thus, allow for compatible deformation of the grains. Distinction is made between these dislocations and statistically stored dislocations which are generated by random encounters of dislocations in the material. In the context of the Kocks and Mecking framework [62,66], this additional storage means that the evolution of dislocation density, p, with shear strain in the grain,  can be written as,  dy  (2.2) LG  12  where k 1 and k 3 are constants and k 2 is a function of temperature, strain rate, and stacking fault energy. This equation is equivalent to Equation (Al .3) in Appendix Al but now, in addition to the statistically determined mean free path of dislocations, i.e. l/J, a second mean free path LG is introduced. This mean free path, dubbed the geometric slip distance [70] or the geometric mean free path, is identified with the spacing of geometric obstacles in the polycrystal (i.e. grain boundaries). Estrin and Mecking [7 1,72] assumed a constant linear relationship between LG and grain size, i.e. =  dy  k p” 2 ‘  —  k,p + K,D’  (2.3)  -  here k, and k 2 are defined as before, K 3 is a constant, and D is the grain size. An important characteristic of this model, referred to as the Kocks-Mecking-Estrin model hereafter, is that although two different dislocation densities are introduced, there is only one evolution equation for the total dislocation density of the polycrystals, i.e. Equation (2.3).  This  approach differs from that of, for example, Ashby [70] and Hansen and Ralph [73] where the two dislocation densities are treated independently. As evident from Equation (2.3), this model predicts that as the grain size of the polycrystal is reduced, its work hardening rate increases for a given value of applied stress. Alternative formulations of the Kocks-Mecking-Estrin model have since been introduced [74-77]. These models in general argue on the validity of the assumption of a constant geometric mean free path. Barlat et al. [74], for example, note that at the early stages of deformation the geometric mean free path is defined by the grain size. But as the deformation progresses, internal dislocation structures (i.e. cells) start to form and cell walls now provide the geometric obstacle in the path of the dislocations. The geometric mean free path should, thus, evolve (decrease) with deformation. Linear [76], exponential [74], and 13  more complicated evolutions determined through finite element calculations of slip plane incompatibility [75,77] have been introduced accordingly. The physical interpretation of the above class of models is that a higher dislocation storage rate is considered for fine-grained polycrystals in comparison to coarse-grained polycrystals.  This is consistent with dislocation density measurements of Narutani and  Takamura [78] on nickel with grain sizes ranging from 20 jim to 90 jim. They showed that dislocation density increases in a linear manner with the reciprocal of grain diameter at a given strain. Similar observations are reported by Hansen [79] in aluminum and by Chia et al. [80] in titanium. However, in the case of very fine grain sizes of the order of 1 jim and smaller, the absence of dislocation substructures [10,11,16,19,20] and lack of evidence of dislocation activity within deformed grains [14,15] indicate that the generalized view portrayed by the Kocks-Mecking-Estrin type models may not hold. At these grain sizes, the geometrical mean free path approaches or reduces below the statistical mean free path [70]. Just as the mechanisms of plastic flow may be modified (Section 2.2.1), the nature of dislocation interaction and ultimately, work hardening of the polycrystal may also evolve. Recently, Sinclair et al. [8 1,82] examined pure copper with grain sizes ranging from 50 jim to 2.2 jim (i.e. grain sizes approaching the ultrafine-grained regime, 100 nm< D <1 jim). They observed a regime of low work hardening rate following yielding for grain sizes below 5 jim. A similar observation has been reported for silver with 2 jim grain size [83,84]. Figure 2.2 shows the work hardening plots of the copper samples from the work of Sinclair et al. [81,82]. Except for the 50 jim sample, an initial steep decrease in work hardening rate followed by a region of increasing work hardening rate is observed. Eventually a peak is reached and thereafter, the hardening rate declines in a linear manner. The work hardening 14  2500 50  2000  a)  C C 0  1000  500  0 0  100  200  300  400  true stress (MPa)  Figure 2.2. Room temperature work hardening rate plotted against flow stress for copper of different grain sizes. Curves replotted using data in [8 1,82].  rate at the point of inflection decreases with decreasing grain size (i.e. in contradiction to the Kock-Mecking-Estrin models introduced earlier) but once the peak is reached, the hardening rate becomes independent of grain size. At this stage, all curves collapse on one curve. To explain the noted work hardening behaviour, Sinclair et al. [82] propose that in addition to the dislocation hardening contribution already accounted for in the Kocks Mecking-Estrin type models, an additional contribution due to build up of back-stresses should be taken into account. In the early stages of deformation, dislocations can freely transverse the grains without any statistical storage and become trapped/stored at grain boundaries; consistent with experiniental observations of lack of dislocation substructure in fine-grained polycrystals [10,11,16,19,20]. The dislocations stored at grain boundaries cause back-stresses to develop in the grains and accordingly, impede the motion of other dislocations. In this way, a kinematic hardening contribution, which increases with decrease 15  in grain size, is developed.  This hypothesis is consistent with recent two-dimensional  dislocation dynamics simulations of Lefebvre et al. [85] which show at the early stages of deformation when the grain size decreases i) the dislocation density blocked at or near grain boundaries increases and ii) the highly stressed areas of the grains which contain dislocations blocked at or near grain boundaries also increases.  As deformation progresses, more  dislocations are stored at the grain boundaries resulting in further increase in this kinematic term. However, a point is reached where the boundary cannot support anymore dislocations and the stored structure at the boundaries start to annihilate (i.e. by activation of a dynamic recovery process).  This causes lowering and ultimately, disappearance of the kinematic  contribution, i.e. the sharp decrease in the work hardening plots observed in Figure 2.2. The hardening of the polycrystal from here on is only attributed to forest dislocations. At this stage, it is also hypothesized that the geometric storage of dislocations also ceases due to dynamic recovery processes and the grain size dependence ceases.  There is limited  experimental data on the behaviour of polycrystals with grain sizes in the range of the aforementioned study, i.e. grain sizes approaching the ultrafine-grained range. As noted by Sinclair et al. [82], more work is still needed to clarify the work hardening behaviour of these metals. In the ultrafine-grained and nanocrystalline (D  <  100 nm) regimes, there is  considerable evidence that the work hardening capability of polycrystals disappears, see e.g. [9].  Figure 2.3 shows the compressive stress-strain curve for a copper sample with  approximately 200 nm grain size. Lack of any work hardening is evident in this metal. In fact, Jia et al. [86] even observe strain softening in their ultrafine-grain iron samples. The  16  500  400 0 300 U) U) 4-.  0  a)  200  L. 4-  100  0 0.0  0.1  0.2  0.3  0.4  true strain  Figure 2.3. Compressive stress-strain curve of copper with approximately 200 nm grain size. Curve plotted using data in [87].  low hardening capability of these metals can be understood in terms of the dislocation evolutions noted in Section 2.2.1.  That is, observation of dislocation free grains is an  indication that storage of dislocations within the grains may not occur.  This notion is  supported by the recent analysis of broadening of the Bragg peaks in X-ray diffraction of nanocrystalline nickel which report on the absence of residual dislocation networks [88]. As noted in Section 2.2.1, grain boundary mediated processes are also believed to play a role in plasticity of polycrystals; particularly in the nanocrystalline regime. Accordingly, work hardening models based on such considerations have been introduced, see e.g. [59,60]. Grain size dependence generally enters into these models in two ways. First, by assuming a Kocks-Mecking-Estrin type dislocation-based work hardening contribution and second, by assuming diffusion-controlled flow of material, the strain rate of which is inversely related to the grain size.  However, activity of diffusional processes at room 17  temperature is still debated [89]. Furthermore, simulations of Kim and Estrin [60] show that at strain rates greater than i0 s dislocation-based mechanisms dominate the work hardening behaviour of polycrystals with grain sizes as small as 20 nm. Thus, the pertinence of work hardening models based on grain boundary mediated processes is still to be substantiated.  2.3. Grain Size Distribution Effects in Polycrystals Compared to the (average) grain size, the impact of grain size distribution on deformation behaviour of polycrystals has received less attention.  A number of recent  experimental studies on ultrafine-grained polycrystals have pointed to manipulation of grain size distribution as a means to engineer the mechanical response of polycrystals [15,90-92]. Furthermore, it is increasingly becoming possible to control the grain size distribution of polycrystals, e.g. through high energy ion bombardment [93] or precise composition control during electrodepostion [94]; increasing the technological relevance of grain size distribution. Below, the literature on the influence of grain size distribution on deformation behaviour of polycrystals is reviewed; both experimental and theoretical studies are examined.  But first, the literature on observed grain size distributions in polycrystals is  briefly reviewed.  2.3.1. Grain Size Distribution in Polycrystalline Metals It is generally accepted, experimentally [95-99] and through theoretical treatment of grain growth [100,101], that the grain size distribution of polycrystals is best represented by the lognonnal distribution.  Grain size distributions formed by normal grain growth, as  opposed to abnormal grain growth [102], are implied here [101]. Consult Appendix A2 for examples of the experimentally observed lognormal grain size distributions in metals. The  18  lognormal distribution is a positively skewed unimodal distribution, Figure 2.4(a), which when plotted on a logarithmic scale has a Gaussian form, Figure 2.4(b).  a  D  LN(D)  Figure 2.4. Schematic representation of grain diameter lognormal distribution on (a) linear and (b) logarithmic scales. The dashed line delineates the mean grain size of the distribution. Note the Gaussian (normal) distribution of grain sizes in (b). P(D)dD denotes the fraction of grains which fall between D and D + dD.  The lognormal probability density function can be written as [101,103],  P(DIM,Sjr  1  f2,rS2D  exp  2S 2  (2.4)  where D is the grain size, and M and S are, respectively, the mean and standard deviation of the logarithms of the grain sizes.  The arithmetic mean grain size, p, and the standard  deviation, o, of the distribution are defined in terms of S and M as,  prexp M+—-  (2.5)  and  =  ([exp (s 2 + 2M)][exp(S2  ) ii) —  1/2  (2.6)  Although the majority of studies favour the lognormal distribution, there are experimental studies [104,1051 which argue otherwise. Accordingly, alternative distribution 19  functions, such as those of Hillert [106] and Louat [107], have been proposed but these functions do not explain the distributions observed in these studies either.  Furthermore,  under special processing conditions, it is possible to obtain non-lognormal grain structures in metals. These are grain structures generally characterized by non-uniform (e.g. bimodal) size distributions.  As an example, Legros et al. [15], through a combination of inert-gas  condensation and annealing, were able to produce copper samples with microstructures consisting of a small volume fraction of 1-5 I.Irn grains embedded in a matrix of nanocrystalline grains. Due to the attractive mechanical properties of these structures, see Section 2.3.2, other processing strategies that also yield such grain structures (for example those involving asymmetric rolling [92], cryogenic rolling [91], cryomilling [108], or hot isostatic pressing [109]) have since been introduced.  2.3.2. Experimental Studies There are a limited number of experimental studies which have examined the effect of grain size distribution on deformation behaviour of polycrystals. The difficulty associated with manipulation of the grain structure of polycrystals but also the challenge to characterize the true grain size distribution of polycrystals have contributed to this lack of experimental data.  In particular, there is a lack of systematic experimental study of the lognormal  distribution and how its characteristics affect the behaviour of a polycrystal.  The  experimental work on this subject has in general been limited to studies comparing unimodal distributions with bimodal distributions under monotonic loading conditions. Bimodal polycrystals demonstrate an attractive balance of strength and ductility as compared to their unimodal (fine-grained or coarse-grained) counterparts [15,91,92,108112]. Figure 2.5, from the work of Wang et al. [91,110] on pure copper, shows an example  20  400  a) a) a) 0, C a)  a, C  0,  100  0  40  20  60  engneorn5J 5tran  80  (%)  0.35  C 0  a:,  0,25  a)  0.20  a, .0  0.15  0 C  0.10 0.05 0.00 50  100  150  200  250  300  350  400  450  grain se (nrn)  0.20 C 0 (.0 a)  0.15  E 0.10  0 a >  0.05  0.00 500  1000 gran 54za)  1500  2000  (rrn)  Figure 2.5. (a) Room temperature tensile stress-strain curves for Cu with different grain structures: fine-grained (curve A), coarse-grained (curve B), and bimodal (curve C). Grain size distributions corresponding to stressstrain curves A and C are shown, respectively, in (b) and (c). Figures replotted using data provided in [91,110].  21  of such an observation. It is evident from Figure 2.5(a) that the fine-grained structure (curve A) demonstrates a higher level of strength but lower ductility compared to the coarse-grained structure (curve B). The most interesting behaviour is, however, that of the bimodal structure (curve C). The bimodal structure, compared to its fine-grained counterpart, has a slightly lower yield strength, but higher ultimate tensile strength and considerably larger ductility. In comparison to the coarse-grained polycrystal, the strength level of the bimodal polycrystal is significantly higher but it shows comparable ductility.  The impact of the grain size  distribution of the polycrystal, particularly when it is changed from unimodal, Figure 2.5(b), to bimodal, Figure 2.5(c), is therefore appreciable. The improved elongation to failure of bimodal polycrystals in comparison to fine grained polycrystals has been attributed to the existence of the coarse grains that tend to deform extensively and contribute to the global uniform elongation of the bimodal polycrystal.  Figure 2.6(a) shows the microstructure of a bimodal aluminum sample  consisting of an ultrafine-grained matrix with embedded micron size grains. Figure 2.6(b) shows the surface of the sample after compression. Accommodation of plasticity in the coarse grain, in the form of slip lines, is evident while the ultrafine-grained matrix does not show any sign of dislocation activity. Similar observation has also been reported for other bimodal structures [11 1]. The implication of grain size distribution on other aspects of the deformation of polycrystals is, comparatively, less explored. In terms of how plasticity propagates through the material, i.e. whether it spreads homogeneously or in a localized manner, Azizi-Alizamini et al. [113] recently showed that a low carbon steel with a bimodal grain structure does not show the undesired Ludering behaviour observed in samples with unimodal distributions. In  22  Figure 2.6. Electron microscope micrographs of a bimodal aluminum specimen in the (a) as-processed and (b) as-deformed states. The ultrafine-grained (ufg) matrix and the embedded microcrystalline (mc) grains are evident in (a). The extensive slip activity in the coarse grain of the deformed sample is clear in (b). Micrographs reproduced with permission from [109].  23  contrast, there are studies that report on extensive shear band formation in bimodal polycrystals, e.g. in Al-Mg alloys [111] and iron [114]. There is also very little information available on the effect of grain size distribution on fracture of polycrystals. Figure 2.7, from the work of Billard and co-workers [109], shows the surface of a bimodal aluminum specimen deformed in compression (same sample shown in Figure 2.6). One crack is visible in this figure which has been stopped at a micron size grain. The coarse constituents of the bimodal microstructures can, thus, act as crack-blunting objects (e.g. by relaxing the stress concentration ahead of a crack [111]), retard crack propagation and ultimately result in prolonged strain to fracture [109,111].  Figure 2.7. Electron microscope micrograph of the surface of a bimodal aluminum sample, same sample shown in Figure 2.6, showing a crack stopped by a micron size grain. Image reproduced with permission from [109].  In summary, despite the relatively small number of the experimental studies on bimodal structures, it is clear that the grain size distribution can have a pronounced effect in modifying the various aspects of mechanical response of a polycrystal. 24  2.3.3. Modelling Studies Unlike experimental data, theoretical studies are not constrained to bimodal structures. In fact, many of the earlier studies have been concerned with understanding the influence of lognormal grain size distributions. Only in recent years have bimodal grain structures received attention. The impact of grain size distribution on the yielding of polycrystals has received considerable attention.  Among the earliest studies to correlate grain size distribution to  yielding is that of Kurzydlowski [115]. In this work, a polycrystalline model based on the Hall-Petch relationship, Equation (2.1), and the assumption that the partitioning of plastic strain is proportional to the grain volume has been used to predict the dependence of yield strength on grain size distribution. The results of this calculation show that as the width of the lognormal grain size distribution increases, the slope of the Hall-Petch plot decreases. This prediction is consistent with recent works where numerical models that explicitly include a distribution of grain sizes and numerical approaches to partition stress and strain amongst the grains have been employed [98,116-118].  Figure 2.8, from the work of  Weertman and co-workers [98,116], is a plot of the predicted yield strength versus the inverse square root of the average grain size for a variety of polycrystals with lognormal size distributions. The decrease in the Hall-Petch slope with increase in the width of the size distribution is evident. The reason for this dependence on grain size distribution is attributed to the fact that an increase in the width of the distribution means an increase in the volume fraction of coarse grains in the microstructure.  These coarse grains have lower yield  strengths and therefore, the polycrystal yield strength is also lowered.  25  1200  lOGO  800  600  4O0  a)  200  0 0.05  0.10  0.15  i,/q’  0.20  0.25  (nmhfZ)  Figure 2.8. Calculated Hall-Petch relationship for copper polycrystals with varying lognormal grain size distributions. The numbers on each line are indicative of the width of the size distribution with larger numbers corresponding to wider distributions. The decrease in the slope of the lines (i.e. the Hall-Petch slope) with the width of the grain size distribution is evident. Lines replotted using data provided in [98,116].  In contrast to yielding, the influence of grain size distribution on work hardening of polycrystals is less documented. This is partly due to uncertainties associated with grain size dependence of the constitutive laws (i.e. the work hardening as discussed in Section 2.2.3). In fact, none of the previously mentioned models have considered work hardening and the few studies that have looked at the interaction of grain size distribution and work hardening, have primarily focused on the special case of bimodal distributions.  Gil Sevillano and  Aldazabal [119] have used a one-dimensional cellular automaton model to simulate the elastic and plastic deformation of a number of polycrystals with uniform and bimodal size distributions. Excluding nano-size grains, work hardening of grains is taken into account by assuming statistical and geometrical storage of dislocations.  However, geometrically-  26  necessary dislocations are assumed to be immune from dynamic recovery. Furthermore, the evolution of the crystallographic orientation of the grains is not considered, limiting the relevance of the model to small plastic deformations. Comparison is made between coarse and fine grained polycrystals with uniform distributions but since different widths of distribution are assigned to the two classes of polycrystals, it is not possible to differentiate between the effect of the mean and the width of the distribution. Bimodal distributions are created by mixing different fractions of the coarse and fine uniform distributions. It is found that the extent of uniform elongation is increased for the bimodal polycrystals. Joshi et al. [120] also developed a model for examining the elastic and plastic behaviour of polycrystals with bimodal size distribution. They used a mean-field approach to simulate the mechanical behaviour of two experimental bimodal polycrystals.  They approximated a bimodal  polycrystal as being composed of a coarse grain phase embedded in a fine grain phase. Similar to Gil Sevillano and Aldazabal [119], they also report on the beneficial effects of bimodal grain size distribution on expanding the uniform elongation of fine grained materials. The gain in uniform elongation is, however, predicted to come at the expense of loss in strength. To the author’s knowledge, the impact of grain size distribution on other aspects of the deformation of polycrystals has not been examined. More importantly, however, is that even for the unimodal size distribution, it is not clear how the character of the size distribution can impact the work hardening behaviour of the material. A detailed analysis of this effect is lacking.  27  2.4. Polycrystal Deformation Models Based on Homogenization Theories Models for deformation of polycrystals are generally constructed to project the local (mesoscopic) behaviour of the grains to the global (macroscopic) level of the polycrystal. Over the years, a comprehensive understanding of the deformation behaviour of single crystals and their constitutive relationships has been established. However, extending such knowledge to polycrystalline systems, as desired in polycrystal deformation models, has been a challenge. Under an applied load, each grain of the polycrystal behaves in a unique manner determined by its characteristics (e.g. size and crystallographic orientation) but more importantly, by the force and displacement boundary conditions imposed upon it by its surrounding environment. These boundary conditions are not explicitly defined, even for the simplest of loading conditions making it difficult to link single crystal behaviour to polycrystalline level as intended in models. As a way to overcome this problem, polycrystal homogenization theories have been introduced. In these theories, a single (i.e. homogenized) behaviour, that of the polycrystal, is extracted from the inhomogeneous and non-uniform response of the individual grains.  