State-space Levy solution for size-dependent static, free vibration and buckling behaviours of functionally graded sandwich plates
Introduction
Functionally graded materials (FGMs) are a class of composite materials in which the material properties vary gradually from one position to the other. The gradation process of this kind of materials can create the industrial products with smooth and continuous properties, hence avoids the stress concentration, cracking and delamination phenomena occurred in the conventional composite materials. These striking features are appealing to the researchers in developing the advanced theories and numerical methods to predict accurate behaviours of FGMs. Their applications can be found in aerospace structures [1], cutting tools [2], actuators, transducers [3] and biomedical installations, etc. An insightful introduction to the applications of FGMs is presented in Ref. [4]. FG-sandwich structures, which are the combinations of FGMs and sandwich structures, have more attractive characteristic since they can tailor material properties and eliminate the delamination, which occurs in conventional sandwich structures.
Recent developments in technology require the knowledge of small-scale structural elements, which are commonly presented in MEMS and NEMS such as thin films, nano-probes, sensors, actuators and other devices. There have been much developments in manufacturing- and measuring-process for these small-scale FG structures in recent years [[5], [6], [7], [8]], which attract more research in investigating the behaviours of such structures. It is evidenced from experiments [[9], [10], [11]] that when the dimensions of these structures are reduced to a certain value, the size effects in structural behaviours can be observed. There are several approaches to investigate these effects including the experimentation, atomistic/molecular dynamics simulation and higher-order continuum mechanics. Although the two former methods can provide more accurate prediction, the latter has been employed widely due to the computational efficiency. The higher-order continuum theories, which are widely known as the non-classical continua, were initiated in the work of Cosserat and Cosserat [12] in 1909. Utilising the concept of directors, which was a triad of vectors, the additional degrees of freedom (DOF) are introduced apart from the classical DOFs of displacements to state the independent microrotation of material particles. This idea has drawn much attention from scholars since 1960s with the development of various assumptions regarding the constitutive laws and the measuring of such additional DOFs. Although a good number of theories have been proposed with respect to these higher-order continuum theories, three major categories can be summarised covering the microcontinua, nonlocal elasticity and the strain gradient family [13]. The microcontinua were developed by Eringen [[14], [15], [16], [17]] for 3M theories which are the micromorphic, microstretch and micropolar with nine, four and three additional DOFs included, respectively [18]. The nonlocal elasticity was firstly proposed by Kroner [19] and further developed by Eringen [[20], [21], [22]]. In these theories, the stress at a reference point is measured through the constitutive law by the strains around its effective area. Therefore, the size effects are captured by introducing a nonlocal parameter to the constitutive equations. The third class of higher-order continua is the strain gradient family, which are composed of the couple stress theory, the strain gradient theory and their modified versions. In the strain gradient family, the strain energy is considered as a function of both strains and strain gradients, which requires additional material constants, i.e. material length scale parameters, compared to the classical continuum. Mindlin [23] proposed an original strain gradient theory considering the first gradient of strains only and developed another version including both the first and second gradients of strains [24]. In order to improve the efficiency and reduce the material parameters required from experiments, various models based on different strain gradients were examined. In the classical couple stress theories, which were proposed by Toupin [25,26], Mindlin and Tiersten [27] and Koiter [28], only the gradients of rotation vectors are included, leading to only two additional material length scale parameters required. Later, the modified couple stress theory (MCST) was developed by Yang et al. [29] with the introduction of an equilibrium condition of moments of couples. This higher-order equilibrium enforces the couple stress tensor to be symmetric, hence only one material length scale parameter is required. An interesting discussion on another approach to derive this symmetry of couple stress tensor can be found in the work of Munch et al. [30]. Using the MCST, a large number of publications were developed to investigate structural behaviours of microplates including