We study elastic anti-plane responses of unidirectional fiber-matrix compos
ites. The fibers are of circular cylinder shape, aligned in the axial direc
tion, and arranged randomly, with no overlap, in the transverse plane. We a
ssume that both fibers and matrix are linear elastic and isotropic. In part
icular. we focus on the effects of scale of observation and boundary condit
ions on the overall anti-plane (axial shear) elastic moduli. We conduct thi
s analysis numerically, using a two-dimensional square spring network, at t
he mesoscale level. More specifically, we consider finite "windows of obser
vation, which we increase in size. We subject these regions to several diff
erent boundary conditions: displacement-controlled, traction-controlled, pe
riodic. and mixed (combination of any of the first three) to evaluate the m
esoscale moduli. The first two boundary conditions give us scale-dependent
bounds on the anti-plane elastic moduli. For each boundary condition case w
e consider many realizations of the random composite to obtain statistics.
In this parametric study we cover a very wide range of stiffness ratios ran
ging from composites with very soft inclusions (approximating holes) to tho
se with very stiff inclusions (approaching rigid fibers), all at several vo
lume fractions.