CONTINUUM MECHANICS for ENGINEERS

( )
ˆn
and because t = σjinj, we can convert the surface integral to a volume
i
integral having the integrand σji,j. By the use of Eq 5.3-11 on the left-hand
side of Eq 5.4-2 we have, after collecting terms,
( ρ ˙vi−σ ji,j −ρbi)
dV = 0
(5.4-3)
where ˙v is the acceleration field of the body. Again, V is arbitrary and so the
i
integrand must vanish, and we obtain
(5.4-4)
which are known as the local equations of motion in Eulerian form.
When the velocity field is zero, or constant so that = 0, the equations
of motion reduce to the equilibrium equations
˙v i
= 0 (5.4-5)
which are important in solid mechanics, especially elastostatics.
5.5 The Piola-Kirchhoff Stress Tensors,
Lagrangian Equations of Motion
∫
V
σ + ρb = ρv˙
ji,j i i
σ ρb
+
ji,j i
As mentioned in the previous section, the equations of motion Eq 5.4-4 are
in Eulerian form. These equations may also be cast in the referential form
based upon the Piola-Kirchhoff tensor, which we now introduce.
Recall that in Section 3.3 we defined the stress components σij of the
t ( eˆ
j )
i
n=e ˆ ˆj
Cauchy stress tensor as the ith component of the stress vector acting
on the material surface having the unit normal . Notice that this unit
normal is defined in the current configuration. It is also possible to define a
stress vector that is referred to a material surface in the reference configuration
and from it construct a stress tensor that is associated with that configuration.
In doing this, we parallel the development in Section 3.3 for the
Cauchy stress tensor associated with the current configuration.
Let the vector p be defined as the stress vector referred to the area
element ∆S° in the plane perpendicular to the unit normal . Just
as we defined the Cauchy stress vector in Eq 3.2-1, we write
(N) o ˆ
N= ˆ N Iˆ
A A
∆f
df
lim = =
∆
o o o
∆S
→0
S dS
( )
o ˆ
p N
(5.5-1)