Analytical

Analytical#

The rotation matrix R is defined as:

\[\begin{split}\mathrm{R}=\begin{bmatrix} \cos(\alpha) & \sin(\alpha) \\ -\sin(\alpha) & \cos(\alpha) \end{bmatrix}\end{split}\]

Rotating a first-order tensor (for example force \(F\)) by an angle \(\alpha\) is defined as:

\[\bar{F} = \mathrm{R} \cdot F \]

The transformation rule for a second order tensor (for example stiffness matrix \(K\)) is defined in matrix notation as:

\[\bar{\mathrm{K}} = \mathrm{R} \cdot \mathrm{K} \cdot \mathrm{R}^T\]

This is treated in chapter 2.1 of the lecture notes Introduction to Continuum Mechanics (Hartsuijker and Welleman, 2007).

The individual terms of \(\bar{\mathrm{K}}\) can be written out as:

\(k_{\bar x \bar x} = \cfrac{1}{2}\left(k_{xx} + k_{yy}\right) + \cfrac{1}{2}\left(k_{xx} - k_{yy}\right) \cos \left(2\alpha\right) + k_{xy} \sin \left(2 \alpha\right)\)
\(k_{\bar y \bar y} = \cfrac{1}{2}\left(k_{xx} + k_{yy}\right) - \cfrac{1}{2}\left(k_{xx} - k_{yy}\right) \cos \left(2\alpha\right) - k_{xy} \sin \left(2 \alpha\right)\)
\(k_{\bar x \bar y} = -\cfrac{1}{2}\left(k_{xx} - k_{yy}\right) \sin \left(2 \alpha \right) + k_{xy} \cos \left(2 \alpha\right)\)

The principal values of the second order tensor \(\bar{\mathrm{K}}\) are defined by:

\[k_{1,2} = \frac{1}{2}\left(k_{xx} + k_{yy}\right) \pm \sqrt{\left(\frac{1}{2}\left(k_{xx} + k_{yy}\right)\right)^2 + k_{xy}^2}\]

The principal directions of the second order tensor \(\bar{\mathrm{K}}\) are defined by:

\[\tan{2\alpha_0} = \frac{k_{xy}}{\frac{1}{2}\left(k_{xx} - k_{yy}\right)}\]

This is treated in chapter 2.3 of the lecture notes Introduction to Continuum Mechanics (Hartsuijker and Welleman, 2007).