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On the calibration of size parameters related to non-classical continuum theories using molecular dynamics simulations. (English) Zbl 07411528

Summary: The topic presented in this research is the calibration of small-scale parameters of non-classical continuum theories such as nonlocal strain gradient theory, strain gradient theory, stress-driven nonlocal elasticity, and strain-driven nonlocal elasticity. Governing equations of vibrational behavior of circular nanoplate and associated boundary conditions for each method derived using Hamilton’s principle. Obtained governing differential equations from non-classical methods were solved using the general differential quadrature rule (GDQR). Then, the first natural frequencies for different radiuses and different size parameters were obtained. On the other hand, the first natural frequencies of circular nanoplates are calculated using molecular dynamics simulation based on AIREBO and Tersoff potentials for different radiuses. Fast Fourier transform (FFT) was utilized to calculate natural frequencies based on the molecular dynamics simulation. Using the accurate size parameter is an important point in the application of non-classical continuum theories. To obtain the size parameters related to different non-classical methods, the results of molecular dynamics compared to those of nan-classical methods and simulated annealing (SA) algorithm optimization technique was utilized. Results show that stress-driven nonlocal, strain-driven nonlocal, and strain gradient methods cannot predict the behavior predicted by molecular dynamics for all ranges of radius. In other words, the responses of these three methods for any value of the size parameters (in some interval radius) and results of the molecular dynamics method are not equal for a few numbers of studied radii. In contrast to these three methods, the nonlocal strain gradient method predicts the results obtained by molecular dynamics well for all radii. The results of this paper are very useful for researchers in the field of non-classical continuum mechanics.

MSC:

74-XX Mechanics of deformable solids
92-XX Biology and other natural sciences

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[1] Eringen, A. C., Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science, 10, 425-435 (1972), 1972/05/01/ · Zbl 0241.73005
[2] Eringen, A. C.; Edelen, D. G.B., On nonlocal elasticity, International Journal of Engineering Science, 10, 233-248 (1972), 1972/03/01/ · Zbl 0247.73005
[3] Lazar, M.; Maugin, G. A.; Aifantis, E. C., Dislocations in second strain gradient elasticity, International Journal of Solids and Structures, 43, 1787-1817 (2006), 2006/03/01/ · Zbl 1120.74343
[4] Zhu, H. T.; Zbib, H. M.; Aifantis, E. C., Strain gradients and continuum modeling of size effect in metal matrix composites, Acta Mechanica, 121, 165-176 (1997), 1997/03/01 · Zbl 0885.73044
[5] Aifantis, E. C., Strain gradient interpretation of size effects, (Bažant, Z. P.; Rajapakse, Y. D.S., Fracture scaling (1999), Springer: Springer DordrechtNetherlands), 299-314
[6] Gurtin, M. E.; Murdoch, A. Ian, A continuum theory of elastic material surfaces, Archive for Rational Mechanics and Analysis, 57, 291-323 (1975), 1975/12/01 · Zbl 0326.73001
[7] Mindlin, R.; Tiersten, H., Effects of couple-stresses in linear elasticity (1962), Columbia Univ New York · Zbl 0112.38906
[8] R. A. Toupin, “Theories of elasticity with couple-stress,” 1964. · Zbl 0131.22001
