Advances in nano research

  • 제17권3호
  • /
  • Pages.275-284
  • /
  • 2024
  • /
  • 2287-237X(pISSN)
  • /
  • 2287-2388(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

The effect of a nonlocal stress-strain elasticity theory on the vibration analysis of Timoshenko sandwich beam theory

  • Mehdi Mohammadimehr (Department of Solid Mechanic, Faculty of Mechanical Engineering, University of Kashan)
  • 투고 : 2020.10.18
  • 심사 : 2024.09.10
  • 발행 : 2024.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this article, a nonlocal stress-strain elasticity theory on the vibration analysis of Timoshenko sandwich beam theory with symmetric and asymmetric distributions of porous core and functionally graded material facesheets is introduced. According to nonlocal elasticity Eringen's theory (nonlocal stress elasticity theory), the stress at a reference point in the body is dependent not only on the strain state at that point, but also on the strain state at all of the points throughout the body; while, according to a new nonlocal strain elasticity theory, the strain at a reference point in the body is dependent not only on the stress state at that point, but also on the stress state at all of the points throughout the body. Also, with combinations of two concepts, the nonlocal stress-strain elasticity theory is defined that can be actual at micro/nano scales. It is concluded that the natural frequency decreases with an increase in the nonlocal stress parameter; while, this effect is vice versa for nonlocal strain elasticity, because the stiffness of Timoshenko sandwich beam decreases with increasing of the nonlocal stress parameter; in which, the nonlocal strain parameter leads to increase the stiffness of structures at micro/nano scale. It is seen that the natural frequency by considering both nonlocal stress parameter and nonlocal strain parameter is higher than the nonlocal stress parameter only and lower for a nonlocal strain parameter only.

키워드

과제정보

The authors would like to thank the referees for their valuable comments. Also, they are thankful to the Iranian Nanotechnology Development Committee for their financial support and the University of Kashan for supporting this work by Grant No. 891238/28.