The application of homogenization theories is  schematically shown in Figure 2.9. Here, the constitutive relationships of an aggregate of grains, g , are reduced to a single relation, P, for the polycrystal using homogenization 1 theory H. Polycrystal deformation models can, therefore, be viewed to be composed of two basic elements: i) a constitutive relationship describing the behaviour of the grains based on their characteristics such as crystallographic orientation and size (i.e. g 1 ‘s) and ii) a framework for linking the mesoscopic and macroscopic boundary conditions (i.e. H). In the following, these two elements are discussed. 28  ( g , 1 d6) o  1 g, P = ]H1[(g  ...,  , 1 g  ...,  ) 1 g,  Figure 2.9. Polycrystal response, P, described in terms of constitutive relationships of its grains, g., using  homogenization theory H. Note that  and s are the grain stress and strain while J and  are the  polycrystal stress and strain values.  2.4.1. Local Constitutive Behaviour The constitutive law for deformation of a grain is a set of equations relating the strain rate of the grain to its stress, rate of change of stress, temperature, and of structure parameters such as dislocation density  through a number  its thermo-mechanical history [5]. In  this section such relationships relevant to plastic slip are presented. The reader is referred to the works of Peirce et al. [121] and Asaro [122] for a more comprehensive treatment of the problem which also involves consideration of elasticity. The plastic constitutive relationships of a grain essentially outline the grain yield surface and describe the evolution of this surface with deformation. These relationships, based on whether they account for strain rate sensitivity of slip [5,123] or not, are of two 29  basic forms: rate-independent and rate-dependent. Rate-independent formulations are based on the Schmid law [1241. According to this law, a grain plastically flows once the projection of the applied stress onto the slip direction of a slip system, i.e. the resolved shear stress, reaches a critical value.  This critical value is called the critical resolved shear stress or  alternatively, the mechanical threshold [5]. In mathematical terms, this translates to mu  (for all systems S)  (2.7)  where mS is the Schmid factor, u the grain stress, and r the threshold stress. Note that the inequality is necessary in order for all inactive system to have a resolved shear stress less than their threshold stress [123]. Use of Schmid law implies that no dislocation movement occurs until the mechanical threshold is reached, after which deformation is instantaneous [125].  This is an inadequate description of plasticity in crystalline systems.  Thermal  activation (a local process on the slip plane) causes straining to occur at finite rates at stresses below the mechanical threshold [122]. Thus, strain rate sensitivity has to be considered. To account for rate-dependent flow in the grain, a power law relationship is often used [122,126,127], i.e. (S/s)  where  S  (rS/r  is the shear strain rate on slip system S, r  reference strain rate  ,  i-  (= mScr)  (2.8) the mechanical threshold at the  the resolved shear stress, and n is the rate-sensitivity  exponent usually assumed independent of slip system.  This equation, referred to as the  viscoplastic relationship, is computationally very attractive as it provides a way to deal with the choice of slip system combinations at vertices of the single crystal yield surface in cubic metals where ambiguities exist [123,127].  Instead of using Equation (2.7) to determine  30  which slip systems are active, it is assumed here that all slip systems are active and the strain rate of each system is defined by Equation (2.8). Irrespective of whether strain rate sensitivity is taken into consideration or not, the evolution of the yield surface with deformation is generally treated the same way. One typical way to do this is to assume that at a given stage of the deformation process, the rates of change of the mechanical thresholds are related to the shear rates by [128], i  =h/s  (2.9)  The h 5 are the hardening moduli; hss is the self-hardening rate on system S and h 5 (S  5’) is the latent-hardening rate of system S (whether it is active or not) caused by slip on  system 5’ [122]. The components of hss’ are defined as [129], h’ where  0s(F)qss  (2.10)  (F) = dr/dF, F being the cumulative strain in the grain, is the instantaneous  hardening rate that can be derived from analyses such as those presented in Section 2.2.3 and qSS  is a matrix describing the latent-hardening behaviour of the grain.  2.4.2. Homogenization Theories The relationships presented in the previous section demonstrate how the stress imposed on a grain can be translated to the strain response of the grain, and vice versa. So, in modelling the polycrystal deformation, it is now only necessary to know how the imposed macroscopic stress (or strain) field can be related to the stress (or strain) field of the grain. In other words, how the imposed stress or strain is partitioned among the grains. Homogenization theories are used for this purpose.  31  The simplest approach to relate the imposed external boundary condition to those of the grains is to assume that it is uniform throughout the polycrystal. Sachs [130] assumed that the same stress as that of the polycrystal is prescribed to the individual grains. The strain response of the polycrystal is then determined as the volume average of the individual strain of the grains. Although equilibrium of forces across the grain boundaries is sustained under this assumption, compatibility of the deformation of grains is violated. This violation of the compatibility condition leads to a lower-bound approximation of the polycrystal response. On the other extreme, Taylor [131] and Bishop and Hill [132] made the assumption that the grains of a polycrystal are fully constrained by their environment (the Full Constraints approach) and they all deform the same amount as the polycrystal. The polycrystal stress is determined from the volume average of the grain stresses. This isostrain assumption satisfies the compatibility condition but violates the equilibrium condition. An upper-bound estimate of the polycrystal response thus results from Full Constraints approach.  In comparison,  however, the isostrain assumption is favoured over the isostress model of Sachs since the fulfillment of compatibility supersedes fulfillment of local equilibrium [133]. Alternatively, a mixture of isostress and isostrain conditions can be used. This is the so-called Relaxed Constraints approach [134,135].  Here, based on grain shape considerations, some  components of the strain tensor are ‘relaxed’, i.e. not prescribed by the surroundings like the isostrain approach. In the flat grain limit, for example, the two shears on (not the one in) the plane of the large grain boundary are not prescribed [136]. Compared to the Full Constraints approach, the Relaxed Constraints assumption yields to an improved force equilibrium but deteriorated compatibility [137].  32  In reality when a polycrystal is deformed, both compatibility and equilibrium conditions are satisfied. To account for both these conditions simultaneously, it is necessary to avoid the assumption of uniformity of stress or strain in the polycrystal and utilize an alternative scheme for partitioning of stress and strain. One way to do this is to utilize the Eshelby [1381 treatment of an elastic ellipsoidal inclusion embedded in an infinite elastic matrix.  Eshelby demonstrated that the elastic stress and strain induced in an elliptical  inclusion as a result of an external field on the matrix can directly be related, based on the characteristics of the inclusion and the matrix, to the far-field stress and strain in the matrix. Using this concept, a polycrystal can be treated as a composite with each grain approximated as an ellipsoidal inclusion embedded in a medium, i.e. the Homogeneous Equivalent Medium (HEM), whose properties are those of the polycrystal itself. This is the basis of the so-called self-consistent homogenization schemes which have successfully been applied to analysis of elastic-plastic [139-141] and plastic [142,143] deformation in polycrystals. A schematic representation of the self-consistent scheme is given in Figure 2.10.  HEM  Figure 2.10. Schematic representation of the (1-site) self-consistent homogenization scheme. The grains are first approximated by their equivalent ellipsoidal inclusions. Each inclusion is then assumed to be embedded in the Homogeneous Equivalent Medium (HEM) whose properties are those of the polycrystal itself  33  In contrast to the Eshelby case where the matrix and inclusion properties are defined in advance and an immediate solution is obtained, for self-consistent methods, the property of the HEM (i.e. the polycrystal) is not defined beforehand.  For this reason, an iterative  approach is adopted where the property of the HEM is adjusted systematically, allowing for a solution to the partitioning problem based on the Eshelby approach.  The iteration is  terminated by requiring the volume average of stress and strain over all grains (in volume element V) to coincide within some small deviation with the overall stress, ,  ,  and strain,  imposed on the polycrystal [144,145], i.e.  (2.11)  and (2.12)  These are the consistency conditions (i.e. global compatibility and equilibrium). In the end, three issues need to be addressed regarding the self-consistent models. First, Eshelby demonstrated that the elastic stress and strain fields within an ellipsoidal inclusion are uniform.  Therefore, the self-consistent approach does not account for the  distribution of stress and strain within the grains and solves the partitioning of stresses and strains in an average sense. Second, self-consistent models as such described above (i.e. 1site models) do not consider the spatial distribution of grains.  The HEM represents the  average neighbourhood of all grains and individual grains do not identify with any of their neighbours.  In other words, direct grain-to-grain interaction is not incorporated in the  approach. Other self-consistent approaches have, however, been formulated [146] and n-site [147] models  —  e.g. the 2-site  which take such local effects into account. Finally, the 34  Eshelby solution is not permitted for non-linear media [148].  However, apart from the  Hookean elastic regime, the stress-strain behaviour of both the grains and the polycrystal are non-linear. For this reason, the constitutive relationships of the grains and the polycrystal are to be linearized based on a linearization approximation. The type of linearization used in the self-consistent calculation will directly impact the nature of the stress and strain distribution in the polycrystal and accordingly, the overall polycrystal stress-strain response. Depending on the choice of linearization, the self-consistent result will fall in between the isostrain Taylor (infinitely stiff matrix-inclusion interaction, upper-bound) and the isostress Sachs (infinitely compliant matrix-inclusion interaction, lower-bound) approximations.  2.4.3. Comparison with Other Modelling Approaches There are other modelling tools and approaches available to study the deformation of polycrystals. Just as the deformation of a polycrystal can be viewed from different scales, e.g. displacement of individual atoms or displacement of an assembly of grains, these deformation models can also be formulated using constitutive relationships which originate from different scales of the microstructure. Molecular dynamics models take into account laws governing interaction of atoms while dislocation dynamics simulations incorporate laws related to dislocation interactions. Table 2.2 lists a number of the commonly used models. For each case, the smallest microstructural feature explicitly accounted for in the model formulations is also tabulated. In comparison to other methods, homogenization methods provide an attractive balance between the detail of the microstructure which can be incorporated into the model and the computational load of the problem.  Similar to finite element models, the full  35  Table 2.2. Commonly used models in studies of polycrystal deformation  molecular dynamics  smallest microstructural feature atom  dislocation dynamics  dislocation  [149]  grain  [150]  gram  [143]  model  continuum mechanics (finite element methods) continuum mechanics (homogenization theories)  .  example  [29]  characteristics of the microstructure of interest is accounted for in these models, i.e. the crystal structure (and its corresponding characteristics), crystallographic orientation, shape (although approximated as ellipsoidal), and size of grains. But at the same time, the model is less taxing in terms of computation. Finite element calculations on the other hand fulfill local compatibility [150] and are therefore well suited for analysis of stress or strain distributions across a grain.  Homogenization methods, as elucidated earlier, fulfill local  compatibility only on an average sense and predict uniform stress-strain distribution within a grain. With regard to molecular dynamics simulations, with current computational power, these models are best suited for analysis of microscopic processes responsible for plasticity [89]. Dislocation dynamics simulations, like molecular dynamics, are computationally very taxing to run; they are intended to address crystal plasticity at the microscopic length scale [149].  They are at the moment best suited for analysis of dislocation interactions and  substructure formation in polycrystals, see e.g. [151].  2.5. Summary: Challenges and Opportunities The complexity of deformation in polycrystalline systems was delineated in this chapter. It was shown that the deformation behaviour of polycrystals, particularly for the 36  case of polycrystals with grain sizes approaching or smaller than 1 tim, still has many unanswered questions. One area which merits further investigation is the examination of grain size distribution effects.  It is clear that compared to other parameters of the  microstructure of a polycrystal, the impact of grain size distribution on mechanical response, specifically work hardening, is less explored.  It was shown that there is growing  experimental evidence pointing to the beneficial effect of grain size distribution manipulation as a means for engineering the response of polycrystals. This surge in experimental evidence is supported by recent technological advancements which has made it possible to exert an increased level of control over the microstructure of a polycrystal. With the current maturity of polycrystal deformation models, a unique opportunity exists to examine grain size distribution effects in a systematic way.  37  CHAPTER 3 Scope and Objectives The present work aims at examining the role of grain size distribution on deformation of polycrystalline metals. For this purpose, tensile deformation of polycrystals with varying grain size distributions is examined using two novel grain size dependent self-consistent formalisms, an elastoplastic and a viscoplastic.  The elastoplastic self-consistent (EPSC)  formalism is used for simulation of the elastoplastic response of the polycrystals while the viscoplastic self-consistent (VPSC) formalism is used for the fully plastic response. The particular novelty of this work is in the inclusion of grain size dependence into the selfconsistent scheme. In the classical self-consistent scheme no scale dependence exists. A scale dependence has been included in this case through the adoption of a grain-level constitutive relationship which allows for each grain to be assigned a size and accordingly, a size dependent yield strength and work hardening response.  The elastoplastic model  developed here is similar to that very recently presented by Berbenni et al. [1181 but the inclusion of grain size dependence in a self consistent scheme for large scale work hardening, to the author’s knowledge, represents a first. The focus of the investigation is on f.c.c. polycrystals with moderate stacking fault energy (such as copper) where plasticity occurs by dislocation-based slip. Local gradients of deformation in the polycrystals are taken into account in an average sense.  That is,  heterogeneity of the stress and strain fields among different grains is modeled but such heterogeneities are not considered across a single grain. Polycrystals with varying grain size distributions, lognormal with average grain sizes in the 0.1-50 tm range and bimodal, are included in the analysis.  The spatial distribution of the grains in the polycrystals is not 38  accounted for and only random spatial distribution is considered. The following objectives are sought in this work: The elastoplastic behaviour  -  The primary goal of this particular analysis is to develop a  quantitative understanding of how plastic deformation progressively spreads through a polycrystal and how the character of the grain size distribution, i.e. its mean grain size and standard deviation, influences this process. This will help shed light on the elastoplastic transition in polycrystals and whether any microstructural information regarding the polycrystal can be extracted from this transitional stage of deformation. The plastic behaviour  —  The focus of the plastic analysis is to understand whether the grain  size distribution can be used as a viable design parameter in engineering the mechanical response of polycrystals.  It is of particular interest to see how grain size distribution,  whether it is unimodal or bimodal, can affect the work hardening behaviour and accordingly, the ultimate tensile strength-uniform elongation relationship in a polycrystal.  The  information gathered from this analysis is useful in constructing property maps for microstructure selection purposes. Such maps allow for an objective oriented approach to the design of polycrystals, thus eliminating or at least reducing the need for a trial-and-error based approach.  39  CHAPTER 4 Model Development 4.1. Introduction The numerical approach utilized in this study consists of two basic components. First, a routine for assembling the model polycrystalline aggregates and second, a routine for simulating the deformation of the model polycrystals. Section 4.2 outlines the procedures used for assembling the model polycrystals. This includes the procedures through which the size and crystallographic orientation of the grains were generated and assigned. In Section 4.3, the two variants of the self-consistent methodology for simulating polycrystal deformation used in the present work, i.e. the elastoplastic self-consistent (EPSC) and the viscoplastic self-consistent (VPSC) formulations, are introduced. The methods developed in this thesis to incorporate grain size into these self-consistent formalisms will also be presented in this section.  4.2. Model Polycrystals The model polycrystals examined in this study are identified by the characteristics of their individual grains. For each grain, its crystal structure (f.c.c.), crystal orientation, elastic stiffness tensor (Ck/), and grain size are assigned. The grains are considered to be spherical in shape and deformation is accommodated by (1 iO){i 11) slip. The size and crystallographic orientation assigned to the individual grains ultimately determine the grain size distribution and texture of the model polycrystals.  The procedures used for assigning these two  characteristics, described below, are implemented in an in-house code programmed in FORTRAN-77. 40  4.2.1. Grain Size Distribution Two types of grain size distribution are examined in this study: unimodal and bimodal. The unimodal distributions follow a lognormal probability density function as  P(DM,Sj==  —(lnD—M) 2 1 exp ..J2,TS2D 2S 2  (4.1)  where D is the grain size, and M and S are, respectively, the mean and standard deviation of the logarithms of the grain sizes.  The arithmetic mean grain size, ,u, and the standard  deviation, o, of the distribution are defined in terms of S and M as,  p=expM+  (4.2)  and  =  ([exp (S 2  +  2M)][exp(52  )  —  i])  (4.3)  The mean grain size of the distributions, p. is varied from 100 nm to 50 m. The standard deviation, o, on the other hand is determined by varying the o /p ratio from 0 to 1 at 0.2 increments; this allows for examination of grain size distributions narrower to slightly wider than those experimentally observed in polycrystals (see Appendix A2). The distributions are discretized by dividing them into 100 size classes.  Each size class is then randomly  populated. In this study only grains larger than 10 nm are allowed in a given grain size distribution.  This condition sets a limit on the width of the distribution that can be  considered for a given value of p. Figure 4.1 shows the generated lognormal grain size distributions in normalized format.  41  1.2  0.05  0.0 1.0  0.04 C  0.8  0  0  0  0.03  0.6 41,  41)  E  E  0.4  0.02  C  0,01  0.2  0.0 0.01  0.1  1  10  0.00 0,01  100  0.1  ‘10  100  10  100  L i,u 0.06  0.05  —  =  0.04  0.04 S  0 0  —  0.4  0 0.03  I  4-  0.02 C  0.01  0.00 0.01  0.03  0.02  0.01  0.00 0.01  A  L  —  0.1  1  10  100  0.1  Dip  1  Dip  0.05  0.05  jg=0.8  =  0.04  [0  0.04  C 0  C 0  0.03  0  0.03  I  0 0.02  E  0.02  C  0.01  0.00 0.01  0.01  A .  0.1  1  Dip  10  100  0.00 0.01  0.1  1  10  100  Dip  Figure 4.1. Generated lognormal grain size distributions. The grain size values, D, are normalized by the mean grain size of the distribution, p. The ob/p ratio noted on each plot is a measure of the width of the distribution.  42  For the bimodal distributions, it is assumed that the grain population consists of 1000 mono-size fine and 1000 mono-size coarse grains. The size of the fine grains is kept constant at either 100, 200, or 300 nm and the size of the coarse grains is either 1, 2, 3, or 10 urn. The volume of the fine grains is multiplied by a factor c such that the volume fraction of the coarse grains in the mixture is allowed to vary from 0.01 to 0.9. Equation (4.4) shows the relationship used for this purpose.  Icoarse = In this equation,  oarse Vcoarse + c  (4.4)  is the volume fraction of the coarse grains, V.oarse and Vf( are the  volume of the coarse and fine grains in the population, respectively, and c is the adjustable factor.  4.2.2. Texture Three Euler angles  1, ct, and 42 [152] are used to define the crystallographic  orientation of each grain with respect to the polycrystal axes, Figure 4.2.  The  crystallographic orientation of the individual grains is chosen in a manner for the polycrystal to have a random texture.  For each grain, this selection process can be summarized as  follows: i) A random unit vector (e.g. that of axis X’), defined by its three randomly selected direction cosines, is selected in the sample coordinate system. ii) A second unit vector (e.g. that of axis Y’), is randomly selected with the restriction that it has to be perpendicular to the first vector. iii) The cross-product of the two previously defined vectors is the unit vector for the third axis, Z’.  