Numerical Simulation of Mechanical Response of Geomaterials from Strain. Hardening to Localized Failure

Transcription

1 Numerical Simulation of Mechanical Response of Geomaterials from Strain Hardening to Localized Failure by MohammadHosein Motamedi B.Sc. (Ferdowsi University of Mashhad) 2005 M.Sc. (Sharif University of Technology) 2008 Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering in the Graduate College of the University of Illinois at Chicago, 2016 Chicago, Illinois Defense Committee: Craig D. Foster, Chair and Advisor Farhad Ansari Ahmed A. Shabana, Department of Mechanical and Indutrial Engineering, UIC Didem Ozevin Eduard Karpov Sheng-Wei Chi

2 Copyright by MohammadHosein Motamedi 2016

3 To my dearest parents and two sweet sisters, Sara and Sanaz. All the support they have provided me over the years was the greatest gift anyone has ever given to me. iii

4 ACKNOWLEDGMENTS First and foremost, I would like to acknowledge Prof. Craig Foster without whom this study and research would not have been possible. Success of this study is due his consistent support and supervision. During my PhD studies, I have taken several courses with him and he has generously shared his knowledge with all of his students including me. For attaining such a solid academic background in the field of computational mechanics, I will owe him for the rest of my professional career. My special thanks also goes to Prof. Ansari who supported me and provided me an opportunity to join his research team for the first two years of my PhD program. I have gained much insightful knowledge in the field of structural health monitoring (SHM) during that time. In addition, I would like to give sincere thanks to my other committee members: Prof. Ahmed A. Shabana, Prof. Eduard Karpov, Prof. Didem Ozevin, and Prof. Sheng-Wei Chi all graciously took time from their busy schedules to review my thesis work and provide insightful suggestions. I have also benefited greatly from my fellow researcher PhD candidate David Weed, Thank you and the best of luck in your future endeavors. iv

5 CONTRIBUTION OF AUTHORS Chapter 1 is a literature review and highlights the significance of my research question. Chapter 2 represents a new modified version of the Sandia GeoModel for geomaterials to enhance the computational tractability, domain of applicability and robustness of the previous model. Chapter 3 represents part of the published manuscript (Motamedi, M.H., and Foster, C.D.: An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropic kinematic hardening for geomaterials, International Journal for Numerical and Analytical Methods in Geomechanics 39 (17): , 2014) for which I was the first author. My research advisor, Prof. Craig Foster contributed to the revising of the manuscript. Chapter 4 represents part of the published manuscript (Motamedi, M.H., and Foster, C.D.: An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropic kinematic hardening for geomaterials, International Journal for Numerical and Analytical Methods in Geomechanics 39 (17): , 2014) for which I was the first author. My research advisor, Prof. Craig Foster contributed to the revising of the manuscript. Chapter 5 represents part of the submitted manuscript (Motamedi, M.H., Weed, D.A., and Foster, C.D.: Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach, International Journal of Solids v

6 and Structures, 2015, In revision.) for which I was the first author. David Weed assisted me for implementing a novel fracture model to simulate the crack propagation, Prof. Craig Foster contributed to the revising of the manuscript. Chapter 6 represents part of the submitted manuscript (Motamedi, M.H., Weed, D.A., and Foster, C.D.: Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach, International Journal of Solids and Structures, 2015, In revision.) for which I was the first author. David Weed assisted me for implementing a novel fracture model to simulate the crack propagation, Prof. Craig Foster contributed to the revising of the manuscript. Chapter 7 represents part of the submitted manuscript (Motamedi, M.H., Weed, D.A., and Foster, C.D.: Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach, International Journal of Solids and Structures, 2015, In revision.) for which I was the first author. David Weed assisted me for implementing a novel fracture model to simulate the crack propagation, Prof. Craig Foster contributed to the revising of the manuscript. Chapter 8 represents part of the submitted manuscript (Motamedi, M.H., Weed, D.A., and Foster, C.D.: Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach, International Journal of Solids and Structures, 2015, In revision.) for which I was the first author. David Weed assisted me for implementing a novel fracture model to simulate the crack propagation, Prof. Craig Foster contributed to the revising of the manuscript. vi

7 Chapter 9 represents a nonlinear dynamic analysis of rail ballast under sinusoidal loads. It also includes a brief research report entitled, Elasto-Viscoplastic Modeling of Rail Ballast and Subgrade. This report was provided for National University Rail Center- NURail. I was the first author. My research advisor, Prof. Craig Foster contributed to the revising of the report. Chapter 10 represents the overall conclusions of this dissertation and includes the future directions of this research. vii

8 TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION ELASTO/VISCOPLASTIC CONSTITUTIVE EQUATIONS FOR A CAP PLASTICITY MODEL: NONASSOCIATED FLOW RULE WITH MIXED HARDENING Basic equations Viscoplasticity equations Application to the modified version of Sandia GeoModel Yield functions and plastic potentials Evolution laws for isotropic/kinematic hardening parameters Rate-dependent model Conclusion NUMERICAL IMPLEMENTATION FOR CAP PLASTICITY MODEL Implicit integration algorithm Efficiency and robustness improvements for numerical computations Uniform dimensionality Return mapping algorithm in principal stress axes A Priori shear/cap surface determination Conclusion SIMULATION RESULTS AND VERIFICATION Uniaxial tensile example Simple shear example Triaxial compression examples Triaxial extension vs. compression Compression/shear example Boundary value problem Conclusion STRONG DISCONTINUITY Kinematics and governing equations Onset of localization: bifurcation analysis Evolution of the displacement jump: post-localization model Conclusion viii

9 TABLE OF CONTENTS (Continued) CHAPTER PAGE 6 FINITE ELEMENT IMPLEMENTATION FOR POST-LOCALIZATION MODEL Assumed enhanced strain (AES) method Implicit integration scheme for post-localization model Conclusion NUMERICAL EFFICIENCY AND ROBUSTNESS FOR POST- LOCALIZATION MODEL Spurious solutions Impl-Ex integration scheme Conclusion NUMERICAL BENCHMARK PROBLEM Cracked Brazilian disc specimen Numerical simulation and discussion Conclusion ELASTO-VISCOPLASTIC MODELING OF RAIL BALLAST AND SUBGRADE Fitting procedure of GeoModel Linear elastic parameters: Young s modulus E and Poisson s ratio ν Shear failure envelope parameters Kinematic hardening parameters Material parameterization of the cap plasticity model for ballast material Dynamic simulation of the rail and substructures Conclusion CONCLUSIONS, AND FUTURE WORK Conclusions Future work Nonlinear elasticity Effect of interaction between specimen and loading platen in CBD tests Three-dimensional (3D) finite element simulation Coupled Multibody and FE Model for Simulating Vehicle- Track-Substructure Interaction APPENDICES Appendix A Appendix B ix

10 TABLE OF CONTENTS (Continued) CHAPTER PAGE CITED LITERATURE VITA x

11 LIST OF TABLES TABLE PAGE I MATERIAL PARAMETERS FOR SALEM LIMESTONE ROCK. 40 II III IV V VI VII VIII IX X CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR A SIMPLE SHEAR TEST CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR A TRIAXIAL COMPRESSION TEST β = CONVERGENCE OF GLOBAL ALGORITHM FOR A SET OF TRIAXIAL COMPRESSION TESTS CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR THE FIRST PLASTIC LOAD STEP OF COMPRESSION/SHEAR TEST CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR THE FIRST SHEAR LOAD STEP OF COMPRESSION/SHEAR TEST CONVERGENCE OF GLOBAL ALGORITHM FOR THE COM- PRESSION/SHEAR TEST CONVERGENCE OF GLOBAL ALGORITHM FOR THE SLOPE STABILITY PROBLEM WITH 400 ELEMENTS CONVERGENCE OF GLOBAL ALGORITHM FOR THE SLOPE STABILITY PROBLEM WITH 1600 ELEMENTS MATERIAL PARAMETERS FOR SALEM LIMESTONE ROCK USED IN THE CBD TEST XI MATERIAL PARAMETERS FOR BALLAST MATERIALS XII RAIL SUBSTRUCTURE MATERIAL PARAMETERS FOR DY- NAMIC LINEAR ELASTIC ANALYSIS [159] xi

12 LIST OF FIGURES FIGURE PAGE 1 Cap plasticity model: (a) three-dimensional view of the yield surface (the exterior free mesh surface) and plastic potential surface (the interior gray solid) in principal stress space; (b) octahedral view, which corresponds to looking down the hydrostatic axis (lines of triaxial compression (TXC), triaxial extension (TXE) marked) Cap function F c Two-dimensional representation of yield and potential surfaces in meridional stress space; deviatoric Modified yield and potential surfaces in meridional stress space ( J 2 versus mean stress I 1 ); dark zone shows the corner region issue tackled with tension cap; constant aspect ratio R = a/b denotes the eccentricity of the cap surface; the hardening compression cap initiates at the point with zero horizontal tangency, g/ I 1 = 0. g/ σ represents the direction of the plastic increment vector under nonassociated plastic flow rule Three-dimensional view of initial yield surface (the interior gray solid) evolution in principal stress space for: (a) isotropic hardening and (b) mixed isotropic-kinematic hardening Schematic interpretation of the implicit return mapping procedure under non-associated plastic flow in 2D stress space; (k + 1) indicates the current iteration Hardening behavior of compression cap in meridional stress space Uniaxial tensile test: mesh and boundary conditions; d t Indicates the vertical nodal displacement Axial stress-strain response for the uniaxial tensile test Stress path in meridional stress space for the uniaxial tensile test: Model with the tension cap (circle markers); Model without the tension cap (diamond markers) xii

13 LIST OF FIGURES (Continued) FIGURE PAGE 10 Uniaxial tensile test: mesh and boundary conditions; d t Indicates the vertical nodal displacement Stress-strain response for selected simple shear tests of a solid element with different values of η indicated by numbers next to each curve. Examples are run for time step t = 0.01 and strain rate of 4% per second Comparison of material response in associative vs. nonassociative models for selected simple shear tests of a single solid element. Results are plotted for inviscid solutions (η = 0) Triaxial compression test: mesh and boundary conditions Stress path in meridional stress space for selected triaxial compression examples. β denotes stress ratio (σ 3 /σ 1 ) Stress path in meridional stress space for the triaxial compression test, stress ratio, σ 3 /σ 1 = Stress path in meridional stress space for the triaxial compression test, stress ratio, σ 3 /σ 1 = Volumetric strain is plotted versus mean stress for selected triaxial compression examples. For reference the hydrostat is plotted as the dashed line. Arrows mark critical stress states (onset of dilatancy C and onset of shear enhanced compaction C. β stands for stress ratio σ 3 /σ Differential stress is plotted versus axial strain for selected triaxial compression examples Residual norm per iteration for the first plastic step in a set of triaxial compression tests Stress path for TXE in principal stress space: showing movement along the hydrostatic axis in the first load step, intersection with the initial yield surface and culminating at the final translated yield surface in the second load step Stress path for TXC in principal stress space: showing movement along the hydrostatic axis in the first load step, intersection with the initial yield surface and culminating at the final translated yield surface in the second load step xiii

14 LIST OF FIGURES (Continued) FIGURE PAGE 22 Axial stress-strain response for the TXE test at the deviatoric plane with pressure P=60 (MPa) Axial stress-strain response for the TXC test at the deviatoric plane with pressure P=60 (MPa) Two-step loading test: mesh and boundary conditions Stress path in meridional stress space for the designed compression/shear test. The letters indicate points on the stress path distinguishing different phases of evolution Stress-strain response for a compression/shear test. The letters indicate points on the stress path distinguishing different phases of evolution Residual norm per iteration for the first plastic step in both loading steps of the compression/shear test. Quadratic convergence is observed Slope stability problem. Gravity load applied before footing displacement u is prescribed Deformed shape for FE mesh with 400 bilinear quadrilateral elements: (a) horizontal displacement dx contours, (b) vertical displacement dy contours Deformed shape for FE mesh with 1600 bilinear quadrilateral elements: (a) horizontal displacement dx contours, (b) vertical displacement dy contours Footing load-displacement plot for two FE meshes FE mesh and selected elements (12 and 32) to draw stress path in meridional stress space Stress path in meridional stress space for the element 12 at the integration point, IP=2. The letters indicate points on the stress path to distinguish different phases of evolution Stress path in meridional stress space for a selected element 32 at the integration point, IP=2. The letters indicate points on the stress path to distinguish different phases of evolution xiv

15 LIST OF FIGURES (Continued) FIGURE PAGE 35 Body Ω with planar strong discontinuity S fixed at the reference configuration, (Ω=Ω + Ω S, Γ= Γ t Γ g ) Cohesive fracture law: (a) isotropic softening of the damage-like surface F = 0 in traction (σ n, τ s ) space (b) equivalent traction-separation relationship with corresponding loading-unloading paths. ζ max indicates the maximum attained equivalent separation Enhancing a CST finite element: (a) element breaks into two parts (triangle with bold lines represents the conforming deformation.) and (b) displacement field with a jump across a discontinuity plane Plot of blending displacement and discontinuous displacement for an enhanced CST finite element: (a) one node at positive side of S, (b)two nodes at positive side of S Flowchart for the numerical integration algorithm within a FE code: (a)implicit scheme, (b)impl-ex scheme Geometry and loading conditions of CCBD specimen subjected to mixed mode I/II loading Strategy for tracing the crack propagation path Crack propagation path simulation for CBD specimen with different inclination angles (β = 10, 30, and 55 ). Initial cracks are represented by the dash thick lines through the finite element mesh Load-displacement plots for CBD specimen with different inclination angles (β = 10, 30, and 55 ) Deformed mesh with enhanced solution for inclination angle (β = 10 ): (a)horizontal displacement contour dx, (b)vertical displacement contour dy Deformed mesh with enhanced solution for inclination angle (β = 30 ): (a)horizontal displacement contour dx, (b)vertical displacement contour dy Deformed mesh with enhanced solution for inclination angle (β = 55 ): (a)horizontal displacement contour dx, (b)vertical displacement contour dy xv

16 LIST OF FIGURES (Continued) FIGURE PAGE 47 Linear fit to the axial stress-axial strain response of the rail ballast, determining Young s modulus E Shear failure surface plotted using series of triaxial compression (TXC) tests conducted to failure [1] Shear failure surface parameters for rail ballast Illustration of Offset Parameter, N [212] Effect of Kinematic Hardening Parameter, c α [212] Shear failure surface, initial yield and plastic potential surfaces in meridional stress space ( J 2 versus mean stress I 1 ) Experimental data and corresponding numerical simulations for triaxial loading tests of ballast material. ɛ deviator and σ deviator denotes to axial strain and axial stress in the second load step of triaxial loading test, respectively Dimension of the rail substructures Railway track structure: (a) 3D model of rails, sleepers and earthen substructure, (b) Load curves applied to sleepers Vertical displacement contours for the rail substructures using dynamic linear elastic FE simulation Vertical displacement contours for the rail substructures using dynamic nonlinear elastoplastic FE simulation Elements with local convergence issues in stress calculation xvi

