Analytical solution with validity analysis for an elliptical void and a rigid inclusion under uniform or nonuniform anti-plane loading

S. Shahzad*, J. Niiranen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

11 Citations (Scopus)
105 Downloads (Pure)


An elliptical inclusion (covering both void and rigid inclusions) embedded in an infinite and finite elastic plane subject to uniform or nonuniform (m-th order polynomial) anti-plane loading conditions is analyzed. An analytical solution in terms of the stress field for an infinite plane is developed through the method of analytic function and conformal mapping. Closed-form complex potentials and analytical expressions for Stress Concentration Factors (SCFs) are obtained. The results show that (i.) the SCF value decreases with an increasing loading order, so that the influence of the non-uniformity of the anti-plane loads on the SCF is revealed to be beneficial from the failure point of view; (ii.) decrease in the SCF value for an infinite plane is monotonic, which does not hold true for a finite plane. The results for an infinite plane are confirmed and extended for finite planes by exploiting the well-known heat–stress analogy and the finite element method. It is worth mentioning that the comparison between the analytical solution for an infinite plane and the numerical solution for finite plane is provided, showing that the analytical solution of an infinite plane can be used as an accurate approximation to the case of a finite plane. Moreover, the proposed heat–stress analogy can be exploited to study the crack–inclusion interaction or multiply connected bodies. The computational efficiency of the proposed methodology makes it an attractive analysis tool for anti-plane problems with respect to the full scale three-dimensional analysis.

Original languageEnglish
Pages (from-to)62-72
Number of pages11
JournalTheoretical and Applied Fracture Mechanics
Publication statusPublished - 1 Oct 2018
MoE publication typeA1 Journal article-refereed


  • Anti-plane elasticity
  • Composites
  • Crack
  • Heat–stress analogy
  • Laplace equation
  • SCF
  • SIF


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