Browse > Article
http://dx.doi.org/10.12989/scs.2018.26.6.663

Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory  

Hadi, Amin (School of Mechanical Engineering, University of Tehran)
Nejad, Mohammad Zamani (Department of Mechanical Engineering, Yasouj University)
Rastgoo, Abbas (School of Mechanical Engineering, University of Tehran)
Hosseini, Mohammad (Department of Mechanical Engineering, Shahid Chamran University of Ahvaz)
Publication Information
Steel and Composite Structures / v.26, no.6, 2018 , pp. 663-672 More about this Journal
Abstract
This paper contains a consistent couple-stress theory to capture size effects in Euler-Bernoulli nano-beams made of three-directional functionally graded materials (TDFGMs). These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-stress tensor is skew-symmetric and energy conjugate to the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson's ratio are assumed to be graded in all three axial, thickness and width directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of minimum potential energy. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of TDFG nano-beam. At the end, some numerical results are performed to investigate some effective parameter on buckling load. In this theory the couple-stress tensor is skew-symmetric and energy conjugate to the skew-symmetric part of the rotation gradients as the curvature tensor.
Keywords
Euler-Bernoulli nano-beams; buckling analysis; consistent couple-stress theory; Three-directional functionally graded materials (TDFGMs); size effect; generalized differential quadrature method (GDQM);
Citations & Related Records
Times Cited By KSCI : 7  (Citation Analysis)
연도 인용수 순위
1 Nejad, M.Z., Jabbari, M. and Ghannad, M. (2017a), "A general disk form formulation for thermo-elastic analysis of functionally graded thick shells of revolution with arbitrary curvature and variable thickness", Acta Mechanica, 228(1), 215-231.   DOI
2 Nejad, M.Z., Jabbari, M. and Hadi, A. (2017b), "A review of functionally graded piezoelectric thick shells", J. Computat. Appl. Mech. DOI: 10.22059/JCAMECH.2017.247963.220   DOI
3 Nejad, M.Z., Hadi, A. and Farajpour, A. (2017c), "Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials", Struct. Eng. Mech., Int. J., 63(2), 161-169.
4 Nguyen, N.-T., Kim, N.-I. and Lee, J. (2014), "Analytical solutions for bending of transversely or axially FG nonlocal beams", Steel Compos. Struct., Int. J., 17(5), 641-665.   DOI
5 Pradhan, S. and Phadikar, J. (2009), "Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory", Struct. Eng. Mech., Int. J., 33(2), 193-213.   DOI
6 Rahmani, O., Refaeinejad, V. and Hosseini, S. (2017), "Assessment of various nonlocal higher order theories for the bending and buckling behavior of functionally graded nanobeams", Steel Compos. Struct., Int. J., 23(3), 339-350.   DOI
7 Sahmani, S. and Aghdam, M.M. (2017), "Nonlinear instability of hydrostatic pressurized hybrid FGM exponential shear deformable nanoshells based on nonlocal continuum elasticity", Compos. Part B: Eng., 114, 404-417.   DOI
8 Shafiei, N., Mirjavadi, S.S., Afshari, B.M., Rabby, S. and Hamouda, A. (2017), "Nonlinear thermal buckling of axially functionally graded micro and nanobeams", Compos. Struct., 168, 428-439.   DOI
9 Shishesaz, M., Hosseini, M., Tahan, K.N. and Hadi, A. (2017), "Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory", Acta Mechanica, 228(12), 4141-4168.   DOI
10 Shu, C. and Chew, Y. (1998), "On the equivalence of generalized differential quadrature and highest order finite difference scheme", Comput. Methods Appl. Mech. Eng,, 155(3), 249-260.   DOI
11 Simsek, M. and Al-shujairi, M. (2017), "Static, free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads", Compos. Part B: Eng., 108, 18-34.
12 Simsek, M. and Yurtcu, H.H. (2013), "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory", Compos. Struct., 97, 378-386.   DOI
13 Steinberg, M.A. (1986), "Materials for aerospace", Sci. Am., (United States), 255(4), 66-73.   DOI
14 Taczala, M., Buczkowski, R. and Kleiber, M. (2017), "Nonlinear buckling and post-buckling response of stiffened FGM plates in thermal environments", Compos. Part B: Eng., 109, 238-247.   DOI
15 Thai, H.-T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64.   DOI
16 Thai, H.-T. and Vo, T.P. (2012), "A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 54, 58-66.
17 Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Rational Mech. Anal., 11(1), 385-414.   DOI
18 Wang, Y.Q. and Zu, J.W. (2017), "Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity", Compos. Part B: Eng., 117, 74-88.   DOI
19 Yang, F., Chong, A., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743.   DOI
20 Wang, C., Zhang, Y., Ramesh, S.S. and Kitipornchai, S. (2006), "Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory", J. Phys. D: Appl. Phys., 39(17), 3904.   DOI
21 Yu, Y. J., Xue, Z.-N., Li, C.-L. and Tian, X.-G. (2016), "Buckling of nanobeams under nonuniform temperature based on nonlocal thermoelasticity", Compos. Struct., 146, 108-113.   DOI
22 Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V. and Darabi, M.A. (2016), "Coupled longitudinal-transverserotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory", Appl. Math. Model., 40(23-24), 9872-9891.   DOI
23 Adeli, M.M., Hadi, A., Hosseini, M. and Gorgani, H.H. (2017), "Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory", Eur. Phys. J. Plus, 132(9), 393.   DOI
24 Afshin, A., Nejad, M.Z. and Dastani, K. (2017), "Transient thermoelastic analysis of FGM rotating thick cylindrical pressure vessels under arbitrary boundary and initial conditions", J. Computat. Appl. Mech., 48(1), 15-26.
25 Ansari, R. and Sahmani, S. (2011), "Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories", Int. J. Eng. Sci., 49(11), 1244-1255.   DOI
26 Sadrabadi, S.A., Rahimi, G.H., Citarella, R., Shahbazi Karami, J., Sepe, R. and Esposito, R. (2017), "Analytical solutions for yield onset achievement in FGM thick walled cylindrical tubes undergoing thermomechanical loads", Compos. Part B: Eng., 116, 211-223.   DOI
27 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.   DOI
28 Apuzzo, A., Barretta, R., Luciano, R., Marotti de Sciarra, F. and Penna, R. (2017), "Free vibrations of Bernoulli-Euler nanobeams by the stress-driven nonlocal integral model", Compos. Part B: Eng., 123, 105-111.   DOI
29 Aydogdu, M. (2009), "A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration", Physica E: Low-dimens. Syst. Nanostruct., 41(9), 1651-1655.   DOI
30 Bahrami, A. and Teimourian, A. (2015), "Nonlocal scale effects on buckling, vibration and wave reflection in nanobeams via wave propagation approach", Compos. Struct., 134, 1061-1075.   DOI
31 Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., Int. J., 20(2), 227-249.   DOI
32 Burlayenko, V.N., Altenbach, H., Sadowski, T., Dimitrova, S.D. and Bhaskar, A. (2017), "Modelling functionally graded materials in heat transfer and thermal stress analysis by means of graded finite elements", Appl. Math. Model., 45, 422-438.   DOI
33 Chen, C., Li, S., Dai, L. and Qian, C. (2014), "Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces", Commun. Nonlinear Sci. Numer. Simul., 19(5), 1626-1637.   DOI
34 Civalek, O . (2017), "Vibration of laminated composite panels and curved plates with different types of FGM composite constituent", Compos. Part B: Eng., 122, 89-108.   DOI
35 Ebrahimi, F. and Barati, M.R. (2017), "A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams", Compos. Struct., 159, 174-182.
36 Ebrahimi, F. and Barati, M.R. (2016a), "Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium", J. Brazil. Soc. Mech. Sci. Eng., 39(3), 937-952.
37 Ebrahimi, F. and Barati, M.R. (2016b), "Thermal Buckling Analysis of Size-Dependent FG Nanobeams Based on the Third-Order Shear Deformation Beam Theory", Acta Mechanica Solida Sinica, 29(5), 547-554.   DOI
38 Ebrahimi, F. and Barati, M.R. (2016c), "Vibration analysis of nonlocal beams made of functionally graded material in thermal environment", Eur. Phys. J. Plus, 131(8), 279.   DOI
39 Ebrahimi, F. and Salari, E. (2015), "Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments", Compos. Struct., 128, 363-380.   DOI
40 Ebrahimi, F., Barati, M.R. and Zenkour, A.M. (2017), "A new nonlocal elasticity theory with graded nonlocality for thermomechanical vibration of FG nanobeams via a nonlocal thirdorder shear deformation theory", Mech. Adv. Mater. Struct., 25(6), 512-522.
