1. Nonlinear Elasticity of Soft Tissues
Soft tissues are not elastic — stress depends on strain and the history of strain
However, the hysteresis loop is only weakly dependent on strain rate
It may be reasonable to assume that tissues in vivo are preconditioned
Fung: elasticity may be suitable for soft tissues, if we use a different stress-strain relation for loading and unloading – the pseudoelasticity concept
a rationale for applying elasticity theory to soft tissues
Unlike in bone, linear elasticity is inappropriate for soft tissues; we need nonlinear finite elasticity
Feb 17, 2000 Completed most of simplification listed below. Still need to improve examples. Best to have fewer examples and to include examples of actual stress-strain relations calculated from them. Maybe even for an incompressible problem?
Feb2000: Need to greatly simplify this for undergraduate class. Coalesce slides 2-8 into one or 2. The use of two different notations for internal energy is crazy. It might be easier just to start right away with the first law as shown on slide 5. It would not be hard to convert slide one to slide five in one or two steps. Don’t need second law, just substitute for dQ and keep going. Don’t need free energy either. Just use slide 6. Might end with current slide 4. Also reduce slide 14 to the essentials.
I made a start on these changes today in this version. You can lose the one called hyperelasticity
Insert some figures of measured and fitted curves for specific laws after the strain-energy function examples. Done for Karen’s ASME paper.
Consider hiding some of the strain energy function slides for lecture: Slide 9,

2. Two Definitions of Elasticity
The work done by the stress producing strain in a hyper-elastic material is stored as potential energy in a thermo-dynamically reversible process.
In words
In an elastic material the stress depends only on the strain.
Mathematically
Example
W is also called the strain energy
A linearly elastic (Hookean) material:

3. Conservation of Energy
Rate of change = Rate of Work + Rate of Heat of Internal Energy Done by Stresses Absorbed

4. Strain Energy
W is the strain-energy function; its derivative with respect to the strain is the stress.
This is equivalent to saying that the stress in a hyperelastic material is independent of the path or history of deformation.
Similarly, when a force vector field is the gradient of a scalar energy function, the forces are said to be conservative; they work they do around a closed path is zero.
The strain energy in an elastic material is stored as internal energy or free energy (related to entropy) …

5. Reversible Process q = d Q S q e ) , ( = I q e ) , ( = S
For a reversible process, the internal entropy is constant and the change in total entropy = change in external entropy q = d Q S e q e ij ) , ( = I q e ij ) , ( = S
Elastic stress arises from an increase in internal energy I or a decrease in specific entropy S with respect to strain; strain energy is stored as either or both of these:
Crystalline materials (e.g. collagen) derive stress from an increase in the internal energy between their bonds, and strain energy is equivalent to rI (a “perfect” material)
Rubbery materials (e.g. elastin) derive stress from a decrease in entropy, and strain-energy is equivalent to rF where F = I - qS is called the Helmholtz Specific Free Energy

6. Cauchy Stress Tensor is Eulerian
Cauchy’s formula: t(n) = n•T
In index notation: e1 e2 e3
Tij = ti•ej
T11
T12
T22
T33
T13
T31
T23
T21
T32 R S n a
Tij is the component in the xj direction of the traction vector t(n) acting on the face normal to the xi axis in the deformed state of the body. The "true" stress.

7. Lagrangian Stress Tensors
The (half) Lagrangian Nominal stress tensor S
SRj is the component in the xj direction of the traction measured per unit reference area acting on the surface normal to the (undeformed) XR axis. Useful experimentally
S = detF.F-1.T  ST
The symmetric (fully) Lagrangian Second Piola-Kirchhoff stress tensor
Useful mathematically but no direct physical interpretation
For small strains differences between T, P, S disappear

8. Example: Uniaxial Stress
undeformed length = L
undeformed area = A
deformed length = l
deformed area = a L A a F l
Cauchy Stress
Nominal Stress
Add diagram – done Jan 2000
Second Piola-Kirchhoff Stress

9. Hyperelastic Constitutive Law for Finite Deformations
Second Piola-Kirchhoff Stress
Cauchy Stress

10. 2-D Example: Exponential Strain-Energy Function
Stress components have interactions

11. 3-D Orthotropic Exponential Strain-Energy Function
From: Choung CJ, Fung YC. On residual stress in arteries. J Biomech Eng 1986;108:

12. 2-D Orthotropic Logarithmic Strain-Energy Function
From: Takamizawa K, Hayashi K. Strain energy density function and uniform strain hypothesis for arterial mechanics. J Biomech 1987;20:7-17

13. Isotropic Strain-Energy Functions
Let, W =W (I1, I2, I3)
where, I1, I2, I3 are the principal invariants of CRS
In component notation:

14. Examples: Isotropic Laws

15. Examples: Anisotropic 2-D Laws

16. Transversely Isotropic Laws for Myocardium

17. Exponential Law for Mitral Valve Leaflets
May-Newman K, Yin FC. A constitutive law for mitral valve tissue. J Biomech Eng 1998;120(1):38-47.

18. Examples: 3-D Orthotropic Laws

19. Incompressible Materials
Stress is not completely determined by the strain because a hydrostatic pressure can be added to Tij without changing CRS. The extra condition is the kinematic incompressibility constraint
To avoid derivative of W tending to 
p is a Lagrange multiplier (a negative stress)

20. Nonlinear Elasticity: Summary of Key Points
Soft tissues have nonlinear material properties
Because strain-rate effects are modest, soft tissues can be approximated as elastic: pseudoelasticity
Strain energy W relates stress to strain in a hyperelastic material; it arises from changes in internal energy or entropy with loading
For finite deformations it is more convenient to use the Lagrangian Second Piola-Kirchhoff stress
Exponential strain-energy functions are common for soft tissues
For isotropic materials, W is a function of the principal strain invariants
Transverse isotropy and orthotropy introduce additional invariants
For incompressible materials an additional pressure enters