MODELING OF MULTIBRANCHED CROSSLIKE CRACK-GROWTH

Multibranched crosslike crack patterns formed in concentrically loaded square plates are studied in terms of fractal geometry, where the associated fractal dimension d(f) is calculated for their characterization. We apply the simplest deterministic and stochastic approaches at a phenomenological level in an attempt to find generic features as guidelines for future experimental and theoretical work. The deterministic model for fracture propagation we apply, which is a variant of the discretized Laplace approach for randomly ramified fractal cracks proposed by Takayasu [Phys. Rev. Lett. 54, 1099 (1985)] reproduces the basic ingredients of observed complex fracture patterns. The stochastic model, although not strictly a model for crack propagation, is based on diffusion-limited aggregation (DLA) for fractal growth and produces a slightly more realistic assessment of the crosslike growth of the cracks in asymmetric multibranches. Nevertheless, this simple ad hoc DLA version for modeling the present phenomena as well as the deterministic approach for fracture propagation give fractal dimensionality for the fracture patterns in accord with our estimations made from recent experimental data. It is found that there is a crossover of two fractal dimensions corresponding to the core (higher d(f)) and multibranched crosslike (lower d(f)) regions that contains loops, which are interpreted as representing different symmetry regions within the square plates of finite size.