Page 54 - Mechanics Analysis Composite Materials

Chapter 2. Fundamentals of mechanics of solids 39
Linear approximation of Eq. (2.26) similar to Eq. (2.21) is

(2.27)

Here, E,. E,, er and yreV,yr,, ylr components are determined with Eqs. (2.22). If we
direct now element LM along the x-axis and element LN along the y-axis putting
I, = 1, I, = I, = 0 and 1: = 1, 1: = ZL = 0, Eq. (2.27) yields y = yr,. Thus, yr,, y,:,
and 7,: are shear strains that are equal to the changes of angles between axes x and
y. x and z, y and z, respectively.

2.6. Transformation of small strains
Consider small strains in Eqs. (2.22) and study their transformation under
rotation of the coordinate frame. Assume that x', y', 1 in Fig. 2.4 form a new
coordinate frame rotated with respect to original frame x, y, z. Because Eqs. (2.22)
are valid for any Cartesian coordinate frame, we have

(2.28)

Here, u,,, u,,. and u? are displacements along the axes 2,y', 2 which can be linked
with displacements u,, u,,, and u, of the same point by the following linear
relations

Similar relations can be written for derivatives of displacement with respect to
variables XI,y', 2 and x, y, z, i.e.,

(2.30)

Substituting displacements, Eqs. (2.29), into Eqs. (2.28),passing to variables I,y. z
with the aid of Eqs. (2.30), and taking into account Eqs. (2.22) we arrive at

These strain transformations are similar to the stress transformations determined
with Eqs. (2.8) and (2.9).