Abstract
According to paradoxical behaviors of common differential nonlocal elasticity, employing the two-phase local/nonlocal elasticity, to consider the size effects of nanostructures, has recently attracted the attentions of nano-mechanics researchers. Now, due to more complexity of the two-phase elasticity problems than the differential nonlocal ones, it is essential to achieve efficient methods for studying the mechanical characteristics of two-phase nanostructures. Therefore, in this work, the exact solution corresponding to the vibrations of two-phase Timoshenko nanobeams is provided for the first time. Furthermore, the shear-locking problem is investigated in the case of two-phase finite-element method (FEM), and since the FE model of local/nonlocal nanobeam is more complex than the classic one, due to coupling of all elements together, one of the main aims of the present work is to create an efficient locking-free local/nonlocal FEM with a simple and efficient beam element. To extract the exact natural frequencies, the basic form of two-phase elasticity is replaced with the equal differential equation and the obtained higher-order governing equations are solved by satisfying additional constitutive boundary conditions. To construct the two-phase FE model, an efficient and simple shear-locking-free Timoshenko beam element is introduced, and next, basic form of two-phase elasticity including local and integral form of nonlocal elasticity is utilized. No need for satisfying higher-order boundary conditions, shear-locking-free, simple shape functions and well convergence are advantages of the present two-phase finite element model. Several convergence and comparison studies are conducted, and the reliability and locking-free properties of the present two-phase finite element model are confirmed. Also, the influences of two-phase elasticity on the natural frequencies of Timoshenko nanobeams with different thickness ratios are studied.
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References
Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1–16
Eringen AC, Edelen D (1972) On nonlocal elasticity. Int J Eng Sci 10(3):233–248
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710
Eringen AC (2002) Nonlocal continuum field theories. Springer, Berlin
Reddy J (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2):288–307
Ece M, Aydogdu M (2007) Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta Mech 190(1):185–195
Thai H-T (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci 52:56–64
Eltaher M, Emam SA, Mahmoud F (2012) Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput 218(14):7406–7420
Ebrahimi F, Barati MR, Civalek Ö (2019) Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures. Eng Comput 2019:1–12
Karami B, Janghorban M, Tounsi A (2019) Galerkin’s approach for buckling analysis of functionally graded anisotropic nanoplates/different boundary conditions. Eng Comput 35(4):1297–1316
Sahmani S, Fattahi A, Ahmed N (2019) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Comput 35(4):1173–1189
Lu P, Lee H, Lu C, Zhang P (2006) Dynamic properties of flexural beams using a nonlocal elasticity model. J Appl Phys 99(7):073510
Wang C, Zhang Y, He X (2007) Vibration of nonlocal Timoshenko beams. Nanotechnology 18(10):105401
Thai S, Thai H-T, Vo TP, Patel VI (2017) A simple shear deformation theory for nonlocal beams. Compos Struct 23:303
Phadikar J, Pradhan S (2010) Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Comput Mater Sci 49(3):492–499
Challamel N, Zhang Z, Wang C, Reddy J, Wang Q, Michelitsch T et al (2014) On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach. Arch Appl Mech 84(9–11):1275–1292
Xu X-J, Deng Z-C, Zhang K, Xu W (2016) Observations of the softening phenomena in the nonlocal cantilever beams. Compos Struct 145:43–57
Fernández-Sáez J, Zaera R, Loya J, Reddy J (2016) Bending of Euler-Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116
Polyanin AD, Manzhirov AV (2008) Handbook of integral equations. CRC Press, Baco Raton
Tuna M, Kirca M (2016) Exact solution of Eringen's nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams. Int J Eng Sci 105:80–92
Tuna M, Kirca M (2016) Exact solution of Eringen's nonlocal integral model for vibration and buckling of Euler-Bernoulli beam. Int J Eng Sci 107:54–67
Romano G, Barretta R (2016) Comment on the paper “Exact solution of Eringen’s nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams” by Meral Tuna & Mesut Kirca. Int J Eng Sci 109:240–242
Wang Y, Zhu X, Dai H (2016) Exact solutions for the static bending of Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. AIP Adv 6(8):085114
Tuna M, Kirca M (2017) Respond to the comment letter by Romano and Barretta on the paper “Exact solution of Eringen's nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams”. Int J Eng Sci 116:141–144
Romano G, Barretta R, Diaco M, de Sciarra FM (2017) Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int J Mech Sci 121:151–156
Ansari R, Rajabiehfard R, Arash B (2010) Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput Mater Sci 49(4):831–838
Mahmoud F, Eltaher M, Alshorbagy A, Meletis E (2012) Static analysis of nanobeams including surface effects by nonlocal finite element. J Mech Sci Technol 26(11):3555–3563
Pradhan S (2012) Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory. Finite Elem Anal Des 50:8–20
Arash B, Wang Q, Liew KM (2012) Wave propagation in graphene sheets with nonlocal elastic theory via finite element formulation. Comput Methods Appl Mech Eng 223:1–9
