Solutions of periodic and doubly periodic bending problems of a thin piezo plate with holes or cracks

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Аннотация

The solutions of periodic and doubly periodic problems of bending of a piezo plate with elliptical holes or cracks are given with an analysis of the results of numerical studies. In this case, complex potentials of the theory of bending of thin electro-magneto-elastic plates are used, holomorphic functions outside the holes are represented by Laurent series in negative powers of variables from the corresponding conformal mappings and, based on the periodicity or doubly periodicity of the electro-magneto-elastic state of the plate, the coefficients of the series from all the holes are expressed through the coefficients of the series from one, the so-called main hole. The determination of the last coefficients is carried out from the boundary conditions on the contour of the main hole using the generalized least squares method. The results of numerical studies for a plate with circular holes or cracks with full or partial consideration of piezo properties, without taking them into account, are described. The patterns of influence on the values of bending moments and their concentration of the geometric characteristics of the discussed plates and the physico-mechanical properties of their materials are established.

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Авторлар туралы

S. Kaloerov

Donetsk State University

Хат алмасуға жауапты Автор.
Email: kaloerov@mail.ru
Ресей, Donetsk, DPR

A. Seroshtanov

Donetsk State University

Email: aleks.serosht@gmail.com
Ресей, Donetsk, DPR

Әдебиет тізімі

  1. Cady W.G. Piezoelectricity: An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals. New York: McGraw-Hill Book Kompany, 1946. 806 p.
  2. Berlincourt D., Curran D.R., Jaffe H. Piezoelectric and Piezomagnetic Materials and Their Function in Transducers. Ed. by W.P. Mason. New York: Academic Press, Physical Acoustics, 1964. P. 169–270.
  3. Bichurin M.I., Petrov V.M., Filippov D.A., et al., Magnetoelectric Composites. Akad. Estestv., Moskow, 2006; Jenny Stanford Publishing, 2019.
  4. Pyatakov A.P., Magnetoelectric Materials and Their Application in Practice // Bul. Ros. Magnit. Obshchestva. 2006. Vol. 5. № 2. P. 1–3.
  5. Nan C.-W., Bichurin M.I., Dong S., Viehland D., Srinivasan G. Multiferroic magnetoelectric composites: Historical perspective, status, and future directions // J. Appl. Phys. 2008. V. 103. № 3. P. 031101. https://doi.org/10.1063/1.2836410
  6. Tian R., Liu J., Liu X. Magnetoelectric properties of piezoelectric-piezomagnetic composites with elliptical nanofibers // Acta Mechanica Solida Sinica. 2020. V. 33. P. 368–380. https://doi.org/10.1007/s10338-019-00129-z
  7. Srinivas S., Jiang Y.L. The effective magnetoelectric coefficients of polycrystalline multiferroic composites // Acta Mater. 2005. V. 53. P. 4135–4142. https://doi.org/10.1016/j.actamat.2005.05.014
  8. Bochkarev S.A., Lekomtsev S.V. Hydroelastic stability of coaxial cylindrical shells made of piezoelectric material // PNRPU Mechanics Bulletin. 2019. № 2. P. 35–48. https://doi.org/10.15593/perm.mech/2019.2.04
  9. Eringen A.C., Maugin G.A. Electrodynamics of Continua I. New York: Springer, 1990. 436 p. https://doi.org/10.1007/978-1-4612-3226-1
  10. Librescu L., Hasanyan D., Ambur DR Electromagnetically conducting elastic plates in a magnetic field: modeling and dynamic implications // International journal of non-linear mechanics. 2004. V. 39. № 5. P. 723–739. https://doi.org/10.1007/s00707-008-0025-7
  11. Shen W., Zhang G., Gu S., Cong Y. A transversely isotropic magneto-electro-elastic circular Kirchhoff plate model incorporating microstructure effect // Acta Mechanica Solida Sinica. 2022. V. 35. № 2. P. 185–197. https://doi.org/10.1007/s10338-021-00271-7
  12. Ieşan D. On the bending of piezoelectric plates with microstructure // Acta Mech. 2008. V. 198. № 3. P. 191–208. https://doi.org/10.1007/s00707-007-0527-8
