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 $\sigma _{h}$ can be given by:

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

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

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

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

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

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