A Numerical and Analytical Solution for the Free Vibration of Laminated Composites Using Different Plate Theories

A Numerical and Analytical Solution for the Free Vibration of Laminated Composites Using Different Plate Theories

M.A. Torabizadeh ^a*, A. Fereidoon ^b

^a Department of Mechanical Engineering, University of Applied Science and Technology, Mashhad, Iran

^b Department of Mechanical Engineering, University of Semnan, Semnan, Iran

1. Introduction

A composite material can be defined as a combination of two or more materials that results in better properties than those of the individual components used alone. In contrast to metallic alloys, each material retains its separate chemical, physical and mechanical properties. The two constituents are reinforcement and a matrix. When composites are compared to bulk materials, the main advantages of composite materials are their high strength and stiffness, combined with low density, allowing for a weight reduction in the finished part. The reinforcing phase provides the strength and stiffness. In most cases, the reinforcement is harder, stronger and stiffer than the matrix. The reinforcement is usually a fiber or a particulate. Particulate composites have dimensions that are approximately equal in all directions. They may be spherical, platelets, or any other regular or irregular geometry. Particulate composites tend to be much weaker and less stiff than continuous-fiber composites, but they are usually much less expensive. Particulate reinforced composites usually contain less reinforcement (up to 40 volume percent to 50 volume percent) due to processing difficulties and brittleness [1].

A fiber’s length is much greater than its diameter. The length-to-diameter (l/d) ratio is known as the aspect ratio and can vary greatly. Continuous fibers have long aspect ratios, whereas discontinuous fibers have short ones. Continuous-fiber composites normally have a preferred orientation, whereas discontinuous fibers generally have a random orientation. Examples of continuous reinforcements include unidirectional, woven cloth and helical winding, whereas examples of discontinuous reinforcements are chopped fibers and random material. Continuous-fiber composites are often made into laminates by stacking single sheets of continuous fibers in different orientations to obtain the desired strength and stiffness properties with fiber volumes as high as 60 percent to 70 percent. Fibers produce high-strength composites because of their small diameter; they contain far fewer defects (normally surface defects) compared to those in the material produced in bulk. As a general rule, the smaller the diameter of the fiber, the higher its strength, but often the cost increases as the diameter becomes smaller. In addition, smaller-diameter/high-strength fibers have greater flexibility and are more amenable to fabrication processes, such as weaving or forming over radius. Typical fibers include glass, aramid and carbon, which may be continuous or discontinuous. The continuous phase is the matrix, which is a polymer, metal or ceramic. Polymers have low strength and stiffness, metals have intermediate strength and stiffness but high ductility, and ceramics have high strength and stiffness but are brittle. The matrix (continuous phase) performs several critical functions, including maintaining the fibers in the proper orientation and spacing and protecting them from abrasion and the environment. In polymer and metal matrix composites that form a strong bond between the fiber and the matrix, the matrix transmits loads from the matrix to the fibers through shear loading at the interface. In ceramics-matrix composites, the objective is often to increase the toughness rather than the strength and stiffness; therefore, a low interfacial strength bond is desirable [1].

