Introduction
The
von Mises stress is often used in determining whether an
isotropic
and
ductile metal will yield when subjected to a complex loading condition.
This is accomplished by calculating the von Mises stress and comparing it to the
material's yield stress, which constitutes the
von Mises Yield Criterion.
The objective is to develop a yield criterion for ductile metals
that works for any complex 3D loading condition, regardless of the
mix of
normal and shear stresses.
The von Mises stress does this by boiling
the complex stress state down into a single scalar number that is compared to
a metal's
yield strength, also a single scalar numerical value determined
from a uniaxial tension test (because that's the easiest) on the material in a lab.
It should be noted that this is not an exact science like, say \(F = m\,a\).
It is an empirical process, with inherent error and deviations. In fact,
there is no hard & fast rule saying that metals must yield according to
von Mises yield criteria. It is as much a coincidence as anything. Nevertheless,
it does work very well and remains the method of choice a full century after it
was first proposed.
History
The defining equation for the von Mises stress was first proposed by Huber
[1] in 1904, but apparently received little attention
until von Mises
[2] proposed it again in 1913.
However, Huber and von Mises' definition was little more than a math equation without
physical interpretation until 1924 when Hencky
[3]
recognized that it is actually related to
deviatoric strain energy.
In 1931, Taylor and Quinney
[4] published results
of tests on copper, aluminum, and mild steel demonstrating that
the von Mises stress is a more accurate predictor of the onset of
metal yielding than the
maximum shear stress
criterion, which had been proposed by Tresca
[5]
in 1864 and was the best predictor of metal yielding to date.
Today, the von Mises stress is sometimes referred to as the HuberMises stress in
recognition of Huber's contribution to its development.
It is also called
Mises effective stress and simply
effective stress.
Technical Background
A complete understanding of the von Mises stress requires an understanding of
hydrostatic and deviatoric components of
stress and
strain tensors,
Hooke's Law, and
strain energy density.
The hydrostatic and deviatoric stresses and strains have already been reviewed.
And Hooke's Law has already been touched on
here and
here, but will need to
be discussed in additional detail on this page as well.
Strain energy density will also be introduced here.
Hydrostatic and Deviatoric Components
Recall that any
stress tensor can be decomposed into the sum of
hydrostatic and deviatoric stresses as follows
\[
\sigma_{ij} = {1 \over 3} \delta_{ij} \sigma_{kk} + \sigma'\!_{ij}
\]
where \( {1 \over 3} \delta_{ij} \sigma_{kk} \) is the hydrostatic term
and \( \sigma' \) is the deviatoric stress.
The same is true for
strain.
\[
\epsilon_{ij} = {1 \over 3} \delta_{ij} \epsilon_{kk} + \epsilon'\!_{ij}
\]
where \( {1 \over 3} \delta_{ij} \epsilon_{kk} \) is the hydrostatic term
and \( \epsilon' \) is the deviatoric strain.
These two will be multiplied together farther down the page.