41 Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F.-A. (2014), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Math. Computat., 229, 283-295.   DOI
42 Emam, S.A. (2013), "A general nonlocal nonlinear model for buckling of nanobeams", Appl. Math. Model., 37(10-11), 6929-6939.   DOI
43 Eringen, A.C. (1972a), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16.   DOI
44 Fatehi, P. and Nejad, M.Z. (2014), "Effects of material gradients on onset of yield in FGM rotating thick cylindrical shells", Int. J. Appl. Mech., 6(4), Article Number: 1450038.
45 Eringen, A.C. (1972b), "Theory of micromorphic materials with memory", Int. J. Eng. Sci, 10(7), 623-641.   DOI
46 Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710.   DOI
47 Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer Science & Business Media.
48 Ghannad, M., Nejad, M.Z., Rahimi, G.H. and Sabouri, H. (2012), "Elastic analysis of pressurized thick truncated conical shells made of functionally graded materials", Struct. Eng. Mech., Int. J., 43(1), 105-126.   DOI
49 Ghannad, M., Rahimi, G.H. and Nejad, M.Z. (2013), "Elastic analysis of pressurized thick cylindrical shells with variable thickness made of functionally graded materials", Compos. Part B-Eng., 45(1), 388-396.   DOI
50 Ghannadpour, S., Mohammadi, B. and Fazilati, J. (2013), "Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method", Compos. Struct., 96, 584-589.   DOI
51 Gharibi, M., Nejad, M.Z. and Hadi, A. (2017), "Elastic analysis of functionally graded rotating thick cylindrical pressure vessels with exponentially-varying properties using power series method of Frobenius", J. Computat. Appl. Mech., 48(1), 89-98.
52 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. Computat. Appl. Mech., 48(1), 123-136.
53 Hosseini, M., Shishesaz, M., Tahan, K.N. and Hadi, A. (2016), "Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials", Int. J. Eng. Sci., 109, 29-53.   DOI
54 Gopalakrishnan, S. and Narendar, S. (2013), Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations, Springer Science & Business Media.
55 Hadi, A., Rastgoo, A., Daneshmehr, A. and Ehsani, F. (2013), "Stress and strain analysis of functionally graded rectangular plate with exponentially varying properties", Indian J. Mater. Sci.
56 Hadjesfandiari, A.R. and Dargush, G.F. (2011), "Couple stress theory for solids", Int. J. Solids Struct., 48(18), 2496-2510.   DOI
57 Hosseini, M., Gorgani, H.H., Shishesaz, M. and Hadi, A. (2017), "Size-Dependent Stress Analysis of Single-Wall Carbon Nanotube Based on Strain Gradient Theory", Int. J. Appl. Mech., 9(6), 1750087.   DOI
58 Jabbari, M., Nejad, M.Z. and Ghannad, M. (2015), "Thermoelastic analysis of axially functionally graded rotating thick cylindrical pressure vessels with variable thickness under mechanical loading", Int. J. Eng. Sci., 96, 1-18.   DOI
59 Jabbari, M., Nejad, M.Z. and Ghannad, M. (2016), "Thermoelastic analysis of axially functionally graded rotating thick truncated conical shells with varying thickness", Compos. Part B-Eng., 96, 20-34.   DOI
60 Jeyakarthikeyan, P.V., Subramanian, G. and Yogeshwaran, R. (2017), "An alternate stable midpoint quadrature to improve the element stiffness matrix of quadrilaterals for application of functionally graded materials (FGM)", Comput. Struct., 178, 71-87.   DOI
61 Li, L. and Hu, Y. (2016), "Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 107, 77-97.   DOI
62 Kashkoli, M.D., Tahan, K.N. and Nejad, M.Z. (2017), "Time-Dependent Thermomechanical Creep Behavior of FGM Thick Hollow Cylindrical Shells Under Non-Uniform Internal Pressure", Int. J. Appl. Mech., 9(6), Article Number: 1750086.
