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| '''Virial stress''' is a measure of [[Stress (physics)|mechanical stress]] on an atomic scale. It is given by
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| :<math>\tau_{ij} = \frac{1}{\Omega} \sum_{k \in \Omega} \left(-m^{(k)} (u_i^{(k)}- \bar{u}_i) (u_j^{(k)}- \bar{u}_j) + \frac{1}{2} \sum_{\ell \in \Omega} ( x_i^{(\ell)} - x_i^{(k)}) f_j^{(k\ell)}\right)</math>
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| where
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| * <math>k</math> and <math>\ell</math> are atoms in the domain,
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| * <math>\Omega</math> is the volume of the domain,
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| * <math>m^{(k)}</math> is the mass of atom ''k'',
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| * <math>u_i^{(k)}</math> is the ''i''<sup>th</sup> component of the velocity of atom ''k'',
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| * <math>\bar{u}_j</math> is the ''j''<sup>th</sup> component of the average velocity of atoms in the volume,
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| * <math>x_i^{(k)}</math> is the ''i''<sup>th</sup> component of the position of atom ''k'', and
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| * <math>f_i^{(k\ell)}</math> is the ''i''<sup>th</sup> component of the force applied on atom <math>k</math> by atom <math>\ell</math>.
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| At [[absolute zero|zero kelvin]], all velocities are zero so we have
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| :<math>\tau_{ij} = \frac{1}{2\Omega} \sum_{k,\ell \in \Omega} ( x_i^{(\ell)} - x_i^{(k)}) f_j^{(k\ell)}</math>.
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| This can be thought of as follows. The τ<sub>11</sub> component of stress is the force in the ''x''<sub>1</sub>-direction divided by the area of a plane perpendicular to that direction. Consider two adjacent volumes separated by such a plane. The 11-component of stress on that interface is the sum of all pairwise forces between atoms on the two sides.
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| ==See also==
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| * [[Virial theorem]]
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| ==External links==
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| * [http://www.eng.fsu.edu/~dommelen/papers/virial/mosaic/index.html Physical Interpretation of the Virial Stress]
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| * [http://adsabs.harvard.edu/abs/2003RSPSA.459.2347Z A new look at the atomic level virial stress: on continuum-molecular system equivalence]
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| [[Category:Continuum mechanics]]
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| {{engineering-stub}}
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