Fracture roughness in three-dimensional beam lattice systems

We study the scaling of three-dimensional crack roughness using large-scale beam lattice systems. Our results for prenotched samples indicate that the crack surface is statistically isotropic, with the implication that experimental findings of anisotropy of fracture surface roughness in directions parallel and perpendicular to crack propagation is not due to the scalar or vectorial elasticity of the model. In contrast to scalar fuse lattices, beam lattice systems do not exhibit anomalous scaling or an extra dependence of roughness on system size. The local and global roughness exponents (${\ensuremath{\zeta}}_{loc}$ and $\ensuremath{\zeta}$, respectively) are equal to each other, and the three-dimensional crack roughness exponent is estimated to be ${\ensuremath{\zeta}}_{loc}=\ensuremath{\zeta}=0.48\ifmmode\pm\else\textpm\fi{}0.03$. This closely matches the roughness exponent observed outside the fracture process zone. The probability density distribution $p[\ensuremath{\Delta}h(\ensuremath{\ell})]$ of the height differences $\ensuremath{\Delta}h(\ensuremath{\ell})=[h(x+\ensuremath{\ell})\ensuremath{-}h(x)]$ of the crack profile follows a Gaussian distribution, in agreement with experimental results.