Abstract
The derivation of an expression of the macroscopic stress tensor in terms of microscopic variables in systems of finite interacting particles is discussed from different points of view.
It is shown that in volume averaging the introduction of a fictitious “interaction stress field”T I with special boundary conditions on the boundary of the averaging volume is needed. In ensemble averaging similar results are obtained by using a multipole expansion of the local stress and force fields. In the appropriate limiting cases, the obtained results are shown to be consistent with the results of kinetic theories of polymer solutions.
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Abbreviations
- A :
-
arbitrary field
- D :
-
rate-of-strain tensor
- f :
-
force density (per unit volume)
- f i :
-
force acting on particlei
- f ij :
-
force acting on particlei due to particlej
- n :
-
unit normal
- r :
-
radius vector with respect to an arbitrary origin
- r i :
-
position of particlei
- r ij :
-
r j −r i
- T :
-
stress tensor
- t :
-
traction
- v :
-
velocity field
- V :
-
averaging volume
- ∂V :
-
boundary ofV
- V i :
-
volume of particlei
- v i :
-
arbitrary volume enclosing particlei
- I :
-
interaction
- H :
-
hydrodynamic
- C :
-
contact
- P :
-
particle
- F :
-
fluid
- \(\overline {( \ldots )} \) :
-
volume average
- 〈⋯〉:
-
ensemble average
- \(\mathop {( \ldots )}\limits^ \sim \) :
-
macroscopic quantity
- \(\vec \nabla or \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{\nabla } \) :
-
spatial differential operator (arrow denotes direction of the quantity to be operated on)
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Jongschaap, R.J.J., Doeksen, D. The average stress tensor in systems with interacting particles. Rheol Acta 22, 4–11 (1983). https://doi.org/10.1007/BF01679824
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DOI: https://doi.org/10.1007/BF01679824