Fractal Pattern Formation at Elastic-Plastic Transition in Heterogeneous Materials

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 nonfractal strict-white-noise field on a $256×256$ 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 static) 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 toward 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 in the model with isotropic grains alone is sufficient to generate fractal patterns at the transition but has a weaker effect than the randomness in yield limits. As the random fluctuations vanish (i.e., the composite becomes a homogeneous body), a sharp elastic-plastic transition is recovered.