43  z  Figure 4.2. Three Euler angles angles 4)1, c1, and 4)2 relating the sample reference frame, XYZ, to crystal reference frame, X’YZ’. 4)1 is the angle between the Xaxis and the projection of Z’ axis onto the XYplane, is simply the angle between the Z and Z’ axes, and 4)2 is the angle between the Y’ axis and the zone axis of the two planes whose normals are the Z and Z’ axes.  iv) Following steps i, ii, and iii, the reference frame of the grain is randomly selected and thereafter the three Euler angles are determined in the manner depicted in Figure 4.2. Noting that 1 and 42 vary between 0 and 2rt and t’ between 0 and  it.  Figure 4.3 shows the equal-area projections (upper hemisphere of the stereographic sphere) of the generated crystal orientations when 2000 discrete orientations are considered. To quantify the randomness of the generated orientations, the equal-area projection is divided into 1296 equal-area elements (corresponding to 5° x 5° elements on the stereographic sphere) and the number of [1111 poles in each element is counted.  This number is  subsequently normalized to the total number of poles in the projection, the result being referred to as density. For each of the area elements the ratio of its density with respect to the average density of the projection, referred to as Multiples of Average Density (MAD), is evaluated.  In Figure 4.4 the distribution of MAD values for a number of the generated  orientations, with total number of orientations varying from 1000 to 5000, is depicted. Note 44  [1001  11101  [1111  Figure 4.3. Equal-area projections of [100], [110], and [111] poles of 2000 discrete orientations generated by the aforementioned approach. Each point on the images represents a single orientation or grain. Xand Y denote the sample axes.  1.0 J)  10.8  0.6  ¶ 0,4 0 0 0.2  0.0 0  2  4  6  8  muftiples of average density (MAD)  Figure 4.4. Density distribution of generated crystal orientations. The peaks on distribution profiles with in excess of 1000 orientations occurs at multiples of average density (MAD) ranging from 0 to 1.5.  45  that the MAD value of all area elements of a randomly textured polycrystal should be one. Figure 4.4 shows that when in excess of 1000 orientations are generated, more than 50 % of the area elements have densities near the average density (i.e. MAD values between 0 and 1.5). As the number of orientations increases, a larger fraction of the area elements will have densities close to the average density.  In other words, the randomness increases as the  number of orientations is increased. Note that this analysis is not affected when alternative poles, other than the [111] pole, are considered. In the end, as evident from Figure 4.4, it is acknowledged that the generated textures are not completely random. In fact, a more appropriate terminology to use instead of random would be pseudo-random. Hereafter, pseudo-random is implied when random is used for characterizing the generated texture.  4.3. Self-Consistent Formulations In this work, two models based on the self-consistent scheme are used for examining the impact of grain size distribution on deformation of polycrystals. The elastoplastic selfconsistent formulation, originally due to Hill [140] and Hutchinson [141], is used to examine the small strain (< 5 %) elastoplastic behaviour of polycrystals while the viscoplastic selfconsistent formulation, originally by Lebensohn and Tome [143] and Lebensohn et al. [153], is used for examining the fully plastic response. The elastoplastic formulation is based on consideration of elasticity and rate-independent plasticity.  It does not account for the  evolution of grain shape and crystallographic orientation with plastic deformation and it is relevant for simulation of small strain behaviour. The viscoplastic formulation on the other hand, disregards the elastic component during loading. It is, therefore, not applicable to small strain regimes where the elastic and the plastic components are comparable. The basic  46  formalisms of the two models are briefly discussed next. For a more detailed description of the two models see Appendix A3.  4.3.1. The Elastoplastic Self-Consistent (EPSC) Model In the EPSC model, the elastic problem treated by Eshelby [138] is extended to the plastic regime by expressing the non-linear response of the polycrystal and the grains in incremental form and further assuming that the stress rates are related to the strain rates through instantaneous moduli. At the grain level this means (4.5)  =  where  and  are the stress rate and strain rate of the grain, respectively, and  Luki  is the  grain instantaneous elastoplastic stiffness tensor. The grain stiffness is dependent on the orientation of the grain, the single crystal elastic stiffness, and the plastic state of the grain [141]. At the macroscopic level, the polycrystal stress rate, d, and strain rate,, , are also 1 linearly related through the polycrystal elastoplastic stiffness, ‘Jk1Ek1  Luki,  as (4.6)  Given the instantaneous (linear) relationships for the polycrystal and the grains, the Eshelby [138] solution is invoked. Accordingly, the deviation of the local magnitudes with respect to the macroscopic values is calculated using what is referred to as the interaction equation [141],  (o —J)=—4, ( kl) where  (4.7)  is the interaction tensor, which is a function of the macroscopic elastoplastic  stiffness L and the shape of the ellipsoid [141]. The partitioning of stresses and strains in  47  the polycrystal can be solved using the above set of equations. To do this, however,  L,Jk/  which is dependent on the response of the individual grains must be found first since it is not known a priori.  For this purpose, the iterative self-consistent approach is implemented.  Guesses regarding the value of problem.  are first made, allowing for a solution to the partitioning  To check whether the correct  L,Jk,  is found, the self-consistent conditions are  tested. This means that the polycrystal stress and strain rates must equal to the weighted stress and strain rate average of all the grains, i.e. (4.8) and (4.9) where  ()  denotes volume average. If the above conditions are satisfied, the problem is  solved and the grain and polycrystal stress and strain are defined. The above described EPSC formalism has been implemented in a FORTRAN-77 program by Tome [154].  4.3.2. The Viscoplastic Self-Consistent (VPSC) Model In this model, the kinematics of plasticity at the grain level is described by the viscoplastic (rate-dependent) equation [142,143], i.e.  -  =  (4.10)  where the superscript S denotes the slip system and the sum is carried over all the slip systems in the grain. Here, shear rate on slip system S,  i  is the deviatoric strain rate of the grain which is related to the through the symmetric traceless Schmid tensor associated 48  with this system, mS. The shear rate the threshold stress r  S  is a function of the deviatoric stress of the grain,  (and its corresponding reference rate  =  1 s’) and the rate  sensitivity exponent n. In this work, the value of n is set to 65. This value ensures that slip is only activated when the resolved shear stress is very close to the threshold stress while maintaining the numerical convergence in the model manageable. In cases where there is a notable difference between the resolved shear stress and the threshold stress, the strain in the grain is negligible and the grain responds as a rigid inclusion. To utilize the Eshelby [138] solution, the non-linear behaviour of the grain represented by Equation (4.10) is linearized as [142], (4.11) where Mkl is the grain compliance. Note that alternative linearization schemes, other than the so-called secant approach shown above, can also be envisioned [142,155]. In a similar manner, the constitutive relationship for the polycrystal is written in linear form as, (4.12) where  ,  .,  and Mkj now correspond to the polycrystal. Having linearized the response  of the grains and the aggregate, the Eshelby [138] solution is adopted, similar to EPSC, to relate the stress and strain rate in the grain to the overall stress and strain rate in the aggregate [142,143], i.e. (_J)=_1uk1(J_5)  The above equation is referred to as the interaction equation and  (4.13) is accordingly, the  interaction tensor. This tensor can be written in terms of a strength controlling parameter, as [155,156], 49  )I fl  MI/k1  where I is the identity tensor and  SUkI  (ia,,,,,  —  Sj,m  Snz,qMpqk1  (4.14)  the Eshelby tensor. The value of n controls the  interaction of the grain with its environment (i.e. the polycrystal). In this work, n’”  10 is  used which gives the polycrystal response in between the infinitely stiff upper-bound Taylor (n = 0) and the infinitely compliant lower-bound Sachs (ne’ = co) approximations  [155,156].  The above relationships allow for a solution to the partitioning problem, however, has to first be determined. For this purpose, the iterative process of the self-consistent theory is implemented; i.e.  is adjusted until the weighted average of the deviatoric  stress and strain-rate over the polycrystal coincides with the macroscopic quantities. Once satisfactory agreement between the weighted averages and the macroscopic values is achieved (within a relative tolerance of 0.001), the grain and polycrystal stress and strain are defined. The VPSC formalism described here has been implemented in a FORTRAN-77 program by Tome and Lebensohn [157].  4.3.3. Incorporating Grain Size Dependence The above described self-consistent models, in their original form, do not account for any type of grain size dependence. To introduce such dependence into the models, two basic avenues may be pursued. An additional equation or set of equations may be added to the grain-level (local) constitutive law used in the model. For example, an equation describing the evolution of a grain size dependent internal stress term in accord with the Sinclair et al. [82] hypothesis.  Alternatively, the local constitutive equations of the model may be 50  preserved and instead, the evolution laws governing these equations may be made grain size dependent. The latter approach is adopted in this study. For this purpose, the EPSC program subroutines G_MODULUS and G_VERJFY, i.e. subroutines where the strain and resolved shear stress increments are determined for each of the active slip systems, are rewritten. In the VPSC program, on the other hand, subroutine UPDATE_CRSS_VOCE is rewritten. The threshold stress associated with each slip system is updated inside the latter subroutine. In the following, details of the grain size dependences implemented in the noted subroutines are discussed. As indicated in Section 2.2, there are two aspects of the grain size dependence which need to be addressed. First, the grain size dependence of yield strength and second, that of work hardening.  For yield strength, in this work it is assumed that the Hall-Petch  relationship is applicable at the grain scale. This means that when the accumulated shear strain in the grain is zero; the threshold stress on any given slip system is given by 112 + 0 r kD  (4.15)  where z0 is the Peierls-Nabarro lattice friction, k a constant, and D the grain size. For work hardening it is assumed that as the grain size is reduced, the probability of storage of dislocations within the grain is reduced  —  in accord with the experimental and  theoretical observations noted in Section 2.2.1. This translates to a lowering of the work hardening capacity of grains as their size is reduced. It is further assumed that this reduction in work hardening with decrease in grain size at the local level, gives rise to a macroscopic/polycrystal work hardening behaviour similar to that reported for copper by Sinclair and co-workers [81 ,82j. That is, excluding the initial transient behaviour (Section  51  2.2.3), the hardening rates of polycrystals with different grain sizes when plotted versus flow stress converge. To account for the above grain size dependences, the threshold stress, r, is set to evolve with the accumulated shear strain in the grain, F, according to a new form of the extended Voce relationship as, o +[r’ where  +  (F+Fo)][l_exP_(F+Fo)]  is the initial hardening rate in the grain, r  stress, 9 is the asymptotic hardening rate, and  +  (4.16)  r is the back-extrapolated threshold  is a grain size dependent characteristic  strain whose size dependence is determined from (r  +  F ) 0 fi  —  0 exp—F  (4.17)  =  L Observe that Equation (4.17) can be derived from Equation (4.16) by setting F and v equal to 0 and r  + kIT” , 2  respectively. To determine the value of F 0 for each grain, Equation  (4.17) is to be solved. To solve this non-linear equation, Equation (4.17) is reduced to a quadratic equation by approximating the exponential term within it with its first order Taylor series about zero. Consequently, a relationship of the form shown below is obtained for F . 0  —  +&  +4L(kDf2)  0 F  +E(D)  (4.18)  2[]  In this expression E(D) represents a grain size dependent term which corrects the value of 0 for the error associated with the Taylor series approximation. The value of this term is F 52  numerically estimated for each grain during the simulation. A schematic plot of Equation (4.16) for different grain size values is shown in Figure 4.5(a).  The hardening law of  Equation (4.16) predicts a Stage III hardening with constant dynamic recovery rate and variable initial hardening rate depending on grain size, followed by a Stage IV hardening characterized by a constant hardening rate 6, see Figure 4.5(b).  I-  (b)  I  dO/dr  0  =  c.onst.  ±  Figure 4.5. Schematic representation of (a) the evolution of threshold stress with accumulated strain in the grain for a number of different grain sizes and (b) the hardening rate (0 = dr /dr) as a function of threshold stress for a given grain sizeD as described by Equation (4.16).  53  For the choice of the parameters in Equation (4.16), the approach proposed by Stout et al. [158] is used. That is, Equation (4.16) is used as input to the VPSC model and the model predictions are compared with a number of experimental stress-strain curves. The parameters of the local law are then iteratively determined once satisfactory agreement (i.e. coefficient of determination, R , greater than 0.99) between the model predictions and the 2 experiments is achieved. For this process, VPSC simulations using polycrystals with random texture and grain size distributions with average grain sizes equal to those reported for the experiments and 0 o/ p ratios equal to 0.4 are used. Note that the choice of 0.4 for the 0 u / p ratio falls within the experimentally determined range of values for this parameter, see Appendix A2. Figure 4.6 shows the fitted curves for the different polycrystal samples.  350 300  250  2  200  ‘i;  150  ‘4-  100 50 0 0.00  0.05  0.10  0.15  0.20  0.25  0,30  0.35  true strain  Figure 4.6. Experimental tensile stress-strain curves from [8 1,82] compared with their corresponding VPSC fit (o). For the VPSC fits, Equation (4.16) is used as grain-level hardening law and polycrystals having grain size distributions with average grain sizes equal to those reported for the experiments and cr 0 /p ratios equal to 0.4 are used. The inset shows the corresponding experimental work hardening rates.  54  The experimental data in this figure are all from the work of Sinclair et al. [81,82]. The determined model parameters are summarized in Table 4.1 and for comparison, the literature value for each of the model parameters is also tabulated.  Table 4.1. Values of parameters in Equation (4.16) used in the analysis  parameter  value  5 MPa 143 MPa  k  literature value <0.48 MPa [2,4,159]  100-l9OMPa[62]  380 MPa  160-480 MPa [62]  7MPa 0.094 MNm 312  1OMPa[160] 0.045-0. 14 MNm 312 [2]  The model parameters, with the exception of r , are all in agreement with the noted 0 literature values. The discrepancy between the model and the literature in the case of r 0 is expected. The noted range from the literature is related to experimental studies, see [161] for details of the experimental procedures, which correspond to conditions of single-slip while in the present model, deformation only occurs through poly-slip rather than single-slip. Similar value to that reported in this work has also been found in the viscoplastic simulations of Tome et al. [162]. Furthermore, from a practical point of view, the sensitivity of the model to this parameter is limited; r 0 only tends to shift the stress-strain curves (slightly) along the stress axis. It should also be noted that the model parameters were determined here using VPSC, though the same hardening law was used in the EPSC simulations. The reason for not utilizing the EPSC model in determination of the parameters is that first, the VPSC model is not limited to small strains like EPSC, allowing estimation of the large-strain parameters (viz. and 1 r ) . Second, the initial transient hardening behaviour observed in the experimental 55  copper stress-strain curves is avoided since the present hardening law does not account for such behaviour. The above described hardening law cannot explain the macroscopic grain size dependences observed in all metals. For example, Figure 4.7 shows the work hardening plots of an interstitial free (IF) steel having three different mean grain sizes [163].  5000 \ 4000  5.5  1m  75gm  3000 tz)  C 0  \ 2000 \  .C 1000  0 0  100  200  300  400  500  600  700  true stress (MPa)  Figure 4.7. Work hardening rate plotted against flow stress for IF steel of different grain sizes tested at room temperature. Rates calculated from stress-strain data in [163].  Unlike copper, the work hardening curves for this material do not converge to a single  curve.  Thus in this work, for comparison purposes a simpler hardening law which assumes that only the onset of plasticity is grain size dependent is also utilized. This behaviour is similar to that observed for the abovementioned IF steel. This new hardening relationship is expressed as, r 2 1 + 0 =r + kD’ (r +F)[l_exP_F]  (4.19) 56  where the parameters are defined as before and the same values used for Equation (4.16), i.e. Table 4.1, are also used in this equation. Finally, to determine the increase in the threshold stress,  r’, due to shear activity in  the grain slip systems, the above hardening laws are implemented in both the EPSC and VPSC models in differential form, i.e. S  where z\F  =  (4.20)  ’ is the accumulated shear strain increment in the grain. In VPSC, this 5 Ay  incremental relationship is found to make the hardening of the grain dependent on the simulation step size [157].  An analytic integration of this relationship is, accordingly,  implemented in the model. Thus, Equation (4.20) is replaced by  &r+I  9  I  (  ) 0 -Q(r+r  ‘  -—r — 1  e  -r  —1  e  lj  9  (4.21)  ) 0 --Q(r÷r  --r  [1+(r+ro+Ar)1_[1+  TI  when Equation (4.16) is used as the local hardening law and by _Lj-  ‘  0  rfl S  &ixr+I  I  e  ‘  ( Ie  -  r  o  r9  --F  -r  1  9  i+—Q-(F+AF) v ]  r  (4.22) 1+—-F  L  when Equation (4.19) is used instead. It should further be noted that the simulation step size, in both EPSC and VPSC models, is chosen such that the change in predicted equivalent stress with reduction in step size at any given strain is kept to  <  1 %.  57  CHAPTER 5 Simulation of Polycrystal Deformation Using the Elastoplastic Self-Consistent Model 5.1. Introduction The elastoplastic self-consistent (EPSC) simulation results are presented in this chapter. As elucidated in the previous chapter, EPSC does not take the evolution of grain shape and grain orientation into account. It is therefore suitable for modelling of small strain deformation. Accordingly, in this chapter EPSC is used for the purpose of examining the role of grain size distribution on the early elastoplastic transient stage of the deformation of polycrystals.  In the following section, the implemented modelling methodology is  introduced. Next in Section 5.3, examples of the model prediction are presented. Note that only polycrystals with lognormal grain size distributions are considered in this analysis. In Section 5.4, the simulation results are discussed. This includes details of how the nature of the grain size distribution of polycrystals affects their stress-strain response and in particular, the evolution of volume fraction of yielded grains with deformation. This information is used to evaluate a new methodology for defining yield strength for polycrystals. Finally, a summary of the chapter is provided in Section 5.5.  5.2. Methodology The model polycrystals, composed of 8000 spherical grains, are deformed in 900 steps to a final strain of 0.03 by subjecting each one to an axisymmetric tensile strain rate tensor of the form  58  where  10  0  0  —0.5  0  0  0  —0.5  (5.1)  is chosen to be 0.00 1 s’. After each deformation step, the macroscopic stress and  strain tensors are calculated for each polycrystal. These tensors are used to calculate the von Mises equivalent stress and strain values for each polycrystal. For each polycrystal, its yield strength and work hardening rate are calculated using the equivalent stress and strain values. The yield strength is defined using the 0.2 % offset method. The work hardening rate, on the other hand, is determined by numerically differentiating the equivalent stress-strain response of the polycrystal. The volume fraction of grains defonning plastically are also reported for each polycrystal after each deformation step.  5.3. Results The effect of grain size distribution on the stress-strain and work hardening behaviour of a number of polycrystals, those with 300 nm and 20 im mean grain sizes, is shown in Figure 5.1. Figure 5.1(a) and (b) demonstrate that the polycrystal loses its strength when the grain size distribution is widened.  This decrease in strength is less pronounced in  polycrystals with 20 urn mean grain size as compared to those with 300 nm mean grain size. For example, at a strain of 0.03 the flow stress of the 300 nm polycrystal with no size distribution (i.e. o /p  =  0) is 43 % larger than that of a 300 nrn polycrystal with the widest  size distribution (i.e. o- /p  =  1). This difference decreases to 18 % when 20 urn mean grain  size is considered. Another important feature of the curves in both these figures is that the transition from elasticity to plasticity is gradual. The above noted features are also apparent on the work hardening curves of the polycrystals, Figure 5.1(c) and (d).  The work 59  600  150  300 nm  (a)  20 .tm  (b)  500  125  100 0..  0.. 75  J  0.2  oie  50  = —  0.4  — -  25  — -  ne”!’ = 0,8  a 0.00  0.01  0.02  0.03  0.00  0.01  0.03  Eeq  eq  140  140  300 nm  (c)  20 m  (d)  120  120  100  100  a  0.. 80  80  0  x  0.02  0  60  -•  x  60  40  40  20  20  0  0 0  100  200  300  400  a (MPa)  500  800  0  25  50  75  100  125  150  (MPa)  Figure 5.1. Calculated von Mises equivalent stress-strain and work hardening rate, 0, curves for polycrystals with 300 nm, (a) and (c), and 20 urn, (b) and (d), mean grain sizes and varying widths of distribution. E denotes Young’s modulus.  hardening rate of the polycrystals with wider size distributions tends to drop below the elastic modulus value (denoted as E in the figures) at lower stresses. Furthermore, this drop is not sharp but gradual. 