Hyperelastic Uniaxial Tension Test

Overview

The aim of this case is to validate the following functions:

Hyperelasticity for uniaxial tension test

The simulation results of SimScale were compared to the results presented in [Tre44]. The mesh used in the study is a hexahedral mesh with only one element imported to SimScale platform.

Import uniaxial tension test validation project into workspace

Geometry

The cube (ABCDEFGH) has the dimension of:

Cube length	1 m
Cube width	1 m
Cube height	1 m

Analysis type and Domain

Tool Type : Calculix (CCX), Code_Aster (CA)

Analysis Type : Static

Mesh and Element types :

Mesh type	Number of nodes	Element type
linear hexahedral	8	3D isoparametric

Simulation Setup

Important

All displacement and load components are referred to the coordinate system in the figure of the geometry section.

Important

All the material model values below are calculated through a curve fitting with Treloar test data presented in [Tre44].

Uniaxial test:

Material:

Material model		$C_{10}$ $C_{10}$	$C_{01}$ $C_{01}$	$C_{20}$ $C_{20}$	$C_{11}$ $C_{11}$	$C_{02}$ $C_{02}$	$C_{30}$ $C_{30}$	$D_{1}$ $D_{1}$	$D_{2}$ $D_{2}$	$D_{3}$ $D_{3}$
$N e o H o o k e a n^{(C C X / C A)}$ $N e o H o o k e a n^{(C C X / C A)}$	–	$2.881331$ $2.881331$	–	–	–	–	–	$1 e - 8^{(C C X)} - 1 e - 6^{(C A)}$ $1 e - 8^{(C C X)} - 1 e - 6^{(C A)}$	–	–
$M o o n e y R i v l i n^{(C C X / C A)}$ $M o o n e y R i v l i n^{(C C X / C A)}$	–	$4.1034451$ $4.1034451$	$- 7.486014$ $- 7.486014$	–	–	–	–	$1 e - 8^{(C C X)} - 1 e - 6^{(C A)}$ $1 e - 8^{(C C X)} - 1 e - 6^{(C A)}$	–	–
$Y e o h^{(C C X)}$ $Y e o h^{(C C X)}$	–	$1.813706$ $1.813706$	–	$- 0.019586$ $- 0.019586$	–	–	$0.000464$ $0.000464$	$1 e - 8$ $1 e - 8$	$1 e - 8$ $1 e - 8$	$1 e - 8$ $1 e - 8$
$R e d P o l y 2^{(C C X)}$ $R e d P o l y 2^{(C C X)}$	–	$0.618107$ $0.618107$	–	$0.026944$ $0.026944$	–	–	–	$1 e - 8$ $1 e - 8$	$1 e - 8$ $1 e - 8$	–
$P o l y 2^{(C C X)}$ $P o l y 2^{(C C X)}$	–	$- 20.465522$ $- 20.465522$	$26.340633$ $26.340633$	$0.185894$ $0.185894$	$- 1.474968$ $- 1.474968$	$9.051008$ $9.051008$	–	$1 e - 8$ $1 e - 8$	$1 e - 8$ $1 e - 8$	–
$S i g n o r i n i^{(C A)}$ $S i g n o r i n i^{(C A)}$	–	$- 1.432189$ $- 1.432189$	$5.578092$ $5.578092$	$0.037811$ $0.037811$	–	–	–	$1 e - 6$ $1 e - 6$	–	–
		$μ_{1}$ $μ_{1}$	$μ_{2}$ $μ_{2}$	$μ_{3}$ $μ_{3}$	$α_{1}$ $α_{1}$	$α_{2}$ $α_{2}$	$α_{3}$ $α_{3}$
$O g d e n 1^{(C C X)}$ $O g d e n 1^{(C C X)}$	–	$0.308812$ $0.308812$	–	–	$3.883934$ $3.883934$	–	–	$1 e - 8$ $1 e - 8$	–	–
$O g d e n 2^{(C C X)}$ $O g d e n 2^{(C C X)}$	–	$2.747783$ $2.747783$	$1.009837 e - 6$ $1.009837 e - 6$	–	$2.199164$ $2.199164$	$10.319457$ $10.319457$	–	$1 e - 8$ $1 e - 8$	$1 e - 8$ $1 e - 8$	–
$O g d e n 3^{(C C X)}$ $O g d e n 3^{(C C X)}$	–	$0.000241$ $0.000241$	$3.234478$ $3.234478$	$2.982735 e - 92$ $2.982735 e - 92$	$7.508369$ $7.508369$	$10.319457$ $10.319457$	$2.982735 e - 92$ $2.982735 e - 92$	$1 e - 8$ $1 e - 8$	$1 e - 8$ $1 e - 8$	$1 e - 8$ $1 e - 8$
	$λ_{m}$ $λ_{m}$	$μ$ $μ$
$A r r u d a - B o y c e^{(C C X)}$ $A r r u d a - B o y c e^{(C C X)}$	$4.441020$ $4.441020$	$2.366097$ $2.366097$	–	–	–	–	–	$1 e - 8$ $1 e - 8$	–	–

