as,

C = »C ’1 — C ’1 + 2(µ ’ » ln J)I (5.30)

where C ’1 — C ’1 = (C ’1 )IJ (C ’1 )KL E I — E J — E K — E L and the

fourth-order tensor I is de¬ned as,

‚C ’1 ‚(C ’1 )IJ

I=’ ; IIJKL = ’ (5.31)

‚C ‚CKL

In order to obtain the coe¬cients of this tensor, recall from Section 2.3.4

that the directional derivative of the inverse of a tensor in the direction of

an arbitrary increment ∆C is,

DC ’1 [∆C] = ’C ’1 (∆C)C ’1 (5.32)

Alternatively, this directional derivative can be expressed in terms of the

partial derivatives as,

‚C ’1

’1

DC [∆C] = : ∆C (5.33)

‚C

Consequently, the components of I can be identi¬ed as,

IIJKL = (C ’1 )IK (C ’1 )JL (5.34)

The Eulerian or spatial elasticity tensor can now be obtained by pushing

forward the Lagrangian tensor using Equation (5.14) to give, after tedious

algebra, c as,

» 2

c= I — I + (µ ’ » ln J)i (5.35)

J J

where i is the fourth-order identity tensor obtained by pushing forward I

and in component form is given in terms of the Kroneker delta as,

i = φ— [I ]; iijkl = FiI FjJ FkK FlL IIJKL = δik δjl (5.36)

I,J,K,L

Note that Equation (5.36) de¬nes an isotropic fourth-order tensor as

discussed in Section 2.2.4, similar to that used in linear elasticity, which can

be expressed in terms of the e¬ective Lam´ moduli » and µ as,

e

c ijkl = » δij δkl + 2µ δik δjl (5.37)

10 HYPERELASTICITY

where the e¬ective coe¬cients » and µ are,

» µ ’ » ln J

»= ; µ= (5.38)

J J

Note that in the case of small strains when J ≈ 1, then » ≈ », µ ≈ µ, and

the standard fourth-order tensor used in linear elastic analysis is recovered.

EXAMPLE 5.3: Pure dilatation (i)

The simplest possible deformation is a pure dilatation case where the

deformation gradient tensor F is,

J = »3

F = »I;

and the left Cauchy“Green tensor b is therefore,

b = »2 I = J 2/3 I

Under such conditions the Cauchy stress tensor for a compressible neo-

Hookean material is evaluated with the help of Equation (5.29) as,

µ 2/3 »

σ= (J ’ 1) + ln J I

J J

which represents a state of hydrostatic stress with pressure p equal to,

µ 2/3 »

p= (J ’ 1) + ln J

J J

EXAMPLE 5.4: Simple shear (i)

The case of simple shear described in Chapter 3 is de¬ned by a defor-

mation gradient and left Cauchy“Green tensors as,

® ®

1 + γ2 γ 0

1γ0

F = °0 1 0»; b=° γ 1 0»

001 0 01

(continued)

11

5.5 INCOMPRESSIBLE MATERIALS

EXAMPLE 5.4 (cont.)

which imply J = 1 and the Cauchy stresses for a neo-Hookean material

are,

®2

γ γ0

σ = µ ° γ 0 0»

000

Note that only when γ ’ 0 is a state of pure shear obtained. Note also

that despite the fact that J = 1, that is, there is no change in volume,

the pressure p = trσ/3 = γ 2 /3 is not zero. This is known as the Kelvin

e¬ect.

5.5 INCOMPRESSIBLE AND NEARLY INCOMPRESSIBLE

MATERIALS

Most practical large strain processes take place under incompressible or near

incompressible conditions. Hence it is pertinent to discuss the constitutive

implications of this constraint on the deformation. The terminology “near

incompressibility” is used here to denote materials that are truly incompress-

ible, but their numerical treatment invokes a small measure of volumetric

deformation. Alternatively, in a large strain elastoplastic or inelastic con-

text, the plastic deformation is often truly incompressible and the elastic

volumetric strain is comparatively small.

5.5.1 INCOMPRESSIBLE ELASTICITY

In order to determine the constitutive equation for an incompressible hyper-

elastic material, recall Equation (5.7a) rearranged as:

1 ‚Ψ ™

:C=0 (5.39)

S’

2 ‚C

™

Previously the fact that C in this equation was arbitrary implied that S =

2‚Ψ/‚C. In the incompressible case, the term in brackets is not guaranteed

™

to vanish because C is no longer arbitrary. In fact, given that J = 1

™

throughout the deformation and therefore J = 0, Equation (3.129) gives the

™

required constraint on C as,

™

1 ’1

2 JC :C=0 (5.40)

12 HYPERELASTICITY

S ‚Ψ J

’ C ’1

’ ’ ’’ =

2 ‚C 2

J

’ C ’1

2

.

