Treolar’s tests on rubber (1944)

1

HYPERELASTICITY

Soft biological tissues: many challenges for

continuum mechanics

Essais de Treloar (1944)

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Lambda-1

Contrainte nominale

Traction simple Equibiaxiale Traction plane

Treolar’s tests on rubber (1944)

Nomi nal st ress F /S0 (MPa )

Stretch λ=L/L

Biaxial tension

Planar tension Uniaxial tension

3

1 2 3 4 5 6 7

Hyperelasticity

Pseudo-hyperelasticity:

Irreversible effects are neglected…

Stretch λ=L/L

1 2 3 4 5 6

No minal str ess ( MPa )

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Contrainte nominale

Lambda-1

Nominal str ess S=F/ A

(MPa )

Stretch λ=L/L

5

Stored energy per unit volume 1

𝐴

𝐿

න 𝐹 ሶ𝐿𝑑𝑡 = න 𝑆 ሶ λ𝑑𝑡

1 2 3 4 5 6 7

Strain energy density

Deformation mapping in 3D

Deformation Gradient

1 2 3

( , , , )

i i i

y   x u x x x t

 

 

( )

or ⁱ _{i i ij} ⁱ _ij

j j j

y u

x u F

x x



    

  

    

  

y x u x F

x u(x)

e₃

e₁ e₂

dx dy

u(x+dx)

Original Configuration

Deformed Configuration

i ik k

d d

dy F dx

 



y F x

e₃

e₁ e₂

l₀

Original Configuration

Deformed Configuration

m

l

1 1

( ) or ( )

2 2

T

E

F F



    

E F F I

 

2 2

0

2 2

0 0 0

2 2

ij i j

l l l l

E m m

l l l

 

      

m E m

7

Green Lagrange strain

Nominal/ 1

Piola-Kirchhoff

Material/2

Piola-Kirchhoff

1 1

ij ik kj

J ^

S

  

S F

σ

1 T 1 1

ij ik kl jl

J ^{ } JF^  F^

 



Σ F

σ

e₃

e₁ e₂

Original Configuration

Deformed Configuration

t

dA₀

dA n

n₀

x

u(x)

dP⁽ⁿ⁾ dP₀⁽ⁿ⁾

True / Cauchy

σ

Stress measures

π π

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

Contrainte nominale

Lambda-1

Nominal str ess S=F/ A

(MPa )

Stretch λ=L/L

9

Stored energy per unit volume න 𝑺: ሶ F𝑑𝑡 = න 𝝅: ሶ E𝑑𝑡 = Ψ(E)

1 2 3 4 5 6 7

Hyperelasticity

compressible hyperelastic behaviour

incompressible hyperelastic behaviour (𝐽=1)

 ?



Strain energy density:

𝛔 = 𝐽𝐅. 𝜕Ψ

𝜕𝐄 . 𝐅 ^𝑇

𝛔 = 𝐅. 𝜕Ψ

𝜕𝐄 . 𝐅 ^𝑇 + 𝑐𝐈



Review: Thermodynamics

e3

e₁ e₂

Original Configuration

Deformed Configuration

S

R R₀

S₀ b

t

Specific Internal Energy

Specific Helmholtz free energy     s Temperature 

Heat flux vector q External heat flux ^q

First Law of Thermodynamics d ( )

KE Q W dt    

ij ij i const i

D q q

t y

  





   

 _x 

Second Law of Thermodynamics ^dS_dt ^{ d}_dt^{ 0} Specific entropy s

( / )

i 0

i

s q q

t y

 



 

  

 

1 0

ij ij i

i

D q s

y t t

  

 



   

        11

Constitutive Laws

General Assumptions:

1. Local homogeneity of deformation

(a deformation gradient can always be calculated) 2. Principle of local action

(stress at a point depends on deformation in

a vanishingly small material element surrounding the point)

Restrictions on constitutive relations:

1. Material Frame Indifference – stress-strain relations must transform consistently under a change of observer

2. Constitutive law must always satisfy the second law of

thermodynamics for any possible deformation/temperature history.

