Double cantilever beam models in adhesive mechanics

There is a growing need to accurately predict failure of adhesive joints. To meet this, mechanics researchers are turning to fracture mechanics [1]. Application of modern fracture mechanics to adhesive joints depends upon a stress analysis and characterization of the singular stress fields within the connection. In the application of fracture mechanics to crack propagation problems this poses only a moderate problem as there have been significant advances in the stress analysis of crack geometries. Unfortunately, the geometry associated with an adhesive joint such as a shear lap joint is extremely complicated. The adherends are flexible and the joint rotates, the adhesive layer has a finite thickness and possibly possesses non-linear and rate-dependent mechanical properties, while the adherend thickness may vary. The stress analysis problems are formidable. Some progress has been made by introducing simplifying assumptions and employing the two-dimensional theory of elasto-statics [2,3]. On the other hand, numerical methods such as finite elements[4] appear promising and, when properly employed, should prove useful. While both of these approaches are important and valuable, they are also difficult, time consuming and expensive. This difficulty in the stress analysis is proving to be a deterrent to the basic understanding of the physics and phenomena of adhesive joint failure. An interesting alternative is to use an approximate structural theory in the stress analysis. In this way it might be possible to extract much of the essential information without extensive mathematical or numerical work. In fact, the first stress analyses of adhesive joints [5-7] were done in this spirit although not with a fracture mechanics analysis as the prime goa\. In the case of an adhesive joint joining two slender members, it is particularly appealing to use plate or beam theories to simplify the stress analysis. For example, the use of simple beam theory by Gilman[8] proved quite useful in the understanding of the double cantilever beam test specimen. Subsequently the approximate predictions were refined by numerical and experimental methods [9-12] but the basic results presented in [8] are still valuable. A similar approach based upon a refined plate theory [13] was used in [14] to solve a related problem. A recent paper by Kanninen [15] has modified Gilman's analysis by approximately accounting for the thickness deformations in the beam. Suitable selection of a somewhat arbitrary parameter in Kanninen's model has led to excellent agreement with established results [9-12]. Kanninen's paper gives a good example of how an approximate structural theory can be successfully employed. As a long range goal it is desirable to develop an approximate but accurate structural theory for adhesive connections. Such a theory would permit approximate analysis of adhesive joints accounting for the effects of inelasticity, joint geometry, and bond line properties. The accuracy of such a theory must, of course, be assessed whenever possible by comparison with elasticity and experimental results. This paper is concerned with two alternative models of the double cantilever specimen and comparison of the responses predicted by the two models. In Section 2 we develop a solution to an idealized model of the double cantilever problem, the solution being exact within the framework of the geometric simplifications and the two dimensional elastostatic theory of plane