[9] Sahmani, S.; Fattahi, A., An anisotropic calibrated nonlocal plate model for biaxial instability analysis of 3D metallic carbon nanosheets using molecular dynamics simulations, Materials Research Express, 4, Article 065001 pp. (2017)
[10] Huang, L. Y.; Han, Q.; Liang, Y. J., Calibration of nonlocal scale effect parameter for bending single-layered graphene sheet under molecular dynamics, Nano, 07, Article 1250033 pp. (2012)
[11] Mehralian, F.; Beni, Y. Tadi; Karimi Zeverdejani, M., Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations, Physica B: Condensed Matter, 521, 102-111 (2017), 2017/09/15/
[12] Mohammadi, K.; Rajabpour, A.; Ghadiri, M., Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation, Computational Materials Science, 148, 104-115 (2018), 2018/06/01/
[13] Ansari, R.; Rouhi, H.; Sahmani, S., Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics, International Journal of Mechanical Sciences, 53, 786-792 (2011), 2011/09/01/
[14] Sahmani, S.; Fattahi, A. M., Development an efficient calibrated nonlocal plate model for nonlinear axial instability of zirconia nanosheets using molecular dynamics simulation, Journal of Molecular Graphics and Modelling, 75, 20-31 (2017), 2017/08/01/
[15] Mehralian, F.; Beni, Y. Tadi; Karimi Zeverdejani, M., Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes, Physica B: Condensed Matter, 514, 61-69 (2017), 2017/06/01/
[16] Ghorbani, K.; Rajabpour, A.; Ghadiri, M., Determination of carbon nanotubes size-dependent parameters: Molecular dynamics simulation and nonlocal strain gradient continuum shell model, Mechanics Based Design of Structures and Machines, 49, 103-120 (2021), 2021/01/02
[17] Nazemnezhad, R.; Hosseini-Hashemi, S., Free vibration analysis of multi-layer graphene nanoribbons incorporating interlayer shear effect via molecular dynamics simulations and nonlocal elasticity, Physics Letters A, 378, 3225-3232 (2014), 2014/09/12/
[18] Sahmani, S.; Fattahi, A. M., Development of efficient size-dependent plate models for axial buckling of single-layered graphene nanosheets using molecular dynamics simulation, Microsystem Technologies, 24, 1265-1277 (2018), 2018/02/01
[19] Islam, M.; Thakur, M. S.H.; Mojumder, S.; Al Amin, A.; Islam, M. M., Mechanical and vibrational characteristics of functionally graded Cu-Ni nanowire: A molecular dynamics study, Composites Part B: Engineering, 198, Article 108212 pp. (2020), /10/01/2020
[20] Ansari, R.; Ajori, S.; Motevalli, B., Mechanical properties of defective single-layered graphene sheets via molecular dynamics simulation, Superlattices and Microstructures, 51, 274-289 (2012), /02/01/2012
[21] Khademolhosseini, F.; Phani, A. S.; Nojeh, A.; Rajapakse, N., Nonlocal continuum modeling and molecular dynamics simulation of torsional vibration of carbon nanotubes, IEEE Transactions on Nanotechnology, 11, 34-43 (2012)
[22] Mählich, D.; Eberhardt, O.; Wallmersperger, T., Numerical simulation of the mechanical behavior of a carbon nanotube bundle, Acta Mechanica, 232, 483-494 (2021), 2021/02/01
[23] Chen, W.-J.; Chang, I. L., The atomistic study on thermal transport of the branched Cnt, Journal of Mechanics, 36, 721-727 (2020)
[24] Ajori, S.; Ansari, R.; Haghighi, S., Vibration characteristics of three-dimensional metallic carbon nanostructures with interlocking hexagons pattern (T6 and T14): A molecular dynamics study, Computational Materials Science, 128, 81-86 (2017), /02/15/2017
[25] Sadeghzadeh, S., Wrinkling C3N nano-grids in uniaxial tensile testing; A molecular dynamics study, Diamond and Related Materials, 92, 130-137 (2019), /02/01/2019
[26] Hemmasizadeh, A.; Mahzoon, M.; Hadi, E.; Khandan, R., A method for developing the equivalent continuum model of a single layer graphene sheet, Thin Solid Films, 516, 7636-7640 (2008), /09/01/2008
