Dimensional analysis and asymptotic expansions of equilibrium equations in nonlinear elasticity. Part I: The membrane model

Abstract

In this paper, we develop a new asymptotic constructive approach in nonlinear plate theory. The dimensional analysis of the three-dimensional equilibrium equations naturally leads to dimensionless numbers which reflect the geometry of the structure and the magnitude of forces. These numbers also define the domain of validity of the two-dimensional models which will later be obtained by asymptotic expansions. For nonlinear plates, we prove that the two-dimensional models we obtain by asymptotic expansions are determined by the magnitude of the forces applied. In this first part, we consider a plate subjected to large loads. In this case, we prove that the nonlinear plate model we obtain by asymptotic expansions is a membrane model. In the second part of this article, we will consider a plate subjected to smaller applied forces.