Homogenization and effective properties of plates weakened by partially penetrating fissures: Asymptotic analysis
This paper is a third approach of the authors to the analysis of fissures in plates. It deals with the overall stiffnesses of elastic plates weakened by periodically distributed part through fissures of constant depth. The fissures are assumed to obey the Signorini-type conditions without friction, so they can open or close. The three-dimensional problem has been solved by E. Sanchez-Palencia (1980). The case of elastic plates cannot be treated analogously because the fissures intersect the boundary surface. As it is usually done in the plate theory, the averaging process (smearing out the fissures) is preceded by the process of reduction of the transverse dimension. The plate behaviour is described by the two-dimensional two-layer plate model. In contrast with approaches based on the Kirchhoff, or the Reissner-like plate models, it is possible in the present approach to describe the opening and the closure of the fissures without the undesired effect of possible interpenetration of the adherent faces of the fissures. This formulation makes it possible to describe deformations of the plates which are weakened by part through cracks in the lower layer. In this frame, the homogenization process is applied to a plate with periodically distributed fissures of constant depth. As usual the homogenized formulae are derived by applying the two-scale asymptotic method. The problem of the local behaviour of the plate (at the scale of a period of the structure) assumes the form of a variational inequality posed on the rescaled cell of periodicity. The effective plate is described by the two-layer plate model that turns out to be nonlinear and hyperelastic. The associated hyperelastic potential of the homogenized plate is exhibited. An example, in which fissures are straight lines, is treated.
scientific article; zbMATH DE number 91141
scientific article