The Effect of Using Different Elastic Moduli on Vibration of Laminated CFRP Rectangular Plates

This paper deals with effects of using different sets of material constants on the natural frequencies of laminated composite rectangular plates. The plate is symmetrically laminated by thin layers composed of recently developed carbon fiber reinforced plastic (CFRP) materials. Numerical experiments are conducted by using a semi-analytical solution based on the thin plate theory and the lamination theory. The displacements are assumed to accommodate any combination of classical boundary conditions. The material property is expressed by a set of four elastic constants, and some typical sets of values are cited from the recent literature. Furthermore, a new standard set of discretized constants is proposed to uncover the underlying characteristics of the existing constants. The convergence study is carried out first, and the lowest five natural frequencies are calculated for five sets of classical boundary conditions including totally free through totally clamped cases. Next, a new definition of frequency parameters is introduced to promote more physically meaningful comparison among the obtained results, and the effect of using slightly different constants is clarified for unified comparison and insights. It is also discussed to derive approximate frequency formulas by linear regression analysis and to test accuracy of the formulas.


Introduction
Structural components in the form of flat plates are found practically in all fields of industry, and the literatures related to free vibration of flat plates are numerous.Leissa [1] compiled in 1969 a monograph "Vibration of Plates" and it is still cited in many papers.For isotropic rectangular plates, he published a comprehensive set of natural frequencies for all possible combinations of classical boundary conditions [2].Since in the 1970's, advanced fibrous composite materials have been developed and served as plate components in many industrial applications, such as airplane, automobile and marine structures.Generally, fiber reinforced plastics have microstructures made of reinforcing fiber and matrix material.There are various fibers, ranging from natural fibers to chemical fibers such as glass, boron and carbon fibers.When such constituents merge into one material, it shows anisotropic characteristics.Among various types of fibers, the use of carbon fibers in reinforced plastics (CFRP) is becoming more increasingly dominant in weight-sensitive structures.For the reason, some books [3][4][5] dealing with mechanics of the composites have been published to serve for analysis and design, and review papers have appeared, for example [6].
Since the publication of journal papers in the 1970's, researchers such as Bert [7] published papers on vibration of laminated composite plates, and it has led to improvement of lamination theories to the first order and higher order plate theories [5,8].For rectangular plates with arbitrary edges, combination of classical boundary conditions became possible to be analyzed [9] in good accuracy.Application of the finite element method has also been active in analyzing laminated plates [10,11].
For defining elastic problem of laminated CFRP plates, four independent elastic constants are necessary, unlike isotropic plates with only two independent constants being needed.Previous literatures have used different values of the constants, because the improvement of fiber stiffness is in progress, and many past researchers have used elastic constants obtained from measurement tests supplied from different chemical companies.This fact makes direct comparison difficult among the published results, and was also obstacles preventing from compiling design date book.
Despite such practical needs, however, there is no literature to discuss the influence caused by using different elastic constants.The present study takes up this problem for studying the effect when slightly different constants are used in the vibration analysis of laminated CFRP plates.In doing so, numerical experiments are conducted to calculate natural frequencies of laminated CFRP rectangular plates by using various sets of constants [3,[12][13][14], and for effective insights, new standard constants are proposed to understand vibration behaviors in the comprehensive way.

Analytical Method
Figure 1 shows a laminated rectangular plate in the coordinate system and in each layer the major and minor principal axes are denoted by the L and T axes.The dimension of the whole plate is given by a×b×h (thickness).The plate considered is limited to symmetric laminate, and the total number of layers is defined as 2N (i.e., N layers in the upper(lower) half cross-section).
Free vibration of a macroscopic model for such thin symmetric plates is governed in the classical lamination (plate) theory by ( ) where w is a deflection and ρ is a mean mass per unit volume of the plate.The Dij (i, j = 1, 2, 6) are the bending stiffness of the symmetric laminate defined by ( ) with   being a thickness coordinate measured from the mid-surface and  ̄ () being elastic constants [3,4] in the k-th layer, obtained from 11 12 11 11 (superscript (k) is omitted) by considering a fiber orientation angle θk in the layer.The EL and ET are moduli of longitudinal elasticity in the L and T directions, respectively, GLT is a shear modulus and νLT is a Poisson ratio.
For the small amplitude (linear) free vibration, the deflection w of a thin plate may be written by ( ) ( ) where W is the amplitude and ω is a radian frequency.Natural frequency is normalized as a frequency parameter where ω is a radian frequency of the laminated plate and 0 D is a reference bending stiffness.
Then, the maximum strain energy due to the bending is expressed by     The maximum kinetic energy is given by 22 1 2 In the Ritz method the amplitude is assumed in the form ( ) ( ) ( ) where Amn are unknown coefficients, and   () and   () are the functions modified so that any kinematical boundary conditions are satisfied at the edges with applying "boundary indices" [9,15,16].To simplify the analysis, the normalized co-ordinate system O-ξη  == (10) and Xm(ξ) and Yn(η) are the displacement functions The bci (i = 1, 2, 3 and 4) is the boundary index which is used to satisfy the kinematical boundary conditions on each edge, and they are defined as on each of four edges.After substituting Eq.( 9) into the sum of energies ( 6) and ( 8), the stationary value is obtained by ( ) The minimizing process gives a set of linear simultaneous equations in terms of the coefficients Amn, and the eigenvalues Ω may be extracted by using existing computer subroutines.This analytical procedure is a standard routine of the Ritz method but the special form of polynomials ( 13) can satisfy any kinematical boundary conditions [9,15,16].

