A revisit to the Williams's solution in bending problem of plates with stress singularities

Abstract

This paper aims to present the details of mathematical derivation for static bending problem of isotropic sectorial plates that involve stress singularities at the vertex of the plates due to geometry and boundary conditions. Following the classical Kirchhoff's plate theory, the general complete solution of 4 th -order partial differential equation governing the plate-bending behaviors can be determined mathematically and expressed explicitly in the form of infinite series through the separation of variables method in terms of polar coordinates. Based on the principle of superposition the solution can be separated into two parts; namely, the particular solution and complementary solution in which the latter describes the local singular behaviors at its sharp vertex exactly. Some cases of circular, annular and sectorial plates with mixed circumferential edge conditions that remain undetermined analytically up to the present time are also suggested.Mathematics Subject Classification: 31A25, 74K20, 35Q74

On singularities of solution of the elasticity system in a bounded domain with angular corner points

“…Let u ∈  (Ω) be the unique weak solution of the boundary value problem (8). The understanding of the singular terms permits us to evaluate the maximal regularity of the weak solution.…”

“…1. Localize the elasticity problem (8) in the neighborhood of an angular corner point and then consider the problem (8) in an infinite wedge. The problem ( 8) is written in local polar coordinates (r, 𝜃), and applying the method of ansatz leads to a boundary eigenvalue problem with the parameter.…”

“…We multiply on the both sides of ( 8) by the smooth cut-off function 𝜒, then substitute v = 𝜒u in (8). The derivatives are considered in the distribution sense.…”

“…6 Dempsey and Sinclair 7 have introduced a new form of Airy stress function to investigate the stress singularities of isotropic elastic plates in extension. The mathematical derivation for the static bending problem of isotropic sectorial plates that contain stress singularities at the vertex of the plates owing to geometry and boundary conditions is described in Boonchareon et al 8 In Hein and Erdogan, 9 the Mellin transform has been used to examine the stress singularities in a two-material wedge. 10 has employed a complex potential approach to analyze the form of eigenvector solutions for a general corner or an opening crack problem.…”

Closed-form solution for a cantilevered sectorial plate subjected to a tip concentrated force

“…He also provided a solution for twisting moments applied at the vertex of an infinite sectorial plate. Lim and Wang ( 2000 ) developed solutions for annular Mindlin sectorial plates using the Kirchhoff solutions, and Boonchareon et al ( 2013 ) revisited the William’s problem for plate bending, providing solutions for problems that were previously unsolved. Huang et al ( 2016 ) solved the infinite sectorial plate problem subjected to tip loads consisting of two twisting moments and a bending moment in a functionally graded plate, as well as concentrated forces in the plane of the plate; these solutions employed complex variable techniques.…”