Abstract
Effective formulas for computing the Green's function of an exponentially graded three dimensional material have been derived In previous work. The expansion approach for evaluating the 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 the Green's function. In turn, the second derivatives are then validated by comparing with finite difference quotients using the first derivatives.