A finite-element programme for the plane-strain analysis of rubber

1975 ◽  
Vol 10 (1) ◽  
pp. 25-31 ◽  
Author(s):  
P B Lindley
Keyword(s):  
Finite Element ◽  
Plane Strain ◽  
Degrees Of Freedom ◽  
Strain Analysis ◽  
Strain Energy Function ◽  
Computation Time ◽  
Incompressible Material ◽  
Quadrilateral Elements ◽  
Triangular Elements ◽  
Classical Elasticity Theory

For a highly incompressible material the use of triangular elements in plane-strain finite-element analyses restricts the number of degrees of freedom. A computer programme is developed which uses quadrilateral elements, and various methods of reducing computation time are employed. A strain-energy function is proposed which will enable solutions to be obtained at strains well beyond those of linear classical elasticity theory.

Plane-stress analysis of rubber at high strains using finite elements

1971 ◽  
Vol 6 (1) ◽  
pp. 45-52 ◽  
Author(s):  
P B Lindley
Keyword(s):  
Strain Energy ◽  
Nodal Point ◽  
Strain Energy Function ◽  
Computation Time ◽  
Classical Elasticity ◽  
Nodal Points ◽  
Triangular Elements ◽  
Classical Elasticity Theory ◽  
Plane Stress Analysis ◽  
Good Agreement

A finite-element programme has been developed for the analysis of stretched rubber sheets. The elastic energy of each of the homogeneously deformed triangular elements into which the sheet is divided can be determined from the displacements of its nodal points and a suitable strain-energy function. Each nodal point of the sheet is considered in turn and moved to a position which minimizes the energy of all the elements local to it. This iterative process is continued until the total energy of the sheet is minimized. A modified classical-elasticity-theory programme provides the first approximation. The boundary conditions are specified displacements, assumed to arise from the use of rigid clamps. The results are in good agreement with experiment for peak strains of about 150 per cent. Computation time on a CDC 6600 computer is about 0.005 × (nodal points)2 seconds.

Crossed patch arrangements of linear triangular elements for upper bound finite-element analysis of plane strain deformation problems

2010 ◽  
Vol 225 (2) ◽  
pp. 280-291 ◽  
Author(s):  
C C Chang
Keyword(s):  
Finite Element Analysis ◽  
Plastic Deformation ◽  
Finite Element ◽  
Velocity Field ◽  
Plane Strain ◽  
Element Analysis ◽  
Quadrilateral Elements ◽  
Plane Strain Deformation ◽  
Incompressibility Constraint ◽  
Triangular Elements

In standard linear finite-element formulations, volumetric locking because of the incompressibility constraint that may occur in computational plasticity is often encountered. This study uses crossed patch arrangements of triangles to form quadrilateral elements in order to overcome the locking in the upper bound finite-element analysis of plane strain deformation problems. The velocity field is described in terms of linear triangular elements, while the incompressibility constraint is imposed by quadrilateral elements. Rigid, perfectly plastic materials, and strain hardening materials that form the von Mises model have been considered. The velocity formulation is presented and has been implemented in a finite-element code. Several examples, some benchmarks problems, are presented to illustrate the applicability of the approach for predicting the load, strain, and velocity field during the plastic deformation. Numerical results show that the crossed patch arrangements of linear triangular elements are free of volumetric locking and achieve well-defined limit loads. This study shows that the presented method can be used to simulate large plastic deformation under plane strain conditions.

Elastic-Plastic Plane Strain Analysis of Compressible Materials

1987 ◽  
Vol 109 (3) ◽  
pp. 357-358
Author(s):  
Hui Fan ◽  
G. E. O. Widera
Keyword(s):  
Plane Strain ◽  
Order Approximation ◽  
Strain Analysis ◽  
Perturbation Parameter ◽  
Incompressible Material ◽  
Perfectly Plastic ◽  
Elastic Perfectly Plastic ◽  
First Order Approximation ◽  
Compressible Materials ◽  
Walled Cylinder

The analysis of elastic, perfectly plastic compressible materials under the assumption of plane strain is a statically indeterminate problem. In the present paper, by introducing a perturbation parameter ε = 1/2−ν, the problem can be changed into a series of statically determinate ones. The first-order approximation yields the solution for the incompressible material. In order to show the details of this method, the second-order approximation for the problem of the thick-walled cylinder under internal pressure is obtained.

