A size-dependent shear deformation beam model based on the strain gradient elasticity theory. (English) Zbl 1423.74452
Summary: A new size-dependent higher-order shear deformation beam model is developed based on modified strain gradient theory. The model captures both the microstructural and shear deformation effects without the need for any shear correction factors. The governing equations and boundary conditions are derived by using Hamilton’s principle. The static bending and free vibration behavior of simply supported microbeams are investigated. Analytical solutions including Poisson effect for deflections under point and uniform loads and for first three natural frequencies are obtained by Navier solution. The results are compared with other beam theories and other classical and non-classical models. A detailed parametric study is carried out to show the influences of thickness-to-material length scale parameter ratio, slenderness ratio and shear deformation on deflections and natural frequencies of microbeams. It is observed that effect of shear deformation becomes more significant for both smaller slenderness ratios and higher modes.
MSC:
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74M25 | Micromechanics of solids |
Keywords:
size effect; modified strain gradient theory; sinusoidal shear deformation theory; bending; vibration; microbeamReferences:
| [1] | Aifantis, E. C., Gradient deformation models at nano, micro, and macro scales, Journal of Engineering Materials and Technology, 121, 189-202, (1999) |
| [2] | Akgöz, B.; Civalek, Ö., Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science, 49, 1268-1280, (2011) · Zbl 1423.74338 |
| [3] | Akgöz, B.; Civalek, Ö., Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory, Archive of Applied Mechanics, 82, 423-443, (2012) · Zbl 1293.74252 |
| [4] | Akgöz, B.; Civalek, Ö., Longitudinal vibration analysis for microbars based on strain gradient elasticity theory, Journal of Vibration and Control, (2013) |
| [5] | Akgöz, B.; Civalek, Ö., Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory, Composite Structures, 98, 314-322, (2013) |
| [6] | Amara, K.; Tounsi, A.; Mechab, I.; Adda-Bedia, E., Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field, Applied Mathematical Modelling, 34, 3933-3942, (2010) · Zbl 1201.74171 |
| [7] | Ansari, R.; Gholami, R.; Sahmani, S., Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Composite Structures, 94, 221-228, (2011) · Zbl 1293.74155 |
| [8] | Asghari, M.; Kahrobaiyan, M. H.; Ahmadian, M. T., A nonlinear Timoshenko beam formulation based on the modified couple stress theory, International Journal of Engineering Science, 48, 1749-1761, (2010) · Zbl 1231.74258 |
| [9] | Asghari, M.; Kahrobaiyan, M. H.; Nikfar, M.; Ahmadian, M. T., A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory, Acta Mechanica, 223, 1233-1249, (2012) · Zbl 1401.74164 |
| [10] | Aydogdu, M., A new shear deformation theory for laminated composite plates, Composite Structures, 89, 94-101, (2009) |
| [11] | Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E, 41, 1651-1655, (2009) |
| [12] | Demir, Ç.; Civalek, Ö.; Akgöz, B., Free vibration analysis of carbon nanotubes based on shear deformable beam theory by discrete singular convolution technique, Mathematical and Computational Applications, 15, 57-65, (2010) · Zbl 1380.74045 |
| [13] | Eringen, A. C., Theory of micropolar plates, Zeitschrift fur Angewandte Mathematik und Physik, 18, 12-30, (1967) |
| [14] | Eringen, A. C., Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1-16, (1972) · Zbl 0229.73006 |
| [15] | Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 4703-4710, (1983) |
| [16] | Faris, W.; Nayfeh, A. H., Mechanical response of a capacitive microsensor under thermal load, Communications in Nonlinear Science and Numerical Simulation, 12, 776-783, (2007) · Zbl 1342.74043 |
| [17] | Farokhi, H.; Ghayesh, M. H.; Amabili, M., Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory, International Journal of Engineering Science, 68, 11-23, (2013) · Zbl 1423.74473 |
| [18] | Fleck, N. A.; Hutchinson, J. W., A phenomenological theory for strain gradient effects in plasticity, Journal of the Mechanics and Physics of Solids, 41, 1825-1857, (1993) · Zbl 0791.73029 |
| [19] | Fleck, N. A.; Hutchinson, J. W., A reformulation of strain gradient plasticity, Journal of the Mechanics and Physics of Solids, 49, 2245-2271, (2001) · Zbl 1033.74006 |
| [20] | Ghayesh, M. H.; Amabili, M.; Farokhi, H., Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory, International Journal of Engineering Science, 63, 52-60, (2013) · Zbl 1423.74392 |
