Nonlinear Elasticity of Soft Tissues

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

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
For a reversible process, the internal entropy is
constant and the change in total entropy change
in external entropy

• 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
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
Cauchy Stress
Nominal Stress
Second Piola-Kirchhoff Stress
9
Hyperelastic Constitutive Law for Finite
Deformations
Second Piola-Kirchhoff Stress
Cauchy Stress
10
2-D ExampleExponential 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 1986108189-192
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 1987207-17
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
1998120(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