On plastic incompressibility within time-adaptive finite elements combined with projection techniques

Abstract

This article treats the interpretation of quasi-static finite elements applied to constitutive equations of evolutionary-type as a solution scheme to solve globally differential-algebraic equations. This concept is applied to finite strain viscoplasticity based on a model with non-linear kinematic hardening under the assumption of plastic incompressibility. The model is based on multiple multiplicative decomposition both for the deformation gradient into an elastic and an inelastic part as well as for the inelastic part into a kinematic hardening (energy storage) and a dissipative part. Both intermediate configurations are described by inelastic right Cauchy–Green tensors satisfying inelastic incompressibility in the theoretical context. The attention in view of the numerical treatment within finite elements is focused on diagonally implicit Runge–Kutta methods which destroy the assumption of plastic incompressibility during the time-integration due to an additive structure of the integration step. In combination with a Multilevel-Newton algorithm these algorithms embed the classical strain-driven radial-return method. To this end, a concept of geometric numerical integration is applied, where the plastic incompressibility condition is taken into account as an additional side-condition. Since the literature states large integration errors if the side-condition is not taken into account, a particular focus lies on the application of a time-adaptive procedure. Accordingly, the article investigates (i) the algorithmic treatment of kinematic hardening within time-adaptive finite elements, (ii) the influence of the Perzyna-type viscoplasticity approach in view of an order reduction phenomenon, and (iii) the influence of taking into account the exact fulfillment of plastic incompressibility using a projection method having the advantage of simple implementation.

Introduction

In the past, models of plasticity and viscoplasticity, which are based on a yield function, frequently are solved by means of the radial-return method or the elastic predictor and plastic corrector scheme within implicit finite element computations. These algorithms are based on Backward-Euler like time-integration methods or exponential-type algorithms. The stress response of the elasticity relation during the inelastic deformation process depends on inelastic variables which evolve according to evolution equations called flow rules (one or more). This equation itself is controlled by additional internal variables, commonly, divided into kinematic and isotropic hardening. Non-linear kinematic hardening in the small strain range was mainly modeled by Armstrong and Frederick-type models where the linear kinematic hardening is modified such that a saturation state is reached, see [4]. The first treatment in the sense of a problem-adapted stress algorithm is found in [29], [6], [31] in the small strain regime. For further applications in view of cyclic plasticity effects, see [39], [30] and the literature cited therein. In this area of models, the Backward-Euler based integration step fulfills exactly the plastic incompressibility condition $\mathrm{tr}{\mathbf{E}}_{\text{p}}=0$. $\mathrm{tr}\mathbf{A}=a_k{}^k$ is the trace operator of a second-order tensor and ${\mathbf{E}}_{\text{p}}$ designates the plastic strain tensor.

In the case of finite deformations different approaches, which are based on the multiplicative decomposition of the deformation gradient, have been proposed to achieve the non-linear kinematic hardening behavior [62], [61], [41], [44], [13], [12], [66], [64], [10], [47]. However, Backward-Euler like integration steps destroy the assumption of plastic incompressibility, $\det {\mathbf{F}}_{\text{p}}=1$. In [59] the necessity to incorporate the assumption of plastic incompressibility was mentioned, which is concretized in [58]. In general, time-integration methods that consider side-conditions of the analytical problem are expected to be superior in long-term simulation, see [22] for a comprehensive survey. The side-conditions may either be incorporated by specially tailored time-integration methods or by the combination of general purpose methods with projection steps to enforce the numerical solution to satisfy the side-condition [55], [15].

In [45] an additional scalar variable was introduced in order to take into account the plastic incompressibility condition. Further investigations were done by Tsakmakis and Willuweit [63], Dettmer and Reese [10], Miehe [48] and Armero and Zambrana-Rojas [3]. The latter within the context of dynamical systems. On the basis of geometry preserving integration procedures a discussion was introduced by Helm [35], see also Vladimirov et al. [65], where extremely large integration errors are recognized if plastic incompressibility is not taken into account. Of course, exponential mapping algorithms are implicitly satisfying the side-condition of plastic incompressibility (see, for example, [67], [57] or [60] for further references). However, there is, currently, no time-adaptive procedure within the solution scheme of DAEs in the finite element framework.

All these investigations have been done in view of local aspects, i.e. on the Gauss-point level within finite elements. This approach can be interpreted as the “local” approach, i.e. the applied time-integration procedure for given deformation is defined on element, i.e. Gauss-point level. For example, a closest-point projection, proposed by Armero and Perez-Foguet [2], see also [51], or the cutting plane algorithm by Ortiz and Simo [49] are related to a locally formulated optimization problem, see also for problems of plasticity [57]. Other algorithms are formulated as higher-order on local level, see, for example, [5], [21], [38] or [37] and the literature cited therein, which is different to the approach proposed here.

