### Finite elasto-plasto-dynamics $\hat a$ challenges \& solutions

*R. Mohr*

*, A. Menzel*

*and P. Steinmann*

#### Abstract

In this contribution, we deal with time-stepping schemes for geometrically nonlinear multiplicative elasto-plasto-dynamics. Thereby, the approximation in space as well as in time rely both on a Finite Element approach, providing a general framework which conceptually includes also higher-order schemes. In this context, the algorithmic conservation properties of the related integrators strongly depend on the numerical computation of time integrals, particularly, if plastic deformations are involved. However, the application of adequate quadrature rules enables a fulfilment of physically motivated balance laws and, consequently, the consistent integration of finite elasto-plasto-dynamics. Using exemplarily linear Finite Elements in time, the resulting integration schemes are analysed regarding the obtained conservation properties and assessed in comparison to classical time-stepping schemes which commonly adopt a time-discretisation procedure based on Finite Differences. 88 On the one hand, computational modelling of materials and structures often demands the incorporation of inelastic and dynamic effects. On the other hand, the performance of classical time integration schemes for structural dynamics, as for instance developed in [HHT77, New59], is strongly restricted when dealing with highly nonlinear systems. In a nonlinear setting, advanced numerical techniques are required to satisfy the classical balance laws as for instance balance of linear and angular momentum or the classical laws of thermodynamics. Nowadays, energy and momentum conserving time integrators for dynamical systems, like multibody systems or elasto-dynamics, are well-established in the computational dynamics community, compare e.g. [BB99, BBT01a, BBT01b, Gonz00, KC99, ST92]. In contrast to the commonly used time discretisation based on Finite Differences, one-step implicit integration algorithms relying on Finite Elements in space and time were developed, for instance, in Betsch and Steinmann [BS00a, BS00b, BS01]. Therein, conservation of energy and angular momentum have been shown to be closely related to quadrature formulas required for numerical integration in time. Furthermore, specific algorithmic energy conserving schemes for hyperelastic materials can be based on the introduction of an enhanced stress tensor for time shape functions of arbitrary order, compare Gross et al. [GBS05]. However, most of the proposed approaches are restricted to conservative dynamical systems. Nevertheless, the consideration of plastic deformations in a dynamical framework, involving dissipation effects, is of cardinal importance for various applications in engineering. In the last years, notable contributions dealing with finite elasto-plasto-dynamics have been published by Meng and Laursen [ML02a, ML02b], Noels et al. [NSP06] and Armero [Arm05, Arm06, AZ06]. In this contribution, we follow the concepts which have been proposed for hyperelasticity in [BS01, GBS05] and pickup the general framework of Galerkin methods in space and time, developing integrators for finite multiplicative elasto-plasto-dynamics with pre-defined conservation properties, compare Mohr et al. [MMS06a, MMS07c, MMS07a, MMS07b]. By means of a representative numerical example, the excellent performance of the resulting schemes, which base on linear Finite Elements in time combined with different quadrature rules, will be demonstrated and compared with the performance of well-accepted standard integrators. 2 Semi-Discrete Dynamics To set the stage, we start with some basic notation of geometrically nonlinear continuum mechanics. First, the nonlinear deformation map $φ$(X, t) : B0 $\times $[0, T ] $\rightarrow $Bt shall be

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