Page 134 - Introduction to Continuum Mechanics

Compatibility Conditions For Rate Of Deformation 119

For example, for the point (Xifafa) = (1,0,0), arctanX 2/Xi - 0,2w, 4rc, etc. It can be
made a single-valued function by the restriction 0 0£&rctanX2/Xi<0 0+2j[ for any 0 0. For a
simply-connected region as that shown in Fig. 3.7, a Q 0 can be chosen so that such a restriction
makes Eq. (vi) a single-valued continuous displacement for the region. But for the body shown
in Fig. 3.8, the function u\ — arctanAV^i* under the same restriction is discontinuous along
the line 0 = 0 0 in the body (in fact, u\ jumps by the value of IM in crossing the line). Thus,
for this so-called doubly-connected region, there does not exist single-valued continuous u\
corresponding to the given E^, even though the compatibility equations are satisfied.

3.17 Compatibility Conditions For Rate Of Deformation

When any three velocity functions v 1,V2» and v-$ are given, one can always determine the six
rate of deformation components in any region where the partial derivatives dv/dXj exist. On
the other hand, when the six components D,y are arbitrarily prescribed in some region, in
general, there does not exist any velocity field v/, satisfying the six equations

The compatibility conditions for the rate of deformation components are similar to those
of the infinitesimal strain components [Eqs. (3.16.7-12)], i.e.,

etc. It should be emphasized that if one deals directly with differentiate velocity functions
r
s
v/(*i,*2»3»0» (as * often the case in fluid mechanics), the question of compatibility does not
arise.