models Common hyperelastic

1

Common hyperelastic

models

2

   

^

^

1

i i

N i

1 i

1 0 i 2

1

J 1

D 3 1

I C

J , I ,

I    

  





Polynomials of the first invariant

Initial shear modulus



 2C

Initial compressibility modulus

1 0

2

k  D

3

Particular case 1: Neo-Hookean behaviour (N=1)

    ^

^

1 10 2

1

J 1

D 3 1

I C

J , I ,

I    



   

^

^

1

i i

N i

1 i

1 0 i 2

1

J 1

D 3 1

I C

J , I ,

I    

  





Polynomials of the first invariant

4

5

In other words, the material law is not material-frame indifferent

6

7

8

9

10

11

12

    ^

^

i i

N i

i

i

J

I D C

J I

I

1 1

1 0 2

1

0

1 1

3 ,

,    

  







Particular case 2: Yeoh behaviour (N=3)

    ^

^

i i

i

i

i

J

I D C

J I

I

3

1 3

1

1 0 2

1

0

1 1

3 ,

,    

  







Forms depending only of the first invariant

polynomials

The usual form of strain energy    ^E    ^I

^{, I}

^{, J} 

ARRUDA-BOYCE

  _{ }   ^{    }



 



  



 







 

  el

2 el 5

1 i

i i 2 1

i 2 m

i 2

1

0

ln J

2 1 J

D 3 1

C I J

, I , I

673750

; 519 7050

; 19 1050

; 11 20

; 1 2 1

5 4

3 2

1  C  C  C  C 

C

Model with 8 chains

Statistical mechanics Gaussian chains





₀

k D2

0 

Forms depending only of the first invariant

The usual form of strain energy    ^E    ^I

^{, I}

^{, J} 



 



 



 







 









 sinh

ln n n

Nk _chain

0



 



  





 ^

and 3 3

I_{1 1 chain}

chain 

13

14

      ^{ }

1 2

01 1

10 2

1

0

1 1

3 3

,

,      

 J

I D C

I C

J I

 I

Particular case: N=1 MOONEY-RIVLIN





0

 2 C  C



1 0

2 k  D

Initial shear modulus

Initial compressibility modulus

Forms depending only of the two invariant

polynomials

The usual form of strain energy    ^E    ^I

^{, I}

^{, J} 

15

OGDEN

    ^

^

N

i i

N

i i

i J

J D I

I ^{i i i 2}

1 1

3 2

2 1 2

1

0 1 1

2 3 ,

,      



 









  

 

 

3 2

1

,  , 





 J





 ^N

i

i 1

0





1 0

2 k  D

It is the best model if we have experimental data for numerous tests in different directions

Rivlin Mooney

; 2

; 2

; 2

N  ₁  ₂    

Hookean Neo

; 2

; 1

N  ₁   

Forms written with the principal stretches

The usual form of strain energy    ^E    ^I

^{, I}

^{, J} 

Prévisions de l'essai biaxial

0 5 10 15 20 25 30

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5

Déformation nominale Contrainte nominale (MPa)

Expérience Neo hooke Mooney-Rivlin

Nominal stress (MPa)

Identification of parameters from all the data Prediction of the planar tensile test

16

Stretch λ=L/L₀

1 2 3 4 5 6 7

Prevision de l'essai de traction plane

0 5 10 15 20 25

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5

déformation nominale

contrainte nominale (MPa)

Expérience Ogden N=1 Ogden N=2 Ogden N=3

Engineering stress (MPa)

Identification of parameters from all the data Prediction of the biaxial tensile test

17

Stretch λ=L/L₀

1 2 3 4 5 6 7

Prevision de l'essai de traction plane

0 5 10 15 20 25

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5

déformation nominale contrainte nominale (MPa)

Expérience Yeoh

Arruda-Boyce Van der waals

Engineering stress (MPa)

Identification of parameters from all the data Prediction of the biaxial tensile test

18

Stretch λ=L/L₀

1 2 3 4 5 6 7

19

20

21

22

23

24