A generalized solution to the combo-crack problem—I. Pressure load on crack surface

Abstract

The axisymmetric elasticity problem of crack combo containing an externally circular crack (ECC) and a coplanar concentric penny-shaped crack (PSC) is mathematically equivalent to the annular contact problem. This problem has been attempted by using Love's strain potential approach, which eventually comes down to solving a pair of simultaneous Fredholm integral equations. Finding the closed-form solutions to the integral equations is difficult, if not impossible. Approximate solutions have been proposed in power series representations, which suffer from two major deficiencies. First, the solutions apply only to a special loading case in which uniform pressure is applied to the whole surface of the interior PSC. Secondly, the accuracy of the solution becomes unsatisfactory when the interior PSC tip is close to the ECC tip. To address these issues, in this paper we revisit this problem by considering a more general loading case in which uniform pressure is applied to a circular region of any size at the center of the PSC's surface. To overcome the lower accuracy caused by power series with limited terms, we numerically solve the pair of simultaneous Fredholm integral equations based on the Gauss-Lobatto quadrature. The high accuracy of our solution in the whole size spectra of the PSC and ECC is verified by finite element simulations. Our paper provides a generalized and more accurate solution to the annular contact problem or the combo crack problem, which deserves to be included in the updated library of the solutions to basic crack problems.