Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

A new three-dimensional model for free vibration analysis of functionally graded nanoplates resting on an elastic foundation

  • Mahsa Najafi (Advanced Materials and Computational Mechanics Lab., Department of Mechanical Engineering, University of Zanjan) ;
  • Isa Ahmadi (Advanced Materials and Computational Mechanics Lab., Department of Mechanical Engineering, University of Zanjan) ;
  • Vladimir Sladek (Institute of Construction and Architecture, Slovak Academy of Sciences)
  • Received : 2023.05.16
  • Accepted : 2024.07.11
  • Published : 2024.08.10
⟨ Previous Next ⟩

Abstract

This paper presents a three-dimensional displacement-based formulation to investigate the free vibration of functionally graded nanoplates resting on a Winkler-Pasternak foundation based on the nonlocal elasticity theory. The material properties of the FG nanoplate are considered to vary continuously through the thickness of the nanoplate according to the power-law distribution model. A general three-dimensional displacement field is considered for the plate, which takes into account the out-of-plane strains of the plate as well as the in-plane strains. Unlike the shear deformation theories, in the present formulation, no predetermined form for the distribution of displacements and transverse strains is considered. The equations of motion for functionally graded nanoplate are derived based on Hamilton's principle. The solution is obtained for simply-supported nanoplate, and the predicted results for natural frequencies are compared with the predictions of shear deformation theories which are available in the literature. The predictions of the present theory are discussed in detail to investigate the effects of power-law index, length-to-thickness ratio, mode numbers and the elastic foundation on the dynamic behavior of the functionally graded nanoplate. The present study presents a three-dimensional solution that is able to determine more accurate results in predicting of the natural frequencies of flexural and thickness modes of nanoplates. The effects of parameters that play a key role in the analysis and mechanical design of functionally graded nanoplates are investigated.

Keywords

References

  1. Abdelbaki, C. (2019), "Free vibration analysis of simply supported P-FGM nanoplate using a nonlocal four variables shear deformation plate theory", Strojnicky casopis-J. Mech. Eng., 69(4), 9-24. https://doi.org/10.2478/scjme-2019-0039.
  2. Aghababaei, R. and Reddy, J. (2009), "Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates", J. Sound Vi., 326(1-2), 277-289. https://doi.org/10.1016/j.jsv.2009.04.044.
  3. Ahmadi, I. (2021), "Vibration analysis of 2D-functionally graded nanobeams using the nonlocal theory and meshless method", Eng. Anal. Bound. Elements, 124, 142-154. https://doi.org/10.1016/j.enganabound.2020.12.010.
  4. Ahmadi, I., Sladek, J. and Sladek, V. (2023), "Size dependent free vibration analysis of 2D-functionally graded curved nanobeam by meshless method", Mech. Adv. Mater. Struct., 1-22. https://doi.org/10.1080/15376494.2023.2195400.
  5. Ahmadi, I., Davarpanah, M., Sladek, J., Sladek, V. and Moradi, M. N. (2024), "A size-dependent meshless model for free vibration analysis of 2D-functionally graded multiple nanobeam system", J. Brazil. Soc. Mech. Sci. En., 46(1), 11. https://doi.org/10.1007/s40430-023-04580-5
  6. Aksencer, T. and Aydogdu, M. (2011), "Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory", Physica E: Low-dimen. Syst. Nanostruct., 43(4), 954-959. https://doi.org/10.1016/j.physe.2010.11.024.
  7. Alibeigloo, A. (2011), "Free vibration analysis of nano-plate using three-dimensional theory of elasticity", Acta Mechanica, 222(1), 149-159. https://doi.org/10.1007/s00707-011-0518-7.
  8. Ansari, R., Ashrafi, M., Pourashraf, T. and Sahmani, S. (2015), "Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory", Acta Astronautica, 109, 42-51. https://doi.org/10.1016/j.actaastro.2014.12.015.
  9. Ansari, R., Sahmani, S. and Arash, B. (2010), "Nonlocal plate model for free vibrations of single-layered graphene sheets", Phys. Lett. A, 375(1), 53-62. https://doi.org/10.1016/j.physleta.2010.10.028.
  10. Ansari, R., Shojaei, M.F., Shahabodini, A. and Bazdid-Vahdati, M. (2015), "Three-dimensional bending and vibration analysis of functionally graded nanoplates by a novel differential quadrature-based approach", Compos. Struct., 131, 753-764. https://doi.org/10.1016/j.compstruct.2015.06.027.
