Fourth-order elastic moduli of polycrystals

The investigation of higher-order elastic moduli of polycrystalline solids is a challenging task. For this purpose the decomposition of the polycrystal Gibbs free energy at hydrostatic pressure in terms of the finite strain tensor components, taking into account the fourth-order contributions, is given. We give the definition of the fourth-order elastic moduli for the polycrystal (the fourth-order Lam\'e coefficients) at an arbitrary pressure. We obtain the relationships between the Lam\'e coefficients of the fourth order at pressure $P$ with the corresponding constants of the single-crystal grains constituting the polycrystal. The case of the arbitrary grain symmetry and, in particular, when the grains have a cubic lattice, is considered. The calculation results for the second-, third-, and fourth-order Lam\'e coefficients of the series metals with cubic structure grains at $P=0$ are analyzed. For polycrystalline tungsten, the dependence of the fourth-order elastic constants on pressure in the range of 0--600 GPa is presented.