Vector Transformation in Two Dimensions

In three dimensions, the vector transformation rule is written as

${\displaystyle v_{i}^{'}=l_{ij}v_{j}}$

where ${\displaystyle \textstyle l_{ij}=\mathbf {e} _{i}^{'}\bullet \mathbf {e} _{j}=\cos(\mathbf {e} _{i}^{'},\mathbf {e} _{j})}$.

In two dimensions, this transformation rule is the familiar

${\displaystyle {\begin{aligned}v_{1}^{'}&=v_{1}\cos \theta +v_{2}\sin \theta \\v_{2}^{'}&=-v_{1}\sin \theta +v_{2}\cos \theta \\\end{aligned}}}$

In matrix form,

${\displaystyle {\begin{bmatrix}v_{1}^{'}\\v_{2}^{'}\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}}$

Since we are using sines, the direction of measurement of ${\displaystyle \textstyle \theta }$ is required. In this case, it is measured counterclockwise.

Tensor Transformation in Two Dimensions

In three dimensions, the second-order tensor transformation rule is written as

${\displaystyle T_{ij}^{'}=l_{ip}l_{jq}T_{pq}}$

where ${\displaystyle \textstyle l_{ij}=\mathbf {e} _{i}^{'}\bullet \mathbf {e} _{j}=\cos(\mathbf {e} _{i}^{'},\mathbf {e} _{j})}$.

The Cauchy stress ${\displaystyle \textstyle {\boldsymbol {\sigma }}}$is a symmetric second-order tensor. In two dimensions, the transformation rule for stress is then written as

${\displaystyle {\begin{aligned}\sigma _{11}^{'}&=\sigma _{11}\cos ^{2}\theta +\sigma _{22}\sin ^{2}\theta +2\sigma _{12}\sin \theta \cos \theta \\\sigma _{22}^{'}&=\sigma _{11}\sin ^{2}\theta +\sigma _{22}\cos ^{2}\theta -2\sigma _{12}\sin \theta \cos \theta \\\sigma _{12}^{'}&=-\sigma _{11}\sin \theta \cos \theta +\sigma _{22}\sin \theta \cos \theta +\sigma _{12}(\cos ^{2}\theta -\sin ^{2}\theta )\end{aligned}}}$

In matrix form,

${\displaystyle {\begin{bmatrix}\sigma _{11}^{'}\\\sigma _{22}^{'}\\\sigma _{12}^{'}\end{bmatrix}}={\begin{bmatrix}\cos ^{2}\theta &\sin ^{2}\theta &2\sin \theta \cos \theta \\\sin ^{2}\theta &\cos ^{2}\theta &-2\sin \theta \cos \theta \\-\sin \theta \cos \theta &\sin \theta \cos \theta &\cos ^{2}\theta -\sin ^{2}\theta \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}}$

Since we are using sines, the direction of measurement of ${\displaystyle \textstyle \theta }$ is required. In this case, it is measured counterclockwise.

Tensor Transformation in two Dimensions, the intrinsic approach

Let construct an orthonormal basis of the second order tensor projected in the first order tensor

${\displaystyle E_{1}=e_{1}\otimes e_{1}}$

${\displaystyle E_{2}=e_{2}\otimes e_{2}}$

${\displaystyle E_{3}=e_{3}\otimes e_{3}}$

${\displaystyle E_{4}={\frac {1}{\sqrt {2}}}(e_{2}\otimes e_{3}+e_{3}\otimes e_{2})}$

${\displaystyle E_{5}={\frac {1}{\sqrt {2}}}(e_{3}\otimes e_{1}+e_{1}\otimes e_{3})}$

${\displaystyle E_{6}={\frac {1}{\sqrt {2}}}(e_{1}\otimes e_{2}+e_{2}\otimes e_{1})}$

The stress and strain tensors are now defined by :

${\displaystyle \left\{\sigma \right\}=\left\{{\begin{aligned}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\{\sqrt {2}}\sigma _{23}\\{\sqrt {2}}\sigma _{31}\\{\sqrt {2}}\sigma _{12}\\\end{aligned}}\right\}}$

and

${\displaystyle \left\{\varepsilon \right\}=\left\{{\begin{aligned}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\{\sqrt {2}}\varepsilon _{23}\\{\sqrt {2}}\varepsilon _{31}\\{\sqrt {2}}\varepsilon _{12}\\\end{aligned}}\right\}}$

