Indentation of geometrically exact adhesive beams

Abstract

We investigate the indentation of thin adhesive beams by a rigid frictionless circular punch in two-dimensions through computations and experiments. A non-linear geometrically exact (GE) beam formulation that allows for arbitrarily large displacements and rotations is employed for the thin beam. The Karush–Kuhn–Tucker condition for unilateral contact is adapted to model adhesive interaction in a nonlinear finite element (FE) framework. In this, contact is enforced through a penalty method and adhesion is incorporated through two different cohesive zone models (CZMs), viz. exponential and bilinear. The indentation of the adhesive GE beam is then studied in terms of the variations of the applied force, the pressure distribution, the cross-sectional rotation and the sizes of the contact area and the adhesive zone for different indentation extents, adhesive strengths, punch radii, beam thicknesses and end supports. We find that the system is largely agnostic to the CZM utilized, as well as the type of end supports unless the contact area spans the beam. At the same time, thinner beams tend to initiate contact with a punch more easily, and are harder to detach. We then demonstrate that thin beams may undergo large rotations even if the indentation is small, which necessitates a nonlinear beam theory. Under such conditions approximating a circular punch by a parabolic one used typically when indentations are small introduces errors. Finally, we conduct experiments on clamped adhesive beams made of two different types of PDMS (polydimethylsiloxane) and various thicknesses, and find a good match with our computations.