Page 150 - Introduction to Continuum Mechanics

Kinematics of a Continuum 135

The meanings for 2E^ and 2#23 can be established in a similar fashion.
We can also express the components of E * in terms of the displacement components. From
Eq. (3.24.1), Eq. (3.23.2) and Eq. (3.18.5), we obtain immediately

in component form,

and in long form,

Other components can be similarly written down.
We note that for small values of displacement gradients these equations reduce to those
of the infinitesimal deformation tensor.

Example 3.24.1
For the simple shear deformation

(a) Compute the Lagrangian Strain tensor [E*]
(b) Referring to Fig. 3.12, by a simple geometrical consideration, find the deformed length for
the element OB in Fig. 3.12.

(c) Compare the results of (b) with E\2
Solution, (a) Using the [C] obtained in Example 3.23.2, we easily obtain from the equation
2E* = C-I that