Details of the integral equation method applied to the analysis of an adhesive layer crack

Abstract

The mathematical model for a crack in an elastic adhesive layer sandwiched between two adherends proposed by Fleck, Hutchinson and Suo ((1991) Crack path selection in a brittle adhesive layer. International Journal of Solids and Structures27(13), 1683–1703) was considered. The elastic mismatch between the adhesive and adherend materials modifies the far-field values of the stress intensity factors and of the T-stress in a manner that depends on the position of the crack inside the layer and on the Dundurs parameters. A complex-potential stress-function formulation, using dislocation distributions represented by truncated Chebyshev series, yielded an integral equation that was solved numerically by the method of collocations. The symbolic derivation of the integral equations was checked, and a few differences from Fleck, Hutchinson and Suo's expressions were identified and reconciled. The computational aspects of the solution were studied in detail using two programming languages, MATHCAD and C++, run on standard PC hardware. A palette of numerical techniques were utilized to study and control the consistency and accuracy of the solution. Symmetry and anti-symmetry arguments were used to identify numerically sensitive regions. Fast Fourier Transform calculation of sine and cosine Fourier integrals was used to increase speed. Convergence of intermediate and final results was examined. It was found that the method is very sensitive to the details of numerical computation, especially for combinations of parameters that lead to nearly singular matrices.