Page 51 - Mechanics Analysis Composite Materials

36 Mechanics and analysis of composite materials

supplement this set with Eq. (2.11). The final equations allowing us to find 1, and
IF,, are

7
2
F7lpx+ 7 Ip-" = 0, lp.x+ l& = 1 .
Solution of these equations yields lp.x= &l/.\/Z and 1, = rl/& and means that
principal planes (or principal axes) make 45" angles with axes x and y. Principal
stresses and principal coordinates XI, x2, x3 are shown in Fig. 2.5.

2.5. Displacements and strains

For any point of a solid (e.g., L or M in Fig. 2.1) introduce coordinate component
displacements u,, uV,and u, specifying the point displacements in the directions of
coordinate axes.
Consider an arbitrary infinitely small element LM characterized with its
directional cosines

dx dY dz
1 --, lv = , Iz=-. (2.14)
"-6 ds
Positions of this element before and after deformation are shown in Fig. 2.6.
Assume that displacements of the point L are u,, u,., and uZ. Then, displacements
of the point M should be

UL') = U, + dux, u.!,')= u,,+ du,,, u!') = U, + duZ . (2.15)

Since uxr uy, and uz are continuous functions of x, y, z

Fig. 2.6. Displacement of an infinitesimal linear element.