A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. (English) Zbl 1171.74367

Summary: A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton’s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson’s ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli-Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli-Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson’s effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74M25 Micromechanics of solids
Full Text: DOI


[1] Anthoine, A., Effect of couple-stresses on the elastic bending of beams, Int. J. Solids Struct., 37, 1003-1018 (2000) · Zbl 0978.74044
[2] Challamel, N.; Wang, C. M., The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology, 19, 345703 (2008), (7pp)
[3] Chong, A. C.M.; Yang, F.; Lam, D. C.C.; Tong, P., Torsion and bending of micron-scaled structures, J. Mater. Res., 16, 1052-1058 (2001)
[4] Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54, 4703-4710 (1983)
[5] Gao, X.-L.; Mall, S., Variational solution for a cracked mosaic model of woven fabric composites, Int. J. Solids Struct., 38, 855-874 (2001) · Zbl 1045.74019
[6] Gao, X.-L.; Park, S. K., Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem, Int. J. Solids Struct., 44, 7486-7499 (2007) · Zbl 1166.74318
[7] Giannakopoulos, A. E.; Stamoulis, K., Structural analysis of gradient elastic components, Int. J. Solids Struct., 44, 3440-3451 (2007) · Zbl 1121.74393
[8] Heireche, H.; Tounsi, A.; Benzair, A.; Maachou, M.; Adda Bedia, E. A., Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity, Physica E, 40, 2791-2799 (2008)
[9] Hutchinson, J. R., Shear coefficients for Timoshenko beam theory, ASME J. Appl. Mech., 68, 87-92 (2001) · Zbl 1110.74489
[10] Jensen, J. J., On the shear coefficient in Timoshenko’s beam theory, J. Sound Vib., 87, 621-635 (1983) · Zbl 0544.73072
[11] Kaneko, T., On Timoshenko’s correction for shear in vibrating beams, J. Phys. D: Appl. Phys., 8, 1927-1936 (1975)
[12] Koiter, W. T., Couple-stresses in the theory of elasticity: I and II, Proc. K. Ned. Akad. Wet. B, 67, 1, 17-44 (1964) · Zbl 0124.17405
[13] Lam, D. C.C.; Yang, F.; Chong, A. C.M.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids, 51, 1477-1508 (2003) · Zbl 1077.74517
[14] Leech, C. M., Beam theories: a variational approach, Int. J. Mech. Eng. Education, 5, 81-87 (1977)
[15] 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)
[16] Li, X.-F.; Wang, B.-L.; Mai, Y.-W., Effects of a surrounding elastic medium on flexural waves propagating in carbon nanotubes via nonlocal elasticity, J. Appl. Phys., 103, 074309 (2008), (pp. 1-9)
[17] Lu, P.; Lee, H. P.; Lu, C.; Zhang, P. Q., Application of nonlocal beam models for carbon nanotubes, Int. J. Solids Struct., 44, 5289-5300 (2007) · Zbl 1124.74029
[18] Maneschy, C. E.; Miyano, Y.; Shimbo, M.; Woo, T. C., Residual-stress analysis of an epoxy plate subjected to rapid cooling on both surfaces, Exp. Mech., 26, 306-312 (1986)
[19] Maranganti, R.; Sharma, P., A novel atomistic approach to determine strain-gradient elasticity constants: tabulation and comparison for various metals, semiconductors, silica, polymers and the (ir) relevance for nanotechnologies, J. Mech. Phys. Solids, 55, 1823-1852 (2007) · Zbl 1173.74003
[20] McFarland, A. W.; Colton, J. S., Role of material microstructure in plate stiffness with relevance to microcantilever sensors, J. Micromech. Microeng., 15, 1060-1067 (2005)