Hooke's Law
We've seen that
Hooke's Law can be written as
\[
\epsilon_{ij} = {1 \over E} \left[ (1 + \nu) \sigma_{ij} 
\nu \, \delta_{ij} \sigma_{kk} \right]
\]
which is shorthand for
\[
\left [
\matrix {
\epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\
\epsilon_{xy} & \epsilon_{yy} & \epsilon_{yz} \\
\epsilon_{xz} & \epsilon_{yz} & \epsilon_{zz}
}
\right ]
= {1 \over E}
\left \{
(1 + \nu)
\left [
\matrix {
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{xz} & \sigma_{yz} & \sigma_{zz}
}
\right ]
 3 \; \nu
\left [
\matrix {
\sigma_{hyd} & 0 & 0 \\
0 & \sigma_{hyd} & 0 \\
0 & 0 & \sigma_{hyd}
}
\right ]
\right \}
\]
which is in turn matrix notation for the following set of equations
\[
\epsilon_{xx} = {1 \over E} \big[ \sigma_{xx}  \nu \, ( \sigma_{yy} + \sigma_{zz} ) \big]
\]
\[
\epsilon_{yy} = {1 \over E} \big[ \sigma_{yy}  \nu \, ( \sigma_{xx} + \sigma_{zz} ) \big]
\]
\[
\epsilon_{zz} = {1 \over E} \big[ \sigma_{zz}  \nu \, ( \sigma_{xx} + \sigma_{yy} ) \big]
\]
for the normal terms, and
\[
\epsilon_{xy} = { 1 + \nu \over E } \sigma_{xy} \qquad
\epsilon_{yz} = { 1 + \nu \over E } \sigma_{yz} \qquad
\epsilon_{xz} = { 1 + \nu \over E } \sigma_{xz}
\]
for the shear terms. The shear terms are more commonly written as
\[
\gamma_{xy} = { \tau_{xy} \over G} \qquad \quad
\gamma_{yz} = { \tau_{yz} \over G} \qquad \quad
\gamma_{xz} = { \tau_{xz} \over G}
\]
where
\[
\gamma_{xy} = 2 \epsilon_{xy} \qquad
\gamma_{yz} = 2 \epsilon_{yz} \qquad
\gamma_{xz} = 2 \epsilon_{xz} \qquad
{\rm and} \qquad
G = {E \over 2 (1 + \nu) }
\]
Return now to Hooke's Law in tensor form
\[
\epsilon_{ij} = {1 \over E} \left[ (1 + \nu) \sigma_{ij} 
\nu \, \delta_{ij} \sigma_{kk} \right]
\]
and multiply both sides by \(\delta_{ij}\).
\[
\delta_{ij} \epsilon_{ij} = {1 \over E} \left[ (1 + \nu) \sigma_{ij} 
\nu \, \delta_{ij} \sigma_{kk} \right] \delta_{ij}
\]
This simplifies to
\[
\epsilon_{kk} = { (1  2 \nu) \over E} \sigma_{kk}
\]
And then multiply both sides by \({ 1 \over 3} \delta_{ij}\) again to get
\[
{ 1 \over 3} \delta_{ij} \epsilon_{kk} = {(1  2 \nu) \over 3 E} \delta_{ij} \sigma_{kk}
\]
This results in an equation relating the
hydrostatic stress
and
strain values.
Now subtract the above equation from the original
Hooke's Law equation to get
\[
\begin{eqnarray}
\epsilon_{ij}  { 1 \over 3} \delta_{ij} \epsilon_{kk}
& \, = \, &
{ (1 + \nu) \over E} \sigma_{ij} 
{ \nu \over E} \, \delta_{ij} \sigma_{kk} 
{(1  2 \nu) \over 3 E} \delta_{ij} \sigma_{kk} \\
\\
\\
\\
\\
& \, = \, &
{ (1 + \nu) \over E} \sigma_{ij} 
{ 1 \over E} \left( \nu + {1  2 \nu \over 3} \right)
\delta_{ij} \sigma_{kk} \\
\\
\\
\\
\\
& \, = \, &
{ (1 + \nu) \over E} \sigma_{ij} 
{ 1 \over 3} {(1 + \nu) \over E}
\delta_{ij} \sigma_{kk} \\
\\
\\
\\
\\
& \, = \, &
{ (1 + \nu) \over E} \big( \sigma_{ij} 
{ 1 \over 3} \delta_{ij} \sigma_{kk} \big) \\
\end{eqnarray}
\]
The remarkable result is that both sides of the equation contain
a deviatoric tensor result. The equation can be summarized as
\[
\epsilon'\!_{ij} = { ( 1 + \nu ) \over E} \sigma'\!_{ij}
\]
But \({ (1 + \nu) \over E }\) is \({1 \over 2G}\), so the equation
can be further simplified to
\[
\epsilon'\!_{ij} = { 1 \over 2 G} \sigma'\!_{ij}
\]
So the
deviatoric stress and
strain
are directly proportional to each other.
