Elastic fields for the ellipsoidal cavity problem

Lur'e (Three-dimensional Problem of the Theory of Elasticity. Interscience, New York, 1964, §6.9) presented an approach to solve the problem of an ellipsoidal cavity in a linear, elastic and isotropic medium loaded by uniform principal stresses at infinity. In this paper we show that the approach by Lur'e may have no solution. Derivation mistakes are first pointed out in his (6.9.22), (6.9.23), (6.9.30) and (6.9.31). With the correct expressions, we then prove the coefficient matrix in his (6.9.32) to be singular. Therefore constants A,A ₄,A ₅ may have no solution. The problem lies in the harmonic functions chosen by Lur'e for the Papkovich-Neuber solution. From the solutions obtained by the Eshelby equivalent inclusion method, the present paper derives new Papkovich-Neuber harmonic functions for the ellipsoidal cavity problem.