  11. Ansari, R., Torabi, J. and Norouzzadeh, A. (2018), "Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method", Physica B: Condens. Mat., 534, 90-97. https://doi.org/10.1016/j.physb.2018.01.025.
  12. Arefi, M. and Civalek, O. (2020), "Static analysis of functionally graded composite shells on elastic foundations with nonlocal elasticity theory", Arch. Civil Mech. Eng., 20(1), 1-17. https://doi.org/10.1007/s43452-020-00032-2.
  13. Arefi, M., Bidgoli, E.M.-R., Dimitri, R. and Tornabene, F. (2018), "Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets", Aerosp. Sci. Tech., 81, 108-117. https://doi.org/10.1016/j.ast.2018.07.036.
  14. Babaei, A. and Ahmadi, I. (2017). "Dynamic vibration characteristics of non-homogenous beam-model MEMS", J. Multidiscip. Eng. Sci. Tech., 4(3), 6807-14.
  15. Baferani, A.H., Saidi, A. and Ehteshami, H. (2011), "Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation", Compos. Struct., 93(7), 1842-1853. https://doi.org/10.1016/j.compstruct.2011.01.020.
  16. Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E. and Mahmoud, S. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct. Int. J., 18(4) 1063-1081. http://dx.doi.org/10.12989/scs.2015.18.4.1063.
  17. Besseghier, A., Houari, M.S.A., Tounsi, A. and Mahmoud, S. (2017), "Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory", Smart Struct. Syst., 19(6) 601-614. https://doi.org/10.12989/sss.2017.19.6.601.
  18. Civalek, O., Uzun, B. and Yayli, M.O. (2022). "A Fourier sine series solution of static and dynamic response of nano/micro-scaled FG rod under torsional effect", Adv. Nano Res., 12(5), 467-482. https://doi.org/10.12989/anr.2022.12.5.467.
  19. Daikh, A.A., Drai, A., Bensaid, I., Houari, M.S.A. and Tounsi, A. (2021), "On vibration of functionally graded sandwich nanoplates in the thermal environment", J. Sandw. Struct. Mater., 23(6), 2217-2244. https://doi.org/10.1177/1099636220909790.
  20. Daneshmehr, A., Rajabpoor, A. and Hadi, A. (2015), "Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories", Int. J. Eng. Sci., 95, 23-35. https://doi.org/10.1016/j.ijengsci.2015.05.011.
  21. Demir, C., Mercan, K., Numanoglu, H.M. and Civalek, O. (2018), "Bending response of nanobeams resting on elastic foundation", J. Appl. Computat. Mech., 4(2), 105-114. https://doi.org/10.22055/jacm.2017.22594.1137.
  22. Dindarloo, M.H. and Li, L. (2019), "Vibration analysis of carbon nanotubes reinforced isotropic doubly-curved nanoshells using nonlocal elasticity theory based on a new higher order shear deformation theory", Compos. Part B: Eng., 175, 107170. https://doi.org/10.1016/j.compositesb.2019.107170.
  23. Eltaher, M., Khairy, A., Sadoun, A. and Omar, F.-A. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Math. Computat., 229, 283-295. https://doi.org/10.1016/j.amc.2013.12.072.
  24. 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.
  25. Eringen, A.C. and Wegner, J. (2003), "Nonlocal continuum field theories", Appl. Mech. Rev., 56(2), B20-B22. https://doi.org/10.1115/1.1553434.
  26. Esmaeilzadeh, M., Golmakani, M. and Sadeghian, M. (2020), "A nonlocal strain gradient model for nonlinear dynamic behavior of bi-directional functionally graded porous nanoplates on elastic foundations", Mech. Based Des. Struct. Mach., 51(1), 1-20. https://doi.org/10.1080/15397734.2020.1845965.
  27. Eyvazian, A., Zhang, C., Civalek, O., Khan, A., Sebaey, T.A. and Farouk, N. (2022), "Wave propagation analysis of sandwich FGM nanoplate surrounded by viscoelastic foundation", Arch. Civil Mech. Eng., 22(4), 159. https://doi.org/10.1007/s43452-022-00474-w.
  28. Fattahi, A. (2019), "The application of nonlocal elasticity to determine vibrational behavior of FG nanoplates", Steel Compos. Struct. Int. J., 32(2), 281-292. https://doi.org/10.12989/scs.2019.32.2.281.
