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A strain gradient functionally graded Euler-Bernoulli beam formulation. (English) Zbl 1423.74488
Summary: A size-dependent functionally graded Euler-Bernoulli beam model is developed based on the strain gradient theory, a non-classical theory capable of capturing the size-effect in micro-scaled structures. The governing equation and both classical and non-classical boundary conditions are obtained using variational approach. To develop the new model, the previously used simplifying assumption which considered the length scale parameter to be constant through the thickness is avoided in this work. As a consequence, equivalent length scale parameters are introduced for functionally graded microbeams as functions of the constituents’ length scale parameters. Moreover, a generally valid closed-form solution is derived for static deflection of the new model. As case studies, the static and free-vibration of the new model are investigated for FG simply supported microbeams in which the properties are varying through the thickness according to a power law and the results of the new model are compared to those of the modified couple stress and the classical continuum theories, noted that the two latter theories are special cases of the strain gradient theory utilized in this paper.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Akgoz, B.; Civalek, O., Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International journal of engineering science, 49, 11, 1268-1280, (2011) · Zbl 1423.74338
[2] Asghari, M.; Ahmadian, M.T.; Kahrobaiyan, M.H.; Rahaeifard, M., On the size-dependent behavior of functionally graded micro-beams, Materials and design, 31, 5, 2324-2329, (2010)
[3] Asghari, M.; Kahrobaiyan, M.H.; Rahaeifard, M.; Ahmadian, M.T., Investigation of the size effects in Timoshenko beams based on the couple stress theory, Archive of applied mechanics, 81, 863-874, (2011) · Zbl 1271.74257
[4] 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
[5] Asghari, M.; Rahaeifard, M.; Kahrobaiyan, M.H.; Ahmadian, M.T., The modified couple stress functionally graded Timoshenko beam formulation, Materials and design, 32, 3, 1435-1443, (2011) · Zbl 1271.74257
[6] Aydogdu, M.; Taskin, V., Free vibration analysis of functionally graded beams with simply supported edges, Materials and design, 28, 1651-1656, (2007)
[7] Craciunescu, C.M.; Wuttig, M., New ferromagnetic and functionally grade shape memory alloys, Journal of optoelectronics and advanced materials, 5, 1, 139-146, (2003)
[8] Fleck, N.A.; Hutchinson, J.W., Phenomenological theory for strain gradient effects in plasticity, Journal of the mechanics and physics of solids, 41, 12, 1825-1857, (1993) · Zbl 0791.73029
[9] Fleck, N.A.; Hutchinson, J.W., Strain gradient plasticity, Advances in applied mechanics, 33, 296-358, (1997) · Zbl 0894.73031
[10] Fleck, N.A.; Hutchinson, J.W., A reformulation of strain gradient plasticity, Journal of the mechanics and physics of solids, 49, 10, 2245-2271, (2001) · Zbl 1033.74006
[11] Fleck, N.A.; Muller, G.M.; Ashby, M.F.; Hutchinson, J.W., Strain gradient plasticity: theory and experiment, Acta metallurgica et materialia, 42, 2, 475-487, (1994)
[12] Fu, Y.Q.; Du, H.J.; Huang, W.M.; Zhang, S.; Hu, M., Tini-based thin films in MEMS applications: a review, Sensors and actuators A, 112, 2-3, 395-408, (2004)
[13] Fu, Y.Q.; Du, H.J.; Zhang, S., Functionally graded tin/tini shape memory alloy films, Materials letters, 57, 20, 2995-2999, (2003)
[14] Fu, Y.; Zhang, J., Modeling and analysis of microtubules based on a modified couple stress theory, Physica E, 42, 1741-1745, (2010)
[15] Gheshlaghi, B.; Hasheminejad, S.M.; Abbasion, S., Size dependent torsional vibration of nanotubes, Physica E, 43, 45-48, (2010)
[16] Jomehzadeh, E.; Noori, H.R.; Saidi, A.R., The size-dependent vibration analysis of micro-plates based on a modified couple stress theory, Physica E, 43, 4, 877-883, (2011)
[17] Kahrobaiyan, M.H.; Asghari, M.; Rahaeifard, M.; Ahmadian, M.T., Investigation of the size dependent dynamic characteristics of AFM microcantilevers, International journal of engineering science, 48, 1985-1994, (2010)
[18] 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
[19] 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
[20] Kapuria, S.; Bhattacharyya, M.; Kumar, A.N., Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation, Composite structures, 82, 390-402, (2008)
[21] 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)
[22] Ke, L-L.; Wang, Y-S.; Yang, J.; Kitipornchai, S., Nonlinear free vibration of size-dependent functionally graded microbeams, International journal of engineering science, 50, 1, 256-267, (2012) · Zbl 1423.74395
[23] 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
[24] 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
[25] 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, 8, 1477-1508, (2003) · Zbl 1077.74517
[26] Li, X-FA., Unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and euler – bernoulli beams, Journal of sound and vibration, 318, 1210-1229, (2008)
[27] 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
[28] McFarland, A.W.; Colton, J.S., Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of micromechanics and microengineering, 15, 5, 1060-1067, (2005)
[29] Mindlin, R.D., Second gradient of strain and surface tension in linear elasticity, International journal of solids and structures, 1, 417-438, (1965)
[30] Park, S.K.; Gao, X.L., Bernoulli – euler beam model based on a modified couple stress theory, Journal of micromechanics and microengineering, 16, 11, 2355-2359, (2006)
[31] Sankar, B.V., An elasticity solution for functionally graded beams, Composites science and technology, 61, 689-696, (2001)
[32] Stolken, J.S.; Evans, A.G., Microbend test method for measuring the plasticity length scale, Acta materialia, 46, 14, 5109-5115, (1998)
[33] Tsiatas, G.C., A new Kirchhoff plate model based on a modified couple stress theory, International journal of solids and structures, 46, 2757-2764, (2009) · Zbl 1167.74489
[34] 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)
[35] Witvrouw, A.; Mehta, A., The use of functionally graded poly-sige layers for MEMS applications, Functionally graded mater VIII, 492-493, 255-260, (2005)
[36] Xiang, H.J.; Yang, J., Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction, Composites part B, 39, 292-303, (2008)
[37] Xia, W.; Wang, L.; Yin, L., Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration, International journal of engineering science, 48, 2044-2053, (2010) · Zbl 1231.74277
[38] Yang, F.; Chong, A.C.M.; Lam, D.C.C.; Tong, P., Couple stress based strain gradient theory for elasticity, International journal of solids and structures, 39, 10, 2731-2743, (2002) · Zbl 1037.74006
[39] Ying, J.; Lu, C.F.; Chen, W.Q., Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations, Composite structures, 84, 3, 209-219, (2008)
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