End stress calculations on elastic cylinders

Abstract

For a semi-infinite circular elastic cylinder z ⩾ 0, r ⩽ 1 deformed solely by a distribution of stress and displacements on its flat end z = 0, the Love stress function can be expanded in a series of eigenfunctions of known form. For problems in which suitable mixed stress and displacements boundary conditions are prescribed on z = 0 the coefficients appearing in the expansion can be determined in an explicit form via sets of biorthogonal functions. When normal and shear stresses are prescribed on z = 0 no such closed expressions for the coefficients exist and approximate methods usually lead to infinite systems of linear equations which are solved by truncation. Stability of solution as the order of truncation is increased can only be guaranteed theoretically when the infinite matrix is diagonally dominated, and this is not the case for existing methods. A Galerkin method has been developed using weighting functions chosen so as to optimise the diagonal dominance of the infinite matrix, and numerical results show that, although the resulting matrix is not completely diagonally dominated, the resulting coefficients show an improvement in stability in the sense that they do not change significantly as the order of truncation is increased.