For single-phase monolayer coatings, macrostress
is connected with microstructural damage caused by high-energy ion bombardment during film growth (internal stresses) and mismatch of the thermal expansion coefficients of the coating and the substrate (thermal stress) occurring during cooling from a deposition to room temperature .
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The exact criterion kind depends on the function S ([[xi].sub.11]) = [PHI] ([[sigma].sub.1], [[sigma].sub.2], [[sigma].sub.3]) determining dependence of root-mean-square deviation on macrostress state.
where [[bar.[xi]].sup.(k).sub.11] is microstress caused by single principal macrostress [[sigma].sub.k], then dispersion D ([[xi].sub.11]) can be found as sum of three dependent random quantities:
where [B.sup.k.sub.11] [equivalent to] [B.sup.kk.sub.11] = d([[bar.[xi]].sup.(k).sub.11]) is dispersions and [B.sup.km.sub.11] = cov ([[bar.[xi]].sup.(k).sub.11] [[bar.[xi]].sup.(m).sub.11]) = <[[bar.[xi]].sup.(k).sub.11] [[bar.[xi]].sup.(m).sub.11]> - <[[bar.[xi]].sup.(k).sub.11]> <[[bar.[xi]].sup.(m).sub.11]> is covariances (correlation moments) of microstresses caused by single macrostresses [[bar.[sigma]].sub.k] and [[bar.[sigma]].sub.m].
For every kind of extension six components of microstresses [bar.[[xi].sup.(1).sub.ij]] and [bar.[[xi].sup.(2).sub.ij]] and deformations caused by single macrostresses were determined.
where [sigma] is the macrostress
(Piola-Kirchhoff stress), [eta] is the microstress and [tau] is the interactive force.
where the macrostress
[sigma], the microstress [eta], and the interactive force [tau] are defined as derivatives of the free energy function
Here u denotes the macrodisplacement, [psi] the microdeformation, [rho] the macrodensity, I the microinertia, [sigma] the macrostress
, [eta] the microstress, [tau] the interactive force, x space coordinate, and t time.
Thus, our ab initio results can serve not only as a guide for interpretation of experimental Raman spectra or symmetry mode assignment in particular titanium borides, but also for estimation of effects connected with macrostresses
and their influence on positions and intensities of the measured Raman peaks.
The proof of this statement seems to be quite evident, because near the boundary surface (under loading of arbitrary type) the representative volume of material is inherent in this material basic property-property of homogeneity of fields of macrostresses
The effect of residual macrostresses
and microstresses on fatigue crack initation, Mater Sci Eng 284: 268-279.