참고문헌

  1. Alizadeh Hamidi, B., Hosseini, S.A., Hassannejad, R. and Khosravi, F. (2020), "An exact solution on gold microbeam with thermoelastic damping via generalized Green-Naghdi and modified couple stress theories", J. Therm. Stress., 43 (2), 157-174. https://doi.org/10.1080/01495739.2019.1666694
  2. Anitescu, C., Atroshchenko, E., Alajlan, N. and Rabczuk, T. (2019), "Artificial neural network methods for the solution of second order boundary value problems", Comput. Mater. Continua, 59(1), 345-359. https://doi.org/10.32604/cmc.2019.06641
  3. Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V. and Sahmani, S. (2012), "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory", Compos. Struct., 100, 385-397. https://doi.org/10.1016/j.compstruct.2012.12.048
  4. Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V. and Sahmani, S. (2013), "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko micro beams based on the most general strain gradient theory", Compos. Struct., 100, 385-397. https://doi.org/10.1016/j.compstruct.2012.12.048
  5. Ansari, R., Gholami, R. and Rouhi, H. (2015), "Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory", Compos. Struct., 126, 216-226. https://doi.org/10.1016/j.compstruct.2015.02.068
  6. Anvari, M., Mohammadimehr, M. and Amiri, A. (2020), "Vibration behavior of a micro cylindrical sandwich panel reinforced by graphene platelet", J. Vib. Control, 26(13-14), 1311-1343. https://doi.org/10.1177/1077546319892730
  7. Babaei, H. and Eslami, M.R. (2019), "Thermally induced large deflection of FGM shallow micro-arches with integrated surface piezoelectric layers based on modified couple stress theory", Acta Mechanica, 230, 2363-2384. https://doi.org/10.1007/s00707-019-02384-0
  8. Bahaadini, R. and Saidi, A.R. (2018), "Aeroelastic analysis of functionally graded rotating blades reinforced with graphene nanoplatelets in supersonic flow", Aerosp. Sci. Technol., 80, 381-391. https://doi.org/10.1016/j.ast.2018.06.035
  9. Bamdad, M., Mohammadimehr, M. and Alambeigi, K. (2019), "Analysis of sandwich Timoshenko porous beam with temperature-dependent material properties: Magneto-electro-elastic vibration and buckling solution", J. Vib. Control, 25(23-24), 2875-2893. https://doi.org/10.1177/1077546319860314
  10. Chen, D., Kitipornchai, S. and Yang, J. (2017), "Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets", Mater. Des., 116, 656-665. https://doi.org/10.1016/j.matdes.2016.12.061
  11. Chu, L., Dui, G. and Ju, C. (2018), "Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory", Compos. Struct., 186, 39-49. https://doi.org/10.1016/j.compstruct.2017.10.083
  12. Civalek, O., Uzun, B. and Ozgur Yayli, M. (2020) "Frequency, bending and buckling loads of nanobeams with different cross sections", Adv. Nano Res., 9(2), 91-104. https://doi.org/10.12989/anr.2020.9.2.091
  13. Danesh, M., Farajpour, A. and Mohammadi, M. (2012), "Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method", Mech. Res. Commun., 39(1), 23-27. https://doi.org/10.1016/j.mechrescom.2011.09.004
  14. Ebrahimi, F., Barati, M.R. and Dabbagh, A. (2016), "A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates", Int. J. Eng. Sci., 107, 169-182. https://doi.org/10.1016/j.ijengsci.2016.07.008
  15. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
  16. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803
  17. Farzam, A. and Hassani, B. (2019), "Size-dependent analysis of FG microplates with temperature-dependent material properties using modified strain gradient theory and isogeometric approach", Compos. Part B Eng., 161, 150-168. https://doi.org/10.1016/j.compositesb.2018.10.028
  18. Ghorbanpour Arani, A., Rahmani, R., Arefmanesh, A. and Golabi, S. (2008), "Buckling analysis of multi-walled carbon nanotubes under combined loading considering the effect of small length scale", J. Mech. Sci. Technol., 22, 429-439. https://doi.org/10.1007/s12206-007-1045-2
  19. Ghorbanpour Arani, A., Abdollahian, M. and Jalaei, M.H. (2015), "Vibration of bioliquid-filled microtubules embedded in cytoplasm including surface effects using modified couple stress theory", J. Theor. Biol., 367, 29-38. https://doi.org/10.1016/j.jtbi.2014.11.019
  20. Guo, H., Zhuang, X. and Rabczuk, T. (2019), "A deep collocation method for the bending analysis of kirchhoff plate", Comput. Mater. Continua, 59(2), 433-456. https://doi.org/ 10.32604/cmc.2019.06660
  21. Hei, X., Chen, W., Pang, G., Xiao, R. and Zhang, C. (2018), "A new visco-elasto-plastic model via time-space fractional derivative", Mech. Time Depend. Mater., 22(1), 129-141. https://doi.org/10.1007/s11043-017-9356-x.
  22. Hussain, M., Naeem, M.N., Asghar, S. and Tounsi, A. (2020) "Eringen's nonlocal model sandwich with Kelvin's theory for vibration of DWCNT", Comput. Concr., 24(4), 343-353. https://doi.org/ 10.12989/cac.2020.25.4.343
