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, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}}$ ( $L$ is the system size) with $\ensuremath{\zeta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{2}{3}$. The 3d ME and SQF results at strong disorder are consistent with the random-bond Ising exponent $\ensuremath{\zeta}(d\ensuremath{\ge}3)\ensuremath{\approx}0.21(5\ensuremath{-}d)$ ( $d$ 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.