Transformation of components of strain (contd...)

If the axes are so chosen that they coincide with the principal axes of the surface then, 1.32 takes the form: , where are the of the new axes.

Principal axes are the roots of the equation:

(1.32)

where is an invariant, since coefficients of 1.32 should be so.

Consider a right parallelepiped having sides OA= , OB = and OC= . Then its volume is . After deformation the right parallelepiped remains a right parallelepiped with sides , , , hence its volume is

=

(1.33)

So that,

(1.34)

Hence is the relative expansion in volume or the cubical expansion.

In a more general form, the ratio of a volume element in strained state to unstrained state is given as

(1.35)

where is the cubical dilation.