LOCALLY ENHANCED VORONOI CELL FINITE ELEMENT MODEL (LE-VCFEM) FOR DUCTILE FRACTURE IN HETEROGENEOUS CAST ALUMINUM ALLOYS

Transcription

1 LOCALLY ENHANCED VORONOI CELL FINITE ELEMENT MODEL (LE-VCFEM) FOR DUCTILE FRACTURE IN HETEROGENEOUS CAST ALUMINUM ALLOYS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Chao Hu, B.S., M.S. * * * * * The Ohio State University 2008 Dissertation Committee: Approved by Prof. Somnath Ghosh, Adviser Prof. Marcelo J. Dapino Prof. Ahmet Kahraman Prof. J.K. Lee Adviser Graduate Program in Mechanical Engineering Dr. Stephen Harris

2 c Copyright by Chao Hu 2008

3 ABSTRACT Ductile heterogeneous materials like cast aluminum alloys, undergo catastrophic ductile failure that initiates with particle fragmentation with evolves with void growth and coalescenece in localized bands of intense plastic deformation and strain softening. Conventional Voronoi Cell finite element model (VCFEM), based on the assumed stress hybrid formulation, is for small deofmration and is unable to account for plastic strain induced softening. To overcome this shortcoming of material softening due to plastic strain localization, this work introduces a locally enhanced Voronoi Cell finite element model (LE-VCFEM) for modeling the very complex phenomenon of ductile failure in heterogeneous metals and alloys. In LE-VCFEM, finite deformation displacement elements are adaptively added to regions of localization in the otherwise assumed stress based hybrid Voronoi celle finite element to locally enhance modeling capabilities for ductile fracture. Adaptive h-refinement is used for the displacement elements to improve accuracy. Damage initiation by particle cracking is trigerred by a Weibull model. The nonlocal Gurson-Tvergaard-Needleman model of porous plasticity is implemented in LE-VCFEM to model matrix cracking. An iterative strain update algorithm is used for the displacement elements. The LE-VCFEM code is validated by comparing with results in the literature and with conventional FE codes. Furthermore, LE-VCFEM simulations of real micrstructures are satisfactorily ii

4 conducted, establishing the high potential of this method. The effect of various microstructural morphological characteristics is also investigated. iii

5 Dedicated to my wife and my parents iv

6 ACKNOWLEDGMENTS I would like to express my sincere gratitude and thanks to my advisor, Prof. Somnath Ghosh, for providing guidance, encouragement and continued financial support throughout the course of this thesis work. I am also grateful to Prof. Marcelo Dapino, Prof. Ahmet Kahraman, Prof. J.K. Lee and Dr. Stephen Harris for serving as a member of my committee. This work has been supported by the National Science Foundation NSF Division of Civil and Mechanical Systems Division through the GOALI Grant No. CMS (Program director: Jorn Larsen-Basse). This sponsorship is gratefully acknowledged. Computer support by the Ohio Supercomputer Center through grant PAS813-2 is also gratefully acknowledged. I would like to thank my wife Zheng Wang and my parents for their continued support and encouragement. I would also like to thank James Giuliani and Pete Carswell from Ohio Supersomputer Center for the help and support on the parallel computing and visualization throughout this thesis work. I would also like to thank Dr. James Boileau from Ford Motor Company and Prof. Bhaskar Majumdar for the experimental data support and discussions. I would also like to thank all my fellow lab research associates Dr. Suresh Moorthy, Dr. Shanhu Li, Dr. Mike Groeber, Jie Bai, Dakshina Murthy Valiveti, Himanshu Bhatnagar, Jayesh R. Jain and Abhijeet Tiwary for the discussions and help throughout the course of my study. v

7 Finally, my earnest special thanks must go to my wife, Zheng Wang, for her love, patience, encouragement and everlasting trust in every sense. vi

8 VITA Born - Jiangxi, China B.S. Mechanical Engineering, North China University of Technology M.S. Engineering Mechanics, Tsinghua University Graduate Research Associate, The Ohio State University. PUBLICATIONS Research Publications S. Ghosh, V.Dakshinamurthy, C. Hu, and J. Bai Multi-Scale Characterization and Modeling of Ductile Failure in Cast Aluminum Alloys, Int. Jour. Comp. Meth. Engng. Mech.. C. Hu and J. Bai and S. Ghosh Micromechanical and Macroscopic Models of Ductile Fracture in Particle Reinforced Aluminum, Modelling Simul. Mater. Sci. Eng., 15:S377-S392,2007. A. Tiwary, C. Hu and S. Ghosh Numerical Conformal Mapping Method Based Voronoi Cell Finite Element Model for Analyzing Microstructures with Irregular Heterogeneities, Finite Elements in Analysis and Design, 43: ,2007 C. Hu and S. Moorthy and S. Ghosh A Voronoi Cell Finite Element Model for Ductile Damage in MMCs, Proceedings of the 8th International Conference on Numerical Methods in Industrial Forming Processes, ,2004 C. Hu and J. Cheng The multimedia simulation experiments in the teaching of theoretical mechanics, Chinese Journal of Mechanics in Engineering, 25:67-69,2003 vii

9 C. Hu, J. Cheng and X. Han Multimedia simulation teaching system for the experiments of material strength, Mechanics in Engineering, Chinese Journal of Mechanics in Engineering, 24:68-70,2002 C. Hu and S. Ghosh Enriched Voronoi Cell Finite Model for Ductile Fracture in Particle Reinforced Metal Matrix Composites, The 9th US National Congress on Computational Mechanics, San Francisco, 2007 C. Hu and S. Ghosh Enhanced Voronoi Cell Finite Model for Ductile Fracture in Multi-phase Materials, The 7th World Congress on Computational Mechanics, Los Angele, 2006 C. Hu and S. Ghosh A Voronoi Cell Finite Element Model for Particle Fracture and Matrix Cracking in Particle Reinforced Metal Matrix Composites, The 8th US National Congress on Computational Mechanics, Austin Texas, 2005 FIELDS OF STUDY Major Field: Mechanical Engineering Studies in Finite Element Analysis: Prof. Somnath Ghosh viii

10 TABLE OF CONTENTS Page Abstract Dedication Acknowledgments Vita List of Tables List of Figures ii iv v vii xii xiii Chapters: 1. Introduction Motivation Scope of Proposed Research Organization of the Thesis VCFEM Formulation for Nonlocal Porous Plasticity in the Absence of Microstructural Damage Introduction Variational Principle with Assumed Stress Hybrid Method Element Formulations and Assumptions Weak Form and Solution Method Constitutive Relations for the Inclusion and Matrix Phases Strain Update Algorithm in VCFEM Formulation Aspects of Numerical Implementation ix

11 2.7.1 Numerical integration in inclusion and matrix phases of the Voronoi cell element Rank sufficiency of [K e ] Constraining rigid-body modes at the interface Error Measure, Adaptivity and Convergence Rate Numerical Examples for Validating the VCFE Model Comparison with results using ABAQUS Convergence rate of VCFEM simulation Comparison of inclusion stresses obtained by Raman spectroscopy Conclusions VCFEM for Inclusion Cracking in a Porous Plastic Matrix Introduction Modified VCFEM Formulation to Account for Cracked Inclusions Numerical Examples on VCFEM with Inclusion Cracking A square RVE with a single cracked inclusion Calibration of Weibull parameters for particle cracking Conclusions VCFEM with Local Enhancement for Matrix Cracking Introduction Criteria for Local Enhancement Constitutive Relations and Stress Update Algorithm Coupling Stress and Displacement Interpolated Regions in the Locally Enhanced VCFE Formulation Weak Form and Matrix Assembly Some Aspects of Numerical Implementation Iterative solver Mapping from stress based domains to post enhancement displacement based regions Adaptive h-refinement for displacement elements Conclusions Numerical Examples of Ductile Fracture with the LE-VCFE Model Introduction Sensitivity analysis with respect to spatial distribution of inclusions Sensitivity analysis with respect to material characteristic length MCL for nonlocal GTN model x

12 5.4 Sensitivity analysis with respect to nucleation parameters in GTN model Effect of ǫ N Effect of f N Sensitivity analysis with respect to void coalescence parameters f c and f f Effect of porosity in plasticity relations on ductile failure Ductile fracture for a real microstructure Conclusions LE-VCFEM for Heterogeneities of Arbitrary Shapes Introduction Numerical Conformal Mapping for Heterogeneities of Arbitrary Shapes Multi-resolution Wavelet Functions for Enhancing Stress functions Numerical integration of element matrices Numerical Examples Comparison with commercial code Real microstructure with multi-arbitrary shaped inclusions case Conclusions Conclusions and Future Studies Bibliography xi

13 LIST OF TABLES Table Page 2.1 Comparison of simulation times by the two approaches Stress-strain slope σyy ǫ yy in the elastic regime Experimentally observed (i) particle volume fraction (VF), (ii) total number of particles(np) and (iii) total number of cracked particles(ncp) for the two micrographs of W319-T7 at a total strain ǫ yy = 6.21% Weibull parameter (σ w ) Si (GPa) calibrated from undamaged VCFEM simulations Number of cracked particles in experiments and by VCFEM simulation using different values of σ w Ductile Matrix and Brittle Inclusion Material Properties Void nucleation parameters in GTN model and Weibull parameters Stress related parameters and pre-enhancement displacement degrees of freedom in each Voronoi cell element Microstructural characterization functions. All the dimensions are in fractions of the size L of the microstructure Equivalent strains to failure for each microstructure xii

14 LIST OF FIGURES Figure Page 2.1 (a) Discretization of a multi-phase material microstructure into Voronoi cells by a tessellation method, (b) A typical Voronoi cell element with a cracked particle Line search in the Regula Falsi algorithm for (a) yield function ˆΦ( f local ) and (b) equivalent plastic work R( ǫ p(i) 0 ) The flow chart of the strain update algorithm for VCFEM (a) Physical domain, (b) transformed domain in the coordinates (ψ, γ), (c) Subdivision of quadrilaterals in the matrix (d) subdivision for a typical undamaged Voronoi cell. Shaded regions are used in the particle domain Two different RVE s and the corresponding Voronoi cell meshes: (a) unit cell with a single circular inclusion, (b) square domain with five circular inclusions Comparison of volume-averaged stress-strain response for: (a) RVE (a) in figure 2.5(a), (b) RVE (b) in figure 2.5(b) Average void volume fraction vs. macroscopic strain for : (a) RVE (a) in figure 2.5(a), (b) RVE (b) in figure 2.5(b) VCFEM convergence rate for the RVE of figure 2.5(a) : (a) Average traction reciprocity error (AT RE) and (b) Average strain energy error (ASEE), plotted as functions of the total degrees of freedom VCFEM convergence rate for the RVE of figure 2.5(b) : (a) Average traction reciprocity error (AT RE) and (b) Average strain energy error (ASEE), plotted as functions of the total degrees of freedom xiii

15 2.10 (a) A micrograph of cast aluminum alloy A356-T6 showing two Si particles interrogated for stresses by Raman spectroscopy; (b) the VCFEM mesh and model for a part of the microstructure Comparison of volume averaged stress σ yy in the two Si particles as a function of the macroscopic applied strain by VCFEM simulation with those by Raman spectroscopy The Voronoi cell mesh with a crack, and the location of nodes on a crack tip A square RVE with a circular cracked inclusion of volume fraction v f = 20%: (a) The Voronoi cell element with boundary conditions, (b) ABAQUS mesh Volume averaged macroscopic stress-strain response for the RVE with a cracked inclusion Contour plot of equivalent plastic strain with the two models: (a) ABAQUS, and (b) VCFEM Scanning electron micrograph images of W319-T7 Aluminum alloy, showing different sections: (a) section 1, (b) section 2; and corresponding VCFEM mesh and model for these sections: (c) section 1, (d) section 2. (The dark shaded ellipses are Si particles, the white ellipses are copper-based intermetallics Number fraction of particle area for (a) section 1 (b) section Comparison of experimental and simulation-based probability density of cracked particle volume with different Weibull modulus, for (a) micrograph 1 (b) micrograph Contour plots of effective plastic strain for (a) micrograph 1, (b) micrograph Distribution of number fraction of damaged particles as functions of the particle size for (a) micrograph 1, (b) micrograph xiv

16 4.1 A locally enhanced Voronoi cell element with superposed displacement based elements Four different microstructures with their Voronoi cell meshes: (a) uniform square edge-packed, (b) uniform square diagonal-packed, (c) hardcore random and (d) hard-core with clustering Distribution of Nearest neighbor distance (NND) for microstructures (C) and (D) Voronoi cell model with locally enhanced displacement elements for the microstructures: (a) C and (b) D at strain to failure Macroscopic stress-strain response for the different microstructures The contour plot of void volume fraction for microstructure: (A), (B), (C), (D). The number in the figures indicate the sequence of particle cracking Microstructural configuration and Voronoi cell meshes for (a) a unit cell containing a single circular inclusion and (b) a representative microstructural element containing 25 inclusions Macroscopic stress-strain response for the unit cell with (a)f in = 0.1 without void nucleation and (b) f in = with nucleation Contour plot of void volume fraction with MCL = 1.0L for (a) f in = 0.1 without nucleation and (b)f in = with void nucleation The number fraction of particles for nearest neighbor distance for 25 Voronoi cell mesh Macroscopic stress-strain response for microstructure with 25 inclusions for different MCL s Contour plots of void volume fraction for the microstructure with 25 inclusions, with nonlocal parameter: (a) MCL = 0, (b) MCL = 0.8L NND,(c) MCL = 1.0L NND, (d) MCL = 1.2L NND and MCL = 1.4L NND. Numbers in the figures indicate the sequence of particle cracking xv

17 5.12 Macroscopic stress-strain response with different ǫ N, for microstructure (D) in figure Macroscopic stress-strain response with different f N, for the microstructure (D) in figure Macroscopic stress-strain response with different void coalescence parameters, f c & f f, for the microstructure (D) shown in figure The macroscopic stress-strain response for GTN model and J 2 plasticity model Contour plot of effective plastic strain for: (a) the GTN model with porosity, (b) J 2 plasticity model (a)experimental micrograph, (b) VCFEM mesh and boundary conditions Macroscopic stress-strain response for the real microstructure with only inclusion cracking and J 2 plasticity model Contour plot of effective plastic strain at (ǫ xx = 8.88%) for the real microstructure with only inclusion cracking and J 2 placticity model under generalized plane strain condition with (ǫ zz = 0.005) Macroscopic stress-strain response for the real microstructure The contour plot of void volume fraction for the real microstructure. The number in the figure indicate the sequence of inclusion cracking A Voronoi cell with a inclusion of multi-sided polygon Conformal transformation of a Voronoi cell with a inclusion of multisided polygon (a) Physical Domain (b) Transformed Domain Generation of multilevel wavelet bases: (a) All the grid points of the uniform rectangular grids are chosen as wavelet bases (b) The finally used wavelet bases for a concave inclusion case The stress σ xx at y = 0.51L/2 for one Voronoi cell with a square shaped inclusion to demonstrate the effect of the multi-resolution waveletenriched stress functions xvi

18 6.5 The division algorithm of the integration scheme for a typical Voronoi cell with concave shaped inclusion in transformation domain (a) mapped Voronoi cell in transformed domain (b) Subdivision of quadrilaterals in the matrix (a) Subdivision for a typical Voronoi cell with concave shaped inclusion in physical domain (b) the adjustment of the discretization because of the neighbor elements A square plate with square shaped inclusion The macroscopic stress-strain response for a square plate with square shaped inclusion The microscopic equivalent plastic strain for square inclusion (a) ABAQUS (b) VCFEM (a) SEM micrograph of a cast aluminum alloy Al319 microstructure with irregular silicon particulates and (b) VCFEM mesh and boundary conditions for 49 inclusions case including 5 arbitrary shaped inclusions Macroscopic stress-strain response for the real microstructure with 5 arbitrary shaped inclusions The contour plot of void volume fraction for the real microstructure with 5 arbitrary shaped inclusions. The number in the figure indicate the sequence of inclusion cracking xvii

19 CHAPTER 1 INTRODUCTION 1.1 Motivation Many metals and alloys widely used in automotive, aerospace and other engineering systems consist of heterogeneities in the form of particulates, fibers, precipitates or voids in their microstructure. For example, cast aluminum alloys such as A 319 used in automotive systems contain microstructural heterogeneities in the form of silicon particulates, intermetallics, precipitates and voids. Processing routes affect morphological variations such as irregularities in spatial dispersion including clustering and alignment, or irregularities in phase shapes and sizes. The presence of these heterogeneities often has adverse effects on their failure properties like ductility and creep resistance. Major microstructural mechanisms that are responsible for deterring the overall properties include particulate fragmentation, interfacial debonding and matrix failure [1, 2, 3, 4]. Ductile failure in many metals and alloys is usually initiated by particle cracking or interfacial debonding. Voids start to grow around the crack tips with deformation, which subsequently interact and coalesce with neighboring voids to result in matrix failure. Evolution of matrix failure cause stress redistribution, 1

20 causing other particles to crack and nucleate new voids, eventually leading to catastrophic failure of the microstructure. Mechanisms of crack initiation and propagation strongly depend on the state of stresses, especially stress triaxiality, and plastic deformation. Particle cracking is essentially dominated by the maximum principal stress. which in turn are affected by morphological variations such as spatial dispersion including clustering and alignment, phase shapes and sizes. Experimental studies on ductile failure in [5, 6, 7, 8] have shown that these morphological variations affect microstructural damage nucleation due to particulate cracking and interfacial decohesion, and subsequently matrix rupture due to void growth and coalescence. Argon [9, 10] has shown that particles in clustered regions have a greater propensity toward cracking than those in regions of dilute concentration, since local stresses increase rapidly with reduced distance between neighboring particles. Experimental work in Caceres et. al. [11, 12] has demonstrated that larger and longer particles are more prone to cracking and damage accumulation increases with higher dendrite arm spacing. Consequently, modeling failure properties like strain to failure, ductility and fracture toughness of these alloys, requires special attention on their microstructural morphology. A large number of analytical, computational and experimental studies have been conducted to model material ductile failure. Early models include a localization condition proposed by Rice in [13] based on a single void in the matrix, without accounting for the interactions between voids. The cohesive zone model, widely used for modeling brittle fracture, has been is used to simulate ductile fracture in [14, 15, 16, 17, 18, 19]. 2

