Hyperelasticity

Tom Scarpas

Continuum Mechanics

Continuum Mechanics 2 / 24

Clausius-Planck inequality

The internal dissipation should always be non-negative:

: 0P F

0 Mechanical process is reversible non-dissipative materials

0 Mechanical process is irreversible dissipative materials

1 2 n, , ,...F X

stored energy F X

internal variables

:P F Helmholtz free energy Work per unit ref. volume

Continuum Mechanics 3 / 24

Clausius-Planck inequality: dissipative materials

: 0P F

n

ii 1 i

: 0

n

ii 1 i

P : : 0FF

Coleman-Noll procedure :

elastic range

i0

inelastic range

PF

i0

i 1 2 n, , ,...F X

Evolution laws:

1 2 n, , ,...F X

Helmholtz free energy n

iii 1

: :FF

Continuum Mechanics 4 / 24

0 Mechanical process is reversible

Clausius-Planck inequality:

: : 0F

P F P FF

FP

FHYPERELASTIC

materials

:P F Work per unit ref. volume

Strain energy function corresponds to

work done by stresses from initial to

final configuration 0

t

t

: dtP F

hyperelastic materials

Continuum Mechanics 5 / 24

Hyperelasticity

Objectivity :

F

Continuum Mechanics 6 / 24

Hyperelasticity

F F

Objectivity

T T

F F R F R RU U

C

QSince is arbitrary: TQ R Objectivity condition

F

F

F QF

a rotation tensor : QF

2since : C U

Continuum Mechanics 7 / 24

Hyperelasticity

???PF

C

: P FF

C

:t

FC

CC

T T:CF F F F

symmetric

T: :S A S A

T: :A BC B A C

T2 : 2 :F F F FC C

T T:

F F F FC

: 2 :F F C FF C

F

2FF C

2P FF

C

C

1 T 1 T J 2JPF F FC

Continuum Mechanics 8 / 24

Isotropic hyperelastic response

Isotropy Response of material same in all directions

Continuum Mechanics 9 / 24

Isotropic hyperelastic response

Isotropy

d dx XF

d dXX QdX

dxd dx F X

dX

F

F

1F F FQ

1F FQ

d d dd X Q FF Xx XF

Isotropic

Continuum Mechanics 10 / 24

Isotropic hyperelastic response

T T T T

T T

C C

F F Q F FQ

Q CQ

TSet :Q R C RCR b

d dx XF

d dXX QdX

dxd dx F X

dX

F F 1F FQ

is an isotropic scalar valued

tensor function

TC RCRi.e.:

Isotropy

TR R

Continuum Mechanics 11 / 24

Representation theorem for invariants

1 2 3 1 2 3I ,I ,I I ,I ,IC C C C b b b

Isotropic scalar valued tensor functions,

are invariant under a rotation

2 2 21 1 2 3 1

2 2 2 2 2 2 2 21 12 22 1 1 2 2 3 1 3 1 2

2 2 2 23 1 2 3 3

I : : I

I I : I : I

I det J det I

C C I b I b

C C C b b b

C C b b

i i 1,...3 principal stretches

TC RCR

Can be expressed in terms of the

principal invariants of their arguments

Continuum Mechanics 12 / 24

Strain energy functions

11

22 1

2 13

II

II I

I J

C

C

C

IC

I CC

C

1 2 2 31

I 1 I I 3 I

11 2 3

2 I I

s s s

S I C C

I C C

1 2 3 1 31

c 1 I ,I ,I I 3 I 12

Neo-Hookean

21 2 3 1 3 3

3

I1 1I , I , I c I 3 I 1 1 c 3 I 1

2 2 I

; c 0,1 ; with & : Lam material constants2 1 2

: Poisson ratio

Blatz-Ko

material

i

3

iIi 1

ICC

C2SC

Continuum Mechanics 13 / 24

Blatz-Ko Neo-Hookean

0

1

3

Ic 1 c

I

3

1 c

I

23 3

3

Ic I 1 c I

I3I

1s

2s

3s

1 2 2 31

I 1 I I 3 I

11 2 3

2 I I

s s s

S I C C

I C C

General stress relation for isotropic hyperelastic materials

1 TJ F FS

Tom Scarpas

Hyperelasticity in principal directions

Continuum Mechanics

Continuum Mechanics 15 / 24

Hyperelasticity in principal directions

2 2 21 1 2 3

2 2 2 2 2 2 212 1 1 2 2 3 1 32

2 2 2 23 1 2 3

I :

I I :

I det J

C C I

C C C

C C

32i i i

i 1

C L L3

1 2i i i

i 1

C L L3

i ii 1

I L L

1 2 2 3

1I 1 I I 3 I2 2 I IS I C CC

1 2 3

32 2 2

I I 1 i I i i ii 1

2 2 I 2JS L L

Continuum Mechanics 16 / 24

Hyperelasticity in principal directions

2 2 21 1 2 3

2 2 2 2 2 2 212 1 1 2 2 3 1 32

2 2 2 23 1 2 3

I :

I I :

I det J

C C I

C C C

C C

1 2 3

32 2 2

I I 1 i I i i ii 1

2 2 I 2JS L L

12i

221 i2

i2

32 2i i

I1

II

I J

2 2 21 2 3i i i

2i

3

I 1 I 2 I 3 i ii 13 3

i i i i ii 1 i 1

2 I 2 I 2 I

2 S

S L L

L L L L

Continuum Mechanics 17 / 24

Hyperelasticity in principal directions

i 2i

S 2

3

i ii 1 i i

1S L L

Material description

Spatial description

1 2 3J

i2

i i i i

12 2

2 i i

1

1 TJ FSF

Continuum Mechanics 18 / 24

Hyperelasticity in principal directions

i 2i

S 2

3

i ii 1 i i

1S L L

Material description

1 TJ F SF

Spatial description

ii

iJ

31

i i iii 1

J l l

i2

i i i i

12 2

2 i i

1

1 2 3J

Continuum Mechanics 19 / 24

Example

23 3 2

1 2 3 i ii 1 i 1

, , ln ln2

Stretch based hyperelastic material

3i

i iii 1 Jl l i

i i i i i

ln 1

ln ln

3

i iii 1

1

J lnl l

3

i i ii i 1

1 1ln 2 ln

J ln J

Continuum Mechanics 20 / 24

Example

23 3 2

1 2 3 i ii 1 i 1

, , ln ln2

Stretch based hyperelastic material

3

i i ii i 1

1 1ln 2 ln

J ln J

3

i i i

i 1

Compare with : 2 : small strain theory

Continuum Mechanics 21 / 24

Example

2

3 3 2

1 2 3 i ii 1 i 1

, ,2

No-tension linear hyperelastic material

3i

i iii 1 Jl l

*i i

* *i i i

3 3 3

i i ii 1 i 1 i 1

with : 1

1

2

1

2

i

Continuum Mechanics 23 / 24

Reference

No-tension non-linear hyperelastic material

A continuum model for granular materials taking

into account the no-tension effect

by Vu-Hieu Nguyen et al.

Mechanics of Materials, Vol. 35, 2003, pp. 955-967

Continuum Mechanics 24 / 24

Hyperelasticity

Tom Scarpas

Continuum Mechanics

Continuum Mechanics 26 / 24

31 T 1 T

i ii ii 1

1J JF SF F FL L

Hyperelasticity in principal directions

A u v Au v

Tu v A u A v

i i iFL l

31 T

i ii ii 1

1J F FL L

31 T

i ii ii 1

1J F FL L

31

i ii ii 1

1J F FL L

31

i i i ii ii 1

1J l l

31

i i iii 1

J l l