Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2

A crystal plasticity finite element code is developed to model lattice strains and texture evolution of HCP crystals. The code is implemented to model elastic and plastic deformation considering slip and twinning based plastic deformation. The model accounts for twinning reorientation and growth. Twinning, as well as slip, is considered to follow a rate dependent formulation. The results of the simulations are compared to previously published in situ neutron diffraction data. Experimental results of the evolution of the texture and lattice strains under uniaxial tension/compression loading along the rolling, transverse, and normal direction of a piece of rolled Zircaloy-2 are compared with model predictions. The rate dependent formulation introduced is capable of correctly capturing the influence of slip and twinning deformation on lattice strains as well as texture evolution.     

Effects of spatial grain orientation distribution and initial surface topography on sheet metal necking

The finite element method is used to numerically simulate localized necking in AA6111-T4 under stretching. The measured EBSD data (grain orientations and their spatial distributions) are directly incorporated into the finite element model and the constitutive response at an integration point is described by the single crystal plasticity theory. We assume that localized necking is associated with surface instability, the onset of unstable growth in surface roughening. It is demonstrated that such a surface instability/necking is the natural outcome of the present approach, and the artificial initial imperfection necessitated by the macroscopic M–K approach [Marciniak and Kuczynski (1967). Int. J. Mech. Sci. 9, 609–620] is not relevant in the present analysis. The effects of spatial orientation distribution, material strain rate sensitivity, texture evolution, and initial surface topography on necking are discussed. It is found that localized necking depends strongly on both the initial texture and its spatial orientation distribution. It is also demonstrated that the initial surface topography has only a small influence on necking.     

Modeling texture evolution in equal channel angular extrusion using crystal plasticity finite element models

A new approach to modeling crystallographic texture evolution in Equal Channel Angular Extrusion (ECAE) is presented in this paper. The proposed approach utilizes an elastic–viscoplastic single crystal constitutive model implemented in a finite element framework. A representative volume element of the polycrystal is subjected to boundary conditions that simulate the approximate deformation history experienced by different regions of the sample (at different through-thickness depths) in both Route A and Route C processing. The proposed approach aims to capture the influence of the complex interactions that ensue among the constituent individual crystals of a polycrystal in controlling the texture evolution in the sample, while capturing the boundary conditions inherent to ECAE deformation. The predictions from the proposed approach are compared against previously reported experimental measurements in ECAE of copper. It is observed that the proposed approach provides significantly better agreement with the measurements when compared against previously reported model predictions.     

Nonlocal damage theory in hybrid-displacement formulations

This paper discusses three hybrid-displacement finite element formulations for the simulation of strain localization based on nonlocal damage theory. An isotropic integral nonlocal damage model is chosen. The hybrid finite element formulations adopted in this work are developed from first principles of Mechanics. The first one defines the domain approximations using the Trefftz functions derived for the linear elastic regime. When damage appears the hybrid-Trefftz displacement formulation degenerates into an hybrid-displacement formulation. The second formulation uses an enriched Trefftz basis with the consideration of local Heaviside functions. The third formulation uses orthonormal Legendre polynomials for the domain approximations. A set of benchmark tests is presented and discussed in order to compare the performance and accuracy of the different models. It is shown that the proposed hybrid-Trefftz formulation allows the reproduction of the general behavior of the structure but does not lead to a correct simulation of the strain tensor evolution. The hybrid-displacement formulation that uses orthonormal Legendre polynomials gives coherent results, so it appears to be a promising field of investigation.     

A model for high temperature creep of single crystal superalloys based on nonlocal damage and viscoplastic material behavior

A model for high temperature creep of single crystal superalloys is developed, which includes constitutive laws for nonlocal damage and viscoplasticity. It is based on a variational formulation, employing potentials for free energy, and dissipation originating from plasticity and damage. Evolution equations for plastic strain and damage variables are derived from the well-established minimum principle for the dissipation potential. The model is capable of describing the different stages of creep in a unified way. Plastic deformation in superalloys incorporates the evolution of dislocation densities of the different phases present. It results in a time dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage. Herein, the nonlocal one is included in order to model strain localization as well as to remove mesh dependence of finite element calculations. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown.     

Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification

The elastic/crystalline viscoplastic constitutive equation, based on a newly proposed hardening-softening evolution equation, is introduced into the dynamic-explicit finite element code “Itas-Dynamic.” In the softening evolution equation, the effective distance and the angle between each slip system of a crystal are introduced to elucidate the interaction between the slip systems, which causes a decrease of dislocation density. The polycrystal sheet is modeled by Voronoi polygons, which correspond to the crystal grains; and by the selected orientations, which can relate to the texture, they are assigned to the integration points of the finite elements. We propose a direct crystal orientation assignment method, which means that each integration point of finite element has an assigned orientation, and its orientation can be rotated independently. Therefore, this inhomogeneous polycrystal model can consider the plastic induced texture development and subsequent anisotropy evolution. The parameters of the constitutive equation are identified by uni-axial tension tests carried out on single crystal sheets. Numerical results obtained for sheet tensions are compared with experimental ones to confirm the validity of our finite element code. Further, we investigate the following subjects: (1) how the initial orientation of single crystal affects slip band formation and strain localization; (2) how the grain size and particular orientations of the grain affect the strain localization in case of a polycrystal sheet. It is confirmed that the orientation of a single crystal can be related to the primary slip system and the deformation induced activation of that system, which in turn can be related to the slip band formation of the single crystal sheet. Further, in case of a polycrystal sheet, the larger the grain size, the more the strain localizes at a specific crystal, which has the particular orientation. It is confirmed through comparisons with experiments that our finite element code can predict the localization of strain in sheets and consequently can estimate the formability of sheet metals.     

Finite element modeling of crystal plasticity with grains shaped as truncated octahedrons

Different modeling strategies are tested for the prediction of texture development and microscopic strain heterogeneity in cold-rolled ULC steel and in multiphase steel under uniaxial tension. The polycrystalline aggregate is represented by a finite element mesh that is loaded under periodic boundary conditions. Grains are shaped as cubes or as truncated octahedrons, defining three levels of mesh refinement. Simulations rely on a simplified implementation of crystal plasticity, in which elastic strains are considered infinitesimal. The constitutive law is integrated fully implicitly in a reference frame bounded to the crystal lattice. Accuracy of the time-integration procedure is assessed by referring to some predictions of the crystal plasticity algorithm developed by Kalidindi et al. [Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solid 40 (1992) 537–569]. Then, results of the micro–macro modeling are compared to experimental data. It is found that the simulations with truncated octahedral grains yield improved predictions compared to those with cuboidal grains.     

A new crystal plasticity scheme for explicit time integration codes to simulate deformation in 3D microstructures: Effects of strain path,strain rate and thermal softening on localized deformation in the aluminum alloy 5754 during simple shear

The predominant deformation mode during material failure is shear. In this paper, a crystal plasticity scheme for explicit time integration codes is developed based on a forward Euler algorithm. The numerical model is incorporated in the UMAT subroutine for implementing rate-dependent crystal plasticity model in LS-DYNA/Explicit. The sheet is modeled as a face centered cubic (FCC) polycrystalline aggregate, and a finite element analysis based on rate-dependent crystal plasticity is implemented to analyze the effects of three different strain paths consisting predominantly of shear. Finite element meshes containing texture data are created with solid elements. The material model can incorporate information obtained from electron backscatter diffraction (EBSD) and apply crystal orientation to each element as well as account for texture evolution. Single elements or multiple elements are used to represent each grain within a microstructure. The three dimensional (3D) polycrystalline microstructure of the aluminum alloy AA5754 is modeled and subjected to three different strain rates for each strain path. The effects of strain paths, strain rates and thermal softening on the formation of localized deformation are investigated. Simulations show that strain path is the most dominant factor in localized deformation and texture evolution.     

Texture predictions using a polycrystal plasticity model incorporating neighbor interactions

A viscoplastic model is presented for distributing the deformation applied to a polycrystal in a non-uniform fashion among the constituent crystals. Interactions with surrounding crystals are incorporated in the calculation of the deformation rate of each crystal through an appropriately defined local neighborhood. A compliance tensor is computed for each crystal based on a viscoplastic constitutive relation for deformation by crystallographic slip. The compliance of the crystal relative to that of its neighborhood provides a means for partitioning the macroscopic deformation rate among the crystals. The deviation of the crystal deformation rate from the macroscopic value is suitably scaled to obtain the crystal spin. Polycrystal simulations of crystallographic texture development using this model are compared to the results of similar calculations using the Taylor model, to finite element simulations of a model polycrystal, and to experimental data. The model incorporating neighbor interactions is shown to result in improved texture predictions, in terms of both the intensity levels and the locations of certain texture components.     

