The general fundamental solution of the sixth-order Reissner and Mindlin plate bending models revisited

Abstract

The sixth-order differential equation system of the Reissner and Mindlin plate bending models describe mathematically the plate problem, where lines normal to the mid-plane before deformation remain straight and inextensible in the deformed configuration, but not necessarily normal to the reference plane anymore. The non-normality condition is due to the consideration of transverse shearing strains, disregarded in the classical bi-harmonic Kirchhoff plate model. As the plate thickness is reduced and/or the transverse shear modulus is increased, the deformed configuration is less dependent of the transverse shear strains and, in the limit as the plate thickness approaches zero and/or the transverse shear modulus approaches ∞, the transverse shear strain effects vanish and the problem is exactly that described by the classical plate model. In this paper, we investigate the fundamental solutions of both the fourth-order Kirchhoff and the sixth-order Reissner and Mindlin plate models. We consider a transversely isotropic material and show that the fundamental solution of the bi-harmonic problem can be obtained directly from the general fundamental solution of the sixth-order plate problem, in the limit as the plate thickness approaches zero and/or the transverse shear modulus approaches ∞. This solution is in agreement with the analytical solution of an infinite thin clamped circular plate submitted to a unitary concentrated load acting at its center.