CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.4
Traction vectors on the three coordinate planes at point P.
t
( eˆ2) eˆ2 eˆ eˆ
eˆ 2 eˆ 2
= t
( )
+ t
( )
+ t
( )
eˆ
or more compactly, using the summation convention
1
1 2 2 3 3
t
( eˆ3) eˆ3 eˆ eˆ
eˆ 3 eˆ 3
= t
( )
+ t
( )
+ t
( )
eˆ
1 1 2 2 3 3
( ) ( )
eˆ i eˆ
i t = t eˆ
i = 123 , ,
j j
( )
(3.3-1b)
(3.3-1c)
(3.3-2)
This equation expresses the stress vector at P for a given coordinate plane
in terms of its rectangular Cartesian components, but what is really needed
is an expression for the coordinate components of the stress vector at P
associated with an arbitrarily oriented plane. For this purpose, we consider
the equilibrium of a small portion of the body in the shape of a tetrahedron
having its vertex at P, and its base ABC perpendicular to an arbitrarily
oriented normal n= ˆ eˆ
as shown by Figure 3.5. The coordinate directions
are chosen so that the three faces BPC, CPA, and APB of the tetrahedron
are situated in the coordinate planes. If the area of the base is assigned the
value dS, the areas of the respective faces will be the projected areas
dSi = dS cos( ), (i = 1,2,3) or specifically,
ni i
ˆ , ˆ n ei for BPC dS 1 = n 1dS (3.3-3a)
for CPA dS 2 = n 2dS (3.3-3b)
for APB dS 3 = n 3dS (3.3-3c)
The stress vectors shown on the surfaces of the tetrahedron of Figure 3.5
represent average values over the areas on which they act. This is indicated
in our notation by an asterisk appended to the stress vector symbols (remember
that the stress vector is a point quantity). Equilibrium requires the vector
sum of all forces acting on the tetrahedron to be zero, that is, for,
t
( )
dS- t
( )
dS − t
( )
dS − t
( )
dS + ρ b dV = 0
* nˆ * eˆ 1 * eˆ 2 * eˆ
3
*
i i 1 i 2 i 3 i
(3.3-4)