Elastic constants of different CFRP materials
For a fixed geometry (a,b,h) and boundary condition (C,S,F) of plate, it is easily understandable that frequency parameters are governed only by the values of elastic constants, and it is the main topic of this work to study the effect from the constants.As shown in Eq. ( 3), there are four independent elastic constants in Qij: EL and ET being modulus of longitudinal elasticity (Young's modulus) in the L (fiber) and T directions, respectively, GLT being a shear modulus and a Poisson's ratio νLT, and there is a relation of EL/νLT =ET/νTL among the constants.
The survey by the authors showed that a little different value of elastic constants has been employed previously, as listed in Table 1.It is observed that the difference is mainly caused by values of EL but not by ET and GLT, because Young modulus of the fibers is a key factor in controlling the stiffness in the longitudinal direction within lamina, while the matrix material is almost the same, mostly epoxy material.In the table, Mat.1 indicates values used by Stanford and Jutte [12] in NASA and shows the lowest anisotropy rate EL/ET=11.6among the listed sets of constants.Mat.2 and 3 are medium values listed in [3] and [13], respectively.Mat.4 indicates those used by Panesar and Weaver [14], a group in Bristle University, and it shows the highest anisotropy EL/ET=18.7.Finally, Mat.5 is introduced to represent the mean values of recently used CFRP materials and used as the standard reference for comparison.This set of discretized values is fictious and not measured in experiment, but it will be shown that it is useful as the representative constants for CFRP.When one measures deviation of listed materials from Mat.5, the differences in the constants show ranges of Table 1.Elastic constants of carbon fiber reinforced plastic (CFRP) materials Table 2. Convergence of the solution for cross-ply and angle-ply square plates (Mat.5)

Convergence and comparison of the solution
Before the natural frequencies are compared to discuss the discrepancy stemming from different CFRP materials, validity of the solution should be established.Table 2 presents frequency parameters of laminated square plates with Mat.5 (i.e., D0= D0* in Es.( 14) (15) ).Number of terms M, N in Eq.( 9) is increased from six to ten for crossply (CSFF, SSSS) and angle-ply (CCCC) plates.They converge well within four significant figures.
For a special case of cross-ply (i.e., D16= D26=0) simply supported rectangular plate (SSSS), the exact solution can be written in the form with m and n being half wave numbers in x and y direction, respectively.The results from this formula exactly agree with the present solution within four significant figures in Table 2.All the frequency parameters listed hereafter will be obtained by using 10×10 solution.