  29. Filiz, S. and Aydogdu, M. (2010), "Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity", Computat. Mater. Sci., 49(3), 619-627. https://doi.org/10.1016/j.commatsci.2010.06.003.
  30. Gholami, R., Ansari, R. and Gholami, Y. (2017), "Size-dependent bending, buckling and vibration of higher-order shear deformable magneto-electro-thermo-elastic rectangular nanoplates", Mater. Res. Exp., 4(6), 065702. http://dx.doi.org/10.1088/2053-1591/aa711c.
  31. Goodarzi, M., Nikkhah Bahrami, M. and Tavaf, V. (2017), "Refined plate theory for free vibration analysis of FG nanoplates using the nonlocal continuum plate model", J. Comput. Appl. Mech., 48(1), 123-136. https://doi.org/10.22059/jcamech.2017.236217.155.
  32. Gul, U., Aydogdu, M. and Karacam, F. (2019). "Dynamics of a functionally graded Timoshenko beam considering new spectrums", Compos. Struct., 207, 273-291. https://doi.org/10.1016/j.compstruct.2018.09.021.
  33. Hadji, L., Avcar, M. and Civalek, O. (2021), "An analytical solution for the free vibration of FG nanoplates", J. Brazil. Soc. Mech. Sci. Eng., 43(9), 1-14. https://doi.org/10.1007/s40430-021-03134-x.
  34. Huang, M., Zheng, X., Zhou, C., An, D. and Li, R. (2021), "On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free", Acta Mechanica, 232(2), 495-513. https://doi.org/10.1007/s00707-020-02829-x.
  35. Jung, W.-Y., Han, S.-C. and Park, W.-T. (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.
  36. Karlicic, D., Kozic, P. and Pavlovic, R. (2014), "Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium", Compo. Struct., 115, 89-99. https://doi.org/10.1016/j.compstruct.2014.04.002.
  37. Ke, L.-L., Liu, C. and Wang, Y.-S. (2015), "Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions", Physica E: Low-Dimen. Syst. Nanostruct., 66, 93-106. https://doi.org/10.1016/j.physe.2014.10.002.
  38. Khorshidi, K. and Fallah, A. (2016), "Buckling analysis of functionally graded rectangular nano-plate based on nonlocal exponential shear deformation theory", Int. J. Mech. Sci., 113, 94-104. https://doi.org/10.1016/j.ijmecsci.2016.04.014.
  39. Li, L. and Hu, Y. (2017), "Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory", Compos. Struct., 172, 242-250. https://doi.org/10.1016/j.compstruct.2017.03.097.
  40. Li, M., Soares, C.G. and Yan, R. (2021), "Free vibration analysis of FGM plates on Winkler/Pasternak/Kerr foundation by using a simple quasi-3D HSDT", Compos. Struct., 264, 113643. https://doi.org/10.1016/j.compstruct.2021.113643.
  41. Malekzadeh, P. and Shojaee, M. (2013), "Free vibration of nanoplates based on a nonlocal two-variable refined plate theory", Compos. Struct., 95, 443-452. https://doi.org/10.1016/j.compstruct.2012.07.006.
  42. Mechab, I., Mechab, B., Benaissa, S., Serier, B. and Bouiadjra, B.B. (2016), "Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories", J. Brazil. Soc. Mech. Sci. Eng., 38(8), 2193-2211. https://doi.org/10.1007/s40430-015-0482-6.
  43. Murmu, T. and Pradhan, S. (2009), "Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM", Physica E: Low-Dimen. Syst. Nanostruct., 41(7), 1232-1239. https://doi.org/10.1016/j.physe.2009.02.004.
  44. Najafi, M. and Ahmadi, I. (2021), "A nonlocal Layerwise theory for free vibration analysis of nanobeams with various boundary conditions on Winkler-Pasternak foundation", Steel Compos. Struct. Int. J., 40(1), 101-119. https://doi.org/10.12989/scs.2021.40.1.101.
  45. Najafi, M. and Ahmadi, I. (2023), "Nonlocal layerwise theory for bending, buckling and vibration analysis of functionally graded nanobeams", Eng. Comput., 39(4), 2653-2675. https://doi.org/10.1007/s00366-022-01605-w.
  46. Najafi, M. and Ahmadi, I. (2022), "A new model to study magnetic-electric fields effects on bending of nano-scale magneto-electro-elastic beams", Eur. J. Mech.-A/Solids, 96, 104712 https://doi.org/10.1016/j.euromechsol.2022.104712.
