Free vibrations of a strain gradient beam by the method of initial values. (English) Zbl 1307.74040

Artan, R.; Batra, R. C.

Acta Mech. 223, No. 11, 2393-2409 (2012).

Summary: We extend the application of the method of initial values (also known as the transfer matrix method) to find frequencies of free vibrations of a strain-gradient-dependent Euler-Bernoulli beam (EBB) under different boundary conditions at the two end faces of the beam. For the classical EBB, we find the exact matricant or the carry-over matrix but it is numerically evaluated for the strain-gradient-dependent EBB. For the numerically evaluated matricant, it is found that ten iterations give converged values of the first six frequencies for the classical and the strain-gradient-dependent EBB. For the strain-gradient EBB, the sixth-order ordinary differential equation for the lateral deflection and three boundary conditions at each end have been derived by using the Hamilton principle. The material characteristic length is found to noticeably affect frequencies of free vibrations. Thus, the difference between frequencies of the classical and the strain-gradient-dependent EBB can be used to determine the value of the material characteristic length for a nanobeam for which length scale effects are believed to be dominant.

MSC:

74H45	Vibrations in dynamical problems in solid mechanics
74K10	Rods (beams, columns, shafts, arches, rings, etc.)

References:

[1]	Toupin R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. 11, 385–414 (1962) · Zbl 0112.16805
[2]	Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(1), 415–448 (1962) · Zbl 0112.38906
[3]	Gurtin M.E.: Thermodynamics and the possibility of spatial interaction on elastic materials. Arch. Ration. Mech. Anal. 19, 339–352 (1965) · Zbl 0146.21106
[4]	Eringen A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972) · Zbl 0229.73006
[5]	Batra R.C.: Thermodynamics of non-simple materials. J. Elasticity 6, 451–456 (1976) · Zbl 0352.73007
[6]	Green A.E., Laws N.: On the entropy production inequality. Arch. Rat. Mech. Anal. 45, 47–53 (1972) · Zbl 0246.73006
[7]	Dillon O.D. Jr, Kratochvil J.: A strain gradient theory of plasticity. Int. J. Solids Struct. 6, 1513–1533 (1970) · Zbl 0262.73036
[8]	Batra R.C.: The initiation and growth of, and the interaction among adiabatic shear bands in simple and dipolar materials. Int. J. Plasticity 3, 75–89 (1987)
[9]	Peddieson J., Buchanan G.R., McNitt R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)
[10]	Lam D.C.C., Fang Y., 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
[11]	Kong S., Zhou S.J., Nie Z.F., Wang K.: Static and dynamic analysis of microbeams based on strain gradient elasticity theory. Int. J. Eng. Sci. 47, 487–498 (2009) · Zbl 1213.74190
[12]	Ashgari M., Kahrobaiyan M.H., Nikfar M., Ahmadian M.T.: A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory. Acta Mech. 223, 1233–1249 (2012) · Zbl 1401.74164
[13]	Marguerre K., Marguerre K.: Mechanics of Vibrations. Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands (1979)
[14]	Magrab E.B.: Vibrations of Elastic Structural Members. Mechanics of Structural Systems 3. Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands (1979)