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~L^ζ ( L is the system size) with ζ0ex0ex=0ex0ex2/3. The 3d ME and SQF results at strong disorder are consistent with the random-bond Ising exponent ζ(d≥3)≈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.
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Räisänen et al. (1998) studied this question.
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