A Dual Hybrid Virtual Element Method for Plane Elasticity Problems

A dual hybrid Virtual Element scheme for plane linear elastic problems is presented and analysed. In particular, stability and convergence results have been established. The method, which is first order convergent, has been numerically tested on two benchmarks with closed form solution, and on a typical microelectromechanical system. The numerical outcomes have proved that the dual hybrid scheme represents a valid alternative to the more classical low-order displacement-based Virtual Element Method

Arbitrary order 2D virtual elements for polygonal meshes: Part II, inelastic problem

The present paper is the second part of a twofold work, whose first part is reported in [3], concerning a newly developed Virtual Element Method (VEM) for 2D continuum problems. The first part of the work proposed a study for linear elastic problem. The aim of this part is to explore the features of the VEM formulation when material nonlinearity is considered, showing that the accuracy and easiness of implementation discovered in the analysis inherent to the first part of the work are still retained. Three different nonlinear constitutive laws are considered in the VEM formulation. In particular, the generalized viscoplastic model, the classical Mises plasticity with isotropic/kinematic hardening and a shape memory alloy (SMA) constitutive law are implemented. The versatility with respect to all the considered nonlinear material constitutive laws is demonstrated through several numerical examples, also remarking that the proposed 2D VEM formulation can be straightforwardly implemented as in a standard nonlinear structural finite element method (FEM) framework

Curvilinear virtual elements for contact mechanics

The virtual element method (VEM) for curved edges with applications to contact mechanics is outlined within this work. VEM allows the use of non-matching meshes at interfaces with the advantage that these can be mapped to a simple node-to-node contact formulation. To account for exact approximation of complex geometries at interfaces, we developed a VEM technology for contact that considers curved edges. A number of numerical examples illustrate the robustness and accuracy of this discretization technique. The results are very promising and underline the advantages of the new VEM formulation for contact between two elastic bodies in the presence of curved interfaces

Modeling And Design Of Periodic Lattices With Tensegrity Architecture And Highly Nonlinear Response

In recent years, the nonlinear response of tensegrity systems has attracted increasing attention in the study of mechanical metamaterials. It has been shown in the literature that geometry and prestress of an elastic tensegrity structure can be designed to obtain different behaviors: stiffening, softening, and snap-through behavior in statics; propagation of solitary waves in dynamics. However, the realization of tensegrity systems is challenging, because of their prestressed state and the presence of tension-only cable members. A design method for periodic lattices with null prestress and no cables is here proposed, in which the repeating unit is at, or close to, a tensegrity configuration, maintaining the nonlinear types of response aforementioned. These structures can be realized by conventional additive manufacturing techniques, while the static and dynamic response can be predicted by means of stick-and-spring models

The G. D. Q. method for the harmonic dynamic analysis of rotational shell structural elements

This paper deals with the modal analysis of rotational shell structures by means of the numerical solution technique known as the Generalized Differential Quadrature (G. D. Q.) method. The treatment is conducted within the Reissner first order shear deformation theory (F. S. D. T.) for linearly elastic isotropic shells. Starting from a non-linear formulation, the compatibility equations via Principle of Virtual Works are obtained, for the general shell structure, given the internal equilibrium equations in terms of stress resultants and couples. These equations are subsequently linearized and specialized for the rotational geometry, expanding all problem variables in a partial Fourier series, with respect to the longitudinal coordinate. The procedure leads to the fundamental system of dynamic equilibrium equations in terms of the reference surface kinematic harmonic components. Finally, a one-dimensional problem, by means of a set of five ordinary differential equations, in which the only spatial coordinate appearing is the one along meridians, is obtained. This can be conveniently solved using an appropriate G. D. Q. method in meridional direction, yielding accurate results with an extremely low computational cost and not using the so-called “delta-point” technique

Static analysis of shear-deformable shells of revolution via G.D.Q. method

This paper deals with a novel application of the Generalized Differential Quadrature (G.D.Q.) method to the linear elastic static analysis of isotropic rotational shells. The governing equations of equilibrium, in terms of stress resultants and couples, are those from Reissner-Mindlin shear deformation shell theory. These equations, written in terms of internal-resultants circular harmonic amplitudes, are first put into generalized displacements form, by use of the strain-displacements relationships and the constitutive equations. The resulting systems are solved by means of the G.D.Q. technique with favourable precision, leading to accurate stress patterns

Static analysis of shear-deformable shells of revolution via G.D.Q. method

This paper deals with a novel application of the Generalized Differential Quadrature (G.D.Q.) method to the linear elastic static analysis of isotropic rotational shells. The governing equations of equilibrium, in terms of stress resultants and couples, are those from Reissner-Mindlin shear deformation shell theory. These equations, written in terms of internal-resultants circular harmonic amplitudes, are first put into generalized displacements form, by use of the strain-displacements relationships and the constitutive equations. The resulting systems are solved by means of the G.D.Q. technique with favourable precision, leading to accurate stress patterns

The G. D. Q. method for the harmonic dynamic analysis of rotational shell structural elements

This paper deals with the modal analysis of rotational shell structures by means of the numerical solution technique known as the Generalized Differential Quadrature (G. D. Q.) method. The treatment is conducted within the Reissner first order shear deformation theory (F. S. D. T.) for linearly elastic isotropic shells. Starting from a non-linear formulation, the compatibility equations via Principle of Virtual Works are obtained, for the general shell structure, given the internal equilibrium equations in terms of stress resultants and couples. These equations are subsequently linearized and specialized for the rotational geometry, expanding all problem variables in a partial Fourier series, with respect to the longitudinal coordinate. The procedure leads to the fundamental system of dynamic equilibrium equations in terms of the reference surface kinematic harmonic components. Finally, a one-dimensional problem, by means of a set of five ordinary differential equations, in which the only spatial coordinate appearing is the one along meridians, is obtained. This can be conveniently solved using an appropriate G. D. Q. method in meridional direction, yielding accurate results with an extremely low computational cost and not using the so-called \u201cdelta-point\u201d technique