Page 62 - Mechanics Analysis Composite Materials

Chapter 2. Fundamentals of mechunics of solids 47
Symmetry conditions. Eqs. (2.50), reduce to

These equations have a simple physical meaning. The higher the stiffness is,
demonstrated by the material in some direction, the less is the strain in this direction
under loading in the orthogonal directions. Taking into account the foregoing
symmetry conditions we can conclude that an orthotropic material is characterized
with nine independent elastic constants.
The simplest material model corresponds to the isotropic material, whose
mechanical properties are the same for any direction or plane of loading. As a
result, subscripts indicating coordinate directions and planes in Eq. (2.53) disap-
pear, and it reduces to

I 1' 1'
- _- __ 0 0 0
E E E

v
__ __ I 0 0 0
1'
E E E
[CI = I (2.54)
0 0 0 - 0 0
G
1
0 0 0 0-0
G
1
0 0 0 00-
G
Compliance matrix, Eq. (2.54), contains three elastic constants, E, G, and Y.
However, only two of them are independent. To show this, consider the problem
of pure shear for a plate discussed in Section 2.4 (see Fig. 2.5). For this problem,
cr, = cr, = cr, = z, = zIz = 0,z,, = z and Eqs. (2.48) and (2.54) yield

Specific strain energy in 33q. (2.51) can be written as

However, as follows from Section 2.4, pure shear can be reduced to tension and
compression in the principal directions (see Fig. 2.5). For these directions,
Eqs. (2.48) and (2.54) give