  Fractal patterns are observed in computational mechanics of elastic-plastic
transitions in two models of linear elastic/perfectly-plastic random
heterogeneous materials: (1) a composite made of locally isotropic grains with
weak random fluctuations in elastic moduli and/or yield limits; and (2) a
polycrystal made of randomly oriented anisotropic grains. In each case, the
spatial assignment of material randomness is a non-fractal, strict-white-noise
field on a 256 x 256 square lattice of homogeneous, square-shaped grains; the
flow rule in each grain follows associated plasticity. These lattices are
subjected to simple shear loading increasing through either one of three
macroscopically uniform boundary conditions (kinematic, mixed-orthogonal or
traction), admitted by the Hill-Mandel condition. Upon following the evolution
of a set of grains that become plastic, we find that it has a fractal dimension
increasing from 0 towards 2 as the material transitions from elastic to
perfectly-plastic. While the grains possess sharp elastic-plastic stress-strain
curves, the overall stress-strain responses are smooth and asymptote toward
perfectly-plastic flows; these responses and the fractal dimension-strain
curves are almost identical for three different loadings. The randomness in
elastic moduli alone is sufficient to generate fractal patterns at the
transition, but has a weaker effect than the randomness in yield limits. In the
model with isotropic grains, as the random fluctuations vanish (i.e. the
composite becomes a homogeneous body), a sharp elastic-plastic transition is
recovered.
