Dynamic homogenization and wave propagation in a nonlinear 1D composite material

Wave propagation in nonlinear elastic media with microstructure is studied. As an illustrative example, a 1D model of a layered composite material is considered. Geometrical nonlinearity is described by the Cauchy–Green strain tensor. For predicting physical nonlinearity the expression of the energy of deformation as a series expansion in powers of the strains is used. The effective wave equation is derived by the higher-order asymptotic homogenization method. An asymptotic solution of the nonlinear cell problem is obtained using series expansions in powers of the gradients of displacements. Analytical expressions for the effective moduli are presented. The balance between nonlinearity and dispersion results in formation of stationary nonlinear waves that are described explicitly in terms of elliptic functions. In the case of weak nonlinearity, an asymptotic solution is developed. A number of nonlinear phenomena are detected, such as generation of higher-order modes and localization. Numerical results are presented and practical significance of the nonlinear effects is discussed.

► Wave propagation in nonlinear elastic media with microstructure is studied.
► Approximate analytical expressions for the effective moduli are derived.
► A number of nonlinear phenomena are detected.
► Numerical results are presented and practical significance of the nonlinear effects is discussed.