Then once constructs the bound matrix in the orthonormal base ${\displaystyle E_{i}\otimes E_{j}}$

${\displaystyle \left[{\hat {R}}(\theta )\right]=\left[{\begin{matrix}R_{11}^{2}&R_{12}^{2}&R_{13}^{2}&{\sqrt {2}}R_{12}R_{13}&{\sqrt {2}}R_{11}R_{13}&{\sqrt {2}}R_{11}R_{12}\\R_{21}^{2}&R_{22}^{2}&R_{23}^{2}&{\sqrt {2}}R_{22}R_{23}&{\sqrt {2}}R_{21}R_{23}&{\sqrt {2}}R_{22}R_{21}\\R_{31}^{2}&R_{32}^{2}&R_{33}^{2}&{\sqrt {2}}R_{33}R_{32}&{\sqrt {2}}R_{33}R_{31}&{\sqrt {2}}R_{31}R_{32}\\{\sqrt {2}}R_{21}R_{31}&{\sqrt {2}}R_{22}R_{32}&{\sqrt {2}}R_{23}R_{33}&R_{22}R_{33}+R_{23}R_{32}&R_{21}R_{33}+R_{31}R_{23}&R_{21}R_{32}+R_{31}R_{22}\\{\sqrt {2}}R_{11}R_{31}&{\sqrt {2}}R_{12}R_{32}&{\sqrt {2}}R_{13}R_{33}&R_{12}R_{33}+R_{32}R_{13}&R_{11}R_{33}+R_{13}R_{31}&R_{11}R_{32}+R_{31}R_{12}\\{\sqrt {2}}R_{11}R_{21}&{\sqrt {2}}R_{12}R_{22}&{\sqrt {2}}R_{13}R_{23}&R_{12}R_{23}+R_{22}R_{13}&R_{11}R_{23}+R_{21}R_{13}&R_{11}R_{22}+R_{21}R_{12}\\\end{matrix}}\right]}$

with

${\displaystyle \left[R(\theta )\right]}$ the rotation matrix in ${\displaystyle e_{i}\otimes e_{j}}$ base.

Four order tensor

The élasticity tensor ${\displaystyle C_{ijkl}}$ in the :${\displaystyle e_{i}\otimes e_{j}\otimes e_{k}\otimes e_{l}}$ is defined in the  :${\displaystyle E_{i}\otimes E_{j}}$ by

${\displaystyle \left[{\overline {C}}\right]=\left[{\begin{aligned}C_{1111}&C_{1122}&C_{1133}&{\sqrt {2}}C_{1123}&{\sqrt {2}}C_{1131}&{\sqrt {2}}C_{1112}\\C_{1122}&C_{2222}&C_{2233}&{\sqrt {2}}C_{2223}&{\sqrt {2}}C_{2231}&{\sqrt {2}}C_{2212}\\C_{1133}&C_{2233}&C_{3333}&{\sqrt {2}}C_{3323}&{\sqrt {2}}C_{3331}&{\sqrt {2}}C_{3312}\\{\sqrt {2}}C_{1123}&{\sqrt {2}}C_{2223}&{\sqrt {2}}C_{2333}&2C_{2323}&2C_{2331}&2C_{2312}\\{\sqrt {2}}C_{1131}&{\sqrt {2}}C_{2231}&{\sqrt {2}}C_{3331}&2C_{2331}&2C_{3131}&2C_{3112}\\{\sqrt {2}}C_{1112}&{\sqrt {2}}C_{2212}&{\sqrt {2}}C_{3312}&2C_{2312}&2C_{3112}&2C_{1212}\end{aligned}}\right]}$

and is rotated by:

${\displaystyle {\left[{\overline {C}}(\theta )\right]}_{g}={\left[{\hat {R}}(\theta )\right]}^{T}\left[{\overline {C}}\right]\left[{\hat {R}}(\theta )\right]}$

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Introduction to Elasticity