Achieving a uniform stress field in a coated non-elliptical inhomogeneity in the presence of a mode III crack

  • Xu WangEmail author
  • Liang Chen
  • Peter SchiavoneEmail author


We establish design criteria which guarantee uniformity of stresses inside a coated non-elliptical inhomogeneity influenced by the presence of a finite mode III crack in a matrix subjected to uniform remote anti-plane shear stresses. We employ a particular conformal mapping function containing an unknown real density function which is obtained via the numerical solution of an associated Cauchy singular integral equation with the aid of the Gauss–Chebyshev integration formula. Interestingly, in contrast to the (non-elliptical) shape of the coated inhomogeneity which is influenced solely by the presence of the nearby crack, the resulting internal uniform stress field remains unaffected by the crack.


Coated inhomogeneity Mode III crack Uniform stress field Anti-plane elasticity Conformal mapping Cauchy singular integral equation 

Mathematics Subject Classification

30D10 45E05 65R32 74B05 74R10 



This work is supported by the National Natural Science Foundation of China (Grant No. 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN—2017-03716115112).


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mechanical and Power EngineeringEast China University of Science and TechnologyShanghaiChina
  2. 2.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada

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