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the components of C to give, after some algebra using Equation (5.22), C
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

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