Legendre Series Solution for Chinese Lantern-Type Stability Loss in a Piezoelectric Cylinder

Öz

In this study, the Chinese lantern-type stability loss of a piezoelectric cylinder under uniformly distributed axial compression is modeled for the axisymmetric case using the exact three-dimensional equations of nonlinear electroelasticity theory. The solution is reduced, through the procedures given in [1], to a series–boundary value problem involving a system of linear partial differential equations. The analytical solution employs Legendre polynomials and window functions to satisfy boundary conditions. The initial imperfection criterion is used to determine the critical parameters [1–4]. Numerical results obtained by MATLAB reveal the effects of problem parameters on the critical force causing stability loss.

Anahtar Kelimeler

• Piezoelectric
• solid cylinder
• stability loss
• Legendre polynomial
• analytical solution

Piezoelektrik Silindirin Çin Feneri Tipi Stabilite Kaybının Legendre Serileri ile Çözümü

Öz

Bu çalışmada, her iki ucundan düzgün yayılı eksenel basınç kuvveti etkisindeki piezoelektrik silindirin Çin feneri tipi stabilite kaybı problemleri lineer olmayan elektroelastisite teorisinin üç boyutlu kesin denklemleri yardımıyla eksenel simetrik durum için modellenmiştir. Bu problemlerin çözümü, detayları [1]’de verilen işlemler yardımıyla, lineer kısmi türevli diferansiyel denklem sistemini içeren seri sınır değer problemlerinin çözümüne indirgenir. Analitik çözüm için Legendre polinomları ile sınır koşullarının sağlatılmasında “pencere fonksiyonlar”ından yararlanılmıştır. Kritik parametrelerin belirlenmesinde, “başlangıç eğinti kriteri” kullanılmıştır [1-4]. MATLAB yardımıyla belirlenen sayısal sonuçlar, problem parametrelerinin stabilite kaybına sebep olan kritik basınç kuvvetine etkilerini göstermektedir. Bu çalışmanın literatürde verilen sayısal sonuçların değerlendirilmesinde önemli bir veri olacağı öngörülmektedir.

Anahtar Kelimeler

• Piezoelektrik
• silindir
• stabilite kaybı
• Legendre polinomu
• analitik çözüm

Kaynakça

1. Akbarov, S.D., Stability Loss and Buckling Delamination: Three-Dimensional Linearized Approach for Elastic and Viscoelastic Composites, Springer, Heidelberg, New York, 2013.
2. Akbarov, S.D., Karakaya, S., 3D analyses of the global stability loss of the circular hollow cylinder made from viscoelastic composite material, European Journal of Mechanics - A/Solids, 33, 48-66, 2012.
3. Yeşil, Ü., Aylıkcı, F., Yahnioğlu, N., Dairesel Piezoelektrik Silindirin Eksenel Basınç Altında Çin Feneri Tipi Stabilite Kaybının Sonlu Elemanlar Yöntemi İle Analizi, Yüksek Ögretim Kurumu Projesi (YTÜ), Proje No: FBA-2023-5998.
4. Aylikci, F., Akbarov, S.D., Yahnioglu, N., Buckling delamination of a PZT/Metal/PZT sandwich rectangular thick plate containing interface inner band cracks. Composite Structures, 202,9-16, 2018.
5. Adegova, L., Bobrysheva, M., Scherbinina, A., Study of stability loss of cylindrical shell made of composite material, Russ Autom Highw Ind J, 18, 342–350, 2021
6. Chanyshev, A.I., Belousova, O.E., Efimenko, L.L., Gutarova, I.V., Frolova, I.V., Torgashova, L.I., The problem of the stability loss of initially anisotropic cylindrical pillar, Interexpo GEO-Sib, 2, 210–218, 2020.
7. Karyakin, M., Obrezkov, L., Stability of a cylinder from murnaghan material under stretching, compression and inflation, Probl Strength Plast, 81, 30–39, 2019.
8. Efimov, A.B., Malyi, V.I., Uteshev, S.A., Loss of stability of a cylindrical shell on longitudinal impact, Naval Surface Weapons Center, Virginia, 1974.

