A consistent second‐order plate theory for monotropic material

Abstract

Mathematical homogenization (or averaging) of composite materials, such as fibre laminates, often leads to non-isotropic homogenized (averaged) materials. Especially the upcoming importance of these materials increases the need for accurate mechanical models of non-isotropic shell-like structures. We present a second-order (or: Reissner-type) theory for the elastic deformation of a plate with constant thickness for a homogeneous monotropic material. It is equivalent to Kirchhoff's plate theory as a first-order theory for the special case of isotropy and, furthermore, shear-deformable and equivalent to R. Kienzler's theory as a second-order theory for isotropy, which implies further equivalences to established shear-deformable theories, especially the Reissner-Mindlin theory and Zhilin's plate theory. Details of the derivation of the theory will be published in a forthcoming paper [3]. For an arbitrary three-dimensional body Ω ⊂ R 3 , which is assumed to be a bounded region, with the boundary regularity ∂Ω ∈ C 0,1 (this assumption allows to treat basically every body in engineering applications) and a boundary decomposition of the type ∂Ω = ∂Ω 0 ∪ ∂Ω N and ∂Ω 0 ∩ ∂Ω N = ∅, with ∂Ω 0 = ∅ relatively open and ∂Ω N relatively open, one can proof that the three-dimensional weak formulation of the mixed-boundary-value problem of linear elasticity (II) has a unique solution under very weak assumptions for the regularity of the given data, i.e., the prescribed displacement field u 0 on ∂Ω 0 , the prescribed traction-vector field g on ∂Ω N and the prescribed field of body force f . If we assume that the component functions of the fourth-order elasticity tensor E ijrs : Ω −→ R are continuous or piecewise constant and fulfill the symmetry relations E ijrs = E jirs , E ijrs = E ijsr and E ijrs = E rsij , so that in addition the associated 6 × 6-elasticity tensor is symmetric positive definite for all x ∈ Ω (which is a basic modeling assumption in continuum solid mechanics), we get:Theorem: Existence and uniqueness of the weak solution of the three-dimensional linear elasticity problemThen there exists an u 0 ∈ X that fulfills the weak displacement-boundary condition S u 0 (x) = u 0 (x) for almost all x ∈ ∂Ω 0 (with respect to the corresponding Lebesgue measure) and the problemsare equivalent and have a unique solution.Here S : W 1,2 (Ω) −→ W 1/2,2 (∂Ω) denotes the trace operator, X := W 1,2 (Ω) 3 and X 0 := {v ∈ X|∀i ∈ {1, 2, 3} :Sv i = 0 on ∂Ω 0 }. Problem (I) corresponds to the "principal of minimal potential energy" in a weak setting, which is usually the starting point of the derivation of consistent theories (e.g. in [1]). We use problem (II) as a starting point instead.We now assume our body to be a plate of constant thickness h. Therefore, let the mid-plane A ⊂ R 2 be a bounded region with ∂A ∈ C 0,1 and Ω :. For the boundary decomposition we assumeFurthermore, we introduce the plate parameter c :=, where a is a characteristic in-plane length of the plate, e.g., the diameter of A. The plate parameter is a dimensionles...

Two multilayered plate models with transverse shear warping functions issued from three dimensional elasticity equations

“…This means that, with respect to the required order of the considered theory, some terms of the three-dimensional strain energy may appear with an erroneous coefficient, or may even be omitted. The consistency of plate theories has been discussed by many authors, one can for example refer to recent works [37,38]. Precise rules for the consistency are formulated for homogeneous materials, but it seems difficult to extrapolate them for inhomogeneous materials like laminates and sandwiches, especially when advanced kinematic and/or static assumptions are made.…”