On large deformation and stability of microcantilevers under follower load

https://doi.org/10.1016/j.ijengsci.2021.103549Get rights and content

Abstract

In the present study, a novel geometrically exact nonlinear beam model based on the nonlinear displacement field, centerline-inextensibility condition, Euler-Bernoulli beam hypothesis, and modified couple stress theory is established to investigate the extremely large deformations and stability characteristics of size-dependent microcantilevers subjected to a tip-concentrated non-conservative (follower type) load with arbitrary inclination angle. The newly developed mathematical formulation explains the size effect phenomenon in the microcantilevers with only one material length scale parameter. The nonlinear shooting method for two-point boundary value problems is utilized to assess the deformed configuration of microsystem based on the quasistatic form of nonlinear mathematical model. Afterwards, the linearized dynamical mathematical model around the nonlinear deformed configurations of microsystem is numerically solved using the Galerkin technique to examine the stability of obtained responses. A comprehensive investigation is then conducted to highlight the role of size-dependency in the critical follower force at which the microsystem undergoes a flutter instability, as well as the nonlinear large deformation of microsystem in the pre-flutter region.

Introduction

Microcantilever beams are extensively utilized as a key component in design and fabrication of microscale devices and machines such as microsensor (Zhao et al., 2021), microactuator (Chen et al., 2020), microresonator (Park et al., 2019), microswitch (Kalafut et al., 2020), microgyroscope (Selvakumar et al., 2021), micro energy harvester (Gupta et al., 2019), and atomic force microscope (Ruppert et al., 2021). Experimental examination (Lam et al., 2003; Lei et al., 2016; Li, He et al., 2019; Li, He et al., 2019; Liebold & Müller, 2016; Tang & Alici, 2011) revealed that their mechanical behavior may be extremely dependent on their sizes. Several higher-order elasticity theories have been developed and employed to interpret the impact of size effect phenomenon on mechanical responses of small-scale structures (Babaei, 2021; Barretta et al., 2020; Barretta et al., 2019; Chen et al., 2019; Darban et al., 2020; Dastjerdi & Akgöz, 2019; Dastjerdi et al., 2020; Farajpour et al., 2020; Ghayesh & Farajpour, 2019; Ghayesh et al., 2019; Gholipour & Ghayesh, 2020; Jalaei & Civalek, 2019; Khaniki, 2019; Malikan et al., 2020; Malikan et al., 2020). (Yang et al., 2002) established a modified couple stress based strain gradient theory which is capable of incorporating the size effect phenomenon in homogeneous isotropic microstructures with only one additional material length scale parameter (Dehrouyeh-Semnani & Nikkhah-Bahrami, 2015; Kwon & Lee, 2020), together with two Lamé constants. This non-classical elasticity theory has been widely used in many researches to analyze the size-dependent behaviour of microstructures (Awrejcewicz et al., 2021; Dehrouyeh-Semnani & Mostafaei, 2021a, b; Enayati & Dardel, 2019; Farokhi & Ghayesh, 2018; Gao et al., 2013; Karimipour et al., 2021; Kim & Reddy, 2013, 2015; Kim et al., 2019; Ma et al., 2011; Reddy & Kim, 2012; Shariati et al., 2020; Taati, 2016; Zhang & Gao, 2019).