The amazing thing here is that this is always true for
Hooke's Law, always,
even for the normal strain components.
For what it's worth, the equation can also be written as
\[
\sigma'\!_{ij} = 2 \, G \, \epsilon'\!_{ij}
\]
Deviatoric Example with Hooke's Law
Suppose you have a material with Poisson's ratio,
\(\nu = 0.5\), and elastic modulus, \(E = 15\;MPa\).
For the stress tensor below, use Hooke's Law to calculate the
strain state. Then get the deviatoric stress and strain tensors
and show that they are proportional to each other by the factor \(2G\).
\[
\boldsymbol{\sigma} \; = \;
\left[ \matrix{
8 & 2 & 4 \\
2 & 6 & 6 \\
4 & 6 & 4 }
\right]
\]
Note that this stress tensor clearly has a significant amount of
hydrostatic stress. It is
\[
\boldsymbol{\sigma}_\text{Hyd} \; = \;
\left[ \matrix{
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 6 }
\right]
\]
Hooke's Law is
\[
\begin{eqnarray}
\left[ \matrix{
\epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\
\epsilon_{xy} & \epsilon_{yy} & \epsilon_{yz} \\
\epsilon_{xz} & \epsilon_{yz} & \epsilon_{zz} }
\right]
& = &
{1 \over 15}
\left\{
(1 + 0.5)
\left[ \matrix{
8 & 2 & 4 \\
2 & 6 & 6 \\
4 & 6 & 4 }
\right]
 3 \; (0.5)
\left[
\matrix {
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 6 }
\right]
\right\} \\
\\
\\
\\
\\
& = &
{1 \over 15}
\left\{
\left[ \matrix{
12 & 3 & 6 \\
3 & 9 & 9 \\
6 & 9 & 6 }
\right]

\left[
\matrix{
9 & 0 & 0 \\
0 & 9 & 0 \\
0 & 0 & 9 }
\right]
\right\} \\
\\
\\
\\
\\
& = &
\left[ \matrix{
0.2 & \;0.2 & \;0.4 \\
0.2 & \;0.0 & \;0.6 \\
0.4 & \;0.6 & \text{}0.2 }
\right]
\end{eqnarray}
\]
Note that this strain tensor is already deviatoric. This is because
we used \(\nu = 0.5\) for the Poisson's Ratio, which is the value used
for incompressible materials. So we obtained an incompressible,
nonhydrostatic strain tensor as a result.
So the question becomes, "Will (\(2 \, G \, \boldsymbol{\epsilon}'\))
give \(\boldsymbol{\sigma}'\)?"
To answer this, first compute \(G\).
\[
\begin{eqnarray}
G
& = &
{E \over 2 ( 1 + \nu) }
& = &
{15 \text{ MPa} \over 2 ( 1 + 0.5) } \\
\\
\\
\\
\\
& = &
5 \text{ MPa}
\end{eqnarray}
\]
So \(2 \, G \, \boldsymbol{\epsilon}'\) equals
\[
2 \, G \, \boldsymbol{\epsilon}' =
\left[ \matrix{
2 & \;2 & \;4 \\
2 & \;0 & \;6 \\
4 & \;6 & \text{}2 }
\right]
\]
And compare this to \(\boldsymbol{\sigma}  \boldsymbol{\sigma}_\text{Hyd}\)
\[
\boldsymbol{\sigma}  \boldsymbol{\sigma}_\text{Hyd}
\; = \;
\left[ \matrix{
8 & 2 & 4 \\
2 & 6 & 6 \\
4 & 6 & 4 }
\right]

\left[
\matrix {
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 6 }
\right]
=
\left[
\matrix {
2 & \;2 & \;4 \\
2 & \;0 & \;6 \\
4 & \;6 & \text{}2 }
\right]
\]
So it does indeed satisfy
\(\boldsymbol{\sigma}' = 2 \, G \, \boldsymbol{\epsilon}'\).
Although this was an example with an incompressible material,
\(\nu = 0.5\), it also works for compressible materials
as well, always.