60  The evolution of volume fraction of yielded grains as a function of the polycrystal strain is shown in Figure 5.2 for the previously examined polycrystals. In both the 300 nm and 20 p.m polycrystals, when there is no size distribution (i.e. o/p  =  0), the volume  fraction of grains deforming plastically rises rapidly from 0 to 1; meaning that the entire polycrystal becomes plastic almost immediately upon yielding. Such rapid transition from elasticity to plasticity is not observed when wider distributions are considered. In fact, the wider the distribution, the longer it takes for the polycrystal to make the transition to the fully plastic state. As before, the grain size distribution effect is stronger when the mean grain size is 300 nm as compared to when it is 20 p.m. For instance, the difference between the strain at which plastic flow is instigated in the polycrystal with the widest grain size distribution and the one with no size distribution is about ten times larger in the case of the 300 nm mean grain size as compared to the 20 p.m mean grain size.  1.2  1.2  20 p.m  (b)  0 C  1.0 0 a)  1.0 a)  0.8  a) >.  o  0 C  0  06  hIi  0 4-.  0  0.4  .4-  a) E  !14  .4-  0.6  4-.  0  0.8  a) >.  0.4  U-  a)  Ez  0.2  0 >  0.2  0.001  0.01  0.1  0.0 0,0001  =0 0.2  0 Ji  =  ...——.‘..  11j’  ——•-  — -  0 >  0.0 0,0001  ‘fA’  n l 1 jf  ‘o/’  0.6  cr//i  0.8 =  l!!ii 0.001  0.4  0.01  I 0.1  eq  Figure 5.2. The evolution of volume fraction of grains yielded as a function of polycrystal von Mises equivalent strain for two polycrystals with (a) 300 nm and (b) 20 urn mean grain sizes and varying widths of distribution.  61  5.4. Discussion 5.4.1. Grain Size Distribution Effect The overall response of a polycrystal is determined from the collective behaviour of its individual grains. Since it is the volume of each grain that is subjected to the load, it is only logical that the volume of the grains rather than their number should be considered in this process. Considering that the grain size distribution of a polycrystal is described by the lognormal probability density function as [101,103], 1  J2.7rS2D  _(lnD_M)2”  exp  2 2S  I  (5.2)  where D is the grain size, and M and S are, respectively, the mean and standard deviation of the logarithms of the grain sizes. It can be shown, assuming grains of same shape, that the probability density function of the volume fraction of grains, F(D), is also lognormal (see Appendix A4). This probability density function can be expressed as  F(D)=  1 exp J2,rS2D  where S and M are defined as before.  2 2S  (5.3)  Figure 5.3 is a plot of the generated lognormal  distributions plotted in terms of the number fraction and volume fraction of grains. It is clear that as the distribution is widened, a larger volume fraction of the polycrystal will consist of grains larger than the mean grain size. This indicates that the mechanical behaviour of the polycrystal will largely be affected by the grains coarser than the mean grain size which occupy a larger fraction of its volume. These coarse grains, based on the assumed Hall-Petch relationship, have low strengths and consequently, the polycrystal loses its strength as wider  62  0.05  0.05 =  0.04  1) 4  0.04  0.03  0.03  0  0  0 —  0.02  0.02  0.01  0.01  0.00 0.01  0.00 0.01  .  —  0.1  1  10  100  0,1  Dip  1  10  100  10  100  Dip  0i.  0.06  —  oJiO.6 0.04  C  0.04  0.03  0.03  0  0  0  C  0.02  0.02  0.01  0.01  0.00 0.01  I  —  0.1  1  10  100  Dip  0.00 — 0.01  -.4  0.1  1  Dip  0,06  0.04  c  0.03  number fraction  0 4  0  *  votume fraction  0.02  0.01  0.00 0.01  Ih  —  0.1  1  10  100  Dip  Figure 5.3. Generated lognormal grain size distributions shown in terms of number fraction and volume fraction of grains. The volume fraction distribution is plotted using Equation (5.3).  63  distributions are considered, Figure 5.1(a) and (b). This effect is, however, balanced against the fact that at a given  ratio, the difference between the mechanical behaviour of the  o /p  small and large grains increases as the mean grain size decreases. This effect is clearly evident on the Hall-Petch plots of the examined polycrystals shown in Figure 5.4. Figure 5.4 illustrates that compared to the finest mean grain sizes, the influence of the width of the grain size distribution is relatively small for the very large mean grain sizes. As a result, the Hall Petch slope decreases from 0.24 MNm 312 to 0.14 MNm 312 when o/p is increased from 0 to 0.8.  Note that this observation is in agreement with the earlier modelling result of  Kurzydlowski [115] and the more recent works of Mitra et al. [98] and Berbenni et al. [118].  800 4  I  tm  044  im  0.25  un  0.16  Lfl1  orn  700 r / 0 p0  600  .‘. “  —500  aI,u  0.2  0’F  0.4  .  °‘o’P -0.6 cri’t0.8 I)  .  -  .  c400  ..:.  •...••  .:•  _a,.  C,  V  ..V.  300  ,::....  ..,.,.. V..  >200 100 0  1  0.0  0.5  1.0  1.5 l?2  2,0  2.5  3.0  ) 2 (im  Figure 5.4. Predicted yield strength as a function of square root of mean grain size for a variety of unimodal polycrystals with varying distribution widths. Yield strength defined as the 0.2 % offset stress. Slope of the lines decreases from 0.24 MNm 2 at = 0 to 0.14 MNm 312 at = 0.8  64  The character of the grain size distribution of the polycrystal also affects the elastoplastic transition of the polycrystal. This effect is depicted in Figure 5.2. It is clear from this figure, as might be expected, that percolation of yielding through the grains of the polycrystal extends over a wider range of strains in polycrystals with wider grain size distributions.  This percolation is less pronounced when larger mean grain sizes are  considered. These results can be understood in terms of the Hall-Petch effect. As discussed in Section 2.2.2, instigation of plasticity/yielding at the grain level, becomes more difficult with refmement of grain size. Accordingly, while keeping the mean grain size constant, as the width of the distribution is increased, the grain population will consist of a larger fraction of coarser grains. These coarse grains yield more readily and therefore, the onset of the elastoplastic transition is shifted to lower strains. The same argument can be used to explain the role of the mean grain size on the elastoplastic transition. At a constant o /p ratio, a larger mean grain size means a population of grains with lower yield strengths. Thus once again, the transition from elasticity to plasticity occurs earlier in the course of the deformation of the polycrystal.  5.4.2. Modelling the Evolution of Volume Fraction of Yielded Grains The elastoplastic transition in a polycrystal is a consequence of the inhomogeneity of yielding among the population of grains. As shown in Figure 5.2, gradients of deformation are imposed by the distribution in size of the grains which ultimately lead to such inhomogeneity. Considering that the grain size distribution of the examined polycrystals is lognonnal, these results can be condensed into a simple expression using the following statistical analysis.  65  Having described the probability density function for the volume fraction of grains, F(D), using Equation (5.3); the volume fraction of yielded grains,  f’, at any instance of the  deformation of the polycrystal can be calculated as, fY..fF(D)i]J  Here D,  is  a  (5.4)  critical grain size where grains with sizes exceeding D. are considered  plastic/yielded. Substituting for F (D) from Equation (5.3) and taking into account that -4 -erf(x)  (5.5)  = _=exp(_x2)  Equation (5.4) can be simplified to  fY  lnD_(M+3S2)  1  =  —erf  (5.6) D.  Substituting for the upper and lower limits, one gets i  i  (lnD._(M+3s2) (5.7)  To complete the evolution expression for  fY,  it is now only necessary to express D in  terms of the macroscopic strain of the polycrystal,  .  Assuming isostrain partitioning of  strain among the grains and that yielding of a grain is governed by the Hall-Petch equation, the relationship between D and the polycrystal strain can be formulated as =  cr’ + KD 2  (5.8)  where E is the Young’s modulus and u’ and K are constants. Substituting for D from Equation (5.8), Equation (5.7) is simplified to  66  1 ’ =_erf (C 3 f  _C,ln(&_c))  (5.9)  where C 1 and C 2 are defined as 21n(K/E)_(M+3S2)  Js and = 7 C  and ç  (= u’/E) is  (5.11)  a characteristic strain. Figure 5.5 shows the evolution of fY with strain  for a number of different grain size distributions as predicted from Equation (5.9).  1.2 C  C  1-a ‘O  0>  0.8 0)  0) >‘  C 0  .4-  0.6  0 C  0 U .4-  0>  B  0.2 0 >  0  >  0.0 0.0000  0.0004  0.0008  0.0012  0.0016  E Figure 5.5. The effect of (a) mean grain size and (b) crc/p ratio of lognormal grain size distribution on the evolution of volume fraction of yielded grains as predicted from Equation (5.9).  For this plot K  =  , E 312 0.094 MNm  123 GPa, and o’  5 MPa are used; in accord with  values reported in Table 4.1. It is clear from this figure that Equation (5.9) does predict the same grain size distribution effects observed in Figure 5.2 and discussed in Section 5.4.1.  67  That is, decreasing the mean grain size and widening the size distribution both lead to a more heterogeneous deformation and accordingly, prolongation of the elastoplastic transition. Figure 5.6 shows Equation (5.9) fitted to a number of the EPSC predicted  f’  profiles. In this process, C 1 and C, are taken as adjustable parameters and for simplicity, ç is assumed to be zero (the latter assumption is justified based on the fact that E is orders of magnitude larger than o’). It is clear that Equation (5.9) fits well to the fY profiles.  1.2  .  1.0  0.8  0.6 C,  0.4  0.2  0.0  cq  1.2  .5  1.0  0.8  o 0 C,  0.4  a,  a  0.2  0.0  Figure 5.6. Fit of Equation (5.9) (dashed lines) to EPSC predicted volume fraction of yielded grains for polycrystals with varying widths of grain size distribution and (a) 300 nm and (b) 20 tm mean grain size.  68  Given that C 1 and C 2 are functions of the grain size distribution of the polycrystal (namely M and S), it is possible to use the C 1 and C, values determined from the fits to backcalculate M and S and from those, the mean and standard deviation of the grain size distribution of the polycrystals. Figure 5.7 shows comparison of the mean grain size and standard deviation of the size distributions predicted from the  ’ 3 f  profile fits and those used  as input values in the corresponding EPSC simulations.  100  100 •  /p0.2  V  7/?J =  0. 6•,  :  10,  . 0.1  *  0.1  (a)  0.1  1  10  (b)  ..  -  0.01 0.01  100  (1.trn)  -‘  0.1  1 input  10  100  (gm.)  Figure 5.7. Comparison of (a) mean grain size and (b) standard deviation determined from fit to model predicted volume fraction of yielded grains (p and and values used as input in the model (p. 1 and Dotted lines represent cases where the input and fitted values are equal.  In this analysis the Hall-Petch exponent and the constant K are initially adjusted for one of the profiles (namely  ji  300 nm and o/p  =  1) so that the correct mean grain size and  standard deviation are deduced from the fitted C and C . These values are then used for the 2 rest of the profiles. The described calibration procedure for the exponent and K is necessary  69  since the isostrain assumption used in derivation of Equations (5.10) and (5.1 1) does not apply to the EPSC calculations. For Figure 5.7, an exponent of 0.63 and K  0.058 MNm 2 ” 3  are used. In Section 5.4.3 it will be shown that the evolution of  f”  with strain can be estimated  directly from the stress-strain curve of a polycrystal. The above methodology of extracting information regarding the grain size distribution of a polycrystal from its therefore be very useful.  f’  profile can  That is, in theory in addition to information regarding the  mechanical response of the polycrystal, the character of the grain structure of the polycrystal may also be accessible through a simple tensile test. However, as evident from Figure 5.7, there is some inconsistency between the predicted and input values.  One source of this  discrepancy is that the simple calibration procedure described above to correct for the isostrain assumption might not be sufficient. The other reason for this disagreement can be traced back to the fact that the grains are not of the same crystallographic orientation. The procedure outlined in this section is purely statistical and only accounts for the grain size distribution being the source of the elastoplastic transition in a polycrystal.  The  inhomogeneity in the yielding of the grains can also stem from the distribution in crystallographic orientation of the grains [164]. Observe that the elastoplastic transitions for polycrystals with o/p  =  0 shown in Figure 5.1 and Figure 5.2 are a consequence of the  distribution of crystallographic orientations among the grains. Figure 5.8 shows an example of how the evolution of yielded grains is affected by the orientation distribution of the grains. The polycrystals delineated by solid line in this figure all have random textures.  For  comparison, polycrystals with the same size distribution but no distribution of  70  12  .E  1,0  0,  •0 0.8  o 0 0  0.4 02  0.0  Ecq  eq  Ec,q  Figure 5.8. The evolution of volume fraction of yielded grains with macroscopic strain for when all the grains have the same crystallographic orientation (dashed lines) and when the grains are randomly orientated (solid lines). The mean grain size of the polycrystals is 300 nm.  crystallographic orientation, i.e. all grains having the same crystallographic orientation, are also shown in this figure. It is clear that the orientation distribution, even if random, does influence the spread of plasticity. The orientation distribution, however, does not affect the early stages of the elastoplastic transition.  Nonetheless, it is difficult to assert a firm  conclusion based on these limited results, particularly given the limited number of grains considered in the present calculations. This represents an. opportunity for future work.  5.4.3. Characterization of Yielding In tests of materials under uniaxial loading, different criteria for the initiation of yielding are generally used: the elastic limit, the proportional limit, and the yield strength [1651. Yield strength, which by definition is the stress required to produce a small specified amount of plastic deformation (usually a plastic strain of 0.002, i.e. the 0.2 % offset yield strength), is contmonly used for design and specification purposes because it avoids the practical difficulties of measuring the elastic limit or proportional limit [165,166]. It is implicitly 71  assumed that the 0.2 % offset yield strength corresponds to the stress at which all grains have started to yield [167-169]. Recent analysis of X-ray diffraction spectra measured during insitu deformation of ultrafine-grained and nanocrystalline Ni has, however, shown that such presumption is not justified when fine-grained polycrystals are considered [169]. In fact, considering  conventional  deformation  mechanisms  at  the  ultrafine-grained  and  nanocrystalline regimes, Saada [167,168] notes that the minimum offset strain should be at least of the order of b / D, with D being the grain size and b the Burgers vector, if each grain of polycrystal is to be swept by at least one dislocation. This criterion is in agreement with the experimental findings of Brandstetter et al. [169] on a 30 nm grain size Ni which demonstrate an offset strain of 0.007 is necessary in order for all the grains to have started yielding. Comparison of the 0.2 % offset yield strength of coarse grained and fine-grained materials, e.g. when examining grain size effects, is thus not appropriate since the comparison of the two classes of materials is not made at the same state. Figure 5.9 shows a series of Hall-Petch plots for a variety of the modeled grain size distributions (the 0 /p ratio of the distributions is noted on each of the plots). In this figure, the 0.2 % offset yield strength of each polycrystal is compared with its true yield strength; the true yield strength being defined as the stress at which 99 % of the volume of the polycrystal is plastic. For comparison, the proportional limit of each polycrystal is also depicted. The proportional limit is defined here as the stress at which first deviation from linear elasticity is observed. It is clear that as the mean grain size is reduced and/or the width of the grain size distribution is increased, the difference between the 0.2 % offset yield and the true yield increases. This is particularly true when polycrystals with p <1 jim and a/p  >  0.4 are considered.  example, in the case of a polycrystal with a mean grain size of 100 nm and u /p 0  =  For  0.6, the 72  1000  1000 1  I  0.25 pm  pm  800  •= —  0.25  pm  600  .  pm  600  C) C  . -  400  400  .  _.w  >‘  200  •  -  200  truc yId sength 02%oIfst ropGrnanimit  ‘7  -  D  ou=fl,2  0  0  o  2  1  3  0  4  1  .t.I/2 (mli ) 2  2  3  4  412 (j.tm) p  800  800 1  0.25 pm  pm  I pm  0.11 pm  0.25 pm  .  :  600 0  0.11 pm  ooo  400  400  —  .  .:  -  ,  0.11  800  C) C  4)  LLm  .,  200  .‘  .  .  ,—  200 1/  /I1O4  W; 0 O  0 4 .112  -112  2 (rn  500  (J.Lm)  300 I pm  0.25 pm  4  Lm  I pm  0.44 pm  250  400  1*  200  •  300 C)  C) ...  160  200 i00  100  >  ,,  --  ç *  0  -I/2  : ) 112 (m  ,.  50  05  10  112 ç  IS  20  ) 112 (jim  Figure 5.9. Hall-Petch plots for polycrystals with varying grain size distributions (u /p ratios noted on each 0 plot). For each case, three different definitions of yield strength are used. True yield strength refers to stress at which 99 % of the volume of the polycrystal is plastic/yielded.  73  offset yield is approximately 10 % smaller than the true yield. In all cases, except for the very coarse grained polycrystals, the significant difference between the proportional limit and the true yield is evident. Another consequence of the discrepancy between the different definitions is that depending on the definition used, a different Hall-Petch slope will be defined. It is therefore useful to devise an alternative methodology for defining the yield strength of polycrystals such that the above noted discrepancies can be avoided. Below, one such methodology, introduced by Saada [1701, is examined with the aid of the present model. Saada and co-workers [167,170] recently demonstrated that the work hardening rate of a polycrystal, ®, during uniaxial tension or compression can be related to the average plastic strain rate, , and the applied strain rate, , of the polycrystal through  (5.12) where E is the Young’s modulus corrected for the rigidity of the test machine. Saada [170] further points out that the yield strength may be defined as the stress for which but close to , i.e. ®/E  —  is smaller  0. In the limit of uniform partitioning of strain amongst grains  (the Full Constraints assumption), the simple statistical approach of Afanasev [171], outlined in Appendix A5, shows that the ratio e/E at any instance of deformation is simply equivalent to the volume fraction of elastic grains. Accordingly, Equation (5.12) can be reformulated as, (5.13)  74  where  f  and  given strain.  f’  are, respectively, the volume fraction of elastic and yielded grains at a  Figure 5.10 shows a comparison of the volume fraction of yielded grains  directly determined from the EPSC simulations and those determined using Equation (5.13).  1.2  ‘U  ‘0  1.0  0.8  >‘  0 C  0.6  U  0.4  0  0.2  0.0  Coq  1.2  .  1.0  a’ ‘0 0.8  o  0,6  C  0 U  i  0.4  4..  E 0.2 >  0.0 0.001  e eq  0.01  eq  Figure 5.10. Comparison of the model predicted volume fraction of yielded grains (solid lines) and that determined using Equation (5.13) (dashed lines) for polycrystals with (a) 300 rim and (b) 20 ,im mean grain size.  75  It is clear from this figure that there is a close correlation between the two  f”  profiles. This  means that the results of the above isostrain approximation can reasonably be extrapolated to the iterative partitioning scheme of the self-consistent approach. Having said that, the Saada definition of yield strength can accordingly be interpreted as indicating that yield strength is the stress at which the fraction of yielded grains approaches unity, i.e. the polycrystal becomes fully plastic. Figure 5.11 shows a comparison between the true yield strength and the yield strength determined using the Saada approach for a number of different polycrystals. The figures are plotted in the form of Hall-Petch plots making comparison of the resultant Hall-Petch slopes also possible. As before, the true yield strength is defined as the stress at which the model predicted volume fraction of yielded grains reaches 0.99. For the Saada yield strength on the other hand, the yield strength is defined as the stress for which  f” (= l—€’/E)  reaches 99 % of the final value.  A comparison between Figure 5.11 and Figure 5.9 clearly shows that the Saada definition of yield strength is more suitable for defining the yield strength of a polycrystal as compared to the 0.2 % offset definition. First, the Saada definition is not arbitrary set as the 0.2 % offset definition. Second, over the grain size distributions examined, the Saada yield strength provides a better estimate of the true yield strength of the polycrystal. This is particularly true for cases where o/p  >  0.2. The same is true for the Hall-Petch slopes. As  shown in Figure 5.12, the evolution of the Hall-Petch slope with the width of the grain size distribution is better approximated by the Saada approach.  76  1000  1000 I rn  0.25 urn  Oil tm  800  1 LN  0.25 urn  0,11 tm  1  2  3  0.25 trn  0.11 im  2  3  800  .  N  cjiO.0  S  •:  600  600  •1  0)  0)  C  C  a, U) •0 C,  0  .  400  0  U)  400  •0 0 >,  i 200  200 tu yiId strergth Saad  •  V 0  0 0  1  2 .i/2  3  0  4  .1f 2  ) 1t2 (pm  800  800 I trn  0.25 im  0.11 trn  I trn  .  600  N 0  1  0  5  0  600  S ‘U  400  C 0  .  400  U)  .#  0  200  ‘p.  0  200  i.  rjt0.4  .  0  0 0  1  2 j1I2  3  0  1 lt2  ) 2 (j.tm”  600  4  ) 112 .i.m’  300 1 pm  0.25 pm  4  .Lm  1  tm  0.44 p111  250  400  ‘-V  N  0  0.. 2  S  200  300 0)  0)  C  0  0  200  160  U)  100  0  0  0  >  50 =  0.8  ‘/1  0  I .0  0 0  1 1I2  2  ) 2 ().Lm”  3  0.0  0.6  1.0 14i2  (p.m  1.6  2.0  .112)  Figure 5.11. Comparison of the Hall-Petch plots for polycrystals with different grain size distributions (go/p ratios noted on each plot) when the true yield strength of the polycrystals and the yield strength determined through the Saada approach is used.  77  0.30 • true yield strength $aada •e 0.2% offset 0’’ proportional limit V  0.25  .  .  E 020  .....••‘•:• .20.15  .,  •0  (j  e>  0.10  0  1005  0.00 0.0  0.2  0.4  0.6  0.8  1,0  1.2  Figure 5.12. Variation of the Hall-Petch slope with o/p when different definitions of yield strength are used.  5.4.4. Impact of the Local Hardening Law The above discussed results are all based on the assumption that a copper-type local hardening law where both the onset and the evolution of plasticity in a grain are dependent on the size of the grain is valid. As noted in Chapter 4, this local hardening law cannot explain the macroscopic grain size behaviour observed in all metals. Thus, to see how the type of hardening law implemented in the EPSC model will affect the observed grain size distribution effects, a simpler model where only the onset of plasticity is made grain size dependent is used here, i.e.  =  2 +(r r +kD” 1 +F)[l_exP_F  (5.14)  where k is the local Hall-Petch constant, D the grain size, and the other parameters are defined as before. 78  Figure 5.13(a) shows the stress-strain behaviour of a number of polycrystals having 300 nm mean grain size but varying distributions. As with the previous hardening law, the effect of widening the width of the grain size distribution is a lowering of the strength of the polycrystal. This decrease in strength also manifests itself in a decrease in the Hall-Petch slope with o/p, as depicted in Figure 5.13(b).  