Development and utilization of size-dependent beam models based on the modified couple stress theory for studying the role of size effect phenomenon in the beam-type microstructures have been the subject of many studies. The size-dependent linear beam modeling based on the modified couple stress theory and different beam hypothesizes for the static, stability, and dynamic analyses of beam-type microstructures can be found in (Akgöz & Civalek, 2011; Dehrouyeh-Semnani et al., 2016; Kong et al., 2008; Ma et al., 2010; Ma et al., 2008; Park & Gao, 2006; Reddy, 2011; Reddy & Arbind, 2012; Zhang & Liu, 2020). Additionally, the size-dependent nonlinear beam modeling for the moderately large-amplitude analysis of beam-type microstructures based on the Von Karman nonlinearity and the modified couple stress theory can be found in (Arbind & Reddy, 2013; Arbind et al., 2014; Babaei & Eslami, 2019; Babaei & Eslami, 2020; Chen et al., 2019; Das, 2019; Ghayesh, 2019; Ghayesh & Farokhi, 2017; Ghayesh et al., 2019; Ghayesh et al., 2016). (Dai et al., 2015) proposed a new nonlinear mathematical model for the microcantilevers by replacing the classical bending rigidity with the non-classical one in the nonlinear classical model. It should be noted that in the aforementioned paper, no mathematical background was reported for the replacement and it seems that it was inspired from the linear model. (Li, Chen, Zhang & Liao, 2019) employed the aforementioned nonlinear model for the large-deformation analysis of microcantilevers. (Kiani, 2017) studied analytically and numerically the buckling (static instability) and large deformation post-buckling of microbeams under uniaxial loading by using a nonlinear inextensible beam model based on a linear displacement field. (Farokhi et al., 2017; Farokhi et al., 2016; Ghayesh & Farokhi, 2018; Ghayesh et al., 2017) investigated the large-amplitude dynamic characteristics of inextensible slender microcantilevers under a harmonic excitation based on an approximated model with third-order nonlinearity derived from an exact geometrically beam model in terms of its longitudinal and transverse displacements. It should be pointed out that in the aforementioned works, additional nonlinear boundary conditions due to the size-dependency were neglected. The aforementioned approximated model was used by (Farokhi et al., 2018, b) to study the electro-mechanical behaviour of electrically actuated small-scale cantilevers with and without geometric imperfection.

The available studies related to the mechanics of size-dependent microbeams concentrated on conservative loading, and the available analyses of microbeams under non-conservative loading didn't take into account the impact of size-dependency. Therefore, the current investigation aims to study the nonlinear mechanics of microcantilevers under non-conservative (follower type) load by incorporating the effect of size-dependency based on the modified couple stress theory, and developing a new mathematical model for the extremely large deformation of microbeams. The outline of this paper is as follows: In the first part of second section, the geometrically exact nonlinear mathematical model of size-dependent inextensible microbeams in terms of its rotation is established by implementation of the modified couple stress theory and the Euler-Bernoulli beam hypothesis. The solution procedures for both the large deformation and stability analyses are then proposed in the second part of this section. In the first part of third section, some verification studies are conducted and accuracy of quasistatic solution for the large deformation is then discussed. In the second part of this section, the role of size-dependency, in conjunction with the follower force's parameters in the large deformation and stability characteristics of microcantilevers are examined in detail. In the last part of this section, accuracy of assumption used in the potential energy of microcantilever is investigated and the impact of size-dependent and size-independent potential energies on the total potential energy of microsystem is then studied. Lastly, the paper is summarized in the fourth section and the main conclusions are presented.

Section snippets

Modeling and solution methods

Fig. 1 illustrates the schematic representation of microsystem including an Euler-Bernoulli microcantilever of length L, cross sectional area A, Young's modulus E, shear modulus G, material length scale parameter l, and mass density ρ, which is subjected to a tip-concentrated non-conservative (follower type) force p. The force's inclination angle α with respect to the beam's deformed axis remains constant during deformation.

Verification study and discussion

(Liebold & Müller, 2016) experimented the static behavior of Su-8 and epoxy microcantilevers exposed to a tip-concentrated force (F) acting in a normal direction to the undeformed axis of the microbeam. The experimental data reported by the aforementioned reference indicated that the static responses of these materials in microscale are size-dependent when the microcantilever's thickness is small enough. The rectangular cross-sectional microcantilever of thickness h, length L=20h, and width b=2h

Conclusions

A new nonlinear mathematical beam model is proposed to analyze the extremely large-amplitude statics and dynamics of the size-dependent inextensible slender microbeams. The governing equation and corresponding boundary conditions are established in terms of the microbeam's rotation by aid of the modified couple stress theory and the Euler-Bernoulli hypothesis. The mechanics of a microcantilever under a tip-concentrated non-conservative (follower type) load is investigated in detail. The

Declaration of Competing Interest

The authors declare no conflict of interest.

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