Strain Energy Density
Strain energy density, W, has units of
Energy / Volume and is
\[
W = \int \boldsymbol{\sigma} : d \boldsymbol{\epsilon}
\]
For linear elastic materials, this equals
\[
W = {1 \over 2} \boldsymbol{\sigma} : \boldsymbol{\epsilon}
\]
which expands out to give
\[
{1 \over 2} \boldsymbol{\sigma} : \boldsymbol{\epsilon} = {1 \over 2} [
\sigma_{xx} \epsilon_{xx} + \sigma_{yy} \epsilon_{yy} + \sigma_{zz} \epsilon_{zz} +
2 ( \sigma_{xy} \epsilon_{xy} + \sigma_{yz} \epsilon_{yz} + \sigma_{xz} \epsilon_{xz} ) ]
\]
But since \( \boldsymbol{\sigma} = \boldsymbol{\sigma}_\text{Hyd} + \boldsymbol{\sigma}' \)
and \( \boldsymbol{\epsilon} = \boldsymbol{\epsilon}_\text{Hyd} + \boldsymbol{\epsilon}' \),
these identities can be substituted into the equation to obtain
\[
W = {1 \over 2} \boldsymbol{\sigma} : \boldsymbol{\epsilon} =
{1 \over 2}
(\boldsymbol{\sigma}_\text{Hyd} + \boldsymbol{\sigma}') :
(\boldsymbol{\epsilon}_\text{Hyd} + \boldsymbol{\epsilon}')
\]
and expanding the multiplication out gives
\[
W = {1 \over 2} \boldsymbol{\sigma} : \boldsymbol{\epsilon} =
{1 \over 2}
\boldsymbol{\sigma}_\text{Hyd} : \boldsymbol{\epsilon}_\text{Hyd}
+ {1 \over 2}
\boldsymbol{\sigma}_\text{Hyd} : \boldsymbol{\epsilon}'
+ {1 \over 2}
\boldsymbol{\sigma}' : \boldsymbol{\epsilon}_\text{Hyd}
+ {1 \over 2}
\boldsymbol{\sigma}' : \boldsymbol{\epsilon}'
\]
But (\(\boldsymbol{\sigma}_\text{Hyd} : \boldsymbol{\epsilon}'\))
and (\(\boldsymbol{\sigma}' : \boldsymbol{\epsilon}_\text{Hyd}\))
are zero! This is because the doubledot product of any hydrostatic
tensor with a deviatoric tensor is always zero.
So the equation reduces to
\[
W = {1 \over 2} \boldsymbol{\sigma} : \boldsymbol{\epsilon} =
\underbrace {
{1 \over 2} \boldsymbol{\sigma}_\text{Hyd} : \boldsymbol{\epsilon}_\text{Hyd}
}_{hydrostatic}
+
\underbrace {
{1 \over 2} \boldsymbol{\sigma}' : \boldsymbol{\epsilon}'
}_{deviatoric}
\]
This shows that strain energy can be partitioned into
hydrostatic and deviatoric components.
Von Mises Stress
The von Mises stress is directly related to the deviatoric strain energy term
in the above equation.
\[
W' = {1 \over 2} \boldsymbol{\sigma}' : \boldsymbol{\epsilon}'
\]
Recall from the section on
Hooke's Law that
\[
\boldsymbol{\epsilon}' = {1 \over 2\, G} \boldsymbol{\sigma}'
\]
Combining the two gives
\[
W' = {1 \over 4 \, G} \boldsymbol{\sigma}' : \boldsymbol{\sigma}'
\]
So the deviatoric part of the
strain energy density
is directly related to the
double dot product of the
deviatoric stress
with itself. Note the similarity
to Kinetic Energy, \(KE = {1 \over 2} M v^2\), a spring's internal energy,
\(E = {1 \over 2} K x^2\), electrical power, \(P = R I^2\), and any
other form one can think of.
