Page 154 - Introduction to Continuum Mechanics

Kinematics of a Continuum 139

By considering two material elements dX^ ' = dSi(R e^) and dXs ' = dS-^R 62) which
deform into dr ' — dsi*n and chr' = dsyn where m and n are unit vectors having an angle
of ft between them, then Eq. (3.23.4) gives

That is

Similarly

and

We can also express the components of B in terms of the displacement components.
Using Eq. (3.18.5), we have

and in component form,

We note that for small displacement gradients,—(5,y - (5,y) reduces to 2Ejj of Eq. (3.7.lOa).

Example 3.25.1

For the simple shear deformation

(a) Obtain the left Cauchy-Green deformation tensor.