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Method of Direct Cutting-Out in the Problems of Elastic Equilibrium of Anisotropic Bodies with Cracks Under Longitudinal Shear

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In the earlier developed method of direct cutting-out, we take into account the anisotropy of material. This method is based on the procedure of modeling of finite or bounded bodies with thin structural defects of any type and boundary conditions on its contour by an infinite space with the same inhomogeneities as in the original problem and additional thin inhomogeneities (cracks or absolutely rigid inclusions), which form the boundary of the investigated body. Thus, loaded cracks are used to model boundary conditions of the first kind, whereas absolutely rigid inclusions embedded in the matrix with certain tension model boundary conditions of the second kind. The developed approach is verified for several problems of longitudinal shear of an anisotropic half space, a layer, and a wedge in the presence of an internal crack under given boundary conditions of the first kind.

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References

  1. K. V. Vasil’ev, Ya. M. Pasternak, and H. T. Sulym, “Antiplane deformation of a square (in plan) body with internal thin inhomogeneity,” Visn. Lviv. Univ., Ser. Mekh.-Mat., Issue 73, 165–176 (2010).

  2. K. V. Vasil’ev and H. T. Sulym, “Application of the method of direct cutting-out to the solution of the problem of longitudinal shear of a wedge with thin heterogeneities of arbitrary orientation,” Mat. Met. Fiz.-Mekh. Polya, 53, No. 3, 117–126 (2010); English translation: J. Math. Sci., 180, No. 2, 122–134 (2012), https://doi.org/10.1007/s10958-011-0634-x.

  3. K. V. Vasil’ev and H. T. Sulym, “Method of direct cutting-out in the problems of piecewise homogeneous bodies with interface cracks under longitudinal shear,” Mat. Met. Fiz.-Mekh. Polya, 59, No. 4, 44–57 (2016); English translation: J. Math. Sci., 238, No. 1, 46–62 (2019), https://doi.org/10.1007/s10958-019-04217-w.

  4. K. Vasil’ev and H. Sulym, “Method of direct cutting-out for modeling of the stress-strain state of isotropic layered media with thin inhomogeneities under antiplane deformation,” Mashynoznavstvo, No. 11-12, 10–17 (2006).

  5. S. G. Lekhnitskii, Theory of Elasticity of Anisotropic Bodies [in Russian], Nauka, Moscow (1977).

    MATH  Google Scholar 

  6. Ia. M. Pasternak, K. V. Vasil’ev, and H. T. Sulym, “Antiplane deformation by concentrated factors of bounded bodies with cracks and rigid inclusions,” Mat. Met. Fiz.-Mekh. Polya, 55, No. 1, 72–83 (2012); English translation: J. Math. Sci., 190, No. 5, 710–724 (2013), https://doi.org/10.1007/s10958-013-1282-0.

  7. H. T. Sulym, Foundations of the Mathematical Theory of Thermoelastic Equilibrium of Deformable Solids with Thin Inclusions [in Ukrainian], Dosl. Vyd. Tsentr NTSh, Lviv (2007).

    Google Scholar 

  8. H. T. Sulym and Ya. M. Pasternak, “Application of the J-integral for the investigation of longitudinal shear of anisotropic bodies with thin-walled strip-like inclusions,” Visn. Donetsk Univ., Ser. A: Pryrod. Nauky, Issue 2, 88–92 (2006).

  9. H. T. Sulym and Ia. M. Pasternak, “Application of the boundary-element method for analysis of the antiplane shear of anisotropic bodies containing thin-walled structures,” Mat. Met. Fiz.-Mekh. Polya, 51, No. 4, 136–144 (2008); English translation: J. Math. Sci., 167, No. 2, 162–172 (2010), https://doi.org/10.1007/s10958-010-9912-2.

  10. H. T. Sulym and S. P. Shevchuk, “Longitudinal shear of layered anisotropic media with band-type inhomogeneities,” Mat. Met. Fiz.-Mekh. Polya, 41, No. 3, 90–97 (1998); English translation: J. Math. Sci., 104, No. 5, 1506–1514 (2001).

  11. S. P. Shevchuk, “Influence of an elastic strip-like inclusion on the deformation of the surface of anisotropic half space in longitudinal shear,” Mat. Met. Fiz.-Mekh. Polya, 49, No. 3, 125–130 (2006).

