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If the stress $\sigma_{xx}$ is applied to an isotropic, three-dimensional body, the following strain tensor results: $$\boldsymbol\epsilon=\left(\begin{matrix}\frac{1}{E}\sigma _{xx} & 0 & 0 \\0 & -\frac{\nu}{E}\sigma _{xx} & 0 \\0 & 0 & -\frac{\nu}{E}\sigma _{xx}\end{matrix}\right)$$ Now the tensor should be rotated around the y-axis with the angle $\alpha$.

figure

The transformation should be carried out with $ A'=QAQ^T $. How would be the transformation matrix $ Q $ in this case?

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  • $\begingroup$ The transformation matrix is $Q=\begin{pmatrix} c\left( \alpha \right) & 0 & s\left( \alpha \right) \\ 0 & 1 & 0 \\ -s\left( \alpha \right) & 0 & c\left( \alpha \right) \end{pmatrix}$ where c is cos and s is sin $\endgroup$
    – Eli
    Sep 16 '21 at 7:37
  • $\begingroup$ Probably $\alpha\mapsto \pi/2-\alpha$ $\endgroup$
    – Eli
    Sep 16 '21 at 12:01
  • $\begingroup$ Many thanks for the answer. I want to look at angle between $0$ and $\pi/2$. I'm not sure if $Q$ fits my sketch to the definition of $ \alpha $. Shouldn't the first and last lines of $Q$ have to be changed? I hope I made no mistakes, but I would get the following result for the transformed strain tensor $\epsilon'$: $\endgroup$
    – Frank
    Sep 16 '21 at 18:36
  • $\begingroup$ $$\boldsymbol\epsilon'=\left(\begin{matrix}cos^2(\alpha)\epsilon _{11}+sin^2(\alpha)\epsilon _{33} & 0 & cos(\alpha)sin(\alpha)(\epsilon _{33}-\epsilon _{11}) \\0 & \epsilon _{22} & 0 \\cos(\alpha)sin(\alpha)(\epsilon _{33}-\epsilon _{11}) & 0 & sin^2(\alpha)\epsilon _{11}+cos^2(\alpha)\epsilon _{33}\end{matrix}\right)$$ $\endgroup$
    – Frank
    Sep 16 '21 at 18:37
  • $\begingroup$ I can understand the behaviour of the main axial strains, but how are the shear strain components explained? I would not have "expected" these here... $\endgroup$
    – Frank
    Sep 16 '21 at 18:37
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I use this notation

the transformation matrix, transformed a vector components from rotate system index B to inertial system index I

rotation about the x-axis angle $~\alpha~$ between y and y'

$${_B^I}\,Q_x=\left[ \begin {array}{ccc} 1&0&0\\ 0&\cos \left( \alpha \right) &-\sin \left( \alpha \right) \\ 0& \sin \left( \alpha \right) &\cos \left( \alpha \right) \end {array} \right] $$

rotation about the y-axis angle $~\alpha~$ between x and x'

$${_B^I}\,Q_y= \left[ \begin {array}{ccc} \cos \left( \alpha \right) &0&\sin \left( \alpha \right) \\ 0&1&0\\ -\sin \left( \alpha \right) &0&\cos \left( \alpha \right) \end {array} \right] $$

rotation about the z-axis angle $~\alpha~$ between x and x'

$${_B^I}\,Q_z=\left[ \begin {array}{ccc} \cos \left( \alpha \right) &-\sin \left( \alpha \right) &0\\ \sin \left( \alpha \right) &\cos \left( \alpha \right) &0\\ 0&0&1\end {array} \right] $$

vector transformation from B to I system

$$\mathbf v_I={_B^I}\mathbf Q\,\mathbf v_B$$

matrix transformation

$$\mathbf M_I=\mathbf ={_B^I}\mathbf Q \,\mathbf M_B\,{_I^B}\mathbf Q =\mathbf Q\,\mathbf M_B\,\mathbf Q^T\\ \mathbf M_B=\mathbf ={_I^B}\mathbf Q \,\mathbf M_I\,{_B^I}\mathbf Q =\mathbf Q^T\,\mathbf M_I\,\mathbf Q $$

your matrix is

$$\mathbf \epsilon_I= \left[ \begin {array}{ccc} \epsilon_{{11}}&0&0\\ 0& \epsilon_{{22}}&0\\ 0&0&\epsilon_{{22}}\end {array} \right] \\ \mathbf \epsilon_B=Q^T\,\mathbf \epsilon_I\,\mathbf Q $$

