Externally funded project

Rate-independent systems in solid mechanics - physical properties, mathematical analysis and efficient numerical implementation

Project Details
Project duration: 06/202009/2023


The mechanical response of many materials subjected to quasi-static
loading can be characterized by rate-independent inelasticity.
Representative examples encompass elastoplastic deformations or the
evolution of cracks in quasi-brittle materials. The energies capturing
rate-independent inelasticity are often non-convex. This is, for
instance, the case for sufficiently large latent hardening in crystal
plasticity or for phase-field models suitable for the analysis of
fracture. Due to this non-convexity, the internal variables capturing
the inelastic properties may jump in time and, thus, the required
mathematical analysis as well as numerical simulations become more
challenging.Rate-independent models have long been investigated from a
modeling/mechanical as well as from a mathematical point of view. If,
however, the contributions in the two different fields are carefully
analyzed, one notices that they have been evolving in an almost
independent manner. Within mathematics, different solution concepts like
global energetic solutions and vanishing viscosity solutions have been
developed in the last twenty years that allow for time-discontinuous
solutions. However, these solution concepts are not equivalent and it is
of vital importance to find the most suitable one from a purely
physical point of view. Furthermore, a large number of the mathematical
convergence results are based on mechanical models with crude
simplifications (as seen from a physical point of view). By way of
contrast, convergence properties or the aforementioned different
solution concepts have not been considered within mechanics. A
physically and mathematically sound general framework for the modeling
of rate-independent systems in solids that interprets the many different
computational approaches in view of mathematical solution concepts is
still missing. It is precisely the goal of this proposal to close this
gap between the two different communities.Since the range of
rate-independent models is very broad, damage and phase field fracture
models are chosen as a prototype class. As a central theme of the
project, models which are established within one of the two communities
will be investigated from the viewpoint of the other. Furthermore,
approaches originally proposed for other rate-independent systems will
be adapted to and analyzed for the modeling of material damage. Many of
the questions fall into the general topic of evolutionary
Gamma-convergence. This theory shall be advanced by focusing on
vanishing viscosity solutions. The composition of the research team
(computational mechanics at Dortmund and applied analysis at Kassel)
reflects the wide range of topics that are addressed in the research
proposal (mechanical modeling, simulation, numerical analysis, purely
mathematical questions).

Funded by DFG


Last updated on 2021-02-07 at 22:21