Other mechanisms of stress evolution

Stress arising from reduction in grain boundary area 

The expressions are conveniently expressed in terms of grain size, rather than grain boundary trace length, by noting that 

The change in surface energy per unit volume due to both the free 

If the boundary surface of the grain in a film is constrained against 

The vacancy volume can be defined in the following way. If an initially perfect crystal is altered by inserting a number density p of vacancies per unit volume throughout its interior without changing the number of atoms in the crystal, then the crystal undergoes a stress-free isotropic expansion if its boundaries are unconstrained. If the change in volume per unit volume is expressed as pfivac, then fivac is the volume of a vacancy its value is commonly assumed to be between one-half and one times the atomic volume of the material. If the distribution is not dilute, then the value of vacancy volume could depend on the density itself. 

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