This chapter constructs and analyzes a model for the dynamic behavior of nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a rigid or reactive foundation underneath it. We use a model, first developed and studied by D.Y. Gao, that allows for the buckling of the beam when the horizontal traction is sufficiently large. In contrast with the behavior of the standard Euler–Bernoulli linear beam, it can have three steady states, two of which are buckled. Moreover, the Gao beam can vibrate about such buckled states, which makes it important in engineering applications. We describe the contact process with either the normal compliance condition when the foundation is reactive, or with the Signorini condition when the foundation is perfectly rigid. We use various tools from the theory of pseudomonotone operators and variational inequalities to establish the existence and uniqueness of the weak or variational solution to the dynamic problem with the normal compliance contact condition. The main step is in the truncation of the nonlinear term and then establishing the necessary a priori estimates. Then, we show that when the viscosity of the material approaches zero and the stiffness of the foundation approaches infinity, making it perfectly rigid, the associated solutions of the problem with normal compliance converge to a solution of the elastic problem with the Signorini condition.
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