Nonlinear modeling and size-dependent vibration analysis of curved microtubes conveying fluid based on modified couple stress theory

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Abstract

In this paper, a nonlinear theoretical model for three-dimensional vibration analysis of curved microtubes conveying fluid with clamped–clamped ends is developed and analyzed based on a modified couple stress theory and the Hamilton’s principle. This new theoretical model contains a material length scale parameter that can capture the size effect. In-plane and out-of-plane bending motions, axial motion and twist angle of the microtube are considered in the proposed model. The Lagrange nonlinear axial strain is adopted to obtain the static deformation induced by internal fluid flow. The derived equations of motion are discretized through the Galerkin method. Linearized equations around the static deformation are obtained from the discretized equations, and then the evolution of in-plane and out-of-plane natural frequencies for the curved microtube with various values of flow velocity and material length scale parameter is investigated. The results show that size effect on the vibration properties is significant when the characteristic size of the microtube is comparable to the internal material length scale parameter, and no instabilities are possible for curved microtubes if the nonlinear axial deformation is considered. Therefore, both the size effect and the axial nonlinearity have to be incorporated in the design of curved microscale beam/tube devices and systems.

Introduction

The dynamics of tubes conveying fluid is of considerable interest in engineering, and has been studied extensively for a long time. In several monographs (Ibrahim, 2010, Ibrahim, 2011, Païdoussis, 1998, Païdoussis and Li, 1993), a perspective review of the available research on this dynamical problem was presented. Païdoussis and Li (1993) showed that this system is fast becoming a new paradigm in dynamics and capable of exhibit richer and more variegated dynamical behaviors than the classical problem of the column subjected to compressive loading. Due to the recent technological developments in science and engineering, the characteristic sizes of tubes could become smaller and smaller. Thus, microscale tubes have held wide applications in micro-electronic-mechanical systems (MEMS) and engineering micro-fluidic devices, with potential applications as nanopipettes, resonators, fluid filtration devices, targeted drug delivery devices and fountain pen nanochemistry for chromium etching (De Boer et al., 2004, Gao and Bando, 2002, Kim et al., 2005, Longhurst and Quirke, 2007, Moser and Gijs, 2007, Pagona and Tagmatarchis, 2006), in view of their hollow geometry and excellent mechanical properties. For example, the hollow geometry can be considered to deliver drugs in targeted cancer therapy which results in a rapid decrease in tumor size (Bhirde et al., 2009). Fluid-conveying microtubes also can be utilized to be a class of microresonators (Najmzadeh, Haasl, & Enoksson, 2007), as shown in Fig. 1(a). It enable resonant sensing in a manner that combines a high mechanical quality factor Q (in excess of 10 000) with the ability to interrogate a fluidic specimen (Burg et al., 2007). Another device, aptly described as a “micromachined fountain pen”, is used for local surface patterning and direct-write lithography (Deladi et al., 2005, Kim et al., 2005). These microtubes are connected to a fluidic reservoir at one end, and eject fluid from the other end (see Fig. 1(b)); hence, preventing any flow-induced instability is a key requirement for successful operation (Rinaldi, Prabhakar, Vengallatore, & Païdoussis, 2010). It is more than likely that more new applications will be found in the near future.

Much of the early work in this area was focused on the behavior of internal fluids flowing through microtubes or microscale channels (Martin and Kohli, 2003, Whitby and Quirke, 2007). Recently, however, it was also reported that preventing any flow-induced vibration and instability is a key requirement for successful operation in micro-fluidic devices (Dai et al., 2014, Rinaldi et al., 2010, Wang, 2009, Wang, 2010, Xia and Wang, 2010). Therefore, the behavior of microtubes with internal flowing fluids is fast becoming an active topic of current interest.

The classical couple stress elasticity theory (Mindlin, 1964, Mindlin and Tiersten, 1962, Toupin, 1962) is a higher order continuum theory that contains four material constants (two classical and two additional) for isotropic elastic materials. Recently, a modified couple stress theory, in which the couple stress tensor is symmetric and only one internal material length scale parameter is involved, was developed by Yang, Chong, Lam, and Tong (2002). The modified couple stress theory has been widely used to study the mechanical and dynamical behavior of microbeams in the past (Ghayesh et al., 2014, Mohammad-Abadi and Daneshmehr, 2014a, Mohammad-Abadi and Daneshmehr, 2014b, Rahmani and Pedram, 2014, Şimşek and Reddy, 2013, Yin et al., 2010, Zeighampour and Beni, 2014). More importantly, the bending rigidity of epoxy polymeric beams predicted by the modified couple stress theory agrees well with that obtained experimentally (Park & Gao, 2006). However, the literature about the size effect of microscale tubes conveying fluid is relatively limited. Until recently, Rinaldi et al. (2010) modeled a class of microresonators that may be characterized as microtubes containing internal fluid flow. They derived the linear equation of motion based on the classical Euler–Bernoulli beam theory. Wang (2010) presented a new linear model for the vibration analysis of fluid-conveying microtubes based on the modified couple stress theory. In that paper, the micro-structure dependent size effect was taken into account. Xia and Wang (2010) formulated the microfluid-induced vibration and instability using the non-classical Timoshenko theory. Yang, Ji, Yang, and Fang (2014) investigated the microfluid-induced nonlinear free vibration of microtubes based on a modified couple stress theory, and the geometric nonlinearity, arising from the mid-plane stretching is taken into account.