600  500  (b) 500  1 aU  4rT1  0.44 ian  400  400 300 300 ‘  200  200 >‘  100  100  0 0,00  0 0.01  0.02  0.03  o.o  o.s  1.0  1.5  2.0  .112  Figure 5.13. (a) Calculated von Mises equivalent stress-strain curves for polycrystals with 300 nm mean grain size and varying width of distribution. (b) Predicted yield strength as a function of square root of mean grain size for a variety of polycrystals with varying distribution widths. Yield strength defined as the stress at which the volume fraction of yielded grains reaches 0.99. Local hardening law where only the onset of plasticity is grain size dependent in each grain is used.  Figure 5.14 shows that given the new local hardening law, the evolution of volume fraction of yielded grains can still be reasonably estimated using Equation (5.13). Furthermore, Figure 5.15 shows that the Saada definition of yield strength does also provide a good estimate for the true yield strength of the polycrystal.  79  12  7...,uO.6 C  1.0  •0 .  X8  1”  o C  0.6  0 C  0.4 r  C 0  02  0.0  ,  10  .1.  ,  0.001  0.01 eq  0.1 10’  0.001  0.01  .._.....__1  0.1 1D  0.001  0.01  0.1  Ceq  1.2  C  .  o  1.0  0.8  0,6  C  0  U C  0.4  C 0 >  02 0.0  teq  Figure 5.14. Comparison of the model predicted volume fraction of yielded grains (solid lines) and that determined using the Saada approach (dashed lines) for polycrystals with (a) 300 nm and (b) 20 im mean grain size. Local hardening law where only the yielding of the grains is size dependent is used.  80  600  4pm  1  pm  4um  0.44 w’  500  1 pm  0.44 pm  V  300 0.  0.  400  300  4,  C  I  200  200 4,  100 100  aije=O2 0.0  0.5  1.0  i.5  o,JjI =06 2,0  0— 0.0  0.6  1.0 -1/2  1.5  2.0  ) 11 (lim’  300  4pm  1pm  0.44prn  250 0.  C a,  200  • v  160’  true yie’d strength Saada  100 4,  C  cr  iiI,O  0 0.0  0.5  1.0 1,-ll2  1.5  2.0  ) 112 (m’  Figure 5.15. Comparison of the Hall-Petch plots for polycrystals with different grain size distributions (o’o/ji ratios noted on each plot) when the true yield strength of the polycrystals and the yield strength determined through the Saada approach are used.  81  One can, therefore, conclude that the details of the local hardening law does not have a notable effect on the trends and analyses discussed in the previous sections. Given the small strains involved in the elastoplastic transition of a polycrystal, the work hardening behaviour of the grains does not have the opportunity to affect the results significantly.  5.5. Summary The influence of lognormal grain size distribution on the elastoplastic transient behaviour of polycrystals is examined in the continuum mechanics framework of an elastoplastic self-consistent model.  It is assumed that instigation of plasticity in a grain  obeys the Hall-Petch relationship. It is found that for a given mean grain size, as the width of the grain size distribution is increased, the polycrystal loses its strength.  This effect  diminishes as larger mean grain sizes are considered. The decrease in strength manifests itself in a reduction in the Hall-Petch slope as wider distributions are considered. The noted effects can be attributed to the increasing volume fraction of grains coarser than the mean grain size of the polycrystal as the width of the grain size distribution is increased. In this work, these coarse grains are assumed to have lower yields. The character of the grain size distribution also affects the nature of percolation of plasticity in the polycrystal, i.e. the elastoplastic transition. Wider distributions and smaller mean grain sizes foster extended elastoplastic transitions.  These are transitions where the evolution of volume fraction of  yielded grains occurs over a wider range of strains. A simple model is put forth for describing the evolution of volume fraction of yielded grains with applied strain. It is shown that the model is capable of describing the results of the EPSC simulations well. The potential of inferring information regarding the character of the grain size distribution of the polycrystal from the evolution of the volume fraction of  82  yielded grains is investigated. It is further shown that these volume fraction profiles can directly be estimated from the tensile stress-strain curves of the polycrystals. The stress strain curve of a polycrystal can thus directly be used to characterize its grain size distribution.  Further work on this particular aspect of the analysis is needed with closer  inspection of textural effects.  Using the volume fraction profiles, a more meaningful  definition of yield strength, as compared to the arbitrary 0.2 % offset definition, is introduced.  83  CHAPTER 6 Simulation of Polycrystal Deformation Using the Viscoplastic Self-Consistent Model 6.1. Introduction The impact of grain size distribution on the elastoplastic behaviour of polycrystals was examined in the previous chapter using the elastoplastic self-consistent model. Given that the elastoplastic model does not account for the evolution of the crystallographic orientation and shape of the grains during deformation, analysis of the grain size distribution effects was limited to only small strains. In this chapter, the results of the viscoplastic selfconsistent simulations, which demonstrate how grain size distribution affects the moderate strain (and fully plastic) response of a polycrystal, are presented and discussed. In particular, the effect of grain size distribution on work hardening behaviour and accordingly, ultimate tensile strength-uniform elongation relationship of polycrystals is examined.  In what  follows, a description of the simulation methodology is first presented. Examples of the predicted plastic response for a number of unimodal and bimodal polycrystals are described next in Section 6.3. In Section 6.4, these results are discussed. In this discussion, the nature of stress-strain partitioning amongst the grains is visited. The impact of the different model assumptions and variables, including the assumed grain-level hardening relationship and homogenization methodology, are also reviewed in this section. At the end of this section, the validity of the developed modelling framework is evaluated through comparison with experimental data from the literature. A summary of the chapter follows in Section 6.5.  84  6.2. Methodology The different model polycrystals, which are composed of 2000 spherical grains, are deformed in 250 strain increments of 0.002 by subjecting them to the following axisymmetric tensile strain rate tensor,  ‘=i  where  10  0  0  —0.5  0  0  0  —0.5  (6.1)  is chosen to be 0.00 1 s’. The shape and crystallographic orientation of the grains  are allowed to evolve (according to the procedures outlined in [143,153]) and they are updated after each deformation step. For each polycrystal the local (grain-level) and global deviatoric stress and strain rate tensors are calculated after each deformation step and then converted to equivalent von Mises values. Expressing the global stress and strain data as von Mises values allows one to directly compare these calculated values with those experimentally determined in uniaxial tension or compression. To compare and assess the plastic response of the different polycrystals, their yield strengths, work hardening rates, uniform elongations, and ultimate tensile strengths are compared.  In this work, the  equivalent stress after the first deformation step is defined as the yield strength. The work hardening rate of a polycrystal is determined by numerically differentiating its equivalent stress-strain curve.  Finally, the extent of uniform elongation is determined from the  Considére criterion (rate independent condition) and the equivalent stress at this point is used to calculate the ultimate tensile strength, i.e. by converting it from the true stress to the engineering stress.  85  6.3. Results 6.3.1. Unimodal Polycrystals Figure 6.1 shows the calculated stress-strain curves for a number of polycrystals with varying mean grain sizes considering that all grains are the same size, i.e. c/,u  0. It is  evident from this figure that as the average grain size increases, the yield strength decreases but the work hardening capability of the polycrystal increases. Figure 6.1 also shows that as the mean grain size increases, the difference between the stress-strain curves of the polycrystals becomes less pronounced.  450  300  --  .-  —  —— /  ,—,  .-—  / ,/  b  ‘  /  150 “  /  /  —‘...— .— —  ./_  z’ /// V  ‘  / 0.5im 0.7iim I  ,/‘  .“  / —..—.-  2im smj  —. — — 50 im  0  I  0.0  0.1  0.2  0.3  0.4  0.5  Figure 6.1. Computed von Mises equivalent stress-strain curves for a number mean grain size. The polycrystals do not have a size distribution (i.e. /u = 0).  0.6  of polycrystals with varying  The effect of increasing the width of grain size distribution on the yield strength is shown in Figure 6.2 for a number of polycrystals with varying mean grain sizes. This Hall Petch plot shows that increasing the width of the distribution results in a lowering of the yield 86  800 4  1 m  m  U.44 m  O2 m  Q.6  up  L11  m  700 ./ii=0  600  (T / 0 Jt  V  0  0.2  (T ? 0 fi:0.4  500  rJu  0  T / 0 p  400  0.6 0.8  0  300 .  200 100 0 0.0  0.5  1.0  1.5  2.0  2,5  3.0  3.5  ) 112 (jim’  Figure 6.2. Predicted yield strength as a function of square root of mean grain size for a variety of unimodal polycrystals with varying distribution widths. Yield strength defined as the von Mises equivalent stress calculated after the first deformation step. The slope of the lines decreases from 0.23 MNm 312 at u /p = 0 to 0 0.13 MNm 312 at = 0.8  strength. This effect is stronger the smaller the mean grain size, resulting in a Hall-Petch slope that decreases from 0.23 MNm’ 312 at /p  0 to 0.13 MNm’ 312 at o/p  0.8.  Observe that the assumed Hall-Petch behaviour at the local level induces a macroscopic Hall Petch effect that depends on the grain size distribution. Figure 6.3 depicts how the width of the grain size distribution affects the stress-strain response (Figure 6.3(a) and (b)) and the work hardening behaviour (Figure 6.3(c) and (d)) of a polycrystal. Two different mean grain size values are selected, 700 nm and 10 p.m. and the stress-strain curves are plotted to the end of uniform elongation (as calculated from the Considère criterion). Figure 6.3(a) shows that as the width of the distribution is increased,  87  450  450  (a)  700 nm  350  350 -—  --  -<— .—. .-—  1.  250  .—:--‘  250  ‘  ‘4’  7fpO  b  0.2  .  150  150  = 0.4  ..  an/p  0.6  50  50  0,0  0.1  0.2  0.3  0.0  0.1  0.2  8 e q  0.3  0.4  eq  2000  2400  (c)  700 nm  10 im 2000  1600  1600 1200 1200 800 800  -— — -‘  400  400  0 0  100  200  300  a (MPa)  400  500  0  100  200  300  400  500  a (MPa)  Figure 6.3. Calculated von Mises equivalent stress-strain curves and work hardening plots of polycrystals with 700 nm, (a) and (c), and 10 m, (b) and (d), mean grain sizes and varying widths of distribution. In (c) and (d) the 0 ueq lines correspond to the Considère criterion. =  the polycrystal loses some strength but its uniform elongation increases. At the same time, the work hardening rate of the polycrystal, €, increases as shown in Figure 6.3(c). The work hardening plots of the wider distributions deviate from linearity close to yield but 88  eventually the work hardening rate becomes a linear function of stress. The polycrystals with 10 .tm mean grain size behave similar to their 700 rim counterparts but the effect of the width of the distribution is much smaller. The evolution of the local plastic stress and strain fields is shown in Figure 6.4 for three polycrystals. Two of these polycrystals have the same average grain size (700 nm) but different distribution widths, (o /p  =  0.2 and o /p = 1). The other polycrystal is chosen to  have an average grain size of 10 im with a wide distribution.  w 0 U h.  U 0  j  2U  E  e U 0  0.0  0.3  0.6  0.9  grain size (tim)  1.2  1.5  0  2  4  6  grain size (j.im)  8  0  20  40  60  80  100  120  grain size (p.m)  Figure 6.4. Predicted relative von Mises equivalent stresses and strains as a function of grain size for three model polycrystals at three different stages of deformation corresponding to macro von Mises equivalent strains of 0.01, 0.10, and 0.50. The mean grain size of the polycrystals and the width of their grain size distribution /p) is indicated in the images. The top row corresponds to the relative strain values and the bottom row to 0 ( the relative stress values. Local refers to grain-level values.  89  For these plots, the grains of each polycrystal are divided into size classes and the volumeweighted average of the equivalent stress and strain of the grains in each of these classes is calculated at different deformation steps.  The ratio of this value to the macroscopic  equivalent stress or strain is plotted. It is evident that the deformation of the polycrystal with the narrow grain size distribution is more homogeneous, this homogeneity increases as deformation progresses. The fine grains of the polycrystals with wider distribution widths tend to deform less and are under more stress than the average. As deformation advances, the polycrystals become less heterogeneous causing the strains and stresses in the grains to approach the farfield average values. The polycrystal with the larger mean grain size loses this heterogeneity faster than the polycrystal with smaller mean grain size. In fact, even at the later stages of deformation, the finer grains of the p distribution retain a higher stress and lower strain.  =  700 nm polycrystal with a wide  This is not the case for the 10 tm  polycrystal. The plots in Figure 6.4 are not smooth, particularly at larger grain size classes. This is a consequence of the crystallographic orientation of the grains. In the large size classes, the number of grains examined is relatively small due to computational limitations. Accordingly, the impact of the crystallographic orientation of the grains on the partitioning is amplified.  6.3.2. Bimodal Polycrystals Figure 6.5 shows the predicted stress-strain curves of a number of bimodal polycrystals constructed from different fractions of 200 nm and 3 pm grains. Also included in this figure are the stress-strain responses for the two limit unimodal polycrystals with 200  90  600  500  0.  J  400  300  200  100  0.00  0.05  0.10  0.15  0,20  0,25  0,30  0.35  eq  Figure 6.5. The effect of adding different volume fractions of coarse 3 urn grains to the mix of fine 200 nm grains as predicted by the model. Percent fractions are noted on the figure. The 0 % and the 100 % indicate polycrystals with unimodal grain sizes of 200 nm and 3 jim, respectively.  nm grains (denoted 0 %) and 3 im grains (denoted 100 %). All the curves are plotted to the end of uniform elongation as determined by the Considère criterion. The polycrystal with a uniform grain size of 200 nm has a high yield stress (524 MPa) but no uniform elongation. In the other extreme, the polycrystal with a grain size of 3 jm has a comparatively low yield strength (144 MPa) but a relatively large uniform elongation (0.28). By adding the 3 jim grains to the 200 nm grains, the uniform elongation of the polycrystal is restored at the expense of its strength. For example, by adding 20 volume percent of 3 p.m grains, the yield strength of the fine-grained material decreases by 22 % but the uniform elongation increases from 0 to 0.064.  Another noteworthy behaviour observed here is that by increasing the  volume fraction of the 3 p.m grains, the initial work hardening rate of the polycrystal increases strongly up to approximately 50 % volume fraction after which it starts to decrease. 91  6.4. Discussion 6.4.1. Grain Size Distribution Effect Consistent trends to those predicted by the elastoplastic model and discussed in Section 5.4.1 are also evident in the present viscoplastic simulations. The same arguments put forth to explain the influence of grain size distribution on the elastoplastic response of polycrystals can, therefore, also be used to explain these results. That is, in the analysis of the deformation of a polycrystal it is necessary to consider the volume of the grains rather than their number. Given that the probability density function of the lognormal distribution is positively skewed, a wider distribution translates to a larger volume fraction of the polycrystal being occupied by grains larger than the mean grain size; see Figure 5.3. The mechanical behaviour of the polycrystal is, thus, more influenced by the coarse grains which occupy a larger fraction of its volume and are assumed to have lower yield strengths and higher work hardening rates.  As in the elastoplastic case, for a given o /jt ratio, the  difference between the mechanical response of the small and large grains increases as the mean grain size decreases. This effect is clearly observed in Figure 6.2 where for large mean grain sizes, the effect of the width of the grain size distribution on the yield strength is relatively small compared to smaller mean grain sizes. The above noted general considerations can also be invoked to understand the effect of the grain size distribution on the work hardening characteristics of the polycrystals. As shown in Figure 6.3, when the grain size is 700 nm, increasing the width of the grain size distribution has a significant effect on the macroscopic stress-strain response since there is a relatively large dependence of yield strength and work hardening behaviour for this range of grain sizes.  On the other hand, for a relatively large mean grain size (e.g. 10 .tm), the 92  difference between small and large grains is much smaller and only a small effect is observed when the grain size distribution is expanded. An important consequence of increasing the work hardening of the polycrystal when the width of the grain size distribution is increased (see Figure 6.3(c) and (d)) is that the uniform elongation of the polycrystal is increased (assuming that this is controlled by the geometric stability of the sample, i.e. the Considère criterion). One reason for this increase is that when the width of a distribution is increased, the polycrystal will include a higher volume fraction of larger grains. These larger grains work harden at a higher rate and hence, the overall work hardening rate of the polycrystal is raised. But there is a more subtle aspect to this increased hardening rate, especially at smaller mean grain sizes. To understand this phenomenon one has to look at how stresses and strains are partitioned among the grains (Figure 6.4).  In a wide distribution, the finer grains in the population have higher yield  strengths and, accordingly, deform less and are under more stress than the average. The volume fraction of these grains is not negligible, e.g. in the 700 nm polycrystal with =  1 shown in Figure 6.4 this volume fraction is -29 %.  These stronger grains  contribute to the strengthening of the polycrystal and increase the work hardening of the aggregate in a manner similar to that observed in composites (note the initial transient part of the work hardening plots in Figure 6.3(c) and (d)). It should be emphasized that although in this model all grains are plastic, the finest grains of the population may be regarded as rigid plastic inclusions since their deformation is negligible even after significant macroscopic deformation. For example in the 700 nm polycrystal with o  =  1, the deformation of these  very fine grains (i.e. grain sizes below ‘l00 nm) is negligible even at an applied strain of 0.5. Another consequence of having a significant volume fraction of the polycrystal deforming  93  less than the average is that the grains in this volume do not exhaust their work hardening capacity at early stages of deformation and in this way also the overall work hardening of the polycrystal is raised. These effects are less pronounced at larger mean grain sizes since there is less dispersion in the yield strength and the mechanical behaviour of the grains as discussed earlier. It is possible to extend the arguments stated above for the case of the increased work hardening of wide unimodal distributions to the bimodal cases. In bimodal polycrystals with high volume fractions of coarse grains, the behaviour of the polycrystal is dictated by the coarse grains (Figure 6.5); indicating that the volumetric effect of the coarser grains is the overriding factor. This is also evident in Figure 6.6(a) and (b) where the evolution of the ultimate tensile strength and uniform elongation of a number of bimodal polycrystals, all having 200 nm fine constituents, is plotted as a function of the volume fraction of their coarse constituents. The situation is reversed at the other extreme when the volume fraction of the fine grains is very high. Figure 6.6(b) shows that the polycrystals do not demonstrate any uniform elongation as long as the volume fraction of their coarse 1 jim grains is kept to below 0.1 (this volume fraction decreases to 0.05 if 10 jim coarse grains are considered). It is at intermediate volume fractions that it is possible to benefit from both the strengthening of the fine grains and the high work hardening capability of the coarse grains. For example as shown in Figure 6.6, by replacing 30 % of the volume of a polycrystal containing only 200 nm grains with 10 jim grains, the uniform elongation of the polycrystal will increase from 0 to 0.l while its ultimate tensile strength will drop by 20 %. It is thus possible to engineer the mechanical behaviour of a polycrystal; its yield strength, work hardening rate, and uniform-elongation (see Figure 6.5); by controlling the volume fraction of these constituents.  94  550  0.35  (b)  (a) 500  1  0.30  ii* COtS5 constttuent  10  450  10  tm  un cosrse Constituent  0.25  coarse constituent  C  400 U, ‘  C  0 0  0.20  E 0.15  350  0  300  0.10  .  ,..  250  0.05  200  I Lm coarse constituent  0.00 0.0  0.2  0.4  0.6  0.8  1,0  volume fraction of coarse constituents  0.0  0.2  0.4  0.6  0.8  1.0  volume fraction of coarse constituents  Figure 6.6. The variation of (a) ultimate tensile strength and (b) uniform elongation as a function of the volume fraction of coarse constituents for a number of bimodal polycrystals having 200 nm fine and 1 or 10 urn coarse constituents.  6.4.2. Comparison of Unimodal and Bimodal Grain Size Distributions To compare the different bimodal and unimodal polycrystals with one another, the results of the simulation of the two classes of polycrystals are plotted together in Figure 6.7 in terms of the ultimate tensile strength versus the uniform elongation.  The unimodal  polycrystals depicted in this figure are of different grain size distributions while the bimodal polycrystals all have 200 rim grains as their fine constituents and varying volume fractions of either 1 u.tm or 10 p.m grains as their coarse constituents (i.e. the same data illustrated in Figure 6.6). This figure shows that new regions of the ultimate tensile strength-uniform elongation space can be exploited when bimodal polycrystals are considered. The ultimate tensile strength-uniform elongation envelope as defined by unimodal metals is shifted favourably towards higher elongations and/or strengths by the bimodal polycrystals. In this 95  550  500 450 C  400  C .  