Linear theory for the bending and extension of a thin, residually stressed, fiber-reinforced lamina

Abstract

The Euler equations of a thickness-wise expansion of the potential energy of a thin body, truncated at a specified order in thickness, furnish a model for the bending and stretching of plates and shells. However, truncated expansions of the energy typically do not lead to well-posed minimization problems. This is related to the fact that the truncations may fail to satisfy the relevant Legendre–Hadamard condition, which is necessary for the existence of minimizers. This lack of well-posedness is thus entirely consistent with well-posedness in the exact theory. However, it is an inconvenience from the viewpoint of analysis. What is desired is an accurate, well-posed truncation that preserves the structure of classical plate theory. The present work is concerned with the development of such a model for a uniform fiber-reinforced lamina.