21 These efforts have concluded that the cohesive energy and strength do not remain constant throughout a crack growth analysis. The use of zone models in ductile fracture with cohesive strength and energy taken as known material properties is questionable. These parameters depend strongly on the applied loading conditions in addition to microstructural variables like void volume fraction, flow strength and hardening. Thus, the determination of the cohesive zone parameters by simple micromechanical simulations is not a straightforward task. A more physical way for theanalysis of ductile fracture is through the use of a phenomenological damage model, described within the framework of continuum damage mechanics. These models introduce damage evolution parameters as internal variables in the constitutive requirements of irreversible thermodynamics processes, with their growth determined by appropriate evolution laws. From the premise that ductile fracture is governed by nucleation, growth and coalescence of voids, early models of ductile failure have been proposed by McClintock[20] and Rice and Tracey [13]. A widely used phenomenological constitutive relation for a progressively cavitating solid has been introduced by Gurson [21]. In this model, the growth of voids is strongly dependent on the tensile hydrostatic stresses developed during the deformation process. This model has been extended by Tvergaard and Needleman [22] to account for the loss of load carrying capacity during void coalescence. Further extensions of the class of Gurson-like models have been made by several authors, viz. void shape effects in [23, 24, 25, 26], rate and temperature-dependent effects in [27], effects of anisotropy on plastic flow in [23, 28]. Models derived from the original Gurson model [21] have proved to be valuable in simulating ductile fracture. These models have attracted a lot of attention because of their physical basis, and have been successfully applied by numerous authors. Among 3

22 the other models, Rousselier [29] has proposed a similar pressure dependent yield surface based on continuum damage mechanics and thermodynamics. Lemaitre [30] has introduced a set of scalar or tensor damage evolution parameters as internal variables in the constitute requirements of irreversible thermodynamics processes, with their growth determined by appropriate evolution laws. The Gurson-Tvergaard-Needleman or GTN models [22] is used in the current work. Void nucleation is first stage of ductile fracture. For Al alloys, void nucleation occurs at second phase particles through particle debonding and/or particle cracking. Two major models have been proposed for void nucleation. The first is the stresscontrolled nucleation criterion. Argon et al.[9] suggested that the void nucleation depends upon the maximum stress transmitted across the particle-matrix interface. Tvergaard [31], Pan et al.[32] and Needleman et al.[33] have suggested the use of tensile flow stress and hydrostatic stress to control void nucleation. The second is strain-controlled nucleation criterion by Chu and Needleman [34], who have proposed a normal distribution function to account for the plastic strain-controlled void nucleation. The use of strain-controlled nucleation also can overcome the inherent shortcoming of the GTN models, where there is no void growth for pure shear stress. Lievers et al. [35] have proposed a technique to calibrate the void nucleation behavior of aluminum sheet alloys for incorporation in FEM models employing the GTN constitutive models. The study by Besson [36] shows that strain controlled nucleation favors cup-cone fracture in 3D, visualized as a zig-zag pattern in 2D. 4

23 Void coalescence is an event that shifts a relatively homogeneous deformation state to a highly localized one in the microstructure. Popular models of coalescence have been suggested by Tvergaard and Needleman [37] using acceleration functions, and by Thomason [38] using the plastic limit-load criterion. Koplik and Needleman [39] has shown that the acceleration function suggested by Tvergaard and Needleman[37] for coalescence works very well for low stress triaxiality, but is less effective for high stress triaxiality. They have also shown that the void volume fraction at coalescence f c strongly depends on initial void volume fraction. Brocks et al [40] have used cell models to simulate the behavior of porous solids for varying stress triaxiality. The study confirms that the void volume fraction at coalescence does not depend significantly on the triaxiality if the initial volume fraction of the primary voids is small and if there are no secondary voids. The strain rate does not affect f c either. Benzerga et al [23, 41] have conducted studies on void coalescence with an elasticviscoplastic GTN model which accounts for void shape evolution, coalescence and post-coalescence micromechanics. Studies by Zhang [42] and Benzerga [43] show that there is no uniqueness of (f c,f f ) in fitting an experimental stress-strain curve. Tvergaard and Hutchinson [44] have studied the two mechanisms of ductile fracture for (i) the interaction of a crack tip with a single neighboring void and (ii) the interaction of multiple voids on the plane ahead of the crack tip, during crack growth initiation and subsequent crack advance. The GTN models do not resolve the growth of individual voids with the accuracy of the approach. 5

24 Mesh sensitivity with continuum damage mechanics models is a pathological problem with conventional finite element models. For avoiding the inherent mesh sensitivity of numerical failure predictions, a type of nonlocal evolution equation for the void volume fraction have been proposed by Leblond et al in conjunction with the GTN models [45, 37]. This work investigates the effect of the material characteristic length in predictions of flow localization into a shear band, failure in metal-matrix composites and void sheet development. The material characteristic length is directly incorporated into the constitutive relation as an additional phenomenological parameter. It have been shown that the nonlocal GTN models have a good control on the localization and provide an accurate phenomenological description of the subsequent response of the ductile failure. A nonlocal GTN model, proposed in [37], is used in this paper to model ductile fracture. Experiments by Caceres et al.[11, 46, 12, 47, 7] have shown that the fracture of aluminum alloys A356 and A357 is initiated by cracking of eutectic silicon due to particle stresses developed by plastic deformation in the aluminum matrix. Microstructural features that affect particle cracking in these alloys are particle size, particle aspect ratio, and the extent of particle clustering. A large volume of computational studies have been conducted for understanding elastic-plastic deformation and damage behavior of discrete heterogeneous materials. Ghosal and Narasimhan [48, 1] have studied the fracture initiation around the notch tip, while Steglich and Brocks [49] have studied the mechanism of void nucleation by interfacial debonding and particle cracking. In a comprehensive discussion of deformation and damage in particle reinforced composites, Llorca [50, 51] has used multi-particle unit cells to investigate the 6

25 influence of particle distribution on mechanical properties and ductile failure. Lim [52] used unit-cell finite element model to model the nucleation and growth of voids in a particle-reinforced metal matrix composite composite material. Negre et. al. [53] have used the GTN model for investigations on stable crack extension with initial cracks present in the material. Huber et al. [54] have developed a micromechanical model for ductility of plastically deforming materials containing a homogeneous distribution of brittle particles. Most of these studies focus on the initial stages of ductile damage and do not consider the effect of microstructural morphology on the evolution of ductile failure by void growth in the matrix and coalescence beyond particle fracture. Various computational models have been proposed for analysis of the mechanical properties and response of multi-phase materials using simplified representations of the microstructural morphology as unit cell models in e.g. [55, 56, 57]. The predictive capabilities of these models for failure properties in nonuniform microstructures are very limited due to simplification of the critical local features that are critical to failure. There is a paucity of studies on failure modeling in the presence of reinforcements of arbitrary shapes, sizes, orientation and non-uniform spatial distribution. The void growth rate in ductility is mainly affected by the stress state, especially stress triaxiality [20, 13]. Ductile fracture depends on extreme values of microstructural characteristics, e.g. nearest neighbor distance, rather than the low order moments such as mean volume fraction. Hence it is particularly important to accurately represent the real microstructural characteristics in the models for predicting ductility. A few studies have focused on modeling more realistic representation of microstructures with non-uniform dispersion of heterogeneities [58, 59] by combining digital image processing with microstructure modeling. The microstructure 7

26 based Voronoi cell finite element model or VCFEM [58, 59, 2, 4, 60] has been shown to offer significant promise in accurate analysis of large microstructural regions with high efficiency. The Voronoi cell finite element method (VCFEM has been developed in [2, 4] for micromechanical analysis of arbitrary heterogeneous microstructures. The method can effectively overcome requirements of large degrees of freedom in conventional finite element models. Morphological arbitrariness in dispersions, shapes and sizes of heterogeneities, as seen in real micrographs are readily modeled by this method. The VCFE model naturally evolves by tessellation of the microstructure into a network of multi-sided Voronoi polygons. Each Voronoi cell with embedded heterogeneities (particle, fiber, void, crack etc.) represents the region of contiguity for the heterogeneity, and is treated as an element in VCFEM. VCFEM elements are considerably larger than conventional FEM elements and incorporate a special assumed stress hybrid FEM formulation. Incorporation of known functional forms from analytical micromechanics substantially enhances its convergence. A high level of accuracy with significantly reduced degrees of freedom has been achieved with VCFEM. Computational efficiency is therefore substantially enhanced compared to conventional displacementbased FE models. Successful applications of 2D-small deformation VCFEM have been made in thermo-elastic-plastic problems of composite and porous materials [61, 62]. An adaptive VCFEM has been developed in [62], where optimal improvement is achieved by h-p adaptation of the displacement field and p-enrichment of the stress field. Mechanisms of microstructural damage inception and growth in the form of particle cracking have also been incorporated in this formulation in [4]. 8

27 1.2 Scope of Proposed Research The motivation of the present work is derived from the need to create a robust VCFE model for arbitrary crack propagation of ductile fracture in ductile metals and alloys. In the present work, the VCFEM is extended to incorporate ductile failure by particle fragmentation followed by matrix cracking in the form of void nucleation, growth and coalescence. Particle cracking is triggered by the Weibull model which is based on weakest link statistics. For ductile fracture, a pressure dependent nonlocal Gurson-Tvergaard-Needleman model [37, 45, 22] is implemented in VCFEM. Onset of localization results in the introduction of a locally enhanced VCFEM (LE-VCFEM) framework for simulating ductile fracture. In LE-VCFEM, the stress-based hybrid VCFEM formulation is enhanced adaptively in narrow bands of localized plastic flow and void growth. This region is overlaid with displacement based elements to accommodate strain softening in the constitutive behavior. In LE-VCFEM, displacement elements in regions undergoing intense plastic strain localization are modeled using finite deformation formulation. The remainder of each VC element is modeled using the assumed stress hybrid formulation for small strains. For considering more complex shaped inclusion, a modified form of Schwarz-Christoffel transformation is introduced into VCFEM to use multi-sided polygons to approximated the inclusions which have complicated shape, and multi-resolution wavelet functions are incorporated to capture the high stress gradient around the corners of multi-sided polygon shaped inclusions. The new LE-VCFEM model is capable of accounting for evolving particle cracking and ductile fracture together in the real heterogeneous materials without user interfacing. The LE-VCFEM is validated through comparisons with traditional computational models and experimental results. Statistical correlations 9

28 of damage evolution with geometric characteristics of the material morphology are also be examined in this work. The locally enhanced VCFEM for ductile fracture provides a natural interface to link quantitative metallography with deformation and fracture analysis. 1.3 Organization of the Thesis The thesis has been divided into 7 chapters. In Chapter 2, variational formulation, element formulations and assumptions, weak form and solution method, a constitutive relation for porous plasticity material, various aspects and numerical validation of the computational scheme for nonlocal porous plasticity in the absence of microstructural damage is presented. A strain update algorithm for the assumed stress VCFEM formulation is presented in this chapter also. VCFEM for porous plastic material with inclusion cracking is presented in Chapter 3. In this chapter, the Weibull model is introduced first which is used to trigger the inclusion cracking. The variational formulation corresponded to the inclusion cracking are presented in this chapter. Numerical validation of the enhanced VCFEM for inclusion cracking is also presented. With the preparation of previous two chapters, the locally enhanced VCFEM for modeling ductile fracture is developed in Chapter 4, which includes the criteria for local enhancement, finite deformation based constitutive relations, the coupling of stress based and displacement based formulation, weak form and matrix assembly and various aspects of numerical implementation. The superposition of displacement elements in the locally enhanced VCFEM is presented in detail. In Chapter 5, the sensitivity analysis with respect to nonlocal parameter, void nucleation parameters and void coalescence parameters are studied. Also, a full ductile fracture analysis 10

29 for a real micrograph is presented. In Chapter 6, the extend of the locally enhanced VCFEM for taking account the complex shaped inclusion is presented, following with some numerical examples. Capabilities of the developed computational model and possible improvements are discussed in the concluding chapter. Each chapter begins with a brief introduction to the essential features analyzed in that chapter. This is followed by main body consisting of theoretical developments and numerical results. A brief set of conclusions at the end of each chapter is used to introduce the reader to the next chapter. 11

30 CHAPTER 2 VCFEM FORMULATION FOR NONLOCAL POROUS PLASTICITY IN THE ABSENCE OF MICROSTRUCTURAL DAMAGE 2.1 Introduction The VCFEM formulation is extended in this section for a porous elastic-plastic matrix, for which the constitutive behavior is represented by a rate independent nonlocal Gurson-Tvergaard-Needleman model [37, 45]. The model incorporates a strain update method for integrating porous elasto-plastic constitutive equations. In this section, the inclusions and particles in each Voronoi cell element are assumed to be uncracked. 2.2 Variational Principle with Assumed Stress Hybrid Method Consider a multi-phase material microstructural section shown in figure 2.1(a). The microstructure is discretized into a network of N Voronoi cells by a modified Dirichlet tessellation [63, 64] that accounts for location, size and shape of the N particles. As shown in figure 2.1(b), the matrix and inclusion phases in each Voronoi cell are designated as Ω m and Ω c respectively. A crack in the inclusion is labeled as Ω cr and this will be discussed in section 3. Each Voronoi cell containing matrix, inclusion 12

31 (a) (b) Figure 2.1: (a) Discretization of a multi-phase material microstructure into Voronoi cells by a tessellation method, (b) A typical Voronoi cell element with a cracked particle and crack phases is designated as an element in the Voronoi cell finite element (VCFE) formulation, i.e. Ω e = Ω m Ω c Ω cr. The element boundary Ω e is assumed to be comprised of a prescribed traction boundary Γ tm, a prescribed displacement boundary Γ um, and an interelement boundary Γ m, i.e. Ω e = Γ tm Γ um Γ m. It is further assumed that these segments are mutually disjoint, i.e. Γ tm Γ um Γ m =. An incremental element energy functional Π ND e is defined for each Voronoi cell element without damage (represented by the superscript N D) in terms of stress increments in the matrix Ω m, and inclusion Ω c phases and displacement increments on the element boundary Ω e, inclusion-matrix interface Ω c. The incremental energy functional Π ND e for each Voronoi cell element is expressed as: 13

32 Π ND e ( σ m, σ c, u, u ) = + B(σ m, σ m )dω ǫ m : σ m dω Ω m Ω m B(σ c, σ c )dω ǫ c : σ c dω Ω c Ω c (σ m + σ m ) n e ud Ω Ω e ( t + t) udγ (2.1) Γ tm (σ m + σ m σ c σ c ) n c u d Ω Ω c where B is the increment of complimentary energy density, and the superscripts m and c correspond to variables associated with the matrix, inclusion and crack phase in each Voronoi cell element. t is the external traction on the traction boundary Γ tm and n is the outward normal on respective boundary segments. The corresponding total energy functional for the entire computational domain is: Π ND = N e=1 Π ND e (2.2) The variation of Π ND e δπ ND is written as: e ( σ m, σ c, u, u ) = ǫ m : δ σ m dω ǫ m : δ σ m dω Ω m Ω m ǫ c : δ σ c dω ǫ c : δ σ c dω Ω c Ω c + δ σ m n e ud Ω Ω e + (σ m + σ m ) n e δ u d Ω Ω e ( t + t) δ udγ Γ tm (δ σ m δ σ c ) n c u d Ω (2.3) Ω c (σ m + σ m σ c σ c ) n c δ u d Ω Ω c 14

33 Setting the first variation of Π ND e with respect to stress increments σ m and σ c to zero, respectively result in the following kinematic conditions as the Euler equations, u m = ǫ m in Ω m and u c = ǫ c in Ω c (2.4) while setting the first variation of Π ND e with respect to the independent boundary displacements u and u to zero, respectively yield the following traction conditions as Euler equations, Element boundary traction reciprocity: (σ + σ) n e+ = (σ + σ) n e on Γ m (2.5) Traction boundary conditions: (σ + σ) n e = t + t on Γ tm (2.6) Inclusion-matrix interface traction reciprocity: (σ c + σ c ) n c = (σ m + σ m ) n c on Ω c (2.7) Superscripts + and - correspond to elements on both sides of the element boundary Ω e. The equilibrated stress field σ ij is interpolated independently in the matrix and particle phases of each Voronoi cell element. Compatible displacement fields u i and u i are independently interpolated on the element and inclusion boundaries using standard shape function for line elements. 2.3 Element Formulations and Assumptions In each Voronoi cell element, independent interpolations are made for equilibrated stress increments σ m and σ c in the matrix and inclusion phases. For 15

34 two-dimensional problems, the Airy s stress function Φ(x,y) is used to derive equilibrated stress-increments in each constituent phase. The corresponding components of σ m/c are expressed as σ m/c xx = 2 Φ m/c, σ m/c y 2 yy = 2 Φ m/c, σ m/c x 2 xy = 2 Φ m/c x y (2.8) Stress functions may be constructed for efficient computations and accuracy. Particularly, two conditions should be accounted for in the choice of matrix stress functions, viz.: They should account for the shape of the inclusion or void, such that the effect of the inclusion shape should be dominant near the matrix-inclusion interface. However this effect should vanish in the far field. The stress function construct should facilitate traction reciprocity across the interface. For two phase composites without any damage, Moorthy and Ghosh [61] have constructed the matrix stress function as the sum of a purely polynomial function Φ m poly and a reciprocal function Φ m rec. The reciprocal stress function accounts for the shape of the interface Ω c through the introduction of a parametric function f(x,y). This analytic function f(x,y) transforms an interface Ω c of any shape to a unit circle by the Schwarz-Christoffel conformal transformation [65]. For example, for an elliptical interface of the form x2 a 2 + y2 b 2 = 1 the Schwarz-Christoffel mapping function may be written as f = ψ 2 + γ 2, where the transformed coordinates (ψ,γ) are obtained as solutions to the complex quadratic equation, (ψ + iγ) 2 2(x + iy)(ψ + iγ) a + b 16 + a b a + b = 0

35 where i = 1 and (a,b) are the principal axes of the ellipse. For arbitrary shaped inclusion, it will be discussed in section 6. It represents a special radial coordinate with the property that 1 f(x,y) = 1 on Ω c and 1 f(x,y) 0 as (x,y). In the absence of a crack in the inclusion, i.e. for elements with the matrix and inclusion phases only, the stress functions are constructed respectively as Φ m = Φ m poly + Φ m rec ; Φ c = Φ c poly, (2.9) in which, Φ m/c poly = m p+q=1 ξ p η q β m/c pq (2.10) Here (ξ = x xc L c Φ m rec = M p+q=1 ξ p η q N i=1 1 f p+q+i 1 βm pqi (2.11),η = y yc L c ) are normalized or scaled element-based local coordinates, where (x c,y c ) are the centroidal coordinates of the element and L c is a maximum element dimension which is given as: L c = (max(x x c )max(y y c )) (x,y) Ω e The scaling to (ξ,η), leads to an approximate range of variation 1 ξ 1 and 1 η 1 in most Voronoi cell elements which avoid the bad conditioning or poor invertability of the [H] matrix. The corresponding element stress increments in the matrix and inclusion phases are obtained by substituting equation (2.9) in equation(2.8) as { σ m } = [P m ]{ β m } (2.12) { σ c } = [P m ]{ β c } (2.13) 17

