Section 4.7
4.7 Linearisation of the Hyperelastic Equations
In this section we linearise the hyperelastic equations for the strain energy and the stress,
showing the conditions which must hold so that the non-linear equations are consistent
with the linear elastic equations.
4.7.1
Isotropic Hyperelastic Constitutive Relations
Isotropic Compressible Hyperelastic Material in terms of Invariants
Consider the non-linear hyperelastic constitutive relation σ   0 I   1b   2 b 2 . Assume
that the material deforms only by a small amount from the reference configuration b  I .
Then a Taylor series gives
σ(b)  σ(I) 
σ
: b  I   o(2)
b b I
(4.7.1)
where b  I is a small number of order o(1) . Taking the material to be stress-free in the
reference configuration and neglecting the terms of order o( 2) , this can be written as
σ(b)  C : ε
(4.7.2)
where ε is the small strain tensor, which is equivalent to the Green-Lagrange tensor E at
small deformations:
ε  E  12 (C  I)  12 (b  I)
(4.7.3)
and C is the elasticity tensor:
C2
σ
b b I
(4.7.4)
The elasticity tensor can be evaluated through
1


C
 0 I   1b   2 b 2 b I
2
b
 I


 
b
b 2 

   I  0  b  1  b 2  2 
(4.7.5)
  0
 1
2
b
b  b I 
b
b
b  b I
 b


 

  1  2 2 b I I   I  0  b  1  b 2  2 
b
b
b  b I

where I is the fourth-order identity tensor,
I   ik  jl e i  e j  e k  e l  e i  e j  e i  e j
Solid Mechanics Part III
399
(4.7.6)
Kelly
Section 4.7
Consider now the case where the stress is a function of the three principal invariants.
Then

  I
 
 II b  0 III b  


 I  0    I   0 b  0
 
b  b I 

 I b b II b b III b b   b I
 


 0

 I   0 I  0 I b I  b  
II b I  I b b  b 2 
II b
III b
 I b
 b I
(4.7.7)
and similarly for the other terms. Also, as
b  I , I b , II b , III b   3,3,1
(4.7.8)
so
 


C  2 1  2 2 (3,3,1) I  2
2

II b III b
 I b


I  I,
 (3,3,1)
   0  1   2
(4.7.9)
Since I  I : ε  ( trε)I and I : ε  ε , the constitutive equation 4.7.2 can be written in the
standard form
σ   ( tre)I  2e
(4.7.10)
where the Lamé constants are
 

 

  2
2

,

I

II

III
b
b  ( 3, 3,1)
 b
   0  1   2
(4.7.11)
   1 (3,3,1)  2 2 (3,3,1)
Isotropic Incompressible Hyperelastic Material in terms of Invariants
In the incompressible case, consider the equation σ   1b   1b 1  pI . Linearising the
first two terms,
1

C
 1b   1b 1 b I
2
b
 b

 
b 1 

   b  1  b 1  1 
  1
  1
b  b I 
b
b  b I
 b


(4.7.12)
Now
Solid Mechanics Part III
400
Kelly
Section 4.7
b 1
1
  bik1blj1  bil1bkj1 e i  e j  e k  e l
b
2
1
b
1
1
   ik  lj   il  kj e i  e j  e k  e l   I  I
b b I
2
2


(4.7.13)
 
where I is the fourth-order identity tensor,
I   il jk e i  e j  e k  e l  e i  e j  e j  e i
(4.7.14)
so
 

 

 1
 
1
C  2 1I   1 I  I  I  I 1  2 1 
 2 1 
2
II b
I b
II b 
 I b

(4.7.15)
Since I  I : ε  ( trε)I and I : ε  ε , I : ε  ε T  ε , the constitutive equation for the
incompressible linear elastic solid can be written in the standard form (with trε  0 )
σ   pI  2e
(4.7.16)
  1 (3,3)   1 (3,3)
(4.7.17)
where the single Lamé constant is
4.7.2
Strain Energy Relations for Compressible Materials
Similar arguments can be used for the strain energy function. In this case
W (b)  W (I) 
W
b
: b  I  
b I
1  2W
2 b 2
: b  I  : b  I   o(3)
(4.7.18)
b I
and the quadratic terms have been retained since the linear stress expressions are obtained
by a differentiation of the strain energy (in fact, the term involving the first derivative
actually turns out to be zero). Then
W (b)  2
W
b
:ε  2
b I
 2W
b 2
:ε:ε
(4.7.19)
b I
Strain Energy in terms of the Principal Stretches
First, recall the derivatives of the stretch, Eqns. 2.3.15. hen, for W a function of the
principal stretches,
Solid Mechanics Part III
401
Kelly
Section 4.7
2
3
W
W i
1 W
2

