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
A particular singular problem in the class of plane linear elasticity theory that plays an important role in the development of linear elastic fracture mechanics (LEFM) is reviewed and mathematically studied. The specific problem is started with consideration of a planar wedge having stress-free on both radial edges but the remainder subjected to in-plane loading at the far-field. Employing the semi-inverse method together with the Fadle eigenfunction expansion technique, its solution can be written in terms of Airy stress function satisfying the two-dimensional biharmonic governing differential equation. Further analysis leads to a special case of crack problems, which is in the field of elastic fracture mechanics.
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