strain energy density function

Strain Energy Density

Hyperelasticity

BME 615

University of Wisconsin

Review of salient information

• Return to finite elasticity and recall:– Stretch – Finite stress – Finite strain

• Note: to simplify models we assume– Incompressibility – Pseudoelastic behavior

Biaxial Stress and Strain(Fung, p. 299, Humphrey & Delange p. 285)

Principal stretches in principal material directions (figure from Michael Sacks)

Recall from previous notes

Principal stretches

(single subscript)

10

11 L

L

20

22 L

L

Figure from Fung “Biomechanics”

Finite Strain

In Lagrangian (material) reference system, define Green (St. Venant) strain

12

1

22

1210

210

21

1

L

LLE

12

1

22

2220

220

22

2

L

LLE

In Eulerian (spatial) reference system, define Almansi (Hamel) strain

21

21

210

21

1

11

2

1

2 L

LLe

22

22

220

22

2

11

2

1

2 L

LLe

Conjugate Stresses(for finite deformation analysis)

thicknesses of deformed and original tissue

0and densities of the deformed and original tissue (assumed equal if tissue is ~incompressible)

Cauchy stress (Eulerian reference system)

hL

Fs

2

1111

hL

Fs

1

2222

Lagrangian stress (or 1st Piola Kirchhoff stress)

2nd Piola Kirchhoff stress (Lagrangian reference system)

0h and h

Little physical meaning

Unloaded shape

“True” stress

Deformation gradient tensor F

1 1 1

1 2 31

2 2 22

1 2 33

3 3 3

1 2 3

0 0

0 0

0 0

x x x

X X X

x x x

X X X

x x x

X X X

F

For incompressibility,

3 1 2det 1 1/or F

For principal stretches

Right Cauchy-Green deformation tensor

For a deformation state in which 1,2,3 are principal axes,

invariants of are identical. They are:

TC F F

TB FF

andB C

2 2 21 1 2 3

2 2 2 2 2 22 1 2 2 3 3 1

2 2 23 1 2 3

I

I

I

Left Cauchy-Green deformation tensor(or Finger tensor)

Strain Energy Density (Hyperelasticity)

• Strain energy per unit of initial (or undeformed) volume W

– Area between the stress strain curve and the strain axis from energy conjugates,

– Often formulated in Lagrangian coordinates.

– (Note that Fung defines strain energy per unit mass Wm so he must multiply by to get strain energy per unit volume.)

• For a purely elastic material,

– Can derive stresses from the stored elastic energy

– Strain energy density is a scalar, so it is objective, i.e. frame invariant, but its effect on stress can easily be computed for any frame of reference.

0

x

Consider the case of a linearly elastic material in 1-D with a modulus of E

xxxx E

The stored energy W is

2

2

1

2

1xxxxxx EW

xx xxxx

dWE

d

Alternatively, area between the stress strain curve and the stress axis is the complementary strain energy density W*

2*

2

1

2

1xxxxxx E

W

* 1xx xx

xx

dW

d E

x

Expand to 3D, linearly elastic system

3,2,1, ji21 1

2 2ij ij ijij ijW C

where

ijijC is the stiffness coefficient in a 4th order constitutive tensor

ijijijij

ij CW

ijij

W

*

If behavior is non-linear, we still take derivatives as above but that will yield a more complicated set of terms for stress and strain

Strain energy of system must be computed from energy conjugates

(or equivalent from other finite metrics)

• Often formulated with Green-Lagrange strains Eij and 2nd Piola-Kirchhoff stresses Sij.

• This approach uses a strain energy density function and its use in mechanics is called “hyperelasticity”.

• For many materials or tissues, linearly elastic models do not accurately describe the observed behavior for large deformations. o Example: Rubber, whose stress-strain relationship can be

defined as non-linearly elastic, isotropic and generally independent of strain rate.

o Hyperelasticity models stress-strain behavior such materials.

o Biolological tissues are also often modeled via hyperelasticity assuming “pseudoelastic” behavior.

