Existence and Stability of Periodic Planar Standing Waves in Phase-Transitional Elasticity with Strain-Gradient Effects I: General Theory

Extending investigations of Antman &Malek-Madani, Schecter & Shearer, Slemrod, Barker & Lewicka & Zumbrun, and others, we investigate phase-transitional elasticity models with strain-gradient effect. We prove the existence of non-constant planar periodic standing waves in these models by variational methods, 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. Previous such results were confined to one-dimensional deformations in models with artificial viscosity–strain-gradient coefficients.