  23. Jomehzadeh, E. and Saidi, A.R. (2011), "Decoupling the nonlocal elasticity equations for three dimensional vibration analysis of nano-plates", Compos. Struct., 93(2), 1015-1020. https://doi.org/10.1016/j.compstruct.2010.06.017
  24. Jung, W., Han, S. and Park, W. (2014), "A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium", Compos. Part B Eng., 60, 746-756. https://doi.org/10.1016/j.compositesb.2013.12.058
  25. Karami, B., Janghorban, M. and Rabczuk, T. (2019), "Static analysis of functionally graded anisotropic nanoplates using nonlocal strain gradient theory", Compos. Struct. 227, 111249. https://doi.org/10.1016/j.compstruct.2019.111249
  26. Karami, B., Janghorban, M. and Rabczuk, T. (2020), "Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory", Compos. Part B Eng., 182, 107622. https://doi.org/10.1016/j.compositesb.2019.107622
  27. Karlicic, D., Cajic, M., Adhikari, S., Kozic, P. and Murmu, T. (2017), "Vibrating nonlocal multi-nanoplate system under inplane magnetic field", Eur. J. Mech. A Solids, 64, 29-45. https://doi.org/10.1016/j.euromechsol.2017.01.013
  28. Ke, L.L. and Wang, Y.S. (2011), "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory", Compos. Struct., 93(2), 342-350. https://doi.org/10.1016/j.compstruct.2010.09.008
  29. Lam, D., Yang, F., Chong, A., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51, 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  30. Lazar, M., Maugin, G.A. and Aifantis, E.C. (2006), "On a theory of nonlocal elasticity of bi-Helmholtz type and some applications", Int. J. Solids Struct., 43(6), 1404-1421. https://doi.org/10.1016/j.ijsolstr.2005.04.027
  31. Lazopoulos, A.K. (2016), "On fractional peridynamic deformations", Archive of Applied Mechanics, 86(12), 1987-1994. https://doi.org/10.1007/s00419-016-1163-3
  32. Li, L. and Hu, Y. (2015), "Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory", Int. J. Eng. Sci., 97, 84-94. https://doi.org/10.1016/j.ijengsci.2015.08.013
  33. Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solids, 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007
  34. Mohammadimehr, M., Saidi, A.R., Ghorbanpour Arani, A., Arefmanesh, A. and Han, Q. (2010), "Torsional buckling of a DWCNT embedded on winkler and pasternak foundations using nonlocal theory", J. Mech. Sci. Technol., 24 (6), 1289-1299. https://doi.org/10.1007/s12206-010-0331-6
  35. Mohammadimehr, M. and Mahmudian-Najafabadi, M. (2013), "Bending and free vibration analysis of nonlocal functionally graded nanocomposite Timoshenko beam model reinforced by SWBNNT based on modified coupled stress theory", J. Nanostruct., 3(4), 483-492. https://doi.org/10.7508/jns.2013.04.014
  36. Mohammadimehr, M. and Monajemi, A.A. (2016), "Nonlinear vibration analysis of MSGT boron-nitride micro ribbon based mass sensor using DQEM", Smart Struct. Syst., 18 (5), 1029-1062. https://doi.org/10.12989/sss.2016.18.5.1029
  37. Mohammadimehr, M., Okhravi, S.V. and Akhavan Alavi, S.M. (2018a), "Free vibration analysis of magneto-electro-elastic cylindrical composite panel reinforced by various distributions of CNTs with considering open and closed circuits boundary conditions based on FSDT", J. Vib. Control, 24(8), 1551-1569. https://doi.org/10.1177/1077546316664022
  38. Mohammadimehr, M., Atifeh, S.J. and Rousta Navi, B. (2018b), "Stress and free vibration analysis of piezoelectric hollow circular FG-SWBNNTs reinforced nanocomposite plate based on modified couple stress theory subjected to thermomechanical loadings", J. Vib. Control, 24(15), 3471-3486. https://doi.org/10.1177/1077546317706887
  39. Mohammadimehr, M., Shabani Nejad, E. and Mehrabi, M. (2018c), "Buckling and vibration analyses of MGSGT double-bonded micro composite sandwich SSDT plates reinforced by CNTs and BNNTs with isotropic foam & flexible transversely orthotropic cores", Struct. Eng. Mech., 65(4), 491-504. https://doi.org/10.12989/sem.2018.65.4.491
  40. Mohammadimehr, M. (2022), "Buckling and bending analyses of a sandwich beam based on nonlocal stress-strain elasticity theory with porous core and functionally graded facesheets", Adv. Mater. Res., 11(4), 279-298. https://doi.org/10.12989/amr.2022.11.4.279
  41. Mousavi, M., Mohammadimehr, M. and Rostami, R. (2019), "Analytical solution for buckling analysis of micro sandwich hollow circular plate", Comput. Concr., 24(3), 185-192. https://doi.org/10.12989/cac.2019.24.3.185
  42. Narendar, S. (2011), "Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects", Compos. Struct., 93(12), 3093-3103. https://doi.org/10.1016/j.compstruct.2011.06.028
  43. Nematzadeh, M. and Baradaran-Nasiri, A. (2019), "Mechanical performance of fiber-reinforced recycled refractory brick concrete exposed to elevated temperatures", Comput. Concr., 24(1), 19-35. https://doi.org/ 10.12989/cac.2019.24.1.019
