Page 52 - Mechanics Analysis Composite Materials

Chapter 2. Fundamentals of mechanics oj's0lid.s 37

au au,
aU,
du, ="dx+-dy+-dz (xly,z) (2.16)
ax 8.Y az
As follows from Fig. 2.6 and Eqs. (2.15) and (2.16)

dxi = dx + u,!.')- u.,= dx + du,
= (1 +g)dx+aydy+--dz aux (x,Y,z) .
aux
az (2.17)

Introduce strain of element LM as

(2.18)

After some rearrangements we arrive at

where

Substituting for dxl, dy1, dzl their expressions from Eqs. (2.17) and taking into
account Eqs. (2.14) we finally get

where

(2.20)
au, at+ au aux aurau au-au,
= -+-+-2- +-2+ 2- (x y z)
ay ax ax ay ax ay ax ay ' *

Assuming that the strain is small we can neglect the second term in the left-hand side
of Eq. (2.19). Moreover, we further assume that the displacements are continuous
functions that change rather slowly with the change of coordinates. This allows us
to neglect the products of derivatives in Eqs. (2.20). As a result, we arrive at the
following equation