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Achieving a uniform stress field in a coated non-elliptical inhomogeneity in the presence of a mode III crack. (English) Zbl 1401.30042

Summary: 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.

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30C35 General theory of conformal mappings
45E05 Integral equations with kernels of Cauchy type
65R32 Numerical methods for inverse problems for integral equations
74B05 Classical linear elasticity
74R10 Brittle fracture
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