for $~\mathbf Q=\mathbf Q_x~$ you obtain

$$\mathbf \epsilon_B=\mathbf \epsilon_I$$

for $~\mathbf Q=\mathbf Q_y~$

$$\mathbf \epsilon_B=\left[ \begin {array}{ccc} \left( \cos \left( \alpha \right) \right) ^{2}\epsilon_{{11}}+\epsilon_{{22}}- \left( \cos \left( \alpha \right) \right) ^{2}\epsilon_{{22}}&0&\cos \left( \alpha \right) \sin \left( \alpha \right) \left( -\epsilon_{{22}}+\epsilon_ {{11}} \right) \\ 0&\epsilon_{{22}}&0 \\ \cos \left( \alpha \right) \sin \left( \alpha \right) \left( -\epsilon_{{22}}+\epsilon_{{11}} \right) &0& \left( \cos \left( \alpha \right) \right) ^{2}\epsilon_{{22}}+\epsilon_{{11} }- \left( \cos \left( \alpha \right) \right) ^{2}\epsilon_{{11}} \end {array} \right] $$

for $~\mathbf Q=\mathbf Q_z~$

$$\mathbf \epsilon_B= \left[ \begin {array}{ccc} \left( \cos \left( \alpha \right) \right) ^{2}\epsilon_{{11}}+\epsilon_{{22}}- \left( \cos \left( \alpha \right) \right) ^{2}\epsilon_{{22}}&-\cos \left( \alpha \right) \sin \left( \alpha \right) \left( -\epsilon_{{22}}+\epsilon_ {{11}} \right) &0\\ -\cos \left( \alpha \right) \sin \left( \alpha \right) \left( -\epsilon_{{22}}+\epsilon_{{11}} \right) & \left( \cos \left( \alpha \right) \right) ^{2}\epsilon_{{ 22}}+\epsilon_{{11}}- \left( \cos \left( \alpha \right) \right) ^{2} \epsilon_{{11}}&0\\ 0&0&\epsilon_{{22}}\end {array} \right] $$

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Tensors are rotated by the same rotation matrices that rotate vectors. So if the basis vectors are rotated by $Q_{ij}$:

$$e'^{(k)}_i = Q_{ij}e^{(k)}_j$$

Then a rank-2 tensor is rotated with:

$$ A'_{il}=Q_{ij}A_{jk}Q^T_{kl}$$

A symmetric tensor has 6 components. There is an invariant trace:

$$ A'_{ii}=Q_{ij}A_{jk}Q^T_{ki}$$ $$ A'_{ii}=Q^T_{ki}Q_{ij}A_{jk}$$ $$ A'_{ii}=\delta_{kj}A_{jk}=A_{kk}$$

and 5 quadrupole components.

With your initial tensor, you can subtract off the trace to get the pure rank-2 part:

$$ N_{ij} = \epsilon_{ij} - \frac 1 3 {\rm Tr}(\epsilon) $$

With

$$ {\rm Tr}(\epsilon) = \frac{\sigma_{xx}}E\big(\frac 1 3 -\frac 2 3\nu\big)$$

$$ N_{xx} = \frac{\sigma_{xx}}E\big(\frac 2 3 + \frac 2 3\nu\big)=\frac 2 3\frac{\sigma_{xx}}E\big(1+\nu\big)$$

$$ N_{yy} = N_{zz}=\frac{\sigma_{xx}}E\big(-\frac 1 3 - \frac 1 3\nu\big) =-\frac 1 3\frac{\sigma_{xx}}E\big(1+\nu\big)=-\frac 1 2 N_{xx}$$

If you transform that into spherical tensors, you get a two non-zero parts:

$$N_{zz} \propto Y_2^0(\theta, \phi)$$

which describes how prolate or oblate the quadrupole moment is (along the $z$-axis), and

$$N_{xx}-N_{yy} \propto (Y_2^2(\theta, \phi) + Y_2^{-2}(\theta, \phi))$$

which describes the lowest degree of azimuthal asymmetry (a bulge like the Earth's tidal bulge).

The

$$N_{xz} \propto (Y_2^1(\theta, \phi) + Y_2^{-1}(\theta, \phi))$$

term is zero. This term is caused by choosing a coordinate system that is not aligned with the principle axes of the quadrupole. It can be diagonalized away.

So: when you transform to $\epsilon'$ you get non-zero $l=2$, $m=\pm 1$ moments, meaning you have not chosen the best coordinates. There is no physical meaning.

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