Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material

Abstract

Size-dependent nonlinear Euler-Bernoulli and Timoshenko beam models, which account for the through-thickness power-law variation of two-constituent functionally graded (FG) materials, are derived to investigate the nonlinear bending and free vibration behaviors in the framework of the nonlocal strain gradient theory. The nonlinearity due to the stretching effect of the mid-plane of the FG beam is the source of nonlinearity of the considered bending and free vibration problems. The size-dependent equations of motion and boundary conditions are derived by employing the Hamilton’s principle. The beam models contain material length scale and nonlocal parameters to consider the effects of both inter-atomic long-range force and microstructure deformation mechanism. In the case of hinged-hinged boundary conditions, the analytical solutions for the nonlinear bending deflection and free vibration frequencies of nonlocal strain gradient Euler-Bernoulli and Timoshenko beams are deduced. The influences of the through-thickness power-law variation of a two-constituent FG material and size-dependent parameters on nonlinear bending deflection and free vibration frequencies are investigated. Due to the intrinsic stiffening effect brought by the stretching effect of the mid-plane of the beam, the nonlinear bending deflections are smaller than their linear counterparts under the action of the same force, while the nonlinear vibration frequencies are higher than their linear counterparts for the same amplitude of the nonlinear oscillator. The nonlinear bending deflections and free vibration frequencies can be affected significantly by the through-thickness grading of FG materials in the beam. When the nonlocal parameter is smaller than the material characteristic parameter, the nonlinear FG beam reveals a stiffness-hardening effect. When the material characteristic parameter is smaller than the nonlocal parameter, the FG beam reveals a stiffness-softening effect.

Introduction

Functionally graded (FG) materials, as a new class of advanced composites, vary their microstructures from one material to another with a chosen gradient, resulting in corresponding changes in their effective material properties (such as the effective Young’s modulus, the effective shear modulus and the effective material density). Traditional laminated composites may cause some unexpected problems (including high shear stress, interface cracking, and interface delamination) due to sudden changes of material properties at the interface of two adjacent layers. The FG materials can show many advantages over the traditional laminated composites due to the smooth variation of material properties and their excellent performances in various engineering fields of application, including higher fracture toughness, enhanced thermal resistance, enhanced corrosion resistance, improved stress spreading and inferior stress intensity factors (Jha, Kant, & Singh, 2013). To improve the performance of composite structures, the development of the FG materials are being accelerated to optimize some certain functional properties of structures by tailoring the material architecture at nano/micro scale. The rapid developments of microelectromechanical systems (MEMS) and nanoelectromechanical (NEMS) make the FG materials possible to be applied in nano/micro scaled systems. Nevertheless, the possible applications rely on a good understanding of the mechanical characteristics (such as bending and vibration) of FG structures. Significant size-dependent effects on the mechanical characteristics have been observed for small-scaled structures. Thus, the study of size-dependent effects on the bending and vibration behaviors of nano/micro-scaled FG beams is always of fundamental significance. The bending and vibration behaviors of small-scaled FG beams may be not predicted adequately by employing the classical continuum theory, some size-dependent elasticity mechanics (such as nonlocal elasticity theory, strain gradient theory and nonlocal strain gradient theory) have been successfully developed and employed to assess the size-dependent effect on the mechanical characteristics of small-scaled structures.

The nonlocal elasticity theory (Eringen, 1983), in which the stress field at a reference point is not only dependent on the strain at the reference point but also dependent on the strains at all other points in the whole body, can account for the inter-atomic long-range force. A lot of nonlocal beam models (Daneshmehr, Rajabpoor, Hadi, 2015, Daneshmehr, Rajabpoor, et al., 2014, Eltaher, Emam, Mahmoud, 2012, Eltaher, Emam, Mahmoud, 2013, Nazemnezhad, Hosseini-Hashemi, 2014, Rahmani, Pedram, 2014, Salehipour, Shahidi, Nahvi, 2015, Şimşek, Yurtcu, 2013, Uymaz, 2013) have been developed to study the static and dynamic behaviors of FG beams based on the nonlocal elasticity theory. Recently, Ghadiri and Shafiei (2015) studied the size-dependent effect on the nonlinear vibration behaviors of a rotating nanobeam based on nonlocal elasticity theory using a differential quadrature method. Nejad, Hadi, 2016a, Nejad, Hadi, 2016b; Nejad, Hadi, and Rastgoo (2016) studied the size-dependent effect on the bending, buckling and free vibration problems of bi-directional FG Euler-Bernoulli nano-beams based on the nonlocal elasticity theory. These studies showed that the size independency nature of the nonlocal elasticity theory potentially plays a very significant role in studying the static and dynamical behaviors of small-scaled FG beams, and a stiffness softening effect has been observed for these nonlocal beam models.