[27] Hosseini-Hashemi, S.; Sepahi-Boroujeni, A.; Sepahi-Boroujeni, S., Analytical and molecular dynamics studies on the impact loading of single-layered graphene sheet by fullerene, Applied Surface Science, 437, 366-374 (2018), /04/15/2018
[28] Azizi, B.; Rezaee, S.; Hadianfard, M. J.; Dehnou, K. H., A comprehensive study on the mechanical properties and failure mechanisms of graphyne nanotubes (GNTs) in different phases, Computational Materials Science, 182, Article 109794 pp. (2020), /09/01/2020
[29] Ghafouri Pourkermani, A.; Azizi, B.; Nejat Pishkenari, H., Vibrational analysis of Ag, Cu and Ni nanobeams using a hybrid continuum-atomistic model, International Journal of Mechanical Sciences, 165, Article 105208 pp. (2020), /01/01/2020
[30] Ebrahimi, F.; Dabbagh, A.; Rabczuk, T., On wave dispersion characteristics of magnetostrictive sandwich nanoplates in thermal environments, European Journal of Mechanics - A/Solids, 85, Article 104130 pp. (2021), /01/01/2021 · Zbl 1476.74072
[31] Abdelrahman, A. A.; Eltaher, M. A., On bending and buckling responses of perforated nanobeams including surface energy for different beams theories, Engineering with Computers (2020), /11/20 2020
[32] Jena, S. K.; Chakraverty, S.; Jena, R. M.; Tornabene, F., A novel fractional nonlocal model and its application in buckling analysis of Euler-Bernoulli nanobeam, Materials Research Express, 6, Article 055016 pp. (2019)
[33] Akbaş, Ş. D., Axially forced vibration analysis of cracked a nanorod, Journal of Computational Applied Mechanics, 50, 63-68 (2019)
[34] Pal, S.; Das, D., Free vibration behavior of rotating bidirectional functionally-graded micro-disk for flexural and torsional modes in thermal environment, International Journal of Mechanical Sciences, 179, Article 105635 pp. (2020), /08/01/2020
[35] Hosseini, S. M., A GN-based modified model for size-dependent coupled thermoelasticity analysis in nano scale, considering nonlocality in heat conduction and elasticity: An analytical solution for a nano beam with energy dissipation, Structural Engineering and Mechanics, 73, 287-302 (2020)
[36] Li, L.; Pan, Y.; Arabmarkadeh, A., Nonlinear finite element study on forced vibration of cylindrical micro-panels based on modified strain gradient theory, Mechanics of Advanced Materials and Structures, 1-20 (2021)
[37] Darban, H.; Fabbrocino, F.; Luciano, R., Size-dependent linear elastic fracture of nanobeams, International Journal of Engineering Science, 157, Article 103381 pp. (2020), /12/01/2020 · Zbl 07278783
[38] Adhikari, S.; Karličić, D.; Liu, X., Dynamic stiffness of nonlocal damped nano-beams on elastic foundation, European Journal of Mechanics - A/Solids, 86, Article 104144 pp. (2021), /03/01/2021 · Zbl 1478.74031
[39] Gul, U.; Aydogdu, M., A micro/nano-scale Timoshenko-Ehrenfest beam model for bending, buckling and vibration analyses based on doublet mechanics theory, European Journal of Mechanics - A/Solids, 86, Article 104199 pp. (2021), /03/01/2021 · Zbl 1479.74074
[40] Sladek, J.; Sladek, V.; Hosseini, S. M., Analysis of a curved Timoshenko nano-beam with flexoelectricity, Acta Mechanica, 232, 1563-1581 (2021), /04/01 2021 · Zbl 1475.74084
[41] Quoc, T. H.; Van Tham, V.; Tu, T. M., Active vibration control of a piezoelectric functionally graded carbon nanotube-reinforced spherical shell panel, Acta Mechanica, 232, 1005-1023 (2021), /03/01 2021 · Zbl 1476.74111
[42] Gupta, A.; Soni, S.; Jain, N. K., A non-classical analytical approach for vibration analysis of isotropic and Fgm plate containing a star shaped crack, Journal of Mechanics, 36, 465-484 (2020)
