Asymptotic simulation of imperfect bonding in periodic fibre-reinforced composite materials under axial shear

An asymptotic approach for simulation of the imperfect interfacial bonding in composite materials is proposed. We introduce between the matrix and inclusions a flexible bond layer of a volume fraction c⁽³⁾ and of a non-dimensional rigidity λ⁽³⁾, derive a solution for such three-component structure, and then set c⁽³⁾→0, λ⁽³⁾→0. In the asymptotic limit depending on the ratio λ⁽³⁾/c⁽³⁾ different degrees of the interface's response can be simulated. A problem of the axial shear of elastic fibre-reinforced composites with square and hexagonal arrays of cylindrical inclusions is considered. The performed analysis is based on the asymptotic homogenization method, the cell problem is solved using the underlying principles of the boundary shape perturbation technique. As a result, we obtain approximate analytical solutions for the effective shear modulus and for the stress field on micro level depending on the degree of the interfacial debonding. Developed solutions are valid for all values of the components’ volume fractions and properties. In particular, they work well in cases of rapid oscillations of local stresses (e.g., in the case of densely packed perfectly rigid inclusions), while many of other commonly used methods may face computational difficulties.