17 SUMMARY The Sandia GeoModel is a continuum elastoplastic constitutive model which captures many features of the mechanical response for geological materials over a wide range of porosities and strain rates. Among the specific features incorporated into the formulation are a smooth compression cap, isotropic/kinematic hardening, nonlinear pressure dependence, strength differential effect, and rate sensitivity. This study attempts to provide enhancements regarding computational tractability, domain of applicability, and robustness of the model. A new functional form is presented for the yield and plastic potential functions. The model is also furnished with a smooth, elliptical tension cap to account for the tensile failure. This reformulation renders a more accurate, robust and efficient model as it eliminates spurious solutions attributed to the original form. In addition, this constitutive model is adopted in bifurcation analysis to track the inception of new localization and crack path propagation. For the post-localization regime, a cohesive-law fracture model, able to address mixed-model failure condition, is implemented to characterize the constitutive softening behavior on the surface of discontinuity. To capture propagating fracture, the Assumed Enhanced Strain (AES) method is invoked. Particular mathematical treatments are incorporated into the simulation concerning numerical efficiency and robustness issues. Finally, the aforementioned modified cap plasticity model is employed to investigate the nonlinear dynamic response of the earthen substructure of the rail. Studying the effects of high-speed trains on the track substructure. xvii

18 CHAPTER 1 INTRODUCTION (Previously published in Motamedi, M.H., Foster, C.D., An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropic kinematic hardening for geomaterials. International Journal for Numerical and Analytical Methods in Geomechanics, 39(17), pages: , 2015.) Geological materials are broadly studied in nature and utilized in engineering applications. Numerical simulation of geotechnical and geological structures, especially in the platform of the finite element method, has attracted much research interest with the advent of modern computational resources and facilities. One crucial part of a finite element (FE) simulation is the selection of an appropriate constitutive material model. Numerical modeling for geomaterials is by no means trivial since it necessitates the incorporation of various aspects of behavior that can be manifested. At low confining pressure, localized deformation in the form of shear and/or dilation bands or fractures may occur due to the growth and coalescence of micro-cracks and pores. At high confining pressure, on the contrary, delocalized irreversible deformation may occur in the form of shear-enhanced compaction and pressure-sensitive yielding. The latter responses, generally accompanied by material hardening, are the results of pore collapse, grain crushing, internal locking and other relevant microphysical mechanisms. Further compounding the complexity of the response to loading, these materials typically contain macroscopic inherent inhomogeneities such as natural flaws, cracks, joint sets, and 1

19 2 bedding planes. The existence of macro-structural heterogeneities in geo-systems renders them vulnerable to catastrophic failures (as for instance observed at the failure of Malpasset Dam in France and the collapse of the Vajont landslide in Italy) at load levels drastically lower than those expected for intact structures. In fact, brittle faulting (or fracturing) can be triggered from these zones and propagate under crack opening (mode I), sliding (mode II) or a mixture of the modes. Notably, in gravity dams most of the observed cracks are created in a mixed-mode fashion [2, 3], and hydraulic fractures propagate in mixed-mode conditions from the wall of the inclined and horizontal wellbores [4]. Particle mechanics models such as the Discrete Element Method (DEM) [5 7] and Lattice Element Method (LEM) [8, 9] have been developed to investigate micromechanical features of complex geomaterials. Despite the remarkable insight those models provide, they need significant computational power to realistically achieve engineering scale problems. In order to reduce computational effort, multiscale computational approaches have been developed. These methods are generally classified as either hierarchical or concurrent schemes [10 14]. However, the micromechanically-based evolution of plastic internal variables (PIVs) adopted in these models is nonsmooth. Accurately handling these nonsmooth evolution laws necessitates special treatments which are numerically cumbersome. A work by Tu et al. [15] makes a clear attempt to address this issue. Further, nonsmooth evolution laws may not be integrable using implicit methods and hence necessitate small step sizes for stability. This limitation underscores its deficiency when the model is applied in numerical simulations for some practical cases such as

20 3 tunneling or excavation in which the load increment could be locally large, and the step size cannot be greatly reduced to achieve required accuracy. On the other hand, recent studies have revealed numerous complexities in the inelastic behavior and failure mode of geomaterials [16, 17]. For instance, sedimentary rocks, including sandstone and limestone, feature a variety of micromechanical processes and basic differences with respect to inelastic compaction. In sandstone, inelastic compaction is most likely associated with an intragranular cracking phenomenon arising from local tensile stress concentration at grain contacts. In limestone, by contrast, pore collapse engendered by stress concentrations at the surface of the pores dominates the compaction processes. A comprehensive review of the different damage processes of porous rocks in both brittle faulting and cataclastic flow has been presented in Ref. [17]. Consequently, micromechanical models, even cast in a multi-scale framework, are unable to integrate the full complexities of geomaterials in a consistent and unified manner with acceptable computational cost. Elastoplasticity is still the most widely acknowledged method to capture material nonlinearities and inelastic behavior in geomaterials. During the last decade, much research work has been devoted in the framework of elastoplasticity to capture one or more complex, yet important, features of geomaterial behavior [18 24]. Recently, Tengattini et al. [24] proposed a thermomechanical-based continuum constitutive model for cemented granular materials. Although the model successfully employed scalar internal variables to describe grain crushing and cement damage process, the authors point out that it is unable to replicate deformation induced anisotropy associated with dilatant failure mode.

21 4 Cap plasticity models such as the one originally proposed by [25], have occupied a prominent place in continuum geomechanics because of their versatility and capability to capture complex features of geomaterials such as soils, rocks, and concrete [1, 26 30]. For traditional cap models, some difficulties arise from the singularities induced by the nonsmooth intersection of the compression cap and shear surface. While specific algorithms have been developed to handle non-smooth intersections, they often introduce significant algorithmic and numerical complexities [31]. To circumvent this issue, a number of smooth variations of the multisurface plasticity models have been proposed [32 36]. Even though recent advances in the development of three-invariant cap plasticity theory endowed models with improved local predictive capability, they still suffer from difficulties in numerical implementation. In explicit [1, 34] and semi-implicit [37] implementations, the lack of unconditional stability may require very small time steps for large-scale problems. On the other hand, fully implicit schemes [38,39] can require complex algorithms that are costly at the local level and may suffer from robustness issues. The Sandia GeoModel is an advanced three-invariant cap plasticity model which has been developed for application to geomaterials [1]. The parameterization procedure has been carried out for many geological materials including frozen soil, limestone, tuff, diatomite, granodiorite, and concrete. To increase computational efficiency, Foster et al. [38] presented an algorithm for the implicit integration of this model in the case of combined isotropic and kinematic hardening using the spectral decomposition of the relative stress. In the aforementioned study, all numerical examples were carried out on a single element using the associative version of

22 5 the model to illustrate the quadratic convergence of the proposed algorithm. Regueiro and Foster [39] conducted a bifurcation analysis on this three-invariant cap plasticity model to investigate the onset and orientation of strain localization. They again provide a number of numerical examples on a single 3D solid element. Sun et al. [37] adopted this three-invariant cap model coupled with a fluid flow model to simulate the hydro-mechanical interactions of fluidinfiltrating porous rocks. To bypass the numerical difficulties, they employ a refined explicit integration algorithm whose accuracy and stability both depend significantly on the chosen time step size. Over the past decades multiple laboratory tests with different configurations (including circular disc [40, 41], semi-circular disc [42, 43], round bar and rectangular-beam [44]) have been designed to investigate the mixed mode fracture behavior in geomaterials. Among these tests is the cracked Brazilian disc (CBD) specimen, which has been frequently utilized for rock materials due to the amenity of test specimen preparation from the rock cores. The other reasons in adopting the CBD specimen in laboratory studies are: relatively straightforward test procedure with regard to the application of compressive load rather than tensile load, and the capability to easily replicate various combinations of mode I and II fracture propagation by varying the initial crack inclination angle relative to the direction of diametral applied load. For many applications, mixed-mode fracture in rocks has been simplified to empirical approaches [45, 46] or linear elastic fracture mechanics (LEFM) in which conventional fracture theories (such as the maximum tangential stress criterion (MTS) [47], the minimum strain energy density criterion [48], the maximum energy release rate criterion [49]) have been used to

23 6 predict resistance to and direction of the fracture growth. LEFM-based methods assume that the energy dissipation is only created in an infinitesimal area at the tip of the crack, where the stress field is singular. To improve the theoretical predictions in accord with experimental results, Smith et al. [50] proposed the generalized MTS (GMTS) criterion by taking into account the effect of the T-stress, the first non-singular terms of the Williams series expansion, in the stress field around the crack tip. This modified criterion has been employed several times for investigation of fracture propagation in CBD specimens ( [42, 51], among others). Later, Ayatollahi and Akbardoost [52] added the second non-singular terms in to the calculations to obtain more accurate results. However, these non-singular stress terms are dependent on the geometry and loading conditions of the cracked body, which is not desirable from practical point of view in solving complex engineering problems. In addition, as Rubin [53] remarked, at depths of geological interest h > 1000m the fracture process zone (FPZ) is large enough to invalidate LEFM and the softening process in the FPZ should not be neglected. In the case of quasi-brittle materials, the theory of the cohesive zone model, which may be traced to the pioneering work of Dugdale [54] and Barenblatt [55], is an appropriate alternative to describe the micromechanical features underlying the initiation and evolution of damage in the FPZ. This model interprets FPZ as a fictitious interface along which the displacement jump (separation) is related to transferring cohesive forces. Cohesive zone models have been proposed in variational settings (see [56] for a review of models) but the most well known are the potential-based laws by [57] and [58] and the linear laws by [59] and [60].

24 7 Numerical simulation of cohesive crack growth with the finite element method may be carried out with different approaches. One of the traditional approach is discrete-crack model employing interface elements so that the crack is allowed to propagate along the boundaries of the finite elements. Though several attempts have been made in a case of crack paths unknown a priori to generate robust and reliable tools for automatic remeshing procedure (see for example [61, 62]), discrete-crack models still suffer from some shortcomings, such as stresslocking (spurious stress transferring across crack surfaces in widely opening mode) and mesh bias. To alleviate these numerical difficulties, fine meshes are needed which lead to large-scale nonlinear systems and in consequence prohibitively expensive computational costs. In recent years, models with cracks embedded in finite elements including either nodal enrichment, the extended or Generalized finite element method (XFEM or GFEM) [63 66], or elemental enrichment, the Embedded finite element method (EFEM) [67 71] have been applied successfully to simulation of fracture propagation. A comparative study provided by [72] has shown that EFEM is computationally more efficient than XFEM, though it cannot accurately capture the crack tip stresses. In addition, by focusing on slip patterns, Borja [73] has elucidated that the extended FE solutions require higher-order crack tip enhancement in order to fully capture the strain singularity at the crack tip and EFEM could predict larger slip (i.e., softer response) compared to the XFEM solutions otherwise. Due to some of the appealing features of the EFEM, computational efficiency being a primary one, the authors of this work have chosen to adopt this method for the simulation of mixed-mode fracture in CBDs. A comprehensive computational package which encompasses

25 8 both inelastic hardening and subsequent material softening is featured. Specifically, the inelastic response is captured by the modified Sandia GeoModel presented in [74], and as for the localized response by the mixed-mode cohesive zone model in [67]. The remainder of this work is organized as follows: Chapter 2 demonstrates in detail a recently modified cap plasticity model for geomaterials, the Sandia GeoModel. As a result, the Sandia GeoModel yield function and relevant plastic potential are reformulated without a need to change material parameters. This new formula eliminates spurious solutions attributed to original form of the yield function. The smooth elliptical tension cap added to the model extends the domain of applicability to the tensile loading. In Chapter 3, the numerical implementation of this model in illustrated. The model is applied to a fully implicit, unconditionally stable time integration algorithm using a returnmapping scheme in principal relative stress (or strain) directions. The yield function reformulation increases computational efficiency for both local and global iterative solutions. A novel a priori shear/cap surface determination algorithm is introduced to reduce the computational cost. Nonassociative models generally provide a more realistic response for low porous geomaterials, particularly with respect to dilation. Hence, the algorithmically consistent elastoplastic modulus is also derived based on nonassociated plasticity flow rule to better describe volumetric material behavior.

26 9 In Chapter 4, various numerical examples including a large-scale boundary value problem (BVP) are presented to demonstrate the fidelity of the modified model and to analyze its numerical performance. In Chapter 5, first, the kinematics of a strong discontinuity are outlined. Second, to capture the fracture initiation and its orientation, bifurcation analysis is provided on the basis of the constitutive model proposed in the previous Chapter. At the end of Section 5, a cohesive fracture model is summarized to describe evolution of the damage with either coupled opening/sliding displacements or solely frictional slip mode. In this post-localization model, the form of the traction-separation law is directly given generating energetically path-dependent model by taking into account different specific fracture energies for modes I and II. In Chapter 6, first, the finite element approximation in the platform of EFEM using assumed enhanced strain (AES) method is briefly discussed. The numerical implementation of the models for softening responses is given at the end of this Chapter. Afterwards in Chapter 7, the numerical efficiency and robustness of the post-localization model are discussed. In particular, we adopt an implicit-explicit integration scheme in the spirit of [75] in order to enhance simulation robustness and numerical efficiency. In Chapter 8, as a benchmark example the CBD specimen is used to investigate the mixed mode fracture behavior of pre-cracked limestone rock and compares finite element solutions with experimental results available in the literature.

27 10 In Chapter 9, the elasto-viscoplastic modeling of the rail ballast is provided using the material constitutive model presented in Chapter 2. Furthermore, the nonlinear dynamic response of the rail substructure under a sinusoidal loading is investigated. Finally, Chapter 10 summarizes the thesis and gives directions for future research.

