A unified formulation for free vibration of functionally graded plates

1 Introduction

Because of their superior thermomechanical properties, functionally graded materials (FGMs) have received wide applications as structural components in modern industries such as mechanical, aerospace, nuclear, reactors, and civil engineering in recent years [1]. Hence, significant efforts have been devoted to study the mechanical behaviors of structural elements made of these materials such as beams, plates, and shells. Also, it is well known that the vibration characteristic of plates made of FGMs is of great interest for better applications, engineering design, and manufacture.

Even if there are different two-dimensional theories for the free vibration analysis of functionally graded (FG) plates such as the classical plate theory (CPT) [2], [3], the first-order shear deformation theory (FSDT) [4], [5], [6], [7], [8], [9], [10], [11], and the higher-order shear deformation theory (HSDT) [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], however, simple and accurate theories with low computational calculations can be still useful for both practical applications and theoretical studies on the free vibration of FG plates.

Because the transverse shear deformation effects are neglected in the CPT, it provides reasonable results for thin plates. For moderately thick plates, it underestimates deflection and overestimates buckling load as well as natural frequency [2], [3]. To overcome the deficiency of the CPT, many shear deformation theories have been used for the free vibration analysis of FG rectangular plates [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]. The basic idea of these theories is based on the explanation of the in-plane and out-of-plane (transverse) displacement components in terms of known functions of the material coordinate in the plate thickness direction with unknown coefficients. The unknown coefficients are only functions of the in-plane material coordinate variables, and hence, the three-dimensional elasticity theory reduces to a two-dimensional theory one. In most of these theories, the polynomial series expansion is used [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. However, other functions such as hyperbolic functions [25], [26] or trigonometric functions [27] have been used by some researchers.

The simplest theory based on the polynomial series expansion is the FSDT. On the basis of this theory, only the linear terms in the series expansion of the in-plane displacement components are considered, and the transverse displacement is assumed to be constant through the thickness. According to these assumptions, the transverse shear stresses become constant through the thickness, but this assumption violates the shear stress-free surface conditions. Hence, shear correction factors are used to compensate for the difference between the actual stress state and the constant stress state [4], [5], [6], [7], [8], [9], [10], [11]. To avoid the use of shear correction factor, usually, the higher-order terms in the series expansions are considered [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. Depending on the numbers of terms in the in-plane displacement components expansion, whether the transverse displacement is assumed to be constant, whether stress-free boundary conditions are satisfied (constraint theory) or not (unconstraint theory), and whether additional assumptions are used to obtain the unknown coefficients, different HSDTs have been developed [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30].

Reddy’s third-order shear deformation theory [31], which has been initially developed for laminated composite plates, has been used by some researchers to study the free vibration of the FG rectangular plates [12], [13], [14], [15]. On the basis of this theory, the stress-free conditions on the top and bottom surface of the plate are satisfied by the displacement components (constraint theory).

Some others used unconstraint shear deformation theories. Roque et al. [16] and Shahrjerdi [17] developed unconstraint third- and second-order shear deformation theories for the free vibration analysis of FG plates, respectively. In both theories, constant transverse displacement was assumed.

To reduce the number of unknown variables, Benachour et al. [18] and Mechab et al. [19] extended the two-variable third-order shear deformation theory of Shimpi [32] to develop the four-variable third-order shear deformation theories for the free vibration of FG plates. The work of Malekzadeh et al. [33] differs from the others for its functions adopted in the in-plane displacement explanation.

Usually, the thickness stretching of the FG plates has been disregarded by neglecting the variation of the transverse displacement component along the thickness direction [12], [13], [14], [15], [16], [17], [18], [19], [25]. However, in some studies, the polynomial series expansion has been used for this displacement component in the thickness direction [20], [21], [22], [23], [24], [26], [27]. Qian et al. [20], [21] applied a higher-order shear and normal deformable plate theory to analyze the static, free, and forced vibrations of a thick rectangular FG plates. On the basis of this theory, the three components of displacement were expanded in terms of Legendre polynomials in the thickness direction. Matsunaga [22] presented a two-dimensional higher-order deformation theory for the free vibration and stability analysis of FG plates. The in-plane and the transverse displacement components were expressed as polynomial power series of the thickness coordinate variable. Fares et al. [23] developed a two-dimensional theory account for the displacements field in which the in-plane displacements vary linearly through the plate thickness, whereas the out-of-plane displacement was a quadratic function of the thickness coordinate variable. Talha and Singh [24] proposed a third-order shear deformation theory, which included the quadratic variation of the transverse displacement through the plate thickness to study the free vibration of FG rectangular plates.

In the previously reviewed studies, the in-plane displacement components have been expanded in terms of the polynomial functions of the thickness coordinate variable. Ait Atmane [25] used the hyperbolic functions to represent through the thickness variation of the in-plane displacement components and implement the stress-free boundary conditions on the top and the bottom of the FG plates. Neves et al. [26], [27] presented sinusoidal and hyperbolic shear deformation theories for the static and free vibration analysis of FG plates by assuming sinusoidal and hyperbolic type variations across the thickness coordinate for the in-plane displacement components, respectively. In both theories, a quadratic variation was considered for the transverse displacement component in the thickness direction.

