Quasistatic evolution problems in plasticity with softening

Gianni Dal Maso S.I.S.S.A.
email: dalmaso@sissa.it

Abstract: In plasticity theory the term softening refers to the
reduction of the yield stress as plastic deformation proceeds. We deal with
this problem in the
quasistatic case, in the framework of small strain associative
elastoplasticity. The presence of a nonconvex term due to the softening
phenomenon requires the extension of a variational framework proposed by Mielke
to the case of a nonconvex energy functional. In this problem the use of global
minimizers in the corresponding incremental problems is not justified from the
mechanical point of view. We analize a different selection criterion for the
solutions of the quasistatic evolution problem, based on a viscous
approximation. In view of the nonconvexity of the problem, taking the limit as
the artificial viscosity parameter tends to zero leads to a weak formulation of
the problem in a space of Young measures. Moreover, since the growth exponent
of the energy is one, we need a suitable notion of generalized Young measure in
order to deal with concentration effects. Finally, the classical notion of
total variation of a time-dependent function on a time interval has to be
extended to time-dependent families of Young measures. This enables us to
define, in this generalized context, a notion of dissipation, which plays a
crucial role in Mielke's variational approach. Some examples show that smooth
initial data may lead, after a critical time, to a Young measure solution with
concentration phenomena.

These results have been obtained in collaboration with Antonio DeSimone,
Maria Giovanna Mora and Massimiliano Morini.