Page 46 - Mechanics Analysis Composite Materials

Chapter 2. Fundumentul.soj’mechunics of’solidr 31

coordinate planes of the Cartesian frame. Internal and external forces acting on this
tetrahedron are shown in Fig. 2.3. The equilibrium equation corresponding, e.g., to
axis-x can be written as

Here, dS, and dV are the elements of the body surface and volume, while
dS, = dS,I,, dS, = dS,l,, and dS, = dS,lZ. When the tetrahedron is infinitely
diminished. the term including dP which is of the order of the cube of the linear
dimensions can be neglected in comparison with terms containing dS which is
of the order of the square of the linear dimensions. The resulting equation is
6,I, + ZI,l, + z:, 1: =p\ (x,y,z) . (2.2)

Symbol (x.y,z) which is widely used in this chapter denotes permutation with the
aid of which we can write two more equations corresponding to the other two axes
changing x for y, y for z, and z for x.
Consider now the equilibrium of an arbitrary finite part C of the body (see
Fig. 2.1). If we single this part out of the body, we should apply to it body forces
qc and surface tractions p, whose coordinate components pr, p,, and p, can be
expressed, obviously, by Eqs. (2.2) in terms of stresses acting inside the volume C.
Because the sum of the components corresponding, e.g., to axis-x must be equal to
zero, we have

where L’ and s are the volume and the surface area of the part of the body under
consideration. Substituting p.y from Eqs. (2.2) we get

(2.3)

i OZ
Fig. 2.3. Forces acting on an elementary tetrahedron.