Existence results for energetic models for rate-independent systems

Abstract

We consider mechanical models which are driven by an external loading on a time scale much slower than any internal time scale (like viscous relaxation times) but still much faster than the time needed to find the thermo-dynamical equilibrium. Typical phenomena involve dry friction, elasto-plasticity, certain hysteresis models for shape-memory alloys and quasistatic delamination or fracture. The main feature is the rate-independency of the system response, which means that a loading with twice (or half) the speed will lead to a response with exactly twice (or half) the speed. We refer to [BrS96, KrP89, Vis94, Mon93] for approaches to these phenomena involving either differential inclusions or abstract hysteresis operators. Our method is different, as we avoid time derivatives and use energy principles instead. As is well-known from dry friction, such systems will not necessarily relax into a complete equilibrium, since friction forces do not tend to 0 for vanishing velocities. One way to explain this phenomenon on a purely energetic basis is via so-called “wiggly energies”, where the macroscopic energy functional has a super-imposed fluctuating part with many local minimizers. Only after reaching a certain activation energy it is possible to leave these local minima and generate macroscopic changes, cf. [ACJ96, Jam96, Men02]. Here we use a different approach which involves a dissipation distance which locally behaves homogeneous of degree 1, in contrast to viscous dissipation which is homogeneous of degree 2. This approach was introduced in [MiT99, MiT03, MTL02, GMH02] for models for shape-memory alloys and is now generalized to many other rate-independent systems. See [Mie03a] for a general setup for rate-independent material models in the framework of “standard generalized materials”. To be more specific we consider the following continuum mechanical model. Let Ω ⊂ R be the undeformed body and t ∈ [0, T ] the slow process time. The deformation or displacement φ(t) : Ω → R is considered to lie in the space F of admissible deformations containing suitable Dirichlet boundary conditions. The internal variable z(t) : Ω → Z ⊂ R describes the internal state which may involve plastic deformations, hardening variables, magnetization or phase indicators. The elastic (Gibbs) stored energy is given