Samira Boddin (Univ. Kassel): Convergence analysis of an approximation scheme for rate-independent systems with state-dependent dissipation

Livestream:  https://uni-kassel.zoom.us/j/96217091997?pwd=RVRaVVBRclFJYU9jczNZSWF3SXI2QT09
Meeting ID: 962 1709 1997
Passcode: cauchy

Convergence analysis of an approximation scheme for rate-independent systems with state-dependent dissipation Samira Boddin (Univ. Kassel)

Abstract:
Many mechanical processes including for example damage, dry friction or elastoplasticity can be described by rate-independent systems driven by an energy functional that contains a time-dependent external loading and by a convex dissipation potential that is positively homogeneous of degree one. The rate-independency is then reflected by the invariance of the solutions under time rescalings, i.e. rescaled external loadings lead to rescaled solutions.

Non-convex energy functionals or state-dependent dissipation potentials give rise to solutions developing jumps in time. Therefore several formulations of rate-independent systems and different solution concepts derived thereof have been proposed. Now one has to decide which concept fits best for a given problem and numerical algorithms are needed that reliably approximate the type of solution one is interested in.

In this talk we adapt an approximation scheme proposed by Efendiev and Mielke to treat also the case of a state-dependent dissipation. On the basis of the existing convergence analysis for the non-adapted scheme and a compactness result for Young measures we show the convergence of the adapted scheme to the physically reasonable notion of balanced viscosity (BV) solutions.

As application of our result we consider non-associative plasticity models. Based on Laborde's generalized principle of maximum dissipation their formulation via a generalized stress constraint and a non-associative flow rule given by a convex and continuous yield function and plastic potential can be transferred into our formulation with an energy functional and a dissipation potential, which then also depends on the states and not only on the rates. Therefore we eventually end up with a problem that fits in our setting by adding a suitable mathematical regularization.

Mit freundlichen Grüßen,
Prof. Dr. Dorothee Knees (AG AAM)