Finite Plane Strain of Incompressible Elastic Solids by the Finite Element Method
TL;DR: In this paper, the finite element method is extended to the problem of finite plane strain of elastic solids, where the displacement fields within each element are approximated by linear functions of the local coordinates.
Abstract: The finite element method is extended to the problem of finite plane strain of elastic solids. A highly elastic body subjected to two-dimensional deformations is represented by an assembly of triangular elements of finite dimension. The displacement fields within each element are approximated by linear functions of the local coordinates. Non-linear stiffness relations involving generalised node forces and displacements are derived from energy considerations. For demonstration purposes, the non-linear stiffness equations are applied to the problems of finite simple shear and generalised shear. For finite simple shear, it is shown that these relations are in exact agreement with finite elasticity theory. Convergence rates of finite element representations of these problems are briefly examined.
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TL;DR: In this article, an incremental and piecewise linear finite element theory is developed for the large displacement, large strain regime with particular reference to elastic-plastic behavior in metals, and the resulting equations, though more complex, are in a similar form to those previously developed for large displacement small strain problems, the only additional term being an initial load stiffness matrix which is dependent on current loads.
312 citations
TL;DR: In this article, a novel three-dimensional finite volume (FV) procedure is described in detail for the analysis of geometrically nonlinear problems, compared with the conventional finite element (FE) Galerkin approach.
85 citations
TL;DR: In this article, a concise survey of the literature related to the large deformation elasto-plasticity problems including unilateral contact and friction is presented together with an extension of the friction law for large deformability analysis.
85 citations
TL;DR: Finite element method applicaions to finite axisymmetric deformations of incompressible elastic solids of revolution have been studied in this article, where the finite element method has been applied to deformations in the case of elastic soliders of revolution.
75 citations
TL;DR: In this paper, a finite element program capable of analyzing finite plane strain deformations of incompressible rubber-like materials has been developed and two problems, namely a long wall loaded uniformly in two directions and a thick-wall cylindrical pressure vessel loaded internally, have been solved.
Abstract: A finite element program capable of analysing finite plane strain deformations of incompressible rubberlike (Mooney–Rivlin) materials has been developed. Two problems, namely a long wall loaded uniformly in two directions and a thick-wall cylindrical pressure vessel loaded internally, have been solved. The computed values of displacements, strains, stresses and hydrostatic pressure agree very closely with their values obtained analytically.
58 citations
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TL;DR: In this article, a variational theorem is presented which is equivalent to the elastic field equations expressed in terms of the displacements and a function of the mean pressure, which is particularly useful in the development of approximate structural analysis techniques for incompressible and nearly-incompressible materials.
Abstract: A variational theorem is presented which is equivalent to the elastic field equations expressed in terms of the displacements and a function of the mean pressure. The variational theorem is particularly useful in the development of approximate structural analysis techniques for incompressible and nearly incompressible materials. An approximate analysis method derived by utilizing the Ritz technique in conjunction with the variational principle is illustrated. The results obtained from the application of this procedure to a sample problem are contrasted with those obtained by a similar technique that employs the theorem of minimum potential energy. The comparison illustrates that the results obtained from the analysis based upon the theorem of minimum potential energy becomes quite inaccurate for nearly incompressible materials whereas the solution technique based upon the new variational theorem yields accurate results.
347 citations
TL;DR: In this article, a continuous membrane is divided into a number of flat triangular elements, and the behavior of a typical element is described in terms of the displacements of its nodes.
133 citations
TL;DR: In this article, the Propagation des ondes Reference Record (PRR) was used to propagate the Physique and Statique Reference Record created on 2004-09-07, modified on 2016-08-08.
Abstract: Keywords: Physique ; Statique ; Propagation des ondes Reference Record created on 2004-09-07, modified on 2016-08-08
50 citations
TL;DR: In this paper, the development of nonlinear stiffness relations for three-dimensional finite elements of an elastic continuum is considered, and consistent finite element representations are formulated for geometrically nonlinear problems involving large displacements and large strains.
Abstract: The development of nonlinear stiffness relations for three-dimensional finite elements of an elastic continuum is considered. On the basis of linear displacement approximations, consistent finite element representations are formulated for geometrically nonlinear problems involving large displacements and large strains. General nonlinear stiffness relations are derived for compressible materials and for incompressible materials of the Mooney-Rivlin and neo-Hookean type. Stiffness relations for triangular plates, membranes, and straight bars are also derived. It is shown that the connection of elements into the assembled system can be accomplished by a series of group transformations. A simple example is provided to demonstrate parts of the theory.
30 citations