Page 59 - Mechanics Analysis Composite Materials

44 Mechanics and analysis of composite materials

Comparing this result with Eq. (2.41) we arrive at Green's formulas

au
(r..-- (2.42)
6J -a&,

that are valid for any elastic material. Function U(eij)can be referred to as specific
strain energy (energy accumulated in the unit of body volume) or elastic potential.
Potential U can be expanded into the Taylor series with respect to strains, Le.,

where

(2.44)

Assume that in the initial state of the body corresponding to zero external forces we
have zij = 0, aij = 0, U = 0. Then, SO = 0 and sij = 0 according to Eq. (2.42). For
small strains, we can neglect high-order terms in Eq. (2.43) and restrict ourselves to
the first system of nonzero terms taking

Then, Eq. (2.42) yields

-
(r.. - s.ijklekl * (2.45)
IJ
These linear equations correspond to a linear elastic model of the material
(see Section 1.1) and, in general, include 34 = 81 coefficients s. However, because
oij= ojf and &ij = eji, we have the following equations Sf$[ =Sjikl = sijlk which
reduce the number of independent coefficientsto 36. Then, taking into account the
fact that the mixed derivative specifying coefficients Sijkl in Eqs. (2.44) does not
depend on the sequence of differentiation we get 15 equations sijkl = skiii (ij# kl).
Thus, Eq. (2.45)contains only 2 1 independent coefficients. Returning to coordinates
x,y, z we can write Eq. (2.45) in the following explicit form

(2.46)
where