[43] Vaccaro, M. S.; Pinnola, F. P.; de Sciarra, F. M.; Canadija, M.; Barretta, R., Stress-driven two-phase integral elasticity for Timoshenko curved beams, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, 0, Article 2397791421990514 pp. (2020)
[44] Romano, G.; Barretta, R., Nonlocal elasticity in nanobeams: The stress-driven integral model, International Journal of Engineering Science, 115, 14-27 (2017), 06/01/2017 · Zbl 1423.74512
[45] Romano, G.; Barretta, R., Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composites Part B: Engineering, 114, 184-188 (2017), 04/01/2017
[46] Apuzzo, A.; Barretta, R.; Luciano, R.; Marotti de Sciarra, F.; Penna, R., Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model, Composites Part B: Engineering, 123, 105-111 (2017), 08/15/2017
[47] Darban, H.; Luciano, R.; Caporale, A.; Fabbrocino, F., Higher modes of buckling in shear deformable nanobeams, International Journal of Engineering Science, 154, Article 103338 pp. (2020), /09/01/2020 · Zbl 07228669
[48] Russillo, A. F.; Failla, G.; Alotta, G.; Marotti de Sciarra, F.; Barretta, R., On the dynamics of nano-frames, International Journal of Engineering Science, 160, Article 103433 pp. (2021), /03/01/2021 · Zbl 07314432
[49] Farajpour, A.; Howard, C. Q.; Robertson, W. S.P., On size-dependent mechanics of nanoplates, International Journal of Engineering Science, 156, Article 103368 pp. (2020), /11/01/2020 · Zbl 07261125
[50] Hosseini, M.; Shishesaz, M.; Tahan, K. N.; Hadi, A., Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, 109, 29-53 (2016), /12/01/2016 · Zbl 1423.74019
[51] Shishesaz, M.; Hosseini, M.; Tahan, K. N.; Hadi, A., Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, 228, 4141-4168 (2017), 2017/12/01 · Zbl 1380.74096
[52] Hosseini, M.; Shishesaz, M.; Hadi, A., Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures, 134, 508-523 (2019), /01/01/2019
[53] Hosseini, M.; Gorgani, H. H.; Shishesaz, M.; Hadi, A., Size-dependent stress analysis of single-wall carbon nanotube based on strain gradient theory, International Journal of Applied Mechanics, 09, Article 1750087 pp. (2017), 2017/09/01
[54] Mohammadi, M.; Hosseini, M.; Shishesaz, M.; Hadi, A.; Rastgoo, A., Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, European Journal of Mechanics - A/Solids, 77, Article 103793 pp. (2019), /09/01/2019 · Zbl 1475.74056
[55] Shishesaz, M.; Hosseini, M., Mechanical behavior of functionally graded nano-cylinders under radial pressure based on strain gradient theory, Journal of Mechanics, 35, 441-454 (2018)
[56] Al-Furjan, M. S.H.; Moghadam, S. A.; Dehini, R.; Shan, L.; Habibi, M.; Safarpour, H., Vibration control of a smart shell reinforced by graphene nanoplatelets under external load: Semi-numerical and finite element modeling, Thin-Walled Structures, 159, Article 107242 pp. (2021), /02/01/2021
[57] Soltani, M.; Atoufi, F.; Mohri, F.; Dimitri, R.; Tornabene, F., Nonlocal elasticity theory for lateral stability analysis of tapered thin-walled nanobeams with axially varying materials, Thin-Walled Structures, 159, Article 107268 pp. (2021), /02/01/2021
[58] Miandoab, E. Maani, Effect of surface on nano-beam mechanical behaviors: A parametric analysis, Microsystem Technologies, 27, 665-672 (2021), /03/01 2021
[59] Ceballes, S.; Abdelkefi, A., Observations on the general nonlocal theory applied to axially loaded nanobeams, Microsystem Technologies, 27, 739-761 (2021), /03/01 2021
[60] Shishesaz, M.; Shariati, M.; Yaghootian, A., Nonlocal elasticity effect on linear vibration of nano-circular plate using Adomian decomposition method, Journal of Applied and Computational Mechanics, 6, 63-76 (2020)