A Simplified Strain Energy Function to Represent the Mechanical Behavior of the Annulus Fibrosus

2009 ◽  
Author(s):  
Jose J. García ◽  
Christian Puttlitz
Keyword(s):  
Finite Element ◽  
Mechanical Behavior ◽  
Energy Function ◽  
Annulus Fibrosus ◽  
Strain Energy ◽  
Degrees Of Freedom ◽  
Strain Energy Function ◽  
Efficient Simulation ◽  
Matrix Interactions ◽  
Complex Functions

Models to represent the mechanical behavior of the annulus fibrosus are important tools to understand the biomechanics of the spine. Many hyperelastic constitutive equations have been proposed to simulate the mechanical behavior of the annulus that incorporate the anisotropic nature of the tissue. Recent approaches [1,2] have included terms into the energy function which take into account fiber-fiber and fiber-matrix interactions, leading to complex functions that cannot be readily implemented into commercial finite element codes for an efficient simulation of nonlinear realistic models of the spine (which are generally composed of 100,000+ degrees of freedom). An effort is undertaken here to test the capability of a relatively simple strain energy function [3] for the description of the annulus fibrosus. This function has already been shown to successfully represent the mechanical behavior of the arterial tissue and can be readily implemented into existing finite element codes.

A Novel Alpha Smoothed Finite Element Method for Ultra-Accurate Solution Using Quadrilateral Elements

2019 ◽  
Vol 17 (02) ◽  
pp. 1845008 ◽  
Author(s):  
Y. H. Li ◽  
M. Li ◽  
G. R. Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Strain Energy ◽  
Solid Mechanics ◽  
Strain Energy Function ◽  
Accurate Solution ◽  
Scale Factor ◽  
Quadrilateral Elements ◽  
Smoothed Finite Element Method ◽  
Element Method

Smoothed finite element method (S-FEM) based on triangular elements has recently been widely used for solving solid mechanics problems. In this paper, a novel [Formula: see text]S-FEM using quadrilateral elements ([Formula: see text]SFEM-Q4) is proposed to obtain ultra-accurate solutions in the displacement and strain energy for solid mechanics problems. This method combines node-based S-FEM (NS-FEM), edge-based S-FEM (ES-FEM) and cell-based S-FEM (CS-FEM) equipped with a scale factor [Formula: see text] that controls contribution from each of these three different S-FEM models. This novel combination makes the best use of the upper bound property of the NS-FEM and the lower bound property of the CS-FEM (with 4 or more sub-smoothing domains for each element), and establishes a continuous strain-energy function of a scale factor [Formula: see text] for obtaining close-to-exact solutions. Our [Formula: see text]SFEM-Q4 also ensures the variational consistence and the compatibility of the displacement field, and hence guarantees to reproduce linear field exactly. Various solid mechanics problems are presented to validate the stability, effectiveness and ultra-accuracy of the proposed method.

Fully Plastic Crack Problems, Part 1: Solutions by a Penalty Method

1984 ◽  
Vol 51 (1) ◽  
pp. 48-56 ◽  
Author(s):  
C. F. Shih ◽  
A. Needleman
Keyword(s):  
Finite Element ◽  
Numerical Method ◽  
Strain Hardening ◽  
Plane Strain ◽  
Penalty Method ◽  
Incompressible Material ◽  
Material Behavior ◽  
Reduced Integration ◽  
Wide Range ◽  
Crack Problems

We formulate a finite-element reduced integration penalty method applicable to plane-strain problems with incompressible material behavior. This numerical method is employed to generate crack solutions for pure power-hardening solids. For two configurations of interest in applications, an edge cracked panel subject to remote tension and an edge-cracked panel subject to remote bending, we obtain solutions for a wide range of crack lengths and strain-hardening behaviors.

scholarly journals A family of C1 quadrilateral finite elements

2021 ◽  
Vol 47 (6) ◽  
Author(s):  
Mario Kapl ◽  
Giancarlo Sangalli ◽  
Thomas Takacs
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Degrees Of Freedom ◽  
Approximation Error ◽  
Mesh Size ◽  
Element Space ◽  
Quadrilateral Elements ◽  
Macro Element ◽  
Quadrilateral Finite Element ◽  
Better Than

