Page 158 - Mechanics Analysis Composite Materials

Chapter 4. Mechanics of a composite layer I43

theory appears if we are to study the interaction of simultaneously acting stresses.
Because for the layer under study this interaction usually takes place for in-plane
stresses 01,a2, and z12 (see Fig. 4.13), we consider further the plane state of stress.
In the simplest but rather useful for practical analysis engineering approach, the
stress interaction is ignored at all, and linear constitutive equations, Eqs. (4.59, are
generalized as

(4.57)

Superscript ‘s’ indicates the corresponding secant characteristics specified by
Eqs. (1 3). These characteristics depend on stresses and are determined using
experimental diagram similar to those presented in Figs. 3.40-3.43. Particularly,
diagrams 01(81) and E~(EI)plotted under uniaxial longitudinal loading yield ,!?](a[)
and vil (a,),secant moduli E;(a2) and Gi2(z12) are determined from experimental
curves a2(~2) and qz(yI2), respectively, while $2 is found from the symmetry
condition in Eqs. (4.53). In a more rigorous model (Jones, 1977), the secant
characteristics of the material in Eqs. (4.57) are also functions but this time of strain
energy U in Eq. (2.5 1) rather than of individual stresses. Models of this type provide
adequate results for unidirectional composites with moderate nonlinearity.
To describe pronounced nonlinear elastic behavior of a unidirectional layer, we
can use Eqs. (4.10). Expanding complementary potential U, into the Taylor series
with respect to stresses we have
I 1 1
+
uc = CO + cijaij + TCijkloijOkl + -C~klmn~ij~kl~mn-Cijklmn~~ij~kl~mr~~
3! 4!
I 1
+
-b
-b -cijklmnwr.~aijakl~m,~~~~~~-Cijklmn~~~‘~ii~kl~mn~~~r.~~t~~~
’
’
’ 7
5! 6!
(4.58)
where

Sixth-order approximation with terms written in Eq. (4.58) (it implies summation
over repeated subscripts) allows us to construct constitutive equations including
stresses in the fifth power. Coefficients ‘e’ should be found from experiments with
material specimens. Because these coefficients are particular derivatives that do
not depend on the sequence of differentiation, the sequence of their subscripts is
not important. As a result, the sixth-order polynomial in Eq. (4.58) includes 84
‘c’-coefficients. Apparently, this is too many for practical analysis of composite
materials. To reduce the number of coefficients, we can first use some general
considerations. Namely, assume that U,= 0 and eij = 0 if there are no stresses
(aii = 0). Then, co = 0 and cii = 0. Second, we should take into account that the