## What is virial stress?

The concept of stress in a body assumed to be a continuum is well known; the Cauchy stress tensor, in such a body, denoted by $\tau_{ij}$ is the force in the $j$-th direction, per unit area, with the normal to the area element is in the $i$-th direction.

However, if suppose the body under investigation cannot be assumed to be a continuum (as, say, in a molecular dynamics simulation); is there a corresponding quantity? Or, in other words, is there a discrete equivalent of the continuum Cauchy stress? The answer is yes, and that quantity is known as the virial stress.

So far, so good. However, there is a controversy in how the virial stress itself is defined: for example, the Wiki entry on virial stress defines it as follows:

$\tau_{ij} = \frac{1}{\Omega} \sum_{k\in\Omega} [\tau_{kin} + \tau_{pot}]$,

where, $\Omega$ is the volume, $k$ is the atoms in the volume $\Omega$, and $\tau_{kin}$ and $\tau_{pot}$ are the kinetic and potential contributions to the virial stress respectively, and are defined as follows:

$\tau_{kin} = -m^{(k)} (u_i^{(k)} - \bar{u_{i}}) (u_j^{(k)} - \bar{u_{j}})$

where, $m^{(k)}$ is the mass of atoms $k$, $u_{\alpha}^{(k)}$ is the $\alpha$-th component of the velocity of the atom $k$, and, $\bar{u_{\alpha}}$ is the $\alpha$-th component of the average velocity of atoms in the volume; and,

$\tau_{pot} = \frac{1}{2} \sum_{\ell \in \Omega} (x_i^{(\ell)} - x_{i}^{(k)}) f_{j}^{(k \ell)}$

where, $\ell$ are the atoms in the volume $\Omega$, $x_i^{(\beta)}$ are the $i$-th component of the position of the $\beta$-th atom, and $f_{i}^{(k\ell)}$ is the $i$-th component of the force between the atoms $k$ and $\ell$.

Calculating stresses in MD simulations is a controversial topic. There are two different schools of thought about the equivalence of the virial stress to the continuum Cauchy stress; for and against. Some argue based on momentum balance, that only the potential contribution to the virial stress should be considered as the continuum Cauchy stress. However, others assert that the total virial stress that contains both the kinetic and potential parts is indeed the quantity that corresponds to the Cauchy stress in continuum mechanics. We used a simple thermo-elastic analysis to verify the validity of using the total virial stress as the continuum Cauchy stress and found that the total virial stress is indeed the continuum Cauchy stress.

In addition to the numerical confirmation reported by Arun, here is another document that explains why the dynamical term is necessary:

The virial stress is rederived using only fundamental physics, in order to better understand its meaning. Despite recent claims to the contrary, it is found that when properly applied, the virial stress produces the desired macroscopic stress, and that the dynamical term in it does represent physical forces between adjacent material regions.

On the other hand, here are several commentators in Arun’s post noting that the kinetic term should be dropped: Gary L Gray; Szhang1; and, Vikastomar. Here is another class notes which defines virial stress without kinetic terms (and also defines virial strain) (pdf).

To summarise, for discrete systems, it is possible to define a stress that is the equivalent of Cauchy stress for continuum systems. Though there seem to be some problems in showing the equivalence between the two (except at zero K), analytically, by including both the kinetic and potential terms, the numerical simulations suggest that the definition of such a stress for discrete systems should have both kinetic and potential components.

### 5 Responses to “What is virial stress?”

1. Shankara Says:

Guru,

Thanx for the refreshing note.

2. Guru Says:

Anna,

Pleasure is mine!

3. Ram Kumar Says:

nice and simple post Guru. keep up the good work.

4. Raghu Says:

Excellent…spreading knowledge is divine…thank you!

5. caikuiyu Says:

Excellent,Thank you for sharing!