Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory

doi:10.1016/j.mechrescom.2014.06.003

Mechanics Research Communications

Volume 60, September 2014, Pages 74-80

https://doi.org/10.1016/j.mechrescom.2014.06.003 Get rights and content

Highlights

•
We present a new model for transverse vibrations of an Euler-Bernoulli beam using the couple stress theory with micro-inertia effects.
•
This study revealed that the results of the couple stress are closer to the experimental results than modified couple stress theory.
•
The results showed that the micro-inertia depends on the ratio of the length scale to the beam length.
•
The effects of the micro-inertia are more important for lower values of the ratio of the length scale to the beam length.

Abstract

This paper presents a new model for the free transverse vibrations of an Euler–Bernoulli beam using the couple stress theory of elasticity with micro-structure. Introducing the kinematic variables, the strain and kinetic energy expressions (involving micro-inertia effect) have been obtained and the Hamilton principle has been used to derive the governing equations and the related boundary conditions of the free vibrations of fixed–fixed and simply supported beams. A numerical solution has been used to study the natural frequencies, mode shapes and free vibrations of the beams. A comparative result has shown that the bending rigidity predicted by the couple stress, is closer to the experiment result than that predicted by the modified couple stress theory. The results have shown that the bending rigidity of the beams depends on the ratio of the length scale to the beam thickness, whereas the micro-inertia term depends on the ratio of the length scale to the beam length.

Introduction

Beams and plates are key components of many engineering structures from nano to macro scales. There are several theories that deal with the mechanical behavior of these components. In the classical theory of linear elasticity, the material is considered as a continuum in mathematical sense. In such continuum the atomic structure of the material is neglected and the material particle is considered simply a geometrical point. This theory is inadequate for describing the mechanical behavior of materials with microstructure, such as polymeric foams, high-toughness ceramics, high strength metal alloys, granular materials or porous bones, because their behavior is characterized by non-local stresses and the existence of an internal length scale. Micro-structural effects are also important when structures have extremely small overall dimensions, which are comparable to the internal length scale of their material (Papargyri-Beskou and Beskos, 2008). Voigt was the first who tried to correct these shortcomings of classical elasticity by taking into account the assumption that interaction between the two parts through an area element inside the body is transmitted not only by a force vector but also by a moment vector giving rise to a ‘couple stress theory’ (Voigt, 1887). The complete theory of asymmetric elasticity was developed by Cosserat and Cosserat (1909), which was non-linear in the beginning. They assumed that each material point of a three dimensional continuum is associated with a ‘rigid triad’ and during the process of deformation; it can rotate independently in addition to the displacement. After a gap of about fifty years, Cosserats theory drew attention of researchers and several Cosserat-type theories were developed independently (e.g., Aero and Kuvshinskii, 1960, Eringen, 1962, Grioli, 1960, Gunther, 1958, Koiter, 1964, Mindlin and Tiersten, 1962, Nowacki, 1974, Palmov, 1964, Rajagopal, 1960, Toupin, 1962), among several others. Later, the general Cosserat continuum theory acquired the name of ‘micropolar continuum theory’ following Eringen, 1966a, Eringen, 1966b, in which the micro-rotation vector is taken independent of displacement vector. Eringen and Suhubi (1964) and Suhubi and Eringen (1964) developed a non-linear theory for ‘micro-elasticity’, in which intrinsic motions of the microelements were taken into account. A further generalization of the continuum with microstructure leads to micromorphic continuum of Eringen, 1966a, Eringen, 1966b. Micromorphic continuum treats a material body as a continuous collection of a large number of deformable particles, with each particle possessing finite size and inner structure. Using assumptions such as infinitesimal deformation and slow motion, micromorphic theory can be reduced to microstructure theory of Mindlin (1964). When the microstructure of the material is considered rigid, it becomes the Eringen's micropolar theory (Eringen, 1966a, Eringen, 1966b). Assuming a constant micro-inertia, Eringen's micropolar theory is identical to the Cosserats theory. Eliminating the distinction of macro-motion of the particle and the micro-motion of its inner structure, it becomes couple stress theory (Mindlin and Tiersten, 1962, Toupin, 1962). Moreover, when the particle reduces to a mass point, all theories reduce to classical or ordinary continuum mechanics.

