Page 171 - Mechanics Analysis Composite Materials

156 Mechanics and analysis of composite materials

calculated with the aid of Eqs. (4.69) and (4.77). As can be seen in this figure, the
best angle is about 10". At this angle, shear strain yI2 is much higher than the normal
strains 81 and ~2,so that material deformation is associated mainly with shear. Off-
axis test with d, = 10" can be also used to evaluate material strength in shear 212
(Chamis, 1979). Stresses acting under off-axis tension in the principal material
coordinates are statically determinate and can be found directly from Eqs. (4.67) as

01 = o,cos 2 4, a2 = oxsin2 4, 212 = --a,sin&cos4 . (4.78)

Thus, applying stress a, and changing d, we can induce a proportional loading with
different combinations of stresses a],62, and 212 to evaluate constitutive or failure
theory for a material under study.
However, the test shown in Fig. 4.23 can hardly be performed because the test
fixture (see Fig. 4.22) restrains the shear deformation of the specimen. This restraint
does not demonstrate itself only if shear is not induced by the off-axis tension,
i.e., if yv = 0 in Eqs. (4.77). This means that qXys = 0 or that in accordance with
Eqs. (4.76),

This equation has two natural solutions 4 = 0 and 4 = 90" corresponding to
tension along and across the fibers. However, it can have one more solution, i.e.,

1+121 I
sin2 d, = El 2G12 (4.79)
I+m I+VlZ 1 .
EI Giz
Because 0 < sin2d, < 1, solution for this equation exists if

(4.80)

Unfortunately, these conditions cannot be satisfied for unidirectional composites.
Indeed, as follows from Eqs. (4.76), there exists the following differential relation
between compliance coefficients (Verchery and Gong, 1999):

VXJV
$(t)= -2G,,

This means that vXJy = 0 if E, reaches its extremum value within the interval
0 < 4 < 90". In other words, function E,(4) should be as shown in Fig. 4.25 with a
solid line. Then, uxsy = 0 at 4 = 4,,. But for unidirectional advanced composites
whose properties are listed in Table 3.5, the curve is similar to the broken line in
Fig. 4.25, and E, reaches its extremum values at d, = 0 and 4 = 90" only.