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:

$\sigma_h = \displaystyle\sum_{i=1}^n \rho_i g h_i$

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

$\sigma_{h,sand} = \rho_w g h_w =1000 \,\text{kg/m}^3 \cdot 9.8 \,\text{m/s}^2 \cdot 10 \,\text{m} =9.8 \cdot {10^4} {kg/ms^2} =9.8 \cdot 10^4 {N/m^2}$

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

$\sigma_h \cdot I_3 = \left[ \begin{array}{ccc} \sigma_h & 0 & 0 \\ 0 & \sigma_h & 0 \\ 0 & 0 & \sigma_h \end{array} \right]$

where $I_3$ is the 3-by-3 identity matrix.