Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Fluid bounding effect on FG cylindrical shell using Hankel's functions of second kind

  • Khaled Mohamed Khedher (Department of Civil Engineering, College of Engineering, King Khalid University) ;
  • Shahzad Ali Chattah (Department of Chemistry, Government College University Faisalabad) ;
  • Mohammad Amien Khadimallah (Department of Civil Engineering, College of Engineering in Al-Kharj, Prince Sattam Bin Abdulaziz University) ;
  • Ikram Ahmad (Department of Chemistry, University of Sahiwal) ;
  • Muzamal Hussain (Department of Mathematics, University of Sahiwal) ;
  • Rana Muhammad Akram Muntazir (Department of Mathematics, Lahore leads University) ;
  • Mohamed Abdelaziz Salem (Department of Mechanical Engineering, College of Engineering, King Khalid University) ;
  • Ghulam Murtaza (Department of Mathematics, University of Sahiwal) ;
  • Faisal Al-Thobiani (Marine Engineering Department, Faculty of Maritime Studies, King Abdulaziz University) ;
  • Muhammad Naeem Mohsin (Institute for Islamic Theological Studies, University of Vienna) ;
  • Abeera Talib (Department of Mathematics, Lahore leads University) ;
  • Abdelouahed Tounsi (Faculty of Technology Civil Engineering Department, Materials and Hydrology Laboratory University of Sidi Bel Abbes)
  • Received : 2021.07.22
  • Accepted : 2024.05.05
  • Published : 2024.06.25
⟨ Previous Next ⟩

Abstract

Vibration investigation of fluid-filled functionally graded cylindrical shells with ring supports is studied here. Shell motion equations are framed first order shell theory due to Sander. These equations are partial differential equations which are usually solved by approximate technique. Robust and efficient techniques are favored to get precise results. Employment of the Rayleigh-Ritz procedure gives birth to the shell frequency equation. Use of acoustic wave equation is done to incorporate the sound pressure produced in a fluid. Hankel's functions of second kind designate the fluid influence. Mathematically the integral form of the Langrange energy functional is converted into a set of three partial differential equations. A cylindrical shell is immersed in a fluid which is a non-viscous one. These shells are stiffened by rings in the tangential direction. For isotropic materials, the physical properties are same everywhere where the laminated and functionally graded materials, they vary from point to point. Here the shell material has been taken as functionally graded material. After these, ring supports are located at various positions along the axial direction round the shell circumferential direction. The influence of the ring supports is investigated at various positions. Effect of ring supports with empty and fluid-filled shell is presented using the Rayleigh - Ritz method with simply supported condition. The frequency behavior is investigated with empty and fluid-filled cylindrical shell with ring supports versus circumferential wave number and axial wave number. Also the variations have been plotted against the locations of ring supports for length-to-radius and height-to-radius ratio. Moreover, frequency pattern is found for the various position of ring supports for empty and fluid-filled cylindrical shell. The frequency first increases and gain maximum value in the midway of the shell length and then lowers down. It is found that due to inducting the fluid term frequency result down than that of empty cylinder. It is also exhibited that the effect of frequencies is investigated by varying the surfaces with stainless steel and nickel as a constituent material. To generate the fundamental natural frequencies and for better accuracy and effectiveness, the computer software MATLAB is used.

Keywords

Acknowledgement

The Authors extend their appreciation to the Deanship Scientific Research at King Khalid University for funding this work through large group Research Project under grant number: RGP2/388/45.

References

  1. Akbas S.D. (2017a), "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", Int. J. Struct. Stabil. Dyn., 17(3), 1750033. https://doi.org/10.1142/S021945541750033X 
  2. Akbas, S.D. (2016a), "Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium", Smart Struct. Syst., 18(6), 1125-1143. https://doi.org/10.12989/sss.2016.18.6.1125 
  3. Akbas, S.D. (2016b), "Analytical solutions for static bending of edge cracked micro beams", Struct. Eng. Mech., 59(3), 579-599. https://doi.org/10.12989/sem.2016.59.3.579 
  4. Akbas, S.D. (2017b), "Forced vibration analysis of functionally graded nanobeams", Int. J. Appl. Mech., 9(7), 1750100. https://doi.org/10.1142/S1758825117501009 
  5. Alwabli, A.S., Kaci, A., Bellifa, H., Bousahla, A.A., Tounsi, A., Alzahrani, D.A., Abulfaraj, A.A., Bourada, F., Benrahou, K.H., Tounsi, A., Mahmoud, S.R. and Hussain, M. (2021), "The nano scale buckling properties of isolated protein microtubules based on modified strain gradient theory and a new single variable trigonometric beam theory", Adv. Nano Res., 10(1), 15-24. https://doi.org/10.12989/anr.2021.10.1.015 
  6. Amabili, M., Pellicano, F. and Paidoussis M.P. (1998), "Nonlinear vibrations of simply Love, A.E.H. (1888), "On the small free vibrations and deformation of thin elastic shell", Phil. Trans. R. Soc. London, A179, 491-549. https://doi.org/10.1098/rsta.1888.0016. 
