The size-dependent natural frequency of Bernoulli–Euler micro-beams
Introduction
Thin beams are one of the major structures used widely in micro–electro-mechanical systems (MEMS). A large number of those applications utilized the dynamic mechanical properties of thin films materials for targeted performance specifications such as those vibration shock sensor [1], [2], atomic force microscopes [3], [4] and resonant testing method [5], [6].
In those applications, the thickness of those beams is typically on the order of microns and sub-microns, and the size dependence of deformation behaviour in this micro scale has been experimentally verified. First, the phenomenon are observed in a lot of metals and polymers deformed plastically [7], [8], [9]. For example, in the micro-torsion test of thin copper wires in 1994, Fleck et al. [10] observed that the torsional hardening increases by a factor of 3 as the wire diameter decreases from 170 to 12 μm. In the micro-indentation experiments in 1995, Ma et al. [11] observed that the measured indentation hardness of silver single crystal increases by a factor of more than two as the penetration depth of the indenter decreases from 2.0 to 0.1 μm. In the micro bending test of thin nickel beams in 1998, Stolken and Evans [12] observed that the plastic work hardening shows a great increase as the beam thickness decreases from 50 to 12.5 μm. Currently, the size dependences are observed in some polymers deformed elastically. For example, in the micro bending testing of epoxy polymeric beams in 2002, Lam et al. [13] observed that the bending rigidity increases about 2.4 times as the beam thickness reduces from 115 to 20 μm. In the micro bending testing of polypropylene micro-cantilevers in 2005, McFarland et al. [14] observed that the measured stiffness values are seen to be at least four times larger than the classical beam theory stiffness predictions and the deformation is also in the range of linear elastic.The above experimental results show that the size dependence is intrinsic to certain materials with non-homogeneous microstructure. Therefore, accurate characterization of these dynamic mechanical properties in micron scale is urgent and vital for the reliable and optimal design of many of these MEMS devices.
Lacking an internal material length scale parameter, conventional strain-based mechanics theories can not interpret and predict those micro-structure-dependent size effects when the structural size is in micron- and sub-micron-scale. However, these size dependences can be successfully modelled in a macroscopic manner by employing higher-order gradient theory and couple stress theory, in which constitutive equations introduce additional material constants besides classical material parameters.
As a higher order continuum theory, the classical couple stress elasticity theory was proposed by Mindlin and others including Toupin and Koiter in 1960s and contains four material constants for isotropic elastic materials [15], [16], [17]. In the classical couple stress elasticity theory, two higher order material length scale parameters are introduced in addition to the two Lame constants. Subsequently, this theory had been used by Mindlin and Tiersten to model the thickness-shear vibrations of a plate and torsional vibrations of a circular cylinder in1963 and had been utilized by Zhou and Li [18] in 2001 to study the length scales in the static and dynamic torsion of a circular cylindrical micro-bar. In 1994, Fleck and Hutchinson [19] extended and reformulated the classical couple stress theory and renamed it the strain gradient theory, in which for homogeneous isotropic and incompressible materials, two additional higher order material length scale parameters are introduced for couple stress theory and three additional higher order material length scale parameters are introduced for stretch and rotation gradient theory.
Taking into account the difficulties in determining the material length scale parameters and the approximate essentiality of beam theories, it is advantageous to model beams with only one additional material length scale parameter. A modified couple stress theory [20] has recently been elaborated by Yang et al. in 2002, in which constitutive equations involve only one additional internal material length scale parameter besides two classical material constants. Park and Gao [21] have studied the static mechanical properties of Bernoulli–Euler beam based on the modified couple stress theory and explained bending test of the epoxy polymeric beams successfully in 2006.
Bernoulli–Euler flexural beam are dynamically analysed by analytic means on the basis of the simple theory of gradient elasticity due to Aifantis [22]. Free and forced flexural vibrations of simple beams are studied and the results show that natural frequencies for all natural modes are size dependence. Recently, the resonant frequency of a micro beam has been analysed theoretically using couple stress theory and the results show that the resonant frequency is size dependence [23].
The purpose of this work is to establish a dynamic model for Bernoulli–Euler beams using both the basic equations of the modified couple stress theory and Hamilton principle. The material is assumed to obey the modified couple stress theory, as described in [20]. The rest of the paper is structured as follows. The basic equations of modified couple stress theory are reviewed in Section 2. Then the displacement field typical for a Bernoulli–Euler beam is established in Section 3. The variational principle is used to obtain the governing equation, initial conditions and boundary conditions for the Bernoulli–Euler beams. The resulting dynamic model involves only one additional internal material length scale parameter and can describe the size effect. To validate the dynamic model, two representative boundary problems (one for simply supported beam and another for cantilever beam) are solved and the size effect on the beam’s dynamic response are assessed in Section 4. The difference between the new dynamic beam model and the classical beam theory are shown and analysed. The paper gives a conclusion in Section 5.
Section snippets
The modified couple stress theory
The modified couple stress theory was presented by Yang et al. in 2002, in which the strain energy density is a function of both strain tensor (conjugated with stress tensor) and curvature tensor (conjugated with couple stress tensor) [20]. Then the strain energy U in a deformed isotropic linear elastic material occupying region Ω is given bywhere the stress tensor, σij, strain tensor, εij, deviatoric part of the couple stress tensor, mij, and symmetric
Dynamic models of Bernoulli–Euler beams
The Cartesian axes for plane beam analysis are established, as shown in Fig. 1. The origin is placed at the leftmost section. The total length (or span) of the beam member is L.
According to the basic hypotheses of Bernoulli–Euler beams and the one-dimensional beam theory, the displacement field can be written as [24]where u, v, w are the x-, y-, and z- components of the displacement vector, and ψ(x) is the rotation angle of the centroidal axis of the beam given
Simply supported beam
Gradient elastic flexural beams are dynamically analysed by analytic means based on the simple gradient elasticity theory due to Aifantis and the governing equation and all possible boundary conditions are obtained [22]. Dynamic analysis of a simply supported and a cantilever beam are presented and size dependence of beams’ natural frequencies has been demonstrated. Now, the problems of simply supported beam and cantilever beam with rectangular and circular cross-sectional shape are solved by
Conclusions
The dynamic problems of Bernoulli–Euler beams are solved analytically on the basis of modified couple stress theory proposed by Yang et al. in 2002. The governing equations of equilibrium and boundary conditions are obtained by a combination of the basic equations and Hamilton’s principle. Two boundary value problems (one for simple supported beam and another for cantilever beam) are solved and the size effect on the beam’s natural frequencies for two kinds of boundary conditions are assessed.
Acknowledgements
The work reported here is funded by the National Natural Science Foundation of China (Grant No. 10572077), the Chinese Ministry of Education University Doctoral Research Fund (Grant No. 20060422013), and the Shandong Province Natural Science Fund (Grant No. Y2007F20).
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