Franz Roters

Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation

A polycrystalline aluminum sample with a quasi-2D single layer of coarse grains is plastically deformed in a channel die plane strain set-up at ambient temperature and low strain rate. The microtexture of the specimen is determined by analysis of electron back scattering patterns obtained in a scanning electron microscope. The spatial distribution of the plastic microstrains at the sample surface is determined by measurement of the 3D plastic displacement field using a photogrametric pixel-based pattern recognition algorithm. The initial microtexture is mapped onto a finite element mesh. Continuum and crystal plasticity finite element simulations are conducted using boundary conditions which approximate those of the channel die experiments. The experimental and simulation data are analyzed with respect to macromechanical and micromechanical effects on grain-scale plastic heterogeneity. The most important contributions among these are the macroscopic strain profile (friction), the kinematic hardness of the crystals (individual orientation factors), the interaction with neighbor grain, and grain boundary effects. Crystallographic analysis of the data reveals two important points. First, the macroscopic plastic strain path is not completely altered by the crystallographic texture, but modulated following soft crystals and avoiding hard crystals. Second, grain- scale mechanisms are strongly superimposed by effects arising from the macroscopic profile of strain. The identification of genuine interaction mechanisms at this scale therefore requires procedures to filter out macro- scopically induced strain gradients. As an analysis tool, the paper introduces a micromechanicalTaylor factor, which differs from the macromechanical Taylor factor by the fact that crystal shear is normalized by the local rather than the global von Mises strain.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

Work hardening in heterogeneous alloys—a microstructural approach based on three internal state variables

Abstract A new work-hardening model for homogeneous and heterogeneous cell-forming alloys is introduced. It distinguishes three internal state variables in terms of three categories of dislocations: mobile dislocations, immobile dislocations in the cell interiors and immobile dislocations in the cell walls. For each dislocation population an evolution law is derived taking into account dislocation generation, annihilation and storage by dipole and lock formation. In particular, these rate equations take into account the number of active glide systems and, thus, introduce texture in the model in addition to the Taylor factor. Microstructure is represented by the dislocation cell structure as well as second-phase particles, which may undergo changes by precipitation and Ostwald ripening. Interaction of mobile dislocations with the microstructure is taken into account through an effective slip length of the mobile dislocations. For the same set of parameters, the predictions are in excellent agreement with measured stress–strain curves of both a precipitation-hardened aluminium alloy (Al–4.16 wt% Cu–1.37 wt% Mg, AlCuMg2) and a precipitation-free model alloy (Al–0.35 wt% Cu–0.25 wt% Mg), the composition of which corresponds to the matrix of the two-phase alloy.

Using texture components in crystal plasticity finite element simulations

We discuss methods to map crystallographic textures in crystal plasticity finite element simulations. Fourier-type series expansion methods which use spherical harmonic library functions as well as the direct pole figure inversion methods are not well suited to reproduce texture information in a sufficiently localized spherical form onto finite element grids. Mathematically compact Gauss-shaped spherical texture components represent a better approach for including textures in finite element models since they represent an excellent compromise between discreteness (spherical localization), compactness (simple functions), mathematical precision (very good approximation also of complex orientation distribution functions already with small sets of texture components), scalability (the number of used texture components can be systematically varied according to the desired exactness of the texture fit), conceptual simplicity (simple mathematical handling), and physical significance (texture components can be directly linked to characteristic metallurgical mechanisms). The use of texture component functions has also advantages over the use of large sets of discrete single orientations with equal scatter and height since they are more compact, practical, and provide better physical insight into microstructural mechanisms and composition sensitive effects. The article presents a new approach for the mathematical reproduction of such crystallographic texture components in crystal plasticity finite element simulations. It explains in some detail why they are particularly suited for this purpose and how they can be used to map and recover textures in/from plasticity simulations. # 2003 Elsevier Ltd. All rights reserved.

