Page 132 - Introduction to Continuum Mechanics

Compatibility Conditions for infinitesimal Strain Components 117

be compatible?
Solution. Yes. There is no need to check, because the displacement u is given (and therefore
exists!)

Example 3.16.2
Does the following strain field:

represent a compatible strain field?
Solution. Since each term of the compatibility equations involves second derivatives of the
strain components with respect to the coordinates, the above strain tensor with each com-
ponent a linear function of Xi. X^. X$ will obviously satisfy them. The given strain components
are obviously continuous functions having continuous second derivatives (in fact continuous
derivatives of all orders) in any bounded region. Thus, the existence of single valued con-
tinuous displacement field in any bounded simply-connected region is ensured by the theorem
stated above. In fact, it can be easily verified that

(to which of course, can be added any rigid body displacements) which is a single-valued
continuous displacement field in any bounded region, including multiply-connected region.

Example 16.3
For the following strain field

does there exist single-valued continuous displacement fields for (a) the cylindrical body with
the normal cross-section shown in Fig. 3.7 and (b) for the body with the normal cross-section
shown in Fig. 3.8 and with the origin of the axis inside the hole of the cross-section.

Solution. Out of the six compatibility conditions, only the first one needs to be checked, the
others are automatically satisfied. Now,