Introduction to Elasticity/Transformation example 1

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Example 1[edit | edit source]

Derive the transformation rule for second order tensors (). Express this relation in matrix notation.

Solution[edit | edit source]

A second-order tensor transforms a vector into another vector . Thus,

In index and matrix notation,

Let us determine the change in the components of with change the basis from () to (). The vectors and , and the tensor remain the same. What changes are the components with respect to a given basis. Therefore, we can write

Now, using the vector transformation rule,

Plugging the first of equation (3) into equation (2) we get,

Substituting for in equation~(4) using equation~(1),

Substituting for in equation (5) using equation (3),

Therefore, if is an arbitrary vector,

which is the transformation rule for second order tensors.

Therefore, in matrix notation, the transformation rule can be written as