Introduction
This page covers standard coordinate transformations, principal strains, and strain invariants.
Everything here applies regardless of the type of strain tensor, so both \(\boldsymbol{\epsilon}\)
and \({\bf E}\) will be used here.
Coordinate transformations of 2nd rank tensors were discussed on this
coordinate transform page.
The transform applies to any strain tensor, or stress tensor for that matter.
It is written as
\[
{\bf E}' = {\bf Q} \cdot {\bf E} \cdot {\bf Q}^T
\]
Everything below follows from two facts: First, the tensors are symmetric.
Second, the above coordinate transformation is used.
2-D Principal Strains
In 2-D, the transformation equations are
\[
\begin{eqnarray}
\epsilon'\!_{xx} & = & \epsilon_{xx} \cos^2 \theta + \epsilon_{yy} \, \sin^2 \theta + 2 \left( {\gamma_{xy} \over 2} \right) \sin \theta \cos \theta \\
\\
\epsilon'\!_{yy} & = & \epsilon_{xx} \sin^2 \theta + \epsilon_{yy} \, \cos^2 \theta - 2 \left( {\gamma_{xy} \over 2} \right) \sin \theta \cos \theta \\
\\
{\gamma'\!_{xy} \over 2} & = & (\epsilon_{yy} - \epsilon_{xx}) \sin \theta \cos \theta + \left( {\gamma_{xy} \over 2} \right) (\cos^2 \theta - \sin^2 \theta)
\end{eqnarray}
\]
The equations are written in terms of \(\gamma / 2\) to emphasize that half of all the
shear values are used in the transformation equations.
This page performs full 3-D tensor transforms,
but can still be used for 2-D problems..
Enter values in the upper left 2x2 positions and rotate in the 1-2 plane to
perform transforms in 2-D. The screenshot below shows a case of pure shear
rotated 45° to obtain the principal strains. Note also how the \({\bf Q}\) matrix
transforms.
The figure below shows the deformed shapes corresponding to the pure shear case in the tensor
transform webpage example. The blue square aligned with the axes clearly undergoes shear.
But the red square inscribed in the larger blue square only sees simple tension and compression.
These are the principal values of the pure shear deformation in the global coordinate system.
In 2-D, the principal strain orientation, \(\theta_P\), can be computed by setting
\(\gamma'\!_{xy} = 0\) in the above shear equation and solving for \(\theta\)
to get \(\theta_P\), the principal strain angle.
\[
0 = (\epsilon_{yy} - \epsilon_{xx}) \sin \theta_P \cos \theta_P + \left( {\gamma_{xy} \over 2} \right) (\cos^2 \theta_P - \sin^2 \theta_P)
\]
This gives
\[
\tan (2 \theta_P) \; = \; {\gamma_{xy} \over \epsilon_{xx} - \epsilon_{yy} } \; = \; {2 \epsilon_{xy} \over \epsilon_{xx} - \epsilon_{yy} }
\]
The transformation matrix, \({\bf Q}\), is
\[
{\bf Q} =
\left[ \matrix{
\;\;\; \cos \theta_P & \sin \theta_P \\
-\sin \theta_P & \cos \theta_P
} \right]
\]
Inserting this value for \(\theta_P\) back into the equations for the normal strains gives
the principal values. They are written as \(\epsilon_{max}\) and \(\epsilon_{min}\),
or alternatively as \(\epsilon_1\) and \(\epsilon_2\).
\[
\epsilon_{max}, \epsilon_{min} = {\epsilon_{xx} + \epsilon_{yy} \over 2} \pm
\sqrt{ \left( {\epsilon_{xx} - \epsilon_{yy} \over 2} \right)^2 +
\left( \gamma_{xy} \over 2 \right)^2 }
\]
They could also be obtained by using \({\bf E}' = {\bf Q} \cdot {\bf E} \cdot {\bf Q}^T\)
with \({\bf Q}\) based on \(\theta_P\).
Principal Strain Notation
Principal strains can be written as \(\epsilon_1\), \(\epsilon_2\),
and \(\epsilon_3\). Only one subscript is usually used in this case to
differentiate the principal strain values from the normal strain components:
\(\epsilon_{11}\), \(\epsilon_{22}\), and \(\epsilon_{33}\).
