Introduction
This page introduces hydrostatic and deviatoric stresses. The two are subsets of
any given
stress tensor, which, when added together, give the original stress
tensor back. The hydrostatic stress is related to
volume change, while
the deviatoric stress is related to
shape change.
Hydrostatic Stress
Hydrostatic stress is simply the average of the three normal stress components
of any stress tensor.
\[
\sigma_{\text{Hyd}} = {\sigma_{11} + \sigma_{22} + \sigma_{33} \over 3}
\]
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
\[
\boldsymbol{\sigma}_{\text{Hyd}} =
\left[ \matrix{\sigma_{\text{Hyd}} & 0 & 0 \\
0 & \sigma_{\text{Hyd}} & 0 \\
0 & 0 & \sigma_{\text{Hyd}}
} \right]
\]
Hydrostatic Stress Example
Given the following stress tensor
\[
\boldsymbol{\sigma} =
\left[ \matrix{
50 & \;\;\; 30 & \;\;\;20 \\
30 & 20 & 10 \\
20 & 10 & \;\;\;10
} \right]
\]
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.
Hydrostatic Stresses and Coordinate Transformations
This could not be simpler. Hydrostatic stresses do not change under
coordinate transformations. This is easily accepted in light of
the fact that \(\sigma_\text{Hyd}\) is a function of \(I_1\). Also
\[
\boldsymbol{\sigma_\text{Hyd}} =
\left[ \matrix{\sigma_\text{Hyd} & 0 & 0 \\
0 & \sigma_\text{Hyd} & 0 \\
0 & 0 & \sigma_\text{Hyd} }
\right]
\]
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.
Hydrostatic Stress and Pressure
Pressure is simply the negative of hydrostatic stress. The
negative
aspect is often confusing. It is why we talk about atmospheric pressure as
30 inches of Hg, a positive number, even though atmospheric pressure is in fact
a negative stress because it is compressive. So using \(P\) for pressure...
\[
P \, = \, \sigma_\text{Hyd} \, = \, {(\sigma_{11} + \sigma_{22} + \sigma_{33}) \over 3}
\]
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.
Hydrostatic Stress in Incompressible Materials
This is almost a nontopic because hydrostatic stresses usually have no
impact on incompressible materials at all, so there is very little to
discuss. This also means that one cannot determine the hydrostatic
stress based on strain or a deformation gradient. And this is why
finite element programs often fudge by making the incompressible
material eversoslightly compressible.
Deviatoric Stress
Deviatoric stress is what's left after subtracting out the hydrostatic stress.
The deviatoric stress will be represented by \(\boldsymbol{\sigma}'\).
For example
\[
\boldsymbol{\sigma}' = \boldsymbol{\sigma}  \boldsymbol{\sigma}_\text{Hyd}
\]
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
\[
\sigma'\!_{ij} = \sigma_{ij} + P \, \delta_{ij}
\]
Deviatoric Stress Example
Given the following stress tensor
\[
\boldsymbol{\sigma} =
\left[ \matrix{
50 & \;\;\; 30 & \;\;\;20 \\
30 & 20 & 10 \\
20 & 10 & \;\;\;10
} \right]
\]
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
traceless. Its first invariant equals zero.
Or put another way, the hydrostatic stress of a deviatoric stress tensor
is zero.
An interesting aspect of a traceless tensor is that it can be formed entirely from
shear components. For example, a
coordinate system transformation can
be found to express the deviatoric stress tensor in the above
example as shear stress exclusively. In the screenshot here, the above
deviatoric stress tensor was input into the webpage, and then the coordinate
system was rotated until the following stress tensor was obtained.
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