[61] Eremeyev, V. A.; Sharma, B. L., Anti-plane surface waves in media with surface structure: Discrete vs. continuum model, International Journal of Engineering Science, 143, 33-38 (2019), /10/01/2019 · Zbl 1476.74076
[62] Eremeyev, V. A.; Rosi, G.; Naili, S., Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses, Mathematics and Mechanics of Solids, 24, 2526-2535 (2019) · Zbl 07254367
[63] Eremeyev, V. A.; Ganghoffer, J.-F.; Konopińska-Zmysłowska, V.; Uglov, N. S., Flexoelectricity and apparent piezoelectricity of a pantographic micro-bar, International Journal of Engineering Science, 149, Article 103213 pp. (2020), /04/01/2020 · Zbl 07261101
[64] Huang, Y.; Chen, J.; Zhao, M.; Feng, M., Responses of multilayered two-dimensional decagonal quasicrystal circular nanoplates with initial stresses and nanoscale interactions, European Journal of Mechanics - A/Solids, 87, Article 104216 pp. (2021), /05/01/2021 · Zbl 1484.74054
[65] Malikan, M.; Uglov, N. S.; Eremeyev, V. A., On instabilities and post-buckling of piezomagnetic and flexomagnetic nanostructures, International Journal of Engineering Science, 157, Article 103395 pp. (2020), /12/01/2020 · Zbl 07278788
[66] Darban, H.; Caporale, A.; Luciano, R., Nonlocal layerwise formulation for bending of multilayered/functionally graded nanobeams featuring weak bonding, European Journal of Mechanics - A/Solids, 86, Article 104193 pp. (2021), /03/01/2021 · Zbl 1479.74071
[67] Huang, M.; Zheng, X.; Zhou, C.; An, D.; Li, R., On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free, Acta Mechanica, 232, 495-513 (2021), /02/01 2021 · Zbl 1458.74095
[68] Khoram, M. M.; Hosseini, M.; Hadi, A.; Shishehsaz, M., Bending analysis of bidirectional FGM timoshenko nanobeam subjected to mechanical and magnetic forces and resting on Winkler-Pasternak foundation, International Journal of Applied Mechanics, 12, Article 2050093 pp. (2020)
[69] Faroughi, S.; Sari, M. S.; Abdelkefi, A., Nonlocal Timoshenko representation and analysis of multi-layered functionally graded nanobeams, Microsystem Technologies, 27, 893-911 (2021), 2021/03/01
[70] Li, L.; Lin, R.; Ng, T. Y., Contribution of nonlocality to surface elasticity, International Journal of Engineering Science, 152, Article 103311 pp. (2020), /07/01/2020 · Zbl 07205509
[71] Boni, C.; Royer-Carfagni, G., A nonlocal elastica inspired by flexural tensegrity, International Journal of Engineering Science, 158, Article 103421 pp. (2021), /01/01/2021 · Zbl 07278802
[72] Varmazyari, S.; Shokrollahi, H., Analytical solution for strain gradient plasticity of rotating functionally graded thick cylinders, International Journal of Applied Mechanics, 12, Article 2050082 pp. (2020), /08/01 2020
[73] Torabi, J.; Niiranen, J.; Ansari, R., Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory, European Journal of Mechanics - A/Solids, 87, Article 104221 pp. (2021), /05/01/2021 · Zbl 1484.74073
[74] Xu, X.; Karami, B.; Shahsavari, D., Time-dependent behavior of porous curved nanobeam, International Journal of Engineering Science, 160, Article 103455 pp. (2021), /03/01/2021 · Zbl 07314436
[75] Hadi, A.; Nejad, M. Z.; Hosseini, M., Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, 128, 12-23 (2018), /07/01/2018 · Zbl 1423.74394
[76] She, G.-L.; Liu, H.-B.; Karami, B., Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets, Thin-Walled Structures, 160, Article 107407 pp. (2021), /03/01/2021
[77] Eremeyev, V. A.; Cazzani, A.; dell’Isola, F., On nonlinear dilatational strain gradient elasticity, Continuum Mechanics and Thermodynamics (2021), /03/08 2021
