On the scatter ranges for the elastic moduli of random aggregates of general anisotropic crystals

On the estimates for the elastic moduli of random Voronoi triclinic polycrystals

“…The general Voigt-Reuss-Hill bounds on the fourth-rank macroscopic (effective) elastic tensor C e f f of a polycrystalline aggregate derived from the minimum energy principles can be given in the form [10]…”

“…Numerical calculations of Hashin-Shtrikman bounds are rather involved, especially for the low symmetry crystal aggregates [12]. The bounds of Pham [10] also have the formal form ( 4)-( 5), where C 0 = C + is chosen to minimize the upper bound in (4) under certain restrictions, including C 0 ≥ C ; while C 0 = C − is chosen to maximize the lower bound in (4) under certain restrictions, including C 0 ≤ C −1 −1 . Numerical calculations of the bounds are also rather involved.…”

“…Those numerical upper Dirichlet and lower Neumann estimates (K D , µ D and K N , µ N ) of [18], Hashin-Shtrikman bounds (K u HS , µ u HS and K l HS , µ l HS ) of [12,13], the bounds (K u , µ u and K l , µ l ) according to [10], the bounds for spherical cell polycrystals (K u s , µ u s and K l s , µ l s ) from ( 15)-( 16), Voigt-Reuss-Hill bounds (K V , µ V and K R , µ R ) from ( 8)- (12), and the selfconsistent approximations (K SC , µ SC ) from ( 17) are compared in Tabs. 1, 2.…”

“…Hence the evaluation of the possible scatter ranges for the macroscopic moduli of the random polycrystals should have the methodological and practical values. From the minimum energy or specialized variational principles, various upper and lower bounds on the effective moduli of random polycrystals have been constructed and calculated for particular crystalline aggregates [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

Estimates for the elastic moduli of 2D aggregate of hexagonal-shape orthorhombic crystals with in-plane random crystalline orientations

“…We still do not know if the effective moduli should converge to unique values as the size of RVE increases indefinitely or they should converge to some finite (though small) scatter intervals. Another interesting question is how the scatter intervals converge toward the known theoretical bounds constructed for the moduli [1][2][3][4][5][6][7][8][9][10] as the sizes of RVE increase, and if the bounds could be improved significantly.…”