The aim of this work is to represent a simple, accurate, and unified formulation for the free vibration analysis of FG plates. New four-variable first- and third-order shear deformation theories, as against five variables in the case of the conventional form of these theories, and also CPT are easily achieved. This theory is free of the assumption of zero in-plane resultant forces used in developing the other four-variable shear deformation theories [18], [19], [34] and hence has the potential to be used for modeling of the nonlinear FG plate problems. In addition, despite the other four-variable theories [18], [19], [34], only the transverse displacement is assumed to consist of bending and shear components. As a result, some new functions are created for the in-plane displacement component explanation. The in-plane displacements cause the parabolic variations of shear strains through the thickness in such a way that the transverse shear stresses vanish on the top and bottom plate surfaces. The theory takes into account the quadratic variation of the transverse shear strains through the thickness of the plate, and hence, it does not require the use of shear correction factors. The equations of motion and the related boundary conditions for the FG plates on two-parameter elastic foundation are derived using Hamilton’s principle. Exact solutions for the simply supported FG plates are extracted. In addition, by extending the application of differential quadrature method (DQM) as an accurate and computationally efficient numerical method [9], [33], [34], [35], [36], [37], [38], approximate solutions for the FG plates with arbitrary boundary conditions are developed. Comparison studies with the other available two- and three-dimensional solutions in the open literature are performed, and excellent agreement is observed.

2 Theoretical formulation

Consider an FG rectangular plate of length L_x=a, width L_y=b, and thickness h, which is supported on a two-parameter elastic foundation as shown in Figure 1. A Cartesian coordinate system (x, y, z) is used to label the material points of the plates in the undeformed reference configuration. The in-plane displacement components u̅ (in the x-direction) and v̅ (in the y-direction) and the transverse displacement component w̅ (in the z-direction) can be approximated as

Figure 1:

The FG plate supported on elastic foundation.

where u and v are the in-plane displacement components of material points on the mid-plane of the FG plate along the x- and y-directions, respectively, and w^b and w^s are the bending and shear transverse displacement components of an arbitrary material point of the FG plate. In addition, φ_i and ψ_i (with i=1, 2, 3) are functions that represent the thickness variations of the in-plane displacement components.

In this study, as a first assumption, the functions due to the linear variation of the in-plane displacement components are approximated as

This assumption together with the zero shear stress components on the top and bottom surface of the FG plate can easily reduce the theory to the CPT one. For an isotropic FG plate, the last aforementioned assumption gives

where γ_iz (with i=x, y) are the transverse shear strain components. On the basis of the linear elasticity theory, Equations (1), (2), and (3) yield

where α=43h2. Using Equations (1), (2), and (4), the nonzero linear strain tensor components in terms of displacement components become

where ε_ii (with i=x, y) and γ_xy are the in-plane normal and shear strain tensor components, respectively.

It is interesting to note that if one insert α=0 in the strain tensor components, the theory reduces to a new first-order shear deformation theory (NFSDT), which has four degrees of freedom instead of five degrees of freedom of the conventional FSDT. The strain components of this new FSDT become

In addition, if the shear component of the transverse displacement is neglected, then the theory reduces to the CPT one, which has the following nonzero strain components:

According to the linear elasticity theory and the present HSDT, the relation between the nonzero stress tensor components and the displacement components can be written as

where E, G[=E2(1+ν)], and v are Young’s modulus, shear rigidity, and Poisson’s ratio of the FG plate, respectively. Also, the material properties of the plate are assumed to vary continuously through the thickness of the plate. In this study, an arbitrary material property P of the FG plate is obtained by using the power law distribution as

where the subscripts c and m refer to the ceramic and metal constituents, respectively; p is the material property graded index or the power law index; and Vf[=(2z+h2h)] is the volume fraction [39], [40], [41], [42], [43], [44], [45].

The equations of motion and the related boundary conditions can be derived in a systematic manner using Hamilton’s principle, which for the free vibration analysis takes the following form:

where δK and δU are the variation of the kinetic energy and strain energy, respectively, and t₁ and t₂ are two arbitrary times.

The variation of strain energy of the FG plate can be stated as

where k_w (N/m³) is the spring or Winkler parameter of the elastic foundation and k_s (N/m) is the shear or Pasternak parameter of the elastic foundation, which are shown in Figure 1 [46], [47], [48]. Also, the variation of kinetic energy is obtained as

Substituting Equations (11) and (12) into Equation (10) and performing the integration by parts, the equations of motion and the related boundary conditions of the proposed plate theory are obtained as follows:

δu:

δv:

δw_b:

δw_s:

In-plane boundary conditions:

Either u_n=n_xu+n_yv is prescribed or

Either u_s=−n_yu+n_xv is prescribed or

Out-of-plane boundary conditions:

Either w^b is prescribed or

Either ∂wb∂n is prescribed or

Either w^s is prescribed or

Either ∂ws∂n is prescribed or

where ∂( )∂n=nx∂( )∂x+ny∂( )∂y;N_ij, M_ij, P_ij, and Q_i with (i, j=x, y) are the stress resultants, and I_i (i=0, 1, …, 4, 6) is the mass inertia, which has the following definitions:

where ρ is the mass density of the FG plate. Substituting for the stress tensor components in terms of the displacement components from Equation (8), one obtains

where the stiffness coefficients are

Different types of classical boundary conditions at the edges of the plate can be obtained by combining the conditions stated in Equations (17–22). Two types of boundary conditions that are considered in this study are as follows:

Simply support (S):

Clamped (C):

Using Equations (8) and (23), the equations of motion [Equations (13–16)] in terms of the displacement components take the following forms, respectively:

Equation (13):

Equation (14):

Equation (15):

Equation (16):

In a similar manner, the natural boundary conditions can be expressed in terms of the displacement components.

3 Solution procedure