General Stress-Strain Relations for HyperelasticityLagrangian Stress (1st Piola-Kirchhoff Stress)

( )W F is the strain energy density function,

T

Thenij

ij

W Wor T

F

T

F

F

In terms of Green strain

ij ikkj

W Wor T F sum on k

E

T F

E

In terms of the right Cauchy-Green deformation tensor

2 2ij ikkj

W Wor T F sum on k

C

T F

C

is the 1st Piola-Kirchhoff stress tensor

is the deformation gradient

Compare to

Cauchy Stress

In terms of Green strain

1 1,T

ij ik jlkl

W Wor s F F sum on k l

J J E

s F F

E

In terms of the right Cauchy-Green deformation tensor

2 2,T

ij ik jlkl

W Wor s F F sum on k l

J J C

s F F

C

Similarly, Cauchy stress is given by

Note: J is known as Jacobian determinant

Cauchy stress in terms of invariants - 1Strain energy (a scalar) must be invariant to reference system. Hence, it can be equivantly formulated from principal stretches or from invariants of the deformation tensors. For isotropic hyperelastic materials, Cauchy stress can be expressed in terms of invariants of left or right Cauchy-Green deformation tensor or principal stretches below.

1 31 2 2 33

ˆ ˆ ˆ ˆ22

W W W WI I

I I I II

s B BB 1

1 1 22/3 4/31 2 1 2 2

2 1 1 12

3

W W W W W WI I I

J J I I I I J I J

s B 1 BB 1

2/3 2 2 21 1 1 1 2 3

4/3 2 2 2 2 2 22 2 2 1 2 2 3 3 1

detI J I I J

I J I I

Fwhere

Equivalent functions but re-parameterized

Cauchy stress in terms of invariants - 2

For isotropic hyperelastic materials, Cauchy stress can be expressed in terms of invariants of left or right Cauchy-Green deformation tensor or principal stretches.

where the diadic product or outer product above is defined as

1 1 1 1 2 1 3

2 1 2 3 2 1 2 2 2 3

3 3 1 3 2 3 3

u u v u v u v

u v v v u v u v u v

u u v u v u v

u v

1 1

1 1 0 0

0 1 0 0 0 0 0

0 0 0 0

n n

Thus,

etc.

Inner product makes vectors into a scalar

Outer product makes vectors into a matrix

Saint Venant-Kirchhoff Model

Simplest hyperelastic model is Saint Venant-Kirchhoff which is extension of the Lame’ linearly elastic, isotropic model for large deformations.

2tr S Ε I E

where S is the 2nd Piola-Kirchhoff stress tensor

E is the Green-Lagrange strain tensor

I is the unit tensor

2 2( )2

W tr tr E E E

ijij

WS

E

2nd Piola-Kirchhoff stress can be derived from the relation

Strain-energy density function for the St. Venant-Kirchhoff model is

are the Lame’ constantsand

Note: this is a scalar!

is right Cauchy-Green deformation tensor

Neo-Hookean Model

1 1 1

1( 3)

2W GI C I

A neo-Hookean solid is isotropic and assumes that the extra stresses due to deformation are proportional to the left Cauchy-Green deformation tensor

B is Finger tensor

s p G I B so that 211 1s p G etc.

where s is Cauchy stress tensor

I is unity tensor

T Tand C F F B FF

C

F is deformation gradient

p is pressure

G is the shear modulus

The strain energy for this model is:

where 2 2 21 1 2 3I tr B

This model has only one coefficient and is used for incompressible media

Note: p doesn’t contribute to SED in incompressible materials but does to stress

Note this is formulated so derivatives of stretch give T stress

Mooney-Rivlin Model

A Mooney-Rivlin solid is a generalization of the neo-Hookean model, where the strain energy W is a linear combination of two invariants of the Finger tensor B

1 1 2 23 3W C I C I

1I and 2I are 1st and 2nd invariants of the Finger tensor

are constants that define the isotropic material.1 2C and C

Note above SED is formulated such that: etc. (- pressure)

11 22 2s p C C I B B

For example, for principal direction 1

1 1 2 23 3W C I C I

Mooney-Rivlin Model

Mooney-Rivilin equation is for 3D. Why? How would it change for 1D & 2D?

3I is associated with compressibility

3 1I for an incompressible medium

3I does not enter the equation unless tissue is assumed compressible

1

1

2C G (where G is shear modulus)

Note:

If , we obtain a neo-Hookean solid as a special case of Mooney-Rivlin

M-R is often formulated for Cauchy stress from Finger tensor

2 0C

Mooney-Rivlin Model

• This model (in the above form) is incompressible.

• It can be modified to admit compressibility if necessary.

