The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (196 KB) | Metadata | Table of Contents | ZAA summary
Volume 33, Issue 1, 2014, pp. 1–19
DOI: 10.4171/ZAA/1496

Published online: 2013-12-27

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

Jinghua Yao[1]

(1) Indiana University, Bloomington, USA

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.

Keywords: Elasticity, strain-gradient effect, periodic wave, Hamiltonian system

Yao Jinghua: Existence and Stability of Periodic Planar Standing Waves in Phase-Transitional Elasticity with Strain-Gradient Effects I: General Theory. Z. Anal. Anwend. 33 (2014), 1-19. doi: 10.4171/ZAA/1496