strain energy density function
DESCRIPTION
strain energy density functionTRANSCRIPT
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