350 300  E 250 200 0.0  0.1  0.2  0.3  0.4  uniform elongation  Figure 6.7. Ultimate tensile strength versus uniform elonga tion predicted for different unimodal and bimodal polycrystals. The unimodal polycrystals are of different averag e grain sizes (p) and grain size distribution widths (o). The bimodal polycrystals, delineated by the dashed lines, all 0 have 200 nm grains as their fine constituents and either 1 or 10 im coarse constituents. Arrow s on the dashed lines show the direction of increase in the volume fraction of the coarse constituents of the bimodal polycrystals (see Figure 6.6 for details). Note that in this plot p values ranging from 100 nm to 50 m are used and for each case, o is determined by varying the o/p ratio from 0 to 0.8 at 0.2 increm ents. Polycrystals with no uniform elongation are not depicted in this plot.  study only bimodal structures with same-size fine and same-s ize coarse grains have been considered and it is anticipated that as the fine and coarse grains start to develop size distributions, the values defined in Figure 6.7 for bimodal polycr ystals should tend towards the unimodal values. Thus, the values depicted in this figure for the bimodal polycrystals represent upper limit estimates.  A noticeable feature in Figure 6.7 is that the unimodal  results show dispersion at smaller uniform elongation values , approximately below 0.15. The source of this dispersion is the improvement observed in uniform elongation of fine grained unimodal polycrystals when the width of their size distribution is increased. An 96  example of such improvement is shown in Figure 6.3(a). With increas ing mean grain size, this type of improvement ceases to be as effective as before and thus the dispersion fades away. The mean grain sizes above which this improvement disappears are 800 nm, 700 nm, 550 nm, 400 nm, and 280 rim for c /,u ratios of 0, 0.2, 0.4, 0.6, and 0.8, respectively. The dashed lines in Figure 6.7 delineate the loci of ultimate tensile strengt h-uniform elongation values for bimodal polycrystals which have fine grains of 200 nm and 1 tm or 10 im grains as their coarse constituents. The arrows on each line indicat e the direction of increase in the volume fraction of the coarse constituents, see Figure 6.6 for details. To examine the effect of varying the size of the fine constituents, Figure 6.8 shows the ultimate tensile strength-uniform elongation plot for two classes of bimod al polycrystals, those with 100 nm grains and those with 300 nm grains as their fine constituents.  750 700 650 600 550 500 450 I)  4-  0)  4-  400  cv$  E 350 4-  Z  309 250 0.00  0.05  0.10  0,15  0,20  0.25  0.30  0.35  uniform elongation  Figure 6.8. Plot of ultimate tensile strength versus uniform elongation for different bimodal polycrystals having fine constituents of 100 or 300 nm with coarse constituents of 1 or 10 tim. Open symbols/solid lines correspond to polycrystals with 10 im coarse grains and solid symbols/dashed lines refer to polycrystals with 1 .tm coarse grains. The arrows delineate the direction of increase in volume fraction of coarse grains.  97  Similar to Figure 6.7, the corresponding loci of the ultima te tensile strength-uniform elongation values are also depicted for each case as a pair of a dashed and a solid line. Aside from shifting either of the lines towards higher ultimate tensile strengt h values, a decrease in the size of the fine constituents also tends to amplify the differe nce between 1 jm and 10 jim loci. This means that as the behaviour of the fine grains approa ches that of the coarse grains, or in other words the strength effect of the fine grains vanishes, there is less improvement to be obtained in ultimate tensile strength or uniform elongation. The above ultimate tensile strength-uniform elongation plots/maps demonstrate the utility of the present model in development of property maps where the most promising of grain structures may be identified for design purposes. For examp le, the dispersion observed in the map shown in Figure 6.7 points to unimodal grain structures where manipulation of the grain size distribution may most effectively be utilized to tailor the ultimate tensile strengthuniform elongation relationship in polycrystals.  In this particular example, lognormal  polycrystals with average grain sizes smaller than 280 nm are found to be most responsive to grain size distribution manipulation. For the case of bimodal polycr ystals, a comparison of Figure 6.7 and Figure 6.8 shows that fine grain constituents smalle r than 300 nm and coarse grain constituents larger than 1 jm are necessary if combinations of ultimate tensile strength and uniform elongation, unattainable in unimodal polycrystals, are to be achieved. In all, these maps serve as a first guide for materials engineers interested in the modification of the mechanical properties of polycrystals through grain size distribution manipu lation.  6.4.3. Model Assumptions and Variables In this section, the assumptions and variables of the model are discussed. Specifically, i) the yield strength definition used in analysis of the stressstrain curves, ii) the  98  assumption of random spatial arrangement of grains, which is implicit in the current selfconsistent methodology, iii) the type of grain-matrix interaction, and iv) the local hardening law considered in the analysis are discussed.  6.4.3.1. Yield Strength Definition In Chapter 5, Section 5.4.3, the ambiguity associated with determining the onset of plasticity or yielding in polycrystals was discussed.  It was demonstrated that the yield  strength may appropriately be defined through an analysis of the work hardening rate of the polycrystal during its elastoplastic transition.  However, since the current viscoplastic  formulation does not account for elastic deformation of grains, such analysis could not be implemented here. Instead, yield strength is simply defined as the equivalent stress after the first deformation increment. Given that in the viscoplastic model all grains are plastic from the onset of the simulation, this simple yield definition is logical. Furthermore, the initial work hardening rate of the polycrystals predicted by the viscoplastic simulations is not high. Thus, use of an offset definition would not affect the outcome either. Figure 6.9 shows a comparison of the Hall-Petch slopes predicted by the elastoplastic and the viscoplastic models for different grain size distributions. It is apparent that in fact, the definition of yield strength used in the viscoplastic simulations correlates with the 0.2 % offset definition in the elastoplastic simulations.  6.4.3.2. Spatial Distribution of Grains Within VPSC each grain is embedded in the average effective medium, and spatial correlations are disregarded. However, the spatial distribution of the grains in a polycrystal can impact the mechanical response of the polycrystal. Figure 6.10 shows an example of how the spatial arrangement of the grains might affect the response of the polycrystal. Here,  99  0.26  0.24  -  E 0.22  EPSC (0.2 % vPsc  offset)  V.  z  0.20 0 0  0.18  (.3  0.16  Dr  0.14  0.12 0.10 0.0  0.2  0.4  0.6  0.8  1.0  1.2  c7 / 0 /4 Figure 6.9. Variation of the Hall-Petch slope with c,/p as predicted from the elastoplastic and viscoplastic simulations. The 0.2 % offset definition is used for the elastoplastic simulations.  400  360  a  300  C  250  200  150  0.00  0.05  0.10  0.15  0.20  0.25  eq Figure 6.10. Predicted von Mises equivalent stress-strain curves for two bimodal polycrystals consisting of 70 volume percent 3 .tm grains and 30 volume percent 200 nm grains with fine and coarse-grained elements of the polycrystal arranged in equal-strain and random configurations. The stress-strain curves are plotted to the end of uniform elongation as calculated with the Considère criterion. The inset schematically shows the equal-strain configuration. The white and gray colors in the inset represent the coarse and fine-grained constituents of the bimodal polycrystal, respectively.  100  the simple case of a bimodal polycrystal consisting of 70 volume percent coarse 3 .irn grains and 30 volume percent fine 200 nm grains is considered. The stress-strain response of the random arrangement of grains, from the VPSC analysis, is compared to that of the simple equal-strain arrangement. For the latter, the VPSC predicted von Mises equivalent stressstrain responses of 3 jim and 200 nm polycrystals (both with cc /p  0) are used to calculate  the stress-strain curve of a 70/30 equal-strain mixture of the two. The stress-strain curves are plotted to the extent of uniform elongation as determined from the Considère criterion. In this particular example, it is evident that the arrangement of the grains primarily influences the yield strength and the initial work hardening of the polycrystal. Interestingly, the equal-strain configuration demonstrates the same level of uniform elongation and ultimate tensile strength while maintaining a higher yield strength. The higher yield strength for the equal-strain configuration can be understood in terms of the equal-strain requirement imposed on the polycrystal.  Under this condition, the fine-grained ‘phase’ is forced to  deform with the polycrystal as a whole. Deformation of this hard phase is only possible at high stress levels, hence raising the yield strength of the polycrystal. Upon yielding, the fine grained phase will continue to deform in accord with the rest of the polycrystal. This phase, however, has a low hardening capacity and consequently, tends to impair the overall work hardening capacity of the polycrystal. In contrast when the grains are distributed randomly, the fine grains do not deform or deform less than the average and the strain is primarily carried by the coarse grains, resulting in a lower yield strength and higher work hardening rate. The above analysis shows that the grain spatial arrangement does provide another degree of freedom in fine-tuning the mechanical response of polycrystals. Jin and Lloyd  101  [92], for example, in their study of duplex AA5754 indicate that the spatial distribution of grain sizes may be used to control work hardening and neck growth. This represents an opportunity for future work.  6.4.3.3. Grain-Matrix Interaction The type of grain-matrix interaction or homogenization used in the self-consistent simulation (Section 2.2) directly impacts the nature of the stress and strain partitioning in the polycrystal and accordingly, the overall polycrystal stress-strain response.  The self-  consistent result falls in between the equal-strain Taylor (infinitely stiff matrix-inclusion interaction, upper-bound) and the equal-stress Sachs (infinitely compliant matrix-inclusion interaction, lower-bound) approximations but where between the bounds does it fall depends on the linearization scheme. The n  =  10 linearization used in this work assumes a uniform  interaction for all the grains, in between the very stiff secant and the very compliant tangent, as illustrated in Figure 6.11. The assumption of same interaction for all grains, irrespective of their relative stiffhess with regard to the matrix, may not be ideal; especially if there is large contrast in the properties of the grains. An alternative is the so-called Second Order approximation recently implemented in VPSC by Lebensohn et a!. [172].  This  approximation introduces explicitly the average stress and the stress fluctuations in the grains in the linearization procedure [172]. This procedure leads to grain-dependent grain-matrix interactions. A number of the model polycrystals were examined using this method and it was found that the predicted response is slightly more compliant as compared to n =10. This is evident in Figure 6.12 where larger deviations between the local and global fields is predicted for the Second Order interaction.  Thus, potentially larger effects than those  102  400  300  0  200 C)  b  100  a 0.0  0.1  0.2  0.3  0,4  0.5  0.6  eq Figure 6.11. The effect of grain-matrix interaction on predicted von Mises equivalent stress-strain response of a polycrystal with mean grain size of 700 nm and cr /p ratio of 1. Note that in this analysis a value of 20 is used 0 for n instead of 65 to keep the numerical convergence of the Second Order interaction manageable.  3.5 3.0 2.5 0 1 0  2.0 1.5  C.,  0  1.0 0.5 0.0  0  1  2  3  4  5  6  7  grain size (pin) Figure 6.12. The effect of grain-matrix interaction on partitioning of strain among grains of a polycrystal with mean grain size of 700 nm and o/p ratio of 1. The plot corresponds to macroscopic von Mises equivalent strain of 0.01.  103  reported here may be anticipated.  However, the observed differences with the n =10  approximation are relatively small and tend to decrease as the farfield strain increases.  6.4.3.4. Local Hardening Relationship The grain-level hardening law considered thus far assumes that both the onset and the evolution of plasticity in a grain are dependent on the size of the grain. To examine how changing this law might affect the observed grain size distribution effects, this constitutive relationship is replaced by the simpler relationship presented in Section 4.3.3 which assumes that only the onset of plasticity is grain size dependent and the work hardening behaviour is grain size independent, i.e.  =  +  kD” +(r  +  l—exp—F & F 1 ) 1 TI ,ij  (6.2)  here k is the local Hall-Petch constant, D the grain size, and the other parameters are defined as before. Figure 6.13 compares the ultimate tensile strength-uniform elongation relationship for the two type of hardening laws; note that two different cr 9 /p ratios are considered in this plot. It is evident that in general, better combination of ultimate tensile strength and uniform elongation values are predicted with the new hardening law. This is no surprise since unlike the previous hardening law, as the grains get smaller, their hardening capability does not diminish. As a result, for the same grain structure, the polycrystal with the new hardening law has a higher work hardening capability and accordingly, better ultimate tensile strength and uniform elongation. It is also clear that the polycrystals with the new hardening law are not sensitive to the grain size distribution; the ultimate tensile strength-uniform elongation loci for cr 0 /p =0 and 0.8 coincide for these polycrystals. To understand this behaviour, one  104  900 0. oa  .  800 700  Equation (4,16) Equation (8.3)  U S  800  ‘  500 400  E  .  300  S  200 0.0  0.1  0.2  0.3  0.4  uniform elongation Figure 6.13. The influence of the local hardening law on the ultimate tensile strength-uniform elongation relationship for a number of unimodal polycrystals. The black and white symbols correspond to oo/p ratios equal to 0 and 0.8, respectively.  can look at how the work hardening rate of a polycrystal is modified when the width of the grain size distribution is increased. Figure 6.14 shows the stress-strain curves and the work hardening rates of two 100 nm and two 50 p.m polycrystals. increasing the width of the grain size distribution from o [p  =  This figure shows that by 0 to o /p  =  0.6, the overall  work hardening of the polycrystal is not altered. This is true for both the fine and coarse grained polycrystals, observe the negligible difference between the dashed and solid work hardening curves for the 100 nm and 50 p.m polycrystals.  Although the grains of these  polycrystals yield at different times, ultimately leading to different hardening profiles for the grains. These differences do not give rise to substantial differences at the macroscopic level.  105  1400 1200 1000  800 0  600  b 400 200  0 0.0  0.1  0.2  0.3  0.4  0.5  0.6  eq  Figure 6.14. Predicted von Mises equivalent stress-strain curves for a number of unimodal polycrystals having Equation (6.2) as their local constitutive behaviour. The corresponding work hardening rate for the curves, ®eq is also depicted.  Hence, the ultimate tensile strength-uniform elongation relationship is independent of the width of the grain size distribution. It should, however, be noted that as in the previous hardening law, increasing the width of the grain size distribution has a more pronounced effect when finer mean grain sizes are considered. Widening the size distribution of the coarse 50 p.m polycrystal essentially has no effect. The above observation that the macroscopic hardening capability of the polycrystal is not affected by the width of the distribution suggests that bimodal grain size distributions should not demonstrate any beneficial property either. Figure 6.15 shows the ultimate tensile strength and uniform elongation values for a number of bimodal grain size distributions superimposed on unimodal polycrystal values.  As anticipated, no improvement in the  106  900 I  800  0  unimodal bimodal  W % 0’  700 C  600 50% ‘  C  .i  500 70 $0  400  E  105%  300 200 0.15  —  0.20  0.25  0.30  0.35  uniform elongation  Figure 6.15. Ultimate tensile strength versus uniform elongation for different unimodal (u /p ratio varying 0 from 0 to 0.8) and bimodal (mixture of 2 tm coarse and 100 nm fine grains) polycrystals. Percent values indicate the percent volume fraction of coarse 2 p.m grains. The 0 % and the 100 % correspond to polycrystals with unimodal grain sizes of 100 nm and 2 jim, respectively. The arrow points to unimodal polycrystal with mean grain size of 240 nm and ob/p = 0.8 whose ultimate tensile strength is approximately the same as 70 % bimodal polycrystal.  ultimate tensile strength-uniform elongation values of the bimodal structures is predicted. In fact, aside from mixture ratios greater than 30 volume percent to smaller than 90 volume percent, the response of the bimodal polycrystals coincides with that of the unimodal polycrystals.  Interestingly, bimodal polycrystals with mixture ratios in the noted range  demonstrate an even worse combination of ultimate tensile strength and uniform elongation in comparison to their unimodal counterparts. To understand this phenomenon, the stressstrain and work hardening curves for the bimodal polycrystal with 70 % mixture ratio and a unimodal polycrystal having the same level of ultimate tensile strength (gray symbol delineated by an arrow in Figure 6.15) are plotted together in Figure 6.16. It is clear from  107  1400 1200 1000  300 0 b  600 400 200 0 :00  0.1  0.2  0.3  0.4  0.5  0,6  eq  Figure 6.16. Comparison of the von Mises equivalent stress-strain curves and work hardening rates, eeq, for unimodal polycrystal with mean grain size of 240 nm and /t = 0.8 and bimodal polycrystal with 70 volume percent 2 im and 30 volume percent 100 nm grains.  this figure that the bimodal polycrystal work hardens at a lower rate over the stress-strain range depicted in this figure, leading to a smaller uniform elongation. The reason for this lowered work hardening capability can be traced back to the nature of stress and strain partitioning in the polycrystal. Figure 6.17 illustrates the evolution of the local plastic strain fields for the above polycrystals at three different stages of their deformation. This figure shows that the coarse constituents of the bimodal polycrystal tend to deform more than the grains of the unimodal polycrystal. Thus, exhausting their work hardening capability earlier than the grains of the unimodal polycrystal, resulting in the lowering of the overall work hardening of the bimodal polycrystal as compared to the unimodal polycrystal.  108  2.0  (a) —  0.0  —  0.5  —  —  1.0  (b)  1.5  grain size (jm)  2.0  (c)  coarse  —  0  0.5  1.0  1.5  grain size (rim)  2.0 0  0,5  1.0  1.5  2.0  grain size (lim)  Figure 6.17. Predicted relative von Mises equivalent strains as a function of grain size for unimodal polycrystal with mean grain size of 240 nm and 0 /p 0.8 at three different stages of deformation corresponding to macro von Mises equivalent strains of(a) 0.01, (b) 0.10, and (c) 0.5. The average strain in the coarse (2 tim) and fine (100 nm) grains of 70 % bimodal polycrystal having approximately the same ultimate tensile strength is also indicated on the plots as dashed lines.  The above analysis demonstrates that the type of hardening assumed for the grains plays an important role in determining the sensitivity of a polycrystal to grain size distribution manipulations.  In particular, care should be taken in generalizing the size  distribution effects. Considering the two hardening relationships examined here, it can be concluded that for a grain size distribution to be an effective and viable design parameter, both the onset of plasticity and its subsequent evolution should be grain size dependent. Whether this hypothesis holds when alternative hardening relationships are examined is of interest for future work. Finally, the approach through which grain size dependence is included into the present local hardening laws is considered. The grain size dependence of the local hardening relationships assumed in this work is controlled by the microscopic Hall-Petch parameter k. Since this parameter is incorporated into the hardening relationships through the kD 2 109  expression, changing the value of this parameter has no effect but to shift the results of the simulation to coarser or finer grain sizes depending on whether it is increased or decreased. That is, the overall predicted trends for the response of the polycrystals will not be altered. For example, the response of a polycrystal having a Hall-Petch parameter of k and a grain size of D is equivalent to that of a polycrystal with a Hall-Petch parameter of 2k and grain size of 4D; in either case kEY 112 has the same value. Figure 6.18 shows the evolution of the ultimate tensile strength-uniform elongation relationship for when k is increased by two-fold from 0.094 to 0.188 MNm . The original hardening law, Equation (4.16), is used in this 312 figure.  400 •o G’P V.,  1 E  vV /p 11 ,  =0 0.8  =  350  300  250  z 200 0.0  0.1  0.2  0.3  0.4  uniform elongation  Figure 6.18. The effect of local Hall-Petch parameter, k, on ultimate tensile strength-uniform elongation relationship for a number of unimodal polycrystals. The black and white symbols correspond to k = 0.094 312 and k = 0.188 MNm MNm , respectively. Equation (4.16) is used as local hardening law, i.e. both the yield 312 strength and work hardening are grain size dependent.  110  As indicated, the loci of the ultimate tensile strength-uniform elongation values at different o /p ratios are independent of k. This is despite the fact that when a higher value of k is used, the same combination of ultimate tensile strength and uniform elongation can be achieved with a coarser grain structure. As an example when o/p  0.8, to attain a ultimate  tensile strength of 327 MPa and a uniform elongation of 0.09, a mean grain size of 800 nm is necessary when k  0.188 312 MNm In contrast when k = 0.094 MNm . , the mean grain size 312  should be decreased to 200 nm.  6.4.4. Model Validation The present viscoplastic model (qualitatively) captures the widely documented paradox of strength and uniform elongation in metals. For example, surveys of the yield strength-uniform elongation relationship for copper [91] and the ultimate tensile strengthuniform elongation relationship for low carbon steels [173] show that these measures of strength vary inversely with uniform elongation. This type of behaviour is predicted in the current work. For example, Figure 6.3(a) shows that an increase to the uniform elongation of a polycrystal (when the width of its grain size distribution is widened) comes at the expense of its strength. Similarly, for the case of bimodal polycrystals, Joshi et al. [120] and Gil Sevillano et al. [119] demonstrate that the uniform elongation of bimodal polycrystals increases with the volume fraction of their coarse constituents while at the same time, their flow stress decreases. This trend is also predicted in the current model, see Figure 6.8 for example. In Figure 6.19, the tensile stress-strain curve of a copper sample with 400 nm average grain size, i.e. the sample without any annealing twins from the work of Shen et al. [174], is  ill  400  300 2500  0  50m  0  U) U,  2000  200  1500  00 0  ‘\J\.  