A Mixed Problem for an Elastic Ring

Abstract

Along the two contours of a doubly connected region, mixed boundary values of slipless nature are prescribed. The problem is formulated on a complex z-plane in terms of a Hilbert functional equation involving a holomorphic function φ(z). Based on plane theory of elasticity, the solution is found in the form of a double series. An effective method of successive approximation is shown for the computation of the coefficients of the series. Numerical examples of the field of stress and deformation are illustrated with diagrams. By superposition, the solution presented can be used to solve another class of mixed problems in elasticity involving frictionless indentation.

On numerical equivalence of parallel generalized solutions of mixed boundary value problem of annulus

“…Among the complicated mixed boundary value problems for arbitrary doubly-connected regions, the mixed boundary value problem in an elastic annulus is the most fundamental model. Yau [12] proposed a particular solution on a bisymmetrical elastic annulus subjected to fixed constraints acting along two opposite quater arcs of the outer periphery and a constant radial pressure acting along the whole inner periphery. Sugiura [13,14] proposed a pair of particular solutions on symmetrical elastic annuli subjected to gravity and axisymmetrically fixed constraints along either periphery.…”

“…The solutions proposed by Yau [12] is elegant by setting Possion's ratio to 0.5 and only considering a symmetrical condition. Sugiura [13,14] also consider symmetrical boundary conditions without displacement solutions.…”

“…Limited by computational techonology, such assumptions and simplifications are advanced back in 1970s. Inspired by these elegant works [12][13][14], and we respectfully seek two parallel sets of more generalized solutions by cancelling these over-simplified assumptions. More importantly, the mutually numerical equivalence between these two parallel solutions is fully dicussed, and several detailed mathematical insights of the application of complex variable method are revealed.…”