Abstract
A new Timoshenko beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and complete boundary conditions for a Timoshenko beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure-and surface energy-dependent size effect. In addition, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, unlike existing Timoshenko beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as limiting cases and recovers the Bernoulli–Euler beam model incorporating the two effects as a special case. Also, the current model reduces to the classical Timoshenko beam model when the microstructure dependence, surface energy and Poisson’s effect are all suppressed. To demonstrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general 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. In addition, 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 given by the classical model, with the difference between them being significantly large for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally.
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References
Cammarata R.C.: Surface and interface stress effects in thin films. Prog. Surf. Sci. 46, 1–38 (1994)
Challamel N.: Higher-order shear beam theories and enriched continuum. Mech. Res. Commun. 38, 388–392 (2011)
Chen J.Y., Huang Y., Ortiz M.: Fracture analysis of cellular materials: a strain gradient model. J. Mech. Phys. Solids 46, 789–828 (1998)
Ellis R.W., Smith C.W.: A thin-plate analysis and experimental evaluation of couple-stress effects. Exp. Mech. 7, 372–380 (1967)
Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)
Gao X.-L.: An expanding cavity model incorporating strain-hardening and indentation size effects. Int. J. Solids Struct. 43, 6615–6629 (2006)
Gao X.-L., Huang J.X., Reddy J.N.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech. 224, 2699–2718 (2013)
Gao X.-L., Ma H.M.: Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797 (2010)
Gao X.-L., Mall S.: Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. 38, 855–874 (2001)
Gao X.-L., Mahmoud F.F.: A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects. Z. Angew. Math. Phys. 65, 393–404 (2014)
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)
Gao X.-L., Zhou S.-S.: Strain gradient solutions of half-space and half-plane contact problems. Z. Angew. Math. Phys. 64, 1363–1386 (2013)
Gurtin M.E., Murdoch A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
Gurtin M.E., Murdoch A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)
Hutchinson J.R.: Shear coefficients for Timoshenko beam theory. ASME J. Appl. Mech. 68, 87–92 (2001)
Hutchinson J.W.: Plasticity at the micron scale. Int. J. Solids Struct. 37, 225–238 (2000)
Kaneko T.: On Timoshenko’s correction for shear in vibrating beams. J. Phys. D: Appl. Phys. 8, 1927–1936 (1975)
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)
Lazar M., Maugin G.A., Aifantis E.C.: On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct. 43, 1404–1421 (2006)
Lazar M., Maugin G.A., Aifantis E.C.: On dislocations in a special class of generalized elasticity. Phys. Stat. Sol. (b) 242, 2365–2390 (2005)
Lazopoulos, K.A., Lazopoulos, A.K.: Bending and buckling of thin strain gradient elastic beams. Euro. J. Mech. A/Solids 29, 837–843 (2010)
Lim C.W., He L.H.: Size-dependent nonlinear response of thin elastic films with nano-scale thickness. Int. J. Mech. Sci. 46, 1715–1726 (2004)
Liu C., Rajapakse R.K.N.D.: Continuum models incorporating surface energy for static and dynamic response of nanoscale beams. IEEE Trans. Nanotech. 9, 422–431 (2010)
Liu C., Rajapakse R.K.N.D., Phani A.S.: Finite element modeling of beams with surface energy effects. ASME J. Appl. Mech. 78, 031014-1–031014-10 (2011)
Lu P., He L.H., Lee H.P., Lu C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43, 4631–4647 (2006)
Lü C.F., Wu D.Z., Chen W.Q.: Nonlinear responses of nanoscale FGM films including the effects of surface energies. IEEE Trans. Nanotech. 10, 1321–1327 (2011)
Ma H.M., Gao X.-L., Reddy J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)
Ma H.M., Gao X.-L., Reddy J.N.: A non-classical Reddy–Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)
Ma H.M., Gao X.-L., Reddy J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)
Mahmoud F.F., Eltaher M.A., Alshorbagy A.E., Meletis E.I.: Static analysis of nanobeams including surface effects by nonlocal finite element. J. Mech. Sci. Tech. 26, 3555–3563 (2012)
Maugin G.A.: A historical perspective of generalized continuum mechanics. In: Altenbach, H., Maugin, G.A., Erofeev, V. (eds.) Mechanics of Generalized Continua, pp. 3–19. Springer, Berlin (2011)
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)
Miller R.E., Shenoy V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)
Mindlin R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3, 1–7 (1963)
Mindlin R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
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)
Park S.K., Gao X.-L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)
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)
Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics, 2nd ed. Wiley, New York (2002)
Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B. 71, 094104-1–094104-11 (2005)
Steigmann D.J., Ogden R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond. A 453, 853–877 (1997)
Steigmann D.J., Ogden R.W.: Elastic surface–substrate interactions. Proc. R. Soc. Lond. A 455, 437–474 (1999)
Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)
Triantafyllou A., Giannakopoulos A.E.: Structural analysis using a dipolar elastic Timoshenko beam. Euro. J. Mech. A/Solids 39, 218–228 (2013)
Wang C.M.: Timoshenko beam-bending solutions in terms of Euler–Bernoulli solutions. ASCE J. Eng. Mech. 121, 763–765 (1995)
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)
Yang F.Q.: Size dependent effective modulus of elastic composite materials: spherical nanocavities at dilute concentrations. J. Appl. Phys. 95, 3516–3520 (2004)
Zhou S.-S., Gao X.-L.: Solutions of half-space and half-plane contact problems based on surface elasticity. Z. Angew. Math. Phys. 64, 145–166 (2013)
Zhou, S.S., Gao, X.-L.: A non-classical model for circular Mindlin plates based on a modified couple stress theory. ASME J. Appl. Mech. 81, 051014-1–051014-8 (2014)
Zhou, S.-S., Gao, X.-L.: Solutions of the generalized half-plane and half-space Cerruti problems with surface effects. Z. angew. Math. Phys. (published online on 16 April 2014). doi:10.1007/s00033-014-0419-4 (2014)
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Gao, XL. A new Timoshenko beam model incorporating microstructure and surface energy effects. Acta Mech 226, 457–474 (2015). https://doi.org/10.1007/s00707-014-1189-y
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DOI: https://doi.org/10.1007/s00707-014-1189-y