28 CHAPTER 2 ELASTO/VISCOPLASTIC CONSTITUTIVE EQUATIONS FOR A CAP PLASTICITY MODEL: NONASSOCIATED FLOW RULE WITH MIXED HARDENING (Previously published in Motamedi, M.H., Foster, C.D., An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropickinematic hardening for geomaterials. International Journal for Numerical and Analytical Methods in Geomechanics, 39(17), pages: , 2015.) In this Chapter, the continuum constitutive equations are formulated for an advanced threeinvariant cap plasticity model within the framework of infinitesimal elastoplasticity. This model is a modified version of the Sandia GeoModel which has been developed for application to geomaterials [1]. Specific features incorporated to this model include a nonlinear pressure sensitive shear surface, smooth compression and tension cap surfaces, strength differential (SD) effect, combined isotropic-kinematic hardening, and plastic flow rule non-associativity. 2.1 Basic equations For the infinitesimal strain case, the strain tensor can be approximated by ɛ = 1 2 [ u + ( u) T ] (2.1) 11

29 12 where u is the displacement field The stress rate constitutive equation for linear isotropic elasticity can be written as: σ = C e : ɛ; C e = λ µI (2.2) where 1 is the second order identity tensor, I (I ijkl = 1/2(δ ik δ jl + δ il δ jk )) is the fourth-order symmetric identity tensor, λ and µ are the Lamé parameters and C e is the fourth-order isotropic elasticity tensor. 2.2 Viscoplasticity equations The hypothesis of small deformations and rotations allows an additive decomposition of the total strain rate ɛ into the elastic and inelastic parts as below ɛ = ɛ e + ɛ p (2.3) The inelastic strain ɛ in will be denoted as plastic strain ɛ p for the classical rate-independent plasticity and viscoplastic strain ɛ vp for the rate-dependent material behavior. The closed convex elastic domain in the stress space is expressed as: E := {σ S : f(σ, q) < 0} (2.4) E := {σ S : f(σ, q) = 0} (2.5)

30 13 where S is stress space and f(σ, q) denotes the yield function whose zero level set provides the boundary of the elastic domain. In this work, the evolution of the stress-like plastic internal variables q is characterized in terms of a phenomenological hardening law as below q = γh q (ɛ p, q) (2.6) Here γ is the plastic consistency parameter, which denotes the magnitude of the plastic flow rate. By using a plastic potential function g, a non-associated plastic flow rule is derived in a form of ɛ p = γ g σ (2.7) The flow rule is associative if the material parameters are chosen such that f = g. In rate form, the consistency condition is utilized to find the non-negative parameter γ f = f σ : σ + f q : q (2.8) then solving the equation for γ concludes to γ = 1 f χ σ : Ce : ɛ (2.9) in which χ = f σ : Ce : g σ f q : hq (2.10)

31 14 Eventually, the continuum elasto-plastic tangent C ep can be derived as the following σ = C ep : ɛ; C ep = ( C e 1 χ Ce : g σ f ) σ : Ce (2.11) 2.3 Application to the modified version of Sandia GeoModel In this section the formulation of a three-invariant, rate-dependent and nonassociative cap plasticity model is described. The model is a modified version of the Sandia GeoModel [1] and comprises a nonlinear shear limit-state surface as well as elliptical compression and tension caps. Figure 1a shows the 3D representation of the model in the principal stress space. The Gudehustype model is adopted using the third stress invariant to replicate the strength differential (SD) effect, which is the difference in strength of geomaterials observed in triaxial compression and extension tests. The rounded triangular shape of the model in deviatoric planes is illustrated in Figure 1b. Triaxial compression (TXC) and triaxial extension (TXE) denote the two main canonical stress paths Yield functions and plastic potentials Assuming that the yielding behavior is isotropic, the yield function f and plastic potential function g can be expressed in terms of stress invariants (I 1, J 2 and J 3 ). In the case of kinematic hardening, a deviatoric backstress tensor α is presented to capture the Bauschinger effect, such that the relative stress tensor can be defined as ξ = σ α. Given a back stress with an

32 Figure 1: Cap plasticity model: (a) three-dimensional view of the yield surface (the exterior free mesh surface) and plastic potential surface (the interior gray solid) in principal stress space; (b) octahedral view, which corresponds to looking down the hydrostatic axis (lines of triaxial compression (TXC), triaxial extension (TXE) marked). 15

33 16 appropriate translation rule [38], the yielding of the material may be expressed in terms of invariants of the relative stress (I 1, J ξ 2 and J ξ 3 ) as below I 1 = tr(ξ) = tr(σ), (tr(α) = 0) (2.12a) J ξ 2 = 1 dev(ξ) : dev(ξ) (2.12b) 2 J ξ 3 = det(dev(ξ)) (2.12c) Many cap plasticity models have been proposed, for example [27,30,36,76]. In this work, we follow a smooth cap formulation initially proposed by [1]. The yield function f and conjugated plastic potential g take the following form: f = Γ(β ξ ) J ξ 2 F c (F f N) (2.13) g = Γ(β ξ ) J ξ 2 Fc g (F g f N) (2.14) where the material parameter N indicates the maximum allowed translation of the initial yield surface during kinematic hardening and Γ accounts for the difference in triaxial extension vs. compression strength. Here Γ(β ξ ) = 1 2 [1 + sin(3βξ ) + 1 ψ ( ) 1 sin(3β ξ ) ] (2.15) in which

34 17 ( β ξ (J ξ 2, J ξ 3 ) = 1 3 ) 3J ξ 3 sin 1 3 2(J ξ 2 )1.5 (2.16) The Lode angle β ξ is the function of second and third invariants of the deviatoric relative stress devξ. The material constant ψ stands for the ratio of triaxial extension strength to compression strength. The strength differential (SD) effect arises from pressure-induced friction on microcrack faces and grain boundaries, which enables the material to withstand higher loads when the most critical surface is under higher compression. function F f and the corresponding plastic potential surface F g f The exponential shear failure are given as F f (I 1 ) = A C exp(b I 1 ) θi 1 (2.17) F g f (I 1) = A C exp(l I 1 ) φi 1 (2.18) The shear failure surface F f captures the pressure dependence of the shear strength of the material where A, B, C and θ are all non-negative material parameters determined from peak stress experimental data, using a procedure described in Fossum and Brannon [1]. L and φ are determined from experimental measurements of volumetric plastic deformation. It is worth mentioning that for geomaterials nonassociated plasticity is usually needed to realistically describe volumetric deformation [77 79]. As pointed out by [80], nonassociativity in geological materials can be described with structural rearrangement. This physical phenomenon has been observed in conjunction with growth of microcracks, propagation of shear

35 18 bands and pressure dependence of frictional shear resistance. The cap function F c, as graphically illustrated in Figure 2, couples hydrostatic (either tensile or compressive) and deviatoric stress-induced deformation. Unlike the previous forms of the model, which only included the compression cap, the new formula, Equation 2.19, generates smooth elliptical tension and compression caps to the yield function. F c (I 1, κ) = 1 H(κ I 1 ) ( ) I1 κ 2 ( H(I 1 I T I1 I T ) ) X(κ) κ 3T I1 T (2.19) where κ stands for the branch point in which combined porous/microcracked yield surface deviates from the nonporous profile for full dense bodies. The function X(κ) is the intersection of the cap surface with the I 1 axis in the J 2 versus I 1 plane and implies the position at which pressure under pure hydrostatic loading would be sufficient to prompt grain crushing and pore collapse mechanisms. X(κ) = κ RF f (κ) (2.20) The material parameter R governs the aspect ratio of the compression cap surface. The corresponding functions for the plastic potential g are written as F g c (I 1, κ) = 1 H(κ I 1 ) ( ) I1 κ 2 ( X g H(I 1 I T I1 I T ) ) (κ) κ 3T I1 T (2.21) X g (κ) = κ QF f (κ) (2.22)

36 19 where Q is a material parameter analogous to R. Parameters I T 1 and T indicate the initiation point of the tension cap and the hydrostatic tensile strength of the material, respectively. A number of standard tension tests have been developed to measure these mechanical parameters [81, 82]. This modification makes the model more applicable to the problems (e.g. retaining walls and hill slopes) in which the stress paths may experience some tension. Figure 2: Cap function F c. Initially, many plasticity models included a tension cut-off type surface to account for tensile failure phenomenon. This simplification introduces an additional singularity problem (corner region problem), at the intersection of shear and tension surfaces. As a result, [35] proposed a model accompanied with a circular tension cap. This circular meridian section may not be as accurate as elliptical forms whose aspect ratio property allows a better representation of experimental data.

37 20 Figure 3: Two-dimensional representation of yield and potential surfaces in meridional stress space; deviatoric Modified yield and potential surfaces in meridional stress space ( J 2 versus mean stress I 1 ); dark zone shows the corner region issue tackled with tension cap; constant aspect ratio R = a/b denotes the eccentricity of the cap surface; the hardening compression cap initiates at the point with zero horizontal tangency, g/ I 1 = 0. g/ σ represents the direction of the plastic increment vector under nonassociated plastic flow rule. The simple, yet important, modification is provided in the format of the yield functions, Equation 2.13 and Equation 2.14, via individually taking the square root of all terms involved in the formula such that no adjustment is needed in material parameters. This modification is a change to previous versions of the model, which used squared versions of each term on the righthand side. While the original form tends to be at an advantage in providing simplified yield function derivatives for user friendly FE implementation, it may trigger spurious solutions in solving the nonlinear optimization problem. The optimization problem for modeling of plasticity is accompanied by set of equalities and inequalities, known as the Kuhn-Tucker conditions and derived based on the rules of Lagrange multipliers method [83, 84]. In addition, this alternative form enhances computational efficiency and robustness in the framework of iterative integration

38 21 algorithm. The comparative results will be presented later in Section 4 to underscore the importance of this reformulation. As shown in Figure 3, the plastic strain increment vector (which is perpendicular to the plastic potential surface for nonassociated plastic flow) is horizontal at two extremities T and X. This characteristic is in accordance with reality in that no incremental deviatoric plastic strain develops for hydrostatic compression and tension load paths Evolution laws for isotropic/kinematic hardening parameters The use of an inadequate number of internal variables, which describe multiple aspects of failure modes, contributes to an inaccurate mechanical response prediction, especially when a model is implemented in a large-scale finite element analysis. The inelastic deformation of geomaterials is commonly categorized into the brittle or ductile fields and the failure mode may undergo a transition from one field to another [17, 85]. Under low confinement, inelastic deformation primarily involves shear deformation along with volumetric dilation induced by the microcracking and frictional sliding of the grain fragments. At the higher level of deviatoric stress, the brittle faulting mode culminates in strain localization which appears in the form of shear or combined shear/dilation bands. As the confining pressure increases, deformation mechanisms including grain crushing and pore collapse will become predominant, leading ultimately to compactant cataclastic flow. This ductile flow commonly operates in a delocalized manner and is accompanied by shear-enhanced compaction and strain hardening phenomena. In this study, the plasticity model is furnished with two internal variables, α and κ. In the shear regime, the deviatoric back stress tensor is adopted to produce directional effects

39 22 on the kinetics of the plastic flow (the so-called Bauschinger effect). Particularly, while stressinduced microcracks begin to propagate through the material, initial yield surface is permitted to translate with respect to the hydrostatic axis toward the failure envelope. The contact of the translated yield surface with this limit may indicate the inception of the softening localization due to growth and coalescence of microcracks. Evolution of the back stress is related to the deviatoric plastic strain and takes the following form: α = γh α ( ) ( = c α G α ɛp g dev = c α G α ɛ p σ g ) 1 I 1 (2.23) Above, γ is the consistency parameter, and c α is the constant that controls the rate of hardening. The scalar function G α limits the growth of the back stress as it approaches the failure surface. ( α : α ) 1 G α 2 = 1 2N 2 (2.24) For the compactant cataclastic flow regime, a scalar hardening parameter κ is defined, associated with the cap surface, to endow the yield surface with isotropic expansion. As extensively discussed in [86], describing the grain crushing through the progressive loss of material integrity is disputable. Hence, this deformation mechanism should be regarded as the evolution to a less porous and stronger state [24]. As a result, evolution of κ should be dependent on plastic volumetric strain ɛ p v. κ = γh κ ; h κ (κ) = ( ɛ tr p ( ɛ p v X ɛ p X κ ) ) = ( ɛ p v X 3 g I 1 X κ ) = 3c κ G κ g/ I 1 (2.25)

40 23 Here, the following form of volumetric plastic strain is used. ɛ p v = W {exp [D 1 (X(κ) X 0 ) D 2 (X(κ) X 0 ) 2 ] 1} (2.26) In the above, W, D 1 and D 2 are material parameters, X 0 = X(κ 0 ) is the end point of initial cap, and κ 0 is the initial value for cap parameter. To be more descriptive, combined isotropic/kinematic hardening of the cap model is visualized by the schematic diagram in Figure 4. It should be mentioned that although the softening-induced localization behavior is also of importance in many geological and geophysical problems [71, 79, 87, 88], it is beyond the scope of the this section and will be discussed in Chapter 5. Figure 4: Three-dimensional view of initial yield surface (the interior gray solid) evolution in principal stress space for: (a) isotropic hardening and (b) mixed isotropic-kinematic hardening.

41 Rate-dependent model The viscoplasticity is taken into account using the type of overstress model originally proposed by Duvaut and Lions [89]. A viscoplastic strain rate, ɛ vp, is adopted rather than ɛ p to express the rate-dependent constitutive equations as below ɛ vp = 1 η Ce 1 : (σ σ) (2.27) α κ = 1 η α α κ κ (2.28) here, σ, α and κ are solutions for the rate-independent case. The relaxation time η (0, + ) is devised to the formula such that for the lower bound η 0 +, the inviscid plastic solution is obtained and we attain the elastic solution when η +. This model is numerically appealing due to its simplicity and ability to exploit the existing framework of classical rate-independent plasticity. 2.4 Conclusion An advanced phenomenological continuum constitutive model is developed in this chapter. The Sandia GeoModel yield function and relevant plastic potential are reformulated without a need to change material parameters. This new formula eliminates spurious solutions attributed to original form of the yield function. The smooth elliptical tension cap added to the model extends the domain of applicability to the tensile loading. The shear yield surface translates using a kinematic hardening law. Moreover, the compression cap surface is furnished with the

42 25 scalar hardening parameter to replicate grain crushing and pore collapse mechanisms in high confining pressure with isotropic expansion of the yield surface.