| [21] | Kahrobaiyan, M. H.; Asghari, M.; Rahaeifard, M.; Ahmadian, M. T., Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory, International Journal of Engineering Science, 48, 1985-1994, (2010) |
| [22] | Kahrobaiyan, M. H.; Asghari, M.; Rahaeifard, M.; Ahmadian, M. T., A nonlinear strain gradient beam formulation, International Journal of Engineering Science, 49, 1256-1267, (2011) · Zbl 1423.74487 |
| [23] | Kahrobaiyan, M. H.; Rahaeifard, M.; Ahmadian, M. T., Nonlinear dynamic analysis of a V-shaped microcantilever of an atomic force microscope, Applied Mathematical Modelling, 35, 5903-5919, (2011) · Zbl 1228.74040 |
| [24] | Kahrobaiyan, M. H.; Rahaeifard, M.; Tajalli, S. A.; Ahmadian, M. T., A strain gradient functionally graded Euler-Bernoulli beam formulation, International Journal of Engineering Science, 52, 65-76, (2012) · Zbl 1423.74488 |
| [25] | Kahrobaiyan, M. H.; Tajalli, S. A.; Movahhedy, M. R.; Akbari, J.; Ahmadian, M. T., Torsion of strain gradient bars, International Journal of Engineering Science, 49, 856-866, (2011) · Zbl 1231.74025 |
| [26] | Karama, M.; Afaq, K. S.; Mistou, S., Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, International Journal of Solids and Structures, 40, 1525-1546, (2003) · Zbl 1087.74579 |
| [27] | Ke, L. L.; Wang, Y. S., Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures, 93, 342-350, (2011) |
| [28] | Koiter, W. T., Couple stresses in the theory of elasticity, I and II, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen (B), 67, 17-44, (1964) · Zbl 0124.17405 |
| [29] | Kong, S.; Zhou, S.; Nie, Z.; Wang, K., The size-dependent natural frequency of Bernoulli-Euler micro-beams, International Journal of Engineering Science, 46, 427-437, (2008) · Zbl 1213.74189 |
| [30] | Kong, S.; Zhou, S.; Nie, Z.; Wang, K., Static and dynamic analysis of micro beams based on strain gradient elasticity theory, International Journal of Engineering Science, 47, 487-498, (2009) · Zbl 1213.74190 |
| [31] | Lam, D. C.C.; Yang, F.; Chong, A. C.M.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51, 1477-1508, (2003) · Zbl 1077.74517 |
| [32] | Lee, H.-L.; Chang, W.-J., Sensitivity of V-shaped atomic force microscope cantilevers based on a modified couple stress theory, Microelectronic Engineering, 88, 3214-3218, (2011) |
| [33] | Levinson, M., A new rectangular beam theory, Journal of Sound and Vibration, 74, 81-87, (1981) · Zbl 0453.73058 |
| [34] | Li, X.; Bhushan, B.; Takashima, K.; Baek, C. W.; Kim, Y. K., Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques, Ultramicroscopy, 97, 481-494, (2003) |
| [35] | Ma, H. M.; Gao, X. L.; Reddy, J. N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 56, 3379-3391, (2008) · Zbl 1171.74367 |
| [36] | McFarland, A. W.; Colton, J. S., Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of Micromechanics and Microengineering, 15, 1060-1067, (2005) |
| [37] | Mindlin, R. D., Second gradient of strain and surface tension in linear elasticity, International Journal of Solids and Structures, 1, 417-438, (1965) |
| [38] | Mindlin, R. D.; Tiersten, H. F., Effects of couple-stresses in linear elasticity, Archives of Rational Mechanics and Analysis, 11, 415-448, (1962) · Zbl 0112.38906 |
| [39] | Moser, Y.; Gijs, M. A.M., Miniaturized flexible temperature sensor, Journal of Microelectromechanical Systems, 16, 1349-1354, (2007) |
| [40] | Movassagh, A. A.; Mahmoodi, M. J., A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory, European Journal of Mechanics A/Solids, 40, 50-59, (2013) · Zbl 1406.74067 |
| [41] | Najar, F.; Choura, S.; El-Borgi, S.; Abdel-Rahman, E. M.; Nayfeh, A. H., Modeling and design of variable-geometry electrostatic microactuators, Journal of Micromechanics and Microengineering, 15, 419-429, (2005) · Zbl 1269.74113 |
| [42] | Nateghi, A.; Salamat-talab, M.; Rezapour, J.; Daneshian, B., Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory, Applied Mathematical Modelling, 36, 4971-4987, (2012) · Zbl 1252.74020 |
| [43] | Park, S. K.; Gao, X. L., Bernoulli-Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 16, 2355-2359, (2006) |
| [44] | Peddieson, J.; Buchanan, G. R.; McNitt, R. P., Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41, 305-312, (2003) |
| [45] | Poole, W. J.; Ashby, M. F.; Fleck, N. A., Micro-hardness of annealed and work- hardened copper polycrystals, Scripta Materialia, 34, 559-564, (1996) |