36 Additionally, displacement increments on the Voronoi cell element boundary segments, as well as on segments of the matrix-inclusion interface are respectively interpolated as { u} = [L e ]{ q} (2.14) { u } = [L c ]{ q } (2.15) [L e ] and [L c ] are interpolation matrices constructed from 1-D shape functions on the boundary segments. 2.4 Weak Form and Solution Method Setting the first variation of the energy functional (2.2) with respect to the stress coefficients β m and β c, respectively, to zero, results in the weak forms of the kinematic relations as [P m ] T {ǫ m + ǫ m } dω = Ω e [P m ] T [n e ] T [L e ]{ q} d Ω Ω m Ω c [P m ] T [n c ] T [L c ]{ q } d Ω (2.16) and Ω c [P c ] T (ǫ c + ǫ c ) dω = Ω c [P c ] T [n c ] T [L c ]{ q }d Ω (2.17) In equation (2.17) the array form of the strain tensor and the matrix of components of the normal vector are used, i.e. ǫ 11 [ ] {ǫ} = ǫ 22 and n1 0 n [ne ] = 2 n 1 0 n 2 ǫ 12 (2.18) Next, setting the first variation of the total energy functional (2.2) with respect to q and q to zero, results in the weak forms of the traction reciprocity conditions: 18

37 N e=1 [ ] { } Ω e [L e ] T [n e ] T [P m ] d Ω 0 Ω c [L c ] T [n e ] T [P m ] d Ω β m + β m Ω c [L c ] T [n c ] T [P c ] d Ω β c + β c = [ ] N Γ tm [L e ] T { t + t} dω e=1 (2.19) 0 Due to the material nonlinearity, the equations (2.16), (2.17) and (2.19) should be solved using an iterative solver. In this process, let {dβ} i correspond to a linearized correction of { β} i s in the i-th iteration. i.e. { } { } β m β m i { } dβ m i β c = β c + dβ c (2.20) Substituting this correction in equations (2.16) and (2.17) yields [ ]{ } Hm 0 dβ m i 0 H c dβ c = [ ]{ } Ge G cm q 0 G cc q { } Ω m [P m ] T {ǫ m + ǫ m } i dω Ω c [P c ] T {ǫ c + ǫ c } i dω (2.21) where, [H m ] = Ω m [P m ] T [S m ][P m ]dω, [H c ] = Ω c [P c ] T [S c ][P c ]dω [G e ] = Ω e [P m ] T [n e ][L e ]d Ω, [G cm ] = Ω c [P m ] T [n e ][L c ]d Ω [G cc ] = Ω c [P c ] T [n c ][L c ]d Ω (2.22) and [S] is the tangent compliance matrix in each phase. Similarly, if {dq} i corresponds to the linearized correction of { q}, i.e. { } q q = { } i q q + { } i dq dq (2.23) 19

38 Substituting in equations (2.19) yields N [ ] { } i K e K mc dq K T e=1 mc K cc dq N [ ] T [ ] [ ] { } i Ge G = mc Hm 0 Ge G mc dq 0 G cc 0 H c 0 G cc dq e=1 N { } = Γ m [L e ] T { t + t}d Ω (2.24) 0 e=1 N [ ] { } Ω e [L e ] T [n e ] T [P m ]d Ω 0 Ω e [L c ] T [n c ] T [P m ]d Ω β m + dβ m Ω c [L c ] T [n c ] T [P c ]d Ω β c + dβ c e=1 The set of equations (2.21) and (2.25) are solved incrementally. 2.5 Constitutive Relations for the Inclusion and Matrix Phases The inclusion phase is assumed to be linear elastic, for which the incremental stress-strain relation is expressed using the generalized Hooke s law as ǫ e = 1 + ν E σ ν σ δ (2.25) E where, E is the Young s modulus, ν is the Possion s ratio and δ is the Kronecker delta. The matrix phase is modeled using a rate independent elastic-plastic constitutive relation for porous ductile materials. Specifically, a nonlocal Gurson-Tvergaard- Needleman or GTN model [37, 45, 22] is used in this work to model ductile failure. This model assumes additive decomposition of the strain rate into elastic and plastic parts, i.e. ǫ = ǫ e + ǫ p (2.26) The elastic part is governed by the generalized Hooke s law of equation (2.25) while the plastic part is modeled using associated flow rule, i.e. ǫ p ij = λ ˆΦ σ ij (2.27) 20

39 In the GTN model, the evolving yield surface ˆΦ = 0 is a function of the homogenized stress tensor σ ij, and equivalent tensile flow stress representing the actual stress state in the matrix material σ 0 and the void volume fraction f as ˆΦ = ( q σ 0 ) 2 + 2f q 1 cosh( 3q 2p 2σ 0 ) (1 + q 3 f 2 ) = 0 (2.28) where q is the Von-Mises stress q = ( 3 2 s ijs ij ) 1/2 with s ij = σ ij 1 3 σ kkδ ij the deviatoric stress, and p = 1 3 σ kk is the hydrostatic pressure. q 1,q 2,q 3 are constants introduced by Tvergaard [66] to bring predictions of the GTN model into closer agreement with full numerical analysis of periodic arrays of voids. The acceleration function f (f) has been introduced in [22] to model the complete loss of material stress carrying capacity due to void coalescence as: { f f fc f = f c + f u f c (2.29) f f f c (f f c ) f > f c Here f c is the critical void volume fraction at which void coalescence first occurs and f f is the value at final failure. When the acceleration function f = 0, the yield surface ˆΦ = 0 corresponds to a cylinder in the stress space, the same as for J 2 plasticity. With increasing f, the effect of the hydrostatic stress on the plastic flow, and the GTN yield surface becomes an ellipsoid in the stress space. Eventually as the void volume fraction f f f, the acceleration function (f f u = 1/q 1 ). At this value, the yield surface shrinks to a point manifesting loss of material load capacity corresponding to ductile failure. To avoid numerical difficulties, f 0.95f f is used instead of f f f in equation (2.29) as proposed in [48]. When f = 0.95f f at a given Gauss quadrature point, the void volume fraction is frozen at this value, implying 21

40 local material failure. The GTN model described above does not incorporate a material length scale and consequently numerical simulations of ductile fracture exhibit inherent mesh sensitivity. To avoid this pathological shortcoming, a nonlocal evolution equation for the void volume fraction has been proposed in [67, 45, 37]. In this formulation, the nonlocal void volume fraction growth rate at any material point x is written as: f = 1 f local (x)w( x x ) dω (2.30) W( x) Ω m where 1 W( x) = w( x x )dω and w( x ) = [ Ω m 1 + (( x )/L) p]q with p = 8, q = 2 and a material characteristic length L > 0. The weighting function w( x ) = 1 at x = 0, w( x ) = 0.25 at x = L, and w( x ) 0 x > L with a narrow transition region. The nonlocality is associated with spatial gradients in f. The explicit introduction of the material length scale parameter L regularizes the localization problem by preventing the matrix cracks from being unreasonably small. In equation (2.30) the local rate of change of the void volume fraction is due to growth of existing voids as well as due to nucleation of new voids, i.e. f local = f growth local + fnucleation local (2.31) Due to the plastic incompressibility of the matrix material, the void growth is f local growth = (1 f) ǫ p kk (2.32) 22

41 Void nucleation is assumed to be by a plastic strain controlled mechanism suggested by Chu and Needleman [34], and its rate is expressed in terms of the effective plastic strain in the matrix ǫ p 0 as: f local nucleation = A( ǫ p 0) ǫ p 0 (2.33) where A( ǫ p 0) is a parameter that is generated using a normal distribution for nucleation parameter and is chosen as: A = f N s N 2π exp[ 1 2 ( ǫp 0 ǫ N s N ) 2 ] (2.34) Here ǫ N is the mean strain for nucleation and s N is its standard deviation, and f N is the volume fraction of void nucleating particles. The effective plastic strain in the matrix ǫ p 0 in equation (2.33) may be obtained by equating the rate of equivalent plastic work in the matrix material to the macroscopic rate of plastic work, incorporating the plastic flow rule as: (1 f)σ 0 ǫ p 0 = σ ij ǫ p ij == λσ ij ˆΦ σ ij (2.35) 2.6 Strain Update Algorithm in VCFEM Formulation Several methods have been proposed in the literature [68, 69, 70, 71, 72] for the integration of the elasto-plastic constitutive equations with porosity. However most of them are stress update algorithms used in conjunction with displacement based FE formulations. In contrast, a strain update algorithm should be used in a stressbased formulation such as VCFEM. Taylor s expansion based iteration algorithms such as the Newton-Raphson methods that assume monotonic increment of independent variables are not suited for this formulation. Ductile failure is proceeded by 23

42 strain softening, where the stress at a point decreases with additional straining, i.e. σ is negative. The non-monotonic behavior of stress in VCFEM formulation necessitates the use of an alternative solution method in the incremental update algorithm. This is circumvented through the use of the bisection method [73], a root-finding algorithm. This method repeatedly bisects an interval and then selects the subinterval in which the root exists, is considered appropriate. In particular, the regula falsi algorithm that retains the estimate for which the function value has opposite sign from the function value at the current best estimate of the root. The increment of the void volume fraction f and the equivalent plastic strain increment in the matrix material ǫ p 0 are used as line search parameters for generality. The incremental update method discussed here is for the generalized plane strain case with a prescribed out of plane strain increment ǫ 33 and in-plane stress increments σ ij, i,j = 1, 2. Three specific variables in GTN model are calculated in each increment using the iterative solution method discussed in this section. These are increments of the void volume fraction f, effective matrix materials plastic strain ǫ p 0 and the out of plane stress σ 33. For calculating the three variables, three condition functions are needed. The three equations that should be solved for these variables are derived from the yield condition, equivalence statement of plastic work and the plane strain condition respectively. These are discussed next. 1. Effective yield condition ˆΦ = 0, for the homogenized porous material discussed in equation (2.28). 2. Equivalence of plastic work in the matrix material to that in homogenized porous material as discussed in equation (2.35). The associated flow rule of equation 24

43 (2.27) may be written in terms of the effective Von-Mises stress q and the hydrostatic pressure p as: ǫ p ij ˆΦ = λ q n ij 1 ˆΦ λ 3 p δ ij and ǫ p ˆΦ kk = λ p (2.36) where δ ij is the Kronecker delta, n ij = q σ ij. The incremental form of the plastic work equivalence equation (2.35) then becomes: (1 f)σ 0 ǫ p 0 = σ ij ǫ p ij ˆΦ ˆΦ = λ p + λ p q q (2.37) Combining equations (2.31), (2.32), (2.33) with the second of equations (2.36), expresses the flow parameter λ as: λ = flocal A( ǫ p 0) ǫ p 0 (1 f) ˆΦ p (2.38) Finally substituting equation (2.38) in equation (2.37) and (2.38) results in a residual form of the equivalent plastic work: R( ǫ p 0, f local, σ 33 ) = (1 f)σ 0 ǫ p + p flocal A ǫ p + 3q flocal A ǫ p (1 f) ˆΦ q ( ˆΦ p (1 f) ) = 0 (2.39) If σ ij and f local are known, equation (2.39) can be used to determine the increment of plastic strain ǫ p 0. 25

44 3. Generalized plane strain condition, with a specified out of plane normal strain increment, i.e. ǫ 33 = ǫ e 33 + ǫ p 33 = ˆǫ constant (2.40) Substituting the expression for ǫ 33 from the generalized Hooke s law into equation(2.40) yields the residual function in terms of σ 33, i.e. G( ǫ p 0, f local, σ 33 ) = 1 E ( σ 33 v σ 11 v σ 22 ) (2.41) + λ( 1 Φ 3 p + 3 Φ 2q q s 33) ˆǫ constant = 0 (2.42) The stress increment σ 33 can be evaluated from equation (2.42), provided σ ij, i,j = 1, 2, f local and ǫ p 0 are known apriori. The above equations may be solved at each integration point of the element corresponding to known values of stress increments σ ij and the function for matrix flow stress σ 0 ( ǫ p 0). The integration method, based on Regula Falsi iteration algorithm, is summarized next. 1. Initialize: For iteration step i = 1, set f local (i) = 0, ǫ p(i) 0 = 0 and σ i 33 = Evaluate ˆΦ. If ˆΦ > 0 then go to step 3 for the Regula Falsi scheme. Otherwise, go to step Start iteration with i = i + 1. The iteration will continue till the yield function ˆΦ( f local (i 1), ǫ p(i 1) 0, σ (i 1) 33 ) Tolerance 1. Set f local (i) = f local (i 1), ǫ p(i) 0 = ǫ p(i 1) 0 and σ i 33 = σ (i 1)

45 Keeping ǫ p(i) and σ i 33 unchanged, assess two increments of void local (i) local (i) volume fraction ( f1 and f2 ( f [0,f f ]), such that local (i) ˆΦ 1 ( f1, ǫ p(i) 0, σ33 i }{{} Fixed ) ˆΦ local (i) 2 ( f 2, ǫ p(i) 0, σ33 i ) < 0 }{{} Fixed This corresponds to two opposite signs for the yield functions. Using the first of the two line searches as shown in figure 2.2(a), obtain the incremental void volume fraction as the intercept on the ˆΦ = 0 line, i.e. f local (i) = f local(i) 1 Set j = 1, ǫ p(j) 0 = ǫ p(i) 0 and σ j 33 = σ i 33. ˆΦ 1 local (i) ˆΦ 1 ˆΦ ( f1 f 2 local (i) 2 ) For a given f local(i), start iteration with j = j + 1. The iteration will continue as long as the plastic work equation: R( f local(i), ǫ p(j 1) 0, σ j 1 33 ) > Tolerance 2. Set ǫ p(j) 0 = ǫ p(j 1) 0 and σ j 33 = σ (j 1) 33. For the known value of f local (i), assess two increments of effective plastic strain ( ǫ p(j) 0(1) and ǫp(j) 0(2) ) such that R 1 ( f local (i), ǫ p(j) 0(1), σj 33) R 2 ( f local (i), ǫ p(j) 0(2), σj 33) < 0 Use the second of the two line searches as shown in figure 2.2(b), obtain the incremental effective plastic strain as the intercept on the R = 0 line, i.e. ǫ p(j) 0 = ǫ p(j) 0(1) R 1 R 1 R 2 ( ǫ p(i) 0(1) ǫp(i) 0(2) ) 27

46 Set k = 1, σ k 33 = σ j 33. With f local (i) and ǫ p(j) 0 known from the previous steps, start iteration with k = k + 1. The iteration will continue as long as the residual function of generalized plane strain condition in equation (2.42 G( f local (i), ǫ p(j) 0, σ (k 1) 33 ) > Tolerance 3. Use the Newton-Raphson method iteration method to determine σ k 33 as: σ33 k = σ (k 1) 33 G( flocal (i), ǫ p(j) 0(1), σ(k 1) 33 ) G σ 33 ( f local (i), ǫ p(j) 0(1), σ(k 1) 33 ) Set σ j 33 = σ33 k and update { R1 = R( f local (i), ǫ p(j) 0, σ33), j ǫ p(j) 0(1) = ǫp(i) 0, if R > 0 R 2 = R( f local (i), ǫ p(j) 0, σ33), j ǫ p(j) 0(2) = ǫp(i) 0, if R < 0. Set ǫ p(i) 0 = ǫ p(j) 0, σ i 33 = σ j 33 and update. {ˆΦ1 = ˆΦ( f local (i), ǫ p(i) 0, σ 33), (i) local (i) f1 = f local (i), if ˆΦ > 0 ˆΦ 2 = ˆΦ( f local (i), ǫ p(i) 0, σ 33), (i) local (i) f2 = f local (i), if ˆΦ < 0 4. Calculate ǫ e ij from equation (2.25), ǫ p ij from the associated flow rule (2.36) and ǫ ij by integrating equation (2.26). An implicit backward Euler time integration scheme is used in the numerical implementation of the stress based VCFEM. It requires an iterative solution process in which tangent operators are obtained through linearized forms of constitutive equations. Solving the linearized equation (2.21) requires evaluation of the instantaneous linearized compliance tensor, defined as the variation of total strain caused by the 28

47 Φ R Φ i 1 R 1 Φ i 0 (i) f 1 f (i) f (i) 2 f 0 R i p [ ε 0 ] 1 ε 0 p(i) p [ ε 0 ] 2 ε 0 p Φ i 2 R 2 (a) (b) Figure 2.2: Line search in the Regula Falsi algorithm for (a) yield function ˆΦ( f local ) and (b) equivalent plastic work R( ǫ p(i) 0 ) variation of the stress. These are evaluated at the end of the step between n and n + 1 as: S ijkl = dǫ ij dσ kl (n+1) The method introduced by Aravas in [69] is used to compute S ijkl = D 1 ijkl. The procedure requires the evaluation ǫ p = λ( Φ p ) (n+1), ǫ q = λ( Φ q ) (n+1) from equation (2.36) with known values of f, ǫ p 0 and ǫ 33 and the flow parameter λ from equation(2.38). It should be noted that the compliance matrix S is positive definite as long as the tangent modulus D is positive definite. Both generalized plane strain and plane stress conditions have been solved by the above algorithm. For plane stress problem, ǫ 33 is updated as: ǫ 33 = v E ( σ 11 + σ 22 ) + ǫ p 33 29

48 2.7 Aspects of Numerical Implementation Special care should be exercised on the numerical stability of the various matrices developed in the VCFEM formulation. Proper conditioning of matrices [H m ] and [H c ] to ensure their invertability, accurate numerical integration of [H] and [G] matrices over their respective domains, judicious choice on the number of stress parameters β m and β m to avoid rank insufficiencies in the stiffness matrix [K e ] and suppression of rigid body modes in the deformation field of the interface, all contribute considerably toward enhancing the accuracy and convergence of the VCFEM solutions. Some of the essential features of numerical implementation are discussed next Numerical integration in inclusion and matrix phases of the Voronoi cell element Numerical integration of the [H] & [G] matrices is performed by subdividing the matrix domain Ω m and the inclusion domain Ω c into quadrilaterals and triangles respectively and applying quadrature formula for corresponding domains. To account for the effect of the high gradients in the stress functions due to the reciprocal terms in equation (2.11) a integration discretization is induced in the matrix domain near the interface. Discretization of the inclusion and matrix domains for domain integration is conducted using the following steps. The essential steps involved in the scheme are: The physical domain is first transformed into a domain that is convenient for discretization. For example an elliptical inclusion of the form x2 a 2 + y2 b 2 = 1 30

