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4 HYPERELASTICITY



where the symmetric fourth-order tensor C, known as the Lagrangian or
material elasticity tensor, is de¬ned by the partial derivatives as,
3
C= CIJKL E I — E J — E K — E L ;
I,J,K,L=1

4 ‚2Ψ
‚SIJ
= = = CKLIJ (5.10)
C IJKL
‚EKL ‚CIJ ‚CKL
For convenience these expressions are often abbreviated as,
4 ‚2Ψ
‚S ‚S
C= =2 = (5.11)
‚E ‚C ‚C‚C

EXAMPLE 5.1: St. Venant“Kirchho¬ Material
The simplest example of a hyperelastic material is the St. Venant“
Kirchho¬ model, which is de¬ned by a strain energy function Ψ as,
1
Ψ(E) = »(trE)2 + µE : E
2
where » and µ are material coe¬cients. Using the second part of Equa-
tion (5.7b), we can obtain the second Piola“Kirchho¬ stress tensor as,
S = »(trE)I + 2µE
and using Equation (5.10), the coe¬cients of the Lagrangian elasticity
tensor emerge as,
CIJKL = »δIJ δKL + 2µδIK δJL
Note that these two last equations are analogous to those used in linear
elasticity, where the small strain tensor has been replaced by the Green
strain. Unfortunately, this St. Venant“Kirchho¬ material has been
found to be of little practical use beyond the small strain regime.



5.3.2 THE SPATIAL OR EULERIAN ELASTICITY TENSOR
It would now be pertinent to attempt to ¬nd a spatial equivalent to Equa-
tion (5.9), and it would be tempting to suppose that this would involve a
relationship between the linearized Cauchy stress and the linearized Almansi
strain. Although, in principle, this can be achieved, the resulting expres-
sion is intractable. An easier route is to interpret Equation (5.9) in a rate
form and apply the push forward operation to the resulting equation. This
is achieved by linearizing S and E in the direction of v, rather than u.
5
5.4 ISOTROPIC HYPERELASTICITY



™ ™
Recalling from Section 3.9.3 that DS[v] = S and DE[v] = E gives,
™ ™
S=C :E (5.12)

Because the push forward of S has been shown in Section 4.5 to be the
Truesdell rate of the Kirchho¬ stress „ —¦ = Jσ —¦ and the push forward of

E is d, namely, Equation (3.91a), it is now possible to obtain the spatial
equivalent of the material linearized constitutive Equation (5.12) as,
σ —¦ =c : d (5.13)
wherec , the Eulerian or spatial elasticity tensor, is de¬ned as the Piola push
forward of C and after some careful indicial manipulations can be obtained
as* ,
3
’1
J ’1 FiI FjJ FkK FlL CIJKL ei — ej — ek — el
c =J φ— [C]; c=
i,j,k,l=1
I,J,K,L=1
(5.14)
Often, Equation (5.13) is used, together with convenient coe¬cients in c , as
the fundamental constitutive equation that de¬nes the material behavior.
Use of such an approach will, in general, not guarantee hyperelastic behav-
ior, and therefore the stresses cannot be obtained directly from an elastic
potential. In such cases, the rate equation has to be integrated in time, and
this can cause substantial di¬culties in a ¬nite element analysis because of
problems associated with objectivity over a ¬nite time increment.

Remark 5.3.1. Using Equations (3.96) and (4.55), it can be observed
that Equation (5.13) can be reinterpreted in terms of Lie derivatives as,
Lφ[„ ] = Jc : Lφ[e] (5.15)


5.4 ISOTROPIC HYPERELASTICITY

5.4.1 MATERIAL DESCRIPTION
The hyperelastic constitutive equations discussed so far are unrestricted in
their application. We are now going to restrict these equations to the com-

* —¦
Using the standard summation convention and noting from Equation (4.54) that σij =
™ ™
J ’1 FiI FjJ SIJ and from Equation (3.91a) that EKL = FkK FlL dkl gives,

σij = J ’1 „ij = J ’1 FiI FjJ CIJKL FkK FlL dkl = c
—¦ —¦
ijkl dkl

= J ’1 FiI FjJ FkK FlL CIJKL .
and, consequently, c ijkl
6 HYPERELASTICITY



mon and important isotropic case. Isotropy is de¬ned by requiring the con-
stitutive behavior to be identical in any material direction† . This implies
that the relationship between Ψ and C must be independent of the material
axes chosen and, consequently, Ψ must only be a function of the invariants
of C as,


Ψ(C(X), X) = Ψ(IC , IIC , IIIC , X) (5.16)
where the invariants of C are de¬ned here as,
IC = trC = C : I (5.17a)
IIC = trCC = C : C (5.17b)
IIIC = det C = J 2 (5.17c)
As a result of the isotropic restriction, the second Piola“Kirchho¬ stress
tensor can be rewritten from Equation (5.7b) as,
‚Ψ ‚Ψ ‚IC ‚Ψ ‚IIC ‚Ψ ‚IIIC
S=2 =2 +2 +2 (5.18)
‚C ‚IC ‚C ‚IIC ‚C ‚IIIC ‚C
The second-order tensors formed by the derivatives of the ¬rst two invariants
with respect to C can be evaluated in component form to give,
3
‚ ‚IC
CKK = δIJ ; =I (5.19a)
‚CIJ ‚C
K=1

