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Asymptotic Modelling of a Contact Problem in Composites



The focus of this paper is to describe the construction of an asymptotically-correct model for a rigid pin inside a cylinder using the Variational Asymptotic Method[1] (VAM). Amongst its many potential applications, the VAM is a well-established analytical tool for obtaining the stress and strain fields for plates and shells. An attempt is made in the present paper to extend VAM to contact mechanics. The development of the present model starts with the formulation of a 3-D strain energy for a short and thick annular composite cylinder. The unknown orders of terms in the strain expressions are estimated, based on the smallness of the maximum permissible absolute strain. Next, VAM is applied to simplify the problem through the retention of only the most dominant terms in the strain energy functional where-as the remaining terms are considered as higher order terms to be included in a series of successively improving approximations. The gap function between the rigid circular pin and the cylinder is derived using contact kinematics[2] in terms of arbitrary global displacements. The next step is to augment the strain energy functional with the gap function employing the Penalty method and to solve the Euler-Lagrange equations to get the displacement field.

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