Page 190 - Introduction to Continuum Mechanics

Stress Tensor 175

Fig. 4.2

Let a small tetrahedron be isolated from the body with the point P as one of its vertices (see
Fig. 4.2). The size of the tetrahedron will ultimately be made to approach zero volume so that,
in the limit, the inclined plane will pass through the point P. The outward normal to the face
PAB is -ej. Thus, the stress vector on this face is denoted by t_ e and the force on the face is

where A^j is the area of PAB. Similarly, the forces acting onPBC, PAC and the inclined face
ABC are

and

respectively. Thus, from Newton's second law written for the tetrahedron, we have

Since the mass m = (density)(volume), and the volume of the tetrahedron is proportional to
the product of three infinitesimal lengths, (in fact, the volume equals to (1/6) AjcjA^A*/}), when
the size of the tetrahedron approaches zero, the right hand side of Eq. (i) will approach zero