It is interesting by the determination of the atomic pressure in solids.
I read some paper about the viral stress which was introduced by Lutsko (J. Appl. Phys. 64 (3), 1988) using the local momentum flux:
dp(r)/dt = - div s(r)
where p(r) is the momentum and s(r) the stress
Following the calculus, it is not clear for me if Lutsko uses the Lagrange of Euler description but I supposed a Lagrangien description. But in this case, I am not sure of the physical meaning of s(r). This point has been discussed by Zhou (Proc. R. Soc. Lond. A (2003) 459, 2347–2392) and in this website :
http://www.eng.fsu.edu/~dommelen/papers/virial/mosaic/index.html
In the absence of volumic forces, in continuum mechanics, the Newton's law is:
\rho d2u(r)/dt2 = - div s(r)
with u(r) and s(r) the displacement and the Cauchy stress. This equation is valid in the Euler description.
I am confused about the right way to get the atomic stress.
Does someone know about that point ? How can I determine the atomic stress properly ?
Thank you for your answers.
Dear Mirko,
Expressing the equation in Lagrangian or Eulerian description is just the matter of pulling back or pushing forward the equation.
As you mentioned, the traditional balance of linear momentum that you have written is in Eulerian description and it can be certainly written in Lagrangian form.
In fact, it depends on how you treat the other variables in your formulations. If the main variables like displacement or velocity is considered in the material configuration and all the derivatives in an equation are with respect to material coordinates, your equation will be Lagrangian. Eulerian descriptions refers to spatial coordinate, of course.
Based on your equation, the evaluated stress will be either Eulerian or Updated Lagrangian or totally Lagrangian.
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