**5. Discussion**

Usually, convenient simplifications are adopted when we develop mathematical descriptions of a physical process, such as the elastic displacements due to boundary forces and internal discontinuities such as circular cylindrical perforations. Examples are the description of two extreme types of boundary conditions—plane stress (thinplate approach) and plane strain (thick-plate approach).

In spite of these model simplifications, the results of the computations based on the thin- and thick-plate approaches are routinely used in many practical applications. However, the disparities between the model results and actual displacements in the natural prototype can rarely be evaluated in detail, but some theoretical extrapolations are still possible (see ref. [10] for some experimental methods).

The present study documents a careful evaluation of the delta's arising in resulting stress concentrations due to a prototype that would behave like a thick medium (plane strain boundary conditions) but is treated with a plane stress boundary condition solution (thin-plate medium). This treatment is basically due to applying Kirsch [1] solutions to quantify the stress concentration on the rim of the hole in a thick plate (where, *σzz* is locally zero likewise everywhere in the thin-plate case). Such loose application of boundary constraints routinely occurs in wellbore stability problems, as detailed in this paper.

This paper specifically discussed the relevance of our findings for borehole stability studies. Traditionally, wellbore stability computations are based on analytical solutions [1] for stresses near a hole in an elastic plate. The Kirsch [1] solution is valid for a plane stress boundary condition; this is a so-called thin-plate solution *<sup>a</sup> <sup>h</sup>* ! <sup>∞</sup> , for which everywhere *σzz* ¼ 0. The opposite end of the spectrum is a plane strain boundary condition (*i:e:*, *<sup>ε</sup>zz* <sup>¼</sup> 0) or a thick plate *<sup>a</sup> <sup>h</sup>* ! <sup>0</sup> approach, where *<sup>σ</sup>zz* is locally zero but in places may be either larger or smaller than zero, as has been quantified in Appendix C of our study. In our present study, we quantified, *σzz*, spatially everywhere normal to both a thick plate and thin plate with thickness 2*h*, perforated by either single or multiple hole(s) of typical hole radius *a* (see Appendix C). Solutions were also given for cases with internal pressure loading.

According to the new results presented in our study, after evaluating the stress concentrations and transverse *σzz* for the perfect plane strain case *<sup>a</sup> <sup>h</sup>* ! <sup>0</sup> not only at the rim of the holes but everywhere in a finite domain around the hole(s), we can confirm that even for the most extreme case of plane strain (as opposed to plane stress, *<sup>a</sup> <sup>h</sup>* ! ∞) where *σ<sup>z</sup>* vanishes in all locations, the difference between the respective solutions remains minimal.

## **6. Conclusion**

Our study articulates that, in fact, any real elastic medium with a finite thickness, for cases involving circular cylindrical holes (single or multiple), will behave in a way intermediate between the plane stress and plane strain end members. We have resorted to [14] scalar *<sup>a</sup> <sup>h</sup>* as a very useful metric to estimate where the real prototype with finite thickness occurs between the end-member solutions. For *<sup>a</sup> <sup>h</sup>* ≪ 1 and *<sup>a</sup> <sup>h</sup>* ! 0, we have tiny holes in a very thick plate. Many prototypes of stress concentrations near boreholes in the geological subsurface will be adequately described by the plane strain boundary condition. For *<sup>a</sup> <sup>h</sup>* ≫ 1 and *<sup>a</sup> <sup>h</sup>* ! ∞, we have large holes in a very thin plate, for which

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*

prototypes exist in riveted wing panels for airplanes. Of course, there exists an unlimited range of prototypes that fall somewhere in between the extremes, and *<sup>a</sup> <sup>h</sup>* provides a metric to estimate how far the solution remains separated from the two end members.

Several specific cases have been analyzed in our study and we have quantified the delta of the displacements, strain, and stresses, as well as the Poisson's ratio for the two end-members (see Appendices to this study). Based on these specific cases, the following conclusions can be drawn (with emphasis on the delta's arising when applying either plane stress or plane strain approximate solutions):