AbstractWe present a novel family of C1 quadrilateral finite elements, which define global C1 spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by Brenner and Sung (J. Sci. Comput. 22(1-3), 83-118, 2005), which is based on polynomial elements of tensor-product degree p ≥ 6, to all degrees p ≥ 3. The proposed C1 quadrilateral is based upon the construction of multi-patch C1 isogeometric spaces developed in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles, developed in Argyris et al. (Aeronaut. J. 72(692), 701–709 1968). Just as for the Argyris triangle, we additionally impose C2 continuity at the vertices. In contrast to Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019), in this paper, we concentrate on quadrilateral finite elements, which significantly simplifies the construction. We present macro-element constructions, extending the elements in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005), for polynomial degrees p = 3 and p = 4 by employing a splitting into 3 × 3 or 2 × 2 polynomial pieces, respectively. We moreover provide approximation error bounds in $L^{\infty }$ L ∞ , L2, H1 and H2 for the piecewise-polynomial macro-element constructions of degree p ∈{3,4} and polynomial elements of degree p ≥ 5. Since the elements locally reproduce polynomials of total degree p, the approximation orders are optimal with respect to the mesh size. Note that the proposed construction combines the possibility for spline refinement (equivalent to a regular splitting of quadrilateral finite elements) as in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 30) with the purely local description of the finite element space and basis as in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005). In addition, we describe the construction of a simple, local basis and give for p ∈{3,4,5} explicit formulas for the Bézier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed C1 quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for p = 5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.

Delamination in the scoring and folding of paperboard

TAPPI Journal ◽  
2012 ◽  
Vol 11 (1) ◽  
pp. 61-66 ◽  
Author(s):  
DOEUNG D. CHOI ◽  
SERGIY A. LAVRYKOV ◽  
BANDARU V. RAMARAO
Keyword(s):  
Finite Element ◽  
Plane Deformation ◽  
Strain Analysis ◽  
Local Strain ◽  
Uniaxial Stress ◽  
Material Model ◽  
Element Model ◽  
Plastic Behavior ◽  
Stress Strain ◽  
Strain Fields

Delamination between layers occurs during the creasing and subsequent folding of paperboard. Delamination is necessary to provide some stiffness properties, but excessive or uncontrolled delamination can weaken the fold, and therefore needs to be controlled. An understanding of the mechanics of delamination is predicated upon the availability of reliable and properly calibrated simulation tools to predict experimental observations. This paper describes a finite element simulation of paper mechanics applied to the scoring and folding of multi-ply carton board. Our goal was to provide an understanding of the mechanics of these operations and the proper models of elastic and plastic behavior of the material that enable us to simulate the deformation and delamination behavior. Our material model accounted for plasticity and sheet anisotropy in the in-plane and z-direction (ZD) dimensions. We used different ZD stress-strain curves during loading and unloading. Material parameters for in-plane deformation were obtained by fitting uniaxial stress-strain data to Ramberg-Osgood plasticity models and the ZD deformation was modeled using a modified power law. Two-dimensional strain fields resulting from loading board typical of a scoring operation were calculated. The strain field was symmetric in the initial stages, but increasing deformation led to asymmetry and heterogeneity. These regions were precursors to delamination and failure. Delamination of the layers occurred in regions of significant shear strain and resulted primarily from the development of large plastic strains. The model predictions were confirmed by experimental observation of the local strain fields using visual microscopy and linear image strain analysis. The finite element model predicted sheet delamination matching the patterns and effects that were observed in experiments.

Stress Analysis of Tire Sections

1996 ◽  
Vol 24 (4) ◽  
pp. 349-366 ◽  
Author(s):  
T-M. Wang ◽  
I. M. Daniel ◽  
K. Huang
Keyword(s):  
Finite Element ◽  
Internal Pressure ◽  
Stress Analysis ◽  
Strain Analysis ◽  
Experimental Stress ◽  
Inflation Pressure ◽  
Substantial Agreement ◽  
Finite Element Stress Analysis ◽  
Strain Components ◽  
Fringe Patterns

Abstract An experimental stress-strain analysis by means of the Moiré method was conducted in the area of the tread and belt regions of tire sections. A special loading fixture was designed to support the tire section and load it in a manner simulating service loading and allowing for Moiré measurements. The specimen was loaded by imposing a uniform fixed deflection on the tread surface and increasing the internal pressure in steps. Moiré fringe patterns were recorded and analyzed to obtain strain components at various locations of interest. Maximum strains in the range of 1–7% were determined for an effective inflation pressure of 690 kPa (100 psi). These results were in substantial agreement with results obtained by a finite element stress analysis.

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