

QuaDynEvoPro 

ERC
Advanced Grant
Quasistatic
and Dynamic Evolution Problems in Plasticity and Fracture
Principal
Investigator: Gianni Dal Maso
This research project deals with nonlinear evolution problems
that arise in the study of the inelastic
behaviour of solids,
in particular in plasticity
and fracture. The
project will focus on selected problems, grouped into three main topics,
namely:
Plasticity with
hardening and softening,
Quasistatic crack
growth,
Dynamic fracture
mechanics.
The analysis of the models of these mechanical problems
leads to deep mathematical questions originated by two common features: the
energies are not convex
and the solutions exhibit discontinuities
both with respect to space and time. In addition, plasticity problems often
lead to concentration
of the strains, whose mathematical description requires singular measures. Most of these
problems have a variational
structure and are governed by partial differential equations. Therefore, the construction of
consistent models and their analysis need advanced mathematical tools from the calculus of variations,
from measure theory and geometric measure theory, and also from the theory of nonlinear elliptic and parabolic partial differential
equations. The models of dynamic crack growth considered in the project
also need results from the theory of linear hyperbolic equations.
Our goal is to develop new mathematical tools in these
areas for the study of the selected problems. Quasistatic evolution problems in
plasticity with hardening and softening will be studied through a vanishing viscosity
approach, that has been successfully used in the study of the CamClay model in
soil mechanics. Quasistatic models of crack growth will be developed under
different assumptions on the elastic response of the material and on the
mechanisms of crack formation. For the problem of crack growth in the dynamic
regime our aim is to develop a model that predicts the crack path as well as the time evolution of the crack
along its path, taking into account all inertial effects.
The numerical aspects of the project will also benefit
of the activity of MathLab, a recently
established SISSA laboratory for mathematical modeling and scientific
computing, directed by Antonio DeSimone and Alfio Quarteroni, and devoted to
the interactions between mathematics and its applications. There will be also a
strong interaction with the PIRE project
ÒScience at the triple point
between mathematics, mechanics and materials scienceÓ.
The starting date of the project is March 1, 2012.