43 CHAPTER 3 NUMERICAL IMPLEMENTATION FOR CAP PLASTICITY MODEL (Previously published in Motamedi, M.H., Foster, C.D., An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropickinematic hardening for geomaterials. International Journal for Numerical and Analytical Methods in Geomechanics, 39(17), pages: , 2015.) In this chapter, the numerical implementation of the proposed model is presented. Since analytical solutions are almost never available for complex boundary value problems, this step is essential for application of the model to engineering problems. In addition to the creation of the tension cap and form of the yield function mentioned previously, we have made some modifications to the numerical algorithm as described below. In particular, we have modified the residual to have uniform units, improving the robustness of the Newton-Raphson procedure, in Section Additionally, we have developed a method to determine whether the stress is on the shear surface or cap surface from the trial state, as discussed in Section This modification improves efficiency as the number of active internal state variables changes depending on which part of the yield surface the stress state lies. Previously, one had to guess which part of the yield surface the stress state was on, and run a second set of iterations if the first guess was incorrect. 26

44 Implicit integration algorithm Let [0, T ] R be the time interval of interest. At time t n [0, T ], the values of plastic strain, stress and internal state variables are known from the previous analysis. Considering a straindriven problem, we attempt to update these variables to t n+1 via the evolution equations for a given strain increment, ɛ n+1. The well established stress integration algorithm recognized as a predictor-corrector method (often called return mapping) is adopted. Consequently, in the first step, the elastic trial stress σ tr n+1 = σ n + C e : ɛ n+1 is estimated by freezing plastic flow during the time step [t n, t n+1 ], and in the second step, the plastic flow rule, Equation 2.7, is applied together with optimality conditions to find a point projection of the trial stress state, (i.e. {σ n+1, α n+1, κ n+1, γ n+1 }, onto the updated curve of the yield surface, f n+1. In addition, fully implicit or backward Euler difference scheme is employed to derive evolution equations and conditions in the discrete form. This approximate numerical technique provides first-order accuracy as well as unconditional stability even for larger strain increments. In comparison, traditional explicit and semi-implicit integration schemes exhibit only conditional stability. The Newton-Raphson (N-R) iterative method is used and hence the system has to be solved as many times as required to converge to the final solution. X k+1 n+1 = Xk n+1 [ ( ) ] DR k 1 R k n+1 (3.1) DX n+1

45 28 in which X and R denote the vectors of unknown variables and residual, respectively; k + 1 refers to the current iteration. The inverse of matrix DR/DX is not explicitly computed, and the equations are solved using Gaussian elimination. X = [σ 11, σ 22, σ 33, σ 23, σ 31, σ 12, α 11, α 22, α 23, α 31, α 12, κ, γ] T (3.2) γc e 11kl ( g/ σ kl) σ 11 + σ tr 11 γc e 22kl ( g/ σ kl) σ 22 + σ tr 22 γc e 33kl ( g/ σ kl) σ 33 + σ tr 33 γc e 23kl ( g/ σ kl) σ 23 + σ tr 23 γc e 31kl ( g/ σ kl) σ 31 + σ tr 31 R = γc e 12kl ( g/ σ kl) σ 12 + σ tr 12 γ(h α ) 11 α 11 + (α 11 ) n = 0 (3.3) γ(h α ) 22 α 22 + (α 22 ) n γ(h α ) 23 α 23 + (α 23 ) n γ(h α ) 31 α 31 + (α 31 ) n γ(h α ) 12 α 12 + (α 12 ) n γh κ κ f 13 1 where subscript n+1 is left off to simplify notation. Here α 33 = (α 11 +α 22 ) can be eliminated since the back stress is deviatoric. Even condensing out α 33, we are left with 13 equations and

46 29 13 unknowns. The linearized system has to be solved several times as we iterate to find the solution. Figure 5 demonstrates the return mapping algorithm used for nonassociated plasticity. Figure 5: Schematic interpretation of the implicit return mapping procedure under nonassociated plastic flow in 2D stress space; (k + 1) indicates the current iteration. It is well known that to obtain a quadratic rate of convergence for global-level iterations, a tangent matrix must be formulated in a manner consistent with the update algorithm, Equation 3.3 [90]. For a converged solution, a variation in strain does not cause a variation of the residuals. In other words: The term ( ) DR = Dɛ n+1 ( ) R + ɛ n+1 ( ) R X n+1 ( ) R ɛ can be obtained and moved to the RHS as below n+1 ( ) X = 0 (3.4) ɛ n+1

47 30 ( ) R X ( ) X = ɛ 13 6 ( ) R = ɛ 13 6 C e (3.5) therefore ( ) X = ɛ 13 6 ( ) R 1 X C e (3.6) which can be rewritten as ( ) X = ɛ 13 6 (C e ) 1 + γ 2 g σ σ γ 2 g σ q g/ σ γ ( h q / σ) 1 γ ( h q / q) h q ( f/ σ) t ( f/ q) t 0 } {{ } A I (3.7) Hence, the consistent tangent operator C n+1 = ( σ/ ɛ) n+1, the upper left 6-by-6 submatrix of ( X/ ɛ) n+1, can be derived from the upper left 6-by-6 submatrix of A. It should be noted that for a nonassociative model, the consistent tangent modulus and resulting stiffness matrix lose their major symmetry. This asymmetric property may lead to more intensive computations. Thus, appropriate algorithms must be applied so that the inverse A 1 can be found much more efficiently.

48 Efficiency and robustness improvements for numerical computations In elastoplastic analyses, the use of an efficient algorithm plays a key role in the numerical implementation to work in a general boundary-value problem. In order to improve computational efficiency particular mathematical treatments may be noted Uniform dimensionality The residual vector R with the lack of uniform dimensionality may considerably reduce efficiency of the iterative equation solver, Equation 3.1, as an increase the condition number (the ratio between the largest and smallest eigenvalues) of the local tangent operator DR/DX may result. However, we have revised Equation 2.13 and Equation 2.14 such that it now has units of stress rather than stress squared. Aligning the units of this equation with the other equations in the residual vector improves the conditioning of the local tangent matrix so that the system of equation can be solved much more effectively. In addition, as described in Box 1, because of the nature of the residual vector, convergence of each component should be fulfilled. As pointed out by [38], in some examples one larger component may hamper the quadratic convergence of other components and consequently increases the number of iterations needed to obtain the final solutions. This drawback is also solved by the use of new format of the yield function Return mapping algorithm in principal stress axes The standard return mapping algorithm in principal stress axes was first adopted for isotropic plasticity models; accordingly, the iterative equation solver needs to evaluate only the principal values of the state variables. Thereafter, Foster et al. [38] extended the appli-

49 32 cation to the models which include kinematic hardening by spectrally decomposing the relative stress ξ. To facilitate computations, an alternative variable, called the plastic corrector σ corr n+1 := σ n+1 σ tr n+1, was defined for the stress increment. Therefore, the reduced form of the unknown variables and residual vector can be obtained as below X = [σi corr, σii corr, σiii corr, α I, α II, κ, γ] T (3.8) γa e 1A ( g/ σ A) + σ corr I γa e 2A ( g/ σ A) + σ corr II R = γa e 3A ( g/ σ A) + σ corr III γ(h α ) I α I = 0 (3.9) γ(h α ) II α II γh κ κ f 7 1 where subscript n + 1 is left off. A=1, 2, 3 indicates the principal direction. α III = α I α II is eliminated since the back stress is deviatoric. a e = λ + 2µ λ λ λ λ + 2µ λ λ λ λ + 2µ (3.10) ( g h α B(α) = c α G α g ) σ B I 1 (3.11)

50 33 The tensor a e is the elasticity tensor projected to principal relative stress space. In the case of kinematic hardening, one can see that g = g ξ B = g (3.12) σ B ξ B σ B ξ B It has been shown that ξ tr n+1, ξ n+1, σ corr n+1 and α n+1 share the same spectral directions (for more details the reader is referred to [38]. These directions may be determined from ξ tr n+1, which is known a priori. Hence, once the principal values of σ corr and α are determined, these tensorial variables can be obtained as follows n (A) σ corr and n(a) α n(a) ξ tr σ corr = 3 A=1 σcorr A n(a) ξ tr n (A) ξ tr α = (3.13) 3 A=1 α An (A) ξ n (A) tr ξ tr where n (A) denotes the corresponding eigenvectors. Afterwards, the updated stress and back stress tensors can be easily computed by σ n+1 = σ corr n+1 + σ tr n+1 (3.14) α n+1 = α n + α n+1 (3.15) A Priori shear/cap surface determination The plastic model hypothesizes that the compression cap characterizes hardening behavior, not softening, graphically illustrated in Figure 6. This assumption is in agreement with the

51 34 physical behavior of geological materials in compactant cataclastic flow regime [86, 91]. Hence, κ is not allowed to increase and remains constant, while the stress path is proceeding along the shear surface in which the kinematic hardening solely operates (with no isotropic hardening participation). Figure 6: Hardening behavior of compression cap in meridional stress space. Accordingly, we utilize the restriction κ 0 to modify the number of unknown variables for the residual vector by differentiating between the compactive compression cap and the shear surface. It is worth remarking here that there is some ambiguity in the literature for smooth cap models as to where the cap surface begins. For this article the cap surface refers to the portion of the yield surface with g/ I 1 < 0, i.e. where compaction hardening may occur. This distinction is different from some authors who begin the cap surface at κ (in non-smooth cap

52 35 models, they are identical). While the yield surface is still affected by the cap function between κ and the point where g/ I 1 = 0, the fact that compaction hardening cannot occur, along with the implementational concerns already discussed, justifies the distinction chosen here. As shown in Equation 3.16, g/ I 1 has the same sign as κ since the rest of factors γ, ɛ p v/ X, and X/ κ are always positive. As a result, the hardening cap is active when g/ I 1 < 0 and otherwise, κ keeps its value from the previous step. κ = ( ( 3 γ g ɛ p v X I 1 ) 0 X κ ) g I 1 < 0 g I 1 0 (3.16) It may be shown that g/ I 1 shares the same sign as g/ I1 tr. This fact is proven in the Appendices A and B. Hence, it can be determined, from the trial stress state, whether or not κ needs to be evaluated as below:

53 36 γa e 1A ( g/ σ A) + σ corr I γa e 2A ( g/ σ A) + σ corr II γa e 3A ( g/ σ A) + σ corr III γ(h α ) I α I g/ I tr 1 < 0 γ(h α ) II α II R = γh κ κ f 7 1 (3.17) γa e 1A ( g/ σ A) + σ corr I γa e 2A ( g/ σ A) + σ corr II γa e 3A ( g/ σ A) + σ corr III g/ I tr 1 0 γ(h α ) I α I γ(h α ) II α II f 6 1 This knowledge may increase efficiency of the stress computation as it eliminates an extra set of N-R iterations. Indeed, in the previous formulation, the full-variable N-R iteration is performed regardless of where the stress path proceeds on the updated yield surface. In the case that the shear surface is activated, it generates a spurious solution including a positive κ. Thus, another N-R iteration needs to be carried out, this time, eliminating κ and its correspondent evolution equation from the system of equations. We outline the implicit stress-integration algorithm in Box 1.

54 37 Box 1. implicit stress-point algorithm for cap plasticity model. Given: σ n, κ n, α n, and ɛ n+1 Goal: σ n+1, κ n+1, α n+1 Step 1: Compute trial state variables: σ tr n+1 = σ n + c e : ɛ n+1, α tr n+1 = α n, κ tr n+1 = κ n. Step 2: Apply spectral decomposition: ξ tr n+1 = σ tr n+1 α n = 3 A=1 ξtr A n(a) ξ n (A) tr ξ and set n (A) tr σ = corr n(a) α = n(a) ξ tr Step 3: Check the yielding condition: f tr n+1 > 0? If no, set σ n+1 = σ tr n+1, α n+1 = α n, κ n+1 = κ n and exit. If yes, Go to step 4 Step 4: Determine whether we are on shear or cap surface: check g/ I tr 1 0 If Yes, Set X 6 1 = 0, κ = 0 and iterate δx (k+1) 6 1 = [DR/DX] R(Xk ) 6 1, then X (k+1) 6 1 = X (k) δx(k+1) 6 1 until (Rσ/Rσ,max) < tolσ, (Rα/Rα,max) < tolα and (R f /R f,max ) < tol f If No, Set X 7 1 = 0 and iterate δx (k+1) 7 1 = [DR/DX] R(Xk ) 7 1, then X (k+1) 7 1 = X (k) δx(k+1) 7 1 until (Rσ/Rσ,max) < tolσ, (Rα/Rα,max) < tolα and (R f /R f,max ) < tol f Step 5: Update state variables and consistency parameter for inviscid plasticity σ n+1 = σ tr n α n+1 = α tr n A=1 σcorr A n(a) ξ tr A=1 αcorr A n(a) ξ tr n (A) ξ tr n (A) ξ tr κ n+1 = κ tr n+1 + κ, γ n+1 = γ n + γ, and exit. Step 6: Obtain the visco-plastic solutions: ( ) n+1 = ( )tr n+1 1+ t η + t η 1+ t η ( ) n+1.

55 Conclusion In this chapter, we addressed the key potential issues that one may confront regarding full-scale non-linear finite element simulation using an advanced phenomenological continuum constitutive model. The model is applied to a fully implicit, unconditionally stable time integration algorithm using a return-mapping scheme in principal relative stress (or strain) directions. The yield function reformulation increases computational efficiency and robustness for local iterative solutions. The novel a priori shear/cap surface determination algorithm is introduced to efficiently reduce the computational cost. To this end, the algorithmically consistent elastoplastic modulus is also derived based on nonassociated plastic flow to better describe volumetric material behavior.

56 CHAPTER 4 SIMULATION RESULTS AND VERIFICATION (Previously published in Motamedi, M.H., Foster, C.D., An improved implicit numerical integration of a non-associated, three-invariant cap plasticity model with mixed isotropickinematic hardening for geomaterials. International Journal for Numerical and Analytical Methods in Geomechanics, 39(17), pages: , 2015.) In order to demonstrate various features of the modified material model, several numerical examples, including a 3D single solid element with different boundary conditions, are presented. In addition, an appropriate large-scale boundary value problem (BVP) is also examined to analyze the fidelity of the model and its numerical performance. The material parameters adopted in the examples are listed in Table I belonging to a Salem limestone rock [39]. The results obtained from the modified model are analyzed and critically compared with numerical results for the form of the Sandia GeoModel presented in [38]. 4.1 Uniaxial tensile example First, a uniaxial tensile test is performed using a single 3D solid element to investigate numerical performance of the model with and without smooth tension cap. The mesh and boundary conditions are shown in Figure 7. As the axial stress-strain response is shown in Figure 8, in the case without tension cap, the model passed over the maximum uniaxial tensile strength, whereas in the modified model the material yielded sooner and axial stress coincided 39

57 40 TABLE I: MATERIAL PARAMETERS FOR SALEM LIMESTONE ROCK Parameter Value Young s Modulus (E) (MPa) Poisson s Ratio (ν) (dimensionless) isotropic tensile strength (T ) 5 (MPa) tension cap parameter (I1 T ) 0.0 (MPa) compression cap parameter (κ 0 ) (MPa) shear yield surface parameter (A) (MPa) shear yield surface parameter (B) 3.94e-4 (1/MPa) shear yield surface parameter (L) 1.0e-4 (1/MPa) shear yield surface parameter (C) (MPa) shear yield surface parameter (θ, φ) 0.0 (rad) aspect ratios (R, Q) 28.0 (dimensionless) isotropic hardening parameter (W ) 0.08 (dimensionless) isotropic hardening parameter (D 1 ) 1.47e-3 (1/MPa) isotropic hardening parameter (D 2 ) 0.0 (1/MPa 2 ) kinematic hardening parameter (C α ) 1e5 (MPa) kinematic hardening parameter (N) 6.0 (MPa) stress triaxiality parameter (ψ) 0.8 (dimensionless) with the prescribed limit. In addition, the stress paths are plotted in J 2 versus mean stress I 1 plane, Figure Simple shear example In this example, a simple shear test is performed. The horizontal nodal displacement was applied on 4 nodes of the top surface and no vertical displacement is allowed, Figure 10. The rate-dependent form of the model presented in Section is investigated for three different η values and results are plotted in Figure 11. The inviscid (elastoplastic) solution is attained for the lower bound (η = 0). The maximum shear stress of the material increases as the element is subjected to the shear displacements d s with higher velocity. In addition, the volumetric response, commonly observed as a dilatant behavior at low pressure regime,

58 41 Figure 7: Uniaxial tensile test: mesh and boundary conditions; d t Indicates the vertical nodal displacement. Figure 8: Axial stress-strain response for the uniaxial tensile test.

