Monte Carlo calculations of the effective shear modulus of a polycrystalline bcc Coulomb solid were performed by Ogata & Ichimaru . The deformation energy, resulting from the application of a specific strain , was evaluated through Monte Carlo sampling.
As we have already mentioned, for an ideal cubic crystal lattice there are only three independent elastic moduli, denoted traditionally as , and (see Chapter 3, pp. 80 – 87 of the book by Kittel ). For pure shear deformation, only two independent elastic moduli are relevant, find .
The definition of an “effective” shear modulus of a bcc polycrystal deserves a comment. In numerous papers, a standard preferred choice was ([43, 319, 285], and references therein). However, replacing by a single maximal elastic modulus of a strongly anisotropic bcc lattice is not correct. An effective value of was calculated by Ogata & Ichimaru . They performed directional averages over rotations of the Cartesian axes. At , they obtained. As expected, the effective shear modulus decreases with increasing temperature.
The formula for , Equation (116), can be rewritten as
Let us remember that the formulae given above hold for the outer crust, where the size of the nuclei is very small compared to the lattice spacing and . For the inner crust these formulae are only approximate.
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