Soft flexural waves and self-localization of wrinkling modes of single- and few-layer graphene under compression in a compliant matrix

Abstract

This Research Letter studies the evolution under external compression of the wrinkling modes of single- and few-layer graphene or another two-dimensional atomic crystal embedded in or placed on a compliant matrix, or on a dense fluid in a gravitational field. An analytical model based on the nonlinear bending elasticity of the graphene is developed, which shows that the compressive surface stress causes spatial localization of the extended sinusoidal wrinkling mode with a solitonlike envelope, with the localization length decreasing with overcritical external strain. The parameters of the extended sinusoidal wrinkling mode are determined from the conditions of the anomalous softening of the flexural surface acoustic wave with predominant out-of-plane and suppressed in-plane surface displacements, propagating along the graphene in a compliant anisotropic matrix. Self-localization of the wrinkling modes finally results in the formation of strongly localized modes with approximately one-period sinusoidal profiles and external-strain-independent wavelengths. Self-localization of the wrinkling modes is governed by the derived Ginzburg-Landau-type nonlinear envelope-function equation with a negative dispersive term, which we relate with the graphene-nonlinearity-induced repulsion between soft flexural surface acoustic waves with negative effective mass. One- and two-dimensional wrinkling patterns and two types of strongly localized graphene wrinkling modes with different symmetry are described.