By Speakman J
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Additional info for An introduction to the electronic theory of valency
These coefficients are always positive, since Ra 0 ≤1− ≤1 ° 1− Mb 0 ≤ Ra ≤ 1 − M b ° . 48] Already being phase-TPCFs, Raa and M bb automatically satisfy the following inequalities dRaa ( r ) ≤ 0 and dr r = 0 dM bb (r ) ≤ 0. 49] into the differentiation of Tqq gives dTqq (r ) dr = r =0 dRaa (r ) § M b · dM bb (r ) § Ra · ¨1 − ¸+ ¨1 − ¸ ≤ 0. 51] where sq must have a true sense of surface area. 52] where sa and sb are the specific surface areas of phase a and b , respectively. 52] yields, dTqq (r ) dr = r =0 dRaa ( r ) § Mb · ¨1 − ¸ dr r = 0 © 1 − Ra ¹ dM bb ( r ) § Ra + ¨1 − 1− Mb dr r =0 © · § sa + sb ¸ ≥ −¨ © α ¹ · ¸.
48] Already being phase-TPCFs, Raa and M bb automatically satisfy the following inequalities dRaa ( r ) ≤ 0 and dr r = 0 dM bb (r ) ≤ 0. 49] into the differentiation of Tqq gives dTqq (r ) dr = r =0 dRaa (r ) § M b · dM bb (r ) § Ra · ¨1 − ¸+ ¨1 − ¸ ≤ 0. 51] where sq must have a true sense of surface area. 52] where sa and sb are the specific surface areas of phase a and b , respectively. 52] yields, dTqq (r ) dr = r =0 dRaa ( r ) § Mb · ¨1 − ¸ dr r = 0 © 1 − Ra ¹ dM bb ( r ) § Ra + ¨1 − 1− Mb dr r =0 © · § sa + sb ¸ ≥ −¨ © α ¹ · ¸.
Garmestani and others [GAR 98, GAR 01, GOK 05, LI 06, LIN 00, SAH 04] have further extended the statistical continuum approach to both composites and polycrystalline materials using two-point functions. Mikdam et al. [MIK 09] have developed an approximation for the three- Approximate Solution for N-Point Correlation Functions for Heterogeneous Materials 45 point correlation functions based on two-point functions. In other research, Mikdam et al. [MIK 09] and Baniassadi et al. [BAN 11b, BAN 11c] have applied the strong-contrast formulation to predict the effective electrical and thermal conductivity of a two-phase composite material where the distribution, shape and orientation of the two phases are taken into account using two-point and three-point correlation functions.