Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

A Review on Recent Advances in Micro and Nano Scale Beam Theories and the Relevant Areas of Applied Solid Mechanics


Affiliations
1 Mechanical Engineering Department, Jadavpur University, Kolkata-700032, India
     

   Subscribe/Renew Journal


The present paper reports a review on several aspects associated with advancement in beam theories to predict size dependent mechanical behavior of micro and nano scale beam. Inherent geometric nonlinearity of large deflection couples with complicated material behavior of micro and nano sized beams, and makes such small scale beam problem very complicated. Hence, proper modeling of constitutive behavior and solution of nonlinear partial differential equation are the two major components in analysis of micro and nano beam. Developments in the two areas from classical to modern higher order approaches are critically reviewed in the paper. Several constitutive relations used to model material behavior of small scale beam and the frequently used analysis methods in association with computational solution schemes to solve nonlinear system governing equation are identified and presented in the paper. Increasing trend of using miniature devices and non-metallic materials leads towards several class of micro and nano beam problems according to application. The problems are not only limited to common beam problem like bending, vibration, static and dynamic stability, but also includes several interdisciplinary areas like fracture mechanics, bio-mechanics, molecular dynamics, material science, etc. Hence in addition to material modeling and analysis methods, type of problem encountered in literature and application areas together with experimental works on micro and nano sized beam like structures are also addressed in the review paper.

Keywords

Micro/Nano Beam, Size Dependent Deformation, Geometric Nonlinearity, Higher Order Continuum Mechanics, Constitutive Relation, Couple Stress Theory, Strain Gradient Theory, Nonlocal Theory.
User
Subscription Login to verify subscription
Notifications
Font Size

