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Z. Suo
ELASTICITY OF RUBBER-LIKE MATERIALS
In the notes on the general theory of finite deformation
(http://imechanica.org/node/538), we have left the free energy function
unspecified. The notes here describe free energy function commonly used to
describe the elasticity of rubber-like materials.
Synopsis of the general theory of elasticity. Let us recall key results
from
the
notes
on
the
general
theory
of
elasticity
(http://imechanica.org/node/538). A homogeneous deformation is specified by
the deformation gradient F, a linear operator that maps a straight segment in the
reference state to a straight segment in the current state. The homogeneous
deformation causes the straight segment to stretch and rotate. The deformation
gradient generalizes the definition of stretch, and is a measure of deformation.
An elastic material is specified by the Helmholtz free energy as a function
of the deformation gradient:
( )
W =W F .
Once this function is known, the nominal stress is given by
siK =
( ).
∂W F
∂FiK
This equation relates the stress and deformation gradient, and is known as the
equation of state.
The free energy is invariant when the block undergoing a rigid-body
rotation. Thus, the free energy depends on F through the deformation tensor
C KL = FiK FiL .
The tensor C is positive-definite and symmetric. In three dimensions, this tensor
has six independent components. Thus, to specify an elastic material model, we
need to specify the free energy as a function of the six variables:
W =W C .
( )
For a given material, such a function is specified by a combination of
experimental measurements and theoretical considerations. Trade off is made
between the amount of effort and the need for accuracy.
A combination of the above expressions gives that
∂W C
.
siK = 2FiL
∂C KL
( )
The true stress relates to the nominal stress as
Jσ ij = siK F jK .
A combination of the above two equations gives that
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Finite Deformation: Special Cases 1
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σ ij = 2FiL F jK
Z. Suo
( ).
∂W C
∂C KL
The true stress is symmetric, σ ij = σ ji .
Isotropy.
For an isotropic material, the free-energy density is
unchanged when the coordinates rotate in the reference state. When the
coordinates rotate, however, the individual components of the tensor C will
change. How do we write the function W (C ) for an isotropic material?
For any symmetric second-rank tensor C, the six components of the
tensor form three scalars:
α = C KK ,
β = C KLC KL ,
γ = C KLC LJ C JK .
These scalars are formed by combining the individual components of the tensor
in ways that make all indices dummy. The three scalars remain unchanged when
the coordinates rotate, and are known as the invariants of the tensor C.
To specify model of an isotropic elastic material, we need to prescribe the
free-energy density as a function of the three invariants of the Green deformation
tensor:
W = f (α , β , γ ) .
Once this function is specified, we can derive the stress by using the chain
rule:
siK =
∂W (F ) ∂f ∂α
∂f ∂β
∂f ∂γ
.
=
+
+
∂FiK
∂α ∂FiK ∂β ∂FiK ∂γ ∂FiK
Derivatives of this kind are recorded in many textbooks on continuum mechanics.
Exercise. Invariants of a symmetric second-rank tensor can be written
in many forms. In many textbooks, the invariants are introduced by using the
characteristic equations. Relate invariants introduced by the characteristic
equations and those written above.
Incompressibility. Subject to external forces, elastomers can undergo
large change in shape, but very small change in volume. A commonly used
idealization is to assume that such materials are incompressible, namely,
det F = 1 .
In arriving at the relation siK = ∂W (F )/ ∂FiK , we have assumed that each
component of FiK can vary independently.
However, the condition of
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Z. Suo
incompressibility places a constraint among the components. To enforce this
constraint, we replace the free-energy function W (F ) by a function
W (F ) − Π(det F − 1) ,
with Π as a Lagrange multiplier. We then allow each component of FiK to vary
independently, and obtain the stress from
∂
[W (F ) − Π (det F − 1)].
siK =
∂FiK
Recall an identity in the calculus of matrix:
∂ det F
= H iK det F ,
∂FiK
where H is defined by H iK FiL = δ KL and H iK F jK = δ ij . For a proof see p. 41 of
Holzapfel.
Thus, for an incompressible material, the stress relates to the deformation
gradient as
∂W (F )
siK =
− ΠH iK .
∂FiK
This equation, along with the constraint det F = 1 , specifies a material model for
incompressible elastic materials. The Lagrange multiplier Π is not a material
parameter. Rather, Π is to be solved as a field for a given boundary-value
problem. When the deformation of the body is inhomogeneous, Π is in general
also inhomogeneous.
