The structure of Airy's stress function in multiply connected regions Page: 3 of 36
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A special decomposition of the Airy function is therefore given;
for simplicity of presentation, the following two different cases will
(a) systems subjected to a plane dislocation and therefore to a
state of stress with zero forces on the boundary
(b) systems subjected to an external stress of a general type, but
in the absence of dislocation (one-valued displacements)
It is evident that from the two compositions corresponding to cases
(a) and (b) the one is immediately obtained that corresponds to the
general case of a plane system in which the state of deformation is the
resultant of those deformations due to a general external stress and a
For a plane dislocation, only the components of the displacement
parallel to the plane of the system are considered herein. Such a dis-
placement results when, after the cuts that decrease the order of con-
nectivity of the system are made, one of the faces of each single cut
is given a displacement relative to the other faces. The displacement
is composed of a translation parallel to the plane of the system and a
rotation about an axis perpendicular to this plane; a certain quantity
of material is then added or subtracted, as the case may be2.
Treatment of the problems of plane dislocations with the aid of the
Airy function seems natural, because the determination of the states of
strain and of stress of a system subjected only to dislocation leads
essentially to the problem of the integration of the equations of elas-
tic equilibrium with assigned forces at the boundary.
In the two decompositions presented, some of the singular parts
figure directly as known elements in that their coefficients may be
expressed in terms of assigned elements characteristic of the state of
stress and strain to which the system is subjected. That is, among the
coefficients of the singular terms, the characteristics of the dis-
location figure in case (a); whereas in case (b), the components of the
resultant forces and the resultant moments of the external stresses
acting on the boundaries of the individual holes are involved.
2Results on the plane dislocations of a circular ring will be found
in references 1 and 2.
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Grioli, Giusippe. The structure of Airy's stress function in multiply connected regions, report, July 1, 1951; (digital.library.unt.edu/ark:/67531/metadc63012/m1/3/: accessed October 17, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.