Three-dimensional deformation process simulation with explicit use of polycrystal plasticity models

The combination of massive parallel processing and polycrystal plasticity theory offers the potential for applying detailed microstructural models to macroscopic deformation processes. In this work the finite element method is used to solve for the three-dimensional deformation of a plastic workpiece. The elemental constitutive response is derived from the microstructural response of a polycrystal aggregate situated in the element. Crystal orientations and their respective weighted contributions to the aggregate response are selected to approximate the orientation distribution derived from experimental pole figure measurements. The interaction of the material symmetry adopted in analysis of pole figures and the boundary conditions posed in the plasticity boundary value problem are examined. Through the introduction of distinct aggregates with decreasing crystal to aggregate ratio, an inhomogenous material response is developed where: (1) the orientation distribution becomes well approximated only by a collection of spatially distinct aggregates, and (2) these aggregates experience deformation paths of increasing variation. It is shown that the use of spatially distinct aggregates in a material experiencing local kinematic inhomogeneities throughout its deformation history leads to texture predictions that compare favorably with experimental measurements.     

Crystal plasticity based analysis of localized necking in aluminum tube under internal pressure

The finite element method is used to numerically simulate localized necking in aluminum alloy tube under internal pressure. The measured electron backscatter diffraction (EBSD) data are directly incorporated into the finite element model and the constitutive response at an integration point is described by the single crystal plasticity theory. The tube is assumed sufficiently long, so that length changes as well as the end effects can be ignored and a plane strain analysis can be performed. Localized necking is assumed to be associated with surface instability, the onset of unstable thinning. It is demonstrated that such a surface instability/necking is the natural outcome of the present approach, and an artificial initial imperfection required by other approaches is not necessary in the present analysis. The effects of spatial grain orientation distribution, material strain rate sensitivity, work hardening, and initial surface topography on necking are discussed. It is found that localized necking depends strongly on both the initial texture and its spatial orientation distribution, while the initial surface topography has a negligible effect on necking.     

Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization

Evolution of properties during processing of materials depends on the underlying material microstructure. A finite element homogenization approach is presented for calculating the evolution of macro-scale properties during processing of microstructures. A mathematically rigorous sensitivity analysis of homogenization is presented that is used to identify optimal forging rates in processes that would lead to a desired microstructure response. Macro-scale parameters such as forging rates are linked with microstructure deformation using boundary conditions drawn from the theory of multi-scale homogenization. Homogenized stresses at the macro-scale are obtained through volume-averaging laws. A constitutive framework for thermo-elastic–viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modelling microstructure evolution. The continuum sensitivity method (CSM) used for designing processes involves differentiation of the governing field equations of homogenization with respect to the processing parameters and development of the weak forms for the corresponding sensitivity equations that are solved using finite element analysis. The sensitivity of the deformation field within the microstructure is exactly defined and an averaging principle is developed to compute the sensitivity of homogenized stresses at the macro-scale due to perturbations in the process parameters. Computed sensitivities are used within a gradient-based optimization framework for controlling the response of the microstructure. Development of texture and stress–strain response in 2D and 3D FCC aluminum polycrystalline aggregates using the homogenization algorithm is compared with both Taylor-based simulations and published experimental results. Processing parameters that would lead to a desired equivalent stress–strain curve in a sample poly-crystalline microstructure are identified for single and two-stage loading using the design algorithm.     

Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals

The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.     

A dislocation density based crystal plasticity finite element model: Application to a two-phase polycrystalline HCP/BCC composites

We present a multiscale model for anisotropic, elasto-plastic, rate- and temperature-sensitive deformation of polycrystalline aggregates to large plastic strains. The model accounts for a dislocation-based hardening law for multiple slip modes and links a single-crystal to a polycrystalline response using a crystal plasticity finite element based homogenization. It is capable of predicting local stress and strain fields based on evolving microstructure including the explicit evolution of dislocation density and crystallographic grain reorientation. We apply the model to simulate monotonic mechanical response of a hexagonal close-packed metal, zirconium (Zr), and a body-centered cubic metal, niobium (Nb), and study the texture evolution and deformation mechanisms in a two-phase Zr/Nb layered composite under severe plastic deformation. The model predicts well the texture in both co-deforming phases to very large plastic strains. In addition, it offers insights into the active slip systems underlying texture evolution, indicating that the observed textures develop by a combination of prismatic, pyramidal, and anomalous basal slip in Zr and primarily {110}〈111〉 slip and secondly {112}〈111〉 slip in Nb.     

