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Transient response of a hollow cylinder with radial and axial material inhomogeneity

The transient response of a hollow finite cylinder with radial and axial material inhomogeneity to a pulse of short duration is analyzed. The problem is solved in two steps. Radial inhomogeneity is treated first by dividing the cylinder into coaxial segments each with a constant modulus E which is allowed to vary from segment to segment. Transfer matrices relating variables at the two radial interfaces of a segment combine to satisfy continuity of stress and displacement at these interfaces. The Galerkin method is then utilized to treat the general case with both radial and axial inhomogeneity adopting the eigenfunctions of the radially inhomogeneous cylinder as trial functions. Features of both static and transient responses resemble that for a weakening material along the axis: displacement increases and stress reduces in proportion to the reduction in modulus. Spatial attenuation of transient stress cannot be replicated when geometric dimensionality is reduced from 3-D axisymmetric to 2-D plane-strain.