Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

“…(4.15) was solved in Lambrecht et al (2003) for a one-dimensional strain-softening elastic-plastic bar. A schematic visualization is given in Fig.…”

“…As pointed out in the recent papers Lambrecht et al (2003) and Miehe and Lambrecht (2003a,b), the incremental variational formulation for the inelastic response opens up the opportunity to resolve a developing microstructure in non-stable standard dissipative solids by a relaxation of the associated non-convex incremental variational problem, in Box 1 denoted by problem (R). If the above-outlined material stability analysis detects a non-convex incremental stress potential, an energy-minimizing deformation microstructure is assumed to develop such as indicated in Fig.…”

“…We refer to Kohn andStrang (1983,1986), Dacorogna (1989), Ä Silhavà y (1997), M uller (1998 and Dolzmann (2003) for a sound mathematical basis. The concept of relaxation has been applied to elastic phase decomposition problems by Kohn (1991), Luskin (1996), Carstensen and Plechà aÄ c (1997), Smyshlyaev and Willis (1999), DeSimone and Dolzmann (2000), Govindjee et al (2002) and just recently on an approximative level to single crystal plasticity by Ortiz and Repetto (1999) and Ortiz et al (2000) and to strain-softening von Mises plasticity by Lambrecht et al (2003) and Miehe and Lambrecht (2003a,b).…”

Existence results for energetic models for rate-independent systems

“…The purpose of this work is to provide an abstract framework for constructing solutions to (S) & (E) without relying on any underlying linear structures in Y = F ×Z. Thus, we hope to provide a basis for applications in genuinely nonlinear mechanical models such as elasto-plasticity with finite strains, see [OrR99,CHM02,Mie02,Mie03,LMD03,Mie04]. Our existence proof is based on the commonly used time-incremental approach which leads to minimization problems.…”

Lower semicontinuity and existence of minimizers in incremental finite‐strain elastoplasticity

“…More precisely, we consider a time-discretized version which leads to a sequence of minimization problems (one for each time step). Such formulations have recently attracted a lot of attention in the engineering literature [OR99,OS99,CHM02,LMD03,Mie03a,NW03]. In the simplest version one considers a multiplicative decomposition of the deformation gradient Dϕ = F el F pl and assumes that the elastic energy depends only on F el and on suitable hardening parameters p ∈ R m .…”