Finite strain viscoplasticity with nonlinear kinematic hardening: Phenomenological modeling and time integration

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Abstract

This article deals with a viscoplastic material model of overstress type. The model is based on a multiplicative decomposition of the deformation gradient into elastic and inelastic part. An additional multiplicative decomposition of inelastic part is used to describe a nonlinear kinematic hardening of Armstrong–Frederick type.

Two implicit time-stepping methods are adopted for numerical integration of evolution equations, such that the plastic incompressibility constraint is exactly satisfied. The first method is based on the tensor exponential. The second method is a modified Euler-backward method. Special numerical tests show that both approaches yield similar results even for finite inelastic increments.

The basic features of the material response, predicted by the material model, are illustrated with a series of numerical simulations.

Introduction

New materials, such as ultrafine-grained-aluminium (see the papers [15], [28]), are of special interest for many practical applications. To promote the innovation of the new materials, the robust numerical simulation of the material response is required. It is desirable to have a phenomenological description of the material which on the one hand takes important phenomena into account, and on the other hand enables stable numerical computations.

In this paper we investigate the simulation of rate-dependent material behavior with equilibrium hysteresis effect (for the general introduction to the theory of viscoplasticity see, for example, [33], [23], [12]).

The Bauschinger effect is observed in most metals under non-monotonic loading. The most popular approach to describe the Bauschinger effect was proposed by Armstrong and Frederick [2] in 1966. Application of the Armstrong–Frederick hardening concept within the framework of Perzyna type viscoplasticity (see [33]) yields the classical material model of overstress type (see [3], [4], [23]). This model has the advantage that it admits simple rheological interpretation (see Fig. 1a). Such phenomena as creep, relaxation and nonlinear kinematic hardening are taken into account by the model. Simple modification of this model is possible to include isotropic hardening as well.1

Several strategies can be adopted for the generalization of this model to finite strains (see, for example, [6], [38], [26], [37], [24], [13], [31]). Some of the generalizations were analysed numerically in [5]. Following the elegant approach of Lion [24], we use the rheological interpretation (Fig. 1a) of the classical model to construct its finite strain counterpart.

The specific assumptions of the material modeling used in this paper are as follows:

  • Multiplicative decomposition of the deformation gradient into elastic and inelastic part: F=F^eFi ([21], [22]).

  • Multiplicative decomposition of the inelastic part into energy storage part and dissipative part: Fi=FˇieFii ([24], [13]).

  • Free energy is a sum of appropriate isotropic strain energy functions ([24], [13]).

The resulting material model takes both kinematic and isotropic hardening into account. The thermodynamic consistency is proved.

The purpose of the present paper is threefold. First, we formulate the material model under consideration. In particular, we transform the constitutive equations to the reference configuration in order to simplify the numerical treatment. Next, two implicit schemes for the numerical integration of evolution equations are developed. Finally, we analyse numerically the basic properties of the material response, predicted by the model.

A global implicit time-stepping procedure in the context of displacement based FEM requires a proper stress algorithm (local integration algorithm) [40]. Such algorithm provides the stresses and the consistent tangent operator as a function of the strain history locally at each integration point. A set of internal variables is used in this paper to describe the history dependence, and the stress algorithm includes implicit integration of a system of differential (evolution) and algebraic equations.

Two most popular implicit schemes for integration of inelastic strains in the context of viscoplasticity/plasticity are:

  • Backward-Euler scheme, also referred as implicit Euler scheme (see, for example, [10], [36], [35], [13], [5]).

  • Exponential scheme, also referred as Euler scheme with exponential map (see, for example, [39], [29], [30], [5]).

The exponential scheme is advantageous since it retains the inelastic incompressibility even for finite time steps. Thus, an important geometric property of the solution is automatically preserved. Moreover, the numerical error of Euler-backward method, related to the violation of incompressibility, tends to accumulate over time (see, for example, [5], [14]). Therefore, even for small time steps, the numerical solution deviates from the exact solution after some period of time.

Helm [14] modified the classical Euler-backward scheme, using a projection on the group of unimodular tensors, to enforce the incompressibility of inelastic flow.

