Mathematical Models with Buckling and Contact Phenomena for Elastic Plates: A Review

A review of mathematical models for elastic plates with buckling and contact phenomena is provided. The state of the art in this domain is presented. Buckling effects are discussed on an example of a system of nonlinear partial differential equations, describing large deflections of the plate. Unilateral contact problems with buckling, including models for plates, resting on elastic foundations, and contact models for delaminated composite plates, are formulated. Dynamic nonlinear equations for elastic plates, which possess buckling and contact effects are also presented. Most commonly used boundary and initial conditions are set up. The advantages and disadvantages of analytical, semi-analytical, and numerical techniques for the buckling and contact problems are discussed. The corresponding references are given.

A priori error for unilateral contact problems with Lagrange multipliers and isogeometric analysis

In this paper we consider a unilateral contact problem without friction between a rigid body and a deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem. For the displacement, we use the pushforward of a nonuniform rational B-spline space of degree $p$ and for the Lagrange multiplier, the pushforward of a B-spline space of degree $p-2$. These choices of space ensure the proof of an inf–sup condition and so on, the stability of the method. We distinguish between contact and noncontact sets to avoid the classical geometrical hypothesis of the contact set. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in two and three dimensions and in small and large deformation frameworks demonstrate the accuracy of the proposed method.