Coordinate transformations

## Contents

* 1 Vector Transformation in Two Dimensions
* 2 Tensor Transformation in Two Dimensions
* 3 Tensor Transformation in two Dimensions, the intrinsic approach
* 3.1 Example
* 4 Four order tensor
* 5 Related Content

## Vector Transformation in Two Dimensions

[edit | edit source]

In three dimensions, the vector transformation rule is written as

v i ′ = l i j v j {\displaystyle v_{i}^{'}=l_{ij}v_{j}}

where  l i j = e i ′ ∙ e j = cos ⁡ ( e i ′ , e j ) {\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

v 1 ′ = v 1 cos ⁡ θ + v 2 sin ⁡ θ v 2 ′ = − v 1 sin ⁡ θ + v 2 cos ⁡ θ {\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,

[ v 1 ′ v 2 ′ ] = [ cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ] [ v 1 v 2 ] {\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  θ {\displaystyle \textstyle \theta } is required. In this case, it is measured counterclockwise.

## Tensor Transformation in Two Dimensions

[edit | edit source]

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

T i j ′ = l i p l j q T p q {\displaystyle T_{ij}^{'}=l_{ip}l_{jq}T_{pq}}

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

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

σ 11 ′ = σ 11 cos 2 ⁡ θ + σ 22 sin 2 ⁡ θ + 2 σ 12 sin ⁡ θ cos ⁡ θ σ 22 ′ = σ 11 sin 2 ⁡ θ + σ 22 cos 2 ⁡ θ − 2 σ 12 sin ⁡ θ cos ⁡ θ σ 12 ′ = − σ 11 sin ⁡ θ cos ⁡ θ + σ 22 sin ⁡ θ cos ⁡ θ + σ 12 ( cos 2 ⁡ θ − sin 2 ⁡ θ ) {\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,

[ σ 11 ′ σ 22 ′ σ 12 ′ ] = [ cos 2 ⁡ θ sin 2 ⁡ θ 2 sin ⁡ θ cos ⁡ θ sin 2 ⁡ θ cos 2 ⁡ θ − 2 sin ⁡ θ cos ⁡ θ − sin ⁡ θ cos ⁡ θ sin ⁡ θ cos ⁡ θ cos 2 ⁡ θ − sin 2 ⁡ θ ] [ σ 11 σ 22 σ 12 ] {\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  θ {\displaystyle \textstyle \theta } is required. In this case, it is measured counterclockwise.

## Tensor Transformation in two Dimensions, the intrinsic approach

[edit | edit source]

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

E 1 = e 1 ⊗ e 1 {\displaystyle E_{1}=e_{1}\otimes e_{1}}

E 2 = e 2 ⊗ e 2 {\displaystyle E_{2}=e_{2}\otimes e_{2}}

E 3 = e 3 ⊗ e 3 {\displaystyle E_{3}=e_{3}\otimes e_{3}}

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

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

E 6 = 1 2 ( e 1 ⊗ e 2 + e 2 ⊗ e 1 ) {\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 :

{ σ } = { σ 11 σ 22 σ 33 2 σ 23 2 σ 31 2 σ 12 } {\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

{ ε } = { ε 11 ε 22 ε 33 2 ε 23 2 ε 31 2 ε 12 } {\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  E i ⊗ E j {\displaystyle E_{i}\otimes E_{j}}

[ R ^ ( θ ) ] = [ R 11 2 R 12 2 R 13 2 2 R 12 R 13 2 R 11 R 13 2 R 11 R 12 R 21 2 R 22 2 R 23 2 2 R 22 R 23 2 R 21 R 23 2 R 22 R 21 R 31 2 R 32 2 R 33 2 2 R 33 R 32 2 R 33 R 31 2 R 31 R 32 2 R 21 R 31 2 R 22 R 32 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 2 R 11 R 31 2 R 12 R 32 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 2 R 11 R 21 2 R 12 R 22 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 ] {\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

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

### Example

[edit | edit source]

[ R ( θ ) ] = [ 1 0 0 0 cos ⁡ θ sin ⁡ θ 0 − sin ⁡ θ cos ⁡ θ ] {\displaystyle \left[R(\theta )\right]=\left[{\begin{matrix}1&0&0\\\0&\cos \theta &\sin \theta \\\0&-\sin \theta &\cos \theta \end{matrix}}\right]}

is the rotation along the axis  e 1 {\displaystyle e_{1}} in the : e i ⊗ e j {\displaystyle e_{i}\otimes e_{j}} base

The associated rotation in the  E i ⊗ E j {\displaystyle E_{i}\otimes E_{j}} base is :

[ R ^ ( θ ) ] = [ 1 0 0 0 0 0 0 cos 2 ⁡ θ sin 2 ⁡ θ 2 sin ⁡ θ cos ⁡ θ 0 0 0 sin 2 ⁡ θ cos 2 ⁡ θ − 2 sin ⁡ θ cos ⁡ θ 0 0 0 − 2 sin ⁡ θ cos ⁡ θ 2 sin ⁡ θ cos ⁡ θ cos 2 ⁡ θ − sin 2 ⁡ θ 0 0 0 0 0 0 cos ⁡ θ − sin ⁡ θ 0 0 0 0 sin ⁡ θ cos ⁡ θ ] {\displaystyle \left[{\hat {R}}(\theta )\right]=\left[{\begin{matrix}1&0&0&0&0&0\\\0&\cos ^{2}\theta &\sin ^{2}\theta &{\sqrt {2}}\sin \theta \cos \theta &0&0\\\0&\sin ^{2}\theta &\cos ^{2}\theta &-{\sqrt {2}}\sin \theta \cos \theta &0&0\\\0&-{\sqrt {2}}\sin \theta \cos \theta &{\sqrt {2}}\sin \theta \cos \theta &\cos ^{2}\theta -\sin ^{2}\theta &0&0\\\0&0&0&0&\cos \theta &-\sin \theta \\\0&0&0&0&\sin \theta &\cos \theta \\\\\end{matrix}}\right]}

The rotation of a second order tensor is now defined by :

{ σ ( θ ) } = [ R ^ ( θ ) ] T { σ } {\displaystyle \left\\{\sigma (\theta )\right\\}={\left[{\hat {R}}(\theta )\right]}^{T}\left\\{\sigma \right\\}}

## Four order tensor

[edit | edit source]

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

[ C ¯ ] = [ C 1111 C 1122 C 1133 2 C 1123 2 C 1131 2 C 1112 C 1122 C 2222 C 2233 2 C 2223 2 C 2231 2 C 2212 C 1133 C 2233 C 3333 2 C 3323 2 C 3331 2 C 3312 2 C 1123 2 C 2223 2 C 2333 2 C 2323 2 C 2331 2 C 2312 2 C 1131 2 C 2231 2 C 3331 2 C 2331 2 C 3131 2 C 3112 2 C 1112 2 C 2212 2 C 3312 2 C 2312 2 C 3112 2 C 1212 ] {\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:

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

## Related Content

[edit | edit source]

Introduction to Elasticity