Three-dimensional dynamic Green’s functions in transversely isotropic bi-materials

Abstract

a b s t r a c tBy virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green's functions for a linear elastic transversely isotropic bi-material full-space is presented. Three-dimensional point-load Green's functions for stresses and displacements are given in complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations, within the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For numerical computation of the integrals, a robust and effective methodology is laid out which gives the necessary account of the presence of singularities including branch points and pole on the path of integration. As illustrations, the present Green's functions are analytically degenerated to the special cases such as half-space, surface and full-space Green's functions. Some typical numerical examples are also given to show the general features of the bi-material Green's functions.

Three-dimensional axisymmetric responses of exponentially graded transversely isotropic tri-materials under interfacial loading

“…The rough-rigid case which resembles the in nite friction capacity at the interface is de ned as: u r (r; h) = u z (r; h) = 0; (28) and the smooth-rigid condition may be described as:…”

“…In this case, by setting I = II = 0, the static Green's functions of a homogeneous transversely isotropic bi-material are obtained as in [28].…”

Equations of anisotropic elastodynamics as a symmetric hyperbolic system: Deriving the time-dependent fundamental solution

“…For the computation of the fundamental solution of SHSE we use formula (35) and then we derive the fundamental solution of ASE by (19). Formula (35) contains 3D integral over the whole space R 3 .…”

“…The computation of fundamental solutions for general linear equations of elastodynamics with three space and one time variables has been obtained only for particular cases of anisotropy (cubic, isotropic, transversely isotropic, orthotropic structures) [29][30][31][32][33][34][35][36][37]. The computation of the fundamental solutions in such anisotropic elastic media as trigonal, monoclinic, triclinic has not been achieved so far.…”

Basic solution of two 3D rectangular cracks in an orthotropic elastic media

“…Zhao [14] proposed the mode I extension of a half plane crack in a transversely isotropic solid under 3-D loading. Khojasteh et al [15] presented the elastodynamic Green's functions for a linear elastic transversely isotropic bi-material full-space. Li [16] addressed three-dimensional exact fundamental solutions of the thermoelastic field in a transversely isotropic elastic medium weakened by a half infinite plane crack subjected to a pair of point thermal loadings symmetrically acting on the crack surface.…”