49 is mapped into a unit circle as ψ 2 + γ 2 = 1, as shown in figure 2.4. The transforming coordinates for the entire Voronoi cell element are ψ = x a, γ = y b. In each transformed cell, each node on the interface (denoted by ) is projected to the boundary of the Voronoi cell along the radial direction. Also each node on the Voronoi cell boundary (denoted by ) is projected to the interface along the radial direction in the transformation domain. All the corresponding projected nodes are denoted by shown in figure 2.4(b). The dashed lines join the nodes as well as the projected points on the interface and cell boundary. The undamaged inclusion domain Ω c is subdivided into triangles in the transformed domain by joining the nodes on the interface to its centroid as shown in 2.4(b). For the matrix domain Ω m, 5 points are identified on each line joining the nodes (denoted by ) with the projected nodes (denoted by ) in the transformed domain. The points are denoted by and a number in figure 2.4(c). The points are positioned such that f 2(M+N+1) i+1 2f 2(M+N+1) i, where i = 1 on the interface and i = 5 on the cell boundary. Points denoted by with the same number on adjacent lines are then joined to create graded regions of 4 quadrilateral marked with roman numerals, as shown in figure 2.4(c). The resulting discretization of a Voronoi cell element is shown in figure 2.4(c). Integration is performed in each subdomain using appropriate Gaussian quadrature rule. The number of Gauss points is determined from the order of the 31

50 integrand in [H]. From equations(2.10,2.8) in equation(2.22), the highest polynomial order for matrix [H c ] is O(2(M 2)) [74]. For triangular subdomains in Ω c, the number of Gauss points is chosen to be n gauss = M 1 from [75]. The [H m ] matrix in Ω m comprises of both polynomial and reciprocal terms. Since the highest gradients are exhibited by the reciprocal terms near the interface, larger number of integration points should concentrate in this region. Substituting equations (2.11 and 2.13) in equation (2.22), the [H m ] matrix is seen to have a maximum of O( 1 f 2(M+N+1)) ). Based on these analysis, the number of integration points in each subdomain is estimated. In the radial direction, n r gauss = 5 for subdomain I and n r gauss = 3 for subdomains II, III and IV. In the tangential direction, n θ gauss = 5 for subdomain I and n θ gauss = 4 for subdomains II,III and IV. This results in 61 integration points for each region between two radial lines in the matrix Rank sufficiency of [K e ] A feature that adversely affects the accuracy of many elements in FEM is the presence of spurious zero energy or hourglass modes in the element stiffness [76, 77]. The internal energy of each element stiffness matrix vanishes for rigid-body deformations imposed on its boundary. However, certain non-rigid body modes can contribute zero energy to the element causing the resulting stiffness matrix to become rank deficient. For a Voronoi cell element, the internal energy IE e corresponding to the linearized stiffness matrix in equation (2.25) written from equation (2.21) as: IE e = 1 2 {dβm } T [H m ]{dβ m } + {dβ c } T [H c ]{dβ c } (2.43) 32

51 where {dβ} = [H m ] 1 ([G e ]{dq} [G cm ]{dq }) and {dβ c } = [H c ] 1 [G cc ]{dq } For positive definite matrices [H m ] and [H c ], the internal energy is zero only if each of the stress coefficients are individually zero. Setting {dβ c } to zero it can be seen that the contribution of the inclusion domain is zero if and only if, [G cc ]{dq } = 0 If the number of nodal degrees of freedom on the interface are denoted by n q (corresponding to the number of columns in [G cc ]) and the number of stress coefficients in the inclusion are denoted by n β c (corresponding to the number of rows in [G cc ]), then the necessary conditions for [G cc ]{dq } = 0 for no more than the 3 in-plane rigid body modes are: n β c 2n q 3 (2.44) Furthermore, for a rigid body displacement {dq } on the interface, the resulting internal energy IE e = 1 2 {dβm } T [H m ]{dβ m } will be zero only if [G e ]{dq} = 0 (2.45) Following an argument similar to equation (2.44), the necessary conditions for equation (2.45) to hold for no more than the 3 in-plane rigid body modes is: n β m 2n q 3 (2.46) where n q are the nodal degrees of freedom on the Voronoi cell element boundary and n β c is the number of stress coefficients in the matrix phase. These two conditions (

52 and 2.46) provide the necessary number of stress coefficients in terms of the number of nodes on the inclusion and cell boundaries in order to suppress hourglass modes in the element stiffness. Sufficiency condition is ensured by numerically ascertaining the invertability of the [H m ] and [H c ] matrices in the VCFEM implementation Constraining rigid-body modes at the interface In the VCFEM formulation, interface nodes are not in general topologically connected to nodes on the element boundary. Consequently, it is necessary to constrain rigid body modes on the interface to be consistent with those on the element boundary. This is achieved through singular value decomposition (SVD) of the matrices [G e ], [G cc ] for Voronoi cell elements to satisfy the discrete L-B-B stability conditions [78]. This condition may expressed as (see [62]) [G]{Q} = [U][λ][V]{Q} = [U][λ]{Q } 0, Q Q rb = (2.47) where Q rb correspond to the rigid body modes of displacement. The matrices [U] and [V], whose columns are the eigenvectors of [G][G] T and [G] T [G] respectively, are orthonormal matrices obtained by singular value decomposition of [G]. [λ] is a rectangular matrix with positive entries on the diagonal corresponding to the square roots of the non-zero eigenvalues of both [G][G] T and [G] T [G]. Positive singular values of [λ] imply that the strain energy associated with the stress field solution for non-rigid body displacement fields is strictly non-zero. The zero singular values in the diagonal of the [λ] matrix correspond to spurious modes resulting from the disparate rigid body modes at the interface and element boundary and these should be constrained. A simple procedure of constraining of selected displacements based on the singular value decomposition of [G mc ] or [G cc ] is carried out. The procedure 34

53 involves re-writing the matrix multiplication as: [G]{q } = [U][λ][V]{q } = [U][λ]{q } alt = [G] alt {q } alt (2.48) Elements in {q alt } corresponding to small eigenvalues in [λ] are pre-constrained to a zero value. Consequently, equation (2.25)is solved for the constrained displacement fields as N [ e=1 K e V T K T mc K mc V 1 V T K cc V 1 ]{ dq dq } i = { fe V T f e } (2.49) 2.8 Error Measure, Adaptivity and Convergence Rate Adaptivity and convergence rate of VCFEM has been studied in [62], where an adaptive methodology has been developed in [62] to enhance convergence characteristics and accuracy of the VCFEM for analyzing heterogeneous material response. Two error measures are introduced and developed to measure the quality of the VCFEM solution, viz.: Traction Reciprocity Error, which is derived a posteriori from the element and interface traction discontinuity in the finite element solution. The average traction reciprocity error for the entire model has been explained in [62] as Average T raction Reciprocity Error (AT RE) = Ne e=1 ee T + N c c=1 ec T N e + N c (2.50) where e e T and ec T are the traction reciprocity error on each boundary and interface segment respectively, and N e and N c are the total number of segments on the element boundary and interface. Strain Energy Error, which may be related to the error in kinematic relationships that is satisfied in a weak sense in each element. The average error in the strain 35

54 energy is calculated in [62] as Average Strain Energy Error (ASEE) = N e=1 (e SE u ) 2 N (2.51) where e SE u is the strain energy error in each element. 2.9 Numerical Examples for Validating the VCFE Model The validation of the Voronoi cell finite element model for elastic-porous plastic heterogeneous materials is done through two sets of examples. The first set compares VCFEM results with those generated by the commercial code ABAQUS. The second set evaluates the convergence rates of the VCFEM solution while the third set compares with experimental results for checking the stress in particles of actual microstructure Comparison with results using ABAQUS The VCFEM results for a local GTN model without void coalescence are compared with those generated by ABAQUS commercial code. The matrix material is assumed to be a ductile Al-3.5% Cu alloy with elastic-plastic behavior with material properties: Young s modulus E = 72GPa, Poisson s ratio v = 0.32; post yield behavior σ m = σ 0 (ǫ p m/ǫ 0 + 1) N, with σ 0 = 175MPa, and N = 0.2. Here ǫ 0 = σ o /E is the uniaxial strain at yield. The brittle inclusion is assumed to be of SiC, with elastic properties: Young s modulus E = 450GPa and Poisson s ratio v = The void nucleation parameters are chosen as ǫ N = 0.1, s N = 0.1 and f N = The problems are solved with two values of initial void volume fraction, viz. f in = and f in =

55 As shown in figure 2.5, two representative volume elements in the microstructure are considered. For both the RVE s, the inclusion volume fraction is V f = 20%. Periodicity conditions are enforced on the surface y = L/2 by constraining it to remain horizontal and straight, and the analysis is conducted under plane strain conditions. The RVE s are loaded in uniaxial tension to a macroscopic tension strain ǫ xx = 2.0%L. The VCFEM model for the square unit cell in figure 2.5(a) consists of a single element with 12 nodes on matrix boundary and 8 nodes on matrix-inclusion interface. The inclusion stress function is generated using 25 polynomial terms (6th order polynomial stress function). The matrix stress function has an additional 36 reciprocal terms due to the inclusion shape (3 reciprocal terms for each polynomial exponent from 2 to 4, i.e. i = p + q p + q + 2 p + q = 2 4). The corresponding ABAQUS mesh is made up of 9828 bilinear QUAD-4 elements. For the RVE with 5 inclusions, the VCFEM mesh is shown in figure 2.5(b). The inclusion stress function is generated using 33 polynomial terms (7th order polynomial stress function). The matrix stress function has an additional 36 reciprocal terms due to the inclusion shape (3 reciprocal terms for each polynomial exponent from 2 to 4, i.e. i = p+q p+q+2 p+q = 2 4) for each element. In this case, the ABAQUS mesh has bilinear QUAD-4 elements. The volume averaged stress-strain response and the average void volume fractionstrain are plotted in figures 2.6 and 2.7 respectively. Excellent match is observed in these results with the two approaches. The efficiency of VCFEM in comparison with conventional FEM codes for the two problems is demonstrated in table 2.1. On an 37

56 IBM System Cluster 1350 with single 2.66HZ AMD Opteron processor, the VCFEM simulations are approximately 200% faster than the ABAQUS simulations. Table 2.1: Comparison of simulation times by the two approaches ABAQUS simulations (sec.) VCFEM simulations (sec.) RVE (a) with f in = RVE (a) with f in = RVE (b) with f in = RVE (b) with f in = Convergence rate of VCFEM simulation The convergence rate of the VCFEM results are examined with respect to the two error measures, viz. (i) average traction reciprocity error or ATRE and (ii)average strain energy error or ASEE, mentioned in section 2.8 and introduced in [62]. The problems in the previous section with the two VCFEM meshes shown in figure 2.5 are considered for this test. The starting values of the degrees of freedom, corresponding to both stress and nodal displacements, are the same as in the previous section. In particular, the total degrees of freedom (D.O.F.) corresponds to the sum of the total displacement degrees of freedom at the interface and element boundary and the number of β parameters, i.e. D.O.F. = 2 N nodes + N β. Figures 2.8 and 2.9 plot the ATRE and ASEE as functions of the inverse of (D.O.F.) for the RVE s of figures 2.5(a) and (b) respectively. For the RVE (a), a total of 85% change in the traction reciprocity error (AT RE) results with a 39% increase in D.O.F.. Also 38

57 the strain energy error (ASEE) is found to drop from 2.76% to 0.65% by enriching the polynomial terms in the element stress function from 6-th order to 9-th order. Linearized forms of the errors as functions of the inverse of D.O.F. are shown of the respective figures. For the RVE (b), a maximum ATRE change of 65% results from a 13% increase in D.O.F.. The ASEE drops from 1.48% to 0.44% by enriching the polynomial terms of stress function in each element from 6-th order to 10-th order. The high accuracy of VCFEM is confirmed by the near linear convergence rates Comparison of inclusion stresses obtained by Raman spectroscopy In this example, experimentally extracted stresses in Si particles of a cast aluminum alloy A356-T6 microstructure are compared with VCFEM simulations. The experimental stresses in the loading direction are evaluated by micro-raman spectroscopy (RS) executed by Harris et. al. [79, 80], in which micro-raman spectroscopy to determine the state of stress of a Si wafer at its surface. Micro-Raman spectroscopy has been widely used to measure local stresses in silicon by taking advantage of the polarization and intensity of the Raman-scattered light. The Raman shifts are calculated using previously published values for silicon phonon deformation potentials. A micrograph of the A356-T6 cast alloy, for which the stresses are determined is shown in figure 2.10(a). Two eutectic Si particles are singled out for interrogation. Experimental results indicate that there are residual stresses in particles due to heat treatment. Since residual stresses are not considered in the VCFEM simulation, the residual stresses are subtracted from the experimentally acquired data shown in figure As shown in figure 2.10(b), the sample is loaded in the y direction by an applied strain. The particle stresses increase linearly at the initial stages of loading 39

58 due to elastic deformation. However, the stress increments reduce substantially with straining after matrix yield due to plastic deformation in the matrix. For the VCFEM simulation, the aluminum matrix is modeled as an elastic-plastic material with properties: Young s modulus E = 72GP a, Poisson s ratio ν = 0.33, and the post yield behavior is represented by Ramberg-Osgood law (σ m = σ 0 (ǫ p m/αǫ 0 ) 1/n ). The initial flow stress of the matrix σ 0 = 200MPa, ǫ 0 = σ 0 /E, α = 3/7 and the strain hardening exponent n = The Si particles are assumed to be linear elastic with properties: Young s modulus E = 170GPa, Poisson s ratio v = The simulation is conducted with plane strain assumptions. Figure 2.11, compares the simulated stresses with the experimental stresses. Table also shows the slope σyy ǫ yy by the two methods, where σ yy is the average stress inside a Si particle and ǫ yy is the average strain in the specimen. The VCFEM results concur well with the experimental results. Stress in the particle #1 is larger than that in particle #2. The corresponding nearest neighbor distance (NND) values in the table 2.9.3, indicate smaller NND s result in larger particle stresses. Thus, stresses in clustered regions can be significantly higher for the same size of particles. Experimental results (Pa) VCFEM results (Pa) NND (µm) Particle # Particle # Average Table 2.2: Stress-strain slope σyy ǫ yy in the elastic regime 40

59 2.10 Conclusions A theoretical framework for the development of VCFEM for nonlocal porous plasticity in the absence of microstructural damage has been presented in this chapter. Assumed stress hybrid variational principle used by VCFEM, element assumptions, constitutive relationships for nonlocal porous plasticity, strain update algorithm, and solution methodology for the resulting weak form and numerical aspects of VCFEM implementation are discussed. The effectiveness of VCFEM to model 2-dimensional porous plasticity in undamaged heterogeneous media has been presented. Results indicate that the VCFEM for nonlocal porous plasticity provides an accurate representation of the deformation in such materials at macroscopic and microscopic levels. In the next chapter, VCFEM is extended to model damage in random heterogeneous materials for nonlocal porous plasticity. 41

60 σ ij, f, ε, p 0 σ ij Φ<0? No Yes End local 1 f 1,... local Φ ( ) Φ 2 ( f 2,... )< 0 First Line Search f local = f 1 local Φ 1 ( ) Φ 1 Φ 2 f 1 local f 2 local R 1 (,... ) R 2 ( ε p 0(2),... ) < 0 ε p 0(1) Second Line Search ε p 0 R 1 = ε p 0(1) ( ε p 0(1) ε p 0(2) ) R 1 R 2 Solve the G( σ 33,... ) function with Newton Raphson Method to calculate σ 33. R( f local,, ) <Tolerance 2? ε p 0 σ 33 No R = R, ε p 0(1) =, if R > 0 1 ε p 0 R = R 2, ε p 0(2) = ε p 0, if R < 0 Yes Φ 1 = Φ, f 1 Φ 2 = Φ, local f 1 local = f local = f local, if, if Φ > 0 Φ < 0 No Φ ( f local, ε p 0, σ 33 ) <Tolerance 2? Yes End Figure 2.3: The flow chart of the strain update algorithm for VCFEM 42

61 Y X (a) (b) I II III 4 IV 5 5 (c) (d) Figure 2.4: (a) Physical domain, (b) transformed domain in the coordinates (ψ, γ), (c) Subdivision of quadrilaterals in the matrix (d) subdivision for a typical undamaged Voronoi cell. Shaded regions are used in the particle domain 43

62 Periodic Boundary y L L x L (a) L (b) Figure 2.5: Two different RVE s and the corresponding Voronoi cell meshes: (a) unit cell with a single circular inclusion, (b) square domain with five circular inclusions. 44

63 350.0 Macroscopic Tensile Stess (MPa) ABAQUS, f in = ABAQUS, f in =0.01 VCFEM, f in = VCFEM, f in = Macroscopic Tensile Strain (%) (a) Macroscopic Tensile Stress (MPa) ABAQUS, f in = ABAQUS, f in =0.01 VCFEM, f in = VCFEM, f in = Macroscopic Tensile Strain (%) (b) Figure 2.6: Comparison of volume-averaged stress-strain response for: (a) RVE (a) in figure 2.5(a), (b) RVE (b) in figure 2.5(b). 45

64 0.015 Average Void Volume Fraction ABAQUS, f in = ABAQUS, f in =0.01 VCFEM, f in = VCFEM, f in = Macroscopic Tensile Strain (%) (a) Average Void Volume Fraction ABAQUS, f in = ABAQUS, f in =0.01 VCFEM, f in = VCFEM, f in = Macroscopic Tensile Strain (%) (b) Figure 2.7: Average void volume fraction vs. macroscopic strain for : (a) RVE (a) in figure 2.5(a), (b) RVE (b) in figure 2.5(b). 46

65 ATRE= *(1/(Total DOF)) ASEE= *(1/(Total DOF)) A.T.R.E A.S.E.E. (%) /(Total D.O.F.) (a) /(Total D.O.F.) (b) Figure 2.8: VCFEM convergence rate for the RVE of figure 2.5(a) : (a) Average traction reciprocity error (AT RE) and (b) Average strain energy error (ASEE), plotted as functions of the total degrees of freedom ATRE= *(1/(Total DOF)) 1.50 ASEE= *(1/(Total DOF)) A.T.R.E A.S.E.E. (%) /(Total D.O.F.) (a) /(Total D.O.F.) (b) Figure 2.9: VCFEM convergence rate for the RVE of figure 2.5(b) : (a) Average traction reciprocity error (AT RE) and (b) Average strain energy error (ASEE), plotted as functions of the total degrees of freedom. 47

66 Fy 1 1 Y 2 10µm Y 2 X X Figure 2.10: (a) A micrograph of cast aluminum alloy A356-T6 showing two Si particles interrogated for stresses by Raman spectroscopy; (b) the VCFEM mesh and model for a part of the microstructure σ yy in particles (MPa) RS: Particle #1 RS: Particle #2 VCFEM: particle #1 VCFEM: particle #2 Macroscopic: A356-T Macroscopic Tensile Strain (%) Figure 2.11: Comparison of volume averaged stress σ yy in the two Si particles as a function of the macroscopic applied strain by VCFEM simulation with those by Raman spectroscopy. 48