59 42 Figure 9: Stress path in meridional stress space for the uniaxial tensile test: Model with the tension cap (circle markers); Model without the tension cap (diamond markers). Figure 10: Uniaxial tensile test: mesh and boundary conditions; d t Indicates the vertical nodal displacement.

60 43 is demonstrated for two associative and non-associative plastic flow rules in Figure 12. As expected, in associative case, the dilation is overestimated compared to nonassociative model. Note that here we do not consider softening and consequently the shear strain extends to four percent. It is well known that modeling of softening requires some form of regularization, and investigation of this behavior is beyond the scope of this study. In every load step, the modified model presents a quadratic rate of convergence for each component of the residual vector, Table II. As discussed earlier, the uniform dimensionality generally leads to components having values in each load step with no significant difference in magnitude order from others and hence improves the overall rate of convergence for the whole residual vector. TABLE II: CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR A SIMPLE SHEAR TEST Local residual vector R 6 1 (MPa) Number of N-R iterations = e e e e e e e e e e e e e e e e e e-13

61 44 Figure 11: Stress-strain response for selected simple shear tests of a solid element with different values of η indicated by numbers next to each curve. Examples are run for time step t = 0.01 and strain rate of 4% per second. Figure 12: Comparison of material response in associative vs. nonassociative models for selected simple shear tests of a single solid element. Results are plotted for inviscid solutions (η = 0).

62 Triaxial compression examples This test is designed to verify applicability of the model to simulate mechanical behavior under confined loading. Again, a single 3D solid element is used and subject to a force-controlled loading test with σ 2 = σ 3 = βσ1, (Figure 13). In this case, we assume that the material obeys the nonassociative plastic flow rule. The stress paths for all five simulations are plotted in J 2 versus mean stress I 1 plane, Figure 14. The comparative results, provided in Figures 15 and 16, demonstrate how the new format of the yield function improves numerical performance of the model. For stress ratio of 0.3, the stress path successfully traverses the yield surface from shear-dilative side to the cap-compactive side. For stress ratios of 0.6, in which the combined isotropic-kinematic hardening operates and the size of the yield surface needs to be updated for each next time step, the modified model completes the load schedule whereas the original version failed to converge after running few plastic steps. Figure 13: Triaxial compression test: mesh and boundary conditions.

63 46 Figure 14: Stress path in meridional stress space for selected triaxial compression examples. β denotes stress ratio (σ 3 /σ 1 ). Additional insights can be gained by comparing the strain response of the element for different stress ratio β = σ 3 /σ 1. The results are computed and displayed in Figures 17 and 18. Associated with the delocalized cataclastic flow regime, the triaxial curves with stress ratio of 0.3 or higher coincide with the elastic line up to a critical stress state C beyond which there is an accelerated decrease in volume in comparison to the hydrostatic loading. These deviations from the hydrostatic loading case would imply that the volumetric strain is not only dependent on the mean stress, but also the deviatoric stresses. This phenomenon is well known as shearenhanced compaction [92]. In contrast, for stress ratios of 0.25 or lower, the compaction reversed beyond critical stress state C, and this corresponds to the onset of dilatancy. Accordingly, these two critical stress states (C and C ) indicate the termination of the elastic regime and the onset

64 47 Figure 15: Stress path in meridional stress space for the triaxial compression test, stress ratio, σ 3 /σ 1 = 0.3 Figure 16: Stress path in meridional stress space for the triaxial compression test, stress ratio, σ 3 /σ 1 = 0.6.

65 48 of inelastic behavior. The differential stresses for two critical stress states (C and C ) show a positive correlation with mean stress and stress ratio because the more confining pressure provides higher yield strength, and thus delays the inelastic deformation. Figure 17: Volumetric strain is plotted versus mean stress for selected triaxial compression examples. For reference the hydrostat is plotted as the dashed line. Arrows mark critical stress states (onset of dilatancy C and onset of shear enhanced compaction C. β stands for stress ratio σ 3 /σ 1. At stress ratios of 0.3 or higher (in Figure 18), the slopes of the differential stress σ 1 σ 3 versus axial strain curves were positive implying the behavior which is a typical of the cataclastic flow regime. On the other hand, at stress ratios of 0.2 and 0.25 (curves marked with dash line) the differential stress became constant after reaching a peak would indicate that the shear surface has reached the failure surface. As can be seen for dashed curves, the peak stress shows

66 49 a positive correlation with the stress ratio and consequently confining pressure. By comparing results in Figures 17 and 18, it is demonstrated that the model can capture the shear failure as a dilatant failure mode (curves for β = 0.2 and 0.25). Overall, these numerical results are qualitatively similar to experimental data compiled in several laboratory studies for porous rocks [85, 93 96]. Similar to the simple shear example, quadratic rates of convergence are obtained for local residual vectors, shown in Table Table III. Moreover, as shown in Table IV and Figure 19, we verify that the global residual vectors converge quadratically. Figure 18: Differential stress is plotted versus axial strain for selected triaxial compression examples.

67 50 Figure 19: Residual norm per iteration for the first plastic step in a set of triaxial compression tests. TABLE III: CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR A TRIAXIAL COMPRESSION TEST β = 0.4 Local residual vector R 7 1 (MPa) Number of N-R iterations = e e e e e e e e e e e e e e e e e e e e e e e e e e e e-11

68 51 TABLE IV: CONVERGENCE OF GLOBAL ALGORITHM FOR A SET OF TRIAXIAL COMPRESSION TESTS Norm of the global residual vector (m) Iteration Number β = 0.2 β = 0.25 β = 0.3 β = 0.4 β = e e e e e e e e e e e e e e e Triaxial extension vs. compression Here, a two-step loading test is adopted to verify that the model exhibits the difference in triaxial extension strength versus triaxial compression strength for ψ = 0.8. The mesh and boundary conditions are the same as designated in Figure 13. Therefore, in the first load step, the element is subject to hydrostatic pressure and subsequently, for triaxial extension case, the confining pressure, P 3 increases while the axial pressure P 1 begins to reduce such that the mean stress keeps the value prescribed at the end of the first load step, i.e. P 3 = P 1 /2. Similarly, we examine the triaxial compression test by increasing the axial pressure rather than confining pressure in the second load step. Figures 20 and 21 provide insightful views in principal stress space about the trajectory of stress evolution during two steps of loading for TXE and TXC tests, respectively. In addition, the 3D representation of initial and translated yield surfaces are depicted. As the axial stress-strain response is shown in Figures 22 and 23, the material yields sooner and undergoes more plastic deformation in the triaxial extension case.

69 52 Figure 20: Stress path for TXE in principal stress space: showing movement along the hydrostatic axis in the first load step, intersection with the initial yield surface and culminating at the final translated yield surface in the second load step. 4.5 Compression/shear example This two-step loading test is designed to investigate numerical implementation of the mod- ified model during changing spectral directions. Mesh and boundary conditions are demonstrated in Figure 24. In the first step of loading (AC), shear displacement is set to zero, i.e. ds = 0 while in the second step (CD) the compression displacement dc and confining force P3 are fixed to the values prescribed at the end of the first load step (C), with P3 = 40 and dc = Again, we assume that the material obeys the nonassociative plastic flow rule. The stress path for the modified model and the Sandia GeoModel is plotted in meridional stress space, Figure 25. As can be seen, different stress paths are obtained due to the fact that the

70 Figure 21: Stress path for TXC in principal stress space: showing movement along the hydrostatic axis in the first load step, intersection with the initial yield surface and culminating at the final translated yield surface in the second load step. 53

71 54 Figure 22: Axial stress-strain response for the TXE test at the deviatoric plane with pressure P=60 (MPa). Figure 23: Axial stress-strain response for the TXC test at the deviatoric plane with pressure P=60 (MPa).

72 55 plastic flow directions of the two models are slightly different. Although the principal directions of the relative stress tensor are rotating, the stress paths remain on the yield surface because the stress invariants (i.e. I 1, J ξ 2, J ξ 3 ), used to formulate the yield functions, are independent of these changes. Moreover, the stress-strain response is plotted in Figure 26. For both models, in the first load step, the axial response begins as elastic (AB) and then becomes plastic (BC), while the shear stress and strain remain zero along the direction marked by d s. On the other hand, in the second load step (CD), the shear response goes to the plastic and concurrently, the compression response appears as a vertical line due to the fact that the related compressive stress drops while no deformation is induced along the d c direction. The key difference realized between the results of two models is that during the plastic deformation (BC and CD), the modified model demonstrates more strain hardening behavior. As seen in Figure 26, for the modified model, the slopes of the stress-strain responses show the greater values compared to Sandia GeoModel. Figure 24: Two-step loading test: mesh and boundary conditions.

73 56 Tables V and VI demonstrate how local iterations for residual vectors quadratically converge for the first and second steps of loading in the compression/shear example test. The resulting norm of the global residual vector is also plotted in Figure 27 and shown in Table VII. As expected, quadratic convergence is observed. Figure 25: Stress path in meridional stress space for the designed compression/shear test. The letters indicate points on the stress path distinguishing different phases of evolution. 4.6 Boundary value problem As a final example, a slope stability problem is presented to investigate the performance of the modified model in a full-scale non-linear finite element simulation. The dimensions and boundary conditions of the problem are depicted in Figure 28. We assume that a gravity load is first applied, then the deformation is reset to zero, and finally a downward displacement

74 57 Figure 26: Stress-strain response for a compression/shear test. The letters indicate points on the stress path distinguishing different phases of evolution. TABLE V: CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR THE FIRST PLASTIC LOAD STEP OF COMPRESSION/SHEAR TEST Local residual vector R 7 1 (MPa) Number of N-R iterations = e e e e e e e e e e e e e e e e e e e e e e e e e e e e-11

75 58 TABLE VI: CONVERGENCE OF INTEGRATION POINT ALGORITHM FOR THE FIRST SHEAR LOAD STEP OF COMPRESSION/SHEAR TEST Local residual vector R 6 1 (MPa) Number of N-R iterations = e e e e e e e e e e e e e e e e e e e e e-13 Figure 27: Residual norm per iteration for the first plastic step in both loading steps of the compression/shear test. Quadratic convergence is observed. TABLE VII: CONVERGENCE OF GLOBAL ALGORITHM FOR THE COMPRES- SION/SHEAR TEST Norm of the global residual vector (m) Iteration Number Loading step 1 (compression) Loading step 2 (shear) e e e e-11

76 59 is loaded at the middle of a rigid footing resting at the crest of the slope. The downward displacement may represent the settlement due to a structure placed at the crest of the slope. Two meshes with 400 and 1600 bilinear quadrilateral elements are used to analyze the problem. Standard 2-by-2 Gaussian integration is used. To get more realistic results, we assume that the material obeys the nonassociative plastic flow rule which its parameters are given in Table I and set ψ = 0.8 to enable SD effect attribute of the model (J ξ 3 dependence). Deformed shapes are scaled by factor of 10. Figure 28: Slope stability problem. prescribed. Gravity load applied before footing displacement u is Figures 29 and 30 illustrate deformed shapes at end of loading for inviscid solutions. As can be seen, displacement contours show the diffused deformation patterns. However, to obtain the

77 60 sharp localized deformation, we need to add a localization capability to the model and advance the solution into the softening-induced localization regime. The load-displacement response of footing is plotted in Figure 31. Slightly smaller values of the reaction force occur in the 1600-element mesh due to the increase in the number of degrees of freedom. The difference between the two meshes is small, however. The global Newton- Raphson convergence profile is illustrated in Table VIII and Table IX. In this problem, an asymptotic rate of quadratic convergence is observed. In order to attain further investigation, the stress path for the second integration point of two elements located under the right corner of the footing (Figure 32), is plotted on J ξ 2 vs. I 1 plane (Figures 33 and 34). As seen in Figure 34, the function G α approaches to zero indicating that for element number 32, the translated yield surface meets the failure envelope. Thus, it is possible for that the point has experienced loss of ellipticity. Though it is beyond the scope of this Chapter, a bifurcation analysis [39] should be performed to ensure that the results not mesh dependent. This criterion has also been used to mark the onset of localized behavior, and can be coupled with an enhanced element [71] or another regularization technique to model the softening portion of the load-displacement curve. TABLE VIII: CONVERGENCE OF GLOBAL ALGORITHM FOR THE SLOPE STABILITY PROBLEM WITH 400 ELEMENTS Norm of the global residual vector d(m) Number of n-r iterations = e e e-09

78 61 Figure 29: Deformed shape for FE mesh with 400 bilinear quadrilateral elements: (a) horizontal displacement dx contours, (b) vertical displacement dy contours. Figure 30: Deformed shape for FE mesh with 1600 bilinear quadrilateral elements: (a) horizontal displacement dx contours, (b) vertical displacement dy contours. 4.7 Conclusion In this Chapter, several numerical examples including a large-scale BVP are presented to validate the robustness of the integration procedure and show that the numerical algorithm exhibits quadratic rate of convergence. According to the results, the new functional form of the model allows us to investigate complex mechanical behaviors of geomaterials under various

79 62 TABLE IX: CONVERGENCE OF GLOBAL ALGORITHM FOR THE SLOPE STABILITY PROBLEM WITH 1600 ELEMENTS Norm of the global residual vector d(m) Number of n-r iterations = e e e-9 Figure 31: Footing load-displacement plot for two FE meshes.

80 63 Figure 32: FE mesh and selected elements (12 and 32) to draw stress path in meridional stress space. Figure 33: Stress path in meridional stress space for the element 12 at the integration point, IP=2. The letters indicate points on the stress path to distinguish different phases of evolution.

81 64 Figure 34: Stress path in meridional stress space for a selected element 32 at the integration point, IP=2. The letters indicate points on the stress path to distinguish different phases of evolution. loading conditions. Notably, previous models have shown some lack of convergence in implicit numerical implementations, especially for large-scale problems. The simulation results demonstrate the ability of the model to capture several behaviors common to geomaterials including strain hardening, shear enhanced compaction, the Bauschinger effect, dilatancy, strain-rate sensitivity, non-associativity, and differences in triaxial extension versus compression strength.

