Concentrated force solutions for an inhomogeneous thick elastic plate

Abstract

The technique developed in refs. [1][2][3][4][5] is applied to the problem of a concentrated line force acting in the interior of an infinite plate. The plate is of arbitrary thickness, is isotropic, but is inhomogeneous in that the elastic moduli are any specified functions, not necessarily continuous, of the through-thickness coordinate. The mechanical properties of the plate are not necessarily symmetric about the mid-surface. The solution is based on the classical solution for a concentrated force in a thin elastic plate. This classical solution is extended to give exact closed form solutions for the displacement and stress in the thick inhomogeneous plate. For a plate that is not symmetric an in-plane force gives rise to bending as well as stretching deformations. Higher order force singularities are also considered, as is the problem of a concentrated force on the boundary of a semi-infinite symmetric plate. Mathematics Subject Classification (2000). 73B27, 73C02, 73K10.

Analysis of fundamental solutions to the equations of statics constructed for transversal-isotropic plates

“…The problem on the concentrated force that acts in the inner part of an infinite plate is solved in article [3]. This publication examines an isotropic plate of arbitrary thickness, in which the moduli of elasticity are any assigned functions.…”

Bending Solutions for Inhomogeneous and Laminated Elastic Plates

“…Note that in equations (2.9) and (2.10) the terms in r 6w and r 8w are the additional contribution due to the enhanced displacement field, compared with that found in [11]. Let us follow the approach adopted in [7,11] and suppose the weight functions satisfy the following system of ordinary differential equations…”

“…In a series of earlier papers, Rogers and Spencer [1], Rogers [2], Spencer [3,4,7,8], Mian and Spencer [5], Spencer and Selvadurai [6] have developed and applied a procedure for deriving exact solutions of the equations of linear elasticity for materials that are isotropic but inhomogeneous in a specified direction. Such inhomogeneous materials occur frequently both naturally and in man-made structures.…”

Three-dimensional elastic deformation of functionally graded isotropic plates under point loading

“…A solution to the problem of a concentrated line force acting in the interior of an infinite plate was developed by Spencer (2000). The plate was assumed to be of arbitrary thickness, isotropic and inhomogeneous, with the elastic moduli being functions, not necessarily continuous, of the through-thickness coordinate.…”