  Extending investigations of Antman & Malek-Madani, Schecter & Shearer,
Slemrod, Barker & Lewicka & Zumbrun, and others, we investigate
phase-transitional elasticity models of strain-gradient effect. We prove the
existence of non-constant planar periodic standing waves in these models with
strain-gradient effects by variational methods and phase-plane analysis, for
deformations of arbitrary dimension and general, physical, viscosity and
strain-gradient terms. Previous investigations considered one-dimensional
phenomenological models with artificial viscosity/strain gradient effect, for
which the existence reduces to a standard (scalar) nonlinear oscillator. For
our variational analysis, we require that the mean vector of the unknowns over
one period be in the elliptic region with respect to the corresponding pure
inviscid elastic model. For our (1-D) phase-plane analysis, we have no such
restriction, obtaining essentially complete information on the existence of
non-constant periodic waves and bounding homoclinic/heteroclinic waves. Our
variational framework has implications also for time-evolutionary stability,
through the link between the action functional for the traveling-wave ODE and
the relative mechanical energy for the time-evolutionary system. Finally, we
show that spectral implies modulational nonlinear stability by using a change
of variables introduced by Kotschote to transform our system to a strictly
parabolic system to which general results of Johnson--Zumbrun apply. Previous
such results were confined to one-dimensional deformations in models with
artificial viscosity--strain-gradient coefficients.