| [46] | Reddy, J. N., A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics, 51, 745-752, (1984) · Zbl 0549.73062 |
| [47] | Reddy, J. N., Energy principles and variational methods in applied mechanicss, (2002), John Wiley & Sons New York |
| [48] | Reddy, J. N., Nonlocal theories for bending, buckling and vibration of nanobeams, International Journal of Engineering Science, 45, 288-307, (2007) · Zbl 1213.74194 |
| [49] | Reddy, J. N., Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, International Journal of Engineering Science, 48, 1507-1518, (2010) · Zbl 1231.74048 |
| [50] | Reddy, J. N., Microstructure-dependent couples stress theories of functionally graded beams, Journal of the Mechanics and Physics of Solids, 59, 2382-2399, (2011) · Zbl 1270.74114 |
| [51] | Reddy, J. N.; Pang, S. D., Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Journal of Applied Physics, 103, 023511, (2008) |
| [52] | Roque, C. M.C.; Ferreira, A. J.M.; Reddy, J. N., Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method, International Journal of Engineering Science, 49, 976-984, (2011) · Zbl 1231.74272 |
| [53] | Roque, C. M.C.; Fidalgo, D. S.; Ferreira, A. J.M.; Reddy, J. N., A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method, Composite Structures, 96, 532-537, (2013) |
| [54] | Sadeghi, H.; Baghani, M.; Naghdabadi, R., Strain gradient elasticity solution for functionally graded micro-cylinders, International Journal of Engineering Science, 50, 22-30, (2012) · Zbl 1423.74087 |
| [55] | Sahmani, S.; Ansari, R., On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory, Composite Structures, 95, 430-442, (2013) |
| [56] | Salamat-talab, M.; Nateghi, A.; Torabi, J., Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory, International Journal of Mechanical Sciences, 57, 63-73, (2012) |
| [57] | Şimşek, M., Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory, International Journal of Engineering Science, 48, 1721-1732, (2010) · Zbl 1231.74275 |
| [58] | Şimşek, M.; Reddy, J. N., Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, International Journal of Engineering Science, 64, 37-53, (2013) · Zbl 1423.74517 |
| [59] | Şimşek, M.; Reddy, J. N., A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory, Composite Structures, 101, 47-58, (2013) |
| [60] | Şimşek, M.; Yurtcu, H. H., Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory, Composite Structures, 97, 378-386, (2013) |
| [61] | Soldatos, K. P., A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica, 94, 195-220, (1992) · Zbl 0850.73130 |
| [62] | Thai, H. T., A nonlocal beam theory for bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, 52, 56-64, (2012) · Zbl 1423.74356 |
| [63] | Thai, H. T.; Kim, S. E., A size-dependent functionally graded reddy plate model based on a modified couple stress theory, Composites Part B: Engineering, 45, 1636-1645, (2013) |
| [64] | Thai, H. T.; Vo, T. P., A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams, International Journal of Engineering Science, 54, 58-66, (2012) · Zbl 1423.74357 |
| [65] | Thai, H. T.; Vo, T. P., A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory, Composite Structures, 96, 376-383, (2013) |
| [66] | Timoshenko, S. P., On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section, Philosophical Magazine, 41, 744-746, (1921) |
| [67] | Tounsi, A.; Semmah, A.; Bousahla, A., Thermal buckling behavior of nanobeam using an efficient higher-order nonlocal beam theory, Journal of Nanomechanics and Micromechanics, (2013), 10.1061/(ASCE)NM.2153-5477.0000057 |
| [68] | Toupin, R. A., Theory of elasticity with couple stresses, Archives of Rational Mechanics and Analysis, 17, 85-112, (1964) · Zbl 0131.22001 |
| [69] | Touratier, M., An efficient standard plate theory, International Journal of Engineering Science, 29, 901-916, (1991) · Zbl 0825.73299 |
| [70] | Vardoulakis, I.; Sulem, J., Bifurcation analysis in geomechanics, (1995), Blackie/Chapman & Hall London · Zbl 0900.73645 |
| [71] | Wang, B.; Zhao, J.; Zhou, S., A micro scale Timoshenko beam model based on strain gradient elasticity theory, European Journal of Mechanics A/Solids, 29, 591-599, (2010) · Zbl 1480.74194 |
| [72] | Wang, B.; Zhou, S.; Zhao, J.; Chen, X., A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory, European Journal of Mechanics A/Solids, 30, 517-524, (2011) · Zbl 1278.74103 |
| [73] | Zhao, J.; Zhou, S.; Wang, B.; Wang, X., Nonlinear microbeam model based on strain gradient theory, Applied Mathematical Modelling, 36, 2674-2686, (2012) · Zbl 1246.74038 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