The general theory of Mindlin includes three equivalent forms which are defined on the basis of three different expressions for the strain energy density. The first expression involves gradients of displacements, the second gradients of strain and the third gradients of rotation. The couple stress theory is based on this third expression of the strain energy density while second form leads to the gradient elastic theory.

The classical couple stress elasticity theory is a higher order continuum theory that contains two higher-order material length-scale parameters appear in addition to the two classical Lame constants. In this classical conception, only the conventional equilibrium relationships of forces and moments (of forces) are enforced and the couple is unconstrained in the absence of higher order equilibrium requirements. The couple stress theory has been applied to model the pure bending of a circular cylinder by Anthoine (2000). He has reported that the bending inertia of a circular cross-section results in higher values than those accepted before, especially when the ratio of the radius of the beam to the characteristic material length is lower than 20.

Yang et al. in 2002 introduced the modified couple stress theory. Beside the two conventional equilibrium relationships in the classical couple stress, they proposed an additional relation to constrain the couple. This relation considers the balance of moment of rotational momentum. This assumption make the couple stress tensor symmetric. Utilizing the modified couple stress theory, Park and Gao (2006) studied the static response of an Euler–Bernoulli beam and interpreted the outcomes of an epoxy polymeric beam bending test. Kong et al., 2008, Kong et al., 2009 derived the governing equation, initial and boundary conditions of an Euler–Bernoulli beam using the modified coupled stress theory and strain gradient elasticity theory. As they reported, the stiffness of beams is size-dependent. In addition, the difference between the stiffness obtained by the classical beam theory and those predicted by the modified couple stress theory is significant when the beam characteristic size is comparable to the internal material length-scale parameter. Recently, Fathalilou et al. (2011) have used the modified couple stress theory to study the pull-in instability of a gold micro-beam switch with the specifications introduced in the experimental work of Ballestra et al. (2010). As they reported, although using the modified couple stress theory leads to better results than the classic theory, yet there is a considerable difference between the results of the experiments and the modified couple stress theory.

Beam bending models based on other non-classical elasticity theories have also been reported. Papargyri-Beskou et al. (2003) have derived and solved the governing equation and corresponding boundary conditions of the beam buckling and bending using the simple strain gradient theory. Lazopoulos and Lazopoulos (2010) have studied the bending and buckling problem of thin beams using strain gradient theory with the terms depending upon the area of the cross-section of the beam.

In spite of mentioned studies about the mechanical behavior of Euler–Bernoulli beams using various elasticity theories, there is no comprehensive modeling of beams in the literature using the couple stress theory of elasticity with micro-structure. The objective of this paper is to introduce a non-classic model for the free vibrations of an Euler–Bernoulli beam using the concepts of the couple stress theory. The present model involves the micro-rotation effects leading to an added inertia in the dynamic motion. The governing equation and boundary conditions for the beams are obtained using the Hamilton principle. Unlike the existence of two non-classic material length scale in the couple stress theory for the general continuum, only one length scale parameter is appeared in the beam model. A comparative result shows that the bending rigidity predicted by the couple stress, is closer to the experiment than that predicted by the modified couple stress theory.

Section snippets

Fundamental equations of the couple stress theory

In the linear couple stress theory, the strain energy, in addition to the strain, is a function of the rotation-gradient (Mindlin, 1964). In the following sub-sections the helpful kinematic variables, strain and kinetic energies and constitutive equations of this theory are presented.

