Analysis of energetic models for rate-independent materials
Alexander Mielke
TL;DR
The paper develops an abstract energetic framework for rate-independent materials defined by an energy-storage functional I and a dissipation distance D, and proves existence of BV solutions via time-discretized incremental minimization. It shows that, under mild compactness and semicontinuity assumptions, discrete solutions converge to a global-stability and energy-balance solution without differentiability. The framework is then instantiated in continuum mechanics, yielding three representative applications: phase transformations in shape-memory alloys, delamination along interfaces, and finite-strain elasto-plasticity, highlighting both successful existence results and fundamental difficulties (e.g., nonconvexity and microstructure) that motivate relaxation. The approach enables robust, time-scale-free modeling of complex material behavior, including phase changes, damage, and plasticity, within a unified variational setting.
Abstract
We consider rate-independent models which are defined via two functionals: the time-dependent energy-storage functional $\calI:[0,T]\ti X\to [0,\infty]$ and the dissipation distance $\calD:X\ti X\to[0,\infty]$. A function $z:[0,T]\to X$ is called a solution of the {energetic model}, if for all $0\leq s<t\leq T$ we have stability: $\mathcal I(t,z(t)) \leq \mathcal I(t,\widetilde z)+ \calD(z(t),\wt z)$ for all $\wt z\in X$; energy inequality: $\mathcal I(t,z(t)) {+} \Diss_\calD(z,[s,t]) \leq \mathcal I(s,z(s)) {+} \int_s^t \partial_τ\mathcal I(τ,z(τ)) \mathrm d τ$. We provide an abstract framework for finding solutions of this problem. It involves time discretization where each incremental problem is a global minimization problem. We give applications in material modeling where $z\in \calZ\subset X$ denotes the internal state of a body. The first application treats shape-memory alloys where $z$ indicates the different crystallographic phases. The second application describes the delamination of bodies glued together where $z$ is the proportion of still active glue along the contact zones. The third application treats finite-strain plasticity where $z(t,x)$ lies in a Lie group.