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elasticity, where the small strain tensor has been replaced by the Green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