67 CHAPTER 3 VCFEM FOR INCLUSION CRACKING IN A POROUS PLASTIC MATRIX 3.1 Introduction The Voronoi cell FEM has been successfully applied to the simulation of brittle cracking of inclusions contained in ductile matrix materials in [2, 4]. Instantaneous splitting of the inclusions is assumed, thereby avoiding the problem of crack propagation inside the inclusion. The crack in the inclusion is represented as an elliptical void with a high aspect ratio a/b = 10. Studies in [2] has found this representation to be near optimal with respect efficiency and accuracy. Subsequently, each Voronoi cell element may consist of three phases, viz. the matrix phase Ω m, the inclusion phase Ω c and the crack phase Ω cr, as shown in figure 2.1. Initiation of an inclusion crack is assumed to be triggered by a Weibull criterion that is based on the local principal stresses, as well as the size of the inclusion. In the model, the probability of fracture F fr for a particle of volume v is: F fr = 1 exp[ v v 0 ( σc I σ w ) m ] (3.1) 49

68 where m and σ w are the Weibull modulus and the characteristic strength respectively, v 0 is a reference volume, and σi c is the maximum principal stress at a point in a particle. The Weibull model, widely used in the literature [56, 81, 82, 59, 83, 50, 84, 85] for predicting brittle particle fracture in composites, is based on weakest link statistics. The criterion suggests that the critical stress or fracture toughness for particle fracture is not only material dependent, but is also influenced by the particle size due to the existence of flaws. Larger particles with large initial flaws will fracture at smaller critical stresses. Implementation of this damage criterion in VCFEM follow the steps outlined next. The crack initiation probability F fr is evaluated for all integration points inside the inclusion phase of each Voronoi cell element. Once F fr reaches a predetermined critical value F critical at an integration point, the element topology is altered by introducing an elliptical crack phase in the inclusion to represent splitting. In the computational procedure, more than one point may satisfy the fracture criterion during increment. Consequently, the location of the crack is determined by a weighted averaging method as: x crack = x σc I (x,y) / σi c(x,y) [F cr > F critical ] σ w σ w y crack = y σc I (x,y) / σi c(x,y) [F cr > F critical ] (3.2) σ w σ w Here σi c (x,y) correspond to the values of maximum tensile principal stress for integration points at which the probability of fracture F fr F critical in the particle. The orientation of the major axis of the elliptical crack is normal to the principal stress directions at (x crack,y crack ). The crack extends to the interface on both sides. 50

69 A high aspect ratio ellipse (a/b = 10) with 12 nodes on the boundary is added to each Voronoi cell element at the onset of inclusion cracking. Shown in figure 3.1, the ellipse extends slightly into the matrix to represent complete splitting. At the crack tip, three additional nodes are generated on the original particle boundary Ω c such that there is no load transfer between the two cracked parts of the particle. The new topology of each Voronoi cell element requires a new set of integration points. Variables at these new integration points are mapped from the precracking integration points. Figure 3.1: The Voronoi cell mesh with a crack, and the location of nodes on a crack tip 3.2 Modified VCFEM Formulation to Account for Cracked Inclusions In addition to the element displacement and stress fields in section 2.2, the displacement field u on the crack boundary Ω cr is now a solution variable the VCFEM 51

70 formulation. The energy functional (2.2) is modified to accommodate the inclusion crack phase as: Π IC e ( σ m, σ c, u, u, u ) = Π ND e ( σ m, σ c, u, u ) (σ c + σ c ) n cr (u + u )d Ω(3.3) Ω cr and the domain energy functional being Π IC = N e=1 ΠIC e. The superscript cr corresponds to variables on the crack boundary and n cr the outward normal on the crack face. The first variation of equation (3.3) may then be expressed as δπ IC e = δπ ND e ( σ m, σ c, u, u ) δ σ c n cr u d Ω (σ c + σ c ) n cr δ u d Ω (3.4) Ω cr Ω cr The first variation of Π IC e with respect to u yields the additional zero traction condition on the crack boundary Ω cr, i.e. (σ c + σ c ) n cr = 0 on Ω cr (3.5) With the addition of the extra crack phase in a Voronoi cell element, stress functions and their associated stress fields are automatically adjusted to account for the altered topology. The crack boundary Ω cr is parametrically as a function f cr (x,y) = 1, through the conformal mapping technique discussed in section 2.3. Again f cr exhibits the property: f cr as (x,y). The corresponding stress functions in equation (2.9 are automatically updated for the matrix and particle phases as: Φ m = Φ m poly + Φm rec + Φ cm rec ; Φ c = Φ c poly + Φcc rec (3.6) 52

71 Additionally, the displacement field u is interpolated from the corresponding values { q } at the nodes on the crack surface as: { u } = [L cr ]{ q } (3.7) Here [L cr ] is taken as a quadratic interpolation matrix. The weak form and the solution procedure are similar to those discussed in section 2.4 and details have been provided in [4]. With the introduction of the crack phase in a Voronoi cell, and new nodes on the inclusion boundary Ω c, The numerical integration scheme discussed in section should be augmented. Additional integration points need to be incorporated in the matrix near the crack tip to capture sharp stress gradients. Also, the integration scheme in the inclusion is changed to one similar to that for the matrix domain integration to accommodate the crack phase. Subdivision of the quadrilaterals for the integration points are based on the mapping function f cr on the crack boundary. The introduction of the new integration points require mapping of evolving variable like stress, strain and other state variables from neighboring old integration points. The superconvergent patch recovery method [86] is implemented, with local patches comprising of the closest 25 integration points, for this variable mapping. 3.3 Numerical Examples on VCFEM with Inclusion Cracking Two numerical examples are solved in this section. In the first example, the VCFEM formulation for a particle with a crack is validated by comparing with results from ABAQUS simulation. The other example, is intended to present a methodology 53

72 for calibrating Weibull parameters by comparing VCFEM simulations with experimental results A square RVE with a single cracked inclusion A square representative volume element containing a circular cracked inclusion is simulated for tensile straining under plane strain conditions. The material properties are the same as those used in example Two initial void volume fraction are chosen, viz. f in = and f in = A single Voronoi cell element with 14 nodes on matrix boundary, 12 nodes on matrix-inclusion interface and 12 nodes on crack boundary shown in figure 3.2(a)is used. The inclusion stress function is generated using 25 polynomial terms (6th order polynomial stress function) and 36 reciprocal terms due to the crack. The matrix stress function has an additional 36 reciprocal terms due to the inclusion shape (3 reciprocal terms for each polynomial exponent from 2 to 4, i.e. i = p + q p + q + 2 p + q = 2 4). The corresponding ABAQUS mesh, made up of 9665 QUAD elements, is shown in figure 3.2(b). The volume averaged stress-strain response in figure 3.3 shows a good match between VCFEM and ABAQUS results. The microscopic equivalent plastic strain distribution, shown in figure 3.4, are also in good concurrence Calibration of Weibull parameters for particle cracking As mentioned earlier in this section, the Weibull probability distribution function is used as a criterion to trigger cracking of inclusions within a Voronoi cell element. The equation (3.1) may be modified to yield the volume fraction of fractured inclusions, corresponding to a known distribution p(v) of their volumes in the 54

73 L Y L X Y Z X (a) (b) Figure 3.2: A square RVE with a circular cracked inclusion of volume fraction v f = 20%: (a) The Voronoi cell element with boundary conditions, (b) ABAQUS mesh. 200 Macroscopic Tensile Stess (MPa) ABAQUS: f in = ABAQUS: f in =0.01 VCFEM: f in = VCFEM: f in = Macroscopic Tensile Strain (%) Figure 3.3: Volume averaged macroscopic stress-strain response for the RVE with a cracked inclusion. 55

74 (a) (b) Figure 3.4: Contour plot of equivalent plastic strain with the two models: (a) ABAQUS, and (b) VCFEM. microstructure as: ρ(v ) = Vmax V min p(v)f fr dv (3.8) where V min V max give the range of particle volumes. A discrete version of this equation is convenient for numerical calculations, and may be expressed as: ρ = N i=1 p(v i )(1 exp[ v i v 0 ( σi I σ w ) m ]) v i (3.9) where the probability density p(v i ) of particle volume v i (area in 2D) in the microstructure can be obtained from experimental micrographs. The entire range of particle volumes is divided into N intervals, such that for the i th interval v i = v i v i 1 (i = 1, 2,,N). σ i I is the average maximum principal stress in particles with sizes in the range of [v i 1, v i ]. 56

75 (a) (b) Y Y X (c) X (d) Figure 3.5: Scanning electron micrograph images of W319-T7 Aluminum alloy, showing different sections: (a) section 1, (b) section 2; and corresponding VCFEM mesh and model for these sections: (c) section 1, (d) section 2. (The dark shaded ellipses are Si particles, the white ellipses are copper-based intermetallics 57

76 The fraction of cracked particles ρ at any applied strain may be obtained from experimental micrographs of a heterogeneous material. Two micrographs of an Aluminum alloy W319-T7, shown in figure 3.5(a,b), are used to calibrate the Weibull parameters σ w and m in this section. The square sections are 81.2µm 81.2µm in dimension. The matrix phase is aluminum and the inclusion phase is eutectic silicon particles in this aluminum alloy. In addition, there is a small volume fraction of brittle, copper-based intermetallics. The silicon particles are modeled with the linear elastic properties: Young s modulus E Si = 170GPa, Poisson s ratio ν Si = 0.17, while the intermetallics have the elastic properties: Young s modulus E In = 128GPa, Poisson s ratio ν In = The aluminum matrix material is ductile and is modeled as porous material with properties: Young s modulus E Al = 72GPa, Poisson s ratio ν Al = 0.33, and the post yield behavior is represented by Ramberg-Osgood law (σ m = σ 0 (ǫ p m/αǫ 0 ) 1/n ). The initial flow stress of the matrix σ 0 = 200MPa, ǫ 0 = σ 0 /E, α = 3/7 and the strain hardening exponent n = The porosity or void volume fraction is assumed to be zero for this material. Two sets of Weibull parameters (σ w,m) characterize damage in the two inclusion materials in the alloy. The parameter m is assumed to be the same for both materials. However, the parameter σ w in the two materials is assumed to vary in the same ratio as the Young s modulus, i.e. (σ w) Si E Si = (σw) In E In. For the stress function, 33 polynomial terms (7th order polynomial stress function) and 36 reciprocal terms due to the inclusion shape (3 reciprocal terms for each polynomial exponent from 2 to 4, i.e. i = p + q p + q + 2 p + q = 2 4) is used for matrix. 33 polynomial terms are used for inclusions. With the introduction of 58

77 cracked inclusions, another 36 reciprocal terms are added to the matrix and inclusion stress functions. The material is subjected to tensile straining to a total strain of ǫ yy = 6.21%. All the simulations are conducted under plane strain condition. Histograms of number fraction, showing the distribution of particle area of the two micrographs are shown in figure 3.6. The particle area fraction (VF), total number of particles(np) and total number of cracked particles (NCP) at this strain for the two sections are tabulated in Table 3.1. For the equivalent microstructures of figures 3.5(c,d), VCFEM simulations are conducted to evaluate the maximum principal stress in the particles that have experimentally cracked at this strain. Important studies on particle cracking using the Weibull model have varied m between 1 and 8. Accordingly, the same variation is tested in this work and the calibration results are tabulated in table 3.2. Figure 3.7 shows the probability density of damaged particles for the two section micrographs with for different values of m. From figure 3.7(b), it is clear that better match between the simulation and experiment is obtained with m between 2 and 3, and a value m = 2.4 is selected for subsequent analysis. The other parameter (σ w ) is obtained by calibrating from VCFEM results involving no particle damage. The value obtained as the average of the two micrographs is (σ w ) Si = 0.625GPa. However, results may vary if particle cracking is also included in the simulations due to stress redistribution. VCFEM simulations with different values of σ w are shown in table 3.3, which lists the number of cracked particles for the two micrographs. The Weibull parameters that yield maximum agreement with experiments are: m = 2.4 and (σ w ) Si = 0.68GPa. Figure 3.8 shows the contour plots of the effective plastic strain and the distribution of the fraction of damaged 59

78 particles as a function of the particle size is given in figure 3.9. The comparison with experimentally observed plots show similar trends. However, there appears to be a small phase shift, especially for the micrograph 2, which may be attributed to the difference between the 2D simulations and 3D experimental observations. Micrograph # VF NP NCP % % Table 3.1: Experimentally observed (i) particle volume fraction (VF), (ii) total number of particles(np) and (iii) total number of cracked particles(ncp) for the two micrographs of W319-T7 at a total strain ǫ yy = 6.21% m micrograph 1 micrograph Table 3.2: Weibull parameter (σ w ) Si (GPa) calibrated from undamaged VCFEM simulations. 60

79 Number Fraction of Particles Particle Area (µm 2 ) 0.30 (a) 0.25 Number Fraction of Particles Particle Area (µm 2 ) (b) Figure 3.6: Number fraction of particle area for (a) section 1 (b) section 2 61

80 Probability Density of Damaged Particles Probability Density of Damaged Particles Exp. m=2 m=3 m=4 m= Particle Area (µm 2 ) (a) Exp. m=2 m=3 m=4 m= Particle Area (µm 2 ) (b) Figure 3.7: Comparison of experimental and simulation-based probability density of cracked particle volume with different Weibull modulus, for (a) micrograph 1 (b) micrograph 2 62

81 (a) (b) Figure 3.8: Contour plots of effective plastic strain for (a) micrograph 1, (b) micrograph 2 63

82 Probability Density of Damaged Particles Probability Density of Damaged Particles Experiment Simulation Particle Area (µm 2 ) (a) Experiment Simulation Particle Area (µm 2 ) (b) Figure 3.9: Distribution of number fraction of damaged particles as functions of the particle size for (a) micrograph 1, (b) micrograph 2 64

83 micrograph # Exp. σ 0 = σ 0 = σ 0 = 0.65 σ 0 = 0.66 σ 0 = Table 3.3: Number of cracked particles in experiments and by VCFEM simulation using different values of σ w. 3.4 Conclusions In this chapter, the VCFEM has been extended to model damage in random heterogeneous materials for nonlocal porous plasticity. The crack initiation criterion, Weibull model, is discussed first. Modifications to the variational principle to account for the new phase introduced in the form of a damaged particle have been discussed. Enrichment of stress functions due to the presence of the new phase are shown to extend naturally from its shape. Numerical examples to validate VCFEM for particle cracking in heterogeneous microstructures have been discussed in this chapter. Comparisons with traditional FEM establish the accuracy of VCFEM in modeling complex deformations in damaging materials. The calibration procedure for Weibull parameters is presented also. In the next chapter, VCFEM is locally enhanced by displacement based formulation to handle material softening, and finally simulate the ductile fracture in random heterogeneous materials. 65

84 CHAPTER 4 VCFEM WITH LOCAL ENHANCEMENT FOR MATRIX CRACKING 4.1 Introduction Ductile fracture in materials with dispersed inclusions often involves inclusion cracking, followed by localized matrix cracking due to void growth and coalescence. Localization in the bands or ligaments of intense void evolution results in a transition of the overall stress-strain response from hardening to the softening behavior with a negative stiffness. The assumed stress based VCFEM formulation discussed in the previous sections faces numerical instabilities in this regime due to non-uniqueness in strains or boundary displacements for given value of stresses. To avert such instabilities and also to provide a high resolution to the strain localization zone, regions of strain softening Ω s are adaptively augmented within each Voronoi cell element with a patch of high-resolution displacement based finite elements. A typical locally enhanced Voronoi cell finite element (LE-VCFE) is shown in figure 4.1. The material rapidly looses its load carrying capacity in Ω s due to a rapid growth of local void volume fraction. A finite deformation formulation for pressure dependent elasto-plastic materials govern the behavior of displacement based elements in Ω s. 66

85 Γ m Ω s Γ s e Γ um Ω s m Ω s m Γ c s UΓ s cr Crack Ω cr Ω m Ω s m Ω c Γ c s UΓ s cr Ω s m Γ tm Ω s Γ s e Figure 4.1: A locally enhanced Voronoi cell element with superposed displacement based elements A typical locally enhanced Voronoi cell element or LE-VCE may consist of four phases: the matrix phase Ω m, the inclusion phase Ω c, the inclusion crack phase Ω cr and a matrix region of localized strain softening Ω s preceding ductile failure. Ω s is the boundary Ω s with an outward normal n s. Ω s may consist of four distinct regions, depending on its overlap with pre-existing topological regions in the Voronoi cell element, i.e. Ω s = Γ e s Γ c s Γ cr s Ω m s. Here Γ e s = Ω e Ω s, Γ c s = Ω c Ω s and Γ cr s = Ω cr Ω s represent the intersections of Ω s with the element, inclusion and crack boundaries respectively. The aggregate of these common boundaries is represented as Γ s = Γ e s Γ c s Γ cr s. Once the displacement based local FE region Ω s is overlaid, a mapping procedure [86] is used to map local stresses, strains and material internal variables from the VCFEM domain to displacement elements in Ω s. Superscript s labels variables associated with Ω s. The mapping should also 67

86 guarantee displacement compatibility on the shared boundaries Γ s between the stress and displacement interpolated regions, as shown in figure 4.1, i.e. u on Γ e s u s = u on Γ c s u on Γ cr s 4.2 Criteria for Local Enhancement Subsequent to particle cracking, the void density increases rapidly in the vicinity of the crack tip due to high stress concentration, often resulting in localized plastic straining. The increased void growth rate further leads to void coalescence, which is associated with a post-peak strain softening. The peak is a bifurcation point, beyond which the solution becomes unstable with the stress-based VCFEM formulation. The Regula-Falsi integration algorithm of the GTN model in section 2.6 will not converge in this regime. Correspondingly, a local transition criterion is devised for opening the region of displacement interpolated elements Ω s in Ω e, as: Ω e Ω e \ Ω s Ω s if: q( ǫ p ) ǫ p 0 and ǫ p ǫ p critical (4.1) where q is the Von-Mises stress q = ( 3s 2 ijs ij ) 1/2 and ǫ p is the effective plastic strain. An enhanced region Ω s is opened if at least 5 neighboring integration points meet the criterion (4.1). As shown in figure 4.1, Ω s is larger than the actual region at which strain softening has set in. This is done to ensure that deformation on the boundary, i.e. Ω s conforms to that of the small strain region Ω e \Ω s at their common interface. This will also not require frequent superposition during the propagation of ductile fracture. 68

