Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 328

2 @

I
@C B

@

II
@C B 1

þ @

IV
@C M

þ @

V
@C N

T ¼ p1 þ : (11.118)

The remainder of the chapter deals with incompressible transversely isotropic
hyperelastic material.

Example 11.11.1
Determine the stress tensor in a rectangular parallelepiped of an incompressible
transversely isotropic hyperelastic material in which the unique direction coincides
with the long dimension of the parallelepiped and the two transverse dimensions are
equal. There is only a force applied to the parallelepiped in the long dimension of
the parallelepiped and denotes the principal stretch in the long dimension. The

stress applied to the parallelepiped is zero in the two transverse dimensions.

l

Solution : For the situation described above Eqs. ( 11.28 ), ( 11.33 ), and ( 11.116 ) may
be used to show that

2
4

3
5 ;

2
4

3
5 ;

2
4

3
5 ;

l 1 00
0

l 00
0

000
000
00

l 1 0
00 l

B 1 B ¼ ¼ l 0 N ¼ M ¼ 0 ;

l 2

2

2

00 l

thus from ( 11.118 ) it follows that

2 @

I

@

II

C l 1

T 11 ¼ T 22 ¼ p þ

C l

;

@ @

=@CÞl 1

and since these two stress are zero, it follows that p ¼ 2 ½ð@ I ð@ II =@ C Þl

and that the axial stress is given by

@ I 1

l

@ II

C þ @

V

l½ðl l 2

T 33 ¼ 2 Þ

C þ C l

:

@ @ @

If the material was isotropic rather than transversely isotropic, then the same
result would apply with @ V
@

0.
A number of solutions for transversely isotropic hyperelastic materials in cylin-
drical coordinates were obtained by Ericksen and Rivlin ( 1954 ); some of these
solutions are contained in the topic by Januzemis ( 1967 ).

C ¼

11.12 Relevant Literature

The developments in nonlinear of nonlinear elasticity are described in many topics,
for example Truesdell ( 1960 ), Truesdell and Toupin ( 1960 ), Green and Adkins
( 1960 ), Januzemis ( 1967 ), Treloar ( 1967 ), Truesdell and Noll ( 1965 ), and Ogden
( 1984 ) among many others.