82 CHAPTER 5 STRONG DISCONTINUITY (Previously submitted in Motamedi, M.H., Weed, D.A., Foster, C.D., Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach. International Journal of Solids and Structures, 2015, In revision.) Standard finite element methods have been successfully utilized for a broad range of geometries, loading, and deformation patterns. However, in modeling of material softening, the governing differential equations may change character, engendering a spurious solution, and thus pathological mesh dependence. This issue must be treated by some regularization technique such as viscous, gradient, or nonlocal enhancement. Material softening is frequently accompanied by intense local deformation, such as shear bands or fractures. Once softening is created in one location in the material, the adjacent regions are allowed to unload. In this case, a strong discontinuity, or displacement jump, may be used to approximate the localized deformation and regularize the solution. 5.1 Kinematics and governing equations For strong discontinuities, the displacement field experiences a spatial jump [u] = u + u across the material surface S separating the subdomains Ω and Ω + of an otherwise continuous body Ω, see Figure 35. The displacement field in the context of strong discontinuity kinematics is given by 65

83 66 u (x, t) := ū (x, t) }{{} + [[u(t)] H S (x) }{{} continuous part jump discontinuity (5.1) in which H S (x) is the Heaviside function across the surface defined by the conditions 1 if x Ω + H S (x) = (5.2) 0 if x Ω Figure 35: Body Ω with planar strong discontinuity S fixed at the reference configuration, (Ω=Ω + Ω S, Γ= Γ t Γ g ). In this study, it is assumed that the jump discontinuity [u] is piecewise constant along surface S (i.e. independent of x) so that the gradient [u] is ignored. The total strain rate

84 67 tensor resulting from this field is the symmetric component of the displacement gradient tensor, which can be derived in compact form as below ( ) ɛ := sym( ū) + sym [u] n δ }{{} S }{{} regular part singular part (5.3) where n is the unit normal vector to the surface S and pointing in the direction of Ω +. The Dirac delta distribution δ s replicates unbounded measure of strain at the discontinuity interface. The local form of quasi-static, isothermal equilibrium for a body with strong discontinuity leads to the following set of governing differential equations σ + b = 0 in Ω σ ν = t σ on Γ t u = g on Γ g (5.4a) (5.4b) (5.4c) [σ ] n = 0 across S (5.4d) where σ is the Cauchy stress tensor, b the body force vector, ν the outward unit normal vector to Γ t, t σ the traction on Γ t, g the prescribed displacement on Γ g, and [σ ] is the jump in stress

85 68 across S. The corresponding variational (weak) form of the problem reads: Ω S w : σdω = Ω w bdω + w t σ dγ + w ([σ ] n)dγ (5.5) Γ t S where w ϑ is the admissible weighting function (or displacement variation) such that ϑ := {w : Ω R ndim ; w = 0 on Γ g } (5.6) In view of Eq. (3.1), the displacement variation can be decomposed in the following w := w + [w ] H S (5.7) Here, w satisfies the usual boundary condition w = 0 on the surface Γ g and [w ] stands for the variation jump on S. By using a Galerkin approach and inserting (Equation 5.7) into (Equation 5.5), the continuous and discontinuous parts of the governing equations can be derived and will be expressed in a discretized form in Section Onset of localization: bifurcation analysis In the material failure mechanics, there exist different approaches to investigate the onset of localized deformation in terms of macrocracks and/or deformation bands. Among them we can point out the local path-independent criterion [97], bifurcation analysis [39, 88], critical plane framework [98], and non-local criteria for pre-existing defects [99]. In this section, the localiza-

86 69 tion condition is derived in terms of bifurcation analysis in conjunction with the cap plasticity model proposed earlier in Chapter 2. This theory is originated at first by the work of [100] to explore the onset of inelastic behavior in solids using the physics of wave propagation through the matter. Later on Rudnicki and Rice [101] adopted this work to develop mathematical framework for detecting shear band localization in pressure-sensitive dilatant materials. It has been shown [39, 102] that continuous bifurcation precedes discontinuous case. For continuous bifurcation, the plastic loading appears outside and within the planar discontinuity band at the instant of localization, thus the plastic consistency parameter reads: γ = γ + γ δ δ S (5.8) Similarly, in order for the stress-like internal state variables (ISVs), including the backstress α and the isotropic hardening parameter κ, to be bounded (and the plastic dissipation to be well-defined [103], the hardening moduli c α and c κ can be obtained in the following (c α ) 1 = ( c α ) 1 + (c α δ ) 1 δ S ( ) g (c α ) 1 α = γg α dev σ ( ) g α = c α G α γ dev = σ h α γ ( ) g α = c α δ Gα γ δ dev = h α δ σ γ δ (5.9)

87 70 (c κ ) 1 = ( c κ ) 1 + (c κ δ ) 1 δ S ( ) g (c κ ) 1 κ = γg κ tr σ ( ) g κ = c κ G κ γ tr = σ h κ γ ( ) g κ = c κ δ Gκ γ δ tr = h κ δ σ γ δ (5.10) Substitution of Equation 5.8, Equation 5.9 and Equation 5.10 in Equation 2.8, and solving for both regular and singular parts of the consistency condition gives γ = 1 χ f σ : Ce : ɛ 0 χ = f σ : Ce : g σ f α : hα f κ h κ (5.11a) γ δ = f σ : Ce : sym([ u] n) f σ : Ce : σ g (5.11b)

88 71 In addition, we can write the stress rate on the discontinuity surface σ 1 and outside that surface σ 0 as below ( σ 1 = C e 1 χ Ce : g σ f ) σ : Ce }{{} C ep ( C ep Ce : σ g σ f : ) Ce f σ : Ce : σ g : sym ([ u] n) δ S }{{} C ep : ɛ 0 + (5.12a) σ 0 = C ep : ɛ 0 (5.12b) where C ep is recognized as the elastic-perfectly plastic tangent modulus. On the basis of Equation 5.4d, imposing the traction continuity across the discontinuity surface ( σ 1 n = σ 0 n) constitutes the classical condition on the localization tensor à already identified in [39]. (n C ep n) [ u] δ }{{} S = 0 à detã = 0 for [ u] 0 (5.13) The above equation states that a nontrivial solution for the traction continuity condition is, of course, possible only when à is singular.

89 72 For discontinuous bifurcation, we assume that the plastic loading (strain softening) is localized to the discontinuity band (x S), whereas the material points immediately adjacent to the band transfer into an elastic loading state. This process can be viewed as an elastic unloading response in the bulk continuum scale of the material. As a result, the consistency parameter takes the form of γ = γ δ δ S (5.14) Again, the hardening moduli bifurcate in order to have well-defined plastic dissipation, such that (c α ) 1 = (c α δ ) 1 δ S ( ) g (c α ) 1 α = γg α dev σ ( ) g α = c α δ Gα γ δ dev = h α δ σ γ δ (5.15) (c κ ) 1 = (c κ δ ) 1 δ S ( ) g (c κ ) 1 κ = γg κ tr σ ( ) g κ = c κ δ Gκ γ δ tr = h κ δ σ γ δ (5.16)

90 73 Then, solving for both regular and singular parts of the consistency condition gives γ δ = σ f : Ce : ɛ 0 f α : hα δ + f κ hκ δ = f σ : Ce : sym([ u] n) f σ : Ce : σ g (5.17) In the same way, by imposing traction continuity requirement across the discontinuity surface, again we can arrive at the localization condition manifested in Equation Following the [71], we use a numerical algorithm to solve detã = 0 for the band normals n which indicates the most critical orientation of the discontinuity surface in the localized element and then à [ u] = 0 for the deformation directions at the inception of localization. 5.3 Evolution of the displacement jump: post-localization model In this section, a post-localization model is introduced to describe the softening response of the material after localization detection. In particular, a novel cohesive traction-separation law recently presented in [67] is utilized to characterize the macro-crack evolution in terms of the displacement jump on the slip surface S. Therefore, similar to the concept of cohesive zone models for quasi-brittle materials (see [59, 104, 105]), a damage-like function F is proposed in two forms of the tensile and compressive regime as below: F tension = (τ s ) 2 + (α σ σ n ) 2 }{{} σ eq c eq (5.18a) F compression = τ s c eq sign (ζ s ) σ n tanφ (5.18b)

91 74 where the normal traction σ n and tangential traction τ s attribute to the slip surface. The notation represents the Macaulay brackets, taking into account the positive portion, α σ is a normal stress factor and φ is the friction angle on the slip surface. In addition, the non-negative parameters σ eq and c eq indicate the equivalent traction and equivalent cohesive strength of the band, respectively. Notably, the factor sign (ζ s ) in Equation 5.18b would imply that the cohesion on the slip surface operates as a restoring force, i.e. the force c eq always acts in the opposite direction of the displacement jump (or separation) vector proceeding. The initial cohesion c 0 is computed in a manner to be balanced with the bulk stress state at the moment of localization. In the literature, various cohesive relations have been proposed for a wide range of materials, such as trapezoidal function for a high-strength-low-alloy (HSLA) steel [106]; exponential function for a C-300 steel [60]; linear softening function for a polycrystalline brittle material [107,108]; linear, bilinear, and exponential softening functions for concrete [109,110]. Here, however, a linear softening curve is adopted for the sake of simplicity. Furthermore, for rock materials like Limestone, a linear model has been found to be adequate [44]. Therefore as depicted in Figure 36, for the tensile regime the elliptical damage surface F = 0 shrinks to the origin while c eq decreases linearly toward zero as the equivalent displacement jump increases. ( c eq = c 0 1 ζ ) max ζ c (5.19) where ζ c denotes to the characteristic slip (or separation) distance, beyond which complete failure occurs in the sense that the crack surface entirely loses its cohesive strength. The scalar ζ eq indicates the equivalent jump magnitude and takes the form of

92 75 ζ eq = ζ 2 s + (α ζ ζ n ) 2 (5.20) in which the parameter α ζ is a coefficient weighing the relative contribution of the opening and sliding modes in the damage process. Following the spirit of damage mechanics in the unloading/reloading case, we assume c eq unloads elastically to the origin. Likewise, the reloading path is also considered elastic until the point of maximum equivalent separation ζ max attained up until the current time. Beyond this point the softening process will resume. The slope of the unloading-reloading curve can be thought of as the stiffness of the cohesion force and derived as k c = c ( eq 1 = c 0 1 ) ζ max ζ max ζ c (5.21) Subsequently, the equivalent stress on the band can be rewritten as σ eq = k c ζ eq (5.22) The variables ζ n and ζ s are normal opening and tangential (in-plane sliding) slip on the localization band. In a number of previous studies, (for example [11, 71, 88]), only one degree of freedom is incorporated into the model, which represents sliding displacement with a jump dilation angle ψ with respect to the discontinuity surface for non-associated plasticity models. It should be remarked that the single sliding jump may cause spurious hardening and/or a geometric locking effect during non-smooth crack propagation. Particularly, in the mixed mode

93 76 fracture where crack kinking phenomena are frequently observed, using the one dimensional separation law could trigger the slippage impedance in deformation band, ultimately resulting in significant convergence problem for numerical implementation. The model proposed in this study tackles this issue allowing the crack surfaces to separate in a coupled opening and sliding mode in the tensile regime. (a) (b) Figure 36: Cohesive fracture law: (a) isotropic softening of the damage-like surface F = 0 in traction (σ n, τ s ) space (b) equivalent traction-separation relationship with corresponding loading-unloading paths. ζ max indicates the maximum attained equivalent separation. The specific fracture energy G is the amount of external energy required to form a unit surface area of the fully-separated (or damaged) crack. In view of this fact, we can assign different specific fracture energies to each of the respective failure modes (I and II) by assigning α σ, α ζ 1.

94 77 α σ α ζ = G II G I (5.23) In this work, we will assume α σ = α ζ, hence α σ = α ζ = ( GII G I ) 1 2 (5.24) It is worth noting that in the compressive case (i.e., crack closure), the frictional resistance always operates on the crack surface independently of the softening process. Indeed, once the cohesion strength completely degrades (ζ max = ζ c ), the cohesive crack surface evolves to a Coulomb friction surface with friction coefficient µ = tanφ. In this work, we consider a static coefficient of friction, though to get more realistic results a variable coefficient, as in [71, 111], can be used. 5.4 Conclusion In this chapter, a general overview of strong discontinuity kinematics in the context of small strain is presented. Furthermore, the modified cap plasticity model, described in Chapter 2, is adopted in bifurcation analysis to track the inception of new localization and crack path propagation. For the post-localization regime, a cohesive-law fracture model, able to address mixed-mode failure condition, is implemented to characterize the constitutive softening behavior on the surface of discontinuity. Notably, the cohesive traction-separation law is formulated to give different strengths in tension and shear allows the model to assign different specific fracture energies for modes I and II.

95 CHAPTER 6 FINITE ELEMENT IMPLEMENTATION FOR POST-LOCALIZATION MODEL (Previously submitted in Motamedi, M.H., Weed, D.A., Foster, C.D., Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach. International Journal of Solids and Structures, 2015, In revision.) Numerical implementation of the fracture model plays a pivotal role in successfully modeling the problems involve propagating cracks. Herein, a well established numerical technique called the Assumed Enhanced Strain (AES) method is described. When localization is detected, this technique inserts a localized surfaces at the critical orientation within the element. The AES method uses a piecewise constant interpolation of slip values which is discontinuous across the boundaries of the elements. This method is numerically appealing technique since the enhancements for discontinuities are condensed out locally and hence no additional global degrees of freedom are imposed to the calculations. The numerical implementation of the models, for softening responses, is discussed at the end of this chapter. 6.1 Assumed enhanced strain (AES) method In order to incorporate strong discontinuity analysis into the platform of finite element simulation, the assumed enhanced strain (AES) method, based on the Hu-Washizu principle, is invoked [71, 73, 103]. In this approach, the displacement discontinuity is conceptualized as 78

96 79 an appropriate incompatible mode and added to the standard FE solution. Note that the AES method is numerically appealing technique since the enhancements for discontinuities are condensed out locally and hence no additional global degrees of freedom are imposed to the calculations [11]. In doing so, the standard static condensation algorithm is considered to confine the enhancement to the element level [11, 71, 73]. Figure 37 demonstrates the underlying idea behind the finite element implementation. As shown, a constant strain triangle (CST) element is traced by a discontinuity surface S e. In addition, we postulate a piecewise constant interpolation of the displacement jump, i.e. [[u] = ζm. In the rest of the section, reparameterized discontinuous kinematics for the AES method are briefly described. Figure 37: Enhancing a CST finite element: (a) element breaks into two parts (triangle with bold lines represents the conforming deformation.) and (b) displacement field with a jump across a discontinuity plane.