    MATH  Google Scholar 

  12. V. Bozhydarnyk, Ia. Pasternak, H. Sulym, and N. Oliyarnyk, “BEM approach for the antiplane shear of anisotropic solids containing thin inhomogeneities,” Acta Mech. Automat., 5, No. 4, 11–16 (2011).

  13. R. T. Faal, S. J. Fariborz, and H. R. Daghyani, “Antiplane deformation of orthotropic strips with multiple defects,” J. Mech. Mater. Struct., 1, No. 7, 1097–1114 (2006).

    Article  Google Scholar 

  14. H. G. Georgiadis, “Complex-variable and integral-transform methods for elastodynamic solutions of cracked orthotropic strip,” Eng. Fract. Mech., 24, No. 5, 727–735 (1986).

    Article  Google Scholar 

  15. M. Ghadiri and A. R. Shahani, “Mode III fracture analysis of an anisotropic finite wedge with an interfacial crack,” Math. Mech. Solids, 18, No. 8, 823–836 (2012).

    Article  MathSciNet  Google Scholar 

  16. W. K. Lee and Y. Y. Earmme, “An interfacial edge crack in anisotropic bimaterial under anti-plane singularity,” Int. J. Fract., 104, No. 1, 15–22 (2000).

    Article  Google Scholar 

  17. X.-F. Li and X.-Y. Duan, “An interfacially cracked orthotropic rectangular bimaterial subjected to antiplane shear loading,” Appl. Math. Comput., 174, No. 2, 1060–1079 (2006).

    MathSciNet  MATH  Google Scholar 

  18. X.-F. Li and K. Y. Lee, “Closed-form solution for an orthotropic elastic strip with a crack perpendicular to the edges under arbitrary anti-plane shear,” ZAMM Z. Angew. Math. Mech., 89, No. 5, 370–382 (2009).

    Article  MathSciNet  Google Scholar 

  19. Ia. Pasternak, R. Pasternak, and H. Sulym, “A comprehensive study on Green’s functions and boundary integral equations for 3D anisotropic thermomagnetoelectroelasticity,” Eng. Anal. Bound. Elem., 64, 222–229 (2016).

    Article  MathSciNet  Google Scholar 

  20. Y. Pasternak and H. Sulym, “Boundary-element analysis of anisotropic thermomagnetoelectroelastic solids with 3D shell-like inclusions,” Acta Mech. Automat., 11, No. 4, 308–312 (2017).

    MATH  Google Scholar 

  21. A. R. Shahani, “Mode III stress intensity factors for edge-cracked circular shafts, bonded wedges, bonded half planes and DCB’s,” Int. J. Solids Struct., 40, No. 24, 6567–6576 (2003).

    Article  Google Scholar 

  22. A. R. Shahani, and M. Ghadiri, “Analysis of anisotropic sector with a radial crack under anti-plane shear loading,” Int. J. Solids Struct., 47, No. 7-8, 1030–1039 (2010).

    Article  Google Scholar 

  23. G. Sulym and S. Shevchuk, “Antiplane problem for anisotropic layered media with thin elastic inclusions under concentrated forces and screw dislocations,” J. Theor. Appl. Mech., 37, No. 1, 47–63 (1999).

    MATH  Google Scholar 

  24. H. Sulym, Ia. Pasternak, and R. Pasternak, Boundary-Element Analysis of Multifield Materials: Recent Studies with Application to Crack and Thin Inclusion Problems, Scientific Thesis No. 274, Library of Mechanics, Printing House of the Białystok University of Technology, Bialystok (2015).

  25. X.-Fa. Wu, Y. A. Dzenis, and W.-S. Zou, “Interfacial edge crack between two bonded dissimilar orthotropic strips under antiplane point loading,” ZAMM Z. Angew. Math. Mech., 83, No. 6, 419–422 (2003).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine

    К. V. Vasil’ev & H. Т. Sulym

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  1. К. V. Vasil’ev
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  2. H. Т. Sulym
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Correspondence to К. V. Vasil’ev.

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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 61, No. 3, pp. 89–100, July–September, 2018.

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Vasil’ev, К.V., Sulym, H.Т. Method of Direct Cutting-Out in the Problems of Elastic Equilibrium of Anisotropic Bodies with Cracks Under Longitudinal Shear. J Math Sci 254, 103–116 (2021). https://doi.org/10.1007/s10958-021-05291-9

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