Page 48 - Mechanics Analysis Composite Materials

Chapter 2. Fundamentals of mechanics of solids 33

located inside the body and that point B coincides with the origin 0 of Cartesian
coordinates x. y, z in Fig. 2.1. Then, the oblique plane of the tetrahedron can be
treated as a coordinate plane z‘ = 0 of a new coordinate frame x’, y’, 1 shown in
Fig. 2.4 and such that the normal element to the oblique plane coincides with the
z’-axis, while axes x’ and y’ are located in this plane. Component P,~of the surface
traction in Eqs. (2.2) can be treated now as the projection on the x-axis of stress (T
acting on plane z’ = 0. Then, Eqs. (2.2) can be presented in the following explicit
form specifying projections of stress (T

Here, I are directional cosines of axis z’ with respect to axes x, y, and z (see Fig. 2.4
in which the corresponding cosines of axes x’ and y’ are also presented). Normal
stress o=,can be found now as

The final result was obtained with the aid of Eqs. (2.6)and (2.7). Changing x’for y’,
y‘ for 2. and z’ for x’, Le., performing permutation in Eq. (2.8) we can write similar
expressions for a.,~and cy!.

Fig. 2.4. Rotation of the coordinate frame.