87 4.3 Constitutive Relations and Stress Update Algorithm A finite deformation element formulation is developed for the domain Ω s within each Voronoi cell element, following the framework given in [87]. The constitutive relations in Ω s are described in a rotated Lagrangian system in terms of the rotated Cauchy stress tensor σ s R and its conjugate, the incremental rotated strain tensor ǫs R [88]. The variables in the rotated system are defined as: σ s R = R T σ s R and ǫ s R = R T ǫ s R (4.2) where σ s is the Cauchy stress tensor and R is a proper orthogonal tensor representing pure rotation obtained from the polar decomposition of the deformation gradient F s = 0 x s i.e. R = F s U 1. The backward Euler integration algorithm has been shown to be unconditionally stable and independent of the smoothness of the yield function in [70]. Consequently, the Gurson Tvergaard-Needleman constitutive model for porous plasticity in Ω s is integrated using the backward Euler formulation using the algorithm presented in [69]. In the (n + 1)th increment between deformed configurations Ω n s and Ω n+1 s the rotated Cauchy stress is updated as: (σ s R) n+1 = (σ s R) n + (D ep R )n+1 : ( ǫ s R) n+1 (4.3) where (D ep R )n+1 is the elastic-plastic tangent stiffness tensor in the rotated configuration that is evaluated at the end of the (n + 1) th increment as: (D ep R ) ijkl = (σs R ) ij (ǫ s R ) (4.4) kl n+1 The integration algorithm in [69] solves the following set of non-linear equations in the rotated Lagrangian coordinates, that are derived from equations ( ). 69

88 Yield Function: ˆΦ(p,q,f, ǫ p ) = ( q σ 0 ) 2 + 2f q 1 cosh( 3q 2p 2σ 0 ) (1 + q 3 f 2 ) = 0 (4.5) Flow Equation: ǫ ˆΦ p q + ǫ ˆΦ q p = 0 (4.6) where p = p e +K ǫ p and q = q e 3G ǫ q are the hydrostatic stress and the equivalent stress respectively. ǫ p = Λ ˆΦ and ǫ p q = Λ ˆΦ q are the primary unknowns in the above equations. The additional variables in the above equations are defined in terms of ǫ p and ǫ q as: ǫ p = p ǫ p + q ǫ q (1 f)σ 0 (4.7) f = (1 f) ǫ p + A ǫ p (4.8) where σ 0 and A are functions of the microscopic equivalent plastic strain ǫ p. These equations are solved iteratively using the Newton-Raphson method with a known incremental strain tensor (ǫ s ij) n+1 = ( us i )n+1. Here u (n+1) corresponds to the x n+1 j displacement increments in the (n + 1) th increment. The rotated strain increment follows from equation (4.2) as ( (ǫ ij ) s R) n+1 = R n+1 ki (ǫ s kl) n+1 R n+1 lj (4.9) Initial and unchanged variables in the iteration process are: (σ s ij) n+1 R = (σ s ij) n R + (D e R) ijkl (ǫ s kl) n+1 R ǫ p = 0 and ǫ q = 0 p e = 1 3 (σs kk) n+1 R and q e = ((σs ij )n+1 R + pe )((σij s )n+1 R + pe )

89 where (D e R ) ijkl is the elastic stiffness. Equations (4.5) and (4.6) are solved iteratively for ǫ p and ǫ q till their corrections have reached a prescribed tolerance. The rotated stress is calculated from the hydrostatic and deviatoric components p n+1 and q n+1, after which the updated Cauchy stress in Ω n+1 s and the corresponding increment are then obtained through the tensor rotation formula (σ s ) n+1 = R n+1 (σ s R) n+1 (R n+1 ) T and (σ s ) n+1 = (σ s ) n+1 (σ s ) n (4.10) 4.4 Coupling Stress and Displacement Interpolated Regions in the Locally Enhanced VCFE Formulation With the addition of displacement elements in Ω s, the energy functional in equations (3.3, 3.4) should be augmented for the LE-VCFEM formulation. It should couple the small deformation assumed stress formulation in Ω e \ Ω s with the finite deformation displacement formulation in Ω s. For the softening region Ω s, all variables are referred to the current configuration. The corresponding incremental element energy functional Π e for (n+1)th increment is defined in terms of stresses, boundary and interface displacement fields, and internal displacements as: Π e ( σ m, σ c, u, u, u, u s ) = Π IC e + A( σ s, ǫ s )dω + σ s : ǫ s dω Ω s Ω s (σ m + σ m ) n s u s d Ω (4.11) Ω s=γ s Ω m s where A is the incremental strain energy density ( A = 1 2 σs ij ǫ s ij), σ s ij and ǫ s ij are the stress and strain increments of with respect to the current configuration. The total energy functional for the domain is Π = N e=1 Π e. The first variational of Π e in equation (4.11) is expressed as 71

90 δπ e ( σ m, σ c, u, u, u, u s ) = ǫ m : δ σ m dω Ω m ǫ m : δ σ m dω Ω m ǫ c : δ σ c dω ǫ c : δ σ c dω + Ω c Ω c δ σ m n e ud Ω Ω e + (σ m + σ m ) n e δ ud Ω Ω e ( t + t) δ udγ Γ tm (δ σ m δ σ c ) n c u d Ω Ω c (σ m + σ m σ c σ c ) n c δ u d Ω δ σ c n cr u d Ω Ω c Ω cr (σ c + σ c ) n cr δ u d Ω + Ω cr (σ s + σ s ) : δ u s dω Ω s δ σ s n s u s d Ω (σ m + σ m ) n s δ u s d Ω Γ s Ω m s (4.12) Γ s Ω m s Applying divergence theorem to the first of the boxed terms yields, (σ s + σ s ) : δ u s dω = (σ s + σ s ) δ u s dω + Ω s Ω s Γ s Ω m s (σ s + σ s ) n s δ u s d Ω (4.13) Setting the first variation of Π with respect to the incremental displacement field u s and incremental stress field σ on the subdomain boundary Γ s Ω m s to zero respectively, yield the following Euler equations, (σ s + σ s ) = 0 in Ω s Equilibrium (4.14) (σ + σ) n s = (σ s + σ s ) n s on Γ s Ω m s Traction reciprocity(4.15) u s = u on Γ e s Boundary displacement continuity (4.16) u s = u on Γ c s Interface displacement continuity (4.17) u s = u on Γ cr s Crack face displacement continuity (4.18) 72

91 The increments of stress σ s in the above equations are obtained from the rotated Cauchy stress equation (4.10). Deformation in Ω s is governed by the weak forms of the traction and displacement boundary conditions in equations (4.15, 4.16, 4.17, 4.18). Displacement compatibility between between the stress and displacement interpolated regions is satisfied in a weak sense as shown in the Euler equations (4.16, 4.17 and 4.18). Traction reciprocity is satisfied in a weak sense on the boundary Ω m s between the two domains. It should be noted that current and initial configurations are not differentiated in the small deformation domain Ω e. 4.5 Weak Form and Matrix Assembly The displacement increments { u s } in the last three terms of of equation (4.12) for each element in the enhanced region Ω e are interpolated as: { u s } = {N s } T { q s e} (4.19) In this work, a 9-noded element with Lagrangian shape functions N s are used for interpolation. Within each Voronoi cell element, the region Ω s may consist of several displacement based elements. Let the union of all the element nodal displacements be represented as the generalized displacement field { q s e} = { q s e}. Furthermore, the generalized displacement field may be subdivided into two groups, one corresponding to nodes on the boundary Ω s and the other corresponding to nodes in the interior Ω s \ Ω s, i.e. q s = q s 1 q s 2 where q s 1 Ω s and q s 2 Ω s \ Ω s ). To avoid duplicity, q, q and q will henceforth correspond to displacement fields in the respective boundary domains that do not overlap with Ω s. 73

92 Setting the first variation of the energy functional (4.11) with respect to the stress coefficients β m and β c, respectively, to zero, results in the weak forms of the kinematic relations as [P m ] T {ǫ m + ǫ m } dω = [P m ] T [n e ] T [L e ]{ q} d Ω Ω m Ω e [P m ] T [n c ] T [L c ]{ q } d Ω Ω c [P m ] T [n s ] T [N s ]{ q s Γ e 1}d Ω (4.20) s Ω m s and [P c ] T {ǫ c + ǫ c } dω = [P c ] T [n c ] T [L c ]{ q }d Ω Ω c Ω c [P c ] T [n cr ] T [L cr ]{ q }d Ω Ω cr [P c ] T [n s ] T [N s ]{ q s Γ c 1}d Ω (4.21) s Γ cr s The boxed terms in the above equations highlight the addition due to the enhanced region Ω s. Next, for the displacement elements in the locally enriched region Ω s, the variational statement for the principle of virtual work may be written by setting the coefficients of δ u s to zero as: δπ s ( u s ) = (σ s + σ m ) : δ( u s )dω Ω s + (σ + σ) n s δ( u s )d Ω = 0 (4.22) Ω s The traction σ n s on the boundary Ω s is imposed from the stresses in the region Ω m \ Ω s on Ω s. In an incremental solution method, the principle of virtual work and 74

93 hence all variables in equation (4.22) are written in the n + 1-th configuration Ω n+1 s. Substituting the displacement interpolation equation (4.19) in equation (4.22) and noting that δ( q s ) is arbitrary, leads to the equation that should be solved for the (n + 1)-th increment: [ B] T {σ s } n+1 dω = [N s ] T [n s ] T [P m ]{β m + β m }d Ω Ω n+1 s (Γ e s) n+1 ( Ω m s ) n+1 + [N s ] T [n s ] T [P c ]{β c + β c }d Ω (4.23) (Γ c s )n+1 (Γ cr s )n+1 Here [ B] is the strain-displacement matrix using selective reduced integration to prevent volumetric locking [89]. Consequently in the solution process, the iterative correction {dq s } i to the displacement solution { q s } i is solved from [K s ] i {dq s } i = {F s } i (4.24) where [K s ] i = {F s } i = + Ω n+1 s [ B] T [D ep ] i [ B]dΩ [N s ] T [n s ] T [P m ]{β m + β m } i d Ω (Γ e s) n+1 ( Ω m s ) n+1 [N s ] T [n s ] T [P c ]{β c + β c } i d Ω (Γ c s )n+1 (Γ cr s )n+1 [ B]({σ s } n+1 ) i dω (4.25) Ω n+1 s Here [D ep ] i is the elasto-plastic tangent stiffness matrix in the i-th iteration. For coupling equations (4.24)in Ω s with those in the VCFEM formulation, the displacement decomposition q s = q s 1 q s 2 mentioned earlier is necessary, in (4.24) i.e.: { } [ ] dq dq s s = 1 K s dq s 11 K s i { } 12 dq s i { } 1 F s i 2 K s 22 K s 22 dq s = 1 2 F s (4.26) 2 Using static condensation, equation (4.26) may be re-written as we have ([K s 11] [K s 12][K s 22] 1 [K s 21]){dq s 1} = {F s 1} [K s 12][K s 22] 1 {F s 2} (4.27) 75

94 Finally, setting the first variation of the total energy functional Π with respect to q, q, q and q s 1 to zero results in the weak form of the traction reciprocity conditions: Ω e [L e ] T [n e ] T [P m ]d Ω 0 N Ω c [L c ] T [n e ] T [P m ]d Ω Ω c [L c ] T [n c ] T [P c { } ]d Ω β m + β m e=1 0 Ω cr [L cr ] T [n cr ] T [P c ]d Ω Γ e s Ω m[n s ] T [n s ] T [P m ]d Ω β c + β c s Γ c s Γ cr[n s ] T [n s ] T [P c ]d Ω s = Γ tm [L e ] T { t + t}dω N 0 e=1 0 (4.28) Ω s [N s ] T [n s ] T {σ s }d Ω Again in the iterative solution process, the elemental equations in the i-th iteration step are obtained by substituting equation (2.20) in equations(4.20) and (4.21), to yield [ ]{ } Hm 0 dβ m i 0 H c dβ c = [ ] i q Ge G cm 0 G sm q 0 G cc G cr G sc q { Ω m\ω s [P m ] T {ǫ + ǫ}dω Ω c [P c ] T {ǫ + ǫ}dω } i q s 1 i (4.29) where [G sm ] = [P m ] T [n s ] T [N s ]d Ω ; [G Γ e sc ] = [P c ] T [n s ] T [N s ]d Ω (4.30) s Ω m s Γ c s Γ cr s All other matrices have been defined in equation (2.22). The iterative solution of equation (4.28), solves the following equation in the j-th iteration step 76

95 j dq N [ [G] T [H] T [G] i] dq Γ N m [L e ] T { t + t}d Ω 0 dq e=1 = 0 dq s e=1 1 Ω s [N s ] T [n s ] T {σ s }d Ω Ω e [L e ] T [n e ] T [P m ]d Ω 0 N Ω e [L c ] T [n c ] T [P m ]d Ω Ω c [L c ] T [n c ] T [P c { } ]d Ω β m + dβ m 0 e=1 Ω cr [L cr ] T [n cr ] T [P c ]d Ω Γ e s Ω m[n s ] T [n s ] T [P m ]d Ω β c + dβ c (4.31) s Γ c s Γ cr[n s ] T [n s ] T [P c ]d Ω s Equation (4.31) may be written in a condensed form as [ ]{ } K11 K 12 dq = K 22 K 22 dq s 1 { F1 F 2 } ; where {dq } = {dq,dq,dq } T (4.32) Equations (4.27) and (4.32) may be combined to finally yield the set of equations to be solved, i.e. [ ] j { } K11 K 12 dq j { } j F K 22 (K 22 + K s 11 K s 12K s 22 1 K s 21) dq s = 1 1 F 2 + F s 1 K s 12K s 22 1 F s (4.33) Some Aspects of Numerical Implementation A few aspects specific to numerical implementation in the locally enhanced VCFEM are discussed here. 4.7 Iterative solver The Newton-Raphson iterative solver is generally used to solve the equations discussed in section 4.5. However this solver is unable to capture phenomena like snapback phenomenon due to material softening, which undergoes reverse loading with instability. The Newton-Raphson solver, where the loading process is monotonically controlled by incremental deformation or load conditions, exhibits a discontinuous drop. The arc-length solver has been proposed in [90, 91] as a method of overcoming 77

96 this shortcoming by introducing an unknown loading parameter (λ + dλ) to govern the load increments. Equation (4.33) is modified with this loading parameter as [K] j dq j = λ j {F ext } {F int } j (4.34) where both dλ j and dq j are unknowns, and dλ j can be either positive or negative. The additional unknown dλ j requires the solution of a constraint equation In this phase, the arc-length method [90, 91] is implemented, with a controlling function: { u} T { u} + λ 2 = L 2 (4.35) Here { u} = {dq,dq,dq,dq s 1} T. The parameter L is initialized as 1 and is adjusted based on the number of iterations for higher efficiency. A similar adaptive load control algorithm, based on the maximum change in porosity over computational load step, is described in [92] to control loading increment for the subsequent load step. 4.8 Mapping from stress based domains to post enhancement displacement based regions The substitution of displacement elements in a portion Ω s of the original stressbased domain in each Voronoi cell domain Ω e requires mapping of variables to integration points of the new elements. Variables to be updated from the old integration points to the new, include stresses, strains, void volume fractions and other state variables. The integration point structure remains the same in remainder of the cell domain Ω e \ Ω s. The superconvergent patch recovery technique proposed in [86] is used to map variables at the integration points of the displacement elements from the original Voronoi cell FE integration points. In this process, a complete 4-order polynomial interpolation is used for all parameters as: ˆv p = [P]{V p },P = [1,x,y,x 2,xy,y 2,x 3,x 2 y,xy 2,y 3,x 4,x 3 y,x 2 y 2,xy 3,y 4 ](4.36) 78

97 and ˆv p may represent each component of the Cauchy stress σ ij, strains ǫ ij, equivalent plastic strain ǫ p and void volume fraction f. {V p } corresponds to a set of unknown parameters to be evaluated for each local patch, comprised of the closest 25 integration points in Ω e Adaptive h-refinement for displacement elements For the ductile fracture problem, a sufficiently refined mesh is needed in the region of the narrow fracture band. The mesh can be coarser away from this localization region. An adaptive h-refinement strategy proposed in [93] is used in this work for local mesh refinement. In this method, additional degrees of freedom due to mesh refinement are statically condensed out at the element level and thus do not change the structure of the assembled stiffness matrix. Special constraint relations are used between the added degrees of freedom and the original ones at the interface of the two domains. The elements to be refined at the end of each solution increment, are determined from a criterion based on the gradient of void volume fraction: Refine element E, if: f g crit at an integration point (4.37) The value of g crit in this paper is set as 0.5 (fc f in) l in, where f c is the critical void volume fraction for coalescence, f i n is the initial void volume fraction and l in distance to the closest integration point. An elements containing this integration point will be split into four sub-elements based on this criterion. The element stiffness matrix K e and force vector F e are assembled using static condensation of the additional degrees of freedom. 79

98 4.9 Conclusions In this chapter, VCFEM is locally enhanced by large deformation displacement based formulation to handle material softening. Modifications to the variational principle to account for the soften phase introduced in the form of displacement based formulation have been discussed. The resulting finite element formulations and the superposition of displacement elements on VCFEM are also presented. In the next chapter, numerical examples are presented to study the nonlocal model, effect of distribution of inclusions and material parameters on ductile fracture. 80

99 CHAPTER 5 NUMERICAL EXAMPLES OF DUCTILE FRACTURE WITH THE LE-VCFE MODEL 5.1 Introduction The locally enhanced VCFE model (LE-VCFEM) is used in this section for simulating ductile fracture of ductile materials containing a dispersion of brittle inclusions. The studies are divided into two groups. In the first group, sensitivity analysis of ductile fracture with respect to various material parameters and geometric configurations are conducted. Sensitivity of the evolution of ductile fracture with respect to various material parameters are examined in this study. Critical material parameters include material characteristic length M CL for nonlocal GTN model in equation (2.30), void nucleation parameters, initial void volume function distribution, and dispersion of inclusions in the microstructure. In these studies, the matrix is assumed to be a ductile porous material with elastic-plastic behavior, while the inclusion (SiC) is brittle with elastic properties. Material parameters are provided in table 5.1. The post yield behavior of the ductile matrix is characterized by the relation σ m = σ 0 (ǫ p m/ǫ 0 + 1) N. Different sets of void nucleation parameters in the GTN constitutive model of equations (2.34) and Weibull 81

100 parameters for the inclusion cracking criterion (3.1) used in the simulations are given in table 5.2. Simulations are conducted under plane strain conditions. As shown in figure 5.6, displacement boundary condition is applied along the x-direction, and a periodicity boundary condition is applied on the top boundary of the model at y = L. For each Voronoi cell element, different stress functions and boundary displacement field interpolations are used and different sets are tabulated in table5.3. The table depicts the order of stress functions, number of terms in the stresses, number of displacement degrees of freedom on Ω c and the maximum number of displacement degrees of freedom on Ω e respectively. The 7th and 8th order polynomial stress functions correspond to 33 and 42 stress terms respectively. The 36 reciprocal terms are due to 3 reciprocal terms for each polynomial exponent from 2 to 4, i.e. i = p +q p +q + 2 p +q = 2 4. After particle cracking, another 36 reciprocal terms are added into the stress function for both of matrix and inclusion. E m (GPa) ν m σ 0 (MPa) N E i (GPa) ν i Table 5.1: Ductile Matrix and Brittle Inclusion Material Properties Set # ǫ N s N f N σ w (GPa) m Table 5.2: Void nucleation parameters in GTN model and Weibull parameters 82

