Quasistatic Cracks and Minimal Energy Surfaces

We compare the roughness of minimal energy (ME) surfaces and scalar “quasistatic” fracture (SQF) surfaces. Two-dimensional ME and SQF surfaces have the same roughness scaling, $w\sim L^{\zeta }$ ( $L$ is the system size) with $\zeta =\frac{2}{3}$. The 3d ME and SQF results at strong disorder are consistent with the random-bond Ising exponent $\zeta (d\ge 3)\approx 0.21(5-d)$ ( $d$ is the bulk dimension). However, 3d SQF surfaces are rougher than ME surfaces due to a larger prefactor. ME surfaces undergo a “weakly rough” to “algebraically rough” transition in 3d, suggesting a similar behavior in fracture.