Robust Finite Elements for 3d‐analysis of Rubber‐like Materials

Bogert, P.A.J. van den; Borst, René de; Luiten, G.T.; Zeilmaker, J

doi:10.1108/eb023823

Cited by 24 publications

(5 citation statements)

References 11 publications

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“…(3) Our pressure equation is derived from the mass balance equation instead of the equation of state. In the traditional two-field variational principle, the equation for the pressure field is often introduced as an algebraic equation of state [25, 45, 81]. We feel that it is unnecessary and improper to put an algebraic equation in a weak form and discretize it by the finite element method (see Remark 5).…”

Section: Discussionmentioning

confidence: 99%

A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction

Marsden

2018

Computer Methods in Applied Mechanics and Engineering

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We develop a unified continuum modeling framework using the Gibbs free energy as the thermodynamic potential. This framework naturally leads to a pressure primitive variable formulation for the continuum body, which is well-behaved in both compressible and incompressible regimes. Our derivation also provides a rational justification of the isochoric-volumetric additive split of free energies in nonlinear elasticity. The variational multiscale analysis is performed for the continuum model to construct a foundation for numerical discretization. We first consider the continuum body instantiated as a hyperelastic material and develop a variational multiscale formulation for the hyper-elastodynamic problem. The generalized-α method is applied for temporal discretization. A segregated algorithm for the nonlinear solver, based on the original idea introduced in [107], is carefully analyzed. Second, we apply the new formulation to construct a novel unified formulation for fluid-solid coupled problems. The variational multiscale formulation is utilized for spatial discretization in both fluid and solid subdomains. The generalized-α method is applied for the whole continuum body, and optimal high-frequency dissipation is achieved in both fluid and solid subproblems. A new predictor multi-corrector algorithm is developed based on the segregated algorithm. The efficacy of the new formulations is examined in several benchmark problems. The results indicate that the proposed modeling and numerical methodologies constitute a promising technology for biomedical and engineering applications, particularly those necessitating incompressible models.

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Section: Discussionmentioning

confidence: 99%

A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction

Marsden

2018

Computer Methods in Applied Mechanics and Engineering

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“…In the subsequent numerical study the proposed FE-formulations are tested on the basis of two academic examples, documented in the literature, including the analysis of a rubber cylinder pressed between two plates 20,26 and of a rubber block 28 .…”

Section: Numerical Studymentioning

confidence: 99%

“…The paper also contains a numerical evaluation of the different finite element formulations including a comparison of numerical results, presented in a recently published paper 28 with those obtained on the basis of the proposed formulations.…”

Section: Introductionmentioning

confidence: 98%

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3D finite element analysis of rubber‐like materials at finite strains

Liu

Hofstetter

Mang

1994

Engineering Computations

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The paper starts with a review of constitutive equations for rubber‐like materials, formulated in the invariants of the right Cauchy—Green deformation tensor. A general framework for the derivation of the stress tensor and the tangent moduli for invariant‐based models, for both the reference and the current configuration, is presented. The free energy of incompressible rubber‐like materials is extended to a compressible formulation by adding the volumetric part of the free energy. In order to overcome numerical problems encountered with displacement‐based finite element formulations for nearly incompressible materials, three‐dimensional finite elements, based on a penalty‐type formulation, are proposed. They are characterized by applying reduced integration to the volumetric parts of the tangent stiffness matrix and the pressure‐related parts of the internal force vector only. Moreover, hybrid finite elements are proposed. They are based on a three‐field variational principle, characterized by treating the displacements, the dilatation and the hydrostatic pressure as independent variables. Subsequently, this formulation is reduced to a generalized displacement formulation. In the numerical study these formulations are evaluated. The results obtained are compared with numerical results available in the literature. In addition, the proposed formulations are applied to 3D finite element analysis of an automobile tyre. The computed results are compared with experimental data.

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“…This example is adopted from van den Bogert et al (1991), Liu et al (1992;1994) dealing with rubber-like materials. Our purpose is to show the application of EC 17,7…”

Section: Compression Of a Blockmentioning

confidence: 99%

On volumetric locking of low‐order solid and solid‐shell elements for finite elastoviscoplastic deformations and selective reduced integration

Doll¹,

Schweizerhof²,

Hauptmann³

et al. 2000

Engineering Computations

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As known from nearly incompressible elasticity, selective reduced integration (SRI) is a simple and effective method of overcoming the volumetric locking problem in 2D and 3D solid elements. This method of finite elastoviscoplasticity is discussed as are its well-known limitations. In this context, an isochoric-volumetric decoupled material behavior is assumed and thus the additive deviatoric-volumetric decoupling of the consistent algorithmic moduli tensor is essential. By means of several numerical examples, the performance of elements using selective reduced integration is demonstrated and compared to the performance of other elements such as the enhanced assumed strain elements. It is shown that a minor modification, with little numerical effort, leads to rather robust element behaviour. The application of this process to so-called solid-shell elements for thin-walled structures is also discussed. IntroductionIt is a well-known fact, that the three-dimensional standard linear eight-node displacement element (here denoted as Q1) exhibits severe locking for nearly or fully incompressible material. Rubber elasticity or metal plasticity are examples for nearly incompressible material behavior. Rubber is characterized by a large ratio of the bulk modulus to the shear modulus. In metal plasticity the plastic deformation is isochoric, i.e. fully incompressible, and the compressible part is a result from elastic deformations only, which remain small in many applications. In both cases reliable results with the standard Q1 element cannot be expected. Due to the volumetric locking behavior of the Q1 element, additional measures are necessary to overcome this problem. Various methods are known from literature and are currently discussed. Among others, a brief review of some of these methods is given in the following as a reference for the developments shown in this contribution. Our focus is particularly on the capabilities concerning finite non-linear deformations:

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Robust Finite Elements for 3d‐analysis of Rubber‐like Materials

Cited by 24 publications

References 11 publications

A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction

A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction

3D finite element analysis of rubber‐like materials at finite strains

On volumetric locking of low‐order solid and solid‐shell elements for finite elastoviscoplastic deformations and selective reduced integration

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