97 80 In view of Equation 5.1, the displacement field can be reformulated as: u(x, t) = ũ(x, t) + [[u(t)] M S (x) (6.1) The scalar function M s (x) generates the discontinuity on the surface S and is given by M S (x) = H S (x) f h (x) with supp[m S (x)] = Ω h Ω h +, (6.2) where supp stands for the support of a function. The function f h (x) is an arbitrary smooth blending function that satisfies the following requirements 1 x S f h + h (Ω + \Ω h +) (x) = 0 x S h (Ω \Ω h ) (6.3) For localized elements, we may conveniently define the function f h (x) as the sum of the shape functions attributed to the active nodes. f h = n en A=1 N A H S (x A ) (6.4) where n en is the number of nodes for a localized element, and N A are the standard finite element shape functions. Figure 38 demonstrates how the functions f h (x) and M S (x) behave in an enhanced CST element. Using such a kinematic description affords the formulation the ability to allow the essential boundary conditions, on Γ g in Figure 35, to be applied exclusively

98 81 on the conforming displacement term ũ(x, t). As a result, the nodal displacements calculated at the global level can be realized as the final displacements. h f (x)[[u(t)]] S ẖ h f (x)[[u(t)]] [[u(t)]] S h + S (a) [[u(t)]] M (x)[[u(t)]] s S ẖ S S M (x)[[u(t)]] s S h + S h + [[u(t)]] S ẖ S S h + (b) S ẖ [[u(t)]] Figure 38: Plot of blending displacement and discontinuous displacement for an enhanced CST finite element: (a) one node at positive side of S, (b)two nodes at positive side of S. The strain rate tensor for infinitesimal deformation is written as ɛ = s u = s ũ }{{} conforming ( + [u] f h) s ( s + [u] n) δs } {{ } enhanced (6.5) which can be regularized in a form of ( ɛ = s u = s ũ + [u] f h) s }{{} ɛ reg + ( s [u] n) δs } {{ } singular (6.6)

99 82 Eventually, the finite element stress for localized elements can be obtained from the regular part of the strain. The relevant mathematical background is discussed in [112]. Thus, we have σ = C e : ɛ reg (6.7) Note that in Equation 6.6 we neglect the gradient of the displacement discontinuity, i.e. s [u] = 0. Nevertheless, it is possible to let the jump displacement vary spatially along the discontinuity surface of the element, as for example, [113] and [114] employ a linear interpolation for it. For further details of the AES method, including its variational and matrix formulation, the reader is referred to [73] and the references therein. 6.2 Implicit integration scheme for post-localization model Regarding the nonlinear formulation proposed for the combined opening-sliding fracture evolution Equation 5.18a, we use the standard N-R algorithm to solve for displacement jump ζ = (ζ n, ζ s ) on the band. As a result, the residual vectors can be defined based on traction balances on the band taking the form of: Φ 1 = σ n k c ζ n = 0 Φ 2 = τ s k c ζ s σ n tanφ = 0 (6.8a) (6.8b)

100 83 It is worth it to mention that in the tensile regime, the residual Φ 2 can be reattained as τ s k c ζ s = 0. On the other hand, in the compression case, we should only solve the second residual Φ 2 with no need for any iterative methods. In the following, two different algorithms are provided for implicit integration of slip values in terms of whether the slip band is newly detected (Box 2) or if the displacement jump evolution has already been activated on the band (Box 3).

101 84 Box 2. Implicit algorithm for a newly localized element. Step 1 : Compute trial state variables: σ tr n+1 = σ n + c e : ɛ reg, σ n tr n+1 = σ tr n+1 : (n n) and τ s tr n+1 = σ tr n+1 : (s n). Step 2 : Check for yielding on the band: F tr n+1 > 0? If no, band is inactive. Set σ n+1 = σ tr n+1 and ζ n+1 = 0 then exit. If yes, band is active. Go to Step 3. Step 3 : Solve for slip values ζ n+1 = (ζ n, ζ s ) n+1 on the band: If σ tr n > 0 (tension), If ζ = 0 then initialize ζ 0 = tol [σ n, τ s ] T n and use N-R scheme to solve for converged solution: δζ (k+1) = [DΦ/Dζ] 1 Φ(ζ k ) ζ (k+1) = ζ (k) + δζ (k+1) Until Φ(ζ) / Φ(ζ 0 ) < tol ζ where k + 1 refers to the current iteration. Step 3.1 : Check to avoid spurious solution (see Section 7.1). Else σ tr n < 0 (crack closure) then solve for ζ s using Equation 6.8b. Step 3.2 : Check to avoid spurious solution (see Section 7.1). Step 4 : Update ζ n+1, k c,n+1, and σ n+1 then exit.

102 85 Box 3. Implicit algorithm for an element which has pre-existing slip on the band. Step 1 : Compute trial state variables: σ tr n+1 = σ n + c e : ɛ reg, σ n tr n+1 = σ tr n+1 : (n n) and τ s tr n+1 = σ tr n+1 : (s n). Step 2 : Assume elastic unloading/reloading phase on the band, hence hold k c as a constant. Set k cn+1 = c 0 ( ) 1 ζ max,n 1 ζ c then: If σ tr n > 0 (tension) solve for slip values ζ n+1 = (ζ n, ζ s ) n+1 using the balance equations Equation 6.8a and Equation 6.8b. Else (σ tr n < 0, hence compression) First, check for slip on the band: If τ s k c ζ s < σ n tanφ No slip on the band due to frictional lock, exit with trial stress. Else Solve for ζ s using: Equation 6.8b. Step 2.1 : After slip value(s) calculated, it necessitates to evaluate elastic unloading/reloading assumption: ζ eff n+1 < ζ max,n? If yes, update σ n+1 and set ζ max,n+1 = ζ max,n and exit. If no, the band is in the softening phase, set k c as a descending variable and use Box 2 to solve for slip value(s). Step 3 : Update ζ n+1, k c,n+1 and σ n+1 then exit.

103 Conclusion The finite element approximation in the platform of embedded finite element method (EFEM) using assumed enhanced strain (AES) method is briefly described. Afterwards, the implicit numerical implementation of the cohesive fracture model, presented in Chapter 5, is embedded in this framework. This platform can be used to investigate propagation of fractures or shear bands under variety of loading conditions as we will demonstrate in future chapters.

104 CHAPTER 7 NUMERICAL EFFICIENCY AND ROBUSTNESS FOR POST-LOCALIZATION MODEL (Previously submitted in Motamedi, M.H., Weed, D.A., Foster, C.D., Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach. International Journal of Solids and Structures, 2015, In revision.) In this chapter, we describe particular mathematical treatments incorporated into the fracture simulation concerning numerical efficiency and robustness issues. First, we introduce a procedure to check for spurious solutions that may arise during calculation of the slip values. Secondly a new form of an Impl-Ex (Implicit-Explicit) integration algorithm, introduced previously by Oliver et al [75], is implemented rather than fully implicit one to increase the robustness of the simulation. This method employs an implicit internal variable calculation at the end of the time step and on the subsequent time step, during the global N-R iteration, these values are used in a semi-implicit calculation of the stresses. This means that the previously obtained implicit values from the prior time step are used as either semi-implicit or explicit values. 7.1 Spurious solutions Since there is no guarantee whether a trial guess for the state of the band (being in tension or compression) stays valid during the integration procedure, the N-R iteration may converge to a spurious solution for slip value(s). To overcome this drawback, once a change in the sign 87

105 88 of the normal traction is detected, the slip on the band should be instead calculated using the appropriate formulation with regard to the new state of the band. Owing to this, a standard linear interpolation is utilized to find a new slip value as the starting point of the subsequent N-R. ( ) ζ s = ζs i σn i ζs f ζs i σn f σn i (7.1) where ( σn, i ζs) i are the initial normal traction and slip values at the beginning of the N-R ( ) iteration, σn, f ζs f are the spurious converged values and ζ s is the interpolated shear slip value corresponding to the critical point (σ n, ζ n ) = (0, 0) in which the sign of the normal traction changes. 7.2 Impl-Ex integration scheme In the finite element simulation of materials undergoing strain softening behavior, even if the nonlinear problem is mathematically well posed and features a unique solution, it is well known that the classical implicit approach may suffer from a lack of robustness during a given iterative procedure. As discussed in detail by [75], if material failure propagates through the solids, the tangent constitutive operator C alg n+1 progressively loses its positive definite character, which is eventually accompanied by the loss of positive definiteness of the global stiffness matrix. In order to ameliorate the shortcomings of the fully implicit schemes, we apply an implicit/explicit (Impl-Ex) integration technique adopted from [75]. Using this semi-implicit al-

106 89 gorithm as the solution is being pursued at time step t n+1, the slip values ζ are explicitly approximated based on their implicitly updated values from prior time steps t n and t n 1. ζ n+1 = ζ n + t n+1 t n ( ζn ζ n 1 ) (7.2) The semi-implicit stress is then calculated σ n+1 = σ n + c e : ε conf n+1 c e : ( ζn+1 f h) s (7.3) since ζ n+1 is postulated as a predetermined vector, we can easily derive the effective algorithmic operator C eff n+1 as below C eff n+1 = σ n+1 ε n+1 = C e (7.4) Hence, for linear elasticity, the tangent modulus is constant. The modification of taking the entire slip vector, rather than the magnitude as used in [75] was proposed in [67]. This approach, at minor cost of accuracy, improved the efficiency of the simulation by creating a linear solution in this part. It should be commented that unconditional stability is lost in the system. At the end of the time step, once the convergent solution of the global displacements is obtained, the stress σ and slip values ζ will be implicitly updated to be used as a reference point for the next time step. A flowchart given in Figure 39 schematically illustrates the difference between the new semi-implicit scheme and the conventional fully implicit algorithm.

107 90 Figure 39: Flowchart for the numerical integration algorithm within a FE code: (a)implicit scheme, (b)impl-ex scheme. 7.3 Conclusion Some mathematical treatments are described in this chapter concerning numerical efficiency and robustness issues in fracture simulation. The implicit-explicit (Impl-Ex) integration technique is incorporated into the model as an alternative for fully implicit algorithm so that it increases the robustness of the simulation in softening response. In particular, for the semi-

108 91 implicit calculation, unlike the fully implicit scheme, a positive definite character of the stiffness matrix is retained and hence affords the simulation significant gains with regard to robustness as a crack propagates through the body.

109 CHAPTER 8 NUMERICAL BENCHMARK PROBLEM (Previously submitted in Motamedi, M.H., Weed, D.A., Foster, C.D. Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach. International Journal of Solids and Structures, 2015, In revision.) The existence of macroscopic flaws in geomaterial structures profoundly in influences their load-carrying capacity and failure patterns. Because of this effect, many laboratory tests have been developed to investigate the propagation of fracture in materials. One such test is the cracked Brazilian disc (CBD), which offers a few advantages for examining mixed mode fracture including simple preparation of the samples, straightforward test procedure and replicating fractures in different modes (i.e. tension, shear, or a mixture) by varying the inclination angle of the initial crack relative to the loading direction. This chapter is devoted to the numerical investigation of mixed mode fracture propagation in a cracked Brazilian disk (CBD) specimen by means of the embedded strong discontinuity approach (SDA). An initial crack is introduced into the model by inserting frictionless interfaces within pre-cracked elements. Assigning a negligible initial cohesion for those elements leads to the immediate appearance of localized deformations in the form of combined shear/opening fractures therein even though they still carry compressive loads as crack closure occurs. To this end, the results obtained from the enhanced FE simulations are analyzed and critically compared with experimental results available in the literature. 92

110 Cracked Brazilian disc specimen The cracked Brazilian disc (CBD) test is one of the most acknowledged methods in evaluating mixed-mode fracture behavior of brittle geomaterials. Among the numerous experimental tests have been conducted in this specimen for different rock materials, we can mention to Keochang granite [115], Yeosan marble [115], Saudi Arabian limestone [40], Guiting limestone [42], Dionysos marble [116], and Neyriz marble [52]. In this study, the material properties listed in Table X were fit to Salem limestone rock by [1] and frequently reported in [39], [37] and [74]. For the specific fracture energy ratio G II /G I, varied values are calculated experimentally dependent on the specific material of the interest and the fracture test chosen. In this paper, we use the empirical value 4.8 given in [40] for a Saudi Arabian limestone using CBD test under ambient conditions. The generic configuration of the test is illustrated in (Figure 40) and the same as one utilized in the experimental test by [40]. The circular disk contains a radius R=49mm as well as the crack length ratio a/r = 0.3. The angle β stands for orientation of the crack with respect to the loading direction u. The displacement-controlled loading test is replicated via applying the vertical displacement on the three nodes of top surface of the disk. The three nodes at the bottom surface are fixed in the vertical direction. In order to maintain the global stability of the example, the two nodes located at the crest and trough of the disk are restricted from lateral movement as well. To investigate the effect of mode mixity on crack growth path and failure pattern, the example is performed with three different crack inclination angles (β = 15, 30, and 55 ). In view of

111 94 TABLE X: MATERIAL PARAMETERS FOR SALEM LIMESTONE ROCK USED IN THE CBD TEST Parameter Value Young s Modulus (E) (MPa) Poisson s Ratio (ν) (dimensionless) isotropic tensile strength (T ) 5 (MPa) tension cap parameter (I1 T ) 0.0 (MPa) compression cap parameter (κ 0 ) (MPa) shear yield surface parameter (A) (MPa) shear yield surface parameter (B) 3.94e-4 (1/MPa) shear yield surface parameter (L) 1.0e-4 (1/MPa) shear yield surface parameter (C) (MPa) shear yield surface parameter (θ, φ) 0.0 (rad) aspect ratios (R, Q) 28.0 (dimensionless) isotropic hardening parameter (W ) 0.08 (dimensionless) isotropic hardening parameter (D 1 ) 1.47e-3 (1/MPa) isotropic hardening parameter (D 2 ) 0.0 (1/MPa 2 ) kinematic hardening parameter (c α ) 1e5 (MPa) kinematic hardening parameter (N) 6.0 (MPa) stress triaxiality parameter (ψ) 0.8 (dimensionless) localized friction angle (φ ) 40 characteristic slip distance (ζ ) 0.4 (mm) specific fracture energy ratio (G II /G I ) 4.8 (dimensionless) the fact that all aforementioned CBD tests captured a clear, continuous fracture surface for rock specimens, we check the localization condition, in particular, only for tip elements. Hence, the new crack surface will be traced from the crack tip at one edge of the tip element to the opposite edge with the orientation obtained from discontinuous bifurcation analysis described in Section 5.2. This algorithm is visualized schematically in Figure 41.

112 95 u R β Figure 40: Geometry and loading conditions of CCBD specimen subjected to mixed mode I/II loading. 8.2 Numerical simulation and discussion To perform the FE simulation, a mesh with 512 CST elements is employed, Figure 42. Deformations are assumed to be infinitesimal. Several theoretical and experimental studies conducted in the past on rock samples with various thicknesses suggest that the specimen thickness has negligible effect on fracture behavior of rock materials (see, for example, [117] and [118]). As a result, this example is analyzed under plane strain condition. The initial central cracks are introduced in the FE model as a fully damaged part of the specimen using the specific embedded discontinuity surfaces inside the elements as graphically shown in Figure 42. This interface includes negligible initial cohesive strength and localized friction angle (c 0 and φ = 0) so that the pre-cracked elements immediately fail when loaded in tension, shear or any of their combinations. These elements can, however, still endure compressive loads.