0 4.3 pm  1  SI,m, t aL  C 1000  I4-  100  25 m .  500  0  .2 pm 100  0 0.00  200  300  400  true stress MPa)  0,05  0.10  0.15  true strain Figure 6.19. Experimental tensile stress-strain curve of copper with an average grain size of 400 nm from the work of Shen et al. [174] compared with its corresponding VPSC simulation (o). The grain size distribution used for the simulation has an average of 400 rim and u / ratio of 0.55. The inset shows the corresponding 0 work hardening rate for the 400 nm copper of Shen et al. plotted along with the work hardening rates of copper reported in [8 1,82]. Young’s modulus of copper is delineated by the dashed line. The point where the work hardening rate of the 400 nm sample collapses on top of the other work hardening curves is demarcated by the arrows on both the stress-strain and work hardening plots.  depicted.  Also depicted in this figure is the model predicted stress-strain response of a  polycrystal having random texture and a grain size distribution characterized by an average grain size of 400 rim and 0 o/ p ratio of 0.55. Shen et al. [174] do not report on the width of the size distribution for this copper specimen. Accordingly, in obtaining the model result for this sample, the width of the distribution is treated as an adjustable parameter. The model determined 0 o/ p ratio for this polycrystal, i.e. 0.55, may thus be used as a measure of the validity of the model.  This value does indeed fall within the range of experimentally  determined o/p ratios, i.e. 0.34-0.94 (consult Appendix A2 for details), validating the model. It is to be pointed that there is close correlation between the model and experiment 112  only beyond a strain of approximately 0.05.  This agreement between the model and  experiment coincides with the stress-strain range over which the work hardening rate of this sample converges to that of the copper with larger grain sizes. The point of this convergence is demarcated by an arrow on both the stress-strain and work hardening plots. For strains smaller than 0.05, there is discrepancy between the model and the experiment.  This  discrepancy is, however, partly justified since the VPSC model does not take elasticity into account while the experiment covers the whole spectrum of deformation from elasticity to plasticity. In fact, the viscoplastic model is not formulated for this stress-strain range and the disagreement is expected. Furthermore, for strains smaller than 0.05, the copper specimen shows a very high hardening rate, compare this hardening rate against the Young’s modulus of copper delineated by the dashed line also depicted in this figure. Such high hardening rate, which can also be observed on the compressive stress-strain curve of the copper with 200 nm average grain size depicted in Figure 2.3, may be attributed, not considering possible artifacts associated with testing of these samples, to the presence of internal stresses (in accord with the Sinclair et al. [82] proposition presented in Section 2.2.3). These convoluted effects are not included in the present self-consistent formulation and may therefore lend themselves to the observed discrepancy between the model and the experiment. Having evaluated the validity of the present model in predicting the stress-strain response of a polycrystal, it is of interest here to extend such validation to the developed ultimate tensile strength-uniform elongation maps.  In Figure 6.20, the ultimate tensile  strengths and uniform elongations of the unimodal (label A) and bimodal (label B) copper samples of Wang et al. [91] are plotted together with their corresponding model predicted values. Unlike the previous example, in this case the experimentally determined grain size  113  500  experiment Q  C  •  B  450  model  /  400 350  0i5  A  0  C  02S  300  020  Ij  250  0.00  .  0  200  100  200  300  400  5000  1000  000  Qfl ze tOrn)  gra,fl  1q00  2000  lrn,)  *  0.00  0.05  0.10  0.15  0.20  0.25  uniform elongation  Figure 6.20. Comparison of experimental results of Wang et al. [91,110] with VPSC model predictions. Data points labeled A correspond to copper sample with unimodal grain size distribution and those labeled B to copper sample with bimodal grain size distribution. The inset shows the experimental grain size distributions for the unimodal and bimodal copper samples. Note that the original grain size distribution corresponding to sample A was provided as number fraction and here, by assuming spherical grains, the number fraction is converted to volume fraction.  distributions are available and they are used as input into the model. These distributions are shown in the inset of the figure. Qualitatively, these results point to the soundness of the present viscoplastic fonnalism. Just as in the experiment, the model also predicts an increase in the uniform elongation of the polycrystal when its grain size distribution is changed from unimodal to bimodal.  Quantitatively, however, there is discrepancy between the model  predictions and the experiments.  This is particularly true when it comes to the uniform  elongation values. In terms of ultimate tensile strength, the model values for the unimodal and bimodal polycrystals fall within 10 % and 3 % of the experiments, respectively. With regard to uniform elongation, on the other hand, the better uniform elongation of the experimental samples as compared to that predicted by the model is apparent.  For the 114  bimodal polycrystal, the model uniform elongation is within 70 % of the experiment. To explain these discrepancies, the assumptions of the model need to be examined; particularly issues related to the mechanisms of deformation and strain-rate sensitivity. In this work, deformation is accommodated by (1 io){i 1 l} slip. Wang et al. [91] point to the presence of deformation twins inside the coarser grains of their bimodal copper samples. A similar observation is made in nanocrystalline cobalt [175]. This is an indication that by creating such inhomogeneous microstructures and increasing the local heterogeneity in plastic flow, alternative and less favourable modes of deformation might be activated to enforce compatibility. Such consideration is not included in the present model. Hence, any improvement in the behaviour of the bimodal polycrystal examined in this work is only related to the way the macroscopic work hardening behaviour of the material is modified by the added volume of coarse grains. In reality, however, in addition to the increased work hardening due to the addition of coarse grains, the work hardening of bimodal structures may further be improved as a result of the twinning.  The better uniform elongation of the  experimental sample as compared to that predicted by the model can be attributed (not considering issues associated with details of the model setup such as spatial distribution of grains) to the above discussed effect. Furthermore, there is growing experimental evidence that strain-rate sensitivity of f.c.c. metals increases noticeably when their grain size is reduced [176-179]. For example, the rate sensitivity parameter of Cu increases from 0.004 at coarse grain sizes [180] to 0.02 when its grain size is 200 nm [179]. In the present model, rate sensitivity is taken into account in the viscoplastic relationship (i.e. the exponent n). However, the same rate sensitivity is used for the coarse and fine-grained polycrystals. Elevated rate sensitivity can translate to a reduced rate of necking during tensile deformation  115  [181]. Thus, better uniform elongations are to be expected for the model polycrystals if such dependence of rate sensitivity on grain size is to be provisioned. This particular grain size dependence is ultimately related to the evolution of mechanisms of deformation with grain refinement [179]. It is therefore necessary to modify the constitutive relationships at the core of the model in order to account for such effects. This is an area where further work is merited.  6.5. Summary A micromechanic polycrystalline model capable of simulating the monotonic plastic deformation of polycrystals has been formulated. It was assumed that plasticity can only occur by dislocation-based crystallographic slip and that the evolution of the threshold stress of each slip system is grain size dependent. The evolution law used for this purpose is based on experimental results on pure copper. The model is used to examine the effect of unimodal and bimodal grain size distributions on uniaxial tensile behaviour of a number of polycrystals.  It is found that as the width of the lognormal grain size distribution of a  polycrystal increases, the work hardening of the polycrystal increases and accordingly, its uniform elongation is improved. This, however, comes at the expense of the strength of the polycrystal. Fine-grained polycrystals with wide grain size distributions demonstrated an initial work hardening behaviour characteristic of composite materials reinforced with hard secondary phases. The origin of this behaviour in these polycrystals are the very fine grains which do not deform and are under more stress than the rest of the polycrystal. They can thus be likened to the hard phases in composites. The examined bimodal grain size distributions demonstrated better overall macroscopic properties as compared to their unimodal counterparts.  The observed  116  improvement in these bimodal structures is only related to the way the macroscopic work hardening behaviour of the material is modified by the added volume of coarse grains. The type of local hardening law assumed for the grains does, however, impact these results. Improved properties are observed when both the onset and the subsequent propagation of plasticity, i.e. work hardening, are made grain size dependent. The present model provides a valuable tool for identifying the most promising of these microstructures and can ultimately be used to construct property maps, such as that depicted in Figure 6.7, for microstructure selection purposes. The challenge of producing the desired microstructure still exists, but with recent technological advancements it is becoming increasingly more feasible to control and engineer the grain structure of materials to very fine details. Examples are selective grain growth through high-energy ion bombardment [93] or grain size control through precise composition control during electrodeposition [94].  117  CHAPTER 7 Concluding Remarks 7.1. Summary of Observations In this work a continuum mechanics approach consisting of two newly developed grain size dependent self-consistent models, an elastoplastic self-consistent (EPSC) and a viscoplastic self-consistent (VPSC), is successfully utilized to evaluate the impact of grain size distribution on uniaxial tensile behaviour of polycrystals. This work represents the first time that the grain size distribution effect on the work hardening rate of materials beyond the first few percent strain has been modeled, and one of only two attempts to include size distribution into an elastoplastic self consistent scheme. It is assumed that deformation is accommodated by dislocation slip and that initiation of plasticity in a grain is governed by the Hall-Petch relationship.  To account for the evolution of the threshold stress with  deformation on each slip plane, a novel grain size dependent constitutive law which assumes an increase in the work hardening rate of a grain with grain size is taken into account. Model polycrystals with random textures and lognormal grain size distributions, having mean grain sizes ranging from 100 rim to 50 ltm, are primarily examined in this work. However, the case of bimodal distributions is also examined for large strain deformations.  The key  findings of these analyses are summarized as follows: .  Increasing the width of the lognormal grain size distribution, while keeping the mean grain size constant, results in a lowering of the strength of the polycrystal. This effect is attributed to the fact that as the grain size distribution is widened, a larger volume fraction of the polycrystal will consist of grains larger than the mean grain size. These  118  coarse grains, according to the assumed grain size dependent constitutive law, have reduced threshold stresses and as a result, lower the overall strength of the polycrystal. The tendency of the polycrystal to lose its strength with increase in width of grain size distribution does, however, diminish as larger mean grain sizes are considered. This leads to a decrease in the Hall-Petch slope with the width of the size distribution. •  The character of the lognormal grain size distribution affects the nature of percolation of plasticity within the polycrystal, i.e. the elastoplastic transition. Wider distributions and smaller mean grain sizes promote extended elastoplastic transitions.  These are  transitions where the evolution of the volume fraction of yielded grains (from 0 in the fully elastic state to 1 in the fully plastic state) occurs over a wider range of strains. •  It is possible to describe the evolution of volume fraction of yielded grains with applied strain using a simple model founded on consideration of size distribution as the sole source of heterogeneity of deformation among grains. The model captures the grain size distribution effects predicted by the EPSC simulations.  Using this model, the  possibility of inferring information regarding the character of the grain size distribution of the polycrystal from its uniaxial stress-strain curve is explored.  It is also  demonstrated that a more meaningful definition of yield strength, as compared to the arbitrary 0.2 % offset definition, may be obtained from the analysis of the elastoplastic transient behaviour of polycrystals. •  The work hardening rate of a polycrystal increases with the width of the lognormal grain size distribution. This is a result of the increasing volume fraction of grains with sizes larger than the mean grain size as the grain size distribution is widened. Considering the assumed grain size dependent constitutive law, these coarse grains  119  work harden at a higher rate than the average, raising the work hardening rate of the polycrystal.  An important implication of this increased work hardening rate is an  improvement in the uniform elongation of the polycrystal. •  Bimodal polycrystals demonstrate better overall macroscopic properties as compared with their unimodal counterparts. In this work, the observed improvements in these structures are related to the way the macroscopic work hardening behaviour of the material is modified by the added volume of coarse grains. The type of local hardening law assumed for the grains does, however, influence these results. Improved properties are observed when both the yielding and work hardening of the grains are made grain size dependent.  •  The simulation results are summarized in the form of a number of ultimate tensile strength-uniform elongation maps. In the case of unimodal (lognormal) polycrystals, it is found that the envelope of achievable ultimate tensile strength-uniform elongation values may be pushed to more desirable values by widening the grain size distribution of the polycrystals. Thus, grain size distribution manipulation may effectively be used to engineer the response of polycrystals. Tailoring the grain size distribution is most effective in polycrystals with average grain sizes smaller than 280 nm. With regard to the bimodal polycrystals, the maps show that new regions of the ultimate tensile strength-uniform elongation space, not attainable using unimodal polycrystals, can be exploited. Fine grain constituents smaller than 300 nm and coarse grain constituents larger than 1 jim are necessary for this purpose.  120  7.2. Future Work There are a number of areas for which further studies could complement and/or advance the present investigation. These areas are the following: •  It is shown that the assumed local (i.e. grain-level) Hall-Petch behaviour induces a macroscopic Hall-Petch effect that depends on the grain size distribution. Similarly, the local work hardening law is also shown to translate to a grain size distribution dependent macroscopic work hardening behaviour.  These observations point to the  value of the present modelling framework for evaluation of different (hypothetical) local constitutive relationships.  That is, trying to work out the local constitutive  relationship for a material by adjusting the input constitutive relationship of the model until the macroscopic behaviour of the material is replicated. This is important since there is still considerable ambiguity surrounding the exact form of the constitutive relationship(s) of a grain. For example, the notion of the increased rate-sensitivity of fine grained polycrystals or grain size dependence of the constitutive relationship of grains may be examined through this process. •  In this work, an analysis of geometric (or textural) hardening has not been included. In order to develop a complete understanding of the deformation of polycrystals, it is necessary to include this hardening in the analyses. On this basis, it is beneficial to examine the extent to which the noted grain size distribution effects are influenced by geometric hardening.  •  It was briefly noted that the 1-site self-consistent approach utilized in this work does not allow for consideration of any spatial correlation between the grains of the polycrystal. The grains are considered to be randomly distributed in the polycrystal. In practice,  121  however, the spatial distribution of grains is generally not random. Depending on the processing route of the material, there is some correlation between certain crystallographic orientations and/or certain grain sizes in the polycrystal. It is therefore beneficial to extend the present analysis to include the spatial distribution of grains. This may pursued using n-site self-consistent models or alternatively, using finiteelement models. •  The potential of extracting information regarding the character of the grain size distribution of a polycrystal from its stress-strain curve was explored in this work. Further work on this aspect of the work is merited. Particularly with regard to the calibration procedure used in the analysis. This is important as characterization of the true (3-dimensional) grain size distribution of polycrystals is currently laborious and complicated. It may, thus, be facilitated using the proposed methodology.  •  The value of the present modelling framework in constructing microstructure-property maps for microstructure selection purposes was demonstrated. In this process, only monotonic loading of the polycrystals was considered.  In practice, however,  polycrystals generally undergo more complicated strain paths. 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Cambridge: Cambridge University Press, 1998. p. . 282 [146] Lebensohn RA, Canova GR. Acta Mater. 1997;45:3687. [147] Canova GR, Wenk H-R, Molinari A. Acta Metall. Mater. 1992;40:1519. [148] Tome CN, Canova GR. Self-Consistent Modeling of Heterogeneous Plasticity. In: Kocks UF, Tome CN, Wenk H-R, editors. Texture and Anisotropy: Preferred Orientations in Polycrystals and their Effect on Materials Properties. Cambridge: Cambridge University Press, 1998. p. . 466 [149] Zbib HM, Hiratani M, Shehadeh M. Multiscale Discrete Dislocation Dynamics Plasticity. In: Raabe D, Roters F, Barlat F, Chen L-Q, editors. Continuum Scale Simulation of Engineering Materials. Weinheim: Wiley-VCH, 2004. p. 20 1. [150] Dawson PR, Beaudoin AJ. Finite Element Modeling of Heterogeneous Plasticity. In: Kocks UF, Tome CN, Wenk H-R, editors. Texture and Anisotropy: Preferred Orientations in Polycrystals and their Effect on Materials Properties. Cambridge: Cambridge University Press, 1998. p. . 512 [151]  Gómez-GarcIa D, Devincre B, Kubin L. Phys. Rev. Lett. 2006;96:125503.  [152]  Kocks UF. The Representation of Orientations and Textures. In: Kocks UF, Tome CN, Wenk H-R, editors. Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties. Cambridge: Cambridge University Press, 1998. p. . 44  [153] Lebensohn RA, Turner PA, Signorelli JW, Canova GR, Tome CN. Modelling Simul. Mater. Sci. Eng. 1998;6:447. [154] Tome CN. Code Elasto-Plastic Self-Consistent (EPSC). Version 3. Los Alamos National Laboratory, USA; 2002. [155] Tome CN. Modelling Simul. Mater. Sci. Eng. 1999;7:723.  130  [156]  Molinari A, Tóth LS. Acta Metall. Mater. 1994;42:2453.  [157] Tome CN, Lebensohn RA. Code Visco-Plastic Self-Consistent (VPSC). Version 7a. Los Alamos National Laboratory, USA; 2006. [158]  Stout MG, Kocks UF. Effect of Texture on Plasticity. In: Kocks UF, Tome CN, Wenk H-R, editors. Texture and Anisotropy. New York: Cambridge University Press, 2000. p.460.  [159]  Hull D, Bacon DJ. Introduction to Dislocations. Oxford: Butterworth-Heinemann; 2001.  [160] Rollett AD, Kocks UF, Embury JD, Stout MG, Doherty RD. In: Kettunen P0, Lepisto TK, Lehtonen ME, editors. International Conference on the Strength of Metals and Alloys. Tampere, Finland: Pergamon Press; 1988. p. . 433 [161] Nabaffo FRN. Phil. Mag. A 1997;75:703. [162] Tome CN, Canova GR, Kocks UF, Christodoulou N, Jonas JJ. Acta Metall. 1984;32: 1637. [163] Bouaziz 0, Buessler P. Rev. Metall. 2002;99:71. [164] Ashby ME, Jones DRR Engineering Materials 1: An Introduction to Their Properties and Applications. Oxford: Butterworth-Heinemann; 1996. [165] Metals Handbook. Volume 8: Mechanical Testing and Evaluation. Ohio: American Society for Metals; 1985. [166] Dieter GE. Mechanical Metallurgy. London: McGraw-Hill; 1988. [167]  Saada G. Phil. Mag. 2005;85:3003.  [168]  Saada G. Mat. Sci. Eng. A 2005;400-40l:146.  [169] Brandstetter 5, Van Swygenhoven H, Van Petegem S, Schmitt B, MaaB R, Derlet PM. Adv. Mater. 2006;18:1545. [170]  Saada G, Verdier M, Dinas GF. Phil. Mag. 2007;87:4875.  [171] Afanasev NN. Statistical Theory of Fatigue Strength of Metals [in Russian]: Izv. Akad. Nauk SSSR; 1953. [172] Lebensohn RA, Tome CN, Castafleda PP. Phil. Mag. 2007;87:4287.  131  [173] Azizi-Alizamini H, Sinclair CW, Militzer M, Mithieux J-D. Mat. Sci. Forum 2007;558-559:1 13. [174]  Shen YF, Lu L, Dao M, Suresh S. Scripta Mater. 2006;55:319.  [175]  Karimpoor AA, Erb U, Aust KT, Palumbo G. Scripta Mater. 2003 ;49:65 1.  [176] Lu L, Li SX, Lu K. Scripta Mater. 2001;45:1163. [177] Dalla Torre F, Van Swygenhoven H, Victoria M. Acta Mater. 2002;50:3957. [178] Wang YM, MaE. Appi. Phys. Lett. 2003;83:3 165. [179] Wei  Q, Cheng 5, Ramesh KT, Ma E.  Mat. Sci. Eng. A 2004;381:71.  [180]  Conrad H. In: Zackay VF, editor. High Strength Materials. New York: Wiley, 1965. p.436.  [181]  Meyers MA, Chawla KK. Mechanical Metallurgy: Principles and Applications. New Jersey: Prentice-Hall; 1984.  132  APPENDIX Al Kocks-Mecking Strain Hardening Model Strain hardening models consist of two parts: i) a model relating the flow stress to the appropriate state variable and ii) a model for the evolution of the state variable with strain. In general, flow stress is related to dislocation density. For this purpose the Taylor formulation [1], given below, offers an adequate framework.  However, more complex relations  incorporating subgrain size and subgrain misorientation can certainly be envisioned. The Taylor formulation can be written as, rrraGbf,  (Al.l)  here r is the flow stress of the single crystal, a a constant, G the shear modulus, b the magnitude of the Burgers vector, and p is the dislocation density. Using the Taylor factor, M, Equation (Al.l) can be modified for polycrystals, i.e. o = MaGbf where uis the flow stress of the polycrystal.  