On the implementation of rate-independent standard dissipative solids at finite strain – Variational constitutive updates

Abstract

This paper is concerned with an efficient, variationally consistent, implementation for rateindependent dissipative solids at finite strain. More precisely, focus is on finite strain plasticity theory based on a multiplicative decomposition of the deformation gradient. Adopting the formalism of standard dissipative solids which allows to describe constitutive models by means of only two potentials being the Helmholtz energy and the yield function (or equivalently, a dissipation functional), finite strain plasticity is recast into an equivalent minimization problem. In contrast to previous models, the presented framework covers isotropic and kinematic hardening as well as isotropic and anisotropic elasticity and yield functions. Based on this approach a novel numerical implementation representing the main contribution of the paper is given. Analogously to the theoretical part, the algorithmic formulation is variationally consistent, i.e., all unknown variables follow naturally from minimizing the energy of the considered system. Extending previously published works on these methods, the advocated model does not rely on any material symmetry and therefore, it can be applied to a broad range of different plasticity theories. As an example, an anisotropic Hill-type model is developed. Numerical examples demonstrate the applicability and the performance of the proposed implementation.

Partially relaxed energy potentials for the modelling of microstructures – application to shape memory alloys

“…In consequence, the use of condensed energies is suitable for the modelling of microstructure formation rather than microstructure evolution. As shown in, e.g., [34,[38][39][40], the continuous evolution of inelastic processes and microstructure evolution can be modeled by approximations for the dissipation distance. In contrast to this approach, an alternative formulation is suggested in [19,26] which leads to the relaxed problem via a partial relaxation.…”

“…Mesoscale: x ∈ B mes ⊂ B mac homogenisation procedure: calculation of the effective material response σ mes according to (40) Macroscale B mac : homogenisation procedure according to (49).…”

Efficient variational constitutive updates by means of a novel parameterization of the flow rule

“…By way of contrast, such algorithms are well-established in case of the return-mapping scheme, see [1,2]. First ideas for a more general framework for variational constitutive updates are discussed in [10]. The second reason why variational constitutive updates have not been frequently applied yet is related to their numerical efficiency.…”

“…While the classical return-mapping algorithm has already reached a certain maturity and thus, its numerical implementation has been highly optimized, the implementation of variational constitutive updates has drawn attention only relatively recently, cf. [6,10,18,19].…”

“…Due to the underlying variational basis, such models are referred to as variational constitutive updates, cf. [7][8][9][10].…”