3
‚IIC

CKL CKL = 2CIJ ; = 2C (5.19b)
‚CIJ ‚C
K,L=1

The derivative of the third invariant is more conveniently evaluated using
the expression for the linearization of the determinant of a tensor given
in Equation (2.119). To this end note that the directional derivative with
respect to an arbitrary increment tensor ∆C and the partial derivatives are
related via,
3
‚IIIC ‚IIIC
DIIIC [∆C] = ∆CIJ = : ∆C (5.20)
‚CIJ ‚C
I,J=1

Rewriting Equation (2.119) as,
DIIIC [∆C] = det C (C ’1 : ∆C) (5.21)

† Note that the resulting spatial behavior as given by the spatial elasticity tensor may be
anisotropic.
7
5.4 ISOTROPIC HYPERELASTICITY



and comparing this equation with Expression (5.20) and noting that both
equations are valid for any increment ∆C yields,
‚IIIC
= J 2 C ’1 (5.22)
‚C
Introducing Expressions (5.19a,b) and (5.22) into Equation (5.18) enables
the second Piola“Kirchho¬ stress to be evaluated as,

S = 2ΨI I + 4ΨIIC + 2J 2 ΨIII C ’1 (5.23)

where ΨI = ‚Ψ/‚IC , ΨII = ‚Ψ/‚IIC , and ΨIII = ‚Ψ/‚IIIC .


5.4.2 SPATIAL DESCRIPTION
In design practice it is obviously the Cauchy stresses that are of engineer-
ing signi¬cance. These can be obtained indirectly from the second Piola“
Kirchho¬ stresses by using Equation (4.45b) as,

σ = J ’1 F SF T (5.24)

Substituting S from Equation (5.23) and noting that the left Cauchy“Green
tensor is b = F F T gives,

σ = 2J ’1 ΨI b + 4J ’1 ΨIIb2 + 2JΨIII I (5.25)

In this equation ΨI , ΨII, and ΨIII still involve derivatives with respect to
the invariants of the material tensor C. Nevertheless it is easy to show
that the invariants of b are identical to the invariants of C, as the following
expressions demonstrate,

Ib = tr[b] = tr[F F T ] = tr[F T F ] = tr[C] = IC (5.26a)
IIb = tr[bb] = tr[F F T F F T ] = tr[F T F F T F ] = tr[CC] = IIC (5.26b)
IIIb = det[b] = det[F F T ] = det[F T F ] = det[C] = IIIC (5.26c)

Consequently, the terms ΨI , ΨII, and ΨIII in Equation (5.25) are also the
derivatives of Ψ with respect to the invariants of b.

Remark 5.4.1. Note that any spatially based expression for Ψ must be
a function of b only via its invariants, which implies an isotropic material.
This follows from the condition that Ψ must remain constant under rigid
body rotations and only the invariants of b, not b itself, remain unchanged
under such rotations.
8 HYPERELASTICITY




EXAMPLE 5.2: Cauchy stresses
It is possible to derive an alternative equation for the Cauchy stresses
directly from the strain energy. For this purpose, note ¬rst that the
time derivative of b is,
™T
™ ™
b = F F T + F F = lb + blT
and therefore the internal energy rate per unit of undeformed volume

w0 = Ψ is,

‚Ψ ™

Ψ= :b
‚b
‚Ψ
: (lb + blT )
=
‚b
‚Ψ
=2 b:l
‚b
If we combine this equation with the fact that σ is work conjugate to l
with respect to the current volume, that is, w = J ’1 w0 = σ : l, gives,
™ ™
‚Ψ
Jσ = 2 b
‚b
It is simple to show that this equation gives the same result as Equa-
tion (5.25) for isotropic materials where Ψ is a function of the invariants
of b.



5.4.3 COMPRESSIBLE NEO-HOOKEAN MATERIAL
The equations derived in the previous sections refer to a general isotropic
hyperelastic material. We can now focus on a particularly simple case known
as compressible neo-Hookean material. This material exhibits characteristics
that can be identi¬ed with the familiar material parameters found in linear
elastic analysis. The energy function of such a material is de¬ned as,
µ »
(IC ’ 3) ’ µ ln J + (ln J)2
Ψ= (5.27)
2 2
where the constants » and µ are material coe¬cients and J 2 = IIIC . Note
that in the absence of deformation, that is, when C = I, the stored energy
function vanishes as expected.
The second Piola“Kirchho¬ stress tensor can now be obtained from
Equation (5.23) as,
S = µ(I ’ C ’1 ) + »(ln J)C ’1 (5.28)
9
5.4 ISOTROPIC HYPERELASTICITY



Alternatively, the Cauchy stresses can be obtained using Equation (5.25) in
terms of the left Cauchy“Green tensor b as,
µ »
σ= (b ’ I) + (ln J)I (5.29)
J J
The Lagrangian elasticity tensor corresponding to this neo-Hookean ma-
terial can be obtained by di¬erentiation of Equation (5.28) with respect to

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