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Unformatted text preview: The fracture energy of brittle crystals S. W. Freiman • J. J. Mecholsky Jr. Received: 1 December 2009 / Accepted: 6 April 2010 / Published online: 22 April 2010 Ó Springer Science+Business Media, LLC 2010 Abstract An expression is derived for the fracture energy, c , in brittle crystals, namely, c = kd E , d being the lattice spacing, E Young’s modulus perpendicular to the fracture plane, and k is a constant. The value of c obtained through this expression is compared to experi mental data for cubic crystals. Despite the fit, we conclude that because the fracture energy is dominated by the elastic constant, comparisons between a computed c and experi mental data cannot be used to distinguish between bonding functions. Introduction The resistance to fracture of a brittle material, i.e., its fracture energy, c , is governed by the energy needed to form the two fracture surfaces. If the fracture process involves no other energyabsorbing mechanisms, i.e., plastic deformation, heat, surface bond reconstruction, etc., the energy required to form these surfaces is that required to separate two atomic planes. Under these conditions, if the stress–strain expression, i.e., bonding function gov erning the behavior of the material is known, one should be able to calculate the fracture energy through the integration of the function as the planes are separated. A number of stress–strain functions have been suggested to apply to inorganic crystals, including the sine function used by Gilman [ 1 ], a Morse type function used by Tromans and Meech [ 2 ] to model the fracture toughness of ionically bonded minerals, a Born model used by these same authors to calculate the fracture energy of covalent materials [ 3 ], and the Universal BindingEnergy Relation [ 4 , 5 ]. Based on dimensional considerations, the form of the final expression for fracture energy is expected to be similar for all such functions. In this article, we will demonstrate that while there is good correspondence between models and experimental data on cubic systems, this agreement alone cannot be used to verify the accuracy of such models. Modeling fracture Fracture can be modeled as two planes being separated by a tensile stress applied perpendicular to them. If this process is carried out reversibly such that no energy is lost to other processes, e.g., dislocation generation, heat loss, etc., the energy needed for plane separation is the fracture energy. Note that the presence of a crack serves to concentrate stress on the bonds at a crack tip, but is not a fundamental requirement of the model. The separation, d , of two planes of atoms from their equilibrium value, d , is given by 2 c ¼ U ð1Þ ÿ U ð d Þ ¼ Z 1 d r R ð Þ d R ð 1 Þ U ( d ) is the equilibrium energy of the crystal, and U ( ? ) is the energy at complete separation. R : d ? d , and r ( R ) is the particular restoring stress governing separation of the planes....
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 Fall '08
 Mecholsky
 Materials Science, Energy, fracture energy, J Mater Sci, J. J. Mecholsky Jr

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