  • Boresi, A.P. and Schmidt, R.J., Advanced mechanics of materials, sixth ed., John Wiley and Sons Incorporated, New York, 2003.
  • Timoshenko, S.P., History of strength of materials, McGraw-Hill Book Company, New York, 1953.
  • Ghuku, S. and Saha, K.N., A review on stress and deformation analysis of curved beams under large deflection, International Journal of Engineering and Technologies, Vol. 11, pp. 13-39,2017.
  • Cosserat, E. and Cosserat, F., Theory of deformable bodies (Translated by D.H. Delphenich), Vol. 6, Scientific Library A Herman and Sons, Rue De La Sorbonne, Paris, 1909.
  • Eringen, A.C. and Edelen, D.G.B., On nonlocal elasticity, International Journal of Engineering Science, Vol. 10, pp. 233-248,1972.
  • Ansari, R., Gholami, R. and Rouhi, H., Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories, Composites Part B: Engineering, Vol. 43, pp. 2985-2989,2012.
  • McFarland, A.W. and Colton, J.S., Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of Micromechanics and Microengineering, Vol. 15, pp. 1060-1067, 2005.
  • Lam, D.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P., Experiments and theory in strain gradient elasticity, Journal ofthe Mechanics and Physics of Solids, Vol. 51, pp. 1477-1508,2003.
  • Sadd, M.H., Elasticity: Theory, applications, and numerics 2005, Burlington, MA: Elsevier Butterworth-Heinemann, 2009.
  • Mindlin, R.D. and Tiersten, H.F., Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and analysis, Vol. 11, pp. 415-448,1962.
  • Toupin, R.A., Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, Vol. 11, pp. 385-414, 1962.
  • Mindlin, R.D., Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, Vol. 16, pp. 51-78, 1964.
  • Mindlin, R.D., Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures, Vol. 1, pp. 417-438,1965.
  • Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of applied physics, Vol. 54, pp. 4703-4710,1983.
  • Hadjesfandiari, A.R. and Dargush, G.F., Couple stress theories: Theoretical underpinnings and practical aspects from a new energy perspective, arXiv preprint arXiv: 1611.10249,2016.
  • Tekofllu, C. and Onck, P.R., Size effects in two-dimensional Voronoi foams: a comparison between generalized continua and discrete models, Journal of the Mechanics and Physics of Solids, Vol. 56, pp. 3541-3564, 2008.
  • Fathalilou, M., Sadeghi, M. and Rezazadeh, G., Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory, Mechanics Research Communications, Vol. 60, pp. 74-80, 2014.
  • Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C. and Tong, P., Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, Vol. 39, pp. 2731-2743,2002.
  • Ansari, R., Torabi, J. and Norouzzadeh, A., Bending analysis of embedded nanoplates based on the integral formulation of Eringen’s nonlocal theory using the finite element method, Physica B: Condensed Matter, Vol. 534, pp. 90-97, 2018.
  • Christensen, R.M., A critical evaluation for a class of micro-mechanics models, Journal ofthe Mechanics and Physics of Solids, Vol. 38, pp. 379-404, 1990.
  • Roscoe, R., The viscosity of suspensions of rigid spheres, British Journal of Applied Physics, Vol. 3, pp. 267-269, 1952.
  • Hashin, Z., The elastic moduli of heterogeneous materials, Journal of Applied Mechanics, Vol. 29, pp. 143-150, 1962.
  • Budiansky, B., On the elastic moduli of some heterogeneous materials, Journal ofthe Mechanics and Physics of Solids, Vol. 13, pp. 223-227, 1965.
  • Hill, R., A self-consistent mechanics of composite materials, Journal ofthe Mechanics and Physics of Solids, Vol. 13, pp. 213-222, 1965.
  • Christensen, R.M. and Lo, K.H., Solutions for effective shear properties in three phase sphere and cylinder models, Journal ofthe Mechanics and Physics of Solids, Vol. 27, pp. 315-330, 1979, Erratum, Vol. 34, pp. 639, 1986.
  • Benveniste, Y., A new approach to the application of Mori-Tanaka’s theory in composite materials, Mechanics of materials, Vol. 6, pp. 147-157,1987.
  • Eraslan, A.N. and Arslan, E., A computational study on the nonlinear hardening curved beam problem, International Journal of Pure and Applied Mathematics, Vol. 43, pp. 129-143,2008.
  • Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W., Strain gradient plasticity: theory and experiment, Acta Metallurgica et Materialia, Vol. 42, pp. 475-487,1994.
  • Stolken, J.S. and Evans, A.G., A microbend test method for measuring the plasticity length scale, Acta Materialia, Vol. 46, pp. 5109-5115,1998.
  • Patel, B.N., Pandit, D. and Srinivasan, S.M., Large elaso-plastic deflection of microbeams using strain gradient plasticity theory, Procedia engineering, Vol. 173, pp.10641070,2017.
  • Mathew, T.V., Natarajan, S. and Martinez-Paneda, E., Size effects in elastic-plastic functionally graded materials, Composite Structures, Vol. 204, pp. 43-51, 2018.
  • Polyzos, D. and Fotiadis, D.I., Derivation of Mindlin’sfirst and second strain gradient elastic theory via simple lattice and continuum models, International Journal of Solids and Structures, Vol. 49, pp. 470-480, 2012.
  • Ansari, R., Rouhi, H. and Sahmani, S., Calibration ofthe analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics, International Journal of Mechanical Sciences, Vol. 53, pp. 786-792,2011.
  • Ghuku, S. and Saha, K.N., A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams, Engineering Science and Technology, an International Journal, Vol. 19, pp. 135-146, 2016.
  • Ghuku, S. and Saha, K.N., Large deflection analysis of curved beam problem with varying curvature and moving boundaries, Engineering Science and Technology, an International Journal, Vol. 21, pp. 408-420, 2018.
  • Gurtin, M.E. and Murdoch, A.I., Surface stress in solids, International Journal of Solids and Structures, Vol. 14, pp. 431-440, 1978.
  • Kiani, K., Thermo-elasto-dynamic analysis of axially functionally graded non-uniform nanobeams with surface energy, International Journal of Engineering Science, Vol. 106, pp. 57-76,2016.
  • Jia, X.L., Yang, J., Kitipornchai, S. and Lim, C.W., Forced vibration of electrically actuated FGM micro-switches, Procedia Engineering, Vol. 14, pp. 280-287, 2011.
  • Nejad, M.Z. and Hadi, A., Eringen’s non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams, International Journal of Engineering Science, Vol. 106, pp. 1-9, 2016.
  • Chen, X. and Meguid, S.A., Snap-through buckling of initially curved microbeam subject to an electrostatic force, Proceedings of the Royal Society of London-Series A/ Mathematical and Physical Sciences, Vol. 471, pp. 20150072, 2015.
  • Al-shujairi, M. and Mollamahmutofllu, Q., Dynamic stability of sandwich functionally graded micro-beam based on the nonlocal strain gradient theory with thermal effect, Composite Structures, Vol. 201, pp. 1018-1030, 2018.
  • Das, D., Nonlinear forced vibration analysis of higher order shear-deformable functionally graded microbeam resting on nonlinear elastic foundation based on modified couple stress theory, Proceedings ofthe Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 2018, doi: 10.1177/ 1464420718789716.
  • Li, L. and Hu, Y., Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory, International Journal of Engineering Science, Vol. 97, pp. 84-94,2015.
  • Rajasekaran, S. and Khaniki, H.B., Bending, buckling and vibration of small-scale tapered beams, International Journal of Engineering Science, Vol. 120, pp. 172-188, 2017.
  • Ke, L.L. and Wang, Y.S., Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures, Vol. 93, pp. 342-350,2011.
  • Talimian, A. and Beda, P., Dynamic stability of a size-dependent micro-beam, European Journal of Mechanics-A/Solids, Vol. 72, pp. 245-251, 2018.
  • Rouhi, H., Ebrahimi, F., Ansari, R. and Torabi, J., Nonlinearfree and forced vibration analysis of Timoshenko nanobeams based on Mindlin’s second strain gradient theory, European Journal of Mechanics-A/Solids, Vol. 73, pp. 268-281, 2019.
  • Challamel, N., Lerbet, J., Wang, C.M. and Zhang, Z., Analytical length scale calibration of nonlocal continuum from a microstructured buckling model, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift fur Angewandte Mathematik und Mechanik, Vol. 94, pp. 402-413,2014.
  • Artan, R. and Tepe, A., The initial values method for buckling of nonlocal bars with application in nanotechnology, European Journal of Mechanics-A/Solids, Vol. 27, pp. 469-477, 2008.
  • Popescu, D.S., Lammerink, T.S. and Elwenspoek, M., Buckled membranes for microstructures, In Micro Electro Mechanical Systems, 1994, MEMS’94, Proceedings of IEEE, pp. 188-192, 1994.
  • Vangbo, M., An analytical analysis of a compressed bistable buckled beam, Sensors and Actuators A: Physical, Vol. 69, pp. 212-216, 1998.
  • Saffari, S., Hashemian, M. andToghraie, D., Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects, Physica B: Condensed Matter, Vol. 520, pp. 97-105, 2017.
  • Zhao, L., Chen, W.Q. and Lu, C.F., Symplectic elasticity for bi-directional functionally graded materials, Mechanics of Materials, Vol. 54, pp. 32-42, 2012.
  • Barretta, R., Faghidian, S.A., Luciano, R., Medaglia, C.M. and Penna, R., Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models, Composites Part B: Engineering, Vol. 154, pp. 20-32, 2018.
  • Torabi, K. and Dastgerdi, J.N., An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model, Thin Solid Films, Vol. 520, pp. 6595-6602, 2012.
  • Shaat, M., Khorshidi, M.A., Abdelkefi, A. and Shariati, M., Modeling and vibration characteristics of cracked nano-beams made of nanocrystalline materials, International Journal of Mechanical Sciences, Vol. 115, pp. 574-585, 2016.
  • Loya, J., Lopez-Puente, J., Zaera, R. and Fernandez-Saez, J., Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model, Journal of Applied Physics, Vol. 105, pp. 044309, 2009.
  • Liu, H.K. and Chao, J.J., Length effect on creep of silicon cantilever microbeams, International Journal of Damage Mechanics, Vol. 24, pp. 947-964, 2015.
  • Shieh, Y.C., Lin, H.Y., Hsu, W. and Lin, Y.H., A Rapid Fatigue Test Method on Micro Structures for High-Cycle Fatigue, IEEE Transactions on Device and Materials Reliability, Vol. 16, pp. 61-68, 2016.
  • Chopra, I., Review of State of Art of Smart Structures and Integrated Systems, AIAA Journal, Vol. 40, pp. 2145-2187, 2002.
  • Liu, H.K., Pan, C.H. and Liu, P.P., Dimension effect on mechanical behavior of silicon micro-cantilever beams, Measurement, Vol. 41, pp. 885-895, 2008.
  • Ballestra, A., Brusa, E., De Pasquale, G., Munteanu, M.G. and Soma, A., FEM modelling and experimental characterization of microbeams in presence of residual stress, Analog Integrated Circuits and Signal Processing, Vol. 63, pp. 477-488, 2010.
  • Tilmans, H.A. and Legtenberg, R., Electrostatically driven vacuum-encapsulated polysilicon resonators: Part II. Theory and performance, Sensors and Actuators A: physical, Vol. 45, pp. 67-84, 1994.

