3D Derivations of Static Plate Theories
Résumé
Plates are three-dimensional structures with a small dimension compared to the other two dimensions. Numerous approaches were suggested in order to replace the three-dimensional problem by a two-dimensional problem while guaranteeing the accuracy of the reconstructed three-dimensional fields. Turning the 3D problem into a 2D plate model is known as dimensional reduction.
The approaches for deriving a plate model from 3D elasticity may be separated in two main categories: axiomatic and asymptotic approaches. Axiomatic approaches start with ad hoc assumptions on the 3D field representation of the plate, separating the out-of-plane coordinate from the in-plane coordinates. The limitation of these approaches comes from the educated guess for the 3D field distribution. Asymptotic approaches are based on the explicit introduction of the plate thickness, which is assumed to go to 0, in the equations of the 3D problem. Following a rather well-established procedure, they enable the derivation of plate models, often justifying a posteriori axiomatic approaches, and are the basis of a convergence result.
The very first and simplest model is the Kirchhoff-Love plate model or thin-plate model (Kirchhoff 1850; Love 1888), where the out-ofplane deflection is the only kinematic degree of freedom. In this model, it is assumed that the fiber normal to the plate mid-surface remains normal during the motion. In order to take into account the influence of shear energy on the deflection, several thick plate models were suggested almost simultaneously (Reissner 1944; Hencky 1947; Bollé 1947). In these models, gathered here under the common denomination ReissnerHencky models, two in-plane rotations are added to the kinematics. Note that the denomination Reissner-Mindlin is also very common in the literature. It comes from Mindlin’s contribution based on dynamic considerations (Mindlin 1951). Whereas all these models were historically derived axiomatically, they also have close relations with asymptotic considerations.
This chapter is dedicated to the case of a homogeneous and linear elastic plate with static loading which was the foundation of many extensions to heterogeneous plates. It recalls in detail the derivation of the thick plate model from (1944). Both approaches are related but yield different plate models. This choice is motivated by the following considerations. First, ReissnerHencky models are the most widely used plate models in engineering applications. Indeed, their boundary conditions seem more natural than those of the Kirchhoff-Love plate model. They displacement required for finite elements implementations. Second, the Kirchhoff-Love model may be directly retrieved from these “Direct Derivation of Plate Theories”.
Two modifications are made with respect to the historical contributions. First, the membrane model is also included in the present derivation at very little price. Second, the applied load is a body force uniformly distributed through the thickness instead of a force per unit surface applied only on the upper face of the plate. This choice leads to a more compact derivation the membrane and bending problems widely ignored in the historical literature. Finally, all mathematical developments are purely formal and the reader is referred to ( “Mathematical Justification of Plate Models” and Ciarlet 1997) for rigorous justifications.