stress tensor is not symmetric, even when there is no distributed body moment
(nonpolar case) on the system.
Stress is defined as a force per unit undeformed area. Keep in mind that stress is a
human construct not a physical measurable quantity; thus, it is possible to define it as
we please. The second Piola-Kirchhoff stress tensor is defined by the following
equation (Fig. 2.35):
n 0 Á σ
0
À
Á ¼
dF
dA 0
where σ
0 is the first Piola-Kirchhoff tensor and dF is the actual force acting on the
deformed configuration dA; however, it is formulated based on the initial
undeformed area dA 0 .
dF can also be written using Cauchy stress tensor σ as dF ¼ (n Á σ)dA. Hence, we
can write the following equation between the first Piola-Kirchhoff stress tensor and
Cauchy stress tensor:
n 0 σ 0
ð
ÞdA 0 ¼ dF ¼ n Á σ
ð
ÞdA
2.13.2 Second Piola-Kirchhoff Stress Tensor e σ
The second Piola-Kirchhoff stress tensor, e σ, uses a force d e
F that is related to the
actual force dF by the inverse of deformation gradient F
À1 :
d e
F ¼ F
À1
Á dF
Remember that material (local) coordinates and spatial coordinates are also
related by
y
A 0
V 0
V
A
dA 0
dA
n0
n
dF
dF
O
Initial
undeformed
deformed
Fig. 2.35 Piola-Kirchhoff
stress definition
2.13 Piola-Kirchhoff Stress Tensors
61