CONTINUUM MECHANICS for ENGINEERS

where ∆f is the resultant force acting on the material surface, which in the
reference configuration was ∆S°.
The principle of linear momentum can also be written in terms of quantities
which are referred to the referential configuration as
( ) ( ) + ( ) = ( )
∫ ∫ ∫
o Nˆ
o o o o o
p X, tdS ρ0b X, tdV ρ0a
X,
tdV
o o o
S
V V
(5.5-2)
where S°, V°, and ρ 0 are the material surface, volume, and density, respectively,
referred to the reference configuration. The superscript zero after the
variable is used to emphasize the fact that the function is written in terms
of the reference configuration. For example,
o
ai(x,t) = ai[(X,t),t)] = (X,t)
Notice that, since all quantities are in terms of material coordinates, we have
moved the differential operator d/dt of Eq 3.2-4 inside the integral to give
rise to the acceleration a°. In a similar procedure to that carried out in
Section 3.2, we apply Eq 5.5-2 to Portions I and II of the body (as defined in
Figure 3.2a) and to the body as a whole to arrive at the equation
∫ o
S
Nˆ Nˆ
⎡
p
( ) −
+ p
( ) ⎤
dS
⎣⎢ ⎦⎥
o o o
= 0 (5.5-3)
This equation must hold for arbitrary portions of the body surface, and so
o N N
p
( ˆ) o − ˆ
=−p
( )
(5.5-4)
which is the analog of Eq 3.2-6.
N
The stress vector p can be written out in components associated with
the referential coordinate planes as
o( ˆ )
o Iˆ Iˆ
o
p
( A ) A
=
( )
eˆ
p i i
(A = 1,2,3) (5.5-5)
This describes the components of the stress vector p with respect to the
referential coordinate planes; to determine its components with respect to
an arbitrary plane defined by the unit vector , we apply a force balance
to an infinitesimal tetrahedron of the body. As we let the tetrahedron shrink
to the point, we have
N o( ˆ )
ˆN
ˆ ˆ
o o A
p
( N) I
=
p
( )
N
i i A
a i
(5.5-6)