CONTINUUM MECHANICS for ENGINEERS

which reduces to
where we have used Eq 5.5-8 and the divergence theorem. Carrying out the
indicated partial differentiation, we obtain
and by Eq 5.5-10b this reduces to
ε ⎡ o o o
ijk ( xP j Ak ) + xρ A
j b⎤ k dV
∫ o
V ⎣⎢ ,
o ⎦⎥
= 0 (5.5-12)
since the term in parentheses is zero on account of the balance of momentum.
Again, this equation must hold for all portions V° of the body, so the integrand
must vanish, giving
= 0 (5.5-13)
Following a similar argument to that presented in Section 3.4, we conclude
that Eq 5.5-13 implies
If we now introduce the definition for s AB
and substitute into Eq 5.5-14, we observe that
(5.5-14)
(5.5-15)
(5.5-16)
which is called the second Piola-Kirchhoff stress tensor, or sometimes the
symmetric Piola-Kirchhoff stress tensor.
The Piola-Kirchhoff stresses can be related to the Cauchy stress by considering
the differential force exerted on an element of deformed surface dS as
= 0
[ ( ) ] =
o o o o
εijk xj,APAk+ xjPAk,A + ρ bk dV
∫ o 0
o
V
∫ ( )
o
V
o o
εijkxj,APAk dV
εijkxj,APAk o
x P = x P
o o
j,A Ak k,A Aj
o
P Ai = xi,BsBA sAB =
sBA
df i = σ jin jdS (5.5-17)