Bending analysis of multiply-connected anisotropic plates with elastic inclusions
Abstract
Relevance. Determining the stress-strain state of thin anisotropic plates with foreign elastic inclusions under transverse bending is an important engineering problem. However, the general case of a plate with multiple, arbitrarily arranged inclusions has lacked an effective numerical or analytical solution due to significant mathematical and computational difficulties.
Objective. The purpose of this work is to develop a new approximate method for determining the stress state of a thin anisotropic plate containing a group of arbitrarily located elliptical or linear elastic inclusions.
Methods. The method is based on the application of S. G. Lekhnitskii's complex potentials. The problem is reduced to determining functions of generalized complex variables for the plate-matrix and the inclusions. These potentials are represented by corresponding Laurent series and Faber polynomials. The generalized least squares method (GLSM) is used to satisfy the contact boundary conditions on the inclusion contours. This reduces the problem to an overdetermined system of linear algebraic equations, which is solved using singular value decomposition (SVD).
Results. The developed method was validated by comparison with the known exact analytical solution for a plate with a single elliptical inclusion, showing perfect agreement. Numerical studies were conducted to analyze the influence of the relative stiffness of the inclusions, the distances between them, and their geometric characteristics on the bending moment values. It was established that the interaction between inclusions is significant and leads to a substantial increase in moments at small distances. Isotropic plates are considered as a special case of anisotropic ones.
Conclusions. It was established for the first time that for linear elastic inclusions, moment singularities, described by moment intensity factors (MIFs), occur only in cases of sufficiently stiff or sufficiently flexible inclusions.
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S.G. Lekhnitskii, S.W. Tsai and T. Cheron, Anisotropic Plates. New York: Gordon and Breach, 1968, 534 p. https://books.google.com.ua/books?id=Ukl9AAAAIAAJ
G.N. Savin, Stress distribution around holes. Washington, D.C.: NASA TT, 1970, 997 p. https://books.google.com.ua/books?id=eC9e0QEACAAJ
C. Hwu and J. Yen Wen, “On the Anisotropic Elastic Inclusions in Plane Elastostatics”. J. Appl Mech, vol. 60 (3), pp. 626–632, 1993. https://doi.org/10.1115/1.2900850
C. Hwu, Anisotropic Elastic Plates. New York: Springer, 2010, 673 p. https://doi.org/10.1007/978-1-4419-5915-7
J. Lee, “Elastic analysis of unbounded solids using volume integral equation method”. J Mech Sci Technol, vol. 22, pp. 450–459, 2008. https://doi.org/10.1007/s12206-007-1215-2
M.C. Hsieh and C. Hwu, “Anisotropic elastic plates with holes / cracks / inclusions subjected to out-of-plane bending moments”. Int. J. Solids and Struct, vol. 39, no. 19, pp. 4905–4925, 2002. https://doi.org/10.1016/S0020-7683(02)00335-9
О.V. Maksymovych, Т.Y. Solyar and Y. Kempa, “Investigation of Bending of Anisotropic Plates with Inclusions with the Help of Singular Integral Equations”. J Math Sci, vol. 254, pp. 129–141, 2021. https://doi.org/10.1007/s10958-021-05293-7
Z. Drmač and K. Veselić, “New fast and accurate Jacobi SVD algorithm. I”. SIAM J. Matrix Anal. Appl, vol. 29, no. 4, pp. 1322–1342, 2008. https://doi.org/10.1137/050639193
G.C. Sih, P.C. Paris and G.R. Irwin, “On cracks in rectilinearly anisotropic bodies”. Int J Fract, vol. 1, pp. 189–203, 1965. https://doi.org/10.1007/BF00186854
S.G. Lekhnitskii, S.W. Tsai and T. Cheron, Anisotropic Plates. New York: Gordon and Breach, 1968, 534 p. https://books.google.com.ua/books?id=Ukl9AAAAIAAJ
G.N. Savin, Stress distribution around holes. Washington, D.C.: NASA TT, 1970, 997 p. https://books.google.com.ua/books?id=eC9e0QEACAAJ
C. Hwu and J. Yen Wen, “On the Anisotropic Elastic Inclusions in Plane Elastostatics”. J. Appl Mech, vol. 60 (3), pp. 626–632, 1993. https://doi.org/10.1115/1.2900850
C. Hwu, Anisotropic Elastic Plates. New York: Springer, 2010, 673 p. https://doi.org/10.1007/978-1-4419-5915-7
J. Lee, “Elastic analysis of unbounded solids using volume integral equation method”. J Mech Sci Technol, vol. 22, pp. 450–459, 2008. https://doi.org/10.1007/s12206-007-1215-2
M.C. Hsieh and C. Hwu, “Anisotropic elastic plates with holes / cracks / inclusions subjected to out-of-plane bending moments”. Int. J. Solids and Struct, vol. 39, no. 19, pp. 4905–4925, 2002. https://doi.org/10.1016/S0020-7683(02)00335-9
О.V. Maksymovych, Т.Y. Solyar and Y. Kempa, “Investigation of Bending of Anisotropic Plates with Inclusions with the Help of Singular Integral Equations”. J Math Sci, vol. 254, pp. 129–141, 2021. https://doi.org/10.1007/s10958-021-05293-7
Z. Drmač and K. Veselić, “New fast and accurate Jacobi SVD algorithm. I”. SIAM J. Matrix Anal. Appl, vol. 29, no. 4, pp. 1322–1342, 2008. https://doi.org/10.1137/050639193
G.C. Sih, P.C. Paris and G.R. Irwin, “On cracks in rectilinearly anisotropic bodies”. Int J Fract, vol. 1, pp. 189–203, 1965. https://doi.org/10.1007/BF00186854