In terms of the true stress, the equation of state is
σ ij = F jK
( ) − Πδ
∂W F
ij
∂FiK
.
The Lagrange multiplier acts like a hydrostatic pressure.
Arruda-Boyce eight-chain model. Next we consider a network of
polymers, crosslinked by covalent bonds. Many models exist in the literature to
derive the behavior of the network from that of the individual polymers. In the
Arruda and Boyce model, the network is represented by many cells, each cell by a
rectangle, and the polymers in the cell by the eight half diagonals. When the
network is unstressed, the cell is a unit cube, and the length of a half diagonal is
3 / 2 . When the network is stretched, the cell is a rectangle with sides λ1 , λ2
(
and λ3 , and the length of a half diagonal is λ21 + λ22 + λ23
end-to-end length of each chain is
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1/2
)
/2. Consequently, the
Finite Deformation: Special Cases 3
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Λ=
Z. Suo
1 2 2
(λ1 + λ2 + λ23 )1/2 .
3
Let v be the volume per link, and nv be the volume per chain. The free
energy per unit volume of the network is
w(Λ )
.
W=
nv
The eight-chain model is described by a free energy function of the form W (Λ ).
• E.M. Arruda and M.C. Boyce, A three dimensional constitutive model for
the large stretch behavior of rubber elastic materials. Journal of the
Mechanics and Physics of Solids 41, 389-412 (1993).
Neo-Hookean material.
free-energy function:
A neo-Hookean material is defined by the
µ
F F −3 .
2 iK iK
Note that C KK = FiK FiK is an invariant of the Green deformation tensor C. The
material is also taken to be incompressible, namely,
det F = 1 .
The stress relates to the deformation gradient as
siK = µFiK − ΠH iK .
W=
(
)
Exercise.
In the notes on special cases of finite deformation
(http://imechanica.org/node/5065), several forms of free energy function are
given in terms of the principal stretches. They can be rewritten in terms of
deformation gradient, following the same procedure outlined above. Go through
this procedure for the Gent model.
Multiaxial stress. We next explore states of multiaxial stress. Of
course, the only way to really know stress-strain relations is to run tests, but tests
alone would be too time-consuming and quickly become impractical. We’ll have
to reduce the number of tests by some approximations. The art of making such
compromise between accuracy and labor is known as formulating constitutive
models. As an example, here we begin to describe this art for rubbers.
Hyperelastic materials. We can represent a material particle by a
rectangular block cut in the orientation of the three principal stresses. Let the
block be stretched in the three directions by λ1 , λ2 and λ3 , and the
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Finite Deformation: Special Cases 4
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corresponding nominal stresses be s1 , s2 and s3 .
(
Z. Suo
The complete stress-strain
) (
)
(
)
relations involve three functions s1 λ1 , λ2 , λ3 , s2 λ1 , λ2 , λ3 and s3 λ1 , λ2 , λ3 .
In
general, we can run tests to determine these functions. However, as we indicated
above, running test alone would be too time-consuming. Instead, we will
formulate a constitutive model on the basis that, in equilibrium, the work done by
the forces is all stored in the block as energy.
Consider a rectangular block of a material, lengths L1 , L2 and L3 in a
reference state. In a current state, the block is subject to forces P1 , P2 and P3 on
its faces, and is deformed into a block of lengths l1 , l2 and l3 . When lengths in
the current state change, each of the forces does work, respectively,
P1δl1 , P2δl2 , P3δl3 . We assume that the sum of the work done by the external forces
equals the change in the free energy. Let F be the Helmholtz free energy of the
block in the current state. When the block and the forces are in a state of
equilibrium, subject to a small variation of the sides of the block, the work done
by the forces equals the change in the free energy:
δF = P1δl1 + P2δl2 + P3δl3 .
This condition of equilibrium holds for arbitrary and independent small
variations δl1 , δl2 , δl3 .
P3
P2
P1
L3
L1
P1
l3
L2
l1
P2
l2
P3
current state
reference state
Let W be nominal density of the free energy, namely, the free energy of
the block in the current state divided by the volume of the block in the reference
state,
F
.
W=
L1 L2 L3
Define the nominal stresses by
s1 =
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P
P1
P
, s2 = 2 , s 3 = 3 .
L2 L3
L3 L1
L1 L2
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Z. Suo
Define the stretches by
λ1 =
l
l1
l
, λ2 = 2 , λ3 = 3 .
