Applied Scientific Research

, Volume 48, Issue 2, pp 193–210

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

• Peter A. A. van Hoogstraten
• Paul M. A. Slaats
• Frank P. T. Baaijens
Article

## 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.

### Keywords

Convection Rubber Element Modelling Finite Element Modelling Test Problem

### Nomenclature

C

material parameter

I1,I2

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

σ

Cauchy stress tensor

τ

stress tensor

Ω

rate of rotation tensor

()T

transpose

(′)

time derivative

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### References

1. 1.
Mooney, M., A theory of large elastic deformation,J. of Appl. Phys. 11 (1940) 582–592.
2. 2.
Rivlin, R.S., Large elastic deformations of isotropic materials — I. Fundamental concepts,Phil. Trans. R. Soc. A 240 (1948) 459–490.Google Scholar
3. 3.
Rivlin, R.S., Large elastic deformations of isotropic materials — IV. Further developments of the general theory,Phil. Trans. R. Soc. A 241 (1948) 379–397.Google Scholar
4. 4.
Treloar, L.R.G., The mechanic of rubber elasticity,Proc. R. Soc. Lond. A 351 (1976) 301–330.Google Scholar
5. 5.
Ogden, R.W., Large deformation isotropic elasticity — On the correlation of theory and experiment for incompressible rubber-like solids,Proc. R. Soc. Lond. A 326 (1972) 565–584.Google Scholar
6. 6.
Pinsky, P.M., Ortiz, M. and Pister, K.S., Numerical integration of rate constitutive equations in finite deformation analysis,Comp. Meth. in Appl. Mech. and Eng. 40 (1983) 137–158.
7. 7.
Morton, K.W., Priestly, A. and Suli, E., Stability analysis of the Lagrange-Galerkin method with nonexact integration, Oxford University Computing Laboratory, Report 86/14.Google Scholar
8. 8.
Donea, J., A Taylor-Galerkin method for convective transport problems,Int. J. for Num. Meth. in Eng. 20 (1984) 101–119.
9. 9.
SEPRAN user manual and SEPRAN programmers guide, ingenieursbureau SEPRA, Leidschendam, The Netherlands.Google Scholar
10. 10.
Cuvelier, C., Segal, A. and van Steenhoven, A.A.,Finite element methods and Navier-Stokes equations, D. Reidel Publishing Company, Dordrecht (1986).Google Scholar
11. 11.
MARC Manuals A, B, C and D, Revision K.3 (July 1988), MARC Analysis Research Corporation.Google Scholar

## Authors and Affiliations

• Peter A. A. van Hoogstraten
• 1
• Paul M. A. Slaats
• 1
• Frank P. T. Baaijens
• 1
• 2
1. 1.Faculty of Mechanical Engineering, Section of Engineering FundamentalsEindhoven University of TechnologyEindhovenThe Netherlands
2. 2.Philips Research LaboratoriesEindhovenThe Netherlands