Size-dependent free flexural vibration behavior of functionally graded nanoplates
Highlights
► Eringen’s differential form of nonlocal elasticity is used. ► The plate kinematics is developed based on first order plate theory. ► A NURBS based iso-geometric FEM is used to study the free vibration of nanoplates. ► The homogenized properties are evaluated based on Mori–Tanaka scheme.
Introduction
Rapid advancement in the application of micro/nano structures in engineering fields, mainly MEMS/NEMS devices, due to their superior mechanical properties, rendered a sudden momentum in modeling the structures of nano and micro length scale. It has been observed that there is a considerable difference in the structural behavior of material at micro-/nano-scale when compared to their bulk counterpart. The difficulty in using the classical theory is that these classical models fails to capture the size effects. The classical model over predicts the response of micro-/nano-structures. Also, in the classical model, the particles influence one another by contact forces. Another way to capture the size effect is to rely on first principle calculations or molecular dynamic (MD) simulations. But even the MD simulation at micron scale is computationally demanding. Hence, there was a need for some theory that can incorporate the length scale effect. Several modifications of the classical elasticity formulation have been proposed in the literature [1], [2], [3], [4], [5]. A common feature of all these theories is that they include one or several intrinsic length scales and the particles influence one another by long range cohesive forces. In this way, the internal length scale can be considered in the constitutive equations simply as a material parameter, also referred to as the ‘nonlocal parameter’ or the ‘internal characteristic length’ in the literature.
Among the various nonlocal theories, Eringen’s differential form of gradient elasticity [3] has received considerable attention to study the static and the dynamic characteristics of nano beams [6], [7] and plates [8], [9], [10]. The theory has been applied by several authors to study the axial vibrations [9], [11], [12], [13] and free transverse vibrations of nanostructures [14], [11], [15], [16]. Reddy and Pang [10] derived governing equations of motion for Euler–Bernoulli beams and Timoshenko beams using the nonlocal differential relations of Eringen [3]. The classical plate theory (CPT) and first order shear deformation theory (FSDT) have been reformulated using the nonlocal differential constitutive relations in [8], [10], [17]. It is observed that increasing the nonlocal parameter decreases the fundamental frequencies of the structure. In all the earlier studies, the nonlocal parameter is kept as a variable and the effect of the nonlocal parameter on the structural response is studied. One approach is to estimate the nonlocal parameter by matching the phonon dispersion relations computed by these theories with the lattice dynamics dispersion relation [18]. Ansari et al. [19] used MD simulations to estimate the appropriate value of the nonlocal parameter. Recently, Eltaher et al. [20] employed Eringen’s differential theory to study the size-effect on the structural response of functionally graded nanobeams. Chang [21] studied the axial vibration characteristics of non-uniform and non-homogeneous nanorods. It was observed that the nonlocal parameter has profound impact on the structural response. The existing numerical approaches are limited to using radial basis functions [22] and differential quadrature method [9], [20], [21] to study the response of nanostructures based on nonlocal elasticity theory. In Eringen’s differential form of gradient elasticity, the nonlocal term appears as additional inertia terms. This implies, the additional inertia terms can be implemented with standard finite element (FE) shape functions.
The main objective of this paper is to solve the nonlocal governing equations of motion using a FE framework. The field variables are approximated by non-uniform rational B-splines (NURBS) [23]. The plate kinematics is described by FSDT. The custom tailorable property of the NURBS functions [24] is exploited to avoid shear locking when the plate becomes very thin. The effect of the nonlocal parameter, the material gradient index, the plate aspect ratio, thickness of the plate and boundary conditions on the fundamental frequencies are numerically studied.
The paper is organized as follows. A brief overview of FG material, Eringen’s nonlocal elasticity and Reissner–Mindlin plate theory is given in the next section. Section 3 presents an overview of NURBS basis functions and spatial discretization. The numerical results and parametric studies are given in Section 4, followed by concluding remarks in the last section.
Section snippets
Functionally graded material
A rectangular plate (length a, width b and thickness h) with functionally graded material (FGM), made by mixing two distinct material phases: a metal and ceramic, is considered with coordinates x, y along the in-plane directions and z along the thickness direction (see Fig. 1). The material on the top surface (z = +h/2) of the plate is ceramic and is graded to metal at the bottom surface of the plate (z = −h/2) by a power law distribution. The homogenized material properties are computed using the
Spatial discretization
In this study, the finite element model has been developed using NURBS basis function.
Non-uniform rational B-splines. The key ingredients in the construction of NURBS basis functions are: the knot vector (a non decreasing sequence of parameter value, ξi ⩽ ξi+1, i = 0,1, ⋯ ,m − 1), the control points, Pi, the degree of the curve p and the weight associated to a control point, w. The ith B-spline basis function of degree p, denoted by Ni,p is defined as [23], [24]:
Numerical Example
In this section, the size-dependent free flexural vibration behavior of functionally graded material plates is studied. Fig. 4 shows the geometry and boundary conditions of the plate. The effect of gradient index n, internal characteristic length , the plate aspect ratio a/b, thickness of the plate h and the boundary conditions on the natural frequency are numerically studied. The assumed values for the parameters considered in the current study are listed in Table 1.
In all cases, we
Conclusion
The size-dependent linear free flexural vibrations of FG plate is numerically studied using NURBS basis functions within a finite element framework. The formulation is based on FSDT and the material is assumed to be graded only in the thickness direction according to the power-law distribution in terms of volume fraction of its constituents. Numerical experiments have been conducted to bring out the effect of the gradient index, the nonlocal parameter, the plate aspect ratio, thickness of the
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