Abstract Views: 9

PDF Views: 1




  • A Review on Recent Advances in Micro and Nano Scale Beam Theories and the Relevant Areas of Applied Solid Mechanics

Abstract Views: 9  |  PDF Views: 1

Authors

Sushanta Ghuku
Mechanical Engineering Department, Jadavpur University, Kolkata-700032, India
Kashi Nath Saha
Mechanical Engineering Department, Jadavpur University, Kolkata-700032, India

Abstract


The present paper reports a review on several aspects associated with advancement in beam theories to predict size dependent mechanical behavior of micro and nano scale beam. Inherent geometric nonlinearity of large deflection couples with complicated material behavior of micro and nano sized beams, and makes such small scale beam problem very complicated. Hence, proper modeling of constitutive behavior and solution of nonlinear partial differential equation are the two major components in analysis of micro and nano beam. Developments in the two areas from classical to modern higher order approaches are critically reviewed in the paper. Several constitutive relations used to model material behavior of small scale beam and the frequently used analysis methods in association with computational solution schemes to solve nonlinear system governing equation are identified and presented in the paper. Increasing trend of using miniature devices and non-metallic materials leads towards several class of micro and nano beam problems according to application. The problems are not only limited to common beam problem like bending, vibration, static and dynamic stability, but also includes several interdisciplinary areas like fracture mechanics, bio-mechanics, molecular dynamics, material science, etc. Hence in addition to material modeling and analysis methods, type of problem encountered in literature and application areas together with experimental works on micro and nano sized beam like structures are also addressed in the review paper.

Keywords


Micro/Nano Beam, Size Dependent Deformation, Geometric Nonlinearity, Higher Order Continuum Mechanics, Constitutive Relation, Couple Stress Theory, Strain Gradient Theory, Nonlocal Theory.

References





DOI: https://doi.org/10.22485/jaei%2F2019%2Fv89%2Fi1-2%2F185670