HOMOGENIZATION AND EFFECTIVE PROPERTIES OF PLATES WEAKENED BY PARTIALLY PENETRATING FISSURES - ASYMPTOTIC ANALYSIS

A method of finding effective properties of elastic plates weakened by periodically distributed part-through-the thickness fissures is presented. The plate behaviour is described by the two-dimensional two-layer plate model. The model involves four kinematical fields standing for: the in-plane displacements of the interface plane, the independent rotations of the cross-sections of the lower and upper layers of the plate and for the common deflection of both layers. Within the framework of this model the bending mode of fissuring is considered. In contrast to the approaches of our previous papers, which were based on the Kirchhoff [J. Elasticity 19 (1988)] and the Reissner-like [Arch. Mech. 40 (1988)] plate models, the present approach enables one to describe the opening and closure of the fissures without the undesired effect of possible interpenetration of the adherent faces of the fissures. The homogenized formulae are derived by applying the two-scale asymptotic method. The effective plate is described by the two-layer plate model that turns out to be non-linear and hyperelastic. The hyperelastic potential involves the solution of the basic cell problem. This problem assumes here the form of a variational inequality posed on a rescaled cell of periodicity. Having found the effective two-layer plate model one can construct the effective model of Kirchhoff type. Also this model turns out to be non-linear and hyperelastic.