CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.5
Free body diagram of tetrahedron element having its vertex at P.
where *bi is an average body force which acts throughout the body. The negative
signs on the coordinate-face tractions result from the outward unit normals on
those faces pointing in the negative coordinate axes directions. (Recall that
t
( −nˆ)
t
( nˆ
=−
)
). Taking into consideration Eq 3.3-3, we can write Eq 3.3-4 as
i i
(3.3-5)
*
if we permit the indices on the unit vectors of the t
( ê j )
i term to participate
in the summation process. The volume of the tetrahedron is given by dV =
1
h dS, where h is the perpendicular distance from point P to the base ABC.
3
Inserting this into Eq 3.3-5 and canceling the common factor dS, we obtain
(3.3-6)
Now, letting the tetrahedron shrink to point P by taking the limit as h → 0
and noting that in this limiting process the starred (averaged) quantities take
on the actual values of those same quantities at point P, we have
j
or, by defining ≡ t
( ê )
,
σ ji i
t
( )
dS- t
( )
n dS + ρ b dV = 0
* nˆ
* eˆ
j
*
i i j i
t
( )
= t
( ) 1
n − bh
3 ρ
* nˆ
* eˆ
j
*
i i j i
t
( nˆ
) eˆ
j
=t
( )
n
i i j
(3.3-7)
t
( ˆn )
= σ n or t
( n ˆ )
= nˆ⋅σ
(3.3-8)
i ji j
which is the Cauchy stress formula. We can obtain this same result for bodies
which are accelerating by using the conservation of linear momentum
instead of a balance of forces on the tetrahedron of Figure 3.5.