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\[ \sigma_{\text{Hyd}} = {\sigma_{11} + \sigma_{22} + \sigma_{33} \over 3} \]

There are many alternative ways to write this.

\[ \sigma_{\text{Hyd}} \; = \; {1 \over 3} \text{tr}(\boldsymbol{\sigma}) \; = \; {1 \over 3} I_1 \; = \; {1 \over 3} \sigma_{kk} \]

It is a scalar quantity, although it is regularly used in tensor form as

\[ \boldsymbol{\sigma}_{\text{Hyd}} = \left[ \matrix{\sigma_{\text{Hyd}} & 0 & 0 \\ 0 & \sigma_{\text{Hyd}} & 0 \\ 0 & 0 & \sigma_{\text{Hyd}} } \right] \]

\[ \boldsymbol{\sigma} = \left[ \matrix{ 50 & \;\;\; 30 & \;\;\;20 \\ 30 & -20 & -10 \\ 20 & -10 & \;\;\;10 } \right] \]

The hydrostatic stress is

\[ \sigma_\text{Hyd} \, = \, {50 + (-20) + 10 \over 3} \, = \, 13.3 \]

which can be written as

\[ \boldsymbol{\sigma}_\text{Hyd} = \left[ \matrix{ 13.3 & 0 & 0 \\ 0 & 13.3 & 0 \\ 0 & 0 & 13.3 } \right] \]

And that's all there is to it.

\[ \boldsymbol{\sigma_\text{Hyd}} = \left[ \matrix{\sigma_\text{Hyd} & 0 & 0 \\ 0 & \sigma_\text{Hyd} & 0 \\ 0 & 0 & \sigma_\text{Hyd} } \right] \]

contains equal amounts of stress in all three directions. And since the tensor does not change under any transformation, this means that no shear stresses ever arise, so every direction is a principal direction with \(\sigma_\text{Hyd}\) stress.

\[ P \, = \, -\sigma_\text{Hyd} \, = \, -{(\sigma_{11} + \sigma_{22} + \sigma_{33}) \over 3} \]

The stress tensor containing pressure, \(P\), is

\[ \boldsymbol{\sigma_\text{Hyd}} = \left[ \matrix{ -P & \;\;0 & \;\;0 \\ \;\;0 & -P & \;\;0 \\ \;\;0 & \;\;0 & -P } \right] \]

Of course, it is rare to talk about pressure unless the hydrostatic stress is compressive, which corresponds to a positive pressure.

Also, unless one is working with boundary layer flows over aircraft, automobiles, etc, then the stress state in the air is one of hydrostatic stress alone, without any shear stresses. And the hydrostatic stress is compressive, which is a positive pressure.

\[ \boldsymbol{\sigma}' = \boldsymbol{\sigma} - \boldsymbol{\sigma}_\text{Hyd} \]

In tensor notation, it is written as

\[ \sigma'\!_{ij} = \sigma_{ij} - {1 \over 3} \delta_{ij} \sigma_{kk} \] And in terms of pressure, it is written as

\[ \sigma'\!_{ij} = \sigma_{ij} + P \, \delta_{ij} \]

\[ \boldsymbol{\sigma} = \left[ \matrix{ 50 & \;\;\; 30 & \;\;\;20 \\ 30 & -20 & -10 \\ 20 & -10 & \;\;\;10 } \right] \]

The hydrostatic stress is

\[ \sigma_\text{Hyd} \, = \, {50 + (-20) + 10 \over 3} \, = \, 13.3 \]

which can be written as

\[ \boldsymbol{\sigma}_\text{Hyd} = \left[ \matrix{ 13.3 & 0 & 0 \\ 0 & 13.3 & 0 \\ 0 & 0 & 13.3 } \right] \]

Subtracting the hydrostatic stress tensor from the total stress gives

\[ \boldsymbol{\sigma}' = \left[ \matrix{ 50 & \;\;\; 30 & \;\;\;20 \\ 30 & -20 & -10 \\ 20 & -10 & \;\;\;10 } \right] - \left[ \matrix{ 13.3 & 0 & 0 \\ 0 & 13.3 & 0 \\ 0 & 0 & 13.3 } \right] = \left[ \matrix{ 36.7 & \;\;\; 30.0 & \;\;\;20.0 \\ 30.0 & -33.3 & -10.0 \\ 20.0 & -10.0 & -3.3 } \right] \]

Note that the result is