Theory of orientation gradients in plastically strained crystals

We suggest a theory of in-grain orientation gradients in plastically strained metals. It is an approach to explain why initially uniformly oriented crystals can—under gradient-free external loadings—build up in-grain orientation gradients during plastic deformation and how this phenomenon depends on intrinsic factors (crystal orientation) and extrinsic factors (neighbor grains). The intrinsic origin (orientation dependence) of in-grain orientation gradients is investigated by quantifying the change in crystal reorientation upon small changes in initial orientation. This part of the approach is formulated by applying a divergence operator to reorientation rate vector fields (in the present paper calculated by using strain-rate homogenization Taylor–Bishop–Hill theory). The obtained scalar divergence function (but not the reorientation vector field itself) quantifies the kinematic stability of grains under homogeneous boundary conditions as a function of their orientation. Positive divergence (source in the reorientation rate vector field) characterizes orientations with diverging non-zero reorientation rates which are kinematically unstable and prone to build up orientation gradients. Zero divergence indicates orientations with reorientation rate identity with the surrounding orientations which are not prone to build up orientation gradients. Negative divergence (sink in the reorientation rate vector field) characterizes orientations with converging non-zero reorientation rates which are kinematically stable and not prone to build up orientation gradients. Corresponding results obtained by use of a crystal plasticity finite element formulation are in good agreement with the reorientation field divergence function derived by homogenization theory. The extrinsic origin of in-grain orientation gradients (influence of grain–neighbor interaction) is addressed using a crystal plasticity finite element bicrystal model. The simulations show that a significant dependence of orientation gradients on the neighbor crystals occurs for grains with high positive divergence. The build-up of orientation gradients in grains with close to zero or negative divergence is in body centered cubic crystals less sensitive to the presence of neighbor orientations than in face centered cubic crystals (Goss and cube orientation).  2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc.

Concepts for integrating plastic anisotropy into metal forming simulations

Modern metal forming and crash simulations are usually based on the finite element method. Aims of such simulations are typically the prediction of the material shape, failure, and mechanical properties during deformation. Further goals lie in the computer assisted lay-out of manufacturing tools used for intricate processing steps. Any such simulation requires that the material under investigation is specified in terms of its respective constitutive behavior. Modern finite element simulations typically use three sets of material input data, covering hardening, forming limits, and anisotropy. The current article is about the latter aspect. It reviews different empirical and physically based concepts for the integration of the elastic-plastic anisotropy into metal forming finite element simulations. Particular pronunciation is placed on the discussion of the crystallographic anisotropy of polycrystalline material rather than on aspects associated with topological or morphological microstructure anisotropy. The reviewed anisotropy concepts are empirical yield surface approximations, yield surface formulations based on crystallographic homogenization theory, combinations of finite element and homogenization approaches, the crystal plasticity finite element method, and the recently introduced texture component crystal plasticity finite element method. The paper presents the basic physical approaches behind the different methods and discusses engineering aspects such as scalability, flexibility, and texture update in the course of a forming simulation.

Crystal Plasticity Finite Element Methods in Materials Science and Engineering

Preface INTRODUCTION TO CRYSTALLINE ANISOTROPY AND THE CRYSTAL PLASTICITY FINITE ELEMENT METHOD PART I: Fundamentals METALLURGICAL FUNDAMENTALS OF PLASTIC DEFORMATION Introduction Lattice Dislocations Deformation Martensite and Mechanical Twinning CONTINUUM MECHANICS Kinematics Mechanical Equilibrium Thermodynamics THE FINITE ELEMENT METHOD The Principle of Virtual Work Solution Procedure - Discretization Non-Linear FEM THE CRYSTAL PLASTICITY FINITE ELEMENT METHOD AS A MULTI-PHYSICS FRAMEWORK PART II: The Crystal Plasticity Finite Element Method CONSTITUTIVE MODELS Dislocation Slip Displacive Transformations Damage HOMOGENIZATION Introduction Statistical Representation of Crystallographic Texture Computational Homogenization Mean-Field Homogenization Grain-Cluster Methods NUMERICAL ASPECTS OF CRYSTAL PLASTICITY FINITE ELEMENT METHOD IMPLEMENTATIONS General Remarks Explicit Versus Implicit Integration Methods Element Types PART III: Application MICROSCOPIC AND MESOSCOPIC EXAMPLES Introduction to the Field of CPFE Experimental Validation Stability and Grain Fragmentation in Aluminum under Plane Strain Deformation Texture and Dislocation Density Evolution in a Bent Single-Crystalline Copper-Nanowire Texture and Microstructure underneath a Nanoindent in a Copper Single Crystal Application of a Nonlocal Dislocation Model Including Geometrically Necessary Dislocations to Simple Shear Tests of Aluminum Single Crystals Application of a Grain Boundary Constitutive Model to Simple Shear Tests of Aluminum Bicrystals with Different Misorientation Evolution of Dislocation Density in a Crystal Plasticity Model Three-Dimensional Aspects of Oligocrystal Plasticity Simulation of Recrystallization Using Micromechanical Results of CPFE Simulations Simulations of Multiphase TRIP Steels Damage Nucleation Example The Grain Size-Dependence in Polycrystal Models MACROSCOPIC EXAMPLES Using Elastic Constants from Ab Initio Simulations for Predicting Textures and Texture-Dependent Elastic Properties of Beta-Titanium Simulation of Earing during Cup Drawing of Steel and Aluminum Simulation of Lankford Values Virtual Material Testing for Sheet Stamping Simulations OUTLOOK AND CONCLUSIONS

