Page 142 - Mechanics Analysis Composite Materials

Chapter 4. Mechanics of a composite layer 127

1
+
1
1
u,= c0 +c~~I~-clzif +-cI31; +-cI4if + ...
2 3! 4!
1 1 I
+ C2li2 + -c2& +--??I; + -c,4i; + ...
2 3! -. 4!
1
1
1
+-CI~ZIZII~+-c1221iji23 +-~112211122
2 3! 3!
1 1 1
+-cI3,1.l:i2 +-c12,21:i; +-c1123i1i;3 + ... , (4.11)
4! 4! 4!
where

Constitutive equations follow from Eq. (4.10) and can be written in the form
au, ai, au, aiz
E.. (4.12)
‘I - ail aoij ai, aoii .
Assuming that for zero stresses U, = 0 and cij = 0 we should take co = 0 and CII = 0
in Eq. (4.11).
Consider a plane stressed state with stresses o.~,o,,,z.~! shown in Fig. 4.5.Stress
invariants in Eqs. (2.13) entering Eq. (4.12) are

Linear elastic material model is described with Eq. (4.1 1) if we take
u, =fC12i; + C2II2 . (4.14)

Using Eqs. (4.12)-(4.14) and engineering notations for stresses and strains, we
arrive at

8.r = c12(o.v +ox) - C,Iql., 4;= c12(o., + oy)- c2lo.r’ y.vj. = 2C21Z,,.
These equations coincide with the corresponding equations in Eqs. (4.6) if we take

1 I +V
c12 = - c,1 = -
E’ E ’
To describe nonlinear stress-strain diagram of the type shown in Fig. 4.6,wc can
generalize Eq. (4.14) as

u.- -c12i; + C2lZ2 + -C14Z14 + -c22z, 2 .
1
1
1
‘-2 4! 2