Arnold's cat map

Virial stress is a measure of mechanical stress on an atomic scale. It is given by

$\tau _{ij}={\frac {1}{\Omega }}\sum _{k\in \Omega }\left(-m^{(k)}(u_{i}^{(k)}-{\bar {u}}_{i})(u_{j}^{(k)}-{\bar {u}}_{j})+{\frac {1}{2}}\sum _{\ell \in \Omega }(x_{i}^{(\ell )}-x_{i}^{(k)})f_{j}^{(k\ell )}\right)$

where

• $k$ and $\ell$ are atoms in the domain,
• $\Omega$ is the volume of the domain,
• $m^{(k)}$ is the mass of atom k,
• $u_{i}^{(k)}$ is the i^th component of the velocity of atom k,
• ${\bar {u}}_{j}$ is the j^th component of the average velocity of atoms in the volume,
• $x_{i}^{(k)}$ is the i^th component of the position of atom k, and
• $f_{i}^{(k\ell )}$ is the i^th component of the force applied on atom $k$ by atom $\ell$.

At zero kelvin, all velocities are zero so we have

$\tau _{ij}={\frac {1}{2\Omega }}\sum _{k,\ell \in \Omega }(x_{i}^{(\ell )}-x_{i}^{(k)})f_{j}^{(k\ell )}$.

This can be thought of as follows. The τ₁₁ component of stress is the force in the x₁-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.

See also

• Virial theorem

External links

• Physical Interpretation of the Virial Stress
• A new look at the atomic level virial stress: on continuum-molecular system equivalence

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