Kroner

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But because of cubic symmetry, c 66 = c 44 , so that Kroner’s coefficients ᾱ, ¯β, ¯γ are consistent with the above coefficients a, b, c, namely: ᾱ = 1 8 (5c 11 + 4c 12 ) = a (228) ¯β = − c 44 8 (7c 11 − 4c 12 ) = −b (229) ¯γ = − c 44 8 (c 11 + 2c 12 )(c 11 − c 12 ) = −c (230) as required, thus completing the derivation of Kroner’s cubic. 13 Comparison with experimental data Hashin and Shtrikman [11], [12] have derived bounds for the isotropic shear modulus of aggregates of cubic crystals. Gschneider [13] provides experimental data, and Simmons [14] provides cubic crystal elastic coefficients in Mbar. Sisodia [15] compares some experimental data with calculated shear modulus and reports the formulation of the Hashin-Shtrikman bounds, G ∗ 1 and G ∗ 2 in terms of the single crystal cubic coefficients c ij as: where ( ) −1 G ∗ 5 1 = G 1 + 3 − 4β 1 (231) G 2 − G 1 ( G ∗ 2 = G 2 + 2 5 G 1 − G 2 − 6β 2 ) −1 (232) β 1 = − 3 K + 2G 1 5 G 1 (3K + 4G 1 ) β 2 = − 3 K + 2G 2 5 G 2 (3K + 4G 2 ) An anisotropy index Ã is defined by: K = c 11 + 2c 12 3 (233) (234) (235) G 1 = c 11 − c 12 2 (236) G 2 = c 44 (237) Ã = G 2 G 1 = 2c 44 c 11 − c 12 (238) which reduces to unity for isotropic material. Table 1 shows the cubic coefficients and the anisotropy index for several materials. Notice that Tungsten (W) and Molybdenum (Mo) come closest to being isotropic. To compare the shear modulus determined by Kroner’s cubic with experimental values, we use a simple iterative Newton’s method, beginning with a trial
value at the arithmetic average of the Hashin-Shtrikman’s bounds, as described in Section 15. Table 1 shows these bounds and compares this calculated value with the experimental value. Notice that for elements which are not in Group IV-A, the error is under 20%, while for Group IV-A, the error varies from 40% to 74%. Sisodia [15] mentions that if the anisotropic index Ã > 1, then G∗ 2 > G ∗ 1 and vice versa. Notice also from Table 1, for those (most) elements where the anisotropy index exceeds unity, it is the case that G ∗ 2 > G ∗ 1. For Molybdenum and Tungsten which are practically isotropic, the bounds coincide. Finally, Niobium is the only element with Ã < 1, and there G∗ 2 < G ∗ 1. Units: kg f cm 2 Gschneider’s experimental values are given in units of × 10 6 . This is converted to GPa by multiplying by gravitational acceleration as 9.80665 m s , so 2 that Gschneider’s units are converted as follows: kg f cm 2 × 106 × 10 4 cm2 m 2 × 9.80665 m s 2 × N kg m s 2 × P a N m 2 × GP a 10 9 P a (239) and 1 kg f cm ×10 6 then corresponds to 98.0665 GPa. For example, with Aluminum, 2 Schneider’s value of 0.271 kg f cm × 10 6 corresponds to 26.6 GPa. 2
Page 1 and 2: Derivation and role of Kroner’s c
Page 3 and 4: 1 Abstract The pioneering work of H
Page 5 and 6: Such a tensor can be decomposed as
Page 7 and 8: For a constant point force F acting
Page 9 and 10: or, defining: we can write: ∫ u i
Page 11 and 12: 7 Symmetry properties and model of
Page 13 and 14: where: [k] = k 2 − k 1 (105) [µ]
Page 15 and 16: Markov now describes the critical c
Page 17 and 18: 11 Rotational averages Before proce
Page 19 and 20: Note that from Eqs. 10, 11, and 174
Page 21: Since I ′ and I ′′ are orthog
Page 25 and 26: 14 Major symmetries of tensor D Wri
Page 27 and 28: Finally, ω represents any D ijkl t
Page 29 and 30: otherwise there is no solution (Mor
Page 31: [15] P. Sisodia and M.P. Verma. She

Kroner

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