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±
β
+ 2 T±i Tβj T±k Tβl (5.106)
»2 ’ »2
± β
±,β=1
±=β



where the derivatives of Ψ for the material under consideration are,
‚2Ψ
1 2µ 2µ
= δ±β ’ (5.107)
J ‚ ln »± ‚ ln »β J 3J


5.6.7 PLANE STRAIN AND PLANE STRESS CASES
The plane strain case is de¬ned by the fact that the stretch in the third di-
rection »3 = 1. Under such conditions, the stored elastic potential becomes,
»
Ψ(»1 , »2 ) = µ[(ln »1 )2 + (ln »2 )2 ] +
(ln j)2 (5.108)
2
where j = det2—2 F is the determinant of the components of F in the n1
and n2 plane. The three stresses are obtained using exactly Equation (5.90)
with »3 = 1 and J = j.
The plane stress case is a little more complicated in that it is the stress in
the n3 direction rather than the stretch that is constrained, that is σ33 = 0.
28 HYPERELASTICITY



Imposing this condition in Equation (5.90) gives,
» 2µ
σ33 = 0 = ln J + ln »3 (5.109)
J J
from which the logarithmic stretch in the third direction emerges as,
»
ln »3 = ’ ln j (5.110)
» + 2µ
Substituting this expression into Equation (5.88) and noting that ln J =
ln »3 + ln j gives,
¯
»
Ψ(»1 , »2 ) = µ[(ln »1 ) + (ln »2 ) ] + (ln j)2
2 2
(5.111)
2
¯
where the e¬ective Lame coe¬cient » is,

¯
» = γ»; γ= (5.112)
» + 2µ
Additionally, using Equation (5.110) the three-dimensional volume ratio J
can be found as a function of the planar component j as,
J = jγ (5.113)
By either substituting Equation (5.110) into Equation (5.88) or di¬erentiat-
ing Equation (5.111) the principal Cauchy stress components are obtained
as,
¯
» 2µ
σ±± = γ ln j + γ ln »± (5.114)
j j
and the coe¬cients of the Eulerian elasticity tensor become,
¯
‚2Ψ
1 » 2µ
= γ + γ δ±β (5.115)
J ‚ ln »± ‚ ln »β j j


5.6.8 UNIAXIAL ROD CASE
In a uniaxial rod case, the stresses in directions orthogonal to the rod, σ22
and σ33 vanish. Imposing this condition in Equation (5.90) gives two equa-
tions as,
» ln J + 2µ ln »2 = 0 (5.116a)
» ln J + 2µ ln »3 = 0 (5.116b)
from which it easily follows that the stretches in the second and third direc-
tions are equal and related to the main stretch via Poisson™s ratio ν as,
»
ln »2 = ln »3 = ’ν ln »1 ; ν= (5.117)
2» + 2µ
29
5.6 ISOTROPIC ELASTICITY IN PRINCIPAL DIRECTIONS



Using Equations (5.89“90) and (5.117) a one-dimensional constitutive equa-
tion involving the rod stress σ11 , the logarithmic strain ln »1 , and Young™s
modulus E emerges as,
E µ(2µ + 3»)
σ11 = ln »1 ; E= (5.118)
J »+µ
where J can be obtained with the help of Equation (5.117) in terms of »1
and ν as,
(1’2ν)
J = »1 (5.119)

Note that for the incompressible case J = 1, Equation (5.118) coincides with
the uniaxial constitutive equation employed in Chapter 1.
Finally, the stored elastic energy given by Equation (5.88) becomes,
E
(ln »1 )2
Ψ(»1 ) = (5.120)
2
and, choosing a local axis in the direction of the rod, the only e¬ective term
in the Eulerian tangent modulus C1111 is given by Equation (5.86) as,
‚2Ψ
1 E
= ’ 2σ11 = ’ 2σ11 (5.121)
c 1111
J ‚ ln »1 ‚ ln »1 J
Again, for the incompressible case J = 1, the term E ’ 2σ11 was already
apparent in Chapter 1 where the equilibrium equation of a rod was linearized
in a direct manner.
Exercises

