# 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:

$\begin{split}r &= \sqrt{\left\{ \tfrac12 \left(T_{xx}-T_{yy}\right)\right\}^2 +T_{xy}^2} \\ \bar{T} &= \tfrac12 \left(T_{xx} + T_{yy}\right)\end{split}$

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:

$(T_{xx}, T_{xy}, T_{yy}) = (-1, 4, 5)$

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:

$\begin{split}r&=5 \\ \bar{T} &= 2 \\ T_{\bar x \bar x 2} &= -3 \\ \varphi_1&\approx 63^\circ\end{split}$

Note

Note the periodicity.

## Passive Transformation

Details

Use:

$\begin{split}\begin{bmatrix} T_{\bar x \bar x} & T_{\bar x\bar y} \\ T_{\bar x\bar y} & T_{\bar y\bar y} \end{bmatrix} = R_\varphi \begin{bmatrix} T_{xx} & T_{xy} \\ T_{xy} & T_{yy} \end{bmatrix} R_\varphi^{\mathsf T}\end{split}$

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

• $$\varphi=15^\circ$$.

$\begin{split}\begin{bmatrix} T_{\bar x \bar x} & \ldots \\ T_{\bar x\bar y} & \ldots \end{bmatrix} \approx \begin{bmatrix} 1.40 & \ldots \\ 4.96 & \ldots \end{bmatrix}\end{split}$

## Active Transformation

Details

Use:

$\begin{split}\begin{bmatrix} T'_{xx} & T'_{xy} \\ T'_{xy} & T'_{yy} \end{bmatrix} = R_\alpha^{\mathsf T} \begin{bmatrix} T_{xx} & T_{xy} \\ T_{xy} & T_{yy} \end{bmatrix} R_\alpha\end{split}$

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

• $$-\alpha=15^\circ$$.

$\begin{split}\begin{bmatrix} T'_{xx} & \ldots \\ T'_{xy} & \ldots \end{bmatrix} \approx \begin{bmatrix} 1.40 & \ldots \\ 4.96 & \ldots \end{bmatrix}\end{split}$