Abstract
In this paper, a first-order virtual element method for Reissner–Mindlin plates is presented. A standard displacement-based variational formulation is employed, assuming transverse displacement and rotations as independent variables. In the framework of the first-order virtual element, a piecewise linear approximation is assumed for both displacement and rotations on the boundary of the element. The consistent term of the stiffness matrix is determined assuming uncoupled polynomial approximations for the generalized strains, with different polynomial degrees for bending and shear parts. In order to mitigate shear locking in the thin-plate limit while keeping the element formulation as simple as possible, a selective scheme for the stabilization term of the stiffness matrix is introduced, to indirectly enrich the approximation of the transverse displacement with respect to that of the rotations. Element performance is tested on various numerical examples involving both thin and thick plates and different polygonal meshes.
Highlights
The Reissner–Mindlin theory is widely used to study the mechanical response of shear-deformable plates
Polygonal meshes may be appealing for a number of applications, e.g. meshing domains with cracks/inclusions, hanging nodes, adaptivity, etc
The present paper aims at proposing simple and efficient first-order virtual elements for Reissner–Mindlin plates based on a standard displacement-based variational formulation, assuming only the transverse displacement and the Computational Mechanics rotations as independent variables in the vertexes, i.e. saving degrees of freedom with respect to existing approaches
Summary
The Reissner–Mindlin theory is widely used to study the mechanical response of shear-deformable plates. Beirão da Veiga et al proposed in [23] a virtual element that assumes the shear strains and the transverse displacement as independent variables while the rotations are obtained by post-processing This stratagem allows to avoid shear locking, while considering for each vertex 5 degrees of freedom plus 1 degree of freedom per side, i.e. basically 6 degrees of freedom per vertex in a first-order approximation framework. The present paper aims at proposing simple and efficient first-order virtual elements for Reissner–Mindlin plates based on a standard displacement-based variational formulation, assuming only the transverse displacement and the Computational Mechanics rotations as independent variables in the vertexes, i.e. saving degrees of freedom with respect to existing approaches. A linear elastic response is assumed for the plate; the constitutive equations are written in the form:
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