  7. Ansari, R. and Rouhi, H. (2015), "Nonlocal Flugge shell model for the axial buckling of single-walled Carbon nanotubes: An analytical approach", Int. J. Nano Dimension, 6(5), 453-462. https://doi.org/10.7508/IJND.2015.05.002 
  8. Asghar, S. Hussain M, and Naeem M. (2019), "Non-local effect on the vibration analysis of double walled carbon nanotubes based on Donnell shell theory", Physica E, 116, 113726. https://doi.org/10.1016/j.physe.2019.113726 
  9. Asghar, S., Khadimallah, M.A., Naeem, M.N., Ghamkhar, M., Khedher, K.M., Hussain, M., Bouzgarrou, S.M., Ali, Z., Mahmoud, S.R., Taj, M. and Tounsi, A. (2020b), "Small scale computational vibration of double-walled CNTs: Estimation of nonlocal shell model", Adv. Concr. Constr., 10(4), 345-355. https://doi.org/10.12989/acc.2020.10.4.34 
  10. Asghar, S., Naeem, M. N., Khadimallah, M. A., Hussain, M., Iqbal, Z. and Tounsi, A. (2020a), "Effect of chiral structure for free vibration of DWCNTs: Modal analysis", Adv. Concr. Constr., 9(6), 577-588. https://doi.org/10.12989/acc.2020.9.6.577 
  11. Avcar M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603 
  12. Benmansour, D.L., Kaci, A., Bousahla, A.A., Heireche, H., Tounsi, A., Alwabli, A.S., Alhebshi, A.M., Al-ghmady, K. and Mahmoud, S.R. (2019), "The nano scale bending and dynamic properties of isolated protein microtubules based on modified strain gradient theory", Adv. Nano Res., 7(6), 443. https://doi.org/10.12989/anr.2019.7.6.443 
  13. Cao, Q., Wang, R., Zhang, T., Wang, Y. and Wang, S. (2022), "Hydrodynamic modeling and parameter identification of a bionic underwater vehicle: RobDact", Cyborg Bionic Syst., 2022. https://doi.org/10.34133/2022/9806328 
  14. Chi, S.H. and Chung, Y.L. (2006), "Mechanical behavior of functionally graded material plates under transverse load-part II: numerical results", Int. J. Solids Struct., 43, 3657-3691. https://doi.org/10.1016/j.ijsolstr.2005.04.010 
  15. Chung, H., Turula, P. Mulcahy, T.M. and Jendrzejczyk, J.A. (1981), "Analysis of cylindrical shell vibrating in a cylindrical fluid region", Nuclear Eng. Des., 63(1), 109-1012. https://doi.org/10.1016/0029-5493(81)90020-0. 
