In this article, the fundamental solutions for anti-plane elasticity are derived using the Fourier transform and the Laplace transform techniques, with the shear modulus and the mass density varying exponentially for functionally graded materials. It has been shown that the transformed fundamental solutions both in the Laplace space and in the time domain have the same order of singularities as that in the static case. The time-dependent variables including the displacement and the shear stresses for anti-plane elasticity are obtained with Durbin’s inversion method for the Laplace transform. The discontinuity displacement method is formulated from the fundamental solutions and applied to the mode III fracture problems.
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