Finite element analysis of rubber‐like materials by a mixed model

Scharnhorst, Thomas; Pian, T. H. H.

doi:10.1002/nme.1620120410

Cited by 50 publications

(8 citation statements)

References 9 publications

(2 reference statements)

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“…However, the incompressibility and nonlinearity of rubber cause many difficulties, one of which is divergence problem, in analytic and numerical studies like the finite element method [1][2][3][4][5][6][7] as noted in the literature [7]. Furthermore, the problems of self-buckling, follower forces, and contact conditions cause additional difficulties in analyzing rubber.…”

Section: Introductionmentioning

confidence: 99%

A Finite Element Procedure with Poisson Iteration Method Adopting Pattern Approach Technique for Near-Incompressible Rubber Problems

Kwon

et al. 2014

Advances in Mechanical Engineering

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A finite element procedure is presented for the analysis of rubber-like hyperelastic materials. The volumetric incompressibility condition of rubber deformation is included in the formulation using the penalty method, while the principle of virtual work is used to derive a nonlinear finite element equation for the large displacement problem that is presented in a total-Lagrangian description. The behavior of rubber deformation is represented by hyperelastic constitutive relations based on a generalized Mooney-Rivlin model. The proposed finite element procedure using analytic differentiation exhibited results that matched very well with those from the well-known commercial packages NISA II and ABAQUS. Furthermore, the convergence of equilibrium iteration is quite slow or frequently fails in the case of near-incompressible rubber. To prevent such phenomenon even for the case that Poisson's ratio is very close to 0.5, Poisson's ratio of 0.49000 is used, first, to get an approximate solution without any difficulty; then the applied load is maintained and Poisson's ratio is increased to 0.49999 following a proposed pattern and adopting a technique of relaxation by monitoring the convergence rate. For a given Poisson ratio near 0.5, with this approach, we could reduce the number of substeps considerably.

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Section: Introductionmentioning

confidence: 99%

A Finite Element Procedure with Poisson Iteration Method Adopting Pattern Approach Technique for Near-Incompressible Rubber Problems

Kwon

et al. 2014

Advances in Mechanical Engineering

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“…For problems corresponding to the state of plane stress in the x 1 x 2 -plane, 13 = 23 = 33 = 0 where r is the Cauchy stress tensor. For a fully incompressible Mooney-Rivlin material,…”

Section: Analysis Of Plane Stress Problemsmentioning

confidence: 99%

“…Many finite element (FE) formulations have been developed to avoid the volumetric locking resulting from the incompressibility constraint. Among them are the mixed formulation [10][11][12], the hybrid methods [13][14][15], the selective reduced integration [16], the perturbed Lagrange formulation [17], and the rank-one filtering method [18]. In addition to difficulties in dealing with incompressibility, FE formulations frequently break down when applied to elastomers where excessive deformations lead to mesh entanglement.…”

Section: Introductionmentioning

confidence: 99%

Analysis of rubber‐like materials using meshless local Petrov–Galerkin (MLPG) method

Batra

Porfiri²

2007

Commun. Numer. Meth. Engng.

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SUMMARYLarge deformations of rubber-like materials are analyzed by the meshless local Petrov-Galerkin (MLPG) method. The method does not require shadow elements or a background mesh and therefore avoids mesh distortion difficulties in large deformation problems. Basis functions for approximating the trial solution and test functions are generated by the moving least-squares (MLS) method. A local mixed total Lagrangian weak formulation of non-linear elastic problems is presented. The deformation gradient is split into deviatoric and dilatational parts. The strain energy density is expressed as the sum of two functions: one is a function of deviatoric strains and the other is a function of dilatational strains. The incompressibility or near incompressibility constraint is accounted for by introducing the pressure field and penalizing the part of the strain energy density depending upon the dilatational strains. Unlike in the mixed finite element formulation, in the MLPG method there is no need for different sets of basis functions for displacement and pressure fields. Results computed with the MLPG method for a few sample problems are found to compare very well with the corresponding analytical solutions.

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“…In the theory of finite elasticity the most widely used strain measures are the Green's strain tensor E=(? ), defined in terms of the Lagrangian coordinates, and the Almansi's strain tensor e = r + ) , (9) defined in terms of the deformed or Eulerian coordinates. The tensor C is the inverse of Finger's tensor, B.…”

Section: Generalized Strain Measurementioning

confidence: 99%

An extension of the penalty function formulation to incompressible hyperelastic solids described by general measure of strain

Kakavas

Chang

1995

J of Applied Polymer Sci

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SYNOPSISThe penalty function formulation for incompressible hyperelastic solids was first proposed about 30 years ago. Since then all studies have been limited to invariant type formulation of the strain energy function, although it is well known that this formulation does not correctly describe the behavior of a real material. On the other hand more realistic constitutive equations, based on general measures of the strain only, have been incorporated to mixed finite element algorithms. In this article, a penalty function formulation is proposed for the analysis of stress field in materials with constitutive equations based on the general measure of strain. The reduced integration method is used to weaken the penalty constraint in order to obtain meaningful numerical results. The incremental equilibrium equations are solved using the regular Newton-Raphson algorithm. The method is applied to evaluate the stress field in materials subjected to plane strain conditions. Satisfactory agreements have been obtained with analytical solutions when available. 0 1995 John Wiley & Sons, Inc.

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Finite element analysis of rubber‐like materials by a mixed model

Cited by 50 publications

References 9 publications

A Finite Element Procedure with Poisson Iteration Method Adopting Pattern Approach Technique for Near-Incompressible Rubber Problems

A Finite Element Procedure with Poisson Iteration Method Adopting Pattern Approach Technique for Near-Incompressible Rubber Problems

Analysis of rubber‐like materials using meshless local Petrov–Galerkin (MLPG) method

An extension of the penalty function formulation to incompressible hyperelastic solids described by general measure of strain

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