Transformation of Strain

Recall that or

To express the components of strain in a new coordinate system, we must express both the displacement and in the new coordinate system. That is,

Therefore, to express in terms of the components of strain in the old coordinate system, we should:

• express the displacement components in the new coordinate system in terms of displacement components in the old coordinate system
• express differentiation with respect to a new coordinate axis in terms of differentiation with respect to the old coordinates.

Transformation of Displacement

The displacement is a vector quantity. Therefore, its components in a new coordinate system follow the transformation of vectors: or

where is the kth component of the ith unit vector for the new coordinate system:

Transformation of Derivatives

It follows from the chain rule that

The rate of change of an old coordinate with respect to a new coordinate is the cosine of the angle between them:

Therefore,

We can write (6) as

and for all new coordinates:

Gradient in New Coordinates

Combining (1) and (6), we get Alternatively, using matrix notation and (2) and (7), we can write

Transformation Law for Strain Tensor

Since we conclude that This is the same transformation formula as for stress:

Special Case: 2D Transformation

In 2D,

Therefore,

This shows that to transform the strain components in a 2D problem, we can use Mohr’s circle, exactly as with stress.