Mohr’s Circle

Videos

Given

• The two frames \((x, y)\) and \((\bar x,\bar y)\).

• The \((x,y)\)-tensor components:

  \[\begin{split}\begin{bmatrix} T_{xx} & T_{xy} \\ \mathsf{sym} & T_{yy} \end{bmatrix}\end{split}\]

  Example:

  \[\begin{split}\begin{bmatrix} T_{xx} & T_{xy} \\ \mathsf{sym} & T_{yy} \end{bmatrix} = \begin{bmatrix} -1 & 4 \\ 4 & 5 \end{bmatrix}\end{split}\]

• The Passive Transformation of tensor components:

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

Unknown

1. The Circle.

2. The components \((T_{\bar x\bar x}, T_{\bar x \bar y})\) for a given angle \(\varphi\).

3. The maximal and minimal \(T_{\bar x\bar x}\) and the respective angles. And the maximal and minimal \(T_{\bar x\bar y}\) and the respective angles.

1. Circle

Axes

• Horizontal Axis: \(T_{\bar x \bar x}\)

• Vertikal Axis: \(T_{\bar x \bar y}\)

Circle from 3 Points

Point	Coordinates	Example	Position
\(P_{0^\circ}\)	\((T_{xx}, T_{xy})\)	\((-1, 4)\)	auf dem Kreis
\(P_{90^\circ}\)	\((T_{yy}, -T_{xy})\)	\((5, -4)\)	gegenüber \(P_{0^\circ}\)
\(P_{M}\)	\((\bar{T}, 0)\)	\((2,0)\)	Kreismittelpunkt

with \(\bar{T} = \tfrac12 \left(T_{xx} + T_{yy}\right)\).

4 more Points

oint	Coordinates	Example	Position
\(P_{\varphi_1}\)	\((\bar{T} + r, 0)\)	\(( 7 , 0 )\)	3 Uhr
\(P_{\varphi_1+90^\circ}\)	\((\bar{T} - r, 0)\)	\((-3 , 0 )\)	9 Uhr
\(P_{\varphi_1-45^\circ}\)	\((\bar{T}, 0 + r)\)	\(( 2 , 5 )\)	12 Uhr
\(P_{\varphi_1+45^\circ}\)	\((\bar{T}, 0 - r)\)	\(( 2 , -5 )\)	6 Uhr

with \(r = \sqrt{\left[ \tfrac12 \left(T_{xx}-T_{yy}\right)\right]^2 +T_{xy}^2}\).

2. Components

Reading the Components

• Draw red radius from \(P_{M}\) and \(P_{0^\circ}\).

• Draw red angle \(2\varphi\).

• Draw new radius and new point.

• Read point coordinates \((T_{\bar x\bar x}, T_{\bar x \bar y})\).

Example

Point	Coordinates	\(\varphi\) (blue)	\(2 \varphi\) (red)
\(P_{\varphi_1}\)	\((-1, 4)\)	\(0^\circ\)	\(0^\circ\)
\(P_{\varphi_1+90^\circ}\)	\((3.96, 4.60)\)	\(30^\circ\)	\(60^\circ\)
\(P_{\varphi_1-45^\circ}\)	\((6.96, 0.60)\)	\(60^\circ\)	\(120^\circ\)
\(P_{\varphi_1+45^\circ}\)	\((5, -4)\)	\(90^\circ\)	\(180^\circ\)

3. Extremal Values

3. Reading Extremal Values and resp. Angles

Point	Coordinates	Extremal Value	Angle
\(P_{\varphi_1}\)	\((\max_\varphi T_{\bar x \bar x} ,0)\)	\(\underset{\varphi}{\max} T_{\bar x \bar x}=\bar T + r\)	\(\underset{\varphi}{\arg\!\max} \, T_{\bar x \bar x} \!=\! \varphi_1\)
\(P_{\varphi_1+90^\circ}\)	\((\min_\varphi T_{\bar x \bar x} ,0)\)	\(\underset{\varphi}{\min} T_{\bar x \bar x}=\bar T - r\)	\(\underset{\varphi}{\arg\!\min} \, T_{\bar x \bar x} \!=\! \varphi_1 \!+\! 90^\circ\)
\(P_{\varphi_1-45^\circ}\)	\((\bar T, \max_\varphi T_{\bar x \bar y})\)	\(\underset{\varphi}{\max} T_{\bar x \bar y}= r\)	\(\underset{\varphi}{\arg\!\max} \, T_{\bar x \bar y} \!=\! \varphi_1 \!-\! 45^\circ\)
\(P_{\varphi_1+45^\circ}\)	\((\bar T, \min_\varphi T_{\bar x \bar y})\)	\(\underset{\varphi}{\min} T_{\bar x \bar y}= -r\)	\(\underset{\varphi}{\arg\!\min} \, T_{\bar x \bar y} \!=\! \varphi_1 \!+\! 45^\circ\)

with \(\varphi_1 =\arctan \tfrac{T_{xy}}{T_{xx}- T_{\bar x \bar x 2}}\).

Example

Point	Coordinates	Extremal Value	Angle
\(P_{\varphi_1}\)	\((7,0)\)	\(\underset{\varphi}{\max} T_{\bar x \bar x}= 7\)	\(\underset{\varphi}{\arg\!\max} \, T_{\bar x \bar x}\! \stackrel{1.0}{\approx}\! 63^\circ\)
\(P_{\varphi_1+90^\circ}\)	\((-3,0)\)	\(\underset{\varphi}{\min} T_{\bar x \bar x}= -3\)	\(\underset{\varphi}{\arg\!\min} \, T_{\bar x \bar x}\! \stackrel{1.0}{\approx}\! 153^\circ\)
\(P_{\varphi_1-45^\circ}\)	\((2, 5)\)	\(\underset{\varphi}{\max} T_{\bar x \bar y}= 5\)	\(\underset{\varphi}{\arg\!\max} \, T_{\bar x \bar y}\! \stackrel{1.0}{\approx}\! 18^\circ\)
\(P_{\varphi_1+45^\circ}\)	\((2, -5)\)	\(\underset{\varphi}{\min} T_{\bar x \bar y}= -5\)	\(\underset{\varphi}{\arg\!\min} \, T_{\bar x \bar y}\! \stackrel{1.0}{\approx}\! 108^\circ\)

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