Coordinate transformations

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[edit] Vector Transformation in Two Dimensions

In three dimensions, the vector transformation rule is written as

v^{'}_i = l_{ij} v_j

where \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

\begin{align} v^{'}_1 & = v_1 \cos\theta + v_2 \sin\theta \\ v^{'}_2 & = -v_1 \sin\theta + v_2 \cos\theta \\ \end{align}

In matrix form,

\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 \textstyle \theta is required. In this case, it is measured counterclockwise.

[edit] Tensor Transformation in Two Dimensions

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

T^{'}_{ij} = l_{ip} l_{jq} T_{pq}

where \textstyle l_{ij} = \mathbf{e}^{'}_i\bullet\mathbf{e}_j = \cos(\mathbf{e}^{'}_i,\mathbf{e}_j).

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

\begin{align} \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{align}

In matrix form,

\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 \textstyle \theta is required. In this case, it is measured counterclockwise.


[edit] Related Content

Introduction to Elasticity

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