L1
L2
L3
Dividing the condition of equilibrium by the volume of the block in the
reference state, L1 L2 L3 , and recalling the definitions of nominal stress and
stretch, we obtain that
δW = s1δλ1 + s2δλ2 + s3δλ3 .
This condition of equilibrium holds for arbitrary and independent variations
δλ1 ,δλ2 ,δλ3 .
As a material model, we assume that the free-energy density is a function
of the three stretches:
W = W λ1 , λ2 , λ3 .
(
)
According calculus,
∂W (λ1 , λ2 , λ3 )
∂W (λ1 , λ2 , λ3 )
∂W (λ1 , λ2 , λ3 )
δW =
δλ1 +
δλ2 +
δλ3 .
∂λ1
∂λ2
∂λ3
A comparison of the two expressions for δW , we obtain that
⎡
∂W (λ1 , λ2 , λ3 )⎤
∂W (λ1 , λ2 , λ3 )⎤
∂W (λ1 , λ2 , λ3 )⎤
⎡
⎡
⎥δλ3 = 0
⎢s1 −
⎥δλ1 + ⎢s2 −
⎥δλ2 + ⎢s3 −
∂λ1
∂λ2
∂λ3
⎢⎣
⎥⎦
⎣
⎦
⎣
⎦
This condition of equilibrium holds for arbitrary and independent variations
δλ1 ,δλ2 ,δλ3 . Consequently, the stresses equal the partial derivatives:
s1 =
∂W (λ1 , λ2 , λ3 )
∂W (λ1 , λ2 , λ3 )
∂W (λ1 , λ2 , λ3 )
.
, s2 =
, s3 =
∂λ1
∂λ2
∂λ3
An elastic material whose stress-strain relation is derivable from a freeenergy function is known as a hyperelastic material. By contrast, an elastic
material whose stress-strain relation is not derivable from a free-energy function
is called a hypoelastic material.
The nominal stress on one face is
P
s1 = 1 .
L2 L3
The true stress on the same face is
σ1 =
P1
.
l2l3
Consequently, the true stress relates to the nominal stress by
s
σ1 = 1 .
λ2λ3
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Z. Suo
The true stress is derivable from the strain-energy function:
∂W (λ1 , λ2 , λ3 )
.
σ1 =
λ2λ3∂λ1
The other two components of true stress can be similarly obtained. Note that the
true stress is not work-conjugate to the stretch.
Incompressible, isotropic, hyperelastic material.
When a
material undergoes large deformation, the amount of volumetric deformation is
often small compared to the overall deformation. Consequently, we may neglect
the volumetric deformation, and assume that the material is incompressible.
A block of a material, of lengths L1 , L2 and L3 in the undeformed state, is
deformed into a rectangle of lengths l1 , l2 and l3 .
If the material is
incompressible, the volume of the block must remain unchanged, namely,
L1 L2 L3 = l1l2l3 , or
λ1λ2λ3 = 1 .
The incompressibility places a constraint among the three stretches: they cannot
vary independently. We may regard λ1 and λ2 as independent variables, so that
−1
λ3 = (λ1 λ2 ) .
−1
Taking differential of the constraint λ3 = (λ1 λ2 ) , we obtain that
δλ3 = λ1−2λ2−1δλ1 + λ1−1 λ2−2δλ2 .
Inserting the assumption of incompressibility into the condition of
equilibrium, δW = s1δλ1 + s2δλ2 + s3δλ3 , we obtain that
δW = (s1 − λ1−2λ2−1 s3 )δλ1 + (s2 − λ2−2λ1−1 s3 )δλ2 .
This condition of equilibrium holds for arbitrary and independent variations δλ1
and δλ2 .
As a material model, we assume that the free-energy density is a function
of the two stretches:
W = W (λ1 , λ2 ) .
According calculus,
δW =
∂W (λ1 , λ2 )
∂W (λ1 , λ2 )
δλ1 +
δλ2 .
∂λ1
∂λ2
A comparison of the two expressions for δW , we obtain that
⎡
⎡
∂W (λ1 , λ2 )⎤
∂W (λ1 , λ2 )⎤
−2 −1
−2 −1
⎢s1 − λ1 λ2 s3 −
⎥δλ1 + ⎢s2 − λ2 λ1 s3 −
⎥δλ2 = 0
∂λ1
∂λ2
⎣
⎦
⎣
⎦
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Z. Suo
This condition of equilibrium holds for arbitrary and independent variations
δλ1 , δλ2 . Consequently, once the function W (λ1 , λ2 ) is determined, the stressstretch relation is given by differentiation:
s1 − λ1−2 λ2−1 s3 =
∂W (λ1 , λ2 )
,
∂λ1
s2 − λ2−2 λ1−1 s3 =
∂W (λ1 , λ2 )
.
