Hydrostatic stress

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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 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.