Green's function expansion for exponentially graded elasticity

Abstract

SUMMARYNew computational forms are derived for Green's function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low-order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second-order derivatives needed in a boundary integral analysis. The new Green's function expressions have been tested by comparing with results from an earlier algorithm.

Evaluation of Green's function derivatives for exponentially graded elasticity

“…and formulas for these quantities can be found in [29]. The i, are relatively simple functions of r , while the A i, depend on x Q through the orthogonal coordinate system vectors {l, m, n}.…”

“…where R L 0, and R U 0, are independent of the component indices; formulas for these functions, for both one-and two-term expansions, were given in [29]. Computing all components of Green's tensor, therefore, only requires re-computing the relatively simple f 0, coefficients.…”

“…The next section therefore reviews the expansion procedure developed in [29], and the subsequent two sections discuss the evaluation of first-and second-order derivatives, respectively. Section 5 contains some concluding remarks.…”

“…The first difficulty has recently been addressed in [29]. An expansion procedure was developed for the inverse Fourier integral defining U g (x Q , x P ), resulting in computational forms expressed as a relatively simple analytic term, plus a single two-dimensional integral to be evaluated numerically.…”

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

“…For a detailed review of studies in the eld of anisotropic and nonhomogeneous isotropic materials, one might refer to [2]. The fundamental solutions concerning the exponentially graded elastic solids may be found in [3][4][5][6][7][8]. Esandari and Shodja [9] obtained fundamental solutions to an exponentially graded transversely isotropic half-space for arbitrary buried static loads.…”