Old and new approaches to rate-independent systems


MR14 - Centre for Mathematical Sciences
Cambridge, United Kingdom

Thursday, 21 May 2015
from 14:00 to 18:00

Dr Filip Rindler
University of Warwick
Many physical systems involve an interplay of elastic (reversible) and dissipative (irreversible) effects. If the amount of energy that is dissipated (removed from the system) only depends on the traveled distance in the state space, but not the rate (speed) by which this movement occurs, then the system is called rate-independent. This is almost always only an approximation to reality, but it turns out to describe a variety of phenomena rather accurately, among them models of plasticity (permanent deformation) in metals, shape-memory alloys and damage/fracture in brittle materials – all of which have numerous applications in engineering.

Mathematically, these models lead to PDEs with unusual features. Most prominently, time derivatives are enclosed in positively $0$-homogeneous functions (such as x/|x|), whereby we can at most expect BV-regularity in time. As jumps may really occur in the time-evolution, we need to specify the behavior of the system over these discontinuities. Many engineering models are under-specified in this area and it is up to mathematicians to find good concepts of what constitutes solutions.

This short course will first explain the physical models in some detail and then consider several approaches to define mathematical solutions concepts and to “find” corresponding solutions. As this is a vastly unfinished area of mathematical analysis, we will end with some future directions and challenges.
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