Page 131 - Introduction to Continuum Mechanics

116 Kinematics of a Continuum

These six equations are known as the equations of compatibility (or integrability condi-
tions).
That these conditions are necessary can be easily proved as follows:
From

we get

Now, since the left-hand sides of the above equations are, by postulate, continuous, therefore,
the right-hand sides are continuous, and so the order of the differentiation is immaterial, so
that

Thus, from Eqs. (iii) and Eq. (3.16.4)

The other five conditions can be similarly established. We omit the proof that the condi-
tions are also sufficient (under the conditions stated in the theorem). In Example 3.16.3 below,
we shall give an instance where the conditions are not sufficient for a region which is not
simply-connected. (A region of space is said to be simply-connected if every closed curve drawn
in the region can be shrunk to a point, by continuous deformation, without passing out of the
boundaries of the region. For example, the solid prismatical bar represented in Fig. 3.7 is
simply-connected whereas, the prismatical tube represented in Fig. 3.8 is not simply-con-
nected).
It is worth noting the following two special cases of strain components where the com-
patibility conditions need not be considered because they are obviously satisfied:
(l)The strain components are obtained from given displacement components.
(2)The strain components are linear functions of coordinates.

Example 3.16.1
Will the strain components obtained from the displacements