Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 314

Piola-Kirchhoff stress tensors in this uniaxial situation. Construct the relationships
between each of these tensors in this particular situation.

Solution : The principal axes of extension are obviously the direction of the applied
load and the two perpendicular transverse directions. The deformation can be
represented in terms of the principal stretches

l I , l II and l III by

x 1 ¼ l I X I ; x 2 ¼ l II X II ; x 3 ¼ l III X III ;

and the deformation gradients F by ( 11.32 ). It follows that the Jacobian J is given
by J ¼ l I l II l III . The area perpendicular to the x 1 -or X I -axis will change with the

deformation. Using the formula of Nanson ( 11.42 ), it follows that

2 3 2 3

l 1 I 00
0

dA I
0
0

T

4 5 4 5 ;

F 1

l 1
II

da ¼ J ð Þ dA ¼ l I l II l III

0

l 1
III

00

or d a 1 ¼ l II l III d A I ,d a 2 ¼ 0 ; thus we can conclude that the relation-

ship between the instantaneous area A and initial area A o is A

0, and d a 3 ¼

¼ l II l III A o . The only

nonzero Cauchy stress is T 11 ¼ l II l III A o ). From ( 11.48 ) the only nonzero

component of the first Piola-Kirchhoff stress tensor is given by T 1 PK
1 I

P/A ¼ P/ (

¼ l II l III T 11

¼ l II l III P A o and, from ( 11.50 ) the only nonzero component of the second

Piola-Kirchhoff stress tensor is given by

= A ¼ P =

T 2 PK
1 I

T 1 PK
1 I

¼ð Þ=l I ¼ðl II l III T 11 Þ=l I ¼ l II l III P =ðl I A Þ¼ P =ðl I A o Þ:

In the special case when the material is incompressible, J ¼ l I l II l III ¼ 1,

and the cross-section transverse to the extension is symmetric, l I ¼ l , l II ¼ l III

2 2 A

¼ √l , then T 2 PK

1 I

¼ð T 1 PK

1 I

Þ=l ¼ T 11 =l ¼ P =ðl Þ¼ P =ðl A o Þ .

1/

Problems

11.6.1. A rectangular parallelepiped with a long dimension a o and a square cross-
section of dimension b o is deformed by an axial tensile force P into a
rectangular parallelepiped with a longer long dimension a and a smaller
square cross-section of dimension b . This problem is a continuation of
Problem 11.4.4.
(a) Record the expressions for the stresses T 2 PK

1 I and T 11 in the deformed
rectangular parallelepiped. Both the x 1 and X I directions are coincident
with the longitudinal axis of the parallelepiped.
(b) If the material of the rectangular parallelepiped is incompressible, what
is the relationship between b and b o ?
(c) Record the expression for the stress T 2 PK

1 I in terms of the stress T 11 if the
rectangular parallelepiped is incompressible.