Modeling of Contact in Orthotropic Materials using Variational Asymptotic Method

Abstract

Composites are materials which cater to the present and future needs of many demanding industries, such as aerospace, as they are weight-sensitive for a given requirement of strength and stiff ness, corrosion resistant, potentially multi-functional and can be tailored according to the application. However, they are in particular difficult to join as they cannot be easily machined, without introducing damages which can eventually grow. Any structure is as strong as its weakest joint. Most of the joints belong to the category of mechanically-fastened joints and they pose enormous challenges in modeling due to contact phenomena, nonlinearity and stress concentration factors. It is therefore a necessity to construct an efficient model that would include all the relevant contact phenomena in the joints, as it has been pointed out in literature that damage typically initiates near the joint holes. The focus of this work is to describe the construction of an asymptotically-correct model using the Variational Asymptotic Method (VAM). Amongst its many potential applications, VAM is a well-established analytical tool for obtaining the stress and strain fields for beams and shells. The methodology takes advantage of the small parameter that is inherent in the problem, such as the ratio of certain characteristic dimensions of the structure. In shells and beams, VAM takes advantage of the dimension-based small parameter(s), thereby splitting the problem into 2-D + 1-D (for beams) and 1-D + 2-D (for shells), in turn offering very high computational efficiency with very little loss of accuracy compared to dimensionally unreduced 3-D models. In this work, the applicability of VAM is extended to two-dimensional (2-D) and three-dimensional (3-D) frictionless contact problems. Since a generalised VAM model for contact has not been pursued before, the `phantom0 step is adopted for both 2-D and 3-D models. The development of the present work starts with the construction of a 2-D model involving a large rectangular plate being pressed against a rigid frictionless pin. The differential equations governing the problem and the associated boundary conditions are obtained by minimizing the reduced strain energy, augmented with the appropriate gap function, by using a penalty method. The model is developed for both isotropic and orthotropic cases. The boundary value problem is solved numerically and the displacement field obtained is compared with the one obtained using commercial software (ABAQUSr) for validation at critical regions such as the contact surfaces. Banking on the validation of the 2-D model, a 3-D model with a pin and a finite annular cylinder was constructed. The strain energy for the finite cylinder was derived using geometrically exact 3-D kinematics and VAM was applied leading to the reduction in the strain energy for isotropic and orthotropic materials in rectangular and cylindrical co-ordinates. As in the 2-D case, the reduced strain energy, subject to the inequality constraint of the gap function, is minimized with respect to the displacement field and the corresponding boundary value problem is solved numerically. The displacements of the contact surface and the top surface of the annular cylinder are compared with those from ABAQUS and thus validated. The displacement fields obtained using the current 2-D and 3-D models show very good agreement with those from commercial finite element software packages. The model could be re ned further by using the gap function derived in this work and applying it to a plate model based on VAM, which could be explored in the future.