  16. Dang, V. H., Sedighi, H. M., Civalek, O . and Abouelregal, A. E. (2021), "Nonlinear vibration and stability of FG nanotubes conveying fluid via nonlocal strain gradient theory", Struct. Eng. Mech., 78(1), 103-116. https://doi.org/10.12989/sem.2021.78.1.103 
  17. Dong S.B. (1977), "A block-stodola eigen solution technique for large algebraic systems with non-symmetrical matrices", Int. J. Numer. Meth. Eng., 11, 247. https://doi.org/10.1002/nme.1620110204 
  18. Dong, Z., Li, X., Yamaguchi, H. and Yu, P. (2024), "Magnetic field effect on the sedimentation process of two non-magnetic particles inside a ferrofluid", J. Magn. Magn. Mater., 589, 171501. https://doi.org/10.1016/j.jmmm.2023.171501 
  19. Ebrahimi, F., Dabbagh, A., Rabczuk, T. and Tornabene, F. (2019), "Analysis of propagation characteristics of elastic waves in heterogeneous nanobeams employing a new two-step porosity-dependent homogenization scheme", Adv. Nano Res., 7(2), 135. https://doi.org/10.12989/anr.2019.7.2.135 
  20. Eltaher, M.A., Almalki, T.A., Ahmed, K.I. and Almitani, K.H. (2019), "Characterization and behaviors of single walled carbon nanotube by equivalent-continuum mechanics approach", Adv. Nano Res., 7(1), 39. https://doi.org/10.12989/anr.2019.7.1.039 
  21. Ergin, A. and Temarel, P. (2002), "Free vibration of a partially liquid-filled and submerged, horizontal cylindrical shell", J. Sound Vib., 254(5), 951-965. https://doi.org/10.1006/jsvi.2001.4139 
  22. Feng, J., Wang, W. and Zeng, H. (2024), "Integral sliding mode control for a class of nonlinear multi-agent systems with multiple time-varying delays", IEEE Access, 12, 10512-10520. https://doi.org/ 10.1109/ACCESS.2024.3354030 
  23. Fu, Y., Liu, Y., Wang, J., Wang, Y., Xu, G. and Wen, J. (2024), "Local resistance characteristics of elbows for supercritical pressure RP-3 flowing in serpentine micro-tubes", Propuls. Power Res., In Press. https://doi.org/10.1016/j.jppr.2023.02.009 
  24. Fu, Z.H., Yang, B.J., Shan, M.L., Li, T., Zhu, Z.Y., Ma, C.P., Zhang, X., Gou, G.Q., Wang, Z.R. and Gao, W. (2020), "Hydrogen embrittlement behavior of SUS301L-MT stainless steel laser-arc hybrid welded joint localized zones", Corros. Sci., 164, 108337. https://doi.org/10.1016/j.corsci.2019.108337 
  25. Gasser LFF. (1987), "Free vibrations on thin cylindrical shells containing liquid", M.S. Thesis, Federal University of Rio de Janerio, Rio de Janerio:
  26. Goncalves, P.B. and Batista (1988), "Non-linear vibration analysis of fluid-filled cylindrical shell", J. Sound Vib., 127(1), 133-143. https://doi.org/10.1006/jsvi.2001.4139 
  27. Gong, Q., Cai, M., Gong, Y., Chen, M., Zhu, T. and Liu, Q. (2024), "Grinding surface and subsurface stress load of nickel-based single crystal superalloy DD5", Precis. Eng., 88, 354-366. https://doi.org/10.1016/j.precisioneng.2024.02.017 
  28. Guo, J., Ding, B., Wang, Y. and Han, Y. (2023), "Co-optimization for hydrodynamic lubrication and leakage of V-shape textured bearings via linear weighting summation", Physica Scripta, 98(12), 125218. https://doi.org/10.1088/1402-4896/ad07be 
  29. Han, Q., Li, X. and Chu, F. (2018), "Skidding behavior of cylindrical roller bearings under time-variable load conditions", Int. J. Mech. Sci., 135, 203-214. https://doi.org/10.1016/j.ijmecsci.2017.11.013 
  30. Iqbal, W., Jalil, M., Khadimallah, M.A., Ayed, H., Naeem, M.N., Hussain, M., Bouzgarrou, S.M., Mahmoud, S.R., Ghandourah, E., Taj, M. and Tounsi, A. (2020), "Runge-Kutta method for flow of dusty fluid along exponentially stretching cylinder", Steel Compos. Struct., 36(5), 603-615. https://doi.org/10.12989/scs.2020.36.5.603 
  31. Iqbal, W., Jalil, M., Khadimallah, M.A., Hussain, M., Naeem, M. N., Al Naim, A.F. and Tounsi, A. (2021), "Interaction of casson nanofluid with Brownian motion: Temperature profile with shooting method", Adv. Nano Res., 10(4), 349-357. https://doi.org/10.12989/anr.2021.10.4.349 
  32. Jiang, J. and Olson, M.D. (1994), "Vibrational analysis of orthogonally stiffened cylindrical shells using super elements", J. Sound Vib., 173, 73-83. https://doi.org/10.1006/jsvi.1994.1218 