  47. Nami, M. and Janghorban, M. (2015), "Free vibration of functionally graded size dependent nanoplates based on second order shear deformation theory using nonlocal elasticity theory", Iran. J. Sci. Tech. Transact. Mech. Eng., 39(M1), 15. https://doi.org/10.22099/IJSTM.2015.2945.
  48. 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.
  49. Narendar, S. and Gopalakrishnan, S. (2012), "Scale effects on buckling analysis of orthotropic nanoplates based on nonlocal two-variable refined plate theory", Acta Mechanica, 223(2), 395-413. https://doi.org/10.1007/s00707-011-0560-5.
  50. Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S. and Rabczuk, T. (2012), "Size-dependent free flexural vibration behavior of functionally graded nanoplates", Computat. Mater. Sci., 65, 74-80. https://doi.org/10.1016/j.commatsci.2012.06.031.
  51. Phadikar, J. and Pradhan, S. (2010), "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates", Computat. Mater. Sci., 49(3), 492-499. https://doi.org/10.1016/j.commatsci.2010.05.040.
  52. Phung-Van, P., Thai, C.H., Nguyen-Xuan, H. and Abdel-Wahab, M. (2019), "An isogeometric approach of static and free vibration analyses for porous FG nanoplates", Eur. J. Mech.-A/Solids, 78, 103851. https://doi.org/10.1016/j.euromechsol.2019.103851.
  53. Pouresmaeeli, S., Ghavanloo, E. and Fazelzadeh, S. (2013), "Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium", Compos. Struct., 96, 405-410. https://doi.org/10.1016/j.compstruct.2012.08.051.
  54. Pradhan, S. and Murmu, T. (2010), "Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory", Physica E: Low-Dimen. Syst. Nanostruct., 42(5), 1293-1301. https://doi.org/10.1016/j.physe.2009.10.053.
  55. Pradhan, S. and Phadikar, J. (2009), "Nonlocal elasticity theory for vibration of nanoplates", J. Sound Vib., 325(1-2), 206-223. https://doi.org/10.1016/j.jsv.2009.03.007.
  56. Pradhan, S. and Raj, R. (2011), "Vibration analyses of nanoplates with various boundary conditions using DQ method", J. Computat. Theor. Nanosci., 8(8), 1432-1436. https://doi.org/10.1166/jctn.2011.1833.
  57. Redddy, B.S., Kumar, J.S., Reddy, C.E. and Reddy, V.K. (2014), "Free vibration behaviour of functionally graded plates using higher-order shear deformation theory", J. Appl. Sci. Eng., 17(3), 231-241. https://doi.org/10.6180/jase.2014.17.3.03.
  58. Reddy, J. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004.
  59. Reddy, J. (2010), "Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates", Int. J. Eng. Sci., 48(11), 1507-1518. https://doi.org/10.1016/j.ijengsci.2010.09.020.
  60. Reddy, J. and Kim, J. (2012), "A nonlinear modified couple stress-based third-order theory of functionally graded plates", Compos. Struct., 94(3), 1128-1143. https://doi.org/10.1016/j.compstruct.2011.10.006.
  61. Salehipour, H., Shahgholian-Ghahfarokhi, D., Shahsavar, A., Civalek, O. and Edalati, M. (2020), "Static deflection and free vibration analysis of functionally graded and porous cylindrical micro/nano shells based on the three-dimensional elasticity and modified couple stress theories", Mech. Based Des. Struct. Mach., 50(6), 1-22. https://doi.org/10.1080/15397734.2020.1775095.
  62. Sari, M.S., Ghaffari, S., Ceballes, S. and Abdelkefi, A. (2020), "Buckling response of functionally graded nanoplates under combined thermal and mechanical loadings", J. Nanopart. Res., 22(4), 1-21. https://doi.org/10.1007/s11051-020-04815-9.
  63. Sidhoum, I.A., Boutchicha, D., Benyoucef, S. and Tounsi, A. (2017), "An original HSDT for free vibration analysis of functionally graded plates", Steel Compos. Struct. Int. J., 25(6), 735-745. https://doi.org/10.12989/scs.2017.25.6.735.
  64. Singh, P.P., Azam, M.S. and Ranjan, V. (2018), "Analysis of free vibration of nano plate resting on Winkler foundation", Vibroeng. Procedia, 21, 65-70. https://doi.org/10.21595/vp.2018.20406.