ni  ni
b
i b i 1 i i
W
2
b
W

i 1 i
3
b I
(4.7.20)
ni  ni
i 1
and
 W  2 i i
 2W
  W i 
  W 




2 2 2 

2



2
b  i b 



b

b


b

b
 i 
 i
2
 2W
b 2
i 1
 W  2 i i
 2W  j 
 2



2
b i  j b 
 i b
3

1 3  1 W
1  2W
   3
ni  ni  ni  ni  
ni  ni  n j  n j 
2 i 1  i i

j 1 i  j i  j


3
1 3  W
 2W

  
ni  ni  ni  ni  
ni  ni  n j  n j 
2 i 1  i  1
j 1 i  j

i , j 1
i

(4.7.21)
Writing ε in terms of the principal strains e i :
3
ε   ei n i  n i ,
(4.7.22)
i 1
contraction with ε gives
2
W
W
:ε 
b
i
ei
(4.7.23)
i 1
and
2
 2W
b 2
Now ei2  ei ei  trε 2 and
:ε:ε  
i 1

ij
1 W
2 i
ei2 
i 1
1 W
2 i  j
ei e j
(4.7.24)
i , j 1
ei e j  ( trε) 2 , so the strain energy can be written in the
standard form
W 

2
trε 2   trε 2
(4.7.25)
by setting
Solid Mechanics Part III
402
Kelly
Section 4.7
W
i
 0,
i 1
 2W
i  j
   2 ij
(4.7.26)
i , j 1
Strain Energy in terms of the Principal Invariants
In terms of the invariants,
2
 W

W
W
I b I  b   W III b b 1 
 2
I
b
II b
III b
 I b

W
2
b
b I
(4.7.27)
 W
W
W 
 2
2

I
II b III b 
 I b
and
2
 W I
 2W
  W
 2
 I  
2
b  I b
b
 I b b
 W I b I  b 
  W
 
 I b I  b   
b
b  II b
 II b
W  III b b 1 
  W

 III b b 1   
III b
b
b  III b



(4.7.28)



so that
 W 
I b b  W  1 III b
 2W
b 1 

2 2  2
 
I 
 b  b  III b b 

II

b

b

III
b



b 
 b
2
2
2
  W I b
 W II b
 W III b 
I



 I b I b b I b II b b I b III b b 
  2W I b
 2W II b
 2W III b 
 I b I  b   



 II b I b b II b II b b II b III b b 
  2W I b
 2W II b
 2W III b 
 III b b 1  



 III b I b b III b II b b III b III b b 


1
 W
I  I  I  W III b  b 1  b 1  b 
 2
III b
b 

 II b
2
  2W

 2W
I b I  b    W III b b 1 
I
I
I b II b
I b III b
 I b I b

  2W

 2W
 2W
I b I  b  
 I b I  b   
I
III b b 1 
II b II b
II b III b
 II b I b


 III b b
1

  2W

 2W
 2W
I b I  b  

I
III b b 1 
III b II b
III b III b
 III b I b

(4.7.29)
Solid Mechanics Part III
403
Kelly
Section 4.7
so that
2
 2W
b 2
 
( 3, 3,1)
 W
1 W
 2 
I
I I

II
2

III
b
b

 W
W
 I  I 

 II b III b
   2W
 2W
 2W
  
2

I b II b I b III b
  I b I b
  2W
 2W
 2W
 2
2

II b II b II b III b
 II b I b



   2W
 2W
 2W 
  

2

III b II b III b III b 
  III b I b
(4.7.30)
Then, since I : ε  trε , contraction with e then gives
2
W
b
( 3, 3,1)
 W
W
W 
: ε  2
2

 trε
II b III b 
 I b
(4.7.31)
Also, since I  I : ε : ε  trε  , I : ε : ε  trε 2 , I : ε  ε T  ε, I : ε : ε  trε 2 ,
2
2
 2W
b 2
 W
W
: ε : ε  2trε 2 

 II b III b
( 3, 3,1)
2
 W

W
  2trε 2 

 II b III b

  2W
 2W
  2  4
I b II b
 I b
 2W
 2W
 2W
 2W 
4

4


I b III b
II b III b III b2 
II b2
(4.7.32)
so that, at (3,3,1),
W
W
W
2

0
I b
II b III b
 W
W 
  
 2

 II b III b 
W
W
 2W
 2W
 2W
 2W
 2W
 2W 

 2 4
2
4 2 4


II b III b
I b II b
I b III b
II b III b III b2 4
I b
II b
(4.7.33)
4.7.3
Problems
1. Show that the linearization conditions for the stress-strain equation for a compressible
material, Eqns. 4.7.11, and for the strain energy function, Eqns. 4.7.33, are satisfied for
the following compressible Neo-Hookean stress-strain equation and corresponding
strain energy function (derive this stress equation from the strain energy function):
Solid Mechanics Part III
404
Kelly
Section 4.7

1
 I b  3
2
2
σ  J 1  ln JI   ( b  I)
2. Repeat Problem 1 for the following compressible Neo-Hookean material:
1
1
2
W   ln J    J 2 / 3 I b  3
2
2

1


σ  J 1   ln JI  J 2 / 3  b  (tr b)I  
3



3. Repeat Problem 1 for the Blatz & Co. model (just verify the strain-energy conditions)

 1


1
  II
W  f I b  3  III b   1  1  f   b  3  III b   1
2

2  III b

 

W
(ln J ) 2   ln J 

Solid Mechanics Part III
405

Kelly