Tensor Components
Circle
Details
The given components \((T_{xx}, T_{xy}, T_{yy})\) are defining points on a circle:
Punkt |
\((T_{\bar x \bar x} , T_{\bar x \bar y})\) |
|---|---|
1 |
\((T_{xx}, T_{xy})\) |
2 |
\((T_{yy}, -T_{xy})\) |
3 |
\((\bar{T}, 0)\) |
4 |
\((\bar{T} + r, 0)\) |
5 |
\((\bar{T} - r, 0)\) |
6 |
\((\bar{T}, 0 + r)\) |
7 |
\((\bar{T}, 0 - r)\) |
with shortcut:
Rules
Find point 1 at the end of the red line. Read the tensor components \((T_{xx}, T_{xy})\) as the point coordinates of point 1.
Find point 2. Read the tensor components \((T_{yy}, -T_{xy})\) as the point coordinates of point 2.
Drag the grey boxes to where they belong.
Check your result by checking the 7 points.
Example = First image:
Point 1: Point coordinates \((-1, 4)\).
Point 2: Point coordinates \((5, -4)\).
So that:
Radius \(r=5\) and:
Punkt |
\((T_{\bar x \bar x} , T_{\bar x \bar y})\) |
|---|---|
1 |
\((-1, 4)\) |
2 |
\((5, -4)\) |
3 |
\((2, 0)\) |
4 |
\((7, 0)\) |
5 |
\((-3, 0)\) |
6 |
\((2, 5)\) |
7 |
\((2, -5)\) |
\(\varphi = \varphi_1 \Leftrightarrow T_{\bar x \bar x}\) maximal
Find the angle \(\varphi=\varphi_1\), for which \(T_{\bar x \bar x}\) is maximal.
Details
Angle |
counted |
from |
to |
|---|---|---|---|
\(2 \varphi_1\) |
\(\circlearrowright\) |
\(\color{red}{｜}\) |
\(\color{red}{￤}\) |
Rules
Read \((T_{xx}, T_{xy})\) as the point coordinates of point 1.
Read:math:(T_{yy}, - T_{xy}) as the point coordinates of point 2.
Find:
\[\begin{split}r &= \sqrt{\left\{ \tfrac12 \left(T_{xx}-T_{yy}\right)\right\}^2 +T_{xy}^2} \\ \bar{T} &= \tfrac12 \left(T_{xx} + T_{yy}\right) \\ T_{\bar x \bar x 2} &= \bar{T} - r\end{split}\]Find \(\varphi_1\) using:
\[\varphi_1 = \arctan \frac{T_{xy}}{T_{xx}-T_{\bar x \bar x 2}}\]Find:
\(2\varphi_1\) counted clockwise
\(-2\varphi_1\) counted counter clockwise
both measured from the solid to the dashed line.
Drag the grey boxes to where they belong.
Example: \((T_{xx}, T_{xy}, T_{yy} ) = (-1, 4, 5)\) leads to:
Note
Note the periodicity.
Passive Transformation
Details
Use:
with \(R_\varphi = \begin{bmatrix}c_\varphi & s_\varphi \\-s_\varphi & c_\varphi\end{bmatrix}\) and \(R_\varphi^{\mathsf T}\) the transpose of \(R_\varphi\).
Angle |
counted |
from |
to |
|---|---|---|---|
\(2 \varphi\) |
\(\circlearrowright\) |
\(\color{red}{｜}\) |
\(\color{red}{￤}\) |
\(\varphi\) |
\(\circlearrowleft\) |
\(\color{blue}{￤}\) |
\(\color{blue}{｜}\) |
Rules
Read \((T_{xx}, T_{xy})\) as the point coordinates of point 1. Also read \(T_{yy}\).
Read the angle \(2\varphi\) counted clockwise from the red solid line to the red dashed line. Also read \(\varphi\) counted counter clockwise from the blue dashed line to the blue solid line.
Find \((T_{\bar x \bar x}, T_{\bar x \bar y})\).
Drag the grey boxes to where they belong.
Example: First image top left:
\((T_{xx}, T_{xy}, T_{yy}) = (-1, 4, 5)\) and
\(\varphi=15^\circ\).
This leads to:
Active Transformation
Details
Use:
with \(R_\alpha = \begin{bmatrix}c_\alpha & s_\alpha \\-s_\alpha & c_\alpha\end{bmatrix}\) and \(R_\alpha^{\mathsf T}\) the transpose of \(R_\alpha\).
Angle |
counted |
from |
to |
|---|---|---|---|
\(-2 \alpha\) |
\(\circlearrowright\) |
\(\color{red}{｜}\) |
\(\color{red}{￤}\) |
\(2 \alpha\) |
\(\circlearrowleft\) |
\(\color{red}{｜}\) |
\(\color{red}{￤}\) |
Anleitung
Read \((T_{xx}, T_{xy})\) as the point coordinates of point 1. Also read \(T_{yy}\).
Read the angle \(-2\alpha\) counted clockwise from the solid red line to the dashed red line. Or read \(2\alpha\) counted counter clockwise from the solid red line to the dashed red line.
Calculate \((T'_{xx}, T'_{xy})\).
Drag the grey boxes to where they belong.
Example: First image top left:
\((T_{xx}, T_{xy}, T_{yy}) = (-1, 4, 5)\) und
\(-\alpha=15^\circ\).
This leads to: