By the method of successive approximations (Schwartz algorithm) and the Fourier integral transform, we construct a Green-type solution for an anisotropic strip with unloaded boundary. This solution is expressed in terms of the Lekhnitskii potentials with isolated poles. On the basis of the constructed solutions, we deduce singular integral equations for anisotropic plates with holes such that the boundary conditions imposed on the sides of the strip are identically satisfied. These equations are numerically solved by the method of mechanical quadratures. We also analyze the stress concentration in the vicinity of holes of different shapes contained in composite plates.
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References
S. G. Lekhnitskii, Anisotropic Plates, Gordon & Breach, New York, etc. (1968).
A. I. Lur’e, Three-Dimensional Problems of the Theory of Elasticity, Interscience, New York (1964).
G. N. Savin, Distribution of Stresses Near Holes [in Russian], Naukova Dumka, Kiev (1968).
H. Sulym and S. Shevchuk, “Plane problem for an anisotropic strip with thin elastic anisotropic inclusion,” Mashynoznavstvo, No. 3 (129), 3–8 (2008).
S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York (1951).
A. M. Baghestani, A. R. Fotuhi, and S. J. Fariborz, “Multiple interacting cracks in an orthotropic layer,” Arch. Appl. Mech., 83, No. 11, 1549–1567 (2013); https://doi.org/10.1007/s00419-013-0761-6.
C. A. Brebbia and J. Dominguez, Boundary Elements: An Introductory Course, WIT Press, Boston (1998).
H. Huang and G. A. Kardomateas, “Stress intensity factors for a mixed mode center crack in an anisotropic strip,” Int. J. Fracture, 108, No. 4, 367–381 (2001); https://doi.org/10.1023/A:1011006225367.
C. Hwu, Anisotropic Elastic Plates, Springer, New York (2010).
R. M. Jones, Mechanics of Composite Materials, Taylor & Francis, Philadelphia (1999).
A. C. Kaya and F. Erdogan, “Stress intensity factors and COD in an orthotropic strip,” Int. J. Fracture, 16, No. 2, 171–190 (1980); https://doi.org/10.1007/BF00012620.
O. V. Maksymovych, A. R. Dzyubyk, Kh. A. Barvinska, and L. V. Dzyubyk, “Determination of stress in composite plates with cracks on the basis of the method of integral equations and solutions by Green,” Nauk. Visn. Nats. Hirnych. Univ., No. 5, 65–73 (2019); https://doi.org/10.29202/nvngu/2019-5/9.
O. Maksymovych and O. Illiushyn, “Stress calculation and optimization in composite plates with holes based on the modified integral equation method,” Eng. Anal. Bound. Elem., 83, 180–187 (2017); https://doi.org/10.1016/j.enganabound.2017.06.009.
M. S. Matbuly and M. Nassar, “Elastostatic analysis of edge cracked orthotropic strips,” Acta Mech., 165, No. 1-2, 17–25 (2003); https://doi.org/10.1007/s00707-003-0031-8.
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 3, pp. 69–77, July–September, 2020.
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Maksymovych, M.O., Kharchenko, Y.V. Determination of Stresses in an Anisotropic Strip with Holes by Using Singular Integral Equations and Green’s Solution. J Math Sci 273, 79–91 (2023). https://doi.org/10.1007/s10958-023-06485-z
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DOI: https://doi.org/10.1007/s10958-023-06485-z