Page 161 - Mechanics Analysis Composite Materials

146 Mechanics and analysis of composite materials

where coefficients'a' are material constants characterizing its plastic behavior. And
finally, we assume the power law in Eq. (4.27) for the plastic potential.
To write constitutive equations for a plane stress state, we return to engineering
notations for stresses and strains and use conditions that should be imposed on an
orthotropic material and were discussed above in application to Eqs. (4.59). Finally,
Eqs. (4.19, (4.27) and (4.61), (4.62), (4.63) yield
1
1
El = alol +d1o2 + no"-![i(biioi +cizo2) +-(diio:+2enoioz +e2141 ,
R;
I
1
E ~ = ~ I C T ~ + ~ ~ Q I+no"-' [&(b2202 + CIm) +7(d22o: +2e2Iwl+ el2o:) ,
R2

(4.64)
where

Deriving Eqs. (4.64) we used new notations for coefficients and restricted ourselves
to the three-term approximation for o as in Eq. (4.63).
For independent uniaxial loading along the fibers, across the fibers, and in pure
shear, Eqs. (4.64) reduce to

If nonlinear material behavior does not depend on the sign of normal stresses, then
dll = d22 =0 in Eqs. (4.65). In the general case, Eqs. (4.65) allow us to describe
material with high nonlinearity and different behavior under tension and compres-
sion.
As an example, consider a boron-aluminum unidirectional composite whose
experimental stress-strain diagrams (Herakovich, 1998) are shown in Fig. 4.17