In this work we implement in a uniform manner both modified Euler-backward method (MEBM) and the exponential method (EM). Both methods result in a nonlinear system of equations with respect to strain-like internal variables Ci=FiTFi, Cii=FiiTFii and ξ=λiΔt.2 This nonlinear system is split into two subproblems:

  • First subproblem: Finding Ci,Cii with a given ξ.

  • Second subproblem: Finding ξ, such that an incremental consistency condition is satisfied.

This adapted strategy is more robust than the straightforward application of a nonlinear solver to the original system of equations. At the same time, this approach is not limited by the special form of the free energy, and finite elastic strains are likewise allowed. Moreover, the stress algorithms are applicable in the limiting case of rate-independent plasticity (as viscosity tends to zero).

Although the material response is anisotropic, it is shown that MEBM as well as EM exactly preserve the symmetry of Ci and Cii. Furthermore, the accuracy and robustness of both integration algorithms is verified with the help of special numerical tests. Both methods provide similar results with almost the same integration error. A common feature of MEBM and EM is that the numerical error is not accumulated over time.

The phenomenological description of each specific material can be schematically subdivided into three steps:

  • Material testing, such that the important phenomena make themselves evident.

  • Choosing an appropriate phenomenological model, that reproduces qualitatively the experimental data.

  • Parameter identification, using the experimental data.

To illustrate the basic characteristics of the material model we simulate a series of material testing experiments. These experiments are uniaxial tension and torsion under monotonic and cyclic loading. In particular, we conclude that the material model can be used (after a proper parameter identification) to describe the mechanical response of an aluminium alloy processed by ECA-pressing [15], [28].

Throughout this article, bold-faced symbols denote first- and second-rank tensors in R3. Expression a:=b means a is defined to be another name for b.

Section snippets

Material model of finite viscoplasticity

The material model is motivated by the rheological diagram in Fig. 1a. This diagram takes the kinematic hardening of Armstrong–Frederick type into account (for the sake of simplicity the isotropic hardening is omitted in the diagram). The total inelastic strains and the inelastic strains of microstructure are used as internal variables. The evolution of these quantities is closely related to the energy dissipation during the inelastic processes. Besides, additional real-valued strain-like

Integration algorithms

The exact solution of (55) has under proper initial conditions the following geometric property: Ci,Cii lie on the manifold M, defined byM:={BSym:detB=1}.Hence, system (55) is a system of differential equations on the manifold (cf. the paper [9]). In this section we analyse two numerical schemes, such that the numerical solution lies exactly on M.

Numerical tests

Now we analyse the robustness and accuracy of the integration methods presented in Section 3. Toward this end, we simulate the material behavior under strain-controlled loading. The loading program in the time interval t[0,300] is defined byF(t)=F(t)¯orF(t)=F(t),where F(t) is a piecewise linear function of time t such that F(0)=F1, F(100)=F2, F(200)=F3, and F(300)=F4 withF1:=1,F2:=2001200012,F3:=110010001,F4:=1200200012.More precisely, we putF(t):=(1-t/100)F1+(t/100)F2ift[0,100],(2-t/

Characterization of the material model

We investigate qualitatively the material response, predicted by the material model. The numerical computations simulate basic material testing experiments. Material parameters from Table 2 and initial conditions (91) are used in this section.

Discussion

The classical material model of viscoplasticity is modified in a thermodynamically consistent manner to incorporate finite elastic and inelastic strains. The model takes rate-dependence (relaxation, creep) and hysteresis effects (nonlinear kinematic and isotropic hardening) into account.

Although the material response is anisotropic, the symmetry of n+1Ci and n+1Cii is a priori preserved by EBM, MEBM and EM. It is shown that no symmetrization procedure is necessary. Moreover, any symmetrization

Acknowledgements

This research was supported by German National Science Foundation (DFG) within the collaborative research center SFB 692 “High-strength aluminium based light weight materials for reliable components”. The authors are grateful to Dr. D. Helm and Dr. P. Neff for fruitful discussions.

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