Equations relating internal force measures to deformation measures are known as Constitutive Relations

e₃

e₁ e₂

Original Configuration

Deformed Configuration

1 0

ij ij i

i

D q s

y t t

  

 



    

        

neglected

13

          ^{x , t   x , t : D x , t   x , t s x , t  0}



  

We go back in the reference configuration:

  ^{X , t : E 0}     ^{X , t s 0 X , t 0}

0

0      

   

   

             

possible E

0 t

, X , t , X e s t

, X , t , X E E

: t , X , t , X e e

0 X 0

X 0

0 X 0

















 



 



   







 

 

 



 



 





 





It can be written:

    ^





 



 E ,

, T E s

0 0 X

X

    _ ^





 



 





 





 E ,

Sym E ,

E

0 X X 0

Compressible materials Constitutive equations:

F E .

sym .

F ^t

0 X 















 



With the Cauchy

tensor:

   

 











 





 



 E X,t , X,t E

0 X 0

because:

Must be antisymmetric for cancelling the dot product of the preceding

slide

15

E s E

h ⁰ _X

0 0

X

0 

 



 

 





enthalpic elasticity

(cristal)

entropic elasticity

(biological tissue)

    ^





 





 E ,

, E E

0 X X 0

We can still choose such as: 

16

0 0,2 0,4 0,6 0,8 1 Dé formation %

Entropie Ene rgie Inte rne

0 100 200 300 400 500 600 Dé formation %

Ene rgie Inte rne Entropie

enthalpic elasticity

(cristal)

entropic elasticity

(biological tissue)

17

incompressible hyperelastic behaviour

  ^C

tr

I ₁  ^I  ^{I tr}  ^{C . C}  

2

1 ₂

1

2  

 ^I ₁ ^{, I} ₂ 



Function of invariants of or E C



Isotropic material so

ISOTROPIC, ISOTHERM

incompressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

E I I

E I I

E

2 2

1

1









 









 







Tensors to calculate independently Scalars

I c F E . sym .

F ^t 















 





whatever c

19

E I I

E I I

E

2 2

1

1









 









 





 2 I I 2 C

E I

1

2

 

 I 

E 2 I

 



I c F

F F I F

F I F

I I

t t

t

 



 









 

 

 









 





 2  2 . 2 . . .

2 2

1 1





I c F E . sym .

F ^t 















 





incompressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

c

whatever

I c F

F F I F

F I F

I I

t t

t

 



 









 

 

 









 





 2  2 . 2 . . .

2 2

1 1





 ^I ₁ ^{, I} ₂ 



Left Cauchy-Green stretch tensor

F . F B 

incompressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

c

whatever

F E .

sym .

F  





 









 



 ^  ^{?, ?, ?} 

Invariants of E compressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

21

  ^F

J  det F J

F

1

 ^det   ^{F  1}

F F

C 

. ^I

^{ tr}   ^{C I}  ^{I tr}   ^{C . C} 

2 1

2





  ^{E  }  ^I ₁ ^{, I} ₂ ^{, J} 



compressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

  ^E    ^I 1 ^{, I} 2 ^{, J} 

 ^{. F}

sym E .

F  





 









 





F E .

J J

E I E I

I . I

F

2 1

1

 





 





 





 













 









 









 





Complex derivations!

compressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

23

s I

p 



 

Hydrostatic pressure related to J

Deviatoric component

Releted to I

and I

 

3

 p   tr

compressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

 





 





 

 









 

 

 









 







 F F F F

F I I F

I I Dév

J

s 2 .

.

. .

2 2

1 1

1



p J





 

 

  ^E    ^I 1 ^{, I} 2 ^{, J} 



compressible hyperelastic behaviour

ISOTROPIC, ISOTHERM

25

Common hyperelastic

models

27

    ^

^

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  ₀  2C ₁₀

Initial compressibility modulus

1 0

2 k  D

Ψ E = Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽)

Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽) ^𝐽

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

    ^

^

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 Ψ E = Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽)

Ψ(ഥ 𝐼 ₁ , ഥ 𝐼 ₂ , 𝐽) ^𝐽