[21] Mindlin, R. D., Influence of couple-stresses on stress concentrations, Exp. Mech., 3, 1-7 (1963)
[22] Mindlin, R. D.; Tiersten, H. F., Effects of couple-stresses in linear elasticity, Arch. Rational Mech. Anal., 11, 415-448 (1962) · Zbl 0112.38906
[23] Papargyri-Beskou, S.; Tsepoura, K. G.; Polyzos, D.; Beskos, D. E., Bending and stability analysis of gradient elastic beams, Int. J. Solids Struct., 40, 385-400 (2003) · Zbl 1022.74010
[24] Park, S. K.; Gao, X.-L., Bernoulli-Euler beam model based on a modified couple stress theory, J. Micromech. Microeng., 16, 2355-2359 (2006)
[25] Park, S. K.; Gao, X.-L., Variational formulation of a modified couple stress theory and its application to a simple shear problem, Z. Angew. Math. Phys., 59, 904-917 (2008) · Zbl 1157.74014
[26] Peddieson, J.; Buchanan, G. R.; McNitt, R. P., Application of nonlocal continuum models to nanotechnology, Int. J. Eng. Sci., 41, 305-312 (2003)
[27] Pei, J.; Tian, F.; Thundat, T., Glucose biosensor based on the microcantilever, Anal. Chem., 76, 292-297 (2004)
[28] Polizzotto, C., Gradient elasticity and nonstandard boundary conditions, Int. J. Solids Struct., 40, 7399-7423 (2003) · Zbl 1063.74015
[29] Reddy, J. N., Energy Principles and Variational Methods in Applied Mechanics (2002), Wiley: Wiley New York
[30] Reddy, J. N., Theory and Analysis of Elastic Plates and Shells (2007), Taylor & Francis: Taylor & Francis Philadelphia · Zbl 1213.74194
[31] Reddy, J. N., Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45, 288-307 (2007) · Zbl 1213.74194
[32] Shames, I. H., Energy and Finite Element Methods in Structural Mechanics (1985), Hemisphere: Hemisphere New York · Zbl 0595.73072
[33] Stephen, N. G.; Levinson, M., A second order beam theory, J. Sound Vib., 67, 293-305 (1979) · Zbl 0422.73047
[34] Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1970), McGraw-Hill: McGraw-Hill New York · Zbl 0266.73008
[35] Toupin, R. A., Theories of elasticity with couple stress, Arch. Rational Mech. Anal., 17, 85-112 (1964) · Zbl 0131.22001
[36] Vardoulakis, I.; Sulem, J., Bifurcation Analysis in Geomechanics (1995), Blackie/Chapman & Hall: Blackie/Chapman & Hall London · Zbl 0900.73645
[37] Volokh, K. Y.; Hutchinson, J. W., Are lower-order gradient theories of plasticity really lower order?, ASME J. Appl. Mech., 69, 862-864 (2002) · Zbl 1110.74730
[38] Wang, C. M., Timoshenko beam-bending solutions in terms of Euler-Bernoulli solutions, ASCE J. Eng. Mech., 121, 763-765 (1995)
[39] Wang, C. M.; Zhang, Y. Y.; Ramesh, S. S.; Kitipornchai, S., Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory, J. Phys. D: Appl. Phys., 39, 3904-3909 (2006)
[40] Wang, C. M.; Zhang, Y. Y.; He, X. Q., Vibration of nonlocal Timoshenko beams, Nanotechnology, 18, 105401 (2007), (9pp)
[41] Wang, Q., Wave propagation in carbon nanotubes via nonlocal continuum mechanics, J. Appl. Phys., 98, 124301 (2005), (pp. 1-6)
[42] Yang, F.; Chong, A. C.M.; Lam, D. C.C.; Tong, P., Couple stress based strain gradient theory for elasticity, Int. J. Solids Struct., 39, 2731-2743 (2002) · Zbl 1037.74006
[43] Yang, J. F.C.; Lakes, R. S., Experimental study of micropolar and couple stress elasticity in compact bone in bending, J. Biomechanics, 15, 91-98 (1982)
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