Pressure
Pressure (\(p\)) is the force per unit area applied on a surface, in a direction perpendicular to that surface, i.e. the scalar part of the stress tensor under equilibrium/hydrosatic conditions.
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[edit] Thermodynamics
In thermodynamics the pressure is given by
\[p = - \left.\frac{\partial A}{\partial V} \right\vert_{T,N} = k_BT \left.\frac{\partial \ln Q}{\partial V} \right\vert_{T,N}\]
where \(A\) is the Helmholtz energy function, \(V\) is the volume, \(k_B\) is the Boltzmann constant, \(T\) is the temperature and \(Q (N,V,T)\) is the canonical ensemble partition function.
[edit] Units
The SI units for pressure are Pascals (Pa), 1 Pa being 1 N/m2, or 1 J/m3. Other frequently encountered units are bars and millibars (mbar); 1 mbar = 100 Pa = 1 hPa, 1 hectopascal. 1 bar is 105 Pa by definition. This is very close to the standard atmosphere (atm), approximately equal to typical air pressure at earth mean sea level: atm, standard atmosphere = 101325 Pa = 101.325 kPa = 1013.25 hPa = 1.01325 bar
[edit] Stress
The stress is given by
\[{\mathbf F} = \sigma_{ij} {\mathbf A}\]
where \({\mathbf F}\) is the force, \({\mathbf A}\) is the area, and \(\sigma_{ij}\) is the stress tensor, given by
\[\sigma_{ij} \equiv \left[{\begin{matrix} \sigma _x & \tau _{xy} & \tau _{xz} \\ \tau _{yx} & \sigma _y & \tau _{yz} \\ \tau _{zx} & \tau _{zy} & \sigma _z \\ \end{matrix}}\right]\]
where where \(\ \sigma_{x}\), \(\ \sigma_{y}\), and \(\ \sigma_{z}\) are normal stresses, and \(\ \tau_{xy}\), \(\ \tau_{xz}\), \(\ \tau_{yx}\), \(\ \tau_{yz}\), \(\ \tau_{zx}\), and \(\ \tau_{zy}\) are shear stresess.
[edit] See also
[edit] References
Related reading
- Aidan P. Thompson, Steven J. Plimpton, and William Mattson "General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions", Journal of Chemical Physics 131 154107 (2009)
- G. C. Rossi and M. Testa "The stress tensor in thermodynamics and statistical mechanics", Journal of Chemical Physics 132 074902 (2010)
- Nikhil Chandra Admal and E. B. Tadmor "Stress and heat flux for arbitrary multibody potentials: A unified framework", Journal of Chemical Physics 134 184106 (2011)
- Takenobu Nakamura, Wataru Shinoda, and Tamio Ikeshoji "Novel numerical method for calculating the pressure tensor in spherical coordinates for molecular systems", Journal of Chemical Physics 135 094106 (2011)