Admissible C plane

.

C

FIGURE 5.1 Incompressibility constraint

™

The fact that Equation (5.39) has to be satis¬ed for any C that complies

with condition (5.40) implies that,

1 ‚Ψ J

= γ C ’1 (5.41)

S’

2 ‚C 2

where γ is an unknown scalar that will, under certain circumstances that

we will discuss later, coincide with the hydrostatic pressure and will be

determined by using the additional equation given by the incompressibility

constraint J = 1. Equation (5.40) is symbolically illustrated in Figure 5.1,

where the double contraction “ : ” has been interpreted as a generalized dot

product. This enables (S/2 ’ ‚Ψ/‚C) and JC ’1 /2 to be seen as being

™

orthogonal to any admissible C and therefore proportional to each other.

From Equation (5.41) the general incompressible hyperelastic constitu-

tive equation emerges as,

‚Ψ(C)

+ γJC ’1

S=2 (5.42)

‚C

The determinant J in the above equation may seem unnecessary in the

case of incompressibility where J = 1, but retaining J has the advantage

that Equation (5.42) is also applicable in the nearly incompressible case.

Furthermore, in practical terms, a ¬nite element analysis rarely enforces

J = 1 in a strict pointwise manner, and hence its retention may be important

for the evaluation of stresses.

Recalling Equation (4.50b) giving the deviatoric“hydrostatic decompo-

sition of the second Piola“Kirchho¬ tensor as S = S + pJC ’1 , it would

13

5.5 INCOMPRESSIBLE MATERIALS

be convenient to identify the parameter γ with the pressure p. With this in

mind, a relationship between p and γ can be established to give,

1

p = J ’1 S : C

3

1 ‚Ψ

= J ’1 2 + γJC ’1 : C

3 ‚C

2 ‚Ψ

= γ + J ’1 :C (5.43)

3 ‚C

which clearly indicates that γ and p coincide only if,

‚Ψ

:C=0 (5.44)

‚C

This implies that the function Ψ(C) must be homogeneous of order 0, that

is, Ψ(±C) = Ψ(C) for any arbitrary constant ±.* This can be achieved by

recognizing that for incompressible materials IIIC = det C = J 2 = 1. We

can therefore express the energy function Ψ in terms of the distortional com-

’1/3

ˆ

ponent of the right Cauchy“Green tensor C = IIIC C to give a formally

ˆ

modi¬ed energy function Ψ(C) = Ψ(C). The homogeneous properties of

the resulting function Ψ(C) are easily shown by,

Ψ(±C) = Ψ[(det ±C)’1/3 (±C)]

= Ψ[(±3 det C)’1/3 ±C]

= Ψ[(det C)’1/3 C]

= Ψ(C) (5.45)

Accepting that for the case of incompressible materials Ψ can be replaced

by Ψ, Condition (5.44) is satis¬ed and Equation (5.42) becomes,

‚ Ψ(C)

+ pJC ’1

S=2 (5.46)

‚C

It is now a trivial matter to identify the deviatoric component of the second

Piola“Kirchho¬ tensor by comparison of the above equation with Equation

* A scalar function f (x) of a k-dimensional vector variable x = [x1 , x2 , ..., xk ]T is said to be

homogeneous of order n if for any arbitrary constant ±,

f (±x) = ±n f (x)

Di¬erentiating this expression with respect to ± at ± = 1 gives,

‚f

· x = nf (x)

‚x

14 HYPERELASTICITY

(4.50b) to give,

‚Ψ

S =2 (5.47)

‚C

Note that the derivative ‚ Ψ(C)/‚C is not equal to the derivative ‚Ψ(C)/‚C,

ˆ

despite the fact that C = C for incompressibility. This is because IIIC re-

ˆ

mains a function of C while the derivative of C is being executed. It is only

after the derivative has been completed that the substitution IIIC = 1 can

be made.

5.5.2 INCOMPRESSIBLE NEO-HOOKEAN MATERIAL

In the case of incompressibility the neo-Hookean material introduced in Sec-

tion 5.4.3 is de¬ned by a hyperelastic potential Ψ(C) given as,

1

Ψ(C) = µ(trC ’ 3) (5.48)

2

The equivalent homogeneous potential Ψ is established by replacing C by

ˆ