Analysis of Energetic Models for Rate-Independent Materials

We consider rate-independent models which are defined via two functionals: the time-dependent energy-storage functional I : [0, T ]×X → [0,∞] and the dissipation distance D : X × X → [0,∞]. A function z : [0, T ] → X is called a solution of the energetic model, if for all 0 ≤ s < t ≤ T we have stability: I(t, z(t)) ≤ I(t, z̃) +D(z(t), z̃) for all z̃ ∈ X; energy inequality: I(t, z(t))+DissD(z, [s, t]) ≤ I(s, z(s))+ ∫ t s ∂τI(τ, z(τ )) dτ . 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 ∈ Z ⊂ 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. 2000 Mathematics Subject Classification: 74C15.