Governing equations of the Euler–Bernoulli beam vibrations

Fig. 1(a and b) shows the fixed–fixed and simply supported Euler–Bernoulli beams, respectively. Primarily, it is mentioned that the i, j and k indices in the previous section varied from 1 to 3, introduce the variables in x, y and z directions in Cartesian coordinates, respectively. Using the coordinate system (x,z) shown in Fig. 1, where x-axis coincides with the centroidal axis of the undeformed beam and z-axis is the symmetry axis, the displacement components in an Euler–Bernoulli beam can

Numerical solution

In order to obtain the natural frequencies and mode shapes of the beam an eigen-value problem should be solved. Hence, the w can be considered as follows: $w (\hat{x}, \hat{t}) = ϕ (\hat{x}) e^{j \hat{ω} \hat{t}}$

Substituting this relation to Eq. (28) leads to $(1 + α - γ {\hat{ω}}^{2}) ϕ^{i v} + β {\hat{ω}}^{2} ϕ^{″} - {\hat{ω}}^{2} ϕ = 0$

Considering the micro-inertia effect make this equation have a second derivative term, so the problem becomes complicated in comparison with no micro-inertia case.

The characteristic equation is written as $(1 + α - γ {\hat{ω}}^{2}) r^{4} + β {\hat{ω}}^{2} r^{2} - {\hat{ω}}^{2} = 0$

Solving this equation

Numerical example

In order to validate the proposed procedure, it is considered a fixed–fixed gold micro-beam switch with the material properties of $E = 98.5 GPa, G = 27 GPa, ν = 0.44$ and geometrical properties of $L = 541.8 μ m, b = 32.2 μ m, h = 2.68 μ m$ as given by Ballestra et al. (2010). Table 1 compares the experimental bending rigidity of the micro-beam with those obtained by the classic, modified couple stress and couple stress theories. As shown, the result predicted by the couple stress theory is closer to the

Conclusion

In the present work, a new model was proposed for the free transverse vibrations of the Euler–Bernoulli beam using the couple stress theory of elasticity with micro-structure. This model involves the micro-inertia effects introduced in Mindlin's general continuum theory. It was shown that the couple stress theory results in one internal length scale parameter for the beam, unlike the general continuum where two material non-classic constants appear. The Hamilton principle was used to derive the

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On size-dependent Timoshenko beam element based on modified couple stress theory
2016, International Journal of Engineering Science
Due to complexity of geometry of small-scale structures as well as size effect phenomenon in such scale, size-dependent finite element method can be employed as a powerful numerical technique to investigate the mechanical behavior of such structures. Recently, two distinct size-dependent Timoshenko beam elements have been proposed based on modified couple stress theory. The first beam element is a two-node element which has 3-DOF (degrees of freedom) at each node. The second beam element is also a two-node element, but it has 2-DOF at each node. Since enough verification and convergence studies have not been performed on the proposed beam elements, the present study aims to examine the accuracy, reliability and stability of aforementioned beam elements in the static bending. To that end, the cantilevered, simply supported and doubly clamped Timoshenko beams are chosen as the case studies. It is observed that the results obtained via the 6-DOF beam element are in excellent agreement with those obtained via the other solutions for the three case studies. In addition, it is found that the rate of convergence increases by ascending the influence of size-dependency. Although the 4-DOF beam element presents stable solutions, the acquired results reveal that the 4-DOF beam element is incapable of verifying the results obtained based on the other solutions for the three case studies. Moreover, it is found that the 4-DOF beam element error has an ascending trend with respect to the size-dependent shear deformation. Finally the reason for inaccuracy of the 4-DOF beam element is discussed.
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Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory

Highlights

Abstract

Introduction

Section snippets

Fundamental equations of the couple stress theory

Governing equations of the Euler–Bernoulli beam vibrations

Numerical solution

Numerical example

Conclusion

Int. J. Solids Struct.

Int. J. Eng. Sci.

Int. J. Solids Struct.

Int. J. Eng. Sci.

Int. J. Eng. Sci.

J. Mech. Phys. Solids

Eur. J. Mech. A/Solids

J. Mech. Phys. Solids

Int. J. Solids Struct.

Int. J. Solids Struct.

Fundamental equations of the theory of elastic media with rotationally interacting particles

Fiz. Tverd. Tela

FEM modelling and experimental characterization of microbeams in presence of residual stress

Analog Integr. Circuits Signal Process.

Theorie des Corps Deformables

Nonlinear Theory of Continuous Media

Linear theory of micropolar elasticity

J. Math. Mech.

Theory of micropolar fluids

J. Math. Mech.