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Mathematics and Mechanics of Solids
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Complex Variable Solutions for an Inhomogeneous Thick Plate Containing a Hole or a Crack

A. H. England

Division of Theoretical Mechanics, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

Exact solutions have been found to the equations of elasticity in a thick plate of inhomogeneous isotropic linearly elastic material in which the elastic moduli are known functions of the coordinate normal to the plane of the plate. The upper and lower surfaces of the plate are assumed to be traction free. In a previous paper these solutions were expressed in terms of four complex potentials that are analytic functions of {zeta} = x + iy, the coordinates in the mid-plane of the plate. Whilst the solutions obtained do not contain enough generality to satisfy the boundary conditions pointwise around the edge of the plate, resultant force and moment conditions or averaged displacements may be specified over the edge of the plate. The intention in this paper is to examine the elastic field in a plate containing a cylindrical hole or a through-thickness line crack under the action of a uniform force field at infinity.

Key Words: Inhomogeneous • elastic • complex-variable methods • cracks

Mathematics and Mechanics of Solids, Vol. 9, No. 5, 445-471 (2004)
DOI: 10.1177/1081286504038670
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A. H. England and A. J. M. Spencer
Complex Variable Solutions for Inhomogeneous and Laminated Elastic Plates
Mathematics and Mechanics of Solids, October 1, 2005; 10(5): 503 - 539.
[Abstract] [PDF]