Continuum Formulations of Conservation Laws - Continuum Mechanics of Anisotropic Materials - page 63

Fig. 3.8 An illustration for
problem 3.3.4

x 1

β

x 2

3.3.3. Repeat the arguments of this section and show that

t ðnÞ 2 ¼ T 21 n 1 þ T 22 n 2 þ T 23 n 3

List each of the arguments and rules or facts used in the proof.
3.3.4. The roughly triangular-shaped region in Fig. 3.8 with the included angle
b

represents the upper portion of a dam. The zigzag lines at the bottom of the
triangular region are an indication that the dam extends beyond those zigzag
lines. Find the stress vectors acting on the face x 1 ¼ 0 and on the slanted face

of the wedge shown in the Fig. 3.8 . The stress matrix at the typical point x 1 ,
x 2 , x 3 of the wedge shown in Fig. 3.8 is given by

x 1 þ x 2 ;

P
tan

g
tan 3

2 g

tan 2

T 11 ¼g x 2 ; T 22 ¼

b b

P

b

T 21 ¼ g

x 1
tan 2

T 12 ¼

b ;

T 33 ¼ T 31 ¼ T 13 ¼ T 23 ¼ T 32 ¼ 0 :

3.4 The Stress Equations of Motion

In continuum mechanics the stress equations of motion are the most useful form of
the principles of balance of linear and angular momentum. The stress equations of
motion are statements of Newton's second law (i.e., that force is equal to mass
times acceleration) written in terms of stress.
The forces that act on the object in Fig. 3.9 are the surface traction t ( x , t ), which
acts at each boundary point, and the action-at-a-distance force d , which represents

forces such as gravity and the effect of electromagnetic forces on charges within the

r