2-D Principal Strain Example
Start with the strain tensor
\[
\boldsymbol{\epsilon} =
\left[ \matrix{
0.50 & \;\;\; 0.30 \\
0.30 & -0.20
} \right]
\]
The principal orientation is
\[
\begin{eqnarray}
\tan (2 \theta_P) & = & {2 * 0.3 \over 0.50 - (\text{-}0.20) } \\
\\
\\
\theta_P & = & 20.3°
\end{eqnarray}
\]
The principal strains are
\[
\begin{eqnarray}
\epsilon_{max}, \epsilon_{min} & = &
{0.50 - 0.20 \over 2} \pm
\sqrt{ \left( {0.50 + 0.20 \over 2} \right)^2 + (0.30)^2 } \\
\\
\\
\epsilon_{max}, \epsilon_{min} & = & 0.611, -0.311
\end{eqnarray}
\]
The only limitation to using these equations for principal values
is that it is not known which one applies to 20.3° and which
applies to 110.3°. That's why an attractive alternative is
\[
\begin{eqnarray}
\left[
\matrix{\epsilon'\!_{11} & \epsilon'\!_{12} \\
\epsilon'\!_{12} & \epsilon'\!_{22} }
\right]
& = &
\left[
\matrix { \;\;\;\cos(20.3^\circ) & \sin(20.3^\circ) \\
-\sin(20.3^\circ) & \cos(20.3^\circ) }
\right]
\left[
\matrix{0.50 & \;\;\;0.30 \\
0.30 & -0.20 }
\right]
\left[
\matrix { \cos(20.3^\circ) & -\sin(20.3^\circ) \\
\sin(20.3^\circ) & \;\;\;\cos(20.3^\circ) }
\right] \\
\\
\\
& = &
\left[
\matrix{0.611 & \;\;\;0.000 \\
0.000 & -0.311 }
\right]
\end{eqnarray}
\]
This confirms that the 0.611 principal strain value
in the \(\epsilon_{11}\) slot is indeed
20.3° from the X-axis. The \(\epsilon_{22}\) value
is 90° from the first.
3-D Principal Strains
Coordinate transforms in 3-D are
\[
\left[
\matrix{E'_{11} & E'_{12} & E'_{13} \\
E'_{12} & E'_{22} & E'_{23} \\
E'_{13} & E'_{23} & E'_{33} }
\right] =
\left[
\matrix { q_{11} & q_{12} & q_{13} \\
q_{21} & q_{22} & q_{23} \\
q_{31} & q_{32} & q_{33} }
\right]
\left[
\matrix{E_{11} & E_{12} & E_{13} \\
E_{12} & E_{22} & E_{23} \\
E_{13} & E_{23} & E_{33} }
\right]
\left[
\matrix { q_{11} & q_{21} & q_{31} \\
q_{12} & q_{22} & q_{32} \\
q_{13} & q_{23} & q_{33} }
\right]
\]
The second \({\bf Q}\) matrix is once again the transpose of the first.
This page performs tensor transforms.
And
this page calculates principal values (eigenvalues)
and principal directions (eigenvectors).
It's important to remember that the inputs to both pages must be symmetric. In fact,
both pages enforce this.
The eigenvalues above can be written in matrix form as
\[
{\bf E} =
\left[ \matrix{
0.243 & 0 & 0 \\
0 & 1.256 & 0 \\
0 & 0 & 4.338
} \right]
\]
Maximum Shear
The maximum amount of shear at any point is easy to calculate from the
principal strains. It is simply
\[
\gamma_{max} = \epsilon_{max} - \epsilon_{min}
\]
This applies in both 2-D and 3-D. The maximum shear always occurs in a
coordinate system orientation that is rotated 45° from the principal
coordinate system. For the principal strain tensor above
\[
{\bf E} =
\left[ \matrix{
0.243 & 0 & 0 \\
0 & 1.256 & 0 \\
0 & 0 & 4.338
} \right]
\]
The max and min principal strains are in the \(E_{33}\) and \(E_{11}\) slots,
respectively. So the max shear orientation is obtained by rotating the
principal coordinate system by 45° in the (\(1-3\)) plane.