• This model and variations of it have been frequently used for biological tissues.

– For example, the ground substance in a ligament/tendon model by Quapp and Weiss (1998) is modeled by these terms. Collagen fibers were added by superposition of typical exponential formulation in fiber direction.

• The above model was proposed by Melvin Mooney and Ronald Rivlin separately in 1952.

1 1 2 23 3W C I C I

Mooney-Rivlin vs. Neo-Hookean Models

figure from work by M. Sacks.

Ogden Model

• Developed by Ray Ogden in 1972• A more general formulation to fit more complex material/mechanical

behaviors. • It is an extension of the previous models and generally considers

materials that can be assumed to be isotropic, incompressible, and strain-rate independent.

• It can be expressed in terms of principal stretches as:

1 2 3 1 2 31

, , 3p p p

Np

p p

W

, ,p pN

Since the material is assumed incompressible the above can be written as:

1 2 1 2 1 21

, 3p p p p

Np

p p

W

are material constants

Ogden Model

When

3N the behavior of rubbers can be described accurately

1, 2N Ogden model reduces to a Neo-Hookean model

Ogden model reduces to a Mooney-Rivilin model

Using Ogden model, Cauchy stresses can be computed as:

ii ii

Ws p

1 2 3 1 2 31

, , 3p p p

Np

p p

W

Fung Model (for large stretches)

Because mechanical behavior for biological tissues is highly non-linear and anisotropic, Fung postulates a useful SED function

11

2QW c e

2 2 2

1 11 2 22 3 33 4 11 22 5 22 33 6 33 11

2 2 2 2 2 27 12 21 8 23 32 9 13 31

2 2 2Q c E c E c E c E E c E E c E E

c E E c E E c E E

ic

c

ijS ijE

where for orthotropic tissues

are Kirchhoff stresses and Green strains

are material constants that govern nonlinearity of the tissue (larger is more nonlinear)

is a scaling constant (larger is stiffer)

Fung Model

12

1 QecW

The relationships for stress and strain from SED still hold i.e.

ijij E

WS

ij

ij S

WE

*

So, for Fung’s strain energy function above

33622411111

11 EcEcEcceE

WS Q

12 7 1212

QWS ce c E

E

2 2 2

1 11 2 22 3 33 4 11 22 5 22 33 6 33 11

2 2 2 2 2 27 12 21 8 23 32 9 13 31

2 2 2Q c E c E c E c E E c E E c E E

c E E c E E c E E

Fung Model

Fung Model(rabbit abdominal skin)

Handles highly non-linear and anisotropic behaviors very well (in pseudoelastic sense).

Complex – requires many constants to fit observed behaviors.

Biaxial Stress and Strain

Figure from Michael Sacks

Structural models with SED

• see Michael Sacks paper and formulation as an example.

Reference configurationand thickness H

pressurized configurationand thickness h

BVP example of SED

Assume a section of a lung is approximately semi-hemispherical and undergoes unrestricted inflation (like a balloon) under internal pressure.

Lung example of BVP with SED – 1

Consider force equilibrium for the pressurized section of lung (under pressure p).The force to the right is:

The force to the left is:

Equating these produces a relationship between pressure and membrane stress

Stretch ratios are equal in all directions and from expanded surface we obtain

where s is the Cauchy stress

Equation 1

Note that right F goes up by square of radius and left goes up linearly

Lung example of BVP with SED – 2Assume tissue incompressibility, hence the volume in the reference configuration V is conserved in the inflated configuration v

From the information given above use the Mooney-Rivlin model to compute and plot both the Cauchy stress and inflation pressure as a function of stretch. Assume plane stress; that is, membrane stress through the thickness is small and assumed to be zero.

Equation 2

Equation 3

Lung example of BVP with SED – 3

From equation 3, you can solve directly for hydrostatic pressure, which in turn, can be used in equation 2 for Cauchy stress in the lung tissue. Once you have an expression for stress, you can solve equation 1 for pressure.

Alternatively, if you know geometry and pressure, you could solve the inverse BVP to find material properties.

Expectationsafter this section

• Know infinitesimal and finite descriptors of stress and strain

• Know what hyperelastic (SED) functions are and how to get stresses or strains from them

• Know simple constitutive formulations for hyperelastic media– St. Venant-Kirchhoff– Neo-Hookean– Mooney Rivlin– Ogden– Fung