Strain and Stress in Polycrystalline Samples

Denoting the volume of this crystallite by Vt the average strain in this crystallite is  

The micro-strain or the Type III strain is the difference between Sj + r, g) and this average  

Obviously, the average over r of the Type III strain is zero. Let us now denote by Ng the total number (presumed large) of crystallites from the group of orientation g and define the following average and difference  

The strain 8, g) defined by Equation (77) is a macroscopic quantity called Type I strain. The difference Equation (78) between the average strain on the crystallite k of the group g and the Type I strain is a sub-macroscopic quantity called Type II strain. Obviously, the average over k of the Type II strain is also zero. Taking account of Equations (76) and (78) we can write the strain at the point r of the crystallite k from the group g as a sum of these three types of strains  

Similar expressions can be written for any strain or stress component in any reference system. In Equation (79) a,- is in fact a placeholder for a,-, e,-, cr,-, Si and also for 8h that gives the diffraction peak shift caused by the strain in a crystallite. To calculate the peak shift for a polycrystalline sample, 8h (R + r, g) given by Equation (79) must be averaged over r, k and g, where g represents those crystallite orientations for which h is parallel to y, the direction in the sample of the scattering vector. Taking account that the sample could be textured this multiple average is the following  

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