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.

a b s t r a c tThe inelastic incompressibility is a typical feature of metal plasticity/viscoplasticity. Over the last decade, there has been a great amount of research related to construction of numerical integration algorithms which exactly preserve this property. In this paper we examine, both numerically and mathematically, the excellent accuracy and convergence characteristics of such integrators. In order to simplify the considerations, we consider strain-driven processes without hardening effects.In terms of a classical model of multiplicative viscoplasticity, we illustrate the notion of exponential stability of the exact solution. We show that this property enables the construction of effective and stable numerical algorithms, if the incompressibility is exactly satisfied. On the other hand, if the incompressibility constraint is violated, spurious degrees of freedom are introduced. In general, this results in the loss of the exponential stability and a dramatic deterioration of convergence of numerical methods.

In this paper we suggest a thermodynamically consistent approach to the simulation of a rate dependent material response at finite strains. The nonlinear mechanical phenomena which are covered by the proposed material model include distortional, kinematic, and isotropic hardening. Firstly, we present a new two‐dimensional rheological model of distortional hardening, which predicts the yield curve in the stress space to be a limaçon of Pascal. Such effects like the distortion of the yield surface in the stress space and its orientation depending on the loading path are captured by the rheological model in a vivid way. Next, the rheological model serves as a guideline for the construction of the constitutive equations. In particular, the kinematic assumptions, the ansatz for the free energy, and the form of the yield function are motivated by the rheological model. Further, two types of flow rules are considered in this study: a normality rule and a radial rule, both thermodynamically consistent. Moreover, we formulate explicitly the constraints on the material parameters, which guarantee the convexity of the yield surface. Furthermore, implicit time‐stepping methods are considered which exactly preserve the incompressibility of the inelastic flow. Finally, the basic features of the predicted material response are illustrated by a series of numerical simulations. In particular, the simulation results are compared to the real experimental data.

We analyze a coordinate‐free tensor setting in ℝ3 within the context of the classical tensor analysis. To this end, we formulate in a basis‐free manner the notions of second‐ and fourth‐rank tensors in ℝ3, and corresponding operations on tensors. Among the large number of different approaches to the tensor setting, we give the preference to the convenient ones, concerning the specific needs of computational solid mechanics. We use the well‐known Fréchet derivative to define the derivative of a function with respect to its tensor argument in a natural way. Furthermore, such aspects as the derivative with respect to a symmetric tensor argument and its uniqueness are covered in this paper. For the sake of completeness we present the coordinate representation of tensors and tensor operations. This representation is obtained in a straight‐forward manner from the coordinate‐free one. In particular, we elaborate the computation of the inverse of a fourth‐rank tensor and the inverse of a linear transformation on the space of symmetric second‐rank tensors. The tensor formalism is applied to the analysis of a nonlinear system of differential and algebraic equations governing visoplastic material response. An implicit time‐stepping algorithm is formulated and the numerical treatment of the algorithm is discussed.

The aim of this contribution is the presentation of an adaptive finite element procedure for the solution of geometrically and physically non-linear problems in structural mechanics. Within this context, the attention is mainly directed on the error estimation and hierarchical strategies for mesh refinement and coarsening in the case of finite elasto-plastic deformations. An important but sensitive aspect of adaptation approaches of the space discretization is the calculation of mechanical field variables for the modified mesh. Procedures of mesh refinement and coarsening imply the determination of strains, stresses and internal variables at the nodes and the Gauss points of new elements based on the transfer of the required data from the former mesh. In order to improve the efficiency as well as the convergence behaviour of the adaptive FE process an approach of data transfer primarily related to nodal values is presented. It is characterized by solving the initial value problem not only at the Gauss points but additionally at the nodes of the elements.