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.

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]

Diagram showing compressive hydrostatic stresses

${\displaystyle \sigma _{h}={\frac {I_{i}}{3}}={\frac {\sigma _{xx}+\sigma _{yy}+\sigma _{zz}}{3}}}$

Its magnitude in a fluid, ${\displaystyle \sigma _{h}}$, can be given by:

${\displaystyle \sigma _{h}=\displaystyle \sum _{i=1}^{n}\rho _{i}gh_{i}}$

where ${\displaystyle i}$ is an index denoting each distinct layer of material above the point of interest, ${\displaystyle \rho _{i}}$ is the density of each layer, ${\displaystyle g}$ is the gravitational acceleration (assumed constant here; this can be substituted with any acceleration that is important in defining weight), and ${\displaystyle h_{i}}$ 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

${\displaystyle \sigma _{h}=\rho _{w}gh_{w}=1000\,{\text{kg m}}^{-3}\cdot 9.8\,{\text{m s}}^{-2}\cdot 10\,{\text{m}}=9.8\cdot {10^{4}}{\text{ kg m}}^{-1}{\text{s}}^{-2}=9.8\cdot 10^{4}{\text{ N m}}^{-2}}$

where the index ${\displaystyle w}$ indicates "water".

Because the hydrostatic stress is isotropic, it acts equally in all directions. In tensor form, the hydrostatic stress is equal to

${\displaystyle \sigma _{h}\cdot I_{3}=\sigma _{h}\left[{\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right]=\left[{\begin{array}{ccc}\sigma _{h}&0&0\\0&\sigma _{h}&0\\0&0&\sigma _{h}\end{array}}\right]}$

where ${\displaystyle I_{3}}$ is the 3-by-3 identity matrix.

Hydrostatic compressive stress is used for the determination of the bulk modulus for materials.

References

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.
2. 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.