bending, vibration and buckling based on various shear deformation theories such as the classical plate theory (CPT), first-order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT). Using the MCST CPT, Asghari and Taati [31] analysed the free vibration of FG microplates with arbitrary shapes. Taati [32] then included the geometric nonlinearity to investigate the buckling and post-buckling behaviours of FG microplates under different boundary conditions (BCs) with an analytical solution. Based on the MCST FSDT, the static, free vibration and buckling behaviours of FG annular microplates with various BCs were investigated by Ke et al. [33]. Thai and Choi [34] developed an analytical solution to linear and nonlinear bending, vibration and buckling behaviours of simply supported FG microplates; later on elastic medium was then included by Jung et al. [35,36]. Ansari et al. [37,38] also adopted the differential quadrature (DQ) method for nonlinear vibration, bending and post-buckling analysis of FG microplates. In recent years, the HSDTs and 3D elasticity have been developed extensively to improve the accuracy in predicting structural behaviours of composite and FG structures [[39], [40], [41], [42], [43], [44]]. They also have been applied to investigate the behaviours of microplates. Thai and Kim [45] examined the bending and free vibration behaviours of FG microplates using analytical solutions while such behaviours for the annular/circular microplates was investigated by Eshraghi using DQ method [46]. The MCST sinusoidal shear deformation model was also developed by Thai and Vo [47] for deflections and natural frequencies of simply supported microplates. Some other refined plate models [48,49] and quasi-3D [[50], [51], [52]] were also employed the MCST for FG microplates. In addition, the thermal effects are examined for the FG microplates in many publications. Using the MSCT CPT, Mirsalehi et al. [53] investigated stability of thin FG microplate under mechanical and thermal load based on spline finite strip method. Ashoori and Vanini [54] also studied thermal buckling of annular FG microplate resting on an elastic medium and extended to geometric nonlinearity effect and snap-through behaviour. Utilising DQ method, Eshaghi et al. [55] analysed static bending and natural frequencies of FG annular/circular employing the MCST CPT, FSDT and HSDT models.
In this paper, a four-variable refined shear deformation theory is developed for static, free vibration and buckling behaviours of FG sandwich microplates. Based on a state space approach, these structural behaviours of micro rectangular plates with two opposite simply-supported sides and arbitrary combinations of boundary conditions on other sides are presented. By this way, the closed-form solutions can be obtained to demonstrate the effect of various boundary conditions to the micro behaviours of FG-sandwich plates for the first time. The effects of geometric parameters, material distribution and graded schemes to the size-dependent behaviours of FG sandwich microplates are also investigated. The governing equations and corresponding boundary conditions together with the tabular results can be used to verify those developed from other numerical methods.
Section snippets
Kinematics and constitutive relations
Consider a FG-sandwich plate with the coordinate and cross-section shown in Fig. 1. By applying the MCST, the variation of strain energy in the body is related to both strain and curvature tensors as [29]:where and are the strain and symmetric microcurvature tensor defined by: and are the corresponding stress and deviatoric part of the symmetric couple stress tensors defined by:in
Variational formulation
The equations of motion are obtained from the variational principle, which stateswhere , and denote the variation of strain, kinetic energy and work done by external forces.
The variation of strain energy is rewritten in terms of mid-plane displacements as:
Numerical examples
In this section, numerical results are presented for the verification and parametric study of the present analytical solution. Unless mentioned otherwise, the material properties are used for metal (Al), Em=70 GPa, ρm=2700 kg/m3, νm=0.3 and for ceramic (Al2O3), Ec=380 GPa, ρc=3800 kg/m3, νc=0.3 and the material length scale parameter is assumed l=17.6 μm based on literature. By using normalized quantities without the inclusion of couple stresses in the strain energy, the displacements,
Conclusions
In this paper, closed-form solutions have been developed to study the static, free vibration and buckling behaviours of FG sandwich microplates. Governing equations are derived from the variational principle based on the framework of the MCST and a refined plate theory. Utilising the state space approach, the deflection, natural frequencies and critical buckling loads of the FG-sandwich microplates with simply supports at two opposite edges and various BCs for the others are analysed
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