An exact dual

We study indentation by a rigid cylindrical punch of finite, free-standing, adhesive elastic layers that are supported only at their ends. The adhesion is considered through an adhesive-zone model. Formulating the boundary-value problem, we obtain two coupled Fredholm integral equations of the first kind, which are solved by a collocation method. Results for non-adhesive contact are obtained when adhesion is zero, and they match well with our own finite element computations and earlier approximate analyses. Additionally, we obtain new results for deeper indentation of non-adhesive contact. In the limit of very adhesive and/or very soft solids, we formulate an approximate model similar to the well-known Johnson–Kendall–Roberts (JKR) model for half-spaces. Our results for adhesive contact match well with preliminary indentation experiments on adhesive layers. Finally, we demonstrate the utility of our approach in modelling structural adhesives through a specific example