Tan and Nie [2] studied free and forced vibration of variable stiffness composite annular thin plates with elastically restrained edges based on the classical plate theory. They found that the transverse mode shapes of the plates with in-plane variable stiffness are different from those with constant stiffness. Zhang et al. [3] analyzed free vibration analysis of triangular CNT-reinforced composite plates subjected to in-plane stresses using the FSDT element-free method. Chakraborty et al. [4] presented a novel approach, referred to as polynomial correlated function expansion (PCFE), for a stochastic free-vibration analysis of a composite laminate. Finally, based on the numerical results, new physical insights had been created on the dynamic behavior of composite laminates. Ganesh et al. [5] studied the free vibration analysis of delaminated composite plates using a finite element method. Mantari and Ore [6] presented a simplified first-order shear deformation theory (FSDT) for a laminated composite and sandwich plates. Their approach had a novel displacement field that includes undetermined integral terms and contains only four unknowns. Su et al. [7] illustrated a modified Fourier series to study the free vibration of a laminated composite and four-parameter functionally graded sector plates with general boundary conditions. Zhang et al. [8] studied the free-vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and the element-free IMLS-Ritz method. They also examined the influence of a carbon nanotube volume fraction, plate thickness-to-width ratio, plate-aspect ratio and a boundary condition on the plate’s vibration behavior. Marjanović and Vuksanović [9] illustrated a layerwise solution to free vibrations and the buckling of a laminated composite and sandwich plates with embedded delamination. The effects of plate geometry, lamination scheme, degree of orthotropy and delamination size or position on the dynamic characteristics of the plate were presented. Boscolo [10] presented an analytical closed-form solution for a free-vibration analysis of multilayered plates by using a layer-wise displacement assumption based on Carrera’s Unified Formulation. A wide range of boundary conditions were analyzed by using a Levy-type solution. Ou et al. [11] presented an efficient method for predicting the free and transient vibrations of multilayered composite structures with parallelepiped shapes, including beams, plates and solids. Rafiee et al. [12] analyzed the geometrically nonlinear free vibration of shear deformable piezoelectric carbon nanotube/fiber/polymer multiscale laminated composite plates. Akhras and Li [13] used a spline finite strip with higher-order shear deformation for stability and a free-vibration analysis of piezoelectric composite plates. Grover et al. [14] assessed a new shear deformation theory for free-vibration-response laminated composite and sandwich plates. They compared the results with finite element and analytical solutions. Jafari et al. [15] presented a free-vibration analysis of a generally laminated composite beam (LCB) based on the Timoshenko beam theory using the method of Lagrange multipliers. They examined some parameters, such as the slenderness ratio, the rotary inertia, the shear deformation, material anisotropy, ply configuration and boundary conditions on the natural frequency and mode shape. Tai and Kim [16] illustrated the free vibration of laminated composite plates using two variable refined plate theories. They applied the Navier technique to obtain the closed-form solutions of anti-symmetric cross-ply and angle-ply laminates. Srinivasa et al. [17] and Ramu and Mohanty [18] used finite element results as a verification method with those obtained from experimental tests on the free vibration of composite plates. Chandrashekhara [19] presented an exact solution for the free vibration of symmetrically laminated composite beams. Ke et al. [20] investigated the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on the Timoshenko beam theory and von Kármán geometric nonlinearity. Also, the free vibration of anisotropic thin-walled composite beams and delaminated composite beams were performed by Song [21] and Lee [22], respectively.

Based on papers reviewed in the literature, few investigations were found that compared the analytical and numerical analyses of different theories and lamination layups. Therefore, in this paper, the analytical and numerical solutions for the free vibration of laminated polymeric composite plates with different layups are compared. Two different theories and layups are selected. Also, finite-element analysis is performed using ANSYS to validate results obtained by analytical methods. The solver is coded in MATLAB. Also investigated are the effects of different parameters, such as the lamination angle, the plate aspect ratio and the modulus ratio on the fundamental natural frequencies of laminated composite. The main objective of this paper is to compare different theories and lamination schemes on the vibration response of laminated composites.

1. Theoretical Formulation

2.1. Classical lamination plate theory (CLPT)

2.1.1 Displacement and strains

A rectangular plate of sides a and b with thickness h is shown in Fig. 1. Based on the classical lamination plate theory, the following displacement field can be assumed [23]:

where  are the displacements along the coordinate lines of a material point on xy-plane.

The von Karman strains associated with the displacement field in static loading can be computed using the strain-displacement relations for small strains:

Note that the transverse strains are identically zero in classical plate theory. The first three strains have the form

2.1.2 Equilibrium equations

By using Eqs. (4) and (5), the constitutive equations are obtained. Equations of equilibrium can be derived using the variational principle, which is not explained in detail here (see [23]). The Euler-Lagrange equations of the theory are obtained as follows,

where, the quantities N_ij are called the in-plane force resultants and M_ij are called the moment resultants and (I₀, I₁, I₂) are the mass moments of inertia.

2.1.3 Navier solution methodology

The displacement fields are assumed by the following form:

where U_mn, V_mn and W_mn are the coefficients that should be determined and  and

The consideration of Eqs. (10) – (12), shows that the mechanical transverse load q should also be expanded in a double sine series. Thus,

Figure 1. The geometry of simply supported rectangular laminated plates used in the analytical solutions.

Substituting expansions (10–12) into expressions given in Eqs. (6-8) without thermal loads yields

where A_ij, D_ij and B_ij are called extensional, bending and bending-extensional coupling stiffness, respectively [23]. Also,  and . Note that the edge shear force is necessarily zero.

Substituting the expansion (13) into (15), we obtain expressions of the form

where a_mn, b_mn and c_mn are coefficients whose explicit form will be given shortly. Since Eq. (16) must hold for any m, n, x and y, it follows that a_mn=0, b_mn=0 and c_mn=0 for every m and n. The explicit forms of these coefficients are given by:

or in matrix form

where is

Eqs. (18) provide three second-order differential equations among the three variables U_mn, V_mn and W_mn for any fixed values of m and n.

