1. Strain Energy Density
Hyperelasticity
BME 615
University of Wisconsin

2. Review of salient information
Return to finite elasticity and recall:
Stretch
Finite stress
Finite strain
Note: to simplify models we assume
Incompressibility
Pseudoelastic behavior

3. Biaxial Stress and Strain
(Fung, p. 299, Humphrey & Delange p. 285)
Principal stretches in principal material directions (figure from Michael Sacks)

4. Recall from previous notes
Principal stretches
(single subscript)
Figure from Fung “Biomechanics”

5. Finite Strain
In Lagrangian (material) reference system, define Green (St. Venant) strain
In Eulerian (spatial) reference system, define Almansi (Hamel) strain

6. Conjugate Stresses (for finite deformation analysis)
thicknesses of deformed and original tissue
densities of the deformed and original tissue
(assumed equal if tissue is ~incompressible)
Cauchy stress (Eulerian reference system)
“True” stress
Lagrangian stress (or 1st Piola Kirchhoff stress)
Unloaded shape
2nd Piola Kirchhoff stress (Lagrangian reference system)
Little physical meaning

7. Deformation gradient tensor F
For principal stretches
For incompressibility,

8. Right Cauchy-Green deformation tensor
Left Cauchy-Green deformation tensor (or Finger tensor)
For a deformation state in which 1,2,3 are principal axes,
invariants of are identical. They are:

9. 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.

10. Consider the case of a linearly elastic material in 1-D with a modulus of E
The stored energy W is
Alternatively, area between the stress strain curve and the stress axis is the complementary strain energy density W*

11. Expand to 3D, linearly elastic system
where
is the stiffness coefficient in a 4th order constitutive tensor
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

12. 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.
Example: Rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic and generally independent of strain rate.
Hyperelasticity models stress-strain behavior such materials.
Biolological tissues are also often modeled via hyperelasticity assuming “pseudoelastic” behavior.

13. General Stress-Strain Relations for Hyperelasticity Lagrangian Stress (1st Piola-Kirchhoff Stress)
is the strain energy density function,
is the deformation gradient
is the 1st Piola-Kirchhoff stress tensor
Then
In terms of Green strain
Compare to
In terms of the right Cauchy-Green deformation tensor

14. Cauchy Stress Similarly, Cauchy stress is given by
In terms of Green strain
In terms of the right Cauchy-Green deformation tensor
Note: J is known as Jacobian determinant

15. Cauchy stress in terms of invariants - 1
Strain 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.
Equivalent functions but re-parameterized
where

16. 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
Inner product makes vectors into a scalar
Outer product makes vectors into a matrix
Thus,
etc.

17. Saint Venant-Kirchhoff Model
Simplest hyperelastic model is Saint Venant-Kirchhoff which is extension of the Lame’ linearly elastic, isotropic model for large deformations.
where
is the 2nd Piola-Kirchhoff stress tensor
is the Green-Lagrange strain tensor
is the unit tensor
are the Lame’ constants
Strain-energy density function for the St. Venant-Kirchhoff model is
Note: this is a scalar!
2nd Piola-Kirchhoff stress can be derived from the relation

18. Neo-Hookean Model p is pressure G is the shear modulus
A neo-Hookean solid is isotropic and assumes that the extra stresses due to deformation are proportional to the left Cauchy-Green deformation tensor
so that
etc.
Note: p doesn’t contribute to SED in incompressible materials but does to stress
p is pressure
where
is Cauchy stress tensor
is unity tensor
G is the shear modulus
is Finger tensor
is deformation gradient
is right Cauchy-Green deformation tensor
The strain energy for this model is:
Note this is formulated so derivatives of stretch give T stress
where
This model has only one coefficient and is used for incompressible media

19. 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 .
and
are 1st and 2nd invariants of the Finger tensor
are constants that define the isotropic material.
Note above SED is formulated such that:
etc. (- pressure)

20. Mooney-Rivlin Model Note:
Mooney-Rivilin equation is for 3D. Why? How would it change for 1D & 2D?
is associated with compressibility
for an incompressible medium
does not enter the equation unless tissue is assumed compressible
(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
For example, for principal direction 1

21. 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.

22. Mooney-Rivlin vs. Neo-Hookean Models
figure from work by M. Sacks.

23. 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:
Still math prof. at U of Glasgow. Published as a grad student.
are material constants
Since the material is assumed incompressible the above can be written as:

24. Ogden Model When the behavior of rubbers can be described accurately
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:

25. Fung Model (for large stretches)
Because mechanical behavior for biological tissues is highly non-linear and anisotropic, Fung postulates a useful SED function
where for orthotropic tissues
are material constants that govern nonlinearity of the tissue (larger is more nonlinear)
is a scaling constant (larger is stiffer)
are Kirchhoff stresses and Green strains

26. Fung Model
The relationships for stress and strain from SED still hold i.e.
So, for Fung’s strain energy function above

27. Fung Model

28. 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.

29. Biaxial Stress and Strain
Figure from Michael Sacks

30. Structural models with SED
see Michael Sacks paper and formulation as an example.

31. BVP example of SED Reference configuration and thickness H
pressurized configuration
and thickness h
Assume a section of a lung is approximately semi-hemispherical and undergoes unrestricted inflation (like a balloon) under internal pressure.

32. 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:
Note that right F goes up by square of radius and left goes up linearly
The force to the left is:
where s is the Cauchy stress
Equating these produces a relationship between pressure and membrane stress
Equation 1
Stretch ratios are equal in all directions and from expanded surface we obtain

33. Lung example of BVP with SED – 2
Assume 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.
Note: surface area of a sphere is 4 pi r squared
Equation 2
Equation 3

34. 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.

35. Expectations after 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