  65. Slimane, M., Hadj Mostefa, A., Boutaleb, S. and Hellal, H. (2020), "Free Vibration Analysis of Functionally Graded FG Nano-Plates with Porosities", J. Nano Res., 64, 61-74. https://doi.org/10.4028/www.scientific.net/JNanoR.64.61.
  66. Sobhy, M. (2014), "Natural frequency and buckling of orthotropic nanoplates resting on two-parameter elastic foundations with various boundary conditions", J. Mech., 30(5), 443-453. https://doi.org/10.1017/jmech.2014.46.
  67. Sobhy, M. (2015), "A comprehensive study on FGM nanoplates embedded in an elastic medium", Compos. Struct., 134, 966-980. https://doi.org/10.1016/j.compstruct.2015.08.102.
  68. Talebizadehsardari, P., Salehipour, H., Shahgholian-Ghahfarokhi, D., Shahsavar, A. and Karimi, M. (2020), "Free vibration analysis of the macro-micro-nano plates and shells made of a material with functionally graded porosity: A closed-form solution", Mech. Based Des. Struct. Mach., 50(3), 1-27. https://doi.org/10.1016/j.compstruct.2017.03.097.
  69. Thai, C.H., Nguyen-Xuan, H. and Phung-Van, P. (2022), "A size-dependent isogeometric analysis of laminated composite plates based on the nonlocal strain gradient theory", Eng. Comput., 1-15. https://doi.org/10.1007/s00366-021-01559-5.
  70. Thai, S., Thai, H.-T., Vo, T.P. and Lee, S. (2018), "Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis", Compos. Struct., 201, 13-20. https://doi.org/10.1016/j.compstruct.2018.05.116.
  71. Torabi, J., Ansari, R., Zabihi, A. and Hosseini, K. (2020), "Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory", Mech. Based Des. Struct. Mach., 50(2), 1-21. https://doi.org/10.1080/15397734.2020.1721298.
  72. Uzun, B. and Yayli, M. O. (2022), "Porosity dependent torsional vibrations of restrained FG nanotubes using modified couple stress theory", Mater. Today Commun., 32, 103969. https://doi.org/10.1016/j.mtcomm.2022.103969
  73. Uzun, B. and Yayli, M.O. (2024), "Porosity and deformable boundary effects on the dynamic of nonlocal sigmoid and power-law FG nanobeams embedded in the Winkler-Pasternak medium", J. Vib. Eng. Tech., 12(3), 3193-3212. https://doi.org/10.1007/s42417-023-01039-8
  74. Uzun, B. and Yayli, M.O. (2024), "Porosity effects on the dynamic response of arbitrary restrained FG nanobeam based on the MCST", Zeitschrift fur Naturforschung A, 79(2), 183-197. https://doi.org/10.1515/zna-2023-0261
  75. Uzun, B., Civalek, O. and Yayli, M.O. (2023), "Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions", Mech. Based Des. Struct. Mach., 51(1), 481-500. https://doi.org/10.1080/15397734.2020.1846560.
  76. Vel, S.S. and Batra, R. (2004), "Three-dimensional exact solution for the vibration of functionally graded rectangular plates", J. Sound Vib., 272(3-5), 703-730. https://doi.org/10.1016/S0022-460X(03)00412-7.
  77. Wang, G., Zhang, Y. and Arefi, M. (2021), "Three-dimensional exact elastic analysis of nanoplates", Arch. Civil Mech. Eng., 21(3), 91. https://doi.org/10.1007/s43452-021-00247-x.
  78. Wang, K. and Wang, B. (2011), "Vibration of nanoscale plates with surface energy via nonlocal elasticity", Physica E: Low-Dimen. Syst. Nanostruct., 44(2), 448-453. https://doi.org/10.1016/j.physe.2011.09.019.
  79. Yan, Z. and Jiang, L. (2012), "Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints", Proc. Royal Soc. A: Math., Phys. Eng. Sci., 468(2147), 3458-3475. https://doi.org/10.1098/rspa.2012.0214.
  80. Zare, M., Nazemnezhad, R. and Hosseini-Hashemi, S. (2015), "Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method", Meccanica, 50(9), 2391-2408. https://doi.org/10.1007/s11012-015-0161-9.
  81. Zargaripoor, A., Daneshmehr, A., Isaac Hosseini, I. and Rajabpoor, A. (2018), "Free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory using finite element method", J. Computat. Appl. Mech., 49(1), 86-101. https://doi.org/10.22059/jcamech.2018.248906.223.