Page 159 - Mechanics Analysis Composite Materials

144 Mechanics and analysis of composite materials

material under study is orthotropic. This means that normal stresses do not induce
shear strain, while shear stress does not cause normal strains. And third, the
direction of shear stresses should influence only shear strains, i.e., shear stresses
should have only even powers in constitutive equations for normal strains, while the
corresponding equation for shear strain should include only odd powers of shear
stresses. As a result, constitutive equations will contain 37 coefficients and acquire
the following form (in new notations for coefficients and stresses):

81 = +azo: +~30:+ ~40: +~5a:+diol
+2d201~2+40:+ 3d4aio2 + d5o: +d601C: +4d76io2 + 3d80:0:
+2dsaio; +dloo: + 5dllofaz+4dl2oia: +3d1340: +2dI4alu:
+dlso: + kla142+k202~:2 + 3k347:~+4k4a:T:z + 2ksalTf2,
82 = b1o2 +b24 +b3a; +b44 +bsa;
+dl al +d28 + 2d3a1o2+d4~:+3ds01o22+dko:a2 +d7al
(4.59)
+ 2dsa:oz +3d90:02 +4dlool0; +dllo: +2d126:~~
+ 3di3o:a$ +2d14a:a; + 5dlsolo; +rn1022:~ +k20lz:~
+ 3rnz~+:~+4rn30:z:~ +2rn4ozz?,,
YU = ~1712+ ~ 2 +c3zi2 +klzi2a: + rn~ma;
4
~
+24znoia2 +2k3z12a: +2rn2212o:
+2k4z12of+4k5zi20: +2rn3zlzo': +4rn47i2a: .

For unidirectional composites, dependence 81(01) is linear which means that we
should put d2 = 1.. dl5 = 0, kl = - -.k5 = 0. Then, the foregoing equations
reduce to

As an example, consider a special two-matrix fiberglass unidirectional composite
with high in-plane transverse and shear deformation (see Section 4.3 where it is
described in details). Stress-strain curves corresponding to transverse tension,
compression, and in-plane shear are shown in Fig. 4.16. Solid lines correspond to
Eqs. (4.60) used to approximate experimental results (circles in Fig. 4.16). Coeffi-
cients a1 and dl in Eqs. (4.60) are found using diagrams EI (01) and ~~(01)which are
linear and not shown here. Coefficients bl .. bs and CI, c2, c3 are determined using
the least-squares method to approximate curves o: (EZ),07 (Q), and z12(yI2). The