113 96 standard element enhanced element tip element Figure 41: Strategy for tracing the crack propagation path. The paths of crack propagation for three inclination angles are depicted in Figure 42. This plot indicates that the fracture propagation initiates from both ends of the pre-existing cracks, and kinks into a new direction in the form of so-called wing cracks. Subsequently the crack growth continues along a curved path toward the direction of loading. This numerical prediction is quite similar to experimental observations in several laboratory studies, see for example [40], [42] and [119], among others. The reaction force-versus-downward displacement response is presented in Figure 43. The numerical procedure adequately converges until the end of the curves, to practically null load u. For inclination angles β equal to 30 and 55 degrees, there is a convergence issue due to the AES method; as pointed out by Chen et al. [11] in the AES method, discontinuities which propagate into essential boundary conditions will have difficulty converging. In order to avoid this, they suggest rotating the discontinuity surface slightly so that there is at least one element between

114 97 the localized elements and the elements which have the assigned boundary conditions. But for β=10 degrees, the entire simulation finishes successfully and a complete softening curve is obtained. Furthermore, the end of the softening curve displays a slight rise in the residual force, which is most likely due to friction effects on the discontinuity surface. There is a noticeable discrepancy between β = 30 degrees and the two other inclinations angles (10 and 55 degrees) with regard to the maximum reaction force. From the simulation results, the 30 degree angle produces a stress state, within the tip elements, which is more critical for bifurcation analysis than the other two angles. Aliha et al [42] carried out different CBD tests for Guiting limestone to investigate the geometry and size effects on fracture trajectory under mixed mode loading. They show that the β = 27 degrees produces a minimum fracture load, which is in close accord with our simulation results. According to their analytical calculations, derived based on GMTS criterion, this difference in reaction forces could be attributed to the magnitude and sign of the T-stress. In the study of the fracture behavior of the disk specimen, additional insights can be gained by comparing the kinematics of deformation. Accordingly, the deformed meshes generated by the AES solution are displayed in Figures 44, 45, and 46. When the disc is loaded under diametral compression, for lower inclination angles of β = 10 and 30 the two faces of the initial coplanar crack simultaneously open and slide relative to each other. For inclination angle β = 55, by contrast, the results demonstrated in-plane sliding of the central crack faces. This numerical observation agrees with experimental data reported by [40] in which a crack extensometer was attached to the pre-existing cracks with a perpendicular position to

115 98 its orientation. As illustrated by the author, for crack angles of β 45, the sensor recorded positive values. This observation validates the presence of crack opening deformation at the crack mouth. Conversely, by increasing the crack inclination, the crack closure becomes more pronounced which would be characterized in our model as a pure shear sliding mode. Figure 42: Crack propagation path simulation for CBD specimen with different inclination angles (β = 10, 30, and 55 ). Initial cracks are represented by the dash thick lines through the finite element mesh.

116 99 Figure 43: Load-displacement plots for CBD specimen with different inclination angles (β = 10, 30, and 55 ). Figure 44: Deformed mesh with enhanced solution for inclination angle (β = 10 ): (a)horizontal displacement contour dx, (b)vertical displacement contour dy

117 100 Figure 45: Deformed mesh with enhanced solution for inclination angle (β = 30 ): (a)horizontal displacement contour dx, (b)vertical displacement contour dy Figure 46: Deformed mesh with enhanced solution for inclination angle (β = 55 ): (a)horizontal displacement contour dx, (b)vertical displacement contour dy

118 Conclusion In this chapter, a finite element simulation of a mixed-mode fracture propagation for CBD is created. Localized failure is detected by a loss of ellipticity condition, and subsequent postlocalization softening is modeled in the framework of an enhanced strain finite element schema, see Chapter 6. These elements include additional internal degrees of freedom which track both opening and shear displacement on a critically orientated surface determined by bifurcation theory. Due to the fact that the accuracy of bifurcation prediction is fundamentally dependent on the constitutive model used in the analysis, the modified cap plasticity model, introduced in Chapter 2, is adopted. The newly added tension cap to the constitutive model enables us to investigate the inception of dilation bands in addition to shear bands. The plane strain simulation of the CBD specimen shows good agreement with empirical results contained in the referenced literature. Specifically, the simulation accurately captures the kinking nature of the crack and the overall crack path orientation.

119 CHAPTER 9 ELASTO-VISCOPLASTIC MODELING OF RAIL BALLAST AND SUBGRADE Settlement of rail ballast, sub-ballast, and sub-grade is a major issue in the rail industry. Differential settlement can lead to decreased ride quality, increased wear, and, if left unchecked, derailment [ ]. In addition, structural discontinuities along the railway could accelerate track geometry degradation and increase differential settlement. Bridge approaches and other areas of non-uniform settlement (for example, passages over culverts and ends of tunnels) are of particular interest [123, 124]. Most simplified models of this rail substructure do not account for permanent deformation (see for example [125, 159] and the references there in). The cost of track maintenance can be significantly reduced if the geotechnical behavior of rail substructure is better understood. Under dynamic train loadings, ballast, sub-ballast and sub-grade layers accumulate both recoverable and unrecoverable deformation vertically. The increase of unrecoverable deformation would impose considerable settlement and probably consequent stability problems for railroad track structure [126]. The Finite Element Method (FEM) is a reliable technique for analysis and performance of the track substructure. In order to more accurately simulate the mechanical behavior of soil layers that interact with rails, the aforementioned modified cap plasticity model is adopted. 102

120 Fitting procedure of GeoModel In this section and following subsections, the fitting procedure for GeoModel is briefly described. For more details and motivation of the material parameterization, the reader is referred to [1]. While many of the material parameters have a clear physical meaning and are easily derivable, others, in particular hardening parameters, are not so tangible Linear elastic parameters: Young s modulus E and Poisson s ratio ν Young s modulus parameter can be derived from the linear initial part of the stress-strain response of the uniaxial or triaxial compression tests (Figure 47). Similarly, Poisson s ratio can also be easily determined from linear strain response of uniaxial stress state, ν = ɛ axial /ɛ laterial. In our case, the volumetric strain data, from which the lateral strain can be readily recovered, was not available. Estimates of Poission s ratio were taken from the literature. Figure 47: Linear fit to the axial stress-axial strain response of the rail ballast, determining Young s modulus E.

121 Shear failure envelope parameters In order to derive the material parameters for the shear failure surface, a set of triaxial compression (TXC) tests are required. Figure 48 represents the shear failure surface along with a series of triaxial loading tests depicted in meridional stress space ( J 2 versus I 1 ). Figure 49 shows the peak stress values for three sets of triaxial test (with confining pressure P=68 KPa, 103 KPa, and 138 KPa) at the ballast material. The figure also includes the linear fitted curve that was made for this data. In order to fit an exponential curve of the form shown in Figure 48 with the high degree of accuracy, more peak data pairs particularly at high confining pressures are required. Therefore, the parameters A and θ were fit which make up the linear portion of the curve. The parameters B and C are taken to be zero. Figure 48: Shear failure surface plotted using series of triaxial compression (TXC) tests conducted to failure [1].

122 105 Figure 49: Shear failure surface parameters for rail ballast Kinematic hardening parameters Using the triaxial compression test affords us deriving two additional parameters incorporated in shearing-induced kinematic hardening of the model. The two parameters of interest are the yield failure surface offset parameter, N, and the scalar decay parameter, c α. As described in detail in Chapter 2, N is the maximum kinematic translation that can occur before reaching the limit (failure) surface. c α assigns the rate at which the initial yield surface translates to the failure envelope or surface. Figure 50 illustrates the effect of the offset parameter, N. The plot shows a simulated triaxial test. The initial elastic response can be seen. The offset parameter can be increased to allow for nearly instantaneous yielding which is often observed in soils.

123 106 Figure 51 demonstrates the influence of the kinematic hardening parameter, c α. Accordingly, as c α increases, the initial yield surface approaches the failure surface more sharply [212]. Giving variety to c α enables us to fit the non-linear yield response to the experimental data more accurately. As a result, for rail ballast materials we used a try and error method to capture acceptable values for these two parameters in accordance with triaxial data plotted in Figure 53. Figure 50: Illustration of Offset Parameter, N [212].

124 Figure 51: Effect of Kinematic Hardening Parameter, c α [212]. 107

125 Material parameterization of the cap plasticity model for ballast material The experimental data for ballast materials is utilized to fit the material parameters of the cap plasticity model. This data is obtained under triaxial loading tests with three levels of confining pressure (68 KPa, 103 KPa, and 138 KPa). The material properties listed in Table XI were fit to this data. A two-dimensional representation of the shear failure surface, initial yield surface, and conjugated plastic potential surface are illustrated in Figure 52. In addition, Figure 53 compares the low rate triaxial loading test data with corresponding numerical simulations. Eventually, we will apply these material parameters to investigate the performance of the rail substructures in bridge approach problems. TABLE XI: MATERIAL PARAMETERS FOR BALLAST MATERIALS Parameter Value Young s Modulus (E) 85 (MPa) Poisson s Ratio (ν) 0.30 (dimensionless) isotropic tensile strength (T ) 0.05 (MPa) tension cap parameter (I1 T ) 0.0 (MPa) compression cap parameter (κ 0 ) -0.5 (MPa) shear yield surface parameter (A) 0.08 (MPa) shear yield surface parameter (B, L) 0.0 (1/MPa) shear yield surface parameter (C) 0.0 (MPa) shear yield surface parameter (θ) 0.22 (rad) shear yield surface parameter (φ) 0.11 (rad) aspect ratios (R, Q) 10.0 (dimensionless) kinematic hardening parameter (C α ) 1e5 (MPa) kinematic hardening parameter (N) (MPa) stress triaxiality parameter (ψ) 0.85 (dimensionless)

126 109 Figure 52: Shear failure surface, initial yield and plastic potential surfaces in meridional stress space ( J 2 versus mean stress I 1 ) 9.3 Dynamic simulation of the rail and substructures In order to investigate the dynamic response of the rail track structure using finite element analysis, the Newmark-β method is utilized, see Box 4. The well-known average acceleration integration algorithm (γ = 0.5 and β = 0.25) is chosen. This technique is more efficient in case of large-scale dynamic systems compared to other ones like linear acceleration or Fox-Goodwin methods [127]. The generic configuration of the full 3D FEM model of the rails, sleepers and their earthen substructures is illustrated in Figure 55a. The dimensions of the different sublayers are shown in Figure 54 as well.

127 Figure 53: Experimental data and corresponding numerical simulations for triaxial loading tests of ballast material. ɛ deviator and σ deviator denotes to axial strain and axial stress in the second load step of triaxial loading test, respectively. 110

128 111 Box 4. The implicit Newmark-β algorithm for nonlinear dynamic analysis. Given: d n, v n, a n, and F extn+1. Goal: d n+1, v n+1, and a n+1 Step 1: Compute predictor values: d n+1 = d n + tv n + t2 2 (1 2β)a n. ṽ n+1 = v n + t(1 γ)a n. d tr n+1 = d n+1 + β t 2 a n+1. v tr n+1 = ṽ n+1 + γ ta n+1. Step 2: Construct the residual vectors based on trial values: R n+1 = F extn+1 F intn+1 Ma n+1 Cv n+1 Step 3: Use N-R scheme to solve for converged solution: δa (k+1) = [DR/Da] 1 R(a k ) a (k+1) = a (k) + δa (k+1) d (k+1) = d n+1 + β t 2 a (k+1) v (k+1) = ṽ n+1 + γ ta (k+1) Until R(a) / R(a 0 ) < tola where k + 1 refers to the current iteration. Step 4: Update state variables d n+1, v n+1, a n+1 then exit. In future, the FEM model will be coupled with a multi-body code to apply dynamic loads to the substructure, and determine outputs such as deformation and contact forces. At the moment, for sake of the simplicity, the sinusoidal loading pattern Figure 55b is assumed to describe the load transmitted to the sleepers from the contact of the wheel and rail. The maximum values on sinusoidal curves correspond to the time at which the contact point of

129 112 Figure 54: Dimension of the rail substructures. Figure 55: Railway track structure: (a) 3D model of rails, sleepers and earthen substructure, (b) Load curves applied to sleepers

130 113 TABLE XII: RAIL SUBSTRUCTURE MATERIAL PARAMETERS FOR DYNAMIC LIN- EAR ELASTIC ANALYSIS [159] Parameter Value Young s Modulus (E) for Sleepers 64e3 (MPa) Poisson s Ratio (ν) for Sleepers 0.25 (dimensionless) Density (γ g ) for Sleepers (MN/m 3 ) Young s Modulus (E) for Ballast 260 (MPa) Poisson s Ratio (ν) for Ballast 0.25 (dimensionless) small Density (γ g ) for Ballast (MN/m 3 ) Young s Modulus (E) for Subballast 200 (MPa) Poisson s Ratio (ν) for Subballast 0.20 (dimensionless) Density (γ g ) for Subballast (MN/m 3 ) Young s Modulus (E) for Subgrade 150 (MPa) Poisson s Ratio (ν) for Subgrade 0.20 (dimensionless) Density (γ g ) for Subgrade (MN/m 3 ) Young s Modulus (E) for Stiff approach 64e3 (MPa) Poisson s Ratio (ν) for Stiff approach 0.25 (dimensionless) Density (γ g ) for Stiff approach (MN/m 3 ) the wheel and rail is exactly located at top of the sleepers. The velocity of the train used is V=30 m/s. First, a linear elastic dynamic analysis of the model is carried out. The linear elastic material parameters used for this analysis is listed in Table XII [159]. The vertical displacement contours of the rail substructures along with the deformed shape of the model is illustrated in Figure 56. Considering the fact that a model with permanent deformation is critical to capture the potential settlement of the material over time, we need to derive the rate-dependent material properties of the track substructure. To get an accurate measure of these properties, we will need the data for high speed loading tests. However, the preliminary results of the model

131 Figure 56: Vertical displacement contours for the rail substructures using dynamic linear elastic FE simulation. 114

132 115 for nonlinear dynamic simulation is fulfilled. In this analysis, the material parameters of the ballast materials listed in Table XI is utilized. The vertical displacement contours of the rail substructure along with the deformed shape of the model is illustrated in Figure 57. Figure 57: Vertical displacement contours for the rail substructures using dynamic nonlinear elastoplastic FE simulation.

133 116 Figure 58: Elements with local convergence issues in stress calculation. 9.4 Conclusion The full 3D dynamic finite element model of the rail substructure is presented in this chapter. For nonlinear dynamic analysis, the simulation failed because of local convergence issues when the stress paths of some elements passed over the tension cap. This incident took place since first the damping effect is not involved in the analysis yet. Secondly, the obtained tension cap for the ballast material is so small and hence elements in tensile regime, shown in Figure 58, quickly reach their ultimate strengths. As a result, the simulation run shows some local convergence issues in stress calculation. We believe by considering the effect of viscous damping on the model, these issues can be solved or reasonably alleviated.