On one-dimensional solitary waves in microstructured solids

Abstract

For describing the longitudinal deformation waves in microstructured solids a Mindlin-type model is used. The embedding of a microstructure in an elastic material is reflected in an inherent length scale causing dispersion of propagating waves. Nonlinear effects, if taken into account, will counteract dispersion. A suitable balance between nonlinearity and dispersion may permit the propagation of solitary waves. Following previous work by Engelbrecht and others the nonlinear hierarchical model is derived in a one-dimensional setting which corresponds to a two-wave equation. The evolution equation as a simplified model, representing a one-wave equation, is able to grasp the essential effects of microinertia and elasticity of the microstructure. It is shown that the nonlinearity in microscale leads to an asymmetry of the wave profile. The nonlinear evolution equation as an extended Korteweg–de Vries equation is solved approximately by a series expansion in a small parameter representing the micro-nonlinearity. Already the first approximation indicates the asymmetry of the solitary waves. It is shown that solitary waves will propagate only if the micro-nonlinearity does not exceed some upper bound. For the limiting case, an analytical solution of the extended Korteweg–de Vries equation can be provided and used as a reference for the approximate solutions.