[78] Hadi, A.; Nejad, M. Z.; Rastgoo, A.; Hosseini, M., Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, 26, 663-672 (2018)
[79] Haghshenas Gorgani, H.; Mahdavi Adeli, M.; Hosseini, M., Pull-in behavior of functionally graded micro/nano-beams for MEMS and NEMS switches, Microsystem Technologies, 25, 3165-3173 (2019), /08/01 2019
[80] Liu, C.; Yu, J.; Xu, W.; Zhang, X.; Wang, X., Dispersion characteristics of guided waves in functionally graded anisotropic micro/nano-plates based on the modified couple stress theory, Thin-Walled Structures, 161, Article 107527 pp. (2021), /04/01/2021
[81] Dehrouyeh-Semnani, A. M.; Mostafaei, H., On the mechanics of microshells of revolution, International Journal of Engineering Science, 161, Article 103450 pp. (2021), /04/01/2021 · Zbl 1497.74055
[82] Xu, X.; Karami, B.; Janghorban, M., On the dynamics of nanoshells, International Journal of Engineering Science, 158, Article 103431 pp. (2021), /01/01/2021 · Zbl 07278807
[83] Salari, E.; Sadough Vanini, S. A., Investigation of thermal preloading and porosity effects on the nonlocal nonlinear instability of FG nanobeams with geometrical imperfection, European Journal of Mechanics - A/Solids, 86, Article 104183 pp. (2021), /03/01/2021 · Zbl 1479.74037
[84] Adeli, M. M.; Hadi, A.; Hosseini, M.; Gorgani, H. H., Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory, The European Physical Journal Plus, 132, 393 (2017), /09/18 2017
[85] Mousavi Khoram, M.; Hosseini, M.; Shishesaz, M., A concise review of nano-plates, Journal of Computational Applied Mechanics, 50, 420-429 (2019)
[86] Hosseini, M.; Hadi, A.; Malekshahi, A.; Shishesaz, M., A review of size-dependent elasticity for nanostructures, Journal of Computational Applied Mechanics, 49, 197-211 (2018)
[87] Wei, H.; Mohammadi, R., Hygro-thermo-mechanical bending and vibration analysis of the CNTRC doubly curved nanoshells with thickness stretching based on nonlocal strain gradient theory, The European Physical Journal Plus, 136, 326 (2021), /03/19 2021
[88] Zenkour, A. M.; Radwan, A. F., A compressive study for porous FG curved nanobeam under various boundary conditions via a nonlocal strain gradient theory, The European Physical Journal Plus, 136, 248 (2021), /02/22 2021
[89] Fan, F.; Safaei, B.; Sahmani, S., Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis, Thin-Walled Structures, 159, Article 107231 pp. (2021), /02/01/2021
[90] Malikan, M.; Krasheninnikov, M.; Eremeyev, V. A., Torsional stability capacity of a nano-composite shell based on a nonlocal strain gradient shell model under a three-dimensional magnetic field, International Journal of Engineering Science, 148, Article 103210 pp. (2020), /03/01/2020 · Zbl 07167846
[91] Faghidian, S. A., Higher-order nonlocal gradient elasticity: A consistent variational theory, International Journal of Engineering Science, 154, Article 103337 pp. (2020), /09/01/2020 · Zbl 07228668
[92] Pinnola, F. P.; Faghidian, S. A.; Barretta, R.; Marotti de Sciarra, F., Variationally consistent dynamics of nonlocal gradient elastic beams, International Journal of Engineering Science, 149, Article 103220 pp. (2020), /04/01/2020 · Zbl 1476.74091
[93] Tong, L. H.; Ding, H. B.; Yan, J. W.; Xu, C.; Lei, Z., Strain gradient nonlocal Biot poromechanics, International Journal of Engineering Science, 156, Article 103372 pp. (2020), /11/01/2020 · Zbl 07261129
[94] Karami, B.; Janghorban, M., On the mechanics of functionally graded nanoshells, International Journal of Engineering Science, 153, Article 103309 pp. (2020), /08/01/2020 · Zbl 07228650