In this manner the first part of the work hardening model is  established. Differentiating Equation (A 1.1) with respect to strain, y, gives dr(aGb) d 2 p 2 dy dy  (A12)  which is the basic work hardening equation for single crystals. A similar equation could be derived for polycrystals using the Taylor factor but for conciseness the single crystal case will be pursued hereafter.  It is evident from this equation that if a description of work  hardening and specifically work hardening rate (6  =  dr / dy) is to be given, it is necessary to  define the rate of dislocation storage dp / dy. This is essentially the second part of the work hardening model and considering the many possibilities that a growing population of  133  dislocations could interact with each other, it is a formidable task. One convenient example of how this intractable problem might be treated is in the approach taken by Kocks and Mecking [2,3], although other approaches are present [4,5]. In their approach, assuming dislocations are not source limited, the evolution of dislocation density with strain is described in terms of dislocation accumulation and annihilation, i.e. (A1.3)  dy  where k 1 is a constant and k 2 is a function of temperature, strain rate, and stacking fault energy. In this equation, the first term represents hardening due to dislocation accumulation (statistical storage only) and the second term, softening due to dislocation elimination by thermally activated processes (setting aside the details of the processes). The constitutive model comprised of Equations (Al .2) and (Al .3) can be integrated analytically resulting in the overall stress-strain relationship of the crystal as,  (Al.4) Ts  here r is the lattice friction stress; and r  ))  aGb/k 1 k ) (2  and 6  (  aGb/2) are, 1 k  respectively, the saturation stress and the initial hardening rate of the crystal. Note that once again by incorporating the Taylor factor M, Equation (Al .4) can be transformed to represent the polycrystal behaviour. This simple model, which is equivalent to the empirical Voce strain hardening model [6], gives remarkably good results [7].  It does, however, require  modifications if very large strains are involved (i.e. in the so-called stage IV of work hardening [8,9]). Taking the large strain behaviour into account, Tome et al. [10] rewrite Equation (A 1.4) as,  134  (Al.5)  where 6 is the asymptotic hardening rate (characteristic of stage IV hardening) and r  +r  is  the back-extrapolated threshold stress. Schematic representations of Equations (Al .4) and (Al.5) are shown in Figure A1.l.  Y Figure A1.1. Schematic representation of (a) Equation (Al .4) and (b) Equation (Al .5). The insets show plots  of the strain hardening rate of each equation as a function of stress.  135  In the end, it should be noted that in crystal plasticity models, the algebraic sum of shears in a grain, F, is commonly used instead of y in the above formulations. The reason for this is that grains generally deform through polyslip and the activity of one slip system affects the activity of the others. To account for this interaction, F is used instead of y.  References [1]  Taylor GI. Proc. Roy. Soc. A 1934;145:362.  [2]  Mecking H, Kocks UF. Acta Metall. 198 l;29:1865.  [3]  Kocks UF, Mecking H. Prog. Mater. Sci. 2003;48:171.  [4]  Kuhlmann-WilsdorfD. Trans. AIME 1962;224: 1047.  [5]  Nes E. Prog. Mater. Sci. l998;41:129.  [6]  Voce B. J. Inst. Met. 1948;74:537.  [7]  Chen SR, Kocks UF. In: Freed AD, editor. High Temperature Constitutive Modeling Theory and Applications. New York: ASME; 1991. p. 1. -  [8]  Rollett AD. Pennsylvania: Drexel University, 1987.  [9]  Rollett AD, Kocks UF, Embury JD, Stout MG, Doherty RD. In: Kettunen P0, Lepisto TK, Lehtonen ME, editors. International Conference on the Strength of Metals and Alloys. Tampere, Finland: Pergamon Press; 1988. p. . 433  [10]  Tome CN, Canova GR, Kocks UF, Christodoulou N, Jonas JJ. Acta Metall. 1984;32:1637.  136  APPENDIX A2 Lognormal Grain Size Distributions in Metals Characterization of the true (3-dimensional) grain size distribution of polycrystals is not a straightforward task; see [1,2] for details of the challenges associated with this task and [3-6] for the current state of research. Nonetheless, a limited number of measurements have been conducted on polycrystalline metals, Table A2. 1.  Table A2.1. Examples of experimentally determined lognormal grain size distributions  measurement technique  measurement  grain extraction  grain weight (volume) grain number and volume grain number and volume grain number  X-ray diffraction, TEM TEM X-ray diffraction  t  reference  0.75  [7]  0.71  [8]  0.90  [91  0.94  [10]  o/P 0.34 0.57 0.78 0.40  —  —  —  —  t based on distribution of grain sizes  Note that the o / p ratios listed in Table A2. 1, which are a measure of the width of the distribution, are provided as a range of values in each case. This is a reflection of the error associated with the measurement techniques, the variability of the measurement from one technique to another, and the dependence of the grain size distribution on processing route of the material (electrodeposition, for example, tends to produce narrow grain size distributions [10,11]). The o- Ip values in Table A2.l are calculated using Jo/p=exp(S2)_l  (A2.l)  137  The value of S used for this purpose is that of the distribution of grain sizes. In instances where this value was not provided directly in the reference (e.g. in [7] the distribution of grain weights is provided), the appropriate conversion (described below) was utilized to obtain S. Kurtz and Carpay [12] demonstrate that if a variate x is distributed lognormally as P (x I M, 52) then  axb  is also lognormally distributed as P (axb a + bM, b252), provided a  and b are constants and a  >  0. Using this property of the lognormal distribution, it is possible  to convert different lognormal distributions. For example, the lognormal distributions of grain weights, P (m M, 1  ,  s),  and grain volumes,  lognormal distribution of grain diameters,  (v I M,  ,  can be converted to  P(D MD,S).  Table A2.2 lists the  corresponding conversion parameters.  Table A2.2. Parameters relating lognorrnal distributions of grain weights (m) and grain volumes (V) to lognormal distribution of grain diameters (D)  x  a  b  MD  SD  1/3  m  D pJ  V  D  +---M,  3  D3” 6 (  (D  1/3  +M,,  References [1]  Underwood EE. Reading, Massachusetts: Addison-Wesley; 1970. p. 104.  [2]  Exner HE. Tnt. Metall. Rev. 1972;17:25.  [3]  Poulsen HF. Three Dimensional X-ray Diffraction Microscopy. Berlin: SpringerVerlag; 2004. 138  [4]  Spowart JE. Scripta Mater. 2006;55:5.  [5]  Brahme A, Alvi MH, Saylor D, Firdy J, Rollett AD. Scripta Mater. 2006;55:75.  [6]  Groeber M, Ghosh S, Uchic MD, Dimiduk D. Acta Mater. 2008;56:1257.  [7]  Rhines FN, Patterson BR. Metall. Trans. A 1982;13:985.  [8]  Krill CE, Birringer R. Phil. Mag. A 1998;77:621.  [9]  Mitra R, Ungar T, Morita T, Sanders PG, Weertman JR. In: Chung Y-W, Dunand DC, Liaw PK, Olson GB, editors. Advanced materials for the 21st century. Ohio: Warrendale: The Minerals, Metals and Materials Society; 1999. p. . 553  [10]  Zhilyaev AP, Gubicza J, Nurislamova G, Revesz A, Surinach S, Baro MD, Ungar T. Phys. Stat. Sol. A 2003;198:263.  [11]  Erb U, Palumbo G, Szpunar B, Aust KT. Nanostr. Mat. 1997;9:261.  [12]  Kurtz SK, Carpay FMA. J. Appi. Phys. 1980;51:5725.  139  APPENDIX A3 EPSC and VPSC Polycrystal Models In this section, first the grain (single crystal) constitutive relationships relevant to the EPSC and VPSC models are introduced. Next, the two models are described.  A31. Grain Constitutive Formulations Considering dislocation slip as the dominant mode of deformation, the grain constitutive law then relates the stress and strain of the grain to the slip activity within the grain. In the following, the set of equations making up the constitutive law of the grain are introduced. Considering the applied stress on a grain is given by slip system S of this grain,  rS,  o,,  the resolved shear stress on  can be written as [1,2], .S  =mcru  (A3.l)  Here, m is the symmetric traceless Schrnid tensor associated with slip system S, which itself is given in terms of the normal, nf, and Burgers vector, L, of this system as,  m =i(nb +nb ) 1  (A3.2)  According to Schmid Law [3], slip system S will be activated once its resolved shear stress reaches a critical value, i.e. vs  =  r. Hence, (A3.3)  140  This critical value, r’, is referred to as the critical resolved shear stress or equivalently, the mechanical threshold [4]. Upon activation, the incremental strain in the grain due to the slip that have occurred along all active slip systems 5, i.e. d, can be expressed as [1,2],  ds  =  mdyS  (A3 .4)  where the summation is taken over all active slip systems and dy is the incremental shear strain in slip system S. Note that an incremental definition of strain is used in the above analysis to denote that strain is not a state variable. Equally, this relationship can be written in terms of strain rates by dividing the two sides of the equation by time increment dt. Thus, (A3.5) where  is now the shear strain rate on slip system S and  is the strain rate of the grain.  The constitutive law of the grain can ultimately be completed by linking o in Equation (A3 .3) to  in Equation (A3.5) at any stage of the deformation process. This is done by  establishing a relationship between r and  in the two equations. One typical way to do  this is to assume that at a given stage of the deformation process, the rates of change of the mechanical thresholds are related to the shear rates by [5], (A3.6)  t  The h’ are the hardening moduli; h is the self-hardening rate on system S and h (S  S’) is the latent-hardening rate of system S (whether it is active or not) caused by slip on  system S 1 [6]. The components of h are defined as [7], ’ 5 h  &S(r)qSS’  (A3.7)  141  where 6 (F) = dz /dF, F being the cumulative strain in the grain, is the instantaneous hardening rate and  q’  is a matrix describing the latent-hardening behaviour of the grain.  The above described constitutive equations; i.e. Equations (A3.3), (A3.5), and (A3.6); do not account for strain rate sensitivity of slip [4]. This type of constitutive law is used in EPSC. Alternatively, a rate-dependent constitutive law can be obtained for the grain using the viscoplastic equation [8,9], i.e.  (A3.8)  =  where  is a reference strain rate and n is the rate sensitivity exponent. Once again, the  evolution of the threshold stress r with deformation is taken into consideration using Equation (A3.6). This rate-dependent constitutive formulation is employed in VPSC.  A3.2. EPSC Formulation In the EPSC model, the elastic problem treated by Eshelby [10] is extended to the plastic regime by expressing the non-linear response of the polycrystal and the grains in incremental form and by assuming instantaneous moduli which relate the stress rates to the strain rates. At the grain level this means =  where  and  1 L  (A3 .9)  are the stress rate and strain rate of the grain, respectively, and L,JkI is the  grain instantaneous elastoplastic stiffness tensor. The grain stiffness, L , is dependent on 1 the orientation of the grain, the single crystal elastic stiffhess, and the plastic state of the grain [11]. It is expressed as [11],  142  Luki  where  (A3.lO)  = jnm ‘nrnkl  is the grain elastic stiffness,  ‘k1  is the identity tensor,  and the sum is over all active slip systems S. The tensor slip system 5,  ,  m  is the Schmid tensor,  solves the shear strain rate of  in terms of the crystal strain rate, i.e. >SfS.  (A3.1l)  It is given by =  1m (X’) C 1  (A3.12)  where the sum runs over the active systems and X is a matrix defined in terms of the hardening matrix, hs’, as, Xn” =hs’  (A3.13)  +mUC,Jklm  Note that in derivation of the above relationships, the rate-independent constitutive laws described previously are used. At the macroscopic level, the polycrystal (HEM) stress rate, d,., and strain rate, ,, are also linearly related through the polycrystal tangent elastoplastic stiffness,  LJkI,  as (A3.14)  Given the instantaneous (linear) constitutive relationships for the polycrystal and the grains, the Eshelby [101 solution is invoked.  Accordingly, the deviation of the local  magnitudes with respect to the macroscopic values is written using what is referred to as the interaction equation (—J)=—tUkl(kl—/)  (A3.15)  143  where  is the interaction tensor which is expressed in terms of the elastoplastic Eshelby  tensor S as =  L,, (Sk,  —  ‘nrnkl)  (A3.16)  is a function of the macroscopic elastoplastic stiffness, LJkI, and the shape of the ellipsoid [12].  Using Equations (A3.9) and (A3.14) in Equation (A3.15), the grain and  polycrystal strain rates can be related as (L,  + Lkl  )  (Lklrnn  )  + Lklmn rnn =  (A3.17)  where Ak/ is the localization tensor. Equation (A3. 17) gives the strain rate in each grain for an imposed strain rate and polycrystal stiffness LJkl. The partitioning problem is thus solved. However, LJkl is dependent on the response of the individual grains and it is not known a priori; it must be found iteratively. To check whether the correct  is found, the self-  consistent condition is tested. This means that the polycrystal stress and strain rates must equal to the weighted stress and strain rate average of all the grains, i.e. (A3.18) and  o=(o) where  ()  (A3.19)  denotes volume average. Using Equations (A3.9), (A3.14), and (A3.17) the above  equations can further be simplified to ‘7kl =  Afllflkl)  (A3 .20)  which is the self-consistent equation.  144  Having established the local and macroscopic constitutive relationships and their corresponding correlations, the response of the aggregate to an imposed external boundary condition can now be estimated. Assuming macroscopic strain rate,  ,  is imposed on the  polycrystal in a given time step, the following algorithm can be used to determine the response of the aggregate. First, a guess is made for the value of J. Using this value, the Eshelby tensor (see [12] for details of the calculation of the Eshelby tensor) and  1 L  (from  Equation (A3.16)) are calculated. Next, another guess is made on the set of active systems within each grain (note that at start of deformation, no active systems are assumed and the grain deforms elastically). With this guess, the grain stiffness, Luki, and localization tensor, are calculated using Equations (A3. 10) and (A3.17), respectively. Knowing  ,  it is  now possible to calculate the strain rate for each grain from Equation (A3.17). This enables evaluation of the stress rate, d, from Equation (A3 .9) and the shear strain rate of each active system,  ,  from Equation (A3.l1). Following Hutchinson [11], a given slip system S  is considered potentially active if the Schmid law, Equation (A3.3), is satisfied. This system is actually active depending on whether t  and  >  0  (A3.21)  Thus, if the above conditions are violated, a new guess for the set of active systems must be made. Otherwise, Equation (A3.20) is used for improving the guess for the macroscopic modulus LJkl.  Once convergence is reached on the value of L, the current local and  macroscopic stress and strain are calculated.  145  A3.2. VPSC Formulation The VPSC model treats the problem of large plastic deformation and for this purpose, the elastic effects in the polycrystal are neglected. Moreover, since plastic deformation is independent of the hydrostatic stress component, the model is formulated only in terms of the deviatoric stress tensor. Just as in the EPSC model, to utilize the Eshelby [10] solution, it is necessary to linearize the non-linear behaviour of both the grains and the polycrystal. Accordingly, the constitutive relationship of the grain relating the grain strain rate,  ,  to the  grain deviatoric stress, u,, can be written as, =  where  1 Mu  (A3.22)  is the grain compliance. Assuming the behaviour of the grains is expressed by  the viscoplastic relationship of Equation (A3.8), it can be shown that [8,9], ‘  M/kI =  S S 7 S mm ki (mu Pq pq  s  (A3 .23)  0  In a similar manner, the constitutive relationship for the HEM can be written as, (A3.24)  =  where  ,  ,  and  now correspond to the HEM. Having linearized the response of the  grains and the aggregate, the Eshelby [10] solution is adopted to relate the stress and strain rate in the grain to the overall stress and strain rate in the aggregate [8,9], i.e.  (  —  =  VI yki  (ui,  3) —  The above equation is referred to as the interaction equation and  (A3 .25) is accordingly, the  interaction tensor defined as [13,14],  146  Mjkl =  n°  —  sd,,,,, ) SnlnpqA/Ipqkl  (A3.26)  where n 41 is a strength controlling parameter, I is the identity tensor, and  is the  Eshelby tensor. The iterative process of the self-consistent theory can now be implemented to determine the behaviour of the polycrystal. The algorithm for this procedure, for when the macroscopic strain rate is imposed on the HEM and the evolution of  is desired, is as  follows. Initially, it is assumed that the strain rate in all grains is that imposed on the HEM, i.e. a Taylor guess. Using the viscoplastic equation, Equation (A3.8), the deviatoric stress for each grain is calculated.  This value is used in Equation (A3.23) to calculate the grain  modulus M . Assuming simple average of grain moduli, M,kl is next estimated. Using 1 this estimate, the initial guess for  follows from Equation (A3 .24). Based on the estimated  and the grain shape, the Eshelby tensor is calculated. The procedure for the calculation of the Eshelby tensor can be found in [15,16]. Subsequently, the interaction tensor calculated from Equation (A3.26).  is  Combining the interaction Equation (A3.25) with the  self-consistent condition, i.e. the condition that the weighted average of the strain rate and stress over the aggregate must coincide with the macroscopic quantity, an improved estimate of  is obtained iteratively. After achieving convergence on MJkl, a new guess for  can be obtained using the interaction equation. If this value for each grain is different from the previously determined value in each grain, the above described algorithm is repeated. Otherwise, the iteration process is complete for this deformation step and the macroscopic and local stress and strain rates are calculated. Before initiating the next deformation step, the new crystallographic orientation and shape of each grain is determined; the procedures used for this stage can be found in [9,15]. 147  References [1]  Kocks UF. In: Miller AK, editor. Unified Con stitutive Equations for Creep and Plasticity. New York: Elsevier; 1987. 1. p.  [2]  Kocks UF. In: Kocks UF, Tome CN, Wenk H-R , editors. Texture and Anisotropy: Preferred Orientations in Polycrystals and Thei r Effect on Materials Properties. Cambridge: Cambridge University Press; 1998 . p. 327.  [3]  Schmid E, Siebel G. Z. Elektrochem. 193 1;37 :447.  [4]  Kocks UF, Argon AS, Ashby MF. Prog. Mate r. Sci. 1975;19:1.  [5]  Hill R. J. Mech. Phys. Solids 1966;14:95.  [6]  Asaro RI. Adv. Appl. Mech. 1983;23:1.  [7]  Li S, Beyerlein IJ, Necker CT. Acta Mate r. 2006;54: 1397.  [8]  Molinari A, Canova GR, Ahzi S. Acta Meta ll. 1987;35:2983.  [9]  Lebensohn RA, Tome CN. Acta Mater. 1993;41:2 61 1.  [10]  Eshelby ID. Proc. Roy. Soc. A 1957;241:376 .  [11]  HutchinsonJW. Proc. Roy. Soc. Lond. A 1970 ;319:247.  [12]  Lebensohn RA, Tome CN. Phil. Mag. A 1993 ;67:187.  [13]  Molinari A, Tóth LS. Acta Metall. Mater. 1994 ;42:2453.  [14]  Tome CN. Modelling Simul. Mater. Sci. Eng. 1999;7:723.  [15]  Lebensohn RA, Turner PA, Signorelli 1W, Cano va GR, Tome CN. Modelling Simul. Mater. Sci. Eng. 1998;6:447.  [16]  Tome CN, Lebensohn RA. In: Raabe D, Rote rs F, Barlat F, Chen L-Q, editors. Continuum Scale Simulation of Engineering Materials. Weinheim: Wiley-VCH; 2004. p. 473.  148  APPENDIX A4 Volume Fraction Probability Density Function for Lognormal Grain Size Distributions Having described the grain size distribution of a polycrystal by the lognormal probability density function, it is the aim here to determine the probability density function for the volume fraction distribution of the grains. The lognormal probability density function can be written as [1,2],  P(D)=  (_(lnD_M)2 i expj .f2,z.S2D 2S 2  (A4.1)  where D is the grain size, and M and S are, respectively, the mean and standard deviation of the logarithms of the grain sizes.  By definition, the number fraction of grains in this  distribution is given by (A4.2) -zvtot  Here, N 0 is the total number of grains in the sample. Assuming all grains are of the same shape, the volume fraction of grains in this distribution can be calculated as dV df(D)=—=  D P 3 (D)dD  (A4.3)  °DP(D)dD where Vis the volume and l’ is the total volume of the sample. Hatch and Choate [3] show that D3P(D)=exp(3M+S2  (A4.4)  Furthermore, it can also be shown that [4]  149  D P 3 (D)  =  1  exp 3M+—S! 2)  2 2S  (A4.5)  Thus, Equation (A4.3) can be simplified to  df(D)=  1 /2,irS2D  (A4.6)  2 2S  and accordingly, the probability density function for the volume fractio n distribution, F (D), can be written as  1  F(D)=  J2,irS2D  exp  —  (  D  ))2 —  (M + 3S 2  2S 2  (A4.7)  Observe that F(D) is lognormal.  References [1]  Aitchison J, Brown JAC. The Lognormal Distribution. Cambridge, UK: Cambridge University Press; 1963.  [2]  Kurtz SK, Carpay FMA. J. Appi. Phys. 1980;51:5725.  [3]  Hatch T, Choate SP. J. Franklin Inst. 1929;207:369.  [4]  Hinds WC. Aerosol Technology: Properties, Behavior, and Measu rement of Airborne Particles. New York: John Wiley and Sons; 1982.  150  APPENDIX A5 Afanesev Theory of Microdeformation in Polycrystals According to the statistical theory of Afanasev [1], as described by Polák and Klesnil [2], a polycrystal is considered to be composed of a population of grains (or elementary volume elements) with different yield strengths. Considering that the volume fraction of grains with yield strength o is described by the probability density function G (o), schematically shown in Figure A5. 1, and that strain is partitioned homogeneously among the  0  EC  Figure A5.1. Schematic illustration of the probability density function describing volume fraction distribution of grains with yield strengths o. For this plot, the polycrystal grain size distribution is assumed to be lognormal and the grain size and grain yield strength are assumed to be related through the Hall-Petch relationship.  grains; the polycrystal stress, u, at any instance of the deformation of the polycrystal can be calculated as,  151  0=  LuiG(Ji)do-+E8fG(ui)dai  (A5.1)  where E is the Young’s modulus and s the polycrystal strain. In this equation, the first and second terms are, respectively, the contributions of plastically and elastically deforming grains. Assuming that the probability density function G (oh) does not vary with straining, the derivative of Equation (A5.l) with respect to strain (i.e. the work hardening rate of the polycrystal, €  =  du/d&) can be calculated as e=EfG(u)do-,  where  (A5.2)  G (u )d0 1 is the volume fraction of elastic grains at applied strain of ; observe  that this integral represents the area under the probability density function G (o) in the range [Es, co[.  Given that by definition of probability density function  G  lo =1,  Equation (A5.2) can be reformulated as  (A5.3)  where  G  lo is now the volume fraction of yielded grains at applied strain of  8.  References [1]  Afanasev NN. Statistical Theory of Fatigue Strength of Metals [in Russian]: Izv. Akad. Nauk SSSR; 1953.  [2]  PolákJ, Kiesnil M. Fatig. Eng. Mater. Struct. 1982;5:19.  152  

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