A similar “local” approach in the framework of elastoplasticity, where the yield condition has to be fulfilled (algebraic constraint) during the evolution of the internal variables described by ordinary differential equations (ODE), is interpreted as a system of differential-algebraic equations (DAE), see [50], [56], [7], [8], [9]. The differential part is defined by the system of ODEs given by the evolution equations (flow and hardening rules) and the algebraic part comes from the yield condition. The semi-explicit structure of the resulting DAE suggests the straightforward application of classical ODE time-integration methods like BDF or (implicit) Runge–Kutta methods. Since the (differential) index of this DAE is two [50], typical problems of time-integration methods for higher index DAEs like numerical instability and order reduction have to be expected, see, for example, [23] for a detailed discussion.

A different interpretation was proposed by [18], where a new interpretation of non-linear (implicit) finite elements, which is applied to constitutive models of rate-type, is proposed going back to an investigation of Fritzen [19]. There, the method of vertical lines leads, after the spatial discretization using finite elements, to a system of DAEs. The algebraic part results from both the discretized weak formulation (equilibrium conditions) and the yield conditions which are currently “active”. The differential part is given by the ODEs described by the evolution equations of the internal variables. From the practical viewpoint, it is important to solve this DAE-system by methods that are tailored to the classical structure of implicit finite elements. Therefore, the entire DAE-system is solved by means of stiffly accurate diagonally implicit Runge–Kutta methods (DIRK, see, for example, [23]) in combination with the Multilevel-Newton algorithm (MLNA, see [53]) having the advantage of obtaining the classical structure of implicit finite elements. This is shown in more detail by Hartmann [25], where it is pointed out that in most situations the Multilevel-Newton algorithm is applied and not the classical Newton–Raphson method (a statement which is independent of the local or global point of view). In this respect see also the discussion in [40]. The application in [18] is curbed to small strain elastoplasticity. In [24] this is extended to finite strain viscoelasticity and in [28] to a problem of pressure-dependent finite strain elasto- and viscoplasticity. This “global” interpretation of finite elements applied to constitutive models of evolutionary-type, however, using different integration algorithms, can be found in [14], [54], [68], [32], where on the global DAE-system BDF or Rosenbrock-type methods are applied.

In [24], [27] the expected order of the applied higher-order DIRK methods are sustained which is shown by numerical examples. The constitutive models are formulated as smooth functions, i.e. there are no case distinctions as in yield function based models. In [18] it was shown that an order reduction is visible in small strain elastoplasticity, i.e. a method of order p does not reach the expected order. In [28] this is numerically observed in a more complex finite strain yield function based model. The reason of this phenomenon is still an open issue, see also [9].

In this article the concept of the global formulation using a time-adaptive finite element program on the basis of stiffly accurate DIRK-methods combined with the MLNA is followed and applied to a constitutive model of finite strain elasto- and viscoplasticity. This model is based on the proposals of Lion [44] and Tsakmakis and Willuweit [64], where, additionally to the multiplicative decomposition of the deformation gradient into an elastic and an inelastic part, the inelastic part is multiplicatively decomposed into one part related to dissipation into heat and another part describing the energy storage (kinematic hardening part). The approach yields two flow-rules of inelastic right Cauchy–Green tensors (inelastic metrics of the two intermediate configurations). Both have to fulfill the side-condition (constant invariants) of inelastic incompressibility. Here, the global structure of the DIRK/MLNA approach is considered and the influence of time-adaptivity and incompressibility is studied. The application of the DIRK/MLNA approach does not exactly fulfill the side-condition. Thus, a geometry preserving algorithm in the sense of a projection method can be applied, see, for example, [22].

The article is structured as follows: first, the proposed constitutive model is described. Afterwards, the applied elastic predictor and plastic corrector scheme within the DIRK/MLNA approach is summarized. A particular focus lies on the incorporation of the inelastic incompressibility. Finally, several examples show the influence of the Perzyna-type extension to viscoplasticity, see [52], onto the time-adaptive procedure as well as the order reduction phenomenon and the influence of the incorporation of the inelastic incompressibility side-condition. The notation in use is defined in the following manner: geometrical vectors are symbolized by $\overrightarrow{a}$, and second-order tensors $\mathbf{A}$ by bold-faced Roman letters. Furthermore, we introduce matrices at global level symbolized by bold-faced italic letters $\mathsf{A}$, and at local (element) level by bold-faced sans-serif letters $\mathsf{A}$.