Euler strain tensor

After combining Equation (124) with Equation (65a) or with (66a) and (64a) into Equation (81) there are 936 integrals to calculate on the Euler space of the products of the functions Py and R. Although an analytical calculation is possible they were calculated numerically. Only 73 are different from zero. The macroscopic strain tensor is the following ... 

There is a variety of methods available to coating, substrate and interface stresses. For metallic films. X-ray diffraction, in which the diffraction angle of X-rays is correlated to the internal strain in the crystalline structure of the coating, can be used to characterize the residual stress state. A common method is the so-called sin method, normally used when anisotropy is expected in the residual stress. The elastic lattice strain is measured via X-ray diffraction with respect to Euler angles, (p and xj/ as related to the Cartesian strain tensor, and finally to the residual stress tensor (see, for example, [69,70]). 

Finally, the Hooke equations in a crystallite can be written by using the eomponents of the strain and stress tensors in the sample reference system. Denoting by g the triplet of Euler angles (q>i, Oq, q>2) and using Equations (64-66) we have ... 

It is noted that within the Euler-Bemoulli theory context, it holds that 6 = v, 6y= — w (the cross section remains perpendicular to the deformed axis) hence, shear strain components Eqs. 3b and 3c vanish. Considering strains to be small and employing the second Piola-Kirchhoff stress tensor, the nonvanishing stress components are defined in terms of the strain ones as... 

---