The max shear value itself is
\[
\begin{eqnarray}
\gamma_{max} & = & \epsilon_{max} - \epsilon_{min} \\
\\
& = & 4.338 - 0.243 \\
\\
& = & 4.095
\end{eqnarray}
\]
The manual way of computing principal strains is to solve a cubic equation
for the three principal values. The equation results from setting the following
determinant equal to zero. The \(\lambda\) values, once computed, will equal
the principal values of the strain tensor.
\[
\left|
\matrix{E_{11}-\lambda & E_{12} & E_{13} \\
E_{12} & E_{22}-\lambda & E_{23} \\
E_{13} & E_{23} & E_{33}-\lambda }
\right|
= 0
\]
This can be expanded out to give
\[
\begin{eqnarray}
(E_{11}-\lambda)\big[(E_{22}-\lambda)(E_{33}-\lambda)-E^2_{23}\big] & - & \\
E_{12}\big[E_{12}(E_{33}-\lambda)-E_{23}E_{13}\big] & + & \\
E_{13}\big[E_{12}E_{23} - (E_{22}-\lambda)E_{13}\big] & = & 0
\end{eqnarray}
\]
Invariants
...and expanded out even further to give
\[
\begin{eqnarray}
\lambda^3 - (E_{11} + E_{22} + E_{33}) \lambda^2 +
(E_{11}E_{22}+E_{22}E_{33}+E_{33}E_{11}-E^2_{12}-E^2_{13}-E^2_{23}) \lambda - \\
(E_{11}E_{22}E_{33}-E_{11}E^2_{23}-E_{22}E^2_{13}-E_{33}E^2_{12}+2E_{12}E_{13}E_{23}) = 0
\end{eqnarray}
\]
"Now consider this..."
No matter what coordinate transformation you apply to the
strain tensor, its principal strains had better be the same three values.
And the only way for this to happen in the above equation is for the
equation itself to always be the same, no matter the transformation.
This means that the combinations of strain components, which serve as
coefficients of the \(\lambda\)'s, must be
invariant under
coordinate transformations. Their values must not change.
And that's why they are called
invariants.
So the equation can be written as
\[
\lambda^3 - I_1 \lambda^2 + I_2 \lambda - I_3 = 0
\]
where
\[
\begin{eqnarray}
I_1 & = & E_{11} + E_{22} + E_{33} \\
\\
I_2 & = & E_{11}E_{22}+E_{22}E_{33}+E_{33}E_{11}-E^2_{12}-E^2_{13}-E^2_{23} \\
\\
I_3 & = & E_{11}E_{22}E_{33}-E_{11}E^2_{23}-E_{22}E^2_{13}-E_{33}E^2_{12}+2E_{12}E_{13}E_{23}
\end{eqnarray}
\]
Although probably not initially apparent, the invariants are actually familiar quantities.
\[
\begin{eqnarray}
I_1 & = & \text{tr}({\bf E}) \\
\\
I_2 & = & \left| \matrix{E_{11} & E_{12} \\ E_{12} & E_{22}} \right| +
\left| \matrix{E_{11} & E_{13} \\ E_{13} & E_{33}} \right| +
\left| \matrix{E_{22} & E_{23} \\ E_{23} & E_{33}} \right| \\
\\
I_3 & = & \text{det}({\bf E})
\end{eqnarray}
\]
They can be written in tensor notation as
\[
\begin{eqnarray}
I_1 & = & E_{kk}
\\
\\
I_2 & = & {1 \over 2} \left[ \left( E_{kk} \right)^2 - E_{ij} E_{ij} \right]
\\
\\
I_3 & = & \epsilon_{ijk} E_{i1} E_{j2} E_{k3}
\end{eqnarray}
\]
\(I_3\) is in tensor notation, but no one should actually calculate
a determinant based on tensor notation rules because it is very inefficient.
Physical Interpretation of Invariants
The physical interpretation of the invariants depends on what
tensor the invariants are computed from. For any stress or strain tensor,
\(I_1\) is directly related to the hydrostatic component of the that tensor.
This is universal.