It is finally time to introduce an equivalent or effective stress that will turn
out to be proportional to the von Mises stress, though about 20% low.
Use the symbol \( \sigma_{Rep} \) for
representative stress to represent this stress value.
And it is a scalar stress value, not a tensor!
The defining equation for
\( \sigma_{Rep} \) is
\[
W' = {1 \over 4\,G} (\sigma_{Rep})^2
\]
The form of the equation is deliberately chosen to be the scalar equivalent of the one above.
Setting them equal to each other (since both are equal to W') gives
\[
W' = {1 \over 4\,G} (\sigma_{Rep})^2 =
{1 \over 4\,G} \boldsymbol{\sigma}' : \boldsymbol{\sigma}'
\]
Clearly \( \sigma_{Rep} \)
is intended to be the scalar stress value that gives the same deviatoric strain energy
as the actual 3D stress tensor. Cancelling \(4\,G\) from both sides gives
\[
\sigma_{Rep} = \sqrt { \boldsymbol{\sigma}' : \boldsymbol{\sigma}' }
\]
The final step is one of simple convenience. It is motivated by the simplest
straightforward case of uniaxial tension. To see it, calculate \(\sigma_\text{Rep}\)
for this case. The stress state for uniaxial tension is
\[
\boldsymbol{\sigma} =
\left[ \matrix{
\sigma & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 }
\right]
\]
The
hydrostatic stress is \({1 \over 3} \sigma\), and the deviatoric stress tensor is
\[
\boldsymbol{\sigma}' =
\left[ \matrix{
{2 \sigma \over 3} & 0 & 0 \\
0 & {\sigma \over 3} & 0 \\
0 & 0 & {\sigma \over 3} }
\right]
\]
So \(\boldsymbol{\sigma}' : \boldsymbol{\sigma}'\) equals \(2 \sigma^2 / 3\). And therefore
\[
\sigma_\text{Rep} = \sqrt { \boldsymbol{\sigma}' : \boldsymbol{\sigma}' } = \sqrt { 2 \over 3} \; \sigma
\]
And therein lies the frustration. The representative stress for uniaxial tension is not equal to
the uniaxial tension stress, \(\sigma\), but is instead about 82% of it. This is terribly
inconvenient, but the fix is simple. Simply scale the representative stress up until it
equals the uniaxial tension stress. This is done by simply multiplying \(\sigma_\text{Ref}\)
by \(\sqrt{3/2}\).
This is acceptable because anything
proportional to \( \sqrt { \boldsymbol{\sigma}' : \boldsymbol{\sigma}' } \) will still reflect
the relationship to deviatoric strain energy. It will just be scaled up some.
The final result is the von Mises stress.
\[
\sigma_\text{VM} = \sqrt { {3 \over 2} \boldsymbol{\sigma}' : \boldsymbol{\sigma}' }
\]
And this is the defining equation for it.
Alternate Forms
Algebraic manipulation of the above equation gives many other equivalent forms.
They are summarized here.
\[
\sigma_\text{VM} = \sqrt{{1\over 2}\left[\left(\sigma_{xx}  \sigma_{yy}\right)^2 +
\left(\sigma_{yy}  \sigma_{zz}\right)^2 +
\left(\sigma_{zz}  \sigma_{xx}\right)^2 \right] +
3 \left(\tau^2_{xy} + \tau^2_{yz} + \tau^2_{zx}\right) }
\]
\[
\sigma_\text{VM} = \sqrt{\sigma^2_{xx} + \sigma^2_{yy} + \sigma^2_{zz} 
\sigma_{xx} \sigma_{yy}  \sigma_{yy} \sigma_{zz} 
\sigma_{zz} \sigma_{xx} +
3 \left(\tau^2_{xy} + \tau^2_{yz} + \tau^2_{zx}\right) }
\]
\[
\sigma_\text{VM} = \sqrt{{3\over 2}\sigma_{ij}\sigma_{ij}  {1\over 2} ( \sigma_{kk} )^2}
\quad \quad \quad
\sigma_\text{VM} = \sqrt{{3\over 2}\sigma'\!_{ij}\sigma'\!_{ij}}
\]
In 2D applications, \( \sigma_{zz} = \tau_{xz} = \tau_{yz} = 0\). This leaves
\[
\sigma_\text{VM} = \sqrt{\sigma^2_{xx} + \sigma^2_{yy} 
\sigma_{xx} \sigma_{yy} +
3 \, \tau^2_{xy} }
\]
Tensor Manipulation of Von Mises Equation
One can (relatively) easily obtain other equations for von Mises stress
thru tensor manipulations of the equation based on deviatoric values.
