# Hydrostatic stress

In continuum mechanics, a hydrostatic stress is an isotropic stress that is given by the weight of water above a certain point. It is often used interchangeably with "pressure" and is also known as confining stress, particularly in the field of geomechanics. Its magnitude ${\displaystyle \sigma _{h}}$ 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.