For free vibration, all applied loads and the in-plane forces are set to zero, and we assume a periodic solution of the form:

where  and is the frequency of natural vibration. Then Eq. (18) reduces to the eigenvalue problem:

For a nontrivial solution, the determinant of the coefficient matrix in (21) should be zero, which yields the characteristic polynomial

where  is the eigenvalue and

The real positive roots of this cubic equation give the square of the natural frequency  associated with mode (m,n). The smallest of the frequencies is called the fundamental frequency. In general,  is not the fundamental frequency; the smallest frequency might occur for values other than m = n = 1.

If the in-plane inertias are neglected (i.e., ), and irrespective of whether the rotary inertia is zero, Eq. (22) will be

Note that if the in-plane inertias are not neglected, the eigenvalue problem cannot be simplified to a single equation, even if the rotary inertia is zero.

2.2. First-order shear deformation theory (FSDT)

2.2.1 Displacement and strains

Under the same assumptions and restrictions as in the classical laminate theory, the displacement field of the first-order theory is of the form:

Where

which indicate that and are the rotations of a transverse normal about the y- and x- axes, respectively.

The nonlinear strains associated with the displacement field (25) are obtained as

Note that the strains are linear through the laminate thickness, whereas the transverse shear strains are constant through the thickness of the laminate in the first-order laminated theory. These strains have the fo

2.2.2 Equilibrium equations

The governing equations of the first-order theory will be derived using the dynamic version of the principle of virtual displacements. The Euler-Lagrange equations are obtained as follows

where Q_x and Q_y are called transverse force resultants and

Parameter K is called a shear correction coefficient and is used because of a discrepancy between the actual stress state and the constant stress state predicted by the first-order theory.

2.2.2 Boundary condition

The natural boundary conditions are obtained by setting the coefficients of and  to zero separately:

where

Thus, the primary and secondary variables of the theory are

The initial conditions of the theory involve specifying the values of the displacements and their first derivatives with respect to time at t = 0.

2.2.3 Equations of motion

The boundary conditions are satisfied by the following expansions

The Navier solution can be calculated from

where

For free vibration, all thermal and mechanical loads are set to zero and substitute to Eq. (29) and obtain

where

and . When rotary inertia is omitted, Eq. (37) can be simplified by eliminating Xmn and Ymn (i.e., using the static condensation method) as follows

where

If the in-plane and rotary inertias are omitted (i.e., ), we have

3. Finite Element Method

     The finite element method (FEM), known as a powerful tool for many engineering problems, has been used to compute such matters as elastic-plastic, residual and thermal stresses, and buckling and vibration analysis. Because of this, ANSYS software that is a commercial FEM program was preferred for the vibration analysis of the laminated composite plates. The Shell 99 element type was selected for the 2D modeling of solid structures in ANSYS. Initially, the plates are to get an initial estimate of the undamped natural frequencies ωn and mode shape n. The element type of Shell 99 may be used for layered applications of a structural shell model. The element has six degrees of freedom at each node; and translations in the nodal x and y directions and rotations about the nodal z-axis. This element is constituted by layers designated by numbers (LN-layer number), increasing from the bottom to the top of the laminate; the last number quantifies the existing total number of layers in the laminate (NL-total number of layers).     The boundary conditions have been applied to the nodes, i.e., the dimensions in the x and y are 400 mm for 2D, and the displacements and rotations of all nodes about the y–z plane are also taken as zero. The model of the laminated plate is generated with a different number of layers (based on different side-to-thickness ratio). The boundary conditions and mesh shape are shown in Fig. 2. It is mentioned that for a free- vibration analysis, the subspace method is applied. The subspace iteration method was described in detail by Bathe [24].

Figure 2: View of a laminated composite plate with boundary conditions and mesh shape.

     After the mesh generation process, a laminated composite plate with six layers has 500 elements and 1,488 nodes. By increasing the number of layers, the numbers of elements and the nodes of the plates increase. The normal penalty stiffnesses of the contact element are chosen between 10⁴ and 10⁹.

4. Results and Discussions

     The nondimensionalized frequencies , of specially orthotropic and antisymmetric cross-ply square laminates are presented in Table 1 for modulus ratios  and 20 (G₁₂ = G₁₃ = 0.5E₂, G₂₃ = 0.2E₂, ʋ₁₂ = 0.25). All layers are of equal thickness. Results are presented for m,n = 1,2,3 and for when the rotary inertia is neglected.

