Wave propagations in exponentially graded transversely isotropic half-space with potential function method

• A vertically graded transversely isotropic, linearly elastic half-space is considered.
• A new set of potential functions is introduced.
• Time-harmonic response to patch and point loads are determined.
• The displacements and stresses are numerically determined and shown graphically.
• The effect of degree of change of material properties is studied.

Time-harmonic response of a vertically graded transversely isotropic, linearly elastic half-space is analytically determined by introducing a new set of potential functions. The potential functions are set in such a way that the governing equations be simple and with physical meaning as well. In addition, the potential functions introduced in this paper are degenerated to a complete set of potential functions used frequently for wave propagations in homogeneous transversely isotropic media. Utilizing Fourier series and Hankel integral transforms, the governing equations for the potential functions are solved, after which the displacements and stresses are presented in the form of line integrals. Both the displacements and stresses determined here are collapsed on the solution previously reported for the constant profile transversely isotropic material. Because of complicated integrand functions, the integrals are evaluated numerically and presented graphically, where the effect of degree of change of material properties plays a major role, which may be recognized easily.