It should be stressed that, most of these studies examined the dynamics of microscale tubes with only straight configurations along the tube length, while in practice this class of systems is usually shaped as curved configurations. For example, microbeams that are fabricated intentionally to be curved were found to be suitable for applications such as micro-shutter positioning, microvalves, microresonators and electrical microrelays (Ouakad & Younis, 2010). Unfortunately, only few papers can be found related to the vibration characteristics of curved microtubes in the literature. Recently, The linear vibration characteristics of fluid-conveying microtubes with curved longitudinal shape are investigated by Wang, Xu, and Ni (2013) and Wang, Liu, Ni, and Wu (2013) by adopting the straight-beam element. However, with the reduction in the radius of curved portions, the stress due to curvature plays more important role (Ibrahim, 2010, Ibrahim, 2011) and the basic assumptions relevant to straight beams become less satisfactory. Thus further studies would be needed to reveal the dynamical behavior of the curved microtube conveying fluid.

To the best of our knowledge, no investigation has been performed on the three-dimensional nonlinear vibration of curved microtubes conveying fluid to date. The objective of the present paper is to establish a framework for investigating the in-plane and out-of-plane vibration analysis of curved microtubes conveying fluid using nonlinear theory. The system is modeled as curved Euler–Bernoulli microscale tubes conveying fluid. The Lagrange axial strain is considered to describe the extensibility of centerline. The strain energy of tube based on the modified couple stress theory, the kinetic energy of tube and internal flow, and their first variations are obtained. Hamilton’s principle is then applied to derive the governing equations and boundary conditions for the fluid-loaded microtube. The in-plane bending, out-of-plane bending deformations, axial deformation and torsion angle are considered. The resulting curved tube model contains an internal material length scale parameter and can capture the size effect. Based on the derived equation of motion, the static deformation and stability of the microtube will be studied. It will be shown that the influences of size effect and the nonlinear axial strain on the natural frequencies are significant.

Section snippets

Derivation of the equations of motion

Consider a semi-circular tube conveying fluid, as shown in Fig. 2, which is clamped at both ends. The tube consists of a centerline radius R, a centerline length L, a cross-sectional outer diameter Do, inner diameter Di, mass per unit length m, conveying incompressible plug fluid of mass per unit length M, flowing axially with velocity U.

Using the curvilinear coordinate system (x, y, z) with the y-axis being coincident with the centroidal axis of the undeformed curved tube, the x-axis is the

Method of solution

The infinite-dimensional tube model is described by four partial differential equations, i.e., Eqs. (15), (16), (17), (18). In order to find the approximate solutions in a finite-dimensional function space, the continuous system is discretized by a Galerkin’s technique. The deformations of the tube can be expressed asu(y,t)=r=1Nur(y)qru(t)=uTqu,v(y,t)=r=1Nvr(y)qrv(t)=vTqv,w(y,t)=r=1Nwr(y)qrw(t)=wTqw,ϕ(y,t)=r=1Nϕr(y)qrϕ(t)=ϕTqϕ,where N is the total number of the basis functions. In this

Results

The three-dimensional nonlinear equations for curved microscale tube are derived for the first time in this study. In this section some representative results are presented in order to assess the influence of Lagrange nonlinear axial strain and size effect.

Before doing so, however, it is of importance to define several necessary parameter values for calculations. For convenience of illustration, the microtube considered here is taken to be made of epoxy (Lam, Yang, Chong, Wang, & Tong, 2003),

Conclusions

In this paper, a nonlinear theoretical model for three-dimensional vibration analysis of curved microtubes conveying fluid has been developed. The governing equations of in-plane and out-of-plane bending motions, axial motion and torsional angle were obtained based on a modified couple stress theory and Hamilton’s principle considering the Lagrange geometric nonlinearity.

Based on the derived equations of motion, the vibration characteristics of the system were studied after the Galerkin

Acknowledgments

The financial support of the National Natural Science Foundation of China (Nos. 11172109 and 11172107) and the Fundamental Research Funds for the Central Universities, HUST (2014YQ007) are gratefully acknowledged.

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