[95] Eyvazian, A.; Shahsavari, D.; Karami, B., On the dynamic of graphene reinforced nanocomposite cylindrical shells subjected to a moving harmonic load, International Journal of Engineering Science, 154, Article 103339 pp. (2020), /09/01/2020 · Zbl 07228670
[96] Yuan, Y.; Xu, K., Postbuckling analysis of axially loaded nanoscaled shells embedded in elastic foundations based on Ru’s surface elasticity theory, Mechanics Based Design of Structures and Machines, 49, 20-40 (2021), /01/02 2021
[97] Zheng, C.; Zhang, G.; Mi, C., On the strength of nanoporous materials with the account of surface effects, International Journal of Engineering Science, 160, Article 103451 pp. (2021), /03/01/2021 · Zbl 07314433
[98] Igumnov, L. A.; Eremeyev, V. A.; Ipatov, A. A.; Bragov, A. M., Surface elasticity for applications to material modelling at small scales, (14th WCCM-ECCOMAS Congress (2020)), 2021
[99] Chaki, M. S.; Eremeyev, V. A.; Singh, A. K., Surface and interfacial anti-plane waves in micropolar solids with surface energy, Mathematics and Mechanics of Solids, 26, 708-721 (2021) · Zbl 07357424
[100] Dabbagh, A.; Rastgoo, A.; Ebrahimi, F., Thermal buckling analysis of agglomerated multiscale hybrid nanocomposites via a refined beam theory, Mechanics Based Design of Structures and Machines, 49, 403-429 (2021), /04/03 2021
[101] Kamarian, S.; Bodaghi, M.; Isfahani, R. B.; Shakeri, M.; Yas, M. H., Influence of carbon nanotubes on thermal expansion coefficient and thermal buckling of polymer composite plates: Experimental and numerical investigations, Mechanics Based Design of Structures and Machines, 49, 217-232 (2021), /02/17 2021
[102] Saadatfar, M.; Zarandi, M. H.; Babaelahi, M., Effects of porosity, profile of thickness, and angular acceleration on the magneto-electro-elastic behavior of a porous FGMEE rotating disc placed in a constant magnetic field, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Article 0954406220938409 pp. (2020)
[103] Haskul, M., Elastic state of functionally graded curved beam on the plane stress state subject to thermal load, Mechanics Based Design of Structures and Machines, 48, 739-754 (2020), 2020/11/01
[104] Bathini, S. R.; Reddy K, V. K.; Ankanna B, C., Free vibration behavior of bi-directional functionally graded plates with porosities using a refined first order shear deformation theory, Journal of Computational Applied Mechanics, 51, 374-388 (2020)
[105] Ma, X.; Wang, S.; Zhou, B.; Xue, S., Study on electromechanical behavior of functionally graded piezoelectric composite beams, Journal of Mechanics, 36, 841-848 (2020)
[106] Salem, T.; Xie, X.; Jiao, P.; Lajnef, N., Maneuverable postbuckling of extensible mechanical metamaterials using functionally graded materials and carbon nanotubes, Thin-Walled Structures, 159, Article 107264 pp. (2021), /02/01/2021
[107] Dastjerdi, S.; Akgöz, B.; Civalek, Ö., On the effect of viscoelasticity on behavior of gyroscopes, International Journal of Engineering Science, 149, Article 103236 pp. (2020), /04/01/2020 · Zbl 1476.74016
[108] Vogl, G. W.; Nayfeh, A. H., A reduced-order model for electrically actuated clamped circular plates, (Proceedings of the international design engineering technical conferences and computers and information in engineering conference (2003)), 1867-1874
[109] Plimpton, S.; Crozier, P.; Thompson, A., LAMMPS-large-scale atomic/molecular massively parallel simulator, Sandia National Laboratories, 18, 43 (2007)
[110] Kınacı, A.; Haskins, J. B.; Sevik, C.; Çağın, T., Thermal conductivity of BN-C nanostructures, Physical Review B, 86, Article 115410 pp. (2012)
