Coordinate transformations

Vector Transformation in Two Dimensions

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In three dimensions, the vector transformation rule is written as

 

where  .

In two dimensions, this transformation rule is the familiar

 

In matrix form,

 

Since we are using sines, the direction of measurement of   is required. In this case, it is measured counterclockwise.

Tensor Transformation in Two Dimensions

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

 

where  .

The Cauchy stress  is a symmetric second-order tensor. In two dimensions, the transformation rule for stress is then written as

 

In matrix form,

 

Since we are using sines, the direction of measurement of   is required. In this case, it is measured counterclockwise.

Tensor Transformation in two Dimensions, the intrinsic approach

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

 
 
 
 
 
 

The stress and strain tensors are now defined by :

 

and

 

Then once constructs the bound matrix in the orthonormal base  

 

with

  the rotation matrix in   base.

Example

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is the rotation along the axis   in the :  base

The associated rotation in the   base is :

 

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

 

Four order tensor

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The élasticity tensor   in the :  is defined in the  :  by

 

and is rotated by:

 
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Introduction to Elasticity