Page 130 - Introduction to Continuum Mechanics

Compatibility Conditions for infinitesimal Strain Components 115

and

where / and g are arbitrary integration functions. Now, since E\2 - 0, we must have, from
Eq.(3.16.4)

Using Eqs. (ii) and (iii), we get from Eq. (iv)

Since the second or third term cannot have terms of the form X^X^ the above equation can
never be satisfied. In other words, there is no displacement field corresponding to this given
Ey. That is, the given six strain components are not compatible with the three displacement-
strain equations.
We now state the following theorem: If EifiX\JtiJQ are continuous functions having
continuous second partial derivatives in a simply connected region, then the necessary and
an U
sufficient conditions for the existence of single-valued continuous solutions #1, #2 ^ 3 °f
the six equation Eq. (3.16.1) to Eq. (3.16.6) are