Liliana Gratie

A NEW APPROACH TO LINEAR SHELL THEORY

We propose a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korns inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincare. The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

On the Existence of Solutions to the Generalized Marguerre-von Kármán Equations

Using techniques from asymptotic analysis, the second author has recently identified equations that generalize the classical Marguerre-von Kármán equations for a nonlinearly elastic shallow shell by allowing more realistic boundary conditions, which may change their type along the lateral face of the shell. We first reduce these more general equations to a single “cubic” operator equation, whose sole unknown is the vertical displacement of the shell. This equation generalizes a cubic operator equation introduced by M. S. Berger and P. Fife for analyzing the von Kármán equations for a nonlinearly elastic plate. We then establish the existence of a solution to this operator equation by means of a compactness method due to J. L. Lions.

Generalized von Kármán equations

Abstract In a previous work, the first author has identified three-dimensional boundary conditions “of von Karmans type” that lead, through a formal asymptotic analysis of the three-dimensional solution, to the classical von Karman equations, when they are applied to the entire lateral face of a nonlinearly elastic plate. In this paper, we consider the more general situation where only a portion of the lateral face is subjected to boundary conditions of von Karmans type, while the remaining portion is subjected to boundary conditions of free edge. We then show that the asymptotic analysis of the three-dimensional solution still leads in this case to a two-dimensional boundary value problem that is analogous to, but is more general than, the von Karman equations. In particular, it is remarkable that the boundary conditions for the Airy function can still be determined solely from the data.

Two-Dimensional Nonlinear Shell Model of Koiter's Type with Variable Thickness

In this paper, we propose a new model “of Koiters type” for nonlinearly elastic shells with variable thickness, which generalizes a model recently proposed by P.G. Ciarlet for shells with constant thickness. We justify this model by means of an asymptotic analysis, by showing that its solution behaves either like that of a “membrane” or like that of a “flexural” shell as the thickness goes to zero.

Unilateral Problems for Nonlinearly Elastic Plates and Shallow Shells

Using the topological degree for pseudo-monotone operators of type (S+), we establish a general existence result for variational inequalities of von Karman type, which model unilateral problems for nonlinearly elastic plates. Then, we give a reduced operatorial form of Marguerre von Karman equations for nonlinearly elastic shallow shells and get a new existence result for this model.