Beam theory models for thin film segments cohesively bonded to an elastic half space

A thin film segment bonded to an elastic half space is modelled. Previous works have considered membrane models which only take into account the in-plane stiffness of the film and ignore its bending stiffness. To include bending stiffness, a beam theory is used to model the film. The beam model calculations are compared to results from membrane theory. Membrane theory is found to agree with beam theory for the stiffest films, but the energy release rates (or J-integrals) are very close for all films. However, membrane theory can never give information on the normal stresses at the interface and consequently on the mode mixity of the loading at the film edge (or crack lip). To include yielding of the interface at the ends of the film, a cohesive zone model is employed. This zone is a shear zone. that is, only tangential slip in the zone is allowed with no normal opening. The cohesive zone results could be used for determining the interface strengths if the size of the cohesive zone was measured. It was also found that the sign of the normal stress at the tip of the cohesive zone depends on the length of the zone. The structure is loaded by an applied uniform compressive strain in the substrate which can also represent a thermal mismatch strain. The method of solution is to reduce the differential equations for a beam to integral equations which are then coupled to the singular integral equations for a half space. The standard technique of expansion in orthogonal polynomials is used. All the integrations required are performed analytically. The only numerical procedures are in the solution of a set of linear equations and a root finding procedure to determine the cohesive zone size at a given value of the yield stress.