Starting with
\[
\sigma_\text{VM} = \sqrt{{3\over 2}\sigma'\!_{ij}\sigma'\!_{ij}}
\]
and expressing \(\sigma'\!_{ij}\) in terms of the full stress tensor as
\[
\sigma'\!_{ij} = \sigma_{ij}  {1 \over 3} \delta_{ij} \sigma_{kk}
\]
gives the following form.
\[
\sigma_\text{VM} = \sqrt{{3\over 2}
\left( \sigma_{ij}  {1 \over 3} \delta_{ij} \sigma_{kk} \right)
\left( \sigma_{ij}  {1 \over 3} \delta_{ij} \sigma_{kk} \right)
}
\]
Multiplying this out gives
\[
\sigma_\text{VM} = \sqrt{{3\over 2} \left( \sigma_{ij} \sigma_{ij} 
{2 \over 3} \delta_{ij} \sigma_{ij} \sigma_{kk} +
{1 \over 9} \delta_{ij} \delta_{ij} ( \sigma_{kk} )^2 \right) }
\]
which simplifies down to
\[
\sigma_\text{VM} = \sqrt{{3\over 2}\sigma_{ij}\sigma_{ij}  {1\over 2} ( \sigma_{kk} )^2}
\]
The other forms listed above can be obtained by expressing this
explicitly in terms of \(\sigma_{xx}\), \(\sigma_{xy}\), \(\sigma_{xz}\),
etc.
Specific Loading Cases
We've already seen during the derivation above that for uniaxial tension, the
von Mises stress equals the uniaxial tension stress. But this is also (almost)
true for compression as well. The only issue is that for compression, the
numerical value of the compressive stress will be negative, but the von Mises
stress is always positive because it is a squareroot of a sum of stress values
squared. So when one is reading a von Mises stress of say, 10 MPa, it is
impossible to know from this alone if the object is undergoing tension or compression.
One can look at the
principal stress values to determine this.
Actually, some FEA postprocessors will make color stress contours of a quantity
call
signed von Mises stress. This has the same absolute value as the
conventional von Mises stress, but the +/ sign is determined by checking
the sign of the
hydrostatic stress.
If it is negative, then the
signed von Mises stress is also negative.
The case of pure shear stress is most interesting. One can see from the equations
above that for a pure shear stress, \(\tau_{xy}\), the von Mises stress is
\[
\sigma_\text{VM} = \sqrt{3} \, \tau
\]
So if a metal yields in uniaxial tension (or compression) at
\(\sigma = \pm 500 \text{ MPa}\), then it will also yield in
shear at a stress that is only 58% of this, or
\(\tau = \pm 290 \text{ MPa}\).
Graphical Representations
Here again is the sketch at the top of the page. It shows a bounding surface
in a 3D principal stress coordinate system where the von Mises stress is a constant
value. (This is the so called
HighWesterguard Space.) It is based on the fact
that any stress state can be converted into its principal values and compared
to this sketch. If the resulting principal stress point in the coordinate system
is within the cylinder, then the material has not yielded. If it is on the surface,
then the material has yielded. And if it is outside the cylinder, it means that
you did an elastic analysis of a situation that cannot in fact be correct because
yielding would have long since taken place.
The remarkable result is that if you look down the \(\sigma_1 = \sigma_2 = \sigma_3\)
axis, the crosssection of the cylinder is a perfect circle. Note that the
hydrostatic stress in this situation does not show up at all.