\(I_2\) tends to be related more to the deviatoric
aspects of stress and strain, although not exclusively.
Finally, \(I_3\) does not seem to have any physical significance as the determinant
of a stress or strain tensor. But do not confuse the determinants of stress and
strain tensors with the determinant of a deformation gradient. In that case,
\(\text{det}({\bf F}) = V_F / V_o\), the ratio of deformed to initial
volume, which is 1 for incompressible materials like rubber.
Invariant Example
This example will start with a random strain tensor and demonstrate that the
invariants are invariant under coordinate transformations.
Start with the strain tensor
\[
{\bf E} =
\left[ \matrix{
0.50 & \;\;\; 0.30 & \;\;\;0.20 \\
0.30 & -0.20 & -0.10 \\
0.20 & -0.10 & \;\;\;0.10
} \right]
\]
Its invariants are
\[
\begin{eqnarray}
I_1 & = & 0.50 + (-0.20) + 0.10 \\
\\
& = & 0.40
\\
\\
I_2 & = & (0.50)(-0.20) + (-0.20)(0.10) + (0.10)(0.50) - (0.30)^2 - (0.20)^2 - (-0.10)^2 \\
\\
& = & -0.21
\\
\\
I_3 & = & \text{det}({\bf E}) \\
\\
& = & -0.028
\end{eqnarray}
\]
Now rotate the coordinate system by some random amount shown in the screenshot.
The transformed strain tensor is now
\[
{\bf E}' =
\left[ \matrix{
\;\;\; 0.308 & -0.106 & 0.273 \\
-0.106 & -0.346 & 0.035 \\
\;\;\; 0.273 & \;\;\;0.035 & 0.437
} \right]
\]
And the "new" invariants are
\[
\begin{eqnarray}
I_1 & = & 0.308 + (-0.346) + 0.437
\\
\\
& = & 0.40
\\
\\
I_2 & = & (0.308)(-0.346) + (-0.346)(0.437) + (0.437)(0.308) - (-0.106)^2 - (0.273)^2 - (0.035)^2
\\
\\
& = & -0.21
\\
\\
I_3 & = & \text{det}({\bf E})
\\
\\
& = & -0.028
\end{eqnarray}
\]
So they are indeed invariant.
Just for grins... Let's compute the principal strains and then recompute the invariants
to demonstrate again that they are invariant.
The principal strains are
\[
{\bf E}' =
\left[ \matrix{
0.655 & 0 & 0 \\
0 & -0.371 & 0 \\
0 & 0 & 0.115
} \right]
\]
Curiously, the two different starting points lead to the principal values being
listed in different orders. This is typical and has no special meaning. The
final, physical answer is the same. The \({\bf Q}\) will accommodate the different
listing orders.
Finally, calculating the invariants one last time using the principal values gives
\[
\begin{eqnarray}
I_1 & = & 0.655 + (-0.371) + 0.115 \\
\\
& = & 0.40
\\
\\
I_2 & = & (0.655)(-0.371) + (-0.371)(0.115) + (0.115)(0.655) \\
\\
& = & -0.21
\\
\\
I_3 & = & \text{det}({\bf E}) \quad = \quad (0.655)(-0.371)(0.115) \\
\\
& = & -0.028
\end{eqnarray}
\]
And so it works again. Note how the invariant calculations are rather simple when
all the off-diagonal terms are zero. The determinant is especially easy.
2-D Invariants
In the 2-D world, there are only two invariants.
\[
\begin{eqnarray}
I_1 & = & \text{tr}(\boldsymbol{\epsilon}) & = & \epsilon_{11} + \epsilon_{22} \\
\\
I_2 & = & \text{det}(\boldsymbol{\epsilon}) & = & \epsilon_{11} * \epsilon_{22} - \epsilon^2_{12} \\
\end{eqnarray}
\]
In 2-D, things are just simple enough to directly prove the
invariance of invariants.
