74A45 Theories of fracture and damage
Complete damage in linear elastic materials — Modeling, weak formulation and existence results
(2013)
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying
Dirichlet boundary conditions. The evolution of the system is
described by a doubly nonlinear differential inclusion for the damage
process and a degenerating quasi-static balance equation for the displacement field
which are strongly nonlinearly coupled.
In our proposed model, the material
may completely disintegrate which is indispensable for a realistic modeling of
damage processes in elastic materials. Complete damage theories
lead to several mathematical problems since, for instance, coercivity properties
of the free energy are lost and, therefore, several difficulties arise.
For the introduced complete damage model, we propose a classical
formulation and a corresponding suitable weak formulation in an
$SBV$-framework. The main aim is to prove existence of weak solutions
for the introduced degenerating model. In addition, we show that the classical
differential inclusion can be regained from the notion of weak solutions under
certain regularity assumptions which is a novelty in the theory of complete damage
models of this type.
For the existence results, we had to handle the following problem:
During the damage process it might occur that not completely damaged material regions are isolated
from the Dirichlet boundary. In this case, the
deformation field cannot be controlled in the transition from incomplete
to complete damage. To tackle this problem, we consider the evolution
process on a time-depending domain. In this context, two major challenges
arise:
Firstly, the time-dependent domain approach leads to jumps in the energy
which have to be accounted for in the energy inequality of the notion of
weak solutions. To handle this problem, several energy estimates are established
by $\Gamma$-convergence techniques. Secondly, the time-depending domain
might have bad smoothness properties such that Korn's inequality cannot be
applied. To this end, a covering result for such sets with smooth
compactly embedded domains has been shown.
Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints
(2012)
A global higher differentiability result in Besov
spaces is proved for the displacement fields of linear elastic models
with self contact.
Domains with cracks are studied, where nonpenetration
conditions/Signorini conditions are imposed on the crack faces.
It is shown that
in a neighborhood of crack tips (in 2D) or
crack fronts (3D) the displacement fields are
$B^{3/2}_{2,\infty}$ regular.
The proof relies on a difference
quotient argument for the directions tangential to the crack. In order
to obtain the regularity estimates also in the normal direction, an
argument due to
Ebmeyer/Frehse/Kassmann is modified.
The methods are then applied to further examples like
contact problems with nonsmooth rigid foundations, to a model with
Tresca friction and
to minimization problems with
nonsmooth energies and constraints as they occur for instance in the modeling of
shape memory alloys.
Based on Falk's approximation Theorem for variational
inequalities, convergence rates for FE-discretizations of contact
problems are derived relying on the proven regularity properties.
Several numerical examples illustrate the theoretical results.