Christian Miehe

Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials

The paper presents a framework for the treatment of a homogenized macro-continuum with locally attached micro-structure, which undergoes non-isothermal inelastic deformations at large strains. The proposed concept is applied to the simulation of texture evolution in polycrystalline metals, where the micro-structure consists of a representative assembly of single crystal grains. The deformation of this micro-structure is coupled with the local deformation at a typical material point of the macro-continuum by three alternative constraints of the microscopic fluctuation field. In a deformation driven process, extensive macroscopic variables, like stresses and dissipation are defined as volume averages of their microscopic counterparts in an accompanying local equilibrium state of the micro-structure. The proposed numerical implementation is based in the general setting on a finite element discretization of the macro-continuum which is locally coupled at each Gauss point with a finite element discretization of the attached micro-structure. In the first part of the paper we set up the two coupled boundary value problems associated with the macro-continuum and the pointwise attached micro-structure and consider aspects of their finite element solutions. The second part presents details of a robust algorithmic model of finite plasticity for single crystals which governs the response of the grains in a typical micro-structure. The paper concludes with some representative numerical examples by demonstrating the performance of the proposed concept with regard to the prediction of texture evolution in polycrystals.

Superimposed finite elastic–viscoelastic–plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation

Abstract The paper presents a phenomenological material model for a superimposed elastic–viscoelastic–plastoelastic stress response with damage at large strains and considers details of its numerical implementation. The formulation is suitable for the simulation of carbon-black filled rubbers in monotonic and cyclic deformation processes under isothermal conditions. The underlying key approach is an experimentally motivated a priori decomposition of the local stress response into three constitutive branches which act in parallel: a rubber–elastic ground–stress response, a rate-dependent viscoelastic overstress response and a rate-independent plastoelastic overstress response. The damage is assumed to act isotropically on all three branches. These three branches are represented in a completely analogous format within separate eigenvalue spaces, where we apply a recently proposed compact setting of finite inelasticity based on developing reference metric tensors. On the numerical side, we propose a time integration scheme which exploits intrinsically the modular structure of the proposed constitutive model. This is achieved on the basis of a convenient operator split of the local evolution system, which we decouple into a stress evolution problem and a parameter evolution problem. The constitutive functions involved in the proposed model are specified for a particular filled rubber on the basis of a parameter identification process. The paper concludes with some numerical examples which demonstrate the overall response of the proposed model by means of a representative set of numerical examples.

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.

Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy

The paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of micro-structures undergoing large-strain deformations. Typically, these micro-structures define representative volumes of nonlinear heterogeneous materials such as inelastic composites, polycrystalline aggregates or particle assemblies. We consider a priori given discretized micro-structures without focussing on details of specific discretization techniques for space and time. The key contribution of the paper is the construction of a family of algorithms and matrix representations of overall properties of discretized micro-structures which are motivated by a minimization of averaged incremental energy. It is shown that the overall stresses and tangent moduli of a typical micro-structure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary. We focus on deformation-driven micro-structures where the overall macroscopic deformation is controlled. In this context, three classical types of boundary conditions are investigated: (i) linear deformation, (ii) uniform tractions and (iii) periodic deformation and antiperiodic tractions. Incorporated by Lagrangian multiplier methods, these conditions generate three classes of constrained minimization problems with associated solution algorithms for the computation of equilibrium states and overall properties of micro-structures. The proposed algorithms and matrix representations of the overall properties are formally independent of the interior spatial structure and the local constitutive response of the micro-structure and are therefore applicable to a broad class of model problems. We demonstrate their performance for some representative model problems including finite elastic–plastic deformations of composites, texture developments in polycrystalline materials and equilibrium states of particle assemblies.

Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity

Abstract An algorithm for the numerical computation of so-called algorithmic or consistent tangent moduli in finite inelasticity is presented. These moduli determine the sensitivity of algorithmic expressions for stresses with respect to the change in total deformation. They serve as iteration operators by application of Newton-type solvers for the iterative solution of non-linear initial-boundary-value problems in finite inelasticity. The underlying concept of the numerical computation is a perturbation technique based on a forward difference approximation which reduces the computation of the tangent moduli to a multiple stress computation. The algorithmic procedure is material-independent and surprisingly simple. It is outlined for a Lagrangian, as well as an Eulerian geometric setting and applied to model problems of finite elasticity and finite elastoplasticity.