  13. Xu S.-P., Wang W. Bending of piezoelectric plates with a circular hole // Acta Mech. 2009. V. 203. P. 127–135. https://doi.org/10.1007/s00707-008-0025-7
  14. Gales C., Baroiu N. On the bending of plates in the electromagnetic theory of microstretch elastity // ZAMM – Journal of Applied Mathematics and Mechanics. 2014. V. 94. № 1–2. P. 55–71. https://doi.org/10.1002/zamm.201200219
  15. Kaloerov S.A. Osnovnye sootnosheniia prikladnoi teorii izgiba tonkikh elektromagnitouprugikh plit [The main relations of the applied theory of bending of thin electro-magneto-elastic plates] // Bulletin of Donetsk National University. Series A: Natural Sciences. 2022. № 1. P. 20–38.
  16. Kaloerov S.A., Parshikova O.A. Thermoviscoelastic state of multiply connected anisotropic plates // International Applied Mechanics. 2012. V. 48. № 3. P. 319–331. https://doi.org/10.1007/s10778-012-0523-0
  17. Kaloerov S.A., Seroshtanov A. V. Solving the problem of electro-magneto-elastic bending of a multiply connected plate // J. Appl. Mech. Tech. Phys. 2022. V. 63. № 4. P. 676–687. https://doi.org/10.1134/S0021894422040150
  18. Kaloerov S. A., Goryanskaya E. S. The two-dimensional stressed state of a multiconnected anisotropic body with cavities and cracks // Journal of Mathematical Sciences. 1997. V. 84. № 6. P. 1497–1504. https://doi.org/10.1007/BF02398809
  19. Voevodin V.V. Vychislitel’nye osnovy lineinoi algebry [Computational Basis of Linear Algebra]. Moskov, Nauka, 1977. 304 p.
  20. Forsythe G.E., Malcolm M.A., Moler C.B. Computer methods for mathematical computations. Englewood Cliffs: Prentice-Hall, 1977. XII, 259 p.
  21. Drmač Z., Veselič K. New fast and accurate Jacobi SVD algorithm. I // SIAM J. Matrix Anal. Appl. 2008. V. 29. № 4. P. 1322–1342. https://doi.org/10.1137/050639193.
  22. Drmač Z., Veselič K. New fast and accurate Jacobi SVD algorithm. II // SIAM J. Matrix Anal. Appl. 2008. V. 29. № 4. P. 1343–1362. https://doi.org/10.1137/05063920X.
  23. Tian W.-Y., Gabbert U. Multiple crack interaction problem in magnetoelectroelastic solids // Europ. J. Mech. Part A. 2004. V. 23. P. 599–614. https://doi.org/10.1016/j.euromechsol.2004.02.002
  24. Yamamoto Y., Miya K. Electromagnetomechanical Interactions in Deformable Solids and Structures. Amsterdam: Elsevier Sci. North Holland, 1987. 450 p.
  25. Hou P.F., Teng G.-H., Chen H.-R. Three-dimensional Greens function for a point heat source in two-phase transversely isotropic magneto-electro-thermo-elastic material // Mech. Materials. 2009. V. 41. P. 329–338. https://doi.org/10.1016/j.mechmat.2008.12.001

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Әрекет
1. JATS XML
Жүктеу
2. Fig. 1. Diagram of a plate with a periodic row of elliptical holes.

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3. Fig. 2. Diagram of a plate with a doubly periodic system of elliptical holes.

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4. Fig. 3. Diagram of a plate with a periodic row of circular holes.

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5. Fig. 4. Graphs of the distribution of moments Ms /my depending on q [rad] near the contour of a circular hole L0 under the action of moments My∞ = my at infinity. Solid, dashed and dotted lines refer to a plate made of materials M1, M2 and M3, respectively. 1 – c/a = 0.5, 1 – c/a = ∞.

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6. Fig. 5. Graphs of the distribution of moments Ms /my depending on q [rad] near the contour of the hole L0 in a plate made of material M3 with circular holes for the problems of EMU (solid line), MU (dashed line), EU (dash-dotted line) and TU (dotted line).

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7. Fig. 6. Diagram of a slab with a periodic row of cracks depending on: (a) – angle j of crack inclination, (b) – distance between cracks at j = 0°, (c) – distance between cracks at j = 90°.

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