[111] Stuart, S. J.; Tutein, A. B.; Harrison, J. A., A reactive potential for hydrocarbons with intermolecular interactions, The Journal of Chemical Physics, 112, 6472-6486 (2000)
[112] Loulijat, H.; Koumina, A.; Zerradi, H., The effect of the thermal vibration of graphene nanosheets on viscosity of nanofluid liquid argon containing graphene nanosheets, Journal of Molecular Liquids, 276, 936-946 (2019)
[113] Sajadi, B.; van Hemert, S.; Arash, B.; Belardinelli, P.; Steeneken, P. G.; Alijani, F., Size-and temperature-dependent bending rigidity of graphene using modal analysis, Carbon, 139, 334-341 (2018)
[114] Klessig, R.; Polak, E., Efficient implementations of the Polak-Ribière conjugate gradient algorithm, SIAM Journal on Control, 10, 524-549 (1972) · Zbl 0244.90034
[115] Hoover, W. G., Canonical dynamics: Equilibrium phase-space distributions, Physical review A, 31, 1695 (1985)
[116] Rafii-Tabar, H., Modelling the nano-scale phenomena in condensed matter physics via computer-based numerical simulations, Physics Reports, 325, 239-310 (2000)
[117] Sajadi, B.; Wahls, S.; van Hemert, S.; Belardinelli, P.; Steeneken, P. G.; Alijani, F., Nonlinear dynamic identification of graphene’s elastic modulus via reduced order modeling of atomistic simulations, Journal of the Mechanics and Physics of Solids, 122, 161-176 (2019)
[118] Thomas, M.; Brehm, M.; Fligg, R.; Vöhringer, P.; Kirchner, B., Computing vibrational spectra from ab initio molecular dynamics, Physical Chemistry Chemical Physics, 15, 6608-6622 (2013)
[119] Kitipornchai, S.; He, X. Q.; Liew, K. M., Continuum model for the vibration of multilayered graphene sheets, Physical Review B, 72, Article 075443 pp. (2005), 08/29/
[120] Ansari, R.; Sahmani, S.; Arash, B., Nonlocal plate model for free vibrations of single-layered graphene sheets, Physics Letters A, 375, 53-62 (2010), /11/15/2010
[121] Leissa, A. W., Vibration of plates vol. 160 (1969), Scientific and Technical Information Division, National Aeronautics and Space Administration
[122] Timoshenko, S. P.; Woinowsky-Krieger, S., Theory of plates and shells (1959), McGraw-hill · Zbl 0114.40801
[123] Barretta, R.; Faghidian, S. A.; De Sciarra, F. M., Stress-driven nonlocal integral elasticity for axisymmetric nano-plates, International Journal of Engineering Science, 136, 38-52 (2019) · Zbl 1425.74055
[124] Shishesaz, M.; Shariati, M.; Yaghootian, A.; Alizadeh, A., Nonlinear vibration analysis of nano-disks based on nonlocal elasticity theory using homotopy perturbation method, International Journal of Applied Mechanics, 11, Article 1950011 pp. (2019)
[125] Lim, C.; Zhang, G.; Reddy, J., A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, 78, 298-313 (2015) · Zbl 1349.74016
[126] Wu, T.; Liu, G., Application of generalized differential quadrature rule to sixth-order differential equations, Communications in Numerical Methods in Engineering, 16, 777-784 (2000) · Zbl 0969.65070
[127] Wu, T.; Liu, G., The generalized differential quadrature rule for fourth-order differential equations, International Journal for Numerical Methods in Engineering, 50, 1907-1929 (2001) · Zbl 0999.74120
[128] Quan, J.; Chang, C., New insights in solving distributed system equations by the quadrature method—I. Analysis, Computers & Chemical Engineering, 13, 779-788 (1989)
[129] Shu, C.; Richards, B. E., Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 15, 791-798 (1992) · Zbl 0762.76085
[130] Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M. P., Optimization by simulated annealing, Science, 220, 671-680 (1983) · Zbl 1225.90162
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