Comparison of finite element and fast Fourier transform crystal plasticity solvers for texture prediction

We compare two full-field formulations, i.e. a crystal plasticity fast Fourier transform-based (CPFFT) model and the crystal plasticity finite element model (CPFEM) in terms of the deformation textures predicted by both approaches. Plane-strain compression of a 1024-grain ensemble is simulated with CPFFT and CPFEM to assess the models in terms of their predictions of texture evolution for engineering applications. Different combinations of final textures and strain distributions are obtained with the CPFFT and CPFEM models for this 1024-grain polycrystal. To further understand these different predictions, the correlation between grain rotations and strain gradients is investigated through the simulation of plane-strain compression of bicrystals. Finally, a study of the influence of the initial crystal orientation and the crystallographic neighborhood on grain rotations and grain subdivisions is carried out by means of plane-strain compression simulations of a 64-grain cluster. (Some figures in this article are in colour only in the electronic version)

Introduction of a texture component crystal plasticity finite element method for anisotropy simulations

The authors present a physically based and time efficient method to include and predict elastic-plastic anisotropy during complex metal forming operations. This new method is based on the direct integration of crystallographic texture components into a nonlinear finite element model, and the approach is designed for performing fast simulations of industry-scale metal forming operations of textiured polycrystalline materials including texture update.

Unraveling the temperature dependence of the yield strength in single-crystal tungsten using atomistically-informed crystal plasticity calculations

Abstract We use a physically-based crystal plasticity model to predict the yield strength of body-centered cubic (bcc) tungsten single crystals subjected to uniaxial loading. Our model captures the thermally-activated character of screw dislocation motion and full non-Schmid effects, both of which are known to play critical roles in bcc plasticity. The model uses atomistic calculations as the sole source of constitutive information, with no parameter fitting of any kind to experimental data. Our results are in excellent agreement with experimental measurements of the yield stress as a function of temperature for a number of loading orientations. The validated methodology is employed to calculate the temperature and strain-rate dependence of the yield strength for 231 crystallographic orientations within the standard stereographic triangle. We extract the strain-rate sensitivity of W crystals at different temperatures, and finish with the calculation of yield surfaces under biaxial loading conditions that can be used to define effective yield criteria for engineering design models.

A novel grain cluster-based homogenization scheme

An efficient homogenization scheme, termed the relaxed grain cluster (RGC), for elasto-plastic deformations of polycrystals is presented. The scheme is based on a generalization of the grain cluster concept. A volume element consisting of eight (= 2 × 2 × 2) hexahedral grains is considered. The kinematics of the RGC scheme is formulated within a finite deformation framework, where the relaxation of the local deformation gradient of each individual grain is connected to the overall deformation gradient by the, so-called, interface relaxation vectors. The set of relaxation vectors is determined by the minimization of the constitutive energy (or work) density of the overall cluster. An additional energy density associated with the mismatch at the grain boundaries due to relaxations is incorporated as a penalty term into the energy minimization formulation. Effectively, this penalty term represents the kinematical condition of deformation compatibility at the grain boundaries.Simulations have been performed for a dual-phase grain cluster loaded in uniaxial tension. The results of the simulations are presented and discussed in terms of the effective stress–strain response and the overall deformation anisotropy as functions of the penalty energy parameters. In addition, the prediction of the RGC scheme is compared with predictions using other averaging schemes, as well as to the result of direct finite element (FE) simulation. The comparison indicates that the present RGC scheme is able to approximate FE simulation results of relatively fine discretization at about three orders of magnitude lower computational cost.