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.

SUMMARYEffective formulas for computing Green's function of an exponentially graded three-dimensional material have been derived in previous work. The expansion approach for evaluating Green's function has been extended to develop corresponding algorithms for its first-and second-order derivatives. The resulting formulas are again obtained as a relatively simple analytic term plus a single double integral, the integrand involving only elementary functions. A primary benefit of the expansion procedure is the ability to compute the second-order derivatives needed for fracture analysis. Moreover, as all singular terms in this hypersingular kernel are contained in the analytic expression, these expressions are readily implemented in a boundary integral equation calculation. The computational formulas for the first derivative are tested by comparing with results of finite difference approximations involving Green's function. In turn, the second derivatives are then validated by comparing with finite difference quotients using the first derivatives.

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