ReviewA review of continuum mechanics models for size-dependent analysis of beams and plates
Introduction
Small-scale structural elements such as beams, plates and shells are commonly used as components in micro- and nano-electromechanical systems (MEMS and NEMS), sensors, actuators and atomic force microscopes. In these applications, size effects were experimentally observed in mechanical properties [1], [2], [3], [4], [5]. These size effects can be captured using either molecular dynamics (MD) simulations or higher-order continuum mechanics. Although the MD method can provide accurate predictions, it is too computationally expensive. Therefore, higher-order continuum mechanics approach was widely used in the modelling of small-scale structures.
The development of higher-order continuum theories can be traced back to the earliest work of Piola on the 19th century as demonstrated in [6], [7] and the work of Cosserat and Cosserat [8] in 1909. However, until 1960s, the Cosserat brothers’ idea was received considerable attention from researchers, and a large number of higher-order continuum theories have been developed. In general, these theories can be categorized into three different classes namely the strain gradient family, microcontinuum and nonlocal elasticity theories. The strain gradient family is composed of the couple stress theory, the first and second strain gradient theories, the modified couple stress theory and the modified strain gradient theory. In the strain gradient family, both strains and gradient of strains are considered in the strain energy, and thus the size effect is accounted for using material length scale parameters. In the couple stress theory initiated by Toupin [9], Mindlin and Tiersten [10] and Koiter [11], only the gradient of rotation vector is considered in the strain energy, and thus only two material length scale parameters are required. The modified couple stress theory was proposed by Yang et al. [12] based on modifying the couple stress theory. By introducing an equilibrium condition of moments of couples to enforce the couple stress tensor to be symmetric, the number of material length scale parameters of the modified couple stress theory is reduced from two to one. The first strain gradient theory initiated by Mindlin [13] considers only the first gradient of strains. One year later, Mindlin [14] derived the second strain gradient theory which is considered as the most general form of strain gradient theory accounting for both the first and second gradients of strains. Lam et al. [15] proposed the modified strain gradient theory with only three material length scale parameters by modifying Mindlin's theory by using a similar approach of Yang et al. [12]. The microcontinuum theory was developed by Eringen [16], [17], [18] consisting of micropolar, microstretch and micromorphic (3M) theories. Micropolar theory which is actually initiated by Cosserat brothers [8] is the simplest one among 3M theories, whilst micromorphic theory is the most general one among 3M theories. In 3M theories, each particle can rotate and deform independently regardless of the motion of the centroid of the particle. More details about the 3M theories as well as their applications can be found in [19], [20], [21], [22], [23], [24], [25]. The nonlocal elasticity theory was originally proposed by Kroner [26] and improved by Eringen [27], [28] and Eringen and Edelen [29]. In this theory, the stress at a reference point in a continuum depends on the strains at all points of the body, and thus the size effect is captured by means of constitutive equations using a nonlocal parameter. Nonlocal elasticity theory was initially formulated in an integral form and later reformulated by Eringen [30] in a differential form by considering a specific kernel function. Compared to the integral model, the differential one is widely used for nanostructures due to its simplicity. In addition, another class of higher-order theory which is called nonlocal strain gradient theory has been recently proposed based on a combination of the nonlocal elasticity theory and the strain gradient theory. The interested reader can refer to [31], [32], [33] for more details on this theory.
Size-dependent models have been widely used for predicting the global behaviour of beam- and plate-like nanostructures such as carbon nanotubes (CNTs) and graphene sheets. CNTs were discovered by Iijima [34] by rolling graphene sheets. Based on synthesis route and reaction parameters, various types of CNTs such as single-walled carbon nanotubes (SWCNTs), double-walled carbon nanotubes (DWCNTs) and multi-walled carbon nanotubes (MWCNTs) can be obtained (see Fig. 1) by rolling single-layered graphene sheets (SLGSs), double-layered graphene sheets (DLGSs) and multi-layered graphene sheets (MLGSs) (see Fig. 2). Nanotube is a key nanostructure and has a wide range of applications in all areas of nanotechnology. Notable among them is conveying fluid [35], [36], [37], [38], [39], [40], [41] and nanofluidic devices and systems.
