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
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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).
Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three-phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330 (1979)CrossRefGoogle Scholar
Wang, X., Chen, L., Schiavone, P.: Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a mode III crack. Proc. Roy. Soc. London A (2018). https://doi.org/10.1098/rspa.2018.0304
Wang, X., Gao, X.L.: On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite. Z. Angew. Math. Phys. 62, 1101–1116 (2011)MathSciNetCrossRefGoogle Scholar
Xiao, Z.M., Chen, B.J.: A screw dislocation interacting with a coated fiber. Mech. Mater. 32, 485–494 (2000)CrossRefGoogle Scholar