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TECHNICAL PAPERS

A Two-Dimensional Linear Assumed Strain Triangular Element for Finite Deformation Analysis

[+] Author and Article Information
Fernando G. Flores

Department of Structures, National University of Cordoba, Casilla de correos 916, 5000 Cordoba, Argentinafflores@efn.uncor.edu

J. Appl. Mech 73(6), 970-976 (Dec 19, 2005) (7 pages) doi:10.1115/1.2173674 History: Received June 08, 2005; Revised December 19, 2005

Abstract

An assumed strain approach for a linear triangular element able to handle finite deformation problems is presented in this paper. The element is based on a total Lagrangian formulation and its geometry is defined by three nodes with only translational degrees of freedom. The strains are computed from the metric tensor, which is interpolated linearly from the values obtained at the mid-side points of the element. The evaluation of the gradient at each side of the triangle is made resorting to the geometry of the adjacent elements, leading to a four element patch. The approach is then nonconforming, nevertheless the element passes the patch test. To deal with plasticity at finite deformations a logarithmic stress-strain pair is used where an additive decomposition of elastic and plastic strains is adopted. A hyper-elastic model for the elastic linear stress-strain relation and an isotropic quadratic yield function (Mises) for the plastic part are considered. The element has been implemented in two finite element codes: an implicit static/dynamic program for moderately non-linear problems and an explicit dynamic code for problems with strong nonlinearities. Several examples are shown to assess the behavior of the present element in linear plane stress states and non-linear plane strain states as well as in axi-symmetric problems.

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Figures

Figure 1

Patch of elements (a) in spatial coordinates, (b) in natural coordinates

Figure 2

Membrane patch test

Figure 3

Shear loaded short cantilever: no contraction allowed at the root. E=30,000, ν=0.25, h=1.

Figure 4

Tip deflections for short cantilever under end load. (a) γ=1, (b) γ=2, (c) γ=4.

Figure 5

Cook’s membrane problem. Geometry and load.

Figure 6

Cook’s membrane problem. Vertical deflections of point C (plane stress).

Figure 7

Cook’s problem in plane strain. Convergence of finite element solutions. (a) quasi-incompressible finite elasticity. (b) Finite Strain J2 flow theory.

Figure 8

Stretching of circular sheet with a hemispherical punch

Figure 9

Stretching of circular sheet with a hemispherical punch. (a) Punch force vs punch travel. (b) Thickness ratio along the radius for different punch travels. (c) Equivalent plastic strain along the radius for different punch travels.

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