A large number of size-dependent models have been proposed based on various beam and plate theories. The simplest models were based on Euler-Bernoulli beam theory (EBT) and classical plate theory (CPT). These models are only appropriate for modelling of slender beams and thin plates because they ignore shear deformation effect. To overcome the limitation of the EBT and CPT, a number of shear deformation theories have been proposed. First-order shear deformation models were based on Timoshenko beam theory (TBT) and first-order shear deformation theory (FSDT). Since the in-plane displacements vary linearly through the thickness in these models, a shear correction factor is required. In order to eliminate the use of the shear correction factor and obtain a better prediction of the responses of thick beams and plates, several higher-order shear deformation theories (HSDTs) have been proposed, notable among them are Reddy beam theory (RBT) and third-order shear deformation theory (TSDT) of Reddy [42]. A comprehensive review on the plate theories can be found in the work by Thai and Kim [43].
The governing equations derived from the aforementioned size-dependent models can be solved using either analytical methods or numerical approaches. However, the application of analytical methods is limited to a particular nanostructure with simple geometry, loading and boundary conditions (BCs). For instance, Navier method is only applied for rectangular plates with simply supported BCs, whilst Levy method is only applied for rectangular plates in which two opposite edges are simply supported and the remaining two edges can have any arbitrary BCs. For the practical problems with general geometry, loading and BCs, seeking their analytical solutions is impossible because of the mathematical complexity of the size-dependent models compared to the classical ones. Therefore, numerical approaches such as finite element method, differential quadrature method, mesh-free method, Ritz method, Galerkin method, etc. become the most suitable ones for solving such problems. Among different numerical techniques, the finite element method is the most powerful tool and commonly used for the analysis of structures, and thus the development of finite element solutions for size-dependent models will be discussed in this review.
Although extensive research on small-scale beams, plates and shells has been made during the past decade, the development of models for capturing the size effect in these structures has not been reviewed. Therefore, this paper aims to provide a comprehensive review on the development of size-dependent models for predicting the behaviour of small-scale beam- and plate-like structures. The review mainly focuses on the beam, plate and shell models which were developed based on the nonlocal elasticity theory of Eringen [30], the modified couple stress theory of Yang et al. [12] and the modified strain gradient theory of Lam et al. [15]. In addition, the development of finite element models of these theories was also highlighted and discussed in details.
Section snippets
Review of the nonlocal elasticity theory
The nonlocal elasticity theory was initially formulated by Eringen [27], [28] and Eringen and Edelen [29] by means of integral constitutive equation aswhere σij and are the components of the nonlocal and local stress tensors, respectively and k is the kernel function determined in terms of nonlocal parameter κ and neighbourhood distance in which κ = e0a and e0 and a are the material constant and the internal characteristic length, respectively, i.e. lattice
Review of the modified couple stress theory
The modified couple stress theory was proposed by Yang et al. [12] by modifying the classical couple stress theory of Toupin [9], Mindlin and Tiersten [10] and Koiter [11]. By introducing an additional equilibrium condition of moments of couples to enforce the couple stress tensor to be symmetric, the number of additional material length scale parameters in the modified couple stress theory is reduced from two to one. This makes the modified couple stress theory more advantageous because the
Review of the modified strain gradient theory
In this theory [15], the strain energy contains two additional gradient parts of the dilatation gradient and the deviatoric stretch gradient in addition to the symmetric curvature . Therefore, the strain energy is written as [15]where the symmetric curvature tensor is defined in Eq. (6). The dilatation gradient vector and the deviatoric stretch gradient tensor are respectively defined in Eqs. (10) and (11a), (11b), (11c) as
Nonlocal elasticity elements
Based on a nonlocal EBT model, Eltaher and his colleagues [381], [382], [383], [384], [385] have developed nonlocal elements for nanobeams made of FG materials [381], [382], [383] and isotropic materials [384], [385]. The EBT element has two nodes with six degrees of freedom (4-DOF) in which the axial and transverse displacements are respectively approximated using Lagrange and Hermite cubic interpolation functions. It is noted that Eltaher et al. [381] dealt with free vibration problems of FG
Concluding remarks and recommendation for future studies
The development of size-dependent models for predicting size effects on the global responses of small-scale beam, plate and shell structures was comprehensively reviewed and discussed in this paper. During the past decade, great efforts have been devoted to the development of size-dependent models based on higher-order continuum mechanics approach. This review mainly focuses on the size-dependent beam, plate and shell models developed based on the nonlocal elasticity theory, modified couple
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2011-0030040) and the School of Engineering and Mathematical Sciences at La Trobe University.
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