Homogenization of a nonlinear elastic fibre-reinforced composite: A second gradient nonlinear elastic material

We study the homogenization of a nonlinear elastic material in contact with periodic parallel nonlinear elastic fibres of higher rigidity. The energy density of both materials is of Saint Venant–Kirchhoff type. The interactions between the matrix and the fibres are described by a local adhesion contact law with interfacial adhesive stiffness parameter depending on the period. Assuming that the Lamé constants in the fibres and the stiffness parameter have appropriate orders of magnitude, we derive a class of nonlinear elastic energies involving displacement gradient of second order.