  33. Khadimallah, M.A., Hussain, M., Khedher, K.M., Naeem, M.N. and Tounsi, A. (2020b), "Backward and forward rotating of FG ring support cylindrical shell", Steel Compos. Struct., 37(2), 137-150. https://doi.org/10.12989/scs.2020.37.2.137 
  34. Khadimallah, M.A., Safeer, M., Taj, M., Ayed, H., Hussain, M., Bouzgarrou, S.M., Mahmoud, S.R and Tounsi, A. (2020a), "The effects of the surrounding viscoelastic media on the buckling behavior of single microfilament within the cell: A mechanical model", Adv. Concr. Constr., 10(2), 141-149. https://doi.org/10.12989/acc.2020.10.2.141 
  35. Koizumi, M. (1997), "FGM Activities in Japan", Compos. Part B Eng., 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9 
  36. Kuang, W., Wang, H., Li, X., Zhang, J., Zhou, Q. and Zhao, Y. (2018), "Application of the thermodynamic extremal principle to diffusion-controlled phase transformations in Fe-C-X alloys: Modeling and applications", Acta Materialia, 159, 16-30. https://doi.org/10.1016/j.actamat.2018.08.008 
  37. Lam, K.Y. and Loy, C.T. (1998), "Influence of boundary conditions for a thin laminated rotating cylindrical shell", Compos. Struct., 41, 215-228. https://doi.org/10.1016/S0263-8223(98)00012-9 
  38. Li, J., Wang, Z., Zhang, S., Lin, Y., Wang, L., Sun, C. and Tan, J. (2023), "A novelty mandrel supported thin-wall tube bending cross-section quality analysis: a diameter-adjustable multi-point contact mandrel", Int. J. Adv. Manuf. Technol., 124(11), 4615-4637. https://doi.org/10.1007/s00170-023-10838-y 
  39. Li, X., Yu, P., Niu, X., Yamaguchi, H. and Li, D. (2020), "Noncontact manipulation of nonmagnetic materials by using a uniform magnetic field: Experiment and simulation", J. Magn. Magn. Mater., 497, 165957. https://doi.org/10.1016/j.jmmm.2019.165957 
  40. Liu, W., Bai, X., Yang, H., Bao, R. and Liu, J. (2024), "Tendon driven bistable origami flexible gripper for high-speed adaptive grasping", IEEE Robot. Auto. Lett., 9(6). https://doi.org/10.1109/LRA.2024.3389413 
  41. Long, X., Chong, K., Su, Y., Du, L. and Zhang, G. (2023), "Connecting the macroscopic and mesoscopic properties of sintered silver nanoparticles by crystal plasticity finite element method", Eng. Fract. Mech., 281, 109137. https://doi.org/10.1016/j.engfracmech.2023.109137 
  42. Loy, C.T. and Lam, K.Y. (1997), "Vibration of cylindrical shells with ring supports", J. Mech. Eng., 39, 455-471. https://doi.org/10.1016/S0020-7403(96)00035-5 
  43. Mousavi, S.M., Rostami, M.N., Yousefi, M. and Dinarvand, S. (2021), "Dual solutions for MHD flow of a water-based TiO2-Cu hybrid nanofluid over a continuously moving thin needle in presence of thermal radiation", Report Mech. Eng., 2(1), 31-40. https://doi.org/10.31181/rme200102031m 
  44. Naeem, M.N., Ghamkhar, M., Arshad, S.H. and Shah, A.G. (2013), "Vibration analysis of submerged thin FGM cylindrical shells", J. Mech. Sci. Technol., 27(3), 649-656. https://doi.org/10.1007/s12206-013-0119-6 
  45. Najafizadeh, M.M. and Isvandzibaei, M.R. (2007), "Vibration of (FGM) cylindrical shells based on higher order shear deformation plate theory with ring support", Acta Mechanica, 191, 75-91. https://doi.org/10.1007/s00707-006-0438-0 
  46. Safaei, B., Khoda, F.H. and Fattahi, A.M. (2019), "Non-classical plate model for single-layered graphene sheet for axial buckling", Adv. Nano Res., 7, 265-275. https://doi.org/10.12989/anr.2019.7.4.265 
  47. Sedighi, H.M. (2020), "Divergence and flutter instability of magneto-thermo-elastic C-BN hetero-nanotubes conveying fluid", Acta Mechanica Sinica, 36, 381-396. https://link.springer.com/article/10.1007/s10409-019-00924-4 
  48. Sedighi, H.M., Ouakad, H.M., Dimitri, R. and Tornabene, F. (2020), "Stress-driven nonlocal elasticity for the instability analysis of fluid-conveying C-BN hybrid-nanotube in a magneto-thermal environment", Physica Scripta, 95(6), 065204. https://doi.org/10.1088/1402-4896/ab793f 
  49. Sewall, J.L. and Naumann, E.C. (1968), An Experimental and Analytical Vibration Study of Thin Cylindrical Shells with and without Longitudinal Stiffeners, National Aeronautic and Space Administration. 