9. Gonik, Y.G., Kibets, A.I., Petrov, M.V., Fyodorova, T.G., Experimentally investigating the elastoplastic deformation and loss of stability of stiffened cylindrical shells with a filling loaded in bending, Probl Strength Plast, 75, 215–220, 20, 2013.
10. Kostka, J., Bocko, J., Frankovský, P., Delyová, I., Kula, T., Varga, P., Stability loss analysis for thinwalled shells with elliptical cross-sectional area, Materials, 14(19), 5636, 2021
11. Cong, F., Zhang, R., Li, W., Jin, Y., Yu, G., Wu, L., Buckling analysis of moderately thick carbon fiber composite cylindrical shells under hydrostatic pressure, Appl Ocean Res, 153, 104272, 2024.
12. Supian, A.B.M., Sapuan, S.M., Zuhri, M.Y.M., Zainudin, E.S., Ya, H.H., Hybrid reinforced thermoset polymer composite in energy absorption tube application: a review, Def Technol, 14(4), 291–305, 2018.
13. Todchuk, V., On the question of the stability of shells, plates and rods, Paper presented at the 4th ISPC scientific trends and trends in the context of globalization, Sweden, August, 2022.
14. Akbarov, S., Kutug, Z., Anwar, M.Y., Non-axisymmetric local stability loss of a hollow cylinder, Synthesis lectures on mechanical engineering. Springer, New York, 2024.
15. Croll, J., Reinterpretation and extensions to the classical theory of axial load buckling of circular cylinders, Thin-Walled Struct, 184, 110379, 2023.
16. Hunt, G.W., Williams, A,K,A,J,, Cowell, R.G., Hidden symmetry concepts in the elastic buckling of axially-loaded cylinders, Int J Solids Struct, 22(12), 1501–1515, 2018.
17. Hoff, N.J., Rehfield, L.W., Buckling of axially compressed circular cylindrical shells at stresses smaller than the classical critical value, ASME J Appl Mech, 32(3), 542–546, 1965.
18. Grabovsky, Y., Harutyunyan, D., Scaling instability in buckling of axially compressed cylindrical shells, J Nonlinear Sci, 26, 83–119, 2016.
19. Smerdov, A., Failure of composite tubes in the Chinese lantern mode under a weight-type load, Mech Compos Mater, 35, 223–226, 1996.
20. Gautschi, G., Piezoelectric Sensorics: Force, Strain, Pressure, Acceleration and Acoustic Emission sensors Materials and Amplifiers. Springer, 2002.
21. Rao, S.S., Sunar, M., Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey, Appl. Mech. Rev., 47, 4, 113–123, 1994.
22. Yang, J.A., Review of a Few Topics in Piezoelectricity, Appl. Mech. Rev., 59, 6, 335–345, 2006.
23. Bisegna, P., Maceri, F., An exact three-dimensional solution for simply supported rectangular piezoelectric plate, ASME J. Appl. Mech., 63, 3, 628–638, 1996.
24. Carrera, E., Boscolo, M., Classical and mixed finite elements for static and dynamic analysis of piezoelectric plates, Int. J. Numer. Methods Eng., 70, 10, 1135–1181, 2007.
25. Guz, A.N., Fundamentals of the Three-Dimensional Theory of Stability of Deformable Bodies, Springer, Berlin, 1999.
26. Kapuria, S., An efficient coupled theory for multilayered beams with embedded piezoelectric sensory and active layers, Int. J. Solids Struct., 38, 50–51, 9179–9199, 2001.
27. Pan, E., Heyliger, P.R., Free vibrations of simply supported and multilayered magneto-electro-elastic plates, J. Sound Vib., 252, 3, 429–442, 2002.
28. Pietrzakowski, M., Piezoelectric control of composite plate vibration: Effect of electric potential distribution, Comput. Struct., 86, 9, 948–954, 2008.
29. Qi, H., Fang, D., Yao, Z., FEM analysis of electro-mechanical coupling effect of piezoelectric materials, Comput. Mater. Sci., 8, 4, 283–290, 1997.
30. Ray, M.C., Bhattacharya, R., Samanta, B., Exact solutions for static analysis of intelligent structures, AIAAJ, 31, 9, 1684–1691, 1993.
31. Sosa, H., Plane problems in piezoelectric media with defects, Int. J. Solids Struct., 28, 491–505, 1991.
32. Tiersten, H.F., Linear Piezoelectric Plate Vibrations, Plenum, New York, 1969.
33. Yesil, U.B., Yahnioglu, N., Free vibration of simply supported piezoelectric plates containing a cylindrical cavity, Arch. Appl. Mech., 92, 2665–2678, 2022.
34. Yesil, U.B., Yazici, G., Free vibration analysis of twin piezoelectric inclusions embedded in elastic medium, Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci., 238, 6, 1954–1967, 2023.
35. Anton, S.R., Sodano, H.A., A Review of power harvesting using piezoelectric materials, Smart Materials and Structures, 16 (3), R1, 2007.
36. Yang, Z., Zhou, S., Zu, J., Inman, D., High-performance piezoelectric energy harvesters and their applications, Joule, 2 (4), 642-697, 2018.
37. Yu, J., Lefebvre, J.E., Guo, Y.Q., Free-ultrasonic waves in multilayered piezoelectric plates: An improvement of the Legendre polynomial approach for multilayered structures with very dissimilar materials. Composites: Part B, 51 260-269, 2013.
38. Jerome, R., Ganesan, N., New generalized plane strain FE formulation for the buckling analysis of piezocomposite beam, Finite Elem. Anal. Des., 46, 10, 896–904, 2010.
39. Shokrgozar, A., Safarpour, H., Habibi, M., Influence of system parameters on buckling and frequency analysis of a spinning cantilever cylindrical 3D shell coupled with piezoelectric actuator, Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci., 234, 2, 512–529, 2019.
40. Yeşil, Ü., Aylıkcı F., Yahnioğlu N., Chinese Lantern-Type Stability Loss of Circular Piezoelectric Cylinder under Unidirectional Pressure, International Applied Mechanics, vol 61, 3, 403-414, 2025.
41. Yeşil, Ü., Aylıkcı F., Yahnioğlu N., Chinese Lantern-Type Stability Loss of a Cylinder Composed of Functionally Graded Materials Under Axial Compression, Journal of Engineering Mathematics, vol 152, 3, 1-20, 2025.
42. Yang, J., An Introduction to the Theory of Piezoelectricity, Springer, 2004.