Experimental Data
The figure here presents experimental data confirming that ductile metals
yield much more consistently at prescribed von Mises stress levels regardless
of the the loading state than at any other criteria.
The graph represents a slice through the \(\sigma_1  \sigma_2\) plane
with \(\sigma_3 = 0\). Since the cylinder is cut at an angle, it appears
to be an ellipse in this situation. It is in fact still a circle.
We are just looking at it at an angle.
Recall that the shear stress criterion was first proposed by
Tresca in 1864,
and this act is considered to represent the birth of the field
of metal plasticity research.
The one exception here is the cast iron metal. It yields, fractures in fact, at
a constant maximum principal stress criterion. This signifies that the
iron is
brittle and behaves more like glass than a ductile metal.
Reference: Dowling, N.E., Mechanical Behavior of Materials, Prentice Hall, 1993.
Note that the correlation here is not perfect. This is a consequence of the
fact that the socalled
von Mises Yield Criterion is NOT a law of
nature. It is more of a convenient coincidence. It is a consequence of
dislocation movement on millions and billions of planes
of atoms sliding over each other at the atomic scale. Those planes of
atoms are all randomly oriented, and the resulting response
at the macroscale is.... the von Mises yield criterion.
Contrasting Stress and Strain
We've seen how the von Mises stress is "the stress" when worrying about
metal yielding and plasticity. Recall that it is
\[
\sigma_\text{VM} = \sqrt{ {3 \over 2} \boldsymbol{\sigma}' : \boldsymbol{\sigma}' }
\]
The next question is, "Is there a
strain analog to the von Mises stress?"
The answer is yes. It is the
effective strain, or sometimes the
Mises effective strain. It is
\[
\epsilon_\text{eff} = \sqrt{ {2 \over 3} \boldsymbol{\epsilon}' : \boldsymbol{\epsilon}' }
\]
Note that it is \(2/3\), not \(3/2\). This arises because the strain tensor for
uniaxial tension of an incompressible material (which includes the plastic part of the
total deformation of a metal) is
\[
\boldsymbol{\epsilon}_\text{True} =
\left[ \matrix{
\epsilon & \;\;0 & \;\;0 \\
0 & {\epsilon \over 2} & \;\;0 \\
0 & \;\;0 & {\epsilon \over 2} }
\right]
\]
and \( \boldsymbol{\epsilon} : \boldsymbol{\epsilon} \) in this case gives
\(3 \, \epsilon^2 / 2\).
So it is necessary to multiply by \(2/3\) in order to make
\( \boldsymbol{\epsilon}_\text{eff} \) equal to the uniaxial tension strain.
This makes it possible to more fairly compare the stress and strain states of
two different deformation modes, say tension versus shear. In fact, in a perfectly
isotropic metal, plots of effective stress versus effective strain will be
indistinguishable in the plastic region regardless of the deformation mode.
Although in reality, metals usually become increasingly anisotropic after yielding.
References
 Huber, M.T. (1904) Czasopismo Techniczne, Lemberg, Austria, Vol. 22, pp. 181.
 Von Mises, R. (1913) "Mechanik der Festen Korper im Plastisch Deformablen Zustand," Nachr. Ges. Wiss. Gottingen, pp. 582.
 Hencky, H.Z. (1924) "Zur Theorie Plasticher Deformationen und der Hierdurch im Material Hervorgerufenen Nachspannungen," Z. Angerw. Math. Mech., Vol. 4, pp. 323.
 Taylor, G.I., Quinney, H. (1931) "The Plastic Distortion of Metals," Phil. Trans. R. Soc., London, Vol. A230, pp. 323.
 Tresca, H. (1864) "Sur l'Ecoulement des Corps Solides Soumis a de Fortes Pressions," C. R. Acad. Sci., Paris, Vol. 59, pp. 754.
 Dowling, N.E. (1993) Mechanical Behavior of Materials, Prentice Hall.
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