22.4. SOME APPLICATIONS OF THE DIVERGENCE THEOREM 417
=∫
DTi j (x(s, t))
∂xα
∂ s∂xβ
∂ t∂xp
∂y j
=ε pαβ det(F)︷ ︸︸ ︷εcab
∂yc
∂xp
∂ya
∂xα
∂yb
∂xβ
dsdt
=∫
D(detF)Ti j (x(s, t))ε pαβ
∂xα
∂ s∂xβ
∂ t∂xp
∂y jdsdt.
Now ∂xp∂y j
= F−1p j and also
ε pαβ
∂xα
∂ s∂xβ
∂ t= (xs×xt)p
so the result just obtained is of the form∫D(detF)F−1
p j Ti j (x(s, t))(xs×xt)p dsdt =
∫D(detF)Ti j (x(s, t))
(F−T )
jp (xs×xt)p dsdt.
This has transformed the integral over Pt to one over P0, the part of ∂V0 which correspondswith Pt . Thus the last integral is of the form∫
P0
det(F)(T F−T )
ip NpdA
Summing these up over the pieces of ∂Vt and ∂V0, yields the last integral in 22.10 equals∫∂V0
det(F)(T F−T )
ip NpdA
and so the balance of momentum in terms of the material coordinates becomes∫V0
ρ0 (x)vt (t,x) dV =∫
V0
b0 (t,x) dV +∫
∂V0
ei det(F)(T F−T )
ip NpdA
The matrix det(F)(T F−T
)ip is called the Piola Kirchhoff stress S. An application of the
divergence theorem yields
∫V0
ρ0 (x)vt (t,x) dV =∫
V0
b0 (t,x) dV +∫
V0
ei
∂
(det(F)
(T F−T
)ip
)∂xp
dV.
Since V0 is arbitrary, a balance law for momentum in terms of the material coordinates isobtained
ρ0 (x)vt (t,x) = b0 (t,x)+ei
∂
(det(F)
(T F−T
)ip
)∂xp
= b0 (t,x)+div(det(F)
(T F−T ))
= b0 (t,x)+divS. (22.13)
As just shown, the relation between the Cauchy stress and the Piola Kirchhoff stress is
S = det(F)(T F−T ) , (22.14)