Hydrostatic stress
In continuum mechanics, hydrostatic stress, also known as volumetric stress,[1] is a component of stress which contains uniaxial stresses, but not shear stresses.[2] A specialized case of hydrostatic stress contains isotropic compressive stress, which changes only in volume, but not in shape.[1] Pure hydrostatic stress can be experienced by a point in a fluid such as water. It is often used interchangeably with "pressure" and is also known as confining stress, particularly in the field of geomechanics.[citation needed]
Hydrostatic stress is equivalent to the average of the uniaxial stresses along three orthogonal axes and can be calculated from the first invariant of the stress tensor:[2]
[math]\displaystyle{ \sigma_h = \frac{I_i}{3}=\frac{\sigma_{xx} + \sigma_{yy} + \sigma_{zz}}{3} }[/math]
Its magnitude in a fluid, [math]\displaystyle{ \sigma_h }[/math], can be given by:
- [math]\displaystyle{ \sigma_h = \displaystyle\sum_{i=1}^n \rho_i g h_i }[/math]
where
- i is an index denoting each distinct layer of material above the point of interest;
- [math]\displaystyle{ \rho_i }[/math] is the density of each layer;
- [math]\displaystyle{ g }[/math] is the gravitational acceleration (assumed constant here; this can be substituted with any acceleration that is important in defining weight);
- [math]\displaystyle{ h_i }[/math] is the height (or thickness) of each given layer of material.
For example, the magnitude of the hydrostatic stress felt at a point under ten meters of fresh water would be
- [math]\displaystyle{ \sigma_h = \rho_w g h_w =1000 \,\text{kg m}^{-3} \cdot 9.8 \,\text{m s}^{-2} \cdot 10 \,\text{m} =9.8 \cdot {10^4} \text{ kg m}^{-1} \text{s}^{-2} =9.8 \cdot 10^4 \text{ N m}^{-2} }[/math]
where the index w indicates "water".
Because the hydrostatic stress is isotropic, it acts equally in all directions. In tensor form, the hydrostatic stress is equal to
- [math]\displaystyle{ \sigma_h \cdot I_3 = \sigma_h \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] = \left[ \begin{array}{ccc} \sigma_h & 0 & 0 \\ 0 & \sigma_h & 0 \\ 0 & 0 & \sigma_h \end{array} \right] }[/math]
where [math]\displaystyle{ I_3 }[/math] is the 3-by-3 identity matrix.
Hydrostatic compressive stress is used for the determination of the bulk modulus for materials.
References
- ↑ 1.0 1.1 Megson, T. H. G. (Thomas Henry Gordon) (2005). Structural and stress analysis (2nd ed.). Amsterdam: Elsevier Butterworth-Heineman. pp. 400. ISBN 0-08-045534-4. OCLC 76822373. https://archive.org/details/structuralstress00megs.
- ↑ 2.0 2.1 Soboyejo, Winston (2003). "3.6 Hydrostatic and Deviatoric Stress". Mechanical properties of engineered materials. Marcel Dekker. pp. 88–89. ISBN 0-8247-8900-8. OCLC 300921090.
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