1. In a plane stress situation the right Cauchy“Green tensor C is,
® 
C11 C12 0
h2
 
C = ° C21 C22 0 »; C33 = 2
H
0 0 C33
where H and h are the initial and current thickness respectively. Show
that incompressibility implies,

C11 C12
C33 = III’1 ; (C ’1 )33 = IIIC ; C=
C C21 C22

Using these equations, show that for an incompressible neo-Hookean ma-
terial the plane stress condition S33 = 0 enables the pressure in Equa-
tion (5.50) to be explicitly evaluated as,
’1
p = 1 µ IC ’ 2IIIC
3
30 HYPERELASTICITY



and therefore the in-plane components of the second Piola“Kirchho¬ and
Cauchy tensors are,
’1
¯
S = µ I ’ IIIC C
¯ ¯
¯
σ = µ(b ’ III¯ I)b

where the overline indicates the 2 — 2 components of a tensor.
2. Show that the Equations in Exercise 1 can also be derived by imposing
the condition C33 = III’1 in the neo-Hookean elastic function Ψ to give,
C
’1
Ψ(C) = 1 µ(IC + IIIC ’ 3)
2

from which S is obtained by di¬erentiation with respect to the in-plane
tensor C. Finally, prove that the Lagrangian and Eulerian in-plane elas-
ticity tensors are,
’1 ’1
’1
C = 2µIIIC (C +I)
—C
¯ ¯i
c = 2µIII’1 (I — I + ¯)
¯ ¯
b

3. Using the push back“pull-forward relationships between E and d and
between C and c show that,
™ ™
E : C : E = Jd : c : d
for any arbitrary motion. Using this equation and recalling Example 5.2,
show that,
‚2Ψ
Jc = 4b b
‚b‚b
Check that using this equation for the compressible neo-Hookean model
you retrieve Equation (5.35).
4. Using the simple stretch-based hyperelastic equations discussed in Sec-
tion 5.6.5, show that the principal stresses for a simple shear test are,
γ
σ11 = ’σ22 = 2µ sinh’1 2

Find the Cartesian stress components.
5. A general type of incompressible hyperelastic material proposed by Ogden
is de¬ned by the following strain energy function:
N
µp ±p ± ±
»1 + »2 p + »2 p ’ 3
Ψ=
±p
p=1

Derive the homogeneous counterpart of this functional. Obtain expres-
sions for the principal components of the deviatoric stresses and elasticity
tensor.
CHAPTER SIX

LINEARIZED EQUILIBRIUM
EQUATIONS




6.1 INTRODUCTION

The virtual work representation of the equilibrium equation presented in
Section 4.3 is nonlinear with respect to both the geometry and the mate-
rial. For a given material and loading conditions, its solution is given by
a deformed con¬guration φ in a state of equilibrium. In order to obtain
this equilibrium position using a Newton“Raphson iterative solution, it is
necessary to linearize the equilibrium equations using the general directional
derivative procedure discussed in Chapter 2. Two approaches are in common
use: some authors prefer to discretize the equilibrium equations and then
linearize with respect to the nodal positions, whereas others prefer to lin-
earize the virtual work statement and then discretize. The latter approach
is more suitable for solid continua and will be adopted herein, although in
some cases where a nonstandard discretization is used this approach may
not be possible.




6.2 LINEARIZATION AND NEWTON “ RAPHSON
PROCESS

The principle of virtual work has been expressed in Chapter 4 in terms of
the virtual velocity as,



δW (φ, δv) = σ : δd dv ’ f · δv dv ’ t · δv da = 0 (6.1)
v v ‚v


1
2 LINEARIZED EQUILIBRIUM EQUATIONS



X 3, x 3
t
n
u
`
δv
δv



p u(xp)

f
v


P time = t
X 1, x 1 ‚v
V
X 2, x 2


time = 0

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. 32
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