End Effects in Prestrained Plates Under Compression

The decay of end perturbations imposed on a rectangular plate subjected to compression is investigated in the context of plane-strain incremental finite elasticity. A separation of variables in the eigenfunction formulation is used for the perturbed field within the plate. Numerical results for the leading decay exponent are given for four rubbers: three compressible and one incompressible. It was found that the lowest decay rate is governed by a symmetric field that exhibits different patterns of dependence on the prestrain for compressible and for nearly incompressible solids. Compressible solids are characterized by low sensitivity of the decay rate to prestrain level up to moderate compression, beyond which an abrupt decrease of decay rate brings it to zero. Nearly incompressible solids, on the other hand, expose a different pattern involving interchange of modes with no decrease of decay rate to zero. Both patterns show that the decay rate obtained from linear elastic analysis can be considered as a good approximation for a prebuckled, slightly compressed plate, which is long enough in comparison to its width. Along with decaying modes, the eigenfunction expansion generates a nondecaying antisymmetric mode corresponding to buckling of the plate. Asymptotic expansion of that nondecaying mode near the stress free state predicts buckling according to the classical Euler formula. A consistent interpretation of end effects in the presence of a nondecaying mode is given.