  50. Shah, A.G., Mahmood, T. and Naeem, M.N. (2009), "Vibrations of FGM thin cylindrical shells with exponential volume fraction law", Appl. Math. Mech., 30(5), 607-615. https://doi.org/10.1007/s10483-009-0507-x 
  51. Shahsavari, D., Karami, B. and Janghorban, M. (2019). "Size-dependent vibration analysis of laminated composite plates", Adv. Nano Res., 7(5), 337-349. https://doi.org/10.12989/anr.2019.7.5.337 
  52. Sharif, H., Naeem, M.N., Khadimallah, M.A., Ayed, H., Bouzgarrou, S.M., Al Naim, A.F., Muzamal, H. and Tounsi, A. (2020), "Energy effects on MHD flow of Eyring's nanofluid containing motile microorganism", Adv. Concr. Constr., 10(4), 357-367. https://doi.org/10.12989/acc.2020.10.4.357 
  53. Sharma, C.B. and Johns, D.J. (1971), "Vibration characteristics of a clamped-free and clamped-ring-stiffened circular cylindrical shell", J. Sound Vib., 14(4), 459-474. https://doi.org/10.1016/0022-460X(71)90575-X 
  54. Sharma, P., Singh, R., & Hussain, M. (2019), "On modal analysis of axially functionally graded material beam under hygrothermal effect", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 234(5), 1085-1101. https://doi.org/10.1177/0954406219888234. 
  55. Sobamowo, G.M., Ogunmola, B.Y. and Osheku, C.A. (2017), "Thermo-mechanical nonlinear vibration analysis of fluid-conveying structures subjected to different boundary conditions using Galerkin-Newton-Harmonic balancing method", J. Appl. Comput. Mech., 3(1), 60-79. https://doi.org/10.22055/JACM.2017.12620 
  56. Sodel W. (1981), Vibration of Shell and Plates, Mechanical Engineering Series, Marcel Dekker, New York, U.S.A. 
  57. Sofiyev, A.H. and Avcar, M. (2010), "The stability of cylindrical shells containing an FGM layer subjected to axial load on the pasternak foundation", Engineering, 2, 228-236. https://doi.org/10.4236/eng.2010.24033 
  58. Sun, L., Liang, T., Zhang, C. and Chen, J. (2023), "The rheological performance of shear-thickening fluids based on carbon fiber and silica nanocomposite", Phys. Fluids, 35(3), 32002. https://doi.org/10.1063/5.0138294 
  59. Sun, L., Wang, G. and Zhang, C. (2024), "Experimental investigation of a novel high performance multi-walled carbon nano-polyvinylpyrrolidone/silicon-based shear thickening fluid damper", J. Intell. Mater. Syst. Struct., 35(6), 661-672. https://doi.org/10.1177/1045389X23122299 
  60. Suresh, S. and Mortensen, A. (1997), "Functionally gradient metals and metal ceramic composites: Part 2 thermo mechanical behavior", Int. Mater. Rev., 42, 85-116. https://doi.org/10.1179/imr.1995.40.6.239 
  61. Taj, M., Khadimallah, M.A., Hussain, M., Khedher, K.M., Shamim, R.A., Ahmad, M. and Tounsi, A. (2020), "Analysis of nonlocal Kelvin's model for embedded microtubules: Via viscoelastic medium", Smart Struct. Syst., 26(6), 809-817. https://doi.org/10.12989/sss.2020.26.6.80 
  62. Taj, M., Khadimallah, M.A., Hussain, M., Mahmood, S., Safeer, M., Al Naim, A.F. and Ahmad, M. (2021), "Confinement effectiveness of Timoshenko and Euler Bernoulli theories on buckling of microfilaments", Adv. Concr. Constr., 11(1), 81-88. https://doi.org/10.12989/acc.2021.11.1.081 
