Dynamics of  solids with non-monotone
                         stress-strain relations

                                   by

                      Andrej Cherkaev and Sasha Balk
             Department of Mathematics, University of Utah

               INSCC 110, 3:30pm  Monday, January 11, 1999


                                Abstract

We discuss the realization of the Gibbs' principle of
the minimal energy in the phase transition process. 
Namely, we are looking for a dynamical process
that could lead to the state of minimal energy. 
We introduce a model of a chain of masses joined by 
springs with a non-monotone strain-stress relation.
Numerical experiments are conducted to find dynamics of that
chain with slow external excitation. 
We find that dynamics leads  either to oscillating
steady state with radiation of the energy, or (if dissipation is
introduced) to hysteresis rather than to unique stress-strain 
dependence that would correspond to energy minimization.
To study details of the dynamical process, we analytically
integrate equations of strongly non-linear waves in the unstable chain. 
We describe the sonic wave, the wave of phase transition, 
and the oscillating state of the chain.
The analytical results are compared with numerical experiments.