101 Set # O(Φ m poly ) σm poly terms σm rec terms O(Φ c poly ) Ω c DOF Ω e DOF Table 5.3: Stress related parameters and pre-enhancement displacement degrees of freedom in each Voronoi cell element In the second class of examples, ductile fracture of a real microstructure is simulated. 5.2 Sensitivity analysis with respect to spatial distribution of inclusions Four different microstructures, (A), (B), (C) and (D) in figure 5.1, are used to investigate the effect of the distribution of inclusions. The different spatial distributions are uniform square edge-packed, uniform square diagonal-packed, hard-core random and hard-core with clustering respectively. All the inclusions have the same size. Microstructures (A), (C) and (D) have an inclusion volume fraction V f = 15%, while (B) has a volume fraction is V f = 13.8%. Table 5.4 lists a few critical microstructural characterization functions, viz. Nearest Neighbor Distance, Cluster index and Contour index. The Nearest Neighbor distance (NND) corresponds to the minimum surface to surface distance between two particles. Cluster index and Contour index are quantities that characterize the level of clustering and have been defined in [94]. Cluster index quantifies the intensity of packing in a cluster represented by the number of inclusions in a prescribed region, while Contour index accounts for the area fraction 83

102 of inclusions within the same region. The distribution of NND for microstructures (C) and (D) is plotted in figure (5.2) showing the higher peak for clusters in (D). A B (a) (b) C D (c) (d) Figure 5.1: Four different microstructures with their Voronoi cell meshes: (a) uniform square edge-packed, (b) uniform square diagonal-packed, (c) hard-core random and (d) hard-core with clustering LE-VCFEM simulations are conducted with void and crack nucleation parameters of set 3 in table 5.2. Porosity parameters f in = and MCL = 0.09L are used. The stress/displacement parameters correspond to set 2 in table 5.3. The boundary conditions are the same as in the previous section. The Voronoi cell model for the 84

103 6 5 Number of Inclusions Microstructure (C) Microstructure (D) Nearest Neighbor Distance Figure 5.2: Distribution of Nearest neighbor distance (NND) for microstructures (C) and (D) 85

104 Micros. Max. NND Min. NND Avg. NND Cluster Index Contour Index (A) (B) (C) E (D) E Table 5.4: Microstructural characterization functions. All the dimensions are in fractions of the size L of the microstructure microstructure with the locally enhanced displacement elements at the final strain are shown in figure 5.3. Figure 5.4 shows the macroscopic stress-strain and ductility response. The strains to failure for each microstructure are listed in Table 5.5. The uniformly distributed square microstructure (A) has the highest ductility and fracture resistance while the clustered microstructure (D) with more inclusions at smaller NND s has the worst fracture properties. Higher local stresses at smaller NND s cause inclusions to crack early. This initiates ductile fracture in the matrix, which accelerates localization of damage to cause catastrophic failure. Microstructure (A) (B) (C) (D) Final equivalent strain Table 5.5: Equivalent strains to failure for each microstructure Figure 5.5 shows contour plots of f at the final stage. For (A) all the inclusions crack at same time because of the uniform distribution and f evolves simultaneously. The final fracture path forms near the loading boundary because of boundary effect at instability. The ductility of microstructure (B) is lower because of the lower volume 86

105 (a) (b) Figure 5.3: Voronoi cell model with locally enhanced displacement elements for the microstructures: (a) C and (b) D at strain to failure fraction and the direction of the loading with respect to the local microstructural symmetries. Evolution of ductile fracture also takes more or less simultaneously in figure 5.5(b). The stresses are not uniformly distributed and are especially low near the boundaries because of the lower local volume fraction. Inclusion cracking is more dispersed for the microstructure (D) that has a smaller minimum NND and higher Cluster index and Contour index. An interesting observation is that though the inclusion 1 in figure (5.5) cracks early, it does not contribute to the dominant damage for the microstructure. The figures 5.5(c,d) show that the inclusion cracking happens mainly along the fracture path after the main ductile fracture path has formed. This example proves that the effect of spatial distribution on ductility, and especially the dominant damage path, is quite complex. Microstructural characterization functions alone are insufficient to predict the damage path. 87

106 Macroscopic Stress (MPa) D C B Macroscopic Strain (%) A Figure 5.4: Macroscopic stress-strain response for the different microstructures 88

107 (a) (b) (c) (d) Figure 5.5: The contour plot of void volume fraction for microstructure: (A), (B), (C), (D). The number in the figures indicate the sequence of particle cracking. 89

108 5.3 Sensitivity analysis with respect to material characteristic length MCL for nonlocal GTN model Two specific microstructural configurations are chosen to study these effects, viz. (a) a unit cell containing a single circular inclusion of volume fraction V f = 20% as shown in figure 5.6(a), and (b) a representative microstructural element containing 25 inclusions of volume fraction V f = 15% as shown in figure 5.6(b). For the unit cell problem, the VCFE model has only one element while the latter microstructure is tessellated into 25 elements. Periodic Boundary y Periodic Boundary L x L Y L (a) X (b) L Figure 5.6: Microstructural configuration and Voronoi cell meshes for (a) a unit cell containing a single circular inclusion and (b) a representative microstructural element containing 25 inclusions. The material characteristic length M CL has very important effect on the localization of material [37, 45]. A numerical study with LE-VCFEM is conducted to understand the effect of MCL on ductile fracture of the two microstructures in figure 90

109 (5.6. Two uniform initial void volume fractions are used for the porous matrix, viz. f in = 0.1 without nucleation and f in = with nucleation. The nucleation parameters are given for set 1 in table 5.2. With the f in = 0.1, the Weibull parameters used are σ w = 0.35GPa and m = 2.4, while for f in = they are σ w = 0.45GPa and m = 2.4. For each initial void volume fraction, four M CL s are examined, viz. MCL = 0L, MCL = 0.3L, MCL = 0.8L and MCL = 1.0L. For the unit cell problem, a comparison of the macroscopic stress-strain response is shown in figure 5.7. The stress for f in = is much higher than that for f in = 0.1. The first drop in stress is caused by particle cracking with subsequent stress reduction caused by matrix cracking leading to total failure of the microstructure. The microstructure with lower value of f in = 0.1 undergoes early particle cracking and complete failure. Particle cracking strain is generally higher for lower M CL values, since faster void growth in the matrix near the interface leads to a reduced particle stress at the same macroscopic strain level. However, the overall strain to failure increases with increasing values of MCL in this range. The growth of void volume fraction is slower with higher values of the MCL due to the non-local effects and hence the ductile failure is delayed. Increasing values of the characteristic length generally lead to less difference in the overall strain to failure. Convergence in the ductility and strain to failure is achieved as MCL L, where L corresponds to the size of the unit cell or the distance between two neighboring inclusions (due to periodicity). The local void growth causing ductile failure is shown in figure 5.8 for the two cases of initial f in s. The brittle inclusion crack is perpendicular to the direction of loading, corresponding to a mode I crack. The matrix crack starts from the inclusion crack tip in a direction that corresponds to the maximum shear stress. With increasing ductile 91

110 500.0 Macroscopic Stress (MPa) MCL=0 MCL=0.3L MCL=0.8L MCL=1.0L Macroscopic Strain (%) (a) Macroscopic Stress (MPa) MCL=0 MCL=0.3L MCL=0.8L MCL=0.9L MCL=1.0L Macroscopic Strain (%) (b) Figure 5.7: Macroscopic stress-strain response for the unit cell with (a)f in = 0.1 without void nucleation and (b) f in = with nucleation. 92

111 failure, the matrix crack slowly re-orients toward the direction perpendicular to the loading. The crack pattern is anti-symmetric at the two ends of the inclusion crack. (a) (b) Figure 5.8: Contour plot of void volume fraction with MCL = 1.0L for (a) f in = 0.1 without nucleation and (b)f in = with void nucleation LE-VCFEM simulations of the microstructure with 25 inclusions are conducted with the set 2 crack parameters in table 5.2 and set 2 stress/displacement parameters in table 5.3. The initial void volume fraction is f in = To estimate the microstructural characteristic length scale, the distribution nearest neighbor distance between inclusions are plotted in figure 5.9. Based on this histogram of the number fraction of nearest neighbor distances, its average L NND for this microstructure is calculated as L NND = 0.073L. Simulations are carried out with 5 different values of the non-local length scale parameter, viz. MCL = 0, MCL = 0.8L NND, MCL = 1.0L NND, MCL = 1.2L NND and MCL = 1.4L NND. Figure 5.10 shows the macroscopic stress-strain response with the different M CL values. This response is more complicated than the unit cell response due to the 93

112 0.3 Number Fraction of Inclusions Nearest Neighbor Distance (NND) Figure 5.9: The number fraction of particles for nearest neighbor distance for 25 Voronoi cell mesh morphological nonuniformity that has an important effect on evolution of the ductile crack. The local GTN model (MCL = 0) has a higher value of strain to life in this case. At the lower strains, particle cracking is mainly determined by the relative location of inclusions and hence the distribution of dispersion is important. These cracked particles initiate matrix cracking in narrow bands of high plastic flow, and subsequently cause more particles to crack due to load shedding. Catastrophic failure is caused when dominant matrix cracks caused by evolution and coalescence of microcracks percolate through the entire microstructure. This complicated interaction causes a delayed ductile failure with local GTN model in this case. The local void growth because of the high stress concentration, causes rapid stress re-distribution that disperses the cracks in this case. This is evident from figure 5.11, which shows different ductile crack growth patterns for different values of MCL. With increasing 94

113 M CL, the stress redistribution is less frequent and a stable dominant crack forms earlier that has a higher dependence on the applied load direction. At higher values of MCL, the dominant cracks are nearly perpendicular to the load direction and lead to lower values of strain to failure. Convergence of the fracture path in a stable narrow band at values MCL = 1.2L NND and higher, point to the fact that the nonlocal characteristic length M CL has a correlation to the average near neighbor distance L NND. The microstructural scale L NND plays an important role in the determination of the nonlocal parameter. This is also corroborated in figure (5.10), where it is seen that results with MCL 1.2L NND yield very similar values of ductility. Larger than optimal M CL may also add other effects which can affect the ductility predictions. This example reiterates that morphology of microstructure has an important effect on plastic flow which is responsible for ductile fracture. 5.4 Sensitivity analysis with respect to nucleation parameters in GTN model The plastic strain controlled nucleation criterion (2.33) has three controlling parameters. Two of these, viz. f N and ǫ N are varied for sensitivity analysis. The microstructure (D) in figure 5.1 is used for these simulations. Material properties are given in table 5.1 within the nonlocal characteristic length MCL = 1.2L NND. The stress/displacement parameters correspond to set 2 in table 5.3. After particle cracking, another 36 reciprocal terms are added to the matrix and inclusion stress functions. Boundary conditions are the as in section

114 500.0 Macroscopic Stress (MPa) MCL=0 MCL=0.8L NND MCL=1.0L NND MCL=1.2L NND MCL=1.4L NND Macroscopic Strain (%) Figure 5.10: Macroscopic stress-strain response for microstructure with 25 inclusions for different MCL s 96

115 (a) (b) (c) (d) (e) Figure 5.11: Contour plots of void volume fraction for the microstructure with 25 inclusions, with nonlocal parameter: (a) MCL = 0, (b) MCL = 0.8L NND,(c) MCL = 1.0L NND, (d) MCL = 1.2L NND and MCL = 1.4L NND. Numbers in the figures indicate the sequence of particle cracking. 97

116 5.4.1 Effect of ǫ N Void and crack nucleation parameters correspond to set 3 in table 5.2 with the exception of ǫ N. For the sensitivity studies, values ǫ N = 0.2, 0.4, 0.6 are used Macroscopic Effective Stress (MPa) ε N =0.2 ε Ν =0.4 ε Ν = Macroscopic Effective Strain (%) Figure 5.12: Macroscopic stress-strain response with different ǫ N, for microstructure (D) in figure 5.1 The macroscopic effective stress-strain responses in figure 5.12 show that failure is delayed with increasing ǫ N. Larger mean strain for void nucleation ǫ N, postpones void nucleation which delays subsequent void growth and coalescence. Also, there are more cracked inclusions with larger ǫ N. 98

117 5.4.2 Effect of f N Void and crack nucleation parameters correspond to set 3 in table 5.2 with the exception of f N. For the sensitivity studies, values f N = 0.2 and f N = 0.8 are used Macroscopic Effective Stress (MPa) f N =0.08 f N = Macroscopic Effective Strain (%) Figure 5.13: Macroscopic stress-strain response with different f N, for the microstructure (D) in figure 5.1 The macroscopic stress-strain response in figure 5.13 shows that the ductile fracture is delayed with the smaller volume fraction of void nucleating particles f N. Studies show that the fracture path can differ widely with different values of f N. Localized fracture occurs in a narrower band with larger values f N. 99

118 5.5 Sensitivity analysis with respect to void coalescence parameters f c and f f Coalescence is an event corresponding to a shift from a relatively homogeneous deformation state to a highly localized straining in the microstructure. The void coalescence parameters f c and f f in equation (2.29) have significant effect on ductile fracture of the microstructure. Analysis in [39] has shown that f c strongly depends on the initial void volume fraction. Benzerga [43] has studied the coalescence by internal necking and has shown that there is no uniqueness of (f c,f f ) to fit an experimental stress-strain curve [43, 42]. All material parameters and boundary conditions are same as the previous examples with f N = Different values of f c and f f, viz. (f c = 0.15, f f = 0.25), (f c = 0.10, f f = 0.15), and (f c = 0.2, f f = 0.35) are used to study its sensitivity for the microstructure (D) of figure 5.1 Figure 5.14 shows that ductility is reduced with increasing values of f c & f f. Microstructural contour plots (not shown ) do not exhibit an significant dependence of the final fracture path on f c and f f, since they do not affect early stress redistribution. 5.6 Effect of porosity in plasticity relations on ductile failure The effect of porosity (void volume fraction) in the plastic flow representation on the material ductility is investigated in this example. Two sets of simulations are conducted with this objective. One is with the pressure dependent GTN model for porous elasto-plastic material. The other uses incompressible J 2 plasticity model. The model in figure 5.1(d) is used in this example. The material properties, crack nucleation properties and VCFEM parameters are given in tables 5.1, 5.2(set 3) and table 5.3(set 2) respectively. Figure 5.15 shows the stress-strain response for the two 100

119 Macroscopic Effective Stress (MPa) f c =0.10, f f =0.15 f c =0.15, f f =0.25 f c =0.25, f f = Macroscopic Effective Strain (%) Figure 5.14: Macroscopic stress-strain response with different void coalescence parameters, f c & f f, for the microstructure (D) shown in figure 5.1 simulations. There is no complete ductile failure with the J 2 plasticity model, with a near stabilized stress value for increasing stress. The strain to failure with the GTN model is around ǫ = From the contour plot of effective plastic strain in figure 5.16, it is seen that only five inclusions cracked for GTN model along the localized path of the dominant crack. Significant drop in the load carrying capacity and instability happens with matrix softening. ON the other hand, almost all inclusions cracked in the J 2 plasticity model. Lack of matrix softening in this model only causes relocation of regions of high inclusion stresses with subsequent cracking. Predictions of ductility made my models like the J 2 plasticity theory can be significantly in error with respect to real experimental observations. 101

120 Macroscopic Effective Stress (MPa) GTN Model J2 Plastic Model Macroscopic Effective Strain (%) Figure 5.15: The macroscopic stress-strain response for GTN model and J 2 plasticity model (a) (b) Figure 5.16: Contour plot of effective plastic strain for: (a) the GTN model with porosity, (b) J 2 plasticity model 102

121 5.7 Ductile fracture for a real microstructure Li et. al. [59] has done the two-dimensional simulation by using VCFEM for only accommodating inclusion cracking for the experimental micrograph shown in figure 5.17(a), which is a section of the extruded commercial X2080 aluminum alloy with 15% volume fraction SiC inclusions. In this work, a full analysis which include inclusion cracking and matrix cracking will be conducted under plane strain condition with monotonically increasing strains for the same experimental micrograph. The size of this experimental micrograph shown in figure 5.17(a) is rectangular µm 2. The equivalent mesh of this experimental micrograph for VCFEM is shown in figure 5.17(b). Periodicity boundary conditions are imposed by requiring edges to remain straight and parallel to the original direction throughout deformation as u x = 0 (on x = 0), u y = 0 (on y = 0), u x = u ap (on x = L x ), u y = D y (on y = L y ), T y = 0 (on x = 0/L x ), T x = 0 (on y = 0/L y ) where u ap is an applied displacement and Dy is determined from the average force condition T X xdx = 0 on y = L y. The reinforcing phase of SiC inclusions is assumed to be brittle and is modeled with the linear elastic properties: Young s modulus E = 427GPa, Poisson s ratio v = The aluminum matrix material is assumed to be ductile and is modeled as porous material with properties: Young s modulus E = 72GPa, Poisson s ratio v = 0.33, the post yield elastic-plastic behavior is obtained from Ref. [59] as shown in Figure 7(a), the initial void volume fraction f in = For the Weibull model, the values (σ w = 1.26GPa, m = 4.2) are calibrated by Li in [59]. For the stress function, 33 polynomial terms (7th order polynomial stress function) with additional 36 reciprocal terms due to the inclusion shape (3 reciprocal 103

122 terms for each polynomial exponent from 2 to 4, i.e. i = p+q p+q+2 p+q = 2 4) is used for matrix, only 33 polynomial terms is used for inclusion. After inclusion cracking, another 36 reciprocal terms due to the crack shape will be added into the stress function for both of matrix and inclusion. Y Periodic Boundary Ly (a) 0 Lx (b) X Figure 5.17: (a)experimental micrograph, (b) VCFEM mesh and boundary conditions The solid line in figure 5.18 shows the macroscopic stress-strain response from experiment. For the early stage, only inclusion cracking is observed, and the ductile fracture which is mainly depended on plastic deformation usually happen later. Once the ductile fracture happen, the whole structure will go to fracture totally very soon. Based on this, for the early stage, the ductile fracture can be ignored, so the J 2 plasticity model is used and only inclusion cracking is considered here for fitting the macroscopic stress-strain response. The LE-VCFEM simulation is only 2D, so the simulations are conducted under plane strain, plane stress and generalized plane strain 104

