Dynamic response of axisymmetric transversely isotropic viscoelastic continuously nonhomogeneous half-space

An analytical derivation is presented for dynamic response of axisymmetric transversely isotropic linear viscoelastic continuously nonhomogeneous half-space subjected to vertical either point or circular patch load. The material coefficients are considered to vary in terms of depth as bounded exponentially functions, and the mass density is assumed to be constant. Hankel integral transforms accompanied with Frobenius series method are applied to solve the boundary value problem. The unknown constants are determined by satisfying boundary conditions and regularity conditions at infinity, after which the displacement and stress fields are specified in Hankel space. The inverse Hankel integral transforms are utilized to specify the displacements and stresses in real domain. It is shown that inhomogeneity parameters affect the dynamic response of the half-space considerably, especially at the vicinity of the free-surface. Moreover, viscoelastic behavior of the half-space is parametrically studied assuming several damping ratios, and it is seen that it can considerably change the dynamic responses especially for long horizontal distances from the excitation point.