An asymptotic approach applied to a longitudinal crack in an adhesive layer

Abstract

The problem of a crack within an adhesive layer which is bonded to two linear elastic half-planes under tensile loading is studied. Two cases are considered. One in which the adhesive is linear elastic and, the second in which it is taken to be elasto-plastic. For the linear elastic layer, the half-planes (adherends) are assumed to be both similar and dissimilar. When the adhesive is considerably more compliant than the adherends, a method of inner and outer asymptotic expansions is employed to determine a relationship between the corresponding stress intensity factors. Expansions are determined in three regions and matched. The inner expansion relates to a region whose distance from the crack tip is much less than the adhesive thickness. The intermediate expansion relates to a region whose size is governed by the decay length of the stress in that part of the adhesive in which its compliancy is significant. The outer expansion relates to a region whose distance from the crack tip is much less than the crack length, for example, but much greater than the adhesive thickness. This method may be employed to determine all field quantities in terms of the outer stress intensity factor. For a layer which is considerably stiffer than the adherends, a similar strategy for solving the problem is sketched. In addition, for dissimilar adherends, energy considerations are employed to verify the relationship between the inner and outer stress intensity factors. It is seen that the two expressions for the stress intensity factors are identical. The problem of oscillatory stress and displacement behavior is addressed.

Next, the problem of a sufficiently compliant elasto-plastic adhesive between dissimilar adherends is examined. Matched asymptotic expansions are employed to determine the plastic zone size, as well as the crack tip opening displacement. Small scale yielding is assumed. A Dugdale-Barenblatt type model is employed with the elasticity of the layer accounted for. The yield stress is taken to be constant throughout the plastic zone.