     The fundamental frequency increases with the modular ratio. The effect of including rotary inertia is to decrease the frequency of vibration. Note that the first four frequencies for an antisymmetric cross-ply plates are (m,n) = (1,1), (1,2), (2,1) and (2,2) and  for antisymmetric laminates. Also, good agreement was found between the analytical solution and the FEM analysis.

     Fig. 3 shows a plot of fundamental frequency  versus aspect ratio a/b for symmetric (0/90)_s cross-ply and antisymmetric (0/90)₂ cross-ply laminates. The material properties used are , G₁₂ = G₁₃= 0.6E₂, G₂₃ = 0.2E₂, ʋ₁₂ = 0.25). Fig. 4 shows the effect of coupling between bending and extension on the fundamental frequencies of antisymmetric cross-ply laminates. The material properties used are , G₁₂ = G₁₃ = 0.5E₂, G₂₃ = 0.2E₂, ʋ₁₂ = 0.25). With an increase in the number of layers, the frequencies approach those of the orthotropic plate. The bending-stretching coupling lowers the vibration frequencies. For example, the two-layer plate has vibration frequencies about 40 percent lower than those of an eight-layer antisymmetric laminate or orthotropic plate with the same total thickness.

Fig. 5 shows a plot of fundamental frequency  versus the modulus ratio E₁/E₂ for antisymmetric (0/90) cross-ply laminates for various values of plate aspect ratios. The plate aspect ratio lowers the vibration frequencies. The rectangle plate has vibration frequencies about 50 percent lower than those of a square plate with the same total thickness.

The nondimensionalized fundamental frequencies  of graphite-epoxy composites with , G₁₂/E₂ = 0.5, ʋ₁₂ = 0.25 and a/b = 1 are shown as a function of the lamination angle in Fig. 6. The bending-stretching coupling due to the presence of B₁₆ and B₂₆ lowers the frequencies. The coupling is the maximum for two-layer plates, and it rapidly decreases with increasing number of layers. At θ = 45^°, the fundamental frequency of the two-layer plate is about 40 percent lower than that of the eight-layer laminate.

The nondimensionalized fundamental frequencies of graphite-epoxy composites with , G₁₂/E₂ = 0.5, ʋ₁₂ = 0.25 are shown as a function of plate aspect ratios in Fig. 7. With an increase in the number of layers, the frequencies approach those of the orthotropic plate. The bending-stretching coupling lowers the vibration frequencies. For example, the two-layer plate has vibration frequencies about 40 percent lower than those of a four-layer antisymmetric angle-ply laminate or orthotropic plate with the same total thickness and aspect ratio. Also, effects of an aspect ratio on the fundamental frequencies of a laminated composite with same total thickness are more significant for values less than 1.

Fig. 8 shows a nondimensionalized fundamental frequency versus a modulus ratio of antisymmetric angle-ply square laminates. The effect of coupling is significant for all modulus ratios and the difference between the two-layer solution and orthotropic solution increases with the modulus ratio.

Table 2 and 3 contain the nondimensionalized fundamental frequencies  for symmetric cross-ply laminates using the FSDT. The effect of the shear correction factor is to decrease the frequencies. The smaller the K, the smaller the frequencies are. The rotary inertia (RI) also decreases the frequencies.

Fig. 9 shows the effect of transverse shear deformation and rotary inertia on the fundamental natural frequencies of orthotropic and symmetric cross-ply (0/90/90/0) square plates with the following lamina properties:

, G₁₂ = G₁₃ = 0.5E₂, G₂₃ = 0.2E₂, ʋ₁₂ = 0.25.

The symmetric cross-ply plate behaves much like an orthotropic plate. The effect of rotary inertia is negligible in the FSDT and, therefore, is not shown in the figure.

Fig. 10 shows the effect of transverse shear deformation, bending-extensional coupling and rotary inertia on the fundamental natural frequencies of two-layer and eight-layer antisymmetric cross-ply laminates ( , G₁₂ = G₁₃ = 0.5E₂, G₂₃ = 0.2E₂, ʋ₁₂ = 0.25).

­The first line corresponds to the shear correction coefficient of K=2/3 and the second and third lines correspond to the shear correction coefficient of K=5/6 and K=1.0, respectively.

­The first line corresponds to the shear correction coefficient of K=1.0 and the second line corresponds to the shear correction coefficient of K=5/6.

. The eight-layer antisymmetric cross-ply plate behaves much like an orthotropic plate. The effect of rotary inertia is negligible in the FSDT and, therefore, is not shown in the figure.

Table 4 contains numerical values of the fundamental frequencies of antisymmetric cross-ply laminated plates for various modular ratios. The results for both two-layer and eight-layer laminated plates for square and rectangular geometries are presented.