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]

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

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

• 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);
• ${\displaystyle h_{i))$ 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 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.