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A Eulerian approach to the finite element modelling of neo-Hookean rubber material

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Abstract

A Eulerian approach is applied to the finite element modelling of neo-Hookean rubber material. Two major problems are encountered. The first problem is the construction of an algorithm to calculate stresses in the rubber material from velocities instead of displacements. This problem is solved with an algorithm based on the definition of the velocity gradient. The second problem is the convection of stresses through the finite element mesh. This problem is solved by adapting the so-called Taylor-Galerkin technique. Solutions for both problems are implemented in a finite element program and their validity is shown by test problems. Results of these implementations are compared with results obtained by a standard Lagrangian approach finite element package and good agreement has been found.

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Abbreviations

C :

material parameter

I 1,I 2 :

first, second invariant ofB

w,q :

weighting function

t :

time

W :

strain energy function

n :

unit outward normal

t :

surface traction vector

u :

velocity vector field

w :

weighting vector

x :

position vector field

α :

interpolation parameter

ε :

penalty parameter

Δt :

timestep

B :

left Cauchy-Green tensor

D :

rate of strain tensor

I :

unity tensor

F :

deformation gradient

σ :

Cauchy stress tensor

τ :

stress tensor

Ω :

rate of rotation tensor

()T :

transpose

(′):

time derivative

∇:

gradient operator

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Section of Engineering Fundamentals, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Peter A. A. van Hoogstraten, Paul M. A. Slaats & Frank P. T. Baaijens

  2. Philips Research Laboratories, P.O. Box 80000, 5600 JA, Eindhoven, The Netherlands

    Frank P. T. Baaijens

Authors
  1. Peter A. A. van Hoogstraten
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  2. Paul M. A. Slaats
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  3. Frank P. T. Baaijens
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van Hoogstraten, P.A.A., Slaats, P.M.A. & Baaijens, F.P.T. A Eulerian approach to the finite element modelling of neo-Hookean rubber material. Appl. Sci. Res. 48, 193–210 (1994). https://doi.org/10.1007/BF02027967

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  • DOI: https://doi.org/10.1007/BF02027967

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