CONTINUUM MECHANICS for ENGINEERS

where S I and S II are the bounding surfaces and V I and V II are the volumes
of Portions I and II, respectively. Also, b i are the body forces, ρ is the density,
and v i is the velocity field for the two portions. We note that S I and S II each
contain S* as part of their total areas.
The linear momentum principle may also be applied to the body B as a
whole, so that
(3.2-4)
If we add Eq 3.2-3a and Eq 3.2-3b and utilize Eq 3.2-4, noting that the normal
to S * for Portion I is ˆn , whereas for Portion II it is – ˆn , we arrive at the
equation
(3.2-5)
since both S I and S II contain a surface integral over S*. This equation must
hold for arbitrary partitioning of the body (that is, for every imaginable
cutting plane through point P) which means that the integrand must be
identically zero. Hence,
(3.2-6)
indicating that if Portion II had been chosen as the free body in Figure 3.2
instead of Portion I, the resulting stress vector would have been ( n)
.
3.3 The Stress Tensor
t dS+ bdV = d
( ˆn )
i ρ i ρvdV
∫ i
S ∫V
dt ∫V
∫
[ i i ]
*
S
t
( nˆ) −nˆ
+ t
( )
dS=0
t
( nˆ) −nˆ
=−t
( )
i i
ˆ
– ti As noted in Section 3.2, the Cauchy stress principle associates with each
direction ˆn at point P a stress vector t
( ˆn )
. In particular, if we introduce a
i
rectangular Cartesian reference frame at P, there is associated with each of
the area elements dSi (i = 1,2,3) located in the coordinate planes and having
unit normals (i = 1,2,3), respectively, a stress vector as shown in
Figure 3.4. In terms of their coordinate components these three stress vectors
associated with the coordinate planes are expressed by
ê j
i
ti t
( eˆ1) eˆ1 eˆ eˆ
eˆ 1 eˆ 1
= t
( )
+ t
( )
+ t
( )
eˆ
1 1 2 2 3 3
( ê )
(3.3-1a)