∂λ2
These relations, together with the incompressibility condition λ1λ2λ3 = 1 , replace
Hooke’s law and serve as the stress-strain relations for incompressible, isotropic
and hyperelastic materials.
For incompressible materials, the true stresses relate to the nominal
stresses as
σ 1 = λ1s1 , σ 2 = λ2s2 , σ 3 = λ3s3 .
Consequently, the stress-strain relations become
∂W (λ1 , λ2 )
,
σ 1 − σ 3 = λ1
∂λ1
σ 2 − σ 3 = λ2
∂W (λ1 , λ2 )
.
∂λ2
Because the material is incompressible, once the stretches are known, the
material model leaves the hydrostatic stress undetermined.
The function W (λ1 , λ2 ) can be determined by subjecting a sheet of a
rubber under various states of biaxial stress. The form of the function is
sometimes inspired by theoretical considerations. Here are some often used
forms.
Neo-Hookean model. This model is specified by the energy-density
function:
W=
µ
(λ
2
2
1
+ λ22 + λ23 − 3) .
The material is also taken to be incompressible, λ1 λ2λ3 = 1 .
Recall that
σ 1 − σ 3 = λ1
∂W (λ1 , λ2 )
.
∂λ1
−1
Inserting the constraint λ3 = (λ1 λ2 ) , we obtain the stress-stretch relation:
σ 1 − σ 3 = µ (λ21 − λ23 ).
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Z. Suo
Similarly, we obtain that
σ 2 − σ 3 = µ (λ22 − λ23 ).
A neo-Hookean material is characterized by a single elastic constant, µ .
The constant may be determined experimentally. For example, consider a bar in
a state of uniaxial stress:
σ 1 = σ , σ 2 = σ 3 = 0.
Let the stretch along the axis of loading be λ1 = λ . Incompressibility dictates that
the stretches in the directions transverse to the loading axis be λ2 = λ3 = λ−1 / 2 .
Inserting into the above stress-stretch relation, we obtain the relation under the
uniaxial stress:
σ = µ (λ2 − λ−1 ).
Recall that stretch relates to the engineering strain as
λ = l/L =1+e.
When the strain is small, namely, e << 1 , the above stress-stretch relation
reduces to
σ = 3µe .
Thus, we interpret 3µ as Young’s modulus and µ the shear modulus.
The form of strain-energy function of Neo-Hookean materials has also
emerged from a model in statistical mechanics. The model gives µ = NkT , where
N is the number of polymer chains per unit volume, and kT the temperature in
units of energy.
• L.R.G. Treloar, The Physics of Rubber Elasticity, Third Edition, Oxford
University Press, 1975.
Gent model. In an elastomer, each individual polymer chain has a finite
contour length. When the elastomer is subject no loads, the polymer chains are
coiled, allowing a large number of conformations. Subject to loads, the polymer
chains become less coiled. As the loads increase, the end-to-end distance of each
polymer chain approaches the finite contour length, and the elastomer
approaches a limiting stretch. On approaching the limiting stretch, the elastomer
stiffens steeply. This effect is absent in the neo-Hookean model, but is
represented by the Gent model:
⎛ λ2 + λ22 + λ23 − 3 ⎞
⎟ .
log⎜⎜ 1 − 1
⎟
2
J
lim
⎝
⎠
where µ is the small-stress shear modulus, and J lim is a constant related to the
W =−
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µJ lim
Finite Deformation: Special Cases 9
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(
(
2
2
)
The stretches are restricted as 0 ≤ λ21 + λ22 + λ23 − 3 / J lim < 1 .
limiting stretch.
2
1
Z. Suo
2
3
)
When λ + λ + λ − 3 / J lim → 0 , the Taylor expansion of the Gent model recovers
(
)
the neo-Hookean mode. When λ21 + λ22 + λ23 − 3 / J lim → 1 , the free energy diverges,
and the elastomer approaches the limiting stretch.
The material is also taken to be incompressible.
relations are
σ1 −σ3 =
1−
σ2 −σ3 =
1−
•
The stress-stretch
µ (λ21 − λ23 )
,
λ21 + λ22 + λ23 − 3
J lim
µ (λ22 − λ23 )
.