The gradient elasticity theory (Aifantis, 1992, Mindlin, 1964) states that small-scaled materials must be modeled as atoms with higher-order deformation mechanism instead of collections of points, and the total stress should consider some additional strain gradient terms. Yang, Chong, Lam, and Tong (2002) presented a modified the gradient elasticity theory (or modified couple stress theory), in which the strain energy density needs to be considered as a function of both the strain tensor conjugated with stress tensor and the curvature tensor conjugated with couple stress tensor. By using the modified couple stress theory, Reddy (2011) developed Euler-Bernoulli and Timoshenko beams models to study the bending, vibration and buckling behaviors of FG beams. Şimşek and Reddy (2013) presented a unified beam theory containing various higher-order shear deformation beam theories as well as the Euler-Bernoulli and Timoshenko beam theories for the static bending and free vibration analysis of FG microbeams based on the modified couple stress theory. Şimşek (2014) investigated the nonlinear size-dependent static and free vibration characteristics of microbeams based on the nonlinear elastic foundation within the framework of the modified couple stress theory. Akgöz, Civalek, 2014a, Akgöz, Civalek, 2014b developed higher-order shear deformation beam models of FG microbeams based on the modified gradient elasticity theories. Lou and He (2015) investigated the nonlinear bending and free vibration characteristics of a simply supported FG microplate with geometry nonlinearity lying on an nonlinear elastic foundation within the framework of the modified couple stress theory and the Kirchhoff and Mindlin plate theories. Taati (2016) performed the buckling and post-buckling analysis of size-dependent FG plates based on the modified couple stress theory. Shafiei, Mousavi, and Ghadiri (2016) studied the transverse vibration of a rotary tapered axially FG microbeam based on the modified couple stress theory in the form of true spatial variation. Shafiei, Kazemi, and Ghadiri (2016) investigated the nonlinear size-dependent free vibration characteristics of a non-uniform axially FG microbeam with geometric nonlinearity based on the Euler-Bernoulli beam theory and the modified couple stress theory. Dehrouyeh-Semnani, Mostafaei, and Nikkhah-Bahrami (2016) investigated the size-dependent free vibration behaviors of FG microbeams with geometric imperfection based on the modified couple stress theory. Additionally, A lot of microstructure-dependent models (Jabbari, Nejad, Ghannad, 2015, Khorshidi, Shariati, Emam, 2016, Lei, He, Zhang, Liu, Shen, Guo, 2015, Lou, He, Du, Wu, 2016, Lou, He, Wu, Du, 2016, Nejad, Fatehi, 2015, Rahaeifard, Kahrobaiyan, Ahmadian, Firoozbakhsh, 2013, Reddy, Romanoff, Loya, 2016, Shafiei, Mousavi, Ghadiri, 2016, Shirazi, Ayatollahi, 2014, Şimşek, Aydın, Yurtcu, Reddy, 2015, Sourki, Hoseini, 2016, Thai, Kim, 2013, Zhang, He, Liu, Gan, Shen, 2014, Zhang, He, Liu, Shen, Lei, 2015) have been also recently developed and employed to study the static and dynamical behaviors of FG beams and plates based on the modified gradient elasticity theory. These studies showed that the size independence nature of the microstructure deformation mechanism potentially plays a very significant role in studying the static and dynamical behaviors of small-scaled FG beams, and a stiffness enhancement effects have been observed in these gradient elasticity models.

From the literature discussed above, it is found that the nonlocal elasticity theory only takes into account the inter-atomic long-range force. However, similar to classical elasticity theory, the nonlocal elasticity theory states the particles are taken as mass points without considering any microstructure deformation mechanism. The gradient elasticity theory takes into account the higher-order microstructure deformation mechanism without considering any inter-atomic long-range force. The nonlocal elasticity and strain gradient theories describe two entirely different size-dependent nano/micro-mechanical characteristics of materials. Therefore, a detail examination of the effects of the inter-atomic long-range force and the microstructure deformation mechanism on the mechanical behaviors of small-scaled structural systems is a topic worth exploring in understanding how the two size-dependent effects interact with the small-scaled structural systems. Some nonlocal elasticity and strain gradient models have been developed and employed to assess the two size-dependent effects on mechanical behaviors of small-scaled structures (Askes, Aifantis, 2011, Attia, Mahmoud, 2016, Challamel, Wang, 2008, Li, Li, Hu, 2016, Lim, Zhang, Reddy, 2015). It was shown by Challamel, Rakotomanana, and Le Marrec (2009); Li and Hu (2016); Li, Hu, and Ling (2016) that an excellent matching of the dispersive relation of the molecular dynamics simulations (or Born-Kármán model of lattice dynamics) can be obtained with that calculated by using the nonlocal strain gradient models for truss- and beam-type structures. The nonlocal strain gradient beam models have been developed and considered to investigate the wave propagating, vibration, bending and buckling problems (Challamel, 2013, Güven, 2014, Huang, Luo, Li, 2013, Wu, Li, Cao, 2013, Zhang, Wang, Challamel, 2009).

