Page 322 - Mechanics Analysis Composite Materials

Chapter 7. Enl'i~JnV7enId. special lotding. and mflnufflcfitring effects 307

As known, heating gives rise to thermal strains which, being restricted, induce
thermal stresses. Assume that the temperature distribution in a composite structure
is known and consider the problem of thermoelasticity.
Consider first a thermoelastic behavior of a unidirectional composite ply studied
in Section 3.3 and shown in Fig. 3.29. The generalized Hooke's law, Eqs. (3.58)
allowing for the temperature effects can be written as:

CIT + El T , &?T = E? + &?. 712T = 712 . (7.12)
T
Here and further subscript "T" shows the strains that belong to the problem of
thermoelasticity, while superscript "T" indicates temperature terms. Elastic strains
cl,cz and *ill in Eqs. (7.12) are linked with stresses by Eqs. (3.58). Temperature
strains. in the first approximation. can be taken as linear functions of the
temperature change, Le.
= XIAT .
~f = !.XIAT, C: (7.13)
where 21 and E: are coefficients of thermal expansion (CTE) along and across the
fibers and AT = T - Ti is the difference between the current temperature T and
some initial temperature TO at which thermal strains are zero. Inverse form of
Eqs. (7.12) is:
T
61 El (CIT + 1?12Fzr) - El (8; + \'1282),
(7.14)
62 = EI(E?T + Y21EII') -E?(&; + \b1&;):
TI: = Gi2Yll.r
where E1.2 = El,2/(l - Y~IV?~).
To describe thermoelastic behavior of a ply, apply the first-order micromechani-
cal model shown in Fig. 3.34. Because CTE (and elastic constants) of some fibers
can be different in longitudinal and transverse directions generalize the first two
equations of Eqs. (3.63) as:

(7.15)

Repeating the derivation of Eqs. (3.76)-(3.79) we arrive at