Hybrid moiré-numerical stress analysis around cutouts in loaded composites

A hybrid experimental-numerical method is presented for evaluating the stresses around holes or notches in orthotropic composites. Moiré-measured displacements and hybrid finite elements are employed. The geometric discontinuity of interest is surrounded by a hybrid element, which is a subregion within the physical structure. Measured displacements provide input information at nodal locations on the external boundary of the hybrid element. Little experimental data are needed, and the approach is accurate and effective. The method is demonstrated for a uniaxially loaded orthotropic laminate containing a central circular hole.     

A peridynamic implementation of crystal plasticity

This paper presents the first application of peridynamics theory for crystal plasticity simulations. A state-based theory of peridynamics is used (Silling et al., 2007) where the forces in the bonds between particles are computed from stress tensors obtained from crystal plasticity. The stress tensor at a particle, in turn, is computed from strains calculated by tracking the motion of surrounding particles. We have developed a quasi-static implementation of the peridynamics theory. The code employs an implicit iterative solution procedure similar to a non-linear finite element implementation. Peridynamics results are compared with crystal plasticity finite element (CPFE) analysis for the problem of plane strain compression of a planar polycrystal. The stress, strain field distribution and the texture formation predicted by CPFE and peridynamics were found to compare well. One particular feature of peridynamics is its ability to model fine shear bands that occur naturally in deforming polycrystalline aggregates. Peridynamics simulations are used to study the origin and evolution of these shear bands as a function of strain and slip geometry.     

An invariant formulation for phase field models in ferroelectrics

This paper introduces an electro-mechanically coupled phase field model for ferroelectric domain evolution based on an invariant formulation for transversely isotropic piezoelectric material behavior. The thermodynamic framework rests upon Gurtin’s notion of a micro-force system in conjunction with an associated micro-force balance. This leads to a formulation of the second law, from which a generalized Ginzburg–Landau evolution equation is derived. The invariant formulation of the thermodynamic potential provides a consistent way to obtain the order parameter dependent elastic stiffness, piezoelectric, and dielectric tensor. The model is reduced to 2d and implemented into a finite element framework. The material constants used in the simulations are adapted to meet the thermodynamic condition of a vanishing micro-force. It is found that the thermodynamic potential taken from the literature has to be extended in order to avoid a loss of positive definiteness of the stiffness and the dielectric tensor. The small-signal response is investigated in the presence and in the absence of the additional regularizing terms in the potential. The simulations show the pathological behavior of the model in case these terms are not taken into account. The paper closes with microstructure simulations concerning a ferroelectric nanodot subjected to an electric field, a cracked single crystal, and a ferroelectric bi-crystal.     

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 (Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.     

On the coupling of plastic slip and deformation-induced twinning in magnesium: A variationally consistent approach based on energy minimization

The present paper is concerned with the analysis of the deformation systems in single crystal magnesium at the micro-scale and with the resulting texture evolution in a polycrystal representing the macroscopic mechanical response. For that purpose, a variationally consistent approach based on energy minimization is proposed. It is suitable for the modeling of crystal plasticity at finite strains including the phase transition associated with deformation-induced twinning. The method relies strongly on the variational structure of crystal plasticity theory, i.e., an incremental minimization principle can be derived which allows to determine the unknown slip rates by computing the stationarity conditions of a (pseudo) potential. Phase transition associated with twinning is modeled in a similar fashion. More precisely, a solid-solid phase transition corresponding to twinning is assumed, if this is energetically favorable. Mathematically speaking, the aforementioned transition can be interpreted as a certain rank-one convexification. Since such a scheme is computationally very expensive and thus, it cannot be applied to the analysis of a polycrystal, a computationally more efficient approximation is elaborated. Within this approximation, the deformation induced by twinning is decomposed into the reorientation of the crystal lattice and simple shear. The latter is assumed to be governed by means of a standard Schmid-type plasticity law (pseudo-dislocation), while the reorientation of the crystal lattice is considered, when the respective plastic shear strain reaches a certain threshold value. The underlying idea is in line with experimental observations, where dislocation slip within the twinned domain is most frequently seen, if the twin laminate reaches a critical volume. The resulting model predicts a stress-strain response in good agreement with that of a rank-one convexification method, while showing the same numerical efficiency as a classical Taylor-type approximation. Consequently, it combines the advantages of both limiting cases. The model is calibrated for single crystal magnesium by means of the channel die test and finally applied to the analysis of texture evolution in a polycrystal. Comparisons of the predicted numerical results to their experimental counterparts show that the novel model is able to capture the characteristic mechanical response of magnesium very well.