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Equivalence of Virial stress to Continuum Cauchy Stress

Arun K. Subramaniyan's picture

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.

Some details of our simulations can be found in the attached document.

PDF icon VirialStress_imechanica.pdf73.89 KB


Pradeep Sharma's picture

Arun, thanks for the pre-print. I will take a look at this.

Gary L. Gray's picture


  1. Pedro C. Andia, Francesco Costanzo, and Gary L. Gray (2006) "A Classical Mechanics Approach to the Determination of the Stress-Strain Response of Particle Systems," Modelling and Simulation in Materials Science and Engineering, 14, pp. 741–757.
  2. Pedro C. Andia, Francesco Costanzo, and Gary L. Gray (2005) "A Lagrangian-Based Continuum Homogenization Approach Applicable to Molecular Dynamics Simulations," International Journal of Solids and Structures, 42(24-25), pp. 6409–6432.
  3. Francesco Costanzo, Gary L. Gray, and Pedro C. Andia (2005) "On the Definitions of Effective Stress and Deformation Gradient for Use in MD: Hill's Macro-homogeneity and the Virial Theorem," International Journal of Engineering Science, 43(7), pp. 533–555.
  4. Francesco Costanzo, Gary L. Gray, and Pedro C. Andia (2004) "On the Notion of Average Mechanical Properties in MD Simulation via Homogenization," Modelling and Simulation in Materials Science and Engineering, 12, pp. S333–S345.

In particular, the first reference has a nice discussion of the relation between the effective Cauchy stress and the virial stress.

All the best,

Arun K. Subramaniyan's picture

Dr. Gray,

Thanks for pointing me to your papers. I was aware of these papers. In fact many of these papers got us interested in the debate on virial stress.

Before getting into the analytical aspects of this problem, we thought it best to see if a simple thermoelastic problem can shed some light on this issue. The outcome of this exercise is what I have posted here. Although there are many alternate measures of stress (without velocity) at the atomic level, I am not sure if they can produce the actual continuum stress in the three simple cases we used. It would be interesting to see if the alternate measures of atomic stress produce the same stress state as that produced by the total virial stress (which in this case is the same as the continuum cauchy stress).

Sulin Zhang's picture

We have recently written a paper in IJNME, in which we proved that the Virial stress is the Cauchy-stress for crystal materials at 0 K. If kinetic part is taken into account, I wonder if we can mathematically approve that. Our paper is now on-line and titled as  

A Bridging Domain and Strain Computation Method for Coupled Atomistic-Continuum Modeling of Solids.

written by Sulin Zhang, Roopam Khare, Qiang Lu, and Ted Belytschko

Arun K. Subramaniyan's picture

Indeed at 0 Kelvin the virial stress reduces to just the potential part. However at elevated temperature, without the velocity term in the stress definition, we cannot match the stress state to a continuum cauchy stress definition. You can check by performing simple MD simulations as detailed above. We have not tried to prove this analytically yet.

Please see the proof in the following reference:

 M. Zhou, A New Look at the Atomic Level Virial Stress--On Continuum-Molecular System Equivalence, Proceedings of the Royal Society of London A; 459, 2347-2392, 2003


The above article can be downloaded from

Arun K. Subramaniyan's picture


The paper by M. Zhou looks at the virial stress from a linear momentum balance formulation. However, the question of the thermal stress for a system at thermal equilibrium is not addressed in that paper. It is in this regard that I said we are not aware of any analytical proof.

An interesting paper by Dommelen commenting on the above paper by M Zhou can be found here :

I really appreciate this wonderful post that you have provided for us. I assure this would be beneficial for most of the people. Looking forward to read more of your post and updates in the future. selfie

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