The Bending-Gradient Theory for Thick Plates: Existence and Uniqueness Results

Abstract

This paper is devoted to the mathematical justification of the Bending-Gradient theory which is considered as the extension of the Reissner-Mindlin theory (or the First Order Shear Deformation Theory) to heterogeneous plates. In order to rigorously assess the well-posedness of the Bending-Gradient problems, we first assume that the compliance tensor related to the generalized shear force is positive definite. We define the functional spaces to which the variables of the theory belong, then state and prove the existence and uniqueness theorems of solutions of the Bending-Gradient problems for clamped and free plates, as well as for simply supported plates. The obtained results are afterward extended to the general case, i.e., when the compliance tensor related to generalized shear forces is not definite.

The Bending-Gradient theory for flexural wave propagation in composite plates

“…To make the presentation self-contained, we now briefly recall the main definitions of the kinematic and static fields of the Bending-Gradient theory as well as the governing equations established by . For more details concerning the Bending-Gradient theory, the reader is referred to , , , , and Bejjani et al (2018).…”

“…In recent papers, the Bending-Gradient theory was justified through asymptotic expansions (Lebée and Sab, 2013) as well as variational methods and a series of existence and uniqueness theorems were formulated and proved (Bejjani et al, 2018). Having mathematically justified this theory, the central aim of this work is to test its validity regarding plane wave propagation in symmetrical anisotropic heterogeneous plates.…”