Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

• A macroscopic wave equation is derived by the higher-order asymptotic homogenization.
• Explicit analytical solutions for stationary nonlinear strain waves are presented.
• Soft and hard nonlinearities lead to strongly different types of localized waves.
• Nonlinearity and dispersion compensate the influence of each other.
• Areas of practical applicability of linear and nonlinear approaches are estimated.

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.