Colloquium
3:30 p.m., Friday
Math 100
Anna Vainchtein
Division of Mechanics and Computation
Department of Mechanical Engineering
Stanford University
Hysteresis and interface
dynamics in mathematical models of phase transitions
Materials undergoing stressinduced martensitic phase transformations
(in particular, shape memory alloys) exhibit a markedly hysteretic
behavior under cyclic loading. The hysteresis loops on the
loadelongation diagrams are often serrated. The serrations are
accompanied by a nonsmooth, ''jerky'' motion of the phase boundaries.
In this talk, we consider two mathematical models describing
the dynamics of phase transitions. In both models, a bar is
subjected to timedependent displacement boundary conditions.
The bar is assumed to be an elastic continuum that can deform only
in the direction along its length. The local deformation
of the bar is described by the displacement field and its spatial
derivative, called strain. The elastic properties of the bar are
determined by an elastic energy density which is a function of strain.
We assume this function to be a nonconvex doublewell potential.
The wells in the elastic energy density represent two different
material phases, austenite and martensite. The dynamic models take
into account both inertia and dissipative viscous terms. The first
model also includes the interfacial energy, modeled by a
straingradient term. In the second model, this term is omitted.
The models result in initialboundary value problems for nonlinear
parabolic PDE. Both models predict hysteresis which is primarily
due to dynamic solution getting locked in local minimizers of the
potential energy functional. The hysteresis loops persist even when
the loading rate is very slow, and viscosity effects are minor. We
find that in the model without the interfacial energy term, the
hysteresis loops are serrated, and a stickslip interface motion
is observed. We show that for a given loading this solution behavior is a
singular limit of the model with interfacial energy term as this
term tends to zero. On the other hand, at fixed straingradient
coefficient and slow enough (quasistatic) loading the model including
the interfacial energy results in a smooth interface motion and
smaller, nonserrated hysteresis loops.
Refreshments will be served in Math Annex Room 1115, 3:15 p.m.
