Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 313

Fig. 11.15 Illustration for a
uniaxial stretching
deformation due to a load P

n , may be extended to the first Piola-Kirchhoff stress tensor T 1PK and
the material coordinate system; thus t ( N ) ¼

by t ( n ) ¼ T

T 1PK N , where N is the unit normal

vector to the plane in the material coordinate system. Using this result, the stress
equations of motion may be rederived in the material coordinates and in terms of
the first Piola-Kirchhoff stress tensor; thus

T

T 1 PK T 1 PK F T T 1 PK

r R € x ¼r o þ r R d ; ¼ F ð Þ ; (11.49)

where the density of the initial configuration r R is used as the reference density and

the divergence is now with respect to material rather than spatial coordinates.
Recall from Chapter 2 (see (2.23)) that the gradient symbol

∇ O with a subscripted
O will indicate a gradient with respect to the material coordinate system X , rather
than the usual gradient symbol used to indicate a gradient with respect to the

spatial coordinate system x . The second of ( 11.49 ) shows that the first
Piola-Kirchhoff stress tensor is not symmetric like the Cauchy stress tensor. To
see that this is the case the second of (11.48) may be substituted twice in the second
of (3.37) to verify.
In order to have a measure of stress referred to the initial configuration that is
symmetric, the second Piola-Kirchhoff stress tensor is introduced; this is also
called the Kirchhoff stress tensor. This new stress tensor is denoted by T 2 PK and
defined as follows:

∇

T

T 2 PK F 1 T 1 PK JF 1 F 1 ¼ T ð Þ :

(11.50)

T 2PK into ( 11.49 ) the equations of motion in terms of the
second Piola-Kirchhoff stress tensor are obtained:

Substituting T 1PK

¼ F

T

T 2 PK T 2 PK T 2 PK r R x ¼r o ð F Þþr R d ; ¼ð Þ : (11.51)

This shows that the second Piola-Kirchhoff stress tensor T 2PK is symmetric.

Example 11.6.1
A solid specimen capable of large deformations is extended by a force of magnitude
P in the x 1 or X I direction (these directions are coincident here). This uniaxial stress
situation is illustrated in Fig. 11.15 . Determine the Cauchy and the first and second