Roughness Scaling of Fracture Surfaces in Polycrystalline Materials
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Roughness Scaling of Fracture Surfaces in Polycrystalline Materials Eira T. Sepp¨al¨a, Bryan W. Reed, Mukul Kumar, Roger W. Minich, and Robert E. Rudd Lawrence Livermore National Laboratory, L-415, Livermore, CA 94551, U.S.A. ABSTRACT The roughness scaling of fracture surfaces in two-dimensional grain boundary networks is studied numerically. Grain boundary networks are created using a Metropolis method in order to mimic the triple junction distributions from experiments. Fracture surfaces through these grain boundary networks are predicted using a combinatorial optimization method of maximum flow − minimum cut type. We have preliminary results from system sizes up to N = 22500 grains suggesting that the roughness scaling of these surfaces follows a random elastic manifold scaling exponent ζ = 2/3. We propose a strong dependence between the energy needed to create a crack and the special boundary fraction. Also the special boundaries at the crack and elsewhere in the system can be tracked. INTRODUCTION Fracture in random media has been of considerable interest in the field of theoretical statistical mechanics for a few decades [1, 2], especially in terms of roughness scaling and toughness properties of crack surfaces. However, the statistical mechanics studies of scaling of fracture surfaces in random media have been conducted using simplified, arbitrary models, such as analytical distributions to describe randomness, and regular lattices for the structure of the material. One attempt to study more realistic systems is a recently published study of scaling laws for manifolds in polycrystalline materials, which used a simple model for transgranular and intergranular cracks, but neglected any dynamical effects [3]. Here we propose that polycrystalline materials with varying grain boundary strengths provide a good example of random media [4, 5, 6], where only intergranular fracture is allowed. The orientation of the grains and boundaries can be essentially random, giving rise to a distribution of strengths among grains. We study fracture in polycrystalline materials by generating grain boundary networks using a Metropolis method in order to mimic the triple junction distributions for special boundaries from experimental samples. The final fracture surface and its properties are predicted from the structure of the material, in particular from the topology of the grain boundary network and the strength properties of the different boundaries in the network. The method used here for predicting the final fracture surface has been developed in statistical mechanics and shown to be very efficient [7, 8]. It finds the weakest (minimum energy) path through the material using a combinatorial optimization method [9]. In this paper we especially wish to study how the properties of the predicted fracture surfaces are dependent on the fraction of special boundaries and whether the scaling of roughness w with respect to the system size L follows the random elastic manifold universality class w ∼ Lζ , with the characteristic r
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