In continuum mechanics, hydrostatic stress, also known as isotropic stress or volumetric stress, is a component of stress which contains uniaxial stresses, but not shear stresses. A specialized case of hydrostatic stress contains isotropic compressive stress, which changes only in volume, but not in shape. Pure hydrostatic stress can be experienced by a point in a fluid such as water. It is often used interchangeably with "mechanical 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, so it is one third of the first invariant of the stress tensor (i.e. the trace of the stress tensor):

${\displaystyle \sigma _{h}={\frac {I_{i}}{3}}={\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})}$

For example in cartesian coordinates (x,y,z) the hydrostatic stress is simply: ${\displaystyle \sigma _{h}={\frac {\sigma _{xx}+\sigma _{yy}+\sigma _{zz}}{3}}}$

In the particular case of an incompressible fluid, the thermodynamic pressure coincides with the mechanical pressure (i.e. the opposite of the hydrostatic stress): ${\displaystyle p=-\sigma _{h}=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})}$

In the general case of a compressible fluid, the thermodynamic pressure ${\displaystyle p}$ is no more proportional to the isotropic stress term (the mechanical pressure), since there is an additional term dependent on the trace of the strain rate tensor:

${\displaystyle p=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})+\zeta \operatorname {tr} ({\boldsymbol {\epsilon }})}$

where the coefficient ${\displaystyle \zeta }$ is the bulk viscosity. The trace of the strain rate tensor corresponds to the flow compression (the divergence of the flow velocity):