  63. Toulokian YS. (1967), Thermo Physical Properties of High Temperature Solid Materials, Macmillan, New York, U.S.A. 
  64. Wang, C. M., Swaddiwudhipong, S. and Tian, J. (1997), "Ritz method for vibration analysis of cylindrical shells with ring-stiffeners", J. Eng. Mech., 123, 134-143. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:2(134) 
  65. Wang, C., Wang, Z., Zhang, S., Liu, X. and Tan, J. (2023), "Reinforced quantum-behaved particle swarm-optimized neural network for cross-sectional distortion prediction of novel variable-diameter-die-formed metal bent tubes", J. Comput. Des. Eng., 10(3), 1060-1079. https://doi.org/10.1093/jcde/qwad037 
  66. Wang, W., Jin, Y., Mu, Y., Zhang, M. and Du, J. (2023a), "A novel tubular structure with negative Poisson's ratio based on gyroid-type triply periodic minimal surfaces", Virtual Phys. Prototyp., 18(1), e2203701. https://doi.org/10.1080/17452759.2023.2203701 
  67. Warburton, G.B. (1965), "Vibration of thin cylindrical shells", J. Mech. Eng. Sci., 7, 399-407. https://doi.org/10.1243/JMES_JOUR_1965_007_062_02 
  68. Wuite J, Adali S. (2005), "Deflection and stress behavior of nanocomposite reinforced beams using a multiscale analysis", Compos. Struct., 71(3-4), 388-396. https://doi.org/10.1016/j.compstruct.2005.09.011 
  69. Xiang, Y., Ma. Y.F., Kitipornchai. S., Lau. C.W.H. (2002), "Exact solutions for vibration of cylindrical shells with intermediate ring supports", Int. J. Mech. Sci., 44(9), 1907-1924. https://doi.org/10.1016/S0020-7403(02)00071-1 
  70. Xuebin, L. (2008), "Study on free vibration analysis of circular cylindrical shells using wave propagation", J. Sound Vib., 311, 667-682. https://doi.org/10.1016/j.jsv.2007.09.023 
  71. Yang, S., Zhang, Y., Sha, Z., Huang, Z., Wang, H., Wang, F. and Li, J. (2022b), "Deterministic manipulation of heat flow via three-dimensional-printed thermal meta-materials for multiple protection of critical components", ACS Appl. Mater. Interf., 14(34), 39354-39363. https://doi.org/10.1021/acsami.2c09602 
  72. Yang, W., Jiang, X., Tian, X., Hou, H. and Zhao, Y. (2023), "Phase-field simulation of nano-α' precipitates under irradiation and dislocations", J. Mater. Res. Technol., 22, 1307-1321. https://doi.org/10.1016/j.jmrt.2022.11.165 
  73. Zhang, G., Yang, Z., Li, X., Deng, S., Liu, Y., Zhou, H., Peng, M., Fu, Z., Chen, R., Meng, D., Zhong, L., Zhou, Q. and Wei, S. (2024), "Gamma-ray irradiation induced dielectric loss of SiO2/Si heterostructures in through-silicon vias (TSVs) by forming border traps", ACS Appl. Electr. Mater., 6(2), 1339-1346. https://doi.org/10.1021/acsaelm.3c01646 
  74. Zhang, X.M. (2002), "Parametric analysis of frequency of rotating laminated composite cylindrical shells with the wave propagation approach", Comput. Meth. Appl. Mech. Eng., 191, 2057-2071. https://doi.org/10.1016/S0045-7825(01)00368-1. 
  75. Zhu, Q., Chen, J., Gou, G., Chen, H. and Li, P. (2017), "Ameliorated longitudinal critically refracted-Attenuation velocity method for welding residual stress measurement", J. Mater. Proc. Technol., 246, 267-275. https://doi.org/10.1016/j.jmatprotec.2017.03.022 
  76. Zhu, S., Li, X., Bian, Y., Dai, N., Yong, J., Hu, Y., Li, J. and Chu, J. (2023), "Inclination-enabled generalized microfluid rectifiers via anisotropic slippery hollow tracks", Adv. Mater. Technol., 8(16), 2300267. https://doi.org/10.1002/admt.202300267