123 conditions respectively. As shown in figure 5.18, both of plane stress condition and plane strain condition do not give a good match between the LE-VCFEM simulation and experiment, and especially the plane stress condition gives a worse match. For minimizing the difference between 2D simulation and 3D experiment which is because of the out-of-plane boundary condition, the generalized plane strain condition is used to match the macroscopic results between 2D LE-VCFEM simulation and experiment. As shown in figure 5.18, the results under generalized plane strain condition with out-of-plane strain (ǫ zz = 0.005) shown as dot-dot-dashed line match well with the experimental results. For the following simulations, the generalized plane strain condition with out-of-plane strain (ǫ zz = 0.005) will be applied Macroscopic Stress (MPa) Experimetal VCFEM: Plane Strain VCFEM: Plane Stress VCFEM: Gen. Plane Strain (ε zz =-0.005) VCFEM: Gen. Plane Strain (ε zz =-0.02) Macroscopic Strain (%) Figure 5.18: Macroscopic stress-strain response for the real microstructure with only inclusion cracking and J 2 plasticity model 105

124 The solid line in figure 5.18 shows that the fracture strain in the experiment is about 8.88%. As shown in previous examples, the void nucleation has big effect on the fracture strain. Based on the contour plot of effective plastic strain for the simulation under generalized plane strain condition with out-of-plane strain (ǫ zz = 0.005) shown in figure 5.19, the effective plastic strain is larger than 0.5 in most areas around the crack tips, so ǫ N = 0.5, s N = and f N = 0.08 are chose for void nucleation. f c = 0.15, f f = 0.25 are used for void coalescence. For the nonlocal model, the average nearest neighbor distance L NND is used as the Material Characteristic Length, which means MCL = 3.2µm. As shown in figure 5.20, the microstructure failure too early with the Weibull parameters (σ w = 1.26GPa, m = 4.2) calibrated by Li [59], comparing with experimetal result. In this case, the inclusion cracking happen too early obviously, which leads the matrix cracking to happen early too. The reason is only the inclusion cracking is considered in Li s work [59] for calibrating Weibull parameters. But based on the previous studies, the matrix cracking can cause the localization of damage, which significantly change the final damage pattern, so new Weibull parameters are calibrated by fitting the macroscopic stress-strain response with the experimental result in this work. For the new calibration, the same Weibull parameter m = 4.2 is used, which is believed to only affect the distribution of damaged inclusions. With the best match, the new Weibull parameter calibrated is : σ w = 2.16GPa. The figure 5.20 shows the macroscopic stress-strain response, with the explicit effect of inclusion cracking followed by ductile matrix fracture. The sequence of inclusion cracking is shown in the stress-strain plot of figure The first 6 inclusions (1-6) cracks in isolation and does not show any major softening in the stress-strain 106

125 Figure 5.19: Contour plot of effective plastic strain at (ǫ xx = 8.88%) for the real microstructure with only inclusion cracking and J 2 placticity model under generalized plane strain condition with (ǫ zz = 0.005) behavior. Softening, manifested by a drop in the stress-strain curve, starts after the seventh inclusion has cracked followed by significant plastic deformation and void growth in the matrix. From the contour plot of void volume fraction shown in figure 5.21, it is evident that the subsequent sequence of inclusion and matrix cracking occur in a very narrow band and hence a dominant fracture path is observed. The dominant fracture gotten by the simulation, the boxed region shown in figure 5.21, is matched with the dominant fracture from experiment, shown in figure 5.17(a). 5.8 Conclusions In this chapter, the sensitivity analysis are studied by using LE-VCFEM for the random heterogeneous materials. In this studies, the spatial distribution of inclusions, 107

126 Macroscopic Stress (MPa) Exp. LE_VCFEM (σ w =1.26) LE_VCFEM (σ w =2.16) Macroscopic Strain (%) Figure 5.20: Macroscopic stress-strain response for the real microstructure Figure 5.21: The contour plot of void volume fraction for the real microstructure. The number in the figure indicate the sequence of inclusion cracking. 108

127 the non-local model, the void nucleation and void coalescence are considered. Finally, a full analysis of the ductile fracture for a real microstructure is conducted and compared with experimental results. All the studies shows the high potential and power of LE-VCFEM on the study of the ductile fracture of heterogeneous materials. Based on this, the next chapter introduced a new numerical conformal mapping method to extend the LE-VCFEM for considering more complicated shaped inclusions. 109

128 CHAPTER 6 LE-VCFEM FOR HETEROGENEITIES OF ARBITRARY SHAPES 6.1 Introduction In conventional VCFEM, the inclusions are approximated by equivalent ellipses in the tessellation. So some geometry information will be lost for the inclusions with complicated shape. To avoid that, a promising method have been implemented to extend the conventional VCFEM in [95], which uses multi-sided polygons to approximated the inclusions with complicated shapes to keep the geometry information to the greatest extent, shown in figure 6.1. Also the multi-resolution wavelet functions are introduced to enrich the stress functions for both matrix phase and inclusion phase. 6.2 Numerical Conformal Mapping for Heterogeneities of Arbitrary Shapes In VCFEM, a reciprocal term is used in the stress function to satisfy the condition that effects of heterogeneity shape for the matrix stress functions should vanish at large distances from the interface. To accomplish this condition, a modified form of 110

129 Γ m Porous Elasto plastic Material Crack Ω cr Γ um Ω c Ω m Γ tm Brittle Material Figure 6.1: A Voronoi cell with a inclusion of multi-sided polygon Schwarz-Christoffel transformation suggested in [65] is used to generate the reciprocal terms in matrix stress function for these inclusions of multi-sided polygon. w = f(z) = A + C z ζ 2 Π N k=1(1 ζ z k ) β k dζ Where, w = x + iy is a point in complex physical domain, z = ψ + iζ is its corresponding point in complex transformed domain, and A,C,z k are unique accessory parameters for a given polygon. And the inverse function z = g(w) of w = f(z) is solved by using Runga-Kutta method. As an example, for a multi-sided interface with a square shaped boundary, this mapping function transfers the inside multi-sided interface Ω c to the outside circle shown in figure 6.2. The function f = 1 g(w) exhibit the necessary properties that the reciprocal term needs: f = 1 on Ω c and 1 f 0 as (x,y). 111

130 Voronoi Cell Boundary Interface Boundary Voronoi Cell Boundary Interface Boundary, f=1 (a) (b) Figure 6.2: Conformal transformation of a Voronoi cell with a inclusion of multi-sided polygon (a) Physical Domain (b) Transformed Domain 6.3 Multi-resolution Wavelet Functions for Enhancing Stress functions Because of the shape effect of the inclusions of multi-sided polygon, the stress distribution becomes more complex for both matrix s and inclusion s phase. The stress field based on high order polynomial function (plus reciprocal function for matrix) is not adequate to capture the stress gradient. Insufficiency of polynomial function to capture this sharp gradient required a superposition of some other function over polynomial functions. Wavelet bases, discussed in [96, 97], are L 2 (R) and generally have compact support. Only the local coefficients in wavelet approximations are affected by abrupt changes in the solution, such as for shock waves. This localization property makes the wavelet basis a desirable tool for problems with a high solution gradients, concentrations or even singularity. Therefore, the multi-resolution wavelet functions are introduced into the stress functions to capture the high stress gradient near the corner. 112

131 A family of Gaussian functions, for which the first and second order derivatives are popular wavelets bases [98], is used by S. Li and S. Ghosh [99] to model multiple cohesive crack propagation using VCFEM. The same function is introduced into the stress functions for both matrix phase and inclusion phase. The wavelet based stress function is constructed in a local orthogonal coordinate system (ξ, η), centered at the inclusion centroid and parallel to global x,y coordinate system. The corresponding stress function Φ a,b,c,d in the Gaussian wavelet basis is given as: ξ b ( Φ a,b,c,d (ξ,η) = e a )2/2 η d ( e c )2/2 β a,b,c,d (6.1) Where a,b,c,d are the dilation and translation parameters that can take arbitrary continuous values. For implementation in multi-resolution analysis involving discrete levels, the translation and dilation parameters should be evaluated for each level. For a particular dilation and translation parameters, the Gaussian wavelet enriched stress function in equation (6.1) becomes: ξ bn ( Φ m,n,k,l (ξ,η) = e am )2/2 e ( η d l ) c 2 /2 k β m,n,k,l (6.2) Where (m,k) correspond to the levels and (n,l) correspond to the discrete translation of the bases in the (ξ, η) directions respectively. The wavelet stress function in VCFEM is written as: Φ wvlt (ξ,η) = And the corresponding stresses are: σ ξξ 2 Φ wvlt η 2 σ ηη = 2 Φ wvlt ξ σ ξη 2 2 Φ = wvlt ξ η m n,k n m=1,n,k=1,l Φ m,n,k,l (ξ,η) (6.3) mn,k n 2 ξ bn ( (e am )2 /2 ( η d l e m=1,n,k=1,l η 2 mn,k n 2 ξ bn ( (e am )2 /2 ( η d l e m=1,n,k=1,l ξ 2 m n,k n m=1,n,k=1,l ξ bn ( (e am )2 /2 ( η d l e ξ η c ) 2 /2 k ) β m,n,k,l c ) 2 /2 k ) β m,n,k,l c ) 2 /2 k ) β m,n,k,l (6.4)

132 Consequently, the enriched stress function can be written as: Φ m Φ c = Φ m poly + Φ m rec + Φ m wlvt = Φ c poly + Φ c wlvt Due to the effect of the shape, there are high stress gradient around each corner of the polygon shaped inclusion. So the wavelets are placed around each corner to capture the high stress gradient. The method of implementation of the multiresolution wavelet-enriched stress functions in VCFEM is described below: For the generation of the wavelet base, uniform rectangular grids denoted by ( ) are generated around each corner, shown in figure 6.3(a). For the lower level wavelet base, circles centered at each corners with radius R 1 (which is decided by the nearest distance between the neighbor corners) are overlay on the rectangular grids, shown in Figure 6.3(a). Each grid point is check if it is inside the circle. All the grid points inside the circle will be check if it is inside the matrix phase or inside the inclusion phase. The grid points inside the matrix phase will become the wavelet bases used to generate the multi-resolution wavelet-enriched stress functions Φ m wlvt. The grid points inside the inclusion phase will become the wavelet bases used to generate the multi-resolution wavelet-enriched stress functions Φ c wlvt, shown in Figure 6.3(b). For the high level (mth.) wavelet base, the circles with radius R m = R 1 / (m) are overlay on the same rectangular grids. The same procedure is repeated to find the wavelet bases for both matrix phase and inclusion phase. 114

133 (a) (b) Figure 6.3: Generation of multilevel wavelet bases: (a) All the grid points of the uniform rectangular grids are chosen as wavelet bases (b) The finally used wavelet bases for a concave inclusion case 50 L 40 y 30 L x σ xx (MPa) ABAQUS VCFEM without wavelet VCFEM with wavelet x Figure 6.4: The stress σ xx at y = 0.51L/2 for one Voronoi cell with a square shaped inclusion to demonstrate the effect of the multi-resolution wavelet-enriched stress functions 115

134 A example of a square shaped inclusion, as shown in Figure 6.4, explains the effect of the multi-resolution Wavelet Functions. Figure 6.4 shows the comparison of the local stress σ xx along the x direction at y = 0.51L/2 between VCFEM and ABAQUS. For VCFEM, the stress functions are constructed with and without multiresolution Wavelet Functions in this example. The ABAQUS s results show there is high stress gradient near the corner. The VCFEM s result with multi-resolution Wavelet Functions shows a good match with ABAQUS s. 6.4 Numerical integration of element matrices The numerical accuracy and convergence of the method significantly depends on the accurate domain integration to evaluate [H] & [G]. In the matrix domain Ω m due to the presence of reciprocal function (f), a discretization scheme is required which can reflect the gradient of (f) and put refined integration in the areas of high gradient. The superposition of displacement elements is also based on the discretization scheme. For the superposition of displacement elements, the discretization scheme would require match along the shared boundaries to ensure the exact match of the nodes along the boundaries shared between two neighbor VC elements. So an integration scheme is developed to achieve this goal. In the integration scheme, the physical domain Ω m is mapped into the complex transformed domain by the inverse function. A more uniform grid is generated in the complex transformed domain and is mapped back to the physical domain by using transformation function w = f(z). The inclusion domain Ω c is simply subdivided into triangles. The essential steps involved in the scheme are: 116

135 The physical domain (shown in figure 6.2(a)) is mapped into the transformation domain. Each node on the interface (denoted by ) is projected to the Voronoi cell boundary along the radial direction in the transformation domain. Also each node on the corner of Voronoi cell boundary (denoted by ) is projected to the interface along the radial direction in the transformation domain. In transformation domain, each node on the interface and Voronoi cell boundary is joined with the corresponding projected node (denoted by ) by a straight dashed line. Shown in Figure 6.5(a). In the transformation domain, identify 5 points on the line joining the nodes with the projected nodes which equally divide the line. Join the points (denoted by ) with the same number to subdivide the original quadrilateral into 4 smaller quadrilateral. The resulting smaller quadrilaterals are marked with roman numerals. Shown in Figure 6.5(b). (Quadrilateral I is on the interface. Quadrilateral IV is on the Voronoi cell boundary.) The grid got in the transformation domain is mapped back into physical domain. The inclusion region is simply subdivided into triangles. Near the corner, a refined grid is used for the integration of inclusion domain Ω c. Shown in Figure 6.6. Along the radial direction, n r gauss = 8 for quadrilateral I and n r gauss = 5 for quadrilateral II,III and IV. Along the polar direction, n θ gauss = 8 for quadrilateral I and n θ gauss = 5 for quadrilateral II,III and IV. For the multi-elements case, the discretization for one element must consider the discretization of its neighbor elements for ensuring the nodes match along 117

136 the shared boundaries for superposition. Once the discretization for all the elements has been done individually, the discretization of the neighbor elements for each element will be checked. Shown in Figure 6.6(b), the solid lines are the discretization based on individual element, but the dashed line is added because of the neighbor elements. Due to the inherent nature of this NCM, a larger number of integration points are concentrated near the interface to capture the sharpest gradients exhibited by the reciprocal terms. 5 4 IV 5 III 4 II I 2 1 (a) (b) Figure 6.5: The division algorithm of the integration scheme for a typical Voronoi cell with concave shaped inclusion in transformation domain (a) mapped Voronoi cell in transformed domain (b) Subdivision of quadrilaterals in the matrix 118

137 1 2 (a) (b) Figure 6.6: (a) Subdivision for a typical Voronoi cell with concave shaped inclusion in physical domain (b) the adjustment of the discretization because of the neighbor elements 6.5 Numerical Examples The enhanced VCFEM for arbitrary shaped inclusion is verified by comparison with conventional finite element analysis. A real microstructure with several arbitrary shaped inclusions is used to do the full simulation for ductile fracture Comparison with commercial code In this example, VCFEM results are compared with those generated by the general purpose FEA code ABAQUS. As shown in figure 6.7, one Voronoi cell element with a square shaped inclusion is used. The matrix material is assumed to be a ductile porous material with elastic-plastic behavior characterized as Young s modulus E = 70GP a, Poisson s ratio v = 0.33; post yield behavior σ m = σ 0 (ǫ p m/ǫ 0 +1) N, with σ 0 = 175MPa, N = 0.2, and ǫ 0 = σ 0 /E is the uniaxial strain at yield. The brittle inclusion is assumed to be of SiC, with elastic properties as Young s modulus E = 450GPa, 119

138 Poisson s ratio v = For the properties of porous matrix material, the initial void volume fractions f in is set to without nucleation. L y L x Figure 6.7: A square plate with square shaped inclusion Real microstructure with multi-arbitrary shaped inclusions case This microstructure has been analyzed by using VCFEM and successfully compared with ABAQUS for elastic-plastic problem without any damage in Tiwary s work [95, 100]. Here, the same microstructure is used, and inclusion cracking and matrix cracking are considered. As shown in figure 6.10(a), it is SEM micrograph of a cast aluminum alloy Al319 microstructure with irregular silicon particulates. The size of this SEM micrograph shown in figure 6.10(a) is rectangular 70 70µm 2. Because of the complex shape of several inclusions in this microstructure, the multi-polygon is 120

139 200 Macroscopic stress (MPa) ABAQUS VCFEM Macroscopic strain (%) Figure 6.8: The macroscopic stress-strain response for a square plate with square shaped inclusion (a) (b) Figure 6.9: The microscopic equivalent plastic strain for square inclusion (a) ABAQUS (b) VCFEM 121

140 used to present some inclusions with complicated shape which can not be simplify presented by ellipses. The microstructure has 49 inclusions, which is tessellated into 49 Voronoi Cell elements including 5 arbitrary shaped inclusions, as shown in figure 6.10(b). The reinforcing phase of Si inclusions is assumed to be brittle and is modeled with the linear elastic properties: Young s modulus E = 165GPa, Poisson s ratio v = The aluminum matrix material is assumed to be ductile and is modeled as porous material with properties: Young s modulus E = 70GPa, Poisson s ratio v = The simulation is conducted under plane strain conditions. The displacement boundary condition is applied along the y-direction, as shown in figure 6.10(b). For the stress function, 88 polynomial terms and 36 reciprocal terms is generally used for matrix, 88 polynomial terms is used for inclusion. But for arbitrary shaped inclusions, additional wavelet terms will be adaptively added into the stress function for both matrix and inclusion around the corner of inclusions. For the inclusion cracking, a ellipse shaped void will still be used to present the crack inside inclusions. Also, another 36 reciprocal terms due to the crack shape will be added into the stress function for both of matrix and inclusion. For the Weibull model, σ w = 0.45GPa, m = 2.4 are used. For the non-local model, the average nearest neighbor distance is used as the Material Characteristic Length: M CL = 2.06µm. The figure 6.11 shows the macroscopic stress-strain response. The contour plot of void volume fraction is shown in figure As shown in these figures, all of the arbitrary shaped inclusions cracked more early than other ellipse shaped inclusions. The reasons are: the shape effect can introduce more load into the arbitrary shaped 122

141 10 µ m (a) (b) Figure 6.10: (a) SEM micrograph of a cast aluminum alloy Al319 microstructure with irregular silicon particulates and (b) VCFEM mesh and boundary conditions for 49 inclusions case including 5 arbitrary shaped inclusions inclusions, or the wavelet used for the stress function of these arbitrary shaped inclusions can also introduced high numerical errors. The figure 6.12 shows clear cracking path through the area with more inclusions. 6.6 Conclusions In this chapter, VCFEM is extend to take account of arbitrary shaped inclusions. The introduction of a new numerical conformal mapping for arbitrary shaped inclusions is discussed. Enhancement of stress functions with multi-resolution wavelet functions due to the presence of arbitrary shaped inclusions are presented. The corresponding numerical integration of element matrices is presented also. Numerical effectiveness of VCFEM for arbitrary shaped inclusions are demonstrated through 123

142 Macroscopic Stress (MPa) Macroscopic Strain (%) Figure 6.11: Macroscopic stress-strain response for the real microstructure with 5 arbitrary shaped inclusions 124

143 Figure 6.12: The contour plot of void volume fraction for the real microstructure with 5 arbitrary shaped inclusions. The number in the figure indicate the sequence of inclusion cracking. 125