We'll prove it for \(I_1\) here. Recall the transformation equations are
\[
\begin{eqnarray}
\epsilon'\!_{xx} & = & \epsilon_{xx} \cos^2 \theta + \epsilon_{yy} \, \sin^2 \theta + 2 \left( {\gamma_{xy} \over 2} \right) \sin \theta \cos \theta \\
\\
\epsilon'\!_{yy} & = & \epsilon_{xx} \sin^2 \theta + \epsilon_{yy} \, \cos^2 \theta - 2 \left( {\gamma_{xy} \over 2} \right) \sin \theta \cos \theta \\
\\
{\gamma'\!_{xy} \over 2} & = & (\epsilon_{yy} - \epsilon_{xx}) \sin \theta \cos \theta + \left( {\gamma_{xy} \over 2} \right) (\cos^2 \theta - \sin^2 \theta)
\end{eqnarray}
\]
So \(\epsilon'\!_{xx} + \epsilon'\!_{yy}\) is
\[
\begin{eqnarray}
\epsilon'\!_{xx} + \epsilon'\!_{yy} & = &
\epsilon_{xx} \cos^2 \theta + \epsilon_{yy} \, \sin^2 \theta + 2 \left( {\gamma_{xy} \over 2} \right) \sin \theta \cos \theta + \\
\\
& & \epsilon_{xx} \sin^2 \theta + \epsilon_{yy} \, \cos^2 \theta - 2 \left( {\gamma_{xy} \over 2} \right) \sin \theta \cos \theta \\
\end{eqnarray}
\]
and the shear terms cancel, while the normal strain terms combine to give
\[
\begin{eqnarray}
\epsilon'\!_{xx} + \epsilon'\!_{yy} & = &
\epsilon_{xx} ( \cos^2 \theta + \sin^2 \theta ) + \epsilon_{yy} ( \sin^2 \theta + \cos^2 \theta ) \\
\\
& = & \epsilon_{xx} + \epsilon_{yy}
\end{eqnarray}
\]
So it works again.
Calculating Principal Values Manually in 3-D
Here's how to compute the roots of a cubic equation... applied to principal values.
Start with
\[
\lambda^3 - I_1 \lambda^2 + I_2 \lambda - I_3 = 0
\]
Recall that the resulting \(\lambda's\) will be the principal values of the stress
or strain tensor... and the invariants will need to be calculated up front.
The first step is to calculate two intermediate quantities, \(Q\) and \(R\).
\[
Q = { 3 I_2 - I^2_1 \over 9 } \qquad R = {2 I^3_1 - 9 I_1 I_2 + 27 I_3 \over 54}
\]
Then calculate another intermediate quantity, \(\theta\).
\[
\theta = \text{cos}^{-1} \left( R \over \sqrt{-Q^3} \right)
\]
And finally, the principal values are
\[
\begin{eqnarray}
\epsilon_1 & = & 2 \, \sqrt{-Q} \, \cos \left( {\theta \over 3} \right) & + & {1\over 3} I_1 \\
\\
\\
\epsilon_2 & = & 2 \, \sqrt{-Q} \, \cos \left( {\theta + 2 \pi \over 3} \right) & + & {1\over 3} I_1 \\
\\
\\
\epsilon_3 & = & 2 \, \sqrt{-Q} \, \cos \left( {\theta + 4 \pi \over 3} \right) & + & {1\over 3} I_1
\end{eqnarray}
\]
Note that minus signs are easily confused here because they are present in the original cubic invariant equation being solved.
Ref:
http://mathworld.wolfram.com/CubicFormula.html
Summary
This principle of invariant quantities under coordinate transformations is
in fact universal across all matrices that are symmetric and being transformed
according to
\[
{\bf A}' = {\bf Q} \cdot {\bf A} \cdot {\bf Q}^T
\]
where \({\bf A}\) is "any symmetric matrix."
And recall that the product of
any matrix with its transpose is always a symmetric result, so this
result would qualify. This is especially relevant to \({\bf F}^T \! \cdot {\bf F}\),
whose invariants are used in the Mooney-Rivlin Law of rubber behavior.
Mooney-Rivlin's Law and coefficients will be discussed on
this page.
As an added teaser, we will see that the 3rd invariant of \({\bf F}^T \! \cdot {\bf F}\)
for rubber always equals 1 because rubber is incompressible. So not only is it a constant,
independent of coordinate transformations, but it is even a constant value,
always equal to 1, independent of coordinate transformations and the state of deformation.
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