  44. Nguyen, H.X., Nguyen, T.N., Abdel-Wahab, M., Bordas, S.P.A., Nguyen-Xuanfg, H. and Vo, T.P. (2017), "A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory", Comput. Meth. Appl. Mech. Eng., 313, 904-940. https://doi.org/10.1016/j.cma.2016.10.002
  45. Pang, G.F., Chen, W. and Fu, Z. (2015), "Space-fractional advection-dispersion equations by the Kansa method", J. Comput. Phys., 293, 280-296. https://doi.org/10.1016/j.jcp.2014.07.020
  46. Polizzotto, C. (2001), "Nonlocal elasticity and related variational principles", Int. J. Solids Struct., 38 (42-43), 7359-7380. https://doi.org/10.1016/S0020-7683(01)00039-7
  47. Raad, M.F., Nader, M.M. and Nadhim, M.F. (2020), "Scale-dependent thermal vibration analysis of FG beams having porosities based on DQM", Adv. Nano Res., 8(4), 283-292. https://doi.org/10.12989/anr.2020.8.4.283
  48. Rahimi, Z., Sumelka, W. and Yang, X.J. (2017), "A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams", Eur. Phys. J. Plus, 132(11), 479. https://doi.org/10.1140/epjp/i2017-11751-x
  49. Rahmati, A.R. and Mohammadimehr, M. (2014), "Vibration analysis of non-uniform and non-homogeneous boron nitride nanorods embedded in an elastic medium under combined loadings using DQM", Physica B, 440, 88-98. https://doi.org/10.1016/j.physb.2014.01.036
  50. Reddy, J.N. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45, 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004.
  51. Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X. and Rabczuk, T. (2020), "An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications", Comput. Meth. Appl. Mech. Eng., 362, 112790. https://doi.org/10.1016/j.cma.2019.112790
  52. Shariati, A., Barati, M.R., Ebrahimi, F., Singhal, A. and Toghroli, A. (2020a), "Investigating vibrational behavior of graphene sheets under linearly varying in-plane bending load based on the nonlocal strain gradient theory", Adv. Nano Res., 8(4), 265-276. https://doi.org/10.12989/anr.2020.8.4.265
  53. Shariati, A., Barati, M.R., Ebrahimi, F. and Toghroli, A. (2020b). "Investigation of microstructure and surface effects on vibrational characteristics of nanobeams based on nonlocal couple stress theory", Adv. Nano Res., 8(3), 191-202. https://doi.org/10.12989/anr.2020.8.3.191
  54. Shen, J.P., Li, C., Fan, X.L. and Jung, C.M. (2017), "Dynamics of silicon nanobeams with axial motion subjected to transverse and longitudinal loads considering nonlocal and surface effects", Smart Struct. Syst., 19(1), 105-113. https://doi.org/10.12989/sss.2017.19.1.105
  55. Sobhy, M. and Zenkour, A.M. (2020), "A comprehensive study on the size-dependent hygrothermal analysis of exponentially graded microplates on elastic foundations", Mech. Adv. Mater. Struct., 1-15, 2020. https://doi.org/10.1080/15376494.2018.1499986
  56. Sumelka, W. (2014), "Thermoelasticity in the framework of the fractional continuum mechanics", J. Therm. Stress., 37(6), 678-706. https://doi.org/10.1080/01495739.2014.885332
  57. Tadi Beni, Y. and Mehralian, F. (2019), "Free vibration of magneto-electro-elastic nanobeams based on modified couple stress theory in thermal environment", Mech. Adv. Mater. Struct., 26(7), 601-613. https://doi.org/10.1080/15376494.2017.1410902
  58. Taj, M., Majeed, A., Hussain, M., Naeem, M.N., Safeer, M., Ahmad, M., Khan, H.U. and Tounsi, A. (2020), "Non-local orthotropic elastic shell model for vibration analysis of protein microtubules", Comput. Concr., 25(3), 245-253. https://doi.org/10.12989/cac.2020.25.3.245
  59. Thai, C.H., Ferreira, A.J.M. and Nguyen, H. (2018), "Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory", Compos. Struct., 192, 274-288. https://doi.org/10.1016/j.compstruct.2018.02.060
  60. Thai, H.T. and Choi, D.H. (2013), "Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory", Compos. Struct., 95, 142-153. https://doi.org/10.1016/j.compstruct.2012.08.023
  61. Timesli, A. (2020), "Buckling analysis of double walled carbon nanotubes embedded in Kerr elastic medium under axial compression using the nonlocal Donnell shell theory", Adv. Nano Res., 9(2), 69-82. https://doi.org/10.12989/anr.2020.9.2.069
  62. Vu-Bac N., Lahmer T., Zhuang X., Nguyen-Thoi T. and Rabczuk T. (2016), "A software framework for probabilistic sensitivity analysis for computationally expensive models", Adv. Eng. Softw., 100, 19-31. https://doi.org/10.1016/j.advengsoft.2016.06.005
  63. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  64. Yazdani, R. and Mohammadimehr, M. (2019), "Double bonded Cooper-Naghdi micro sandwich cylindrical shells with porous core and CNTRC face sheets: Wave propagation solution", Comput. Concr., 24 (6), 499-511. https://doi.org/10.12989/cac.2019.24.6.499
  65. Zheng, H., Zhou, C., Yan, D.J., Wang, Y.S. and Zhang, C. (2020), "A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory", J. Comput. Phys., 408, 109268. https://doi.org/10.1016/j.jcp.2020.109268