Eltaher M, Emam SA, Mahmoud F (2013) Static and stability analysis of nonlocal functionally graded nanobeams. Compos Struct 96:82–88
Civalek Ö, Demir C (2016) A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl Math Comput 289:335–352
Demir Ç, Civalek Ö (2017) A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos Struct 168:872–884
Aria A, Friswell M (2019) A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos B Eng 166:233–246
Pisano A, Sofi A, Fuschi P (2009) Nonlocal integral elasticity: 2D finite element based solutions. Int J Solids Struct 46(21):3836–3849
Taghizadeh M, Ovesy H, Ghannadpour S (2016) Beam buckling analysis by nonlocal integral elasticity finite element method. Int J Struct Stab Dyn 16(06):1550015
Khodabakhshi P, Reddy J (2015) A unified integro-differential nonlocal model. Int J Eng Sci 95:60–75
Tuna M, Kirca M (2017) Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Compos Struct 179:269–284
Eptaimeros K, Koutsoumaris CC, Tsamasphyros G (2016) Nonlocal integral approach to the dynamical response of nanobeams. Int J Mech Sci 115:68–80
Naghinejad M, Ovesy HR (2017) Free vibration characteristics of nanoscaled beams based on nonlocal integral elasticity theory. J Vib Control. 2017:1077546317717867
Fakher M, Rahmanian S, Hosseini-Hashemi S (2019) On the carbon nanotube mass nanosensor by integral form of nonlocal elasticity. Int J Mech Sci 150:445–457
Norouzzadeh A, Ansari R (2017) Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Physica E 88:194–200
Norouzzadeh A, Ansari R, Rouhi H (2017) Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach. Appl Phys A 123(5):330
Norouzzadeh A, Ansari R, Rouhi H (2018) Isogeometric vibration analysis of small-scale Timoshenko beams based on the most comprehensive size-dependent theory. Sci Iran 25(3):1864–1878
Norouzzadeh A, Ansari R, Rouhi H (2018) Isogeometric analysis of Mindlin nanoplates based on the integral formulation of nonlocal elasticity. Multidisc Model Mater Struct 14(5):810–827
Ansari R, Torabi J, Norouzzadeh A (2018) Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method. Phys B 534:90–97
Faraji-Oskouie M, Norouzzadeh A, Ansari R, Rouhi H (2019) Bending of small-scale Timoshenko beams based on the integral/differential nonlocal-micropolar elasticity theory: a finite element approach. Appl Math Mech 40(6):767–782
Zhu X, Li L (2017) Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity. Int J Mech Sci 133:639–650
Fernández-Sáez J, Zaera R (2017) Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory. Int J Eng Sci 119:232–248
Zhu X, Wang Y, Dai H-H (2017) Buckling analysis of Euler-Bernoulli beams using Eringen’s two-phase nonlocal model. Int J Eng Sci 116:130–140
Eptaimeros K, Koutsoumaris CC, Dernikas I, Zisis T (2018) Dynamical response of an embedded nanobeam by using nonlocal integral stress models. Compos B Eng 150:255–268
Wang Y, Huang K, Zhu X, Lou Z (2018) Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model. Math Mech Solids. 2018:1081286517750008
Draiche K, Bousahla AA, Tounsi A, Alwabli AS, Tounsi A, Mahmoud S (2019) Static analysis of laminated reinforced composite plates using a simple first-order shear deformation theory. Comput Concr 24(4):369–378
Semmah A, Heireche H, Bousahla AA, Tounsi A (2019) Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT. Adv Nano Res 7(2):89
Draoui A, Zidour M, Tounsi A, Adim B (2019) Static and dynamic behavior of nanotubes-reinforced sandwich plates using (FSDT). J Nano Res 2019:117–135
Khiloun M, Bousahla AA, Kaci A, Bessaim A, Tounsi A, Mahmoud S (2019) Analytical modeling of bending and vibration of thick advanced composite plates using a four-variable quasi 3D HSDT. Eng Compute. 2019:1–15
Zaoui FZ, Ouinas D, Tounsi A (2019) New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Compos B Eng 159:231–247
Karami B, Janghorban M, Tounsi A (2019) On pre-stressed functionally graded anisotropic nanoshell in magnetic field. J Braz Soc Mecha Sci Eng 41(11):495
Patel M, Shabana AA (2018) Locking alleviation in the large displacement analysis of beam elements: the strain split method. Acta Mech 229(7):2923–2946
Zienkiewicz O, Owen D, Lee K (1974) Least square-finite element for elasto-static problems Use of ‘reduced’integration. Int J Numer Methods Eng 8(2):341–358
Taylor R, Filippou F, Saritas A, Auricchio F (2003) A mixed finite element method for beam and frame problems. Comput Mech 31(1–2):192–203
Heyliger P, Reddy J (1988) A higher order beam finite element for bending and vibration problems. J Sound Vib 126(2):309–326
Lee PG, Sin HC (1994) Locking-free curved beam element based on curvature. Int J Numer Meth Eng 37(6):989–1007
Thomas D, Wilson J, Wilson R (1973) Timoshenko beam finite elements. J Sound Vib 31(3):315–330
Zhuang X, Huang R, Zhu H, Askes H, Mathisen K (2013) A new and simple locking-free triangular thick plate element using independent shear degrees of freedom. Finite Elem Anal Des 75:1–7
Monterrubio L, Ilanko S (2012) Sets of admissible functions for the Rayleigh-Ritz method. Civ Comp Proc 99:2012
Fakher M, Hosseini-Hashemi S (2017) Bending and free vibration analysis of nanobeams by differential and integral forms of nonlocal strain gradient with Rayleigh-Ritz method. Mater Res Express 4(12):125025
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Fakher, M., Hosseini-Hashemi, S. Vibration of two-phase local/nonlocal Timoshenko nanobeams with an efficient shear-locking-free finite-element model and exact solution. Engineering with Computers 38, 231–245 (2022). https://doi.org/10.1007/s00366-020-01058-z
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DOI: https://doi.org/10.1007/s00366-020-01058-z