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By virtue of a complete set of two displacement potentials, an analytical derivation of the elastostatic Green’s functions of an exponentially graded transversely isotropic bi-material full-space was presented. Three-dimensional point-load Green’s functions for stresses and displacements were given in line-integral representations. The formulation included a complete set of transformed stress-potential and displacement-potential relations, with the utilization of Fourier series and Hankel transform. As illustrations, the present Green’s functions were analytically degenerated into special cases, such as exponentially graded half-space and homogeneous full-space bi-material Green’s functions. Owing to the complicated integrand functions, the integrals were evaluated numerically, and in computing the integrals numerically, a robust and effective methodology was laid out which provided the necessary account of the presence of singularities of integration. Some typical numerical examples were also illustrated to demonstrate the general features of the exponentially graded bi-material Green’s functions which will be recognized by the effect of degree of variation of material properties.

Sometimes, suddenly or gradually structural or nonstructural damages are created in structure’s members due to natural or unnatural reasons like earthquake or corrosion. One of the methods used from the early in last decade to recognize damage is wavelet transform method. In this method, instead of examination of natural frequencies and the rate of its changes, vibration response or static response of a structure in various points of a structure is recorded at the same time, which in fact is a place-domain signal. In this paper, some factors have been investigated on a plate such as the effect of crack depth reduction, effect of changing support conditions, and the effect of approaching two rectangular cracks; and the responses obtained from finite element analysis, have been studied and the following results are gained: the ability of wavelet method to recognize the locality and severity of cracks, identification of tiny cracks up to 0.5mm depth and increasing the intensification of cracks’ effect on plate by reducing their distance from each other.