Material model

C_{10}

C_{10}

C_{01}

C_{01}

C_{20}

C_{20}

C_{11}

C_{11}

C_{02}

C_{02}

C_{30}

C_{30}

D_{1}

D_{1}

D_{2}

D_{2}

D_{3}

D_{3}

N e o H o o k e a n^{(C C X / C A)}

N e o H o o k e a n^{(C C X / C A)}

–

2.881331

2.881331

–

1 e - 8^{(C C X)} - 1 e - 6^{(C A)}

1 e - 8^{(C C X)} - 1 e - 6^{(C A)}

–

M o o n e y R i v l i n^{(C C X / C A)}

M o o n e y R i v l i n^{(C C X / C A)}

–

4.1034451

4.1034451

- 7.486014

- 7.486014

–

1 e - 8^{(C C X)} - 1 e - 6^{(C A)}

1 e - 8^{(C C X)} - 1 e - 6^{(C A)}

–

Y e o h^{(C C X)}

Y e o h^{(C C X)}

–

1.813706

1.813706

–

- 0.019586

- 0.019586

–

0.000464

0.000464

1 e - 8

1 e - 8

1 e - 8

1 e - 8

1 e - 8

1 e - 8

R e d P o l y 2^{(C C X)}

R e d P o l y 2^{(C C X)}

–

0.618107

0.618107

–

0.026944

0.026944

–

1 e - 8

1 e - 8

1 e - 8

1 e - 8

–

P o l y 2^{(C C X)}

P o l y 2^{(C C X)}

–

- 20.465522

- 20.465522

26.340633

26.340633

0.185894

0.185894

- 1.474968

- 1.474968

9.051008

9.051008

–

1 e - 8

1 e - 8

1 e - 8

1 e - 8

–

S i g n o r i n i^{(C A)}

S i g n o r i n i^{(C A)}

–

- 1.432189

- 1.432189

5.578092

5.578092

0.037811

0.037811

–

1 e - 6

1 e - 6

–

μ_{1}

μ_{1}

μ_{2}

μ_{2}

μ_{3}

μ_{3}

α_{1}

α_{1}

α_{2}

α_{2}

α_{3}

α_{3}

O g d e n 1^{(C C X)}

O g d e n 1^{(C C X)}

–

0.308812

0.308812

–

3.883934

3.883934

–

1 e - 8

1 e - 8

–

O g d e n 2^{(C C X)}

O g d e n 2^{(C C X)}

–

2.747783

2.747783

1.009837 e - 6

1.009837 e - 6

–

2.199164

2.199164

10.319457

10.319457

–

1 e - 8

1 e - 8

1 e - 8

1 e - 8

–

O g d e n 3^{(C C X)}

O g d e n 3^{(C C X)}

–

0.000241

0.000241

3.234478

3.234478

2.982735 e - 92

2.982735 e - 92

7.508369

7.508369

10.319457

10.319457

2.982735 e - 92

2.982735 e - 92

1 e - 8

1 e - 8

1 e - 8

1 e - 8

1 e - 8

1 e - 8

λ_{m}

λ_{m}

μ

μ

A r r u d a - B o y c e^{(C C X)}

A r r u d a - B o y c e^{(C C X)}

4.441020

4.441020

2.366097

2.366097

–

1 e - 8

1 e - 8

–

Constraints:

Face ABFE zero x-displacement
Face AEHD zero y-displacement
Face ABCD zero z-displacement
Face DCGH 7m x-displacement

Results

Comparison of the nominal stress vs. the nominal strain calculated on the node of edge AB and AE. The values of the reference [Tre44] in all figures were extracted with WebPlotDigitizer.

References

[Tre44]

(1, 2, 3) L. R. G. Treloar. Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc., 40:59–70, 1944.

Last updated: November 25th, 2019

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Hyperelastic Uniaxial Tension Test

Overview

Geometry

Analysis type and Domain

Simulation Setup

Results

References

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