λ21 + λ22 + λ23 − 3
J lim
Gent, A. N., A new constitutive relation for rubber. Rubber Chemistry
and Technology, 1996, 69: 59-61.
Model a polymer as a freely-jointed chain. First consider an
individual polymer. Let Λ be the stretch of the polymer, namely, the end-to-end
length of the stretched polymer r divided by that of the released polymer r0 ,
namely,
r
Λ=
r0
The Helmholtz free energy of the polymer, w, is a function of the stretch, namely,
w = w(Λ ) .
Consider the model of freely-jointed chain of Kuhn and Grun. In this
model, a polymer chain is modeled by a sequence of links capable of free rotation
relative to each other. At a finite temperature, the relative rotations of the links
allow the polymer chain to fluctuate among a large number of conformations.
The statistics of the conformations gives the force-stretch behavior of the polymer
chain
⎛ 1
1 ⎞
Λ = n ⎜⎜
− ⎟⎟
⎝ tanh ζ ζ ⎠
where n is the number of links per polymer chain, and ζ the normalized force in
the polymer chain.
The Helmholtz free energy of the polymer is
⎛ ζ
ζ
w = nkT ⎜⎜
− 1 + log
sinh ζ
⎝ tanh ζ
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⎞
⎟⎟
⎠
Finite Deformation: Special Cases 10
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Z. Suo
where kT is the temperature in the unit of energy.
• W. Kuhn and F. Grun, Kolloidzschr. 101, 248 (1942).
• L.R.G. Treloar, The Physics of Rubber Elasticity, Third Edition, Oxford
University Press, 1975.
Next we represent each chain by using the freely-jointed chain model.
The Helmholtz free energy per unit volume of the elastomer is
kT
v
⎛ ζ
ζ ⎞
⎜⎜
⎟
− 1 + log
sinh ζ ⎟⎠
⎝ tanh ζ
These Equations defines the function W (λ1 , λ2 ) using two intermediate
W=
parameters: the stretch Λ in each chain and the normalized force ζ in each
chain. In one limit, ζ → ∞ , the chain approaches the limiting stretch, Λ → n .
In the other limit, ζ → 0 , the chain coils much below the limiting stretch,
Λ << n , and the model reduces to W = (kT / 6v)ζ 2 and Λ =
( n / 3)ζ , which
recovers the neo-Hookean model, with the small-strain shear modulus
µ = kT /(vn ) .
The material is also taken to be incompressible, λ1 λ2λ3 = 1 . Recall that
σ 1 − σ 3 = λ1
∂W (λ1 , λ2 )
.
∂λ1
According to calculus, we write
−1
dW (ζ ) ⎛ dΛ(ζ ) ⎞ ∂Λ(λ1 , λ2 )
.
⎜
⎟
σ 1 − σ 3 = λ1
dζ ⎜⎝ dζ ⎟⎠
∂λ1
A direct calculation gives that
kTζ (λ21 − λ23 )
.
σ1 −σ3 =
3v n Λ
We can similarly obtain that
kTζ (λ22 − λ23 )
.
σ2 −σ3 =
3v n Λ
Like the Gent model, the model of freely-joined chain can also capture the
extension limit.
Mooney model. In this model, the free energy is fit to the expression
W = c1 (λ21 + λ22 + λ23 − 3)+ c2 (λ1−2 + λ2−2 + λ−32 − 3).
•
M. Mooney (1940) A theory of large elastic deformation, Journal of
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Z. Suo
Applied Physics 11, 582-592.
Ogden model. In this model, the free energy density is fit to a series of
more terms
W=
µ
∑ α (λ
n
1
n
where
αn
)
+ λα2 + λα3 − 3 ,
n
n
n
α n may have any values, positive or negative, and are not necessarily
integers, and
µ n are constants. The stress-stretch relations are
σ 1 − σ 3 = ∑ µn (λα1 − λα3 ),
n
n
n
σ 2 − σ 3 = ∑ µn (λα2 − λα3 ).
n
n
n
•
R.W. Odgen (1972) Large deformation isotropic elasticity – on the
correlation of theory and experiment for incompressible rubberlike solids.
Proceedings of the Royal Society of London A326, 567-583.
Reading. Erwan Verron (http://imechanica.org/node/4167),
bibliography of the literature on the mechanics of rubbers up to 2003.
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a
Finite Deformation: Special Cases 12