Fracture mechanics of bi-material interface cracks

It is well known that the elastic stress distribution near a crack tip in an isotropic two dimensional deformation field exhibits an inverse square root singularity. The nature of the stress singularity at the tip of a plane crack terminating at an interface of two materials is, however, dependent on both the crack geometry and the elastic constants of the two materials. The order of singularity, for such problems, is represented by r^-P. A procedure based on the use of elements with numerically integrated shape functions has been proposed in recent literature in order to generate elements with a general r^-P strain singularity.

The purpose of this investigation was to study the behavior of bi-metallic interface cracks. A plate composed of two dissimilar materials bonded together along a straight line containing a center crack was studied. This problem represented an idealization of an adhesive joint or a composite structure with an interfacial flaw or crack.

Solutions obtained using three different approaches were compared in order to develop a recommended procedure for analyzing interface crack problems. The first method was based on the hybrid crack tip element method. Such elements cannot be conveniently used by common analysts nor can they be utilized with ease directly in standard computer codes. Standard eight node isoparametric elements, with modifications to account for singularity, were also used to analyze the problem. Two distinct techniques using this general approach were studied - one based on the mapping technique and the other based on the displacement technique. The mapping technique involved the use of two dimensional isoparametric elements with the midside nodes shifted to the quarter point location. The effectiveness of the quadratic isoparametric elements with quarter point node locations for the two dimensional elastic analysis of cracked bodies was studied. However, this procedure was limited to singularity of the order r ^-½ The displacement technique involved the use of two dimensional isoparametric elements with numerically integrated shape functions. This technique could be used for deriving singularity of the order r^-P.

Comparison of the solutions, based on these three methods, indicated that the displacement technique yielded more accurate results. The singular elements did not pass the patch test. The error involved was, however, very small.

In conclusion, based on the results of this study, the use of isoparametric elements with singularity correction is recommended in the analysis of bi-material interface cracks.