Computational micro–macro transitions and overall moduli in the analysis of polycrystals at large strains

Abstract This paper presents a numerical procedure for the computation of the overall moduli of polycrystalline materials based on a direct evaluation of a micro–macro transition. We consider a homogenized macro-continuum with locally attached representative micro-structure, which consists of perfectly bonded single crystal grains. The deformation of the micro-structure is assumed to be coupled with the local deformation at a typical point on the macro-continuum by three alternative constraints of the microscopic fluctuation field. The underlying key approach is a finite-element discretization of the boundary value problem for the fluctuation field on the micro-structure of the polycrystal. This results in a new closed-form representation of the overall elastoplastic tangent moduli or so-called generalized Prandtl–Reuss-tensors in terms of a Taylor-type upper bound term and a characteristic softening term which depends on global fluctuation stiffness matrices of the discretized micro-structure.

Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials

The paper presents a modular formulation and computational implementation of a class of anisotropic plasticity models at finite strains based on incremental minimization principles. The modular kinematic setting consists of a constitutive model in the logarithmic strain space that is framed by a purely geometric pre- and postprocessing. On the theoretical side, the point of departure is an a priori six-dimensional approach to finite plasticity based on the notion of a plastic metric. In a first step, a geometric preprocessor defines a total and a plastic logarithmic strain measure obtained from the current and the plastic metrics, respectively. In a second step, these strains enter in an additive format a constitutive model of anisotropic plasticity that may have a structure identical to the geometrically linear theory. The model defines the stresses and consistent tangents work-conjugate to the logarithmic strain measure. In a third step these objects of the logarithmic space are then mapped back to nominal, Lagrangian or Eulerian objects by a geometric postprocessor. This geometric three-step-approach defines a broad class of anisotropic models of finite plasticity directly related to counterparts of the geometrically linear theory. It is specified to a model problem of anisotropic metal plasticity. On the computational side we develop an incremental variational formulation of the above outlined constitutive structure where a quasi-hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables. It is shown that this minimization problem is exclusively restricted to the logarithmic strain space in a structure identical to the small-strain theory. The minimization problem determines the internal state of the material for finite increments of time. We develop a discrete formulation in terms of just one scalar parameter for the amount of incremental flow. The existence of the incremental stress potential provides a natural basis for the definition of the geometric postprocessor based on function evaluations. Furthermore, the global initial-boundary-value-problem of the elastic–plastic solid appears in the incremental setting as an energy minimization problem. Numerical examples show that the results obtained are surprisingly close to those obtained by a reference framework of multiplicative plasticity.

EXPONENTIAL MAP ALGORITHM FOR STRESS UPDATES IN ANISOTROPIC MULTIPLICATIVE ELASTOPLASTICITY FOR SINGLE CRYSTALS

This paper presents a new stress update algorithm for large-strain rate-independent single-crystal plasticity. The theoretical frame is the well-established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts, resulting in a very convenient representation of the Schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of Eulerian vectors. The key contribution of this paper is an algorithmic formulation of the exponential map exp: sl(3) SL(3) for updating the special linear group SL(3) of unimodular plastic deformation maps. This update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration. The resulting fully implicit stress update algorithm treats the possibly redundant constraints of single-crystal plasticity by means of an active set search. It exploits intrinsically the simple representation of the Schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of Eulerian vectors updates. The performance of the proposed algorithm is demonstrated by means of a representative numerical example.

A multi-variant martensitic phase transformation model: formulation and numerical implementation

The development of models for shape memory alloys and other materials that undergo martensitic phase transformations has been moving towards a common generalized thermodynamic framework. Several promising models utilizing single martensitic variants and some with multiple variants have appeared recently. In this work we develop a model in a general multi-variant framework for single crystals that is based upon lattice correspondence variants and the use of dissipation arguments for the generation of specialized evolution equations. The evolution equations that appear are of a unique nature in that not only are the thermodynamic forces restricted in range but so are their kinematic conjugates. This unusual situation complicates the discrete time integration of the evolution equations. We show that the trial elastic state method that is popular in metal plasticity is inadequate in the present situation and needs to be replaced by a non-linear programming problem with a simple geometric interpretation. The developed integration methodology is robust and leads to symmetric tangent moduli. Example computations show the behavior of the model in the pseudoelastic range. Of particular interest is the fact that the model can predict the generation of habit plane-like variants solely from the lattice correspondence variants; this is demonstrated through a comparison to the experimental work of Shield [J. Mech. Phys. Solids 43 (1995) 869].

A quadratically convergent procedure for the calculation of stability points in finite element analysis

In this paper a new finite element formulation is given for the analysis of nonlinear stability problems. The introduction of extended systems opens the possibility to compute limit and bifurcation points directly. Here, the use of the directional derivative yields a quadratically convergent iteration scheme. The combination with arc-length and branch-switching procedures leads to a global algorithm for path-following.