63 Keivani, M., Koochi, A. and Abadyan, M. (2016), "Coupled effects of surface energy and size dependency on the stability of nanotweezers using GDQ method", Microsyst. Technol., 23(5), 1295-1308.
64 Lam, D., Yang, F., Chong, A., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508.   DOI
65 Li, L. and Hu, Y. (2017a), "Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects", Int. J. Mech. Sci., 120, 159-170.   DOI
66 Li, L. and Hu, Y. (2017b), "Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory", Compos. Struct., 172, 242-250.   DOI
67 Li, L., Li, X. and Hu, Y. (2016), "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 102, 77-92.   DOI
68 Li, X., Li, L., Hu, Y., Ding, Z. and Deng, W. (2017), "Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory", Compos. Struct., 165, 250-265.   DOI
69 Lu, C., Chen, W., Xu, R. and Lim, C.W. (2008), "Semi-analytical elasticity solutions for bi-directional functionally graded beams", Int. J. Solids Struct., 45(1), 258-275.   DOI
70 Mazarei, Z., Nejad, M.Z. and Hadi, A. (2016), "Thermo-elastoplastic analysis of thick-walled spherical pressure vessels made of functionally graded materials", Int. J. Appl. Mech., 8(4), Article Number: 1650054.
71 Mindlin, R. and Tiersten, H. (1962), "Effects of couple-stresses in linear elasticity", Arch. Rational Mech. Anal, 11(1), 415-448.   DOI
72 Najibi, A. and Talebitooti, R. (2017), "Nonlinear transient thermoelastic analysis of a 2D-FGM thick hollow finite length cylinder", Compos. Part B: Eng., 111, 211-227.   DOI
73 Nejad, M.Z. and Fatehi, P. (2015), "Exact elasto-plastic analysis of rotating thick-walled cylindrical pressure vessels made of functionally graded materials", Int. J. Eng. Sci., 86, 26-43.   DOI
74 Nejad, M.Z. and Hadi, A. (2016a), "Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 106, 1-9.
75 Nejad, M.Z. and Hadi, A. (2016b), "Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 105, 1-11.
76 Nejad, M.Z. and Rahimi, G.H. (2009), "Deformations and stresses in rotating FGM pressurized thick hollow cylinder under thermal load", Sci. Res. Essays, 4(3), 131-140.
77 Nejad, M.Z. and Rahimi, G.H. (2010), "Elastic analysis of FGM rotating cylindrical pressure vessels", J. Chinese Inst. Engr., 33(4), 525-530.   DOI
78 Li, L., Li, X. and Hu, Y. (2018), "Nonlinear bending of a twodimensionally functionally graded beam", Compos. Struct., 184, 1049-1061.   DOI
79 Nejad, M., Rastgoo, A. and Hadi, A. (2014a), "Effect of Exponentially-Varying Properties on Displacements and Stresses in Pressurized Functionally Graded Thick Spherical Shells with Using Iterative Technique", J. Solid Mech., 6(4), 366-377.
80 Nejad, M.Z., Rahimi, G.H. and Ghannad, M. (2009), "Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system", Mechanika, 77(3), 18-26.
81 Nejad, M.Z., Rastgoo, A. and Hadi, A. (2014b), "Exact elastoplastic analysis of rotating disks made of functionally graded materials", Int. J. Eng. Sci., 85, 47-57.   DOI
82 Nejad, M.Z., Jabbari, M. and Ghannad M. (2015a), "Elastic analysis of FGM rotating thick truncated conical shells with axially-varying properties under non-uniform pressure loading", Compos. Struct., 122, 561-569.   DOI
83 Nejad, M.Z., Jabbari, M. and Ghannad, M. (2015b), "Elastic analysis of axially functionally graded rotating thick cylinder with variable thickness under non-uniform arbitrarily pressure loading", Int. J. Eng. Sci., 89, 86-99.   DOI
84 Nejad, M.Z., Hadi, A. and Rastgoo, A. (2016a), "Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory", Int. J. Eng. Sci., 103, 1-10.   DOI
85 Nejad, M.Z., Abedi, M., Lotfian, M.H. and Ghannad, M. (2016b), "Exact and numerical elastic analysis for the FGM thick-walled cylindrical pressure vessels with exponentially-varying properties", Arch. Metal. Mater., 61(3), 1303-1308.   DOI