In the context of thermodynamics framework, Lim et al. (2015) stated that the stress accounts for not only non-gradient nonlocal stress field (Eringen, 1983), but also the higher-order pure strain gradient stress field (Aifantis, 1992), and cast the nonlocal elasticity and strain gradient models into a single theory. The nonlocal strain gradient theory considers the influences of the microstructure deformations in conjunction with the lower- and higher-order nonlocal stress field. The nonlocal strain gradient models are probably the most successful of the many size-dependent models to investigate the small-scaled effect since these models consider the effects of both the higher-order microstructure deformation mechanism and the inter-atomic long-range force. Based on the nonlocal strain gradient theory, Li and Hu (2015) examined the size-dependent effects on the post-buckling behaviors of nonlinear Euler-Bernoulli beams and showed that the stiffness softening or enhancement effects are dependent on the values of the nonlocal parameter and the material length scale parameter. Farajpour, Yazdi, Rastgoo, and Mohammadi (2016) studied the effects of the nonlocal parameter and the material length scale parameter on the buckling behavior of graphene sheets by using the nonlocal strain gradient theory. Based on the nonlocal strain gradient theory, Li and Hu (2016); Li et al. (2016); Tang, Liu, and Zhao (2016) carried out the wave propagating analyses in carbon nanotubes (CNTs) by considering the effects of the nonlocal parameter and the material length scale parameter. Based on the nonlocal strain gradient theory, Li, Hu, and Li (2016) investigated the size-dependent effects on the longitudinal vibration analysis of small-scaled rods with different classical and non-classical boundary conditions by using analytical solutions and a finite element method. It is shown that the high-order vibration frequencies are more sensitive to the non-classical (high order) boundary conditions in comparison with the low-order frequencies. Li, Hu, Li, and Ling (2016) developed Timoshenko and Euler-Bernoulli beam models for studying the mechanical behaviors of fluid-conveying microtubes based on the nonlocal strain gradient theory, and investigated the size-dependent effects on critical flow velocity of fluid-conveying microtubes.

The FG materials are gaining broad applications in various branches of engineering structures with a view to designing the potential material properties in the best possible way. In fact, the FG material characteristics can be also observed in some specific structures found in nature (e.g., Bamboo Tree, bones, Human skin and sea shells), and a better understanding of the mechanical behaviors of FG materials may be very helpful for synthesizing and designing new advance materials used in MEMS, NEMS, spacecraft structures, nuclear components, rocket casing and high temperature thermal coatings, etc. To investigate the size-dependent effects on the mechanical characteristics, Li, Hu, and Ling (2015) investigated the size-dependent effects on the flexural wave propagation in small-scaled FG beams via the nonlocal strain gradient theory. Li et al. (2016) deduced the equations of motion and boundary conditions for FG Timoshenko and Euler-Bernoulli beams models within the framework of the nonlocal strain gradient theory, and gave the analytical vibration frequency solutions for FG Timoshenko and Euler-Bernoulli beams models. These two works were carried out based on linear theory and showed that the results regarding wave propagation and natural frequencies of FG material at small-scale are size-dependent. Although the linear theory can be applied for some specific extent, the effect of nonlinearities on the static and dynamical characteristics of FG beams at small scale may be pronounced. Under such case, these models developed based on linear theory cannot be applied for providing reliable results, which have to be reassessed by using these models based on nonlinear theory. Şimşek (2016) examined the size-dependent effects on the nonlinear free vibration behaviors of FG Euler-Bernoulli beams based on the nonlocal strain gradient theory and showed that the two size-dependent parameters are more effective on the nonlinear vibration frequencies at a relatively lower vibration amplitude. However, the Euler-Bernoulli model neglected the effects of shear deformation and rotational inertia, making it unsuitable for describing the nonlinear free vibration behaviors of short beams and beams subject to high-frequency. Under such case, the Timoshenko beam theory may be used to reassess these results. In applied structural mechanics, the bending needs to be considered when the FG beam subjected to an external force applied perpendicularly to the longitudinal axis of the beam. The nonlinear bending may take place when the beam subjected to a large force and should be analyzed by nonlinear theory. This work is motivated by considering these facts.

In the previous work, the Timoshenko beam theory is employed to carry out the nonlinear free vibration analysis so that the effect of shear deformation can be considered and high-order vibration frequencies can be calculated where the Euler-Bernoulli beam theory is inadequate. Size-dependent nonlinear Euler-Bernoulli and Timoshenko beam models, which account for the through-thickness power-law variation of two-constituent FG materials, will be derived to investigate the nonlinear bending and free vibration behaviors within the framework of the nonlocal strain gradient theory. The nonlinearity due to the stretching effect of the mid-plane of the FG beam is the source of nonlinearity of the considered bending and free vibration problems. The size-dependent equations of motion and boundary conditions of the Euler-Bernoulli and Timoshenko beams will be derived by employing the Hamilton’s principle. In the case of hinged-hinged boundary conditions, the analytical solutions for the nonlinear bending deflection and free vibration frequencies of nonlocal strain gradient Euler-Bernoulli and Timoshenko beams will be deduced. These analytical bending deflection and free vibration frequencies of the Euler-Bernoulli and Timoshenko FG beams can serve as the benchmarks for future numerical computations. The influences of the through-thickness power-law variation of a two-constituent FG material and size-dependent parameters on nonlinear bending deflection and free vibration frequencies will be studied. Finally, some concluding remarks will be summarized.