dr ¼ F
À1
Á dx
Then we can write
n 0 Á e σ
ð
Þ¼
d e
F
dA 0
Using the relation we defined above, we can write
n 0 Á e σ
ð
Þ¼
F
À1
Á dF
dA 0
At the same time, dF can be also defined in terms of Cauchy stress tensor as
n 0 Á e σ
ð
Þ¼
F
À1
Á n Á σ
ð
ÞdA
dA 0
¼
n Á σ
ð
ÞÁ F
2 1
À
Á T dA
dA 0
2.14 Direct Relation Between Cauchy Stress Tensor
and Piola-Kirchhoff Stress Tensors
The following relation can be given:
e σ ¼
ρ 0
ρ
F
2 1
Á σ Á F
À1
À
Á T and e σ ¼ σ
0
Á F
À1
À
Á T
Details of the long derivations for these relations are provided by Malvern (1969).
The second Piola-Kirchhoff stress tensor is usually preferred in finite strain
elasticity problems.
Equations of motions in undeformed (referenced) state first Piola-Kirchhoff stress
tensor is given by
Z
A 0
n 0 Á σ
0 dA 0 þ
Z
V
ρ 0 b 0 dV 0 ¼
Z
V 0
ρ
d
2 r
dt 2 dV 0
Transforming the surface integral to volume integral by Gauss’s theorem (divergence theorem) in indicial notation, we can write
62
2 Stress and Strain in Continuum
À1
Á dx
Then we can write
n 0 Á e σ
ð
Þ¼
d e
F
dA 0
Using the relation we defined above, we can write
n 0 Á e σ
ð
Þ¼
F
À1
Á dF
dA 0
At the same time, dF can be also defined in terms of Cauchy stress tensor as
n 0 Á e σ
ð
Þ¼
F
À1
Á n Á σ
ð
ÞdA
dA 0
¼
n Á σ
ð
ÞÁ F
2 1
À
Á T dA
dA 0
2.14 Direct Relation Between Cauchy Stress Tensor
and Piola-Kirchhoff Stress Tensors
The following relation can be given:
e σ ¼
ρ 0
ρ
F
2 1
Á σ Á F
À1
À
Á T and e σ ¼ σ
0
Á F
À1
À
Á T
Details of the long derivations for these relations are provided by Malvern (1969).
The second Piola-Kirchhoff stress tensor is usually preferred in finite strain
elasticity problems.
Equations of motions in undeformed (referenced) state first Piola-Kirchhoff stress
tensor is given by
Z
A 0
n 0 Á σ
0 dA 0 þ
Z
V
ρ 0 b 0 dV 0 ¼
Z
V 0
ρ
d
2 r
dt 2 dV 0
Transforming the surface integral to volume integral by Gauss’s theorem (divergence theorem) in indicial notation, we can write
62
2 Stress and Strain in Continuum