Pure deformation and rigid body motion

Point to be noted here is that if the quantities are given, the transformation of the coordinates of a point will not be completely defined, because, will still remain undetermined. But these quantities do not influence the deformations because they define only the rigid translation of the body.

Let us now define the quantities which are components of vector , or the displacement vector. Then

(1.8)

We want to know the constraints on the tensor that will result in rigid body motion and no deformation.

Basically we want to find out the condition that the magnitude of the vector or the square of its modulus remains unaltered due to the transformation. Restricting ourselves to infinitesimal transformation, that is neglecting higher order terms,

(1.9)

In order that , it is imperative that

and

(1.10)

which implies that . Then equation 1.8 can be written as,

(1.11)

where . These quantities are infinitesimal angles of rotation about the coordinate axes and are called the components of rotation.