Unified parameter for different CFRP
Table 3 presents lowest five frequency parameters Ω, defined in Eq.( 14) by using ET of each material, for simply supported (SSSS) square plate.The results are given for Mat.1 ～ Mat.5, and the differences ( ％ ) in frequency measured from Mat.5 are also written in the table.The frequency parameter Ω is non-dimensional and it does not include the plate geometry (a, b, h) and mass density ρ.So for the fixed geometry and density, values of the frequency parameters depend on ω itself and ET and νLT.
From observation in the table, no significant differences (％) exist among the frequency orders (i.e., from the first to fifth frequency) and between the lamination schemes of cross-ply and angle-ply.The increasing (or decreasing) trend of the frequencies is unclear with respect to the constant EL, which should be the most influential factor on the frequency values.Next, the degree of influence by EL will be quantitatively focused.
The idea here is that by using the identical elastic constants ET* and νLT* in the frequency parameter, the differences are caused only by ω in the new frequency parameters.Actually, it is observed in Tables 4-8 that the increasing order of EL (Mat.1<Mat.2≒ Mat.3<Mat.4)among the materials basically reflects the order in the difference, except for FFFF plate, and the differences only for Mat.4 are positive, i.e. session Ω* (Mat.5)<Ω* (Mat.4) due to EL (Mat.5)<EL (Mat.4),unlike unnatural findings in Table 3.
The frequency parameters listed in Tables 5 (FFFF), 6 (CFFF), 7 (CSFF) and 8 (CCCC) are based on the same idea, and Ω* uses the identical values of ET*=10GPa and νLT*=0.3.In Table 5 (FFFF), it is seen that the differences for Mat.2 take alternatively positive and negative values, but those of Mat 3 show only all negative differences.This unpredicted behavior may be caused by the fact that in the FFFF case, there are three rigid body motions (elastic frequencies become zero) and the vibrations take complicated mode shapes to satisfy the self-equilibrium.Such strange behavior is found when the plate edges involve a number of free edges, such as FFFF, CFFF and CSFF, and it decreases as the number of free edges (F) diminished.
For Mat.1 with EL being the smallest in Table 1, all the differences are negative in Tables 4-8 for ten cases (cross-ply and angle-ply plates with 5 different combinations of boundary conditions), and the average differences of the lowest five Ω for the ten cases stay within the range of -6.8～-4.6％.Similarly, for Mat.5 with EL being the largest in the table, all the differences are positive for the ten cases.
By using the data in Tables 4 and 8, Figure 2 presents comparison of percentage differences for two lamination types (cross-ply, angle-ply) and two sets of boundary conditions (SSSS, CCCC).The lowest five frequency parameters are presented for the four cases with material constants for Mat.4.These differences are located between 3.0 and 5.5 percentages, and it is observed that there are no significant differences for different modes, laminations and boundary conditions.
Figure 3 presents also the lowest five frequency parameters for different sets of boundary conditions.The differences stay within the range of 3.0 and 5.5 percent, and no significant difference is observed, except for the FFFF case.As previously mentioned, the FFFF plate shows three rigid body motions and the vibration modes presented take very different shapes from the other sets (CFFF, CSFF, SSSS, CCCC).9. Frequency parameters Ω* (Eq.( 15)) of symmetric 8-layer rectangular plate (a/b=2, SSSS) Table 10.Frequency parameters Ω* (Eq.( 15)) of symmetric 8-layer rectangular plates (a/b=2, CCCC)

Effect of aspect ratio on frequency parameters
Numerical experiment is extended to include the frequency parameters of rectangular plate (a/b=2) to see the effect of aspect ratio.Table 9 present the lowest five frequency parameters of cross-ply and angle-ply rectangular plates with SSSS boundary conditions.The results in the same format is given in Table 10 for CCCC boundary condition.The data for Mat.4 is plotted in Fig. 4 with the same format in Fig. 2. When one compares results in Fig. 4 with those in Fig. 2, there is no difference globally and one can conclude that the differences do not exist for different aspect ratios.

Effect of each material constant on frequency parameters
From the results in Tables 4-8, it is obvious that Young's modulus EL in the fiber direction is the most decisive controlling factor among the four constants to determine the frequency parameters.Therefore, numerical experiment is done in Fig. 5 to calculate the lowest five frequencies Ω1～Ω5 for the angle-ply square plate (SSSS) [(30/-30)2]s by changing independently each of the four elastic constants in the range of 80 ％～120 ％ from the reference value of Mat.5 (i.e., EL=150GPa, ET=10GPa, GLT=5GPa, νLT=0.3).In the figure, the values with change of EL are denoted by black solid columns and those with other three constants ET, GLT, νLT are by lighter grey columns.As clearly seen, the frequency values increase significantly in linear fashion with change of EL, while those values for change of the other three constants stay almost unchanged.Thus, the straight line can be used to approximate the frequency values, as indicated by red straight lines in the figure, for laminated plates with different CFRP materials under specified lamination condition of fixed aspect ratio and lamination scheme (in this case, cross-ply or angle-ply laminates).
A simple linear regression is applied by using ( ) where C1 and C0 indicate slope (sensitivity) and constant (intercept), respectively.Those constants are listed in

Conclusions
It has been demonstrated that, for symmetrically laminated CFRP rectangular plates having slightly different elastic constants, there is a underlying relation among calculated values of frequency parameters when the frequency parameters are properly defined.The Ritz method was used as an analytical tool in numerical experiments for solving free vibration of the plates.In numerical results, examples were given for cross-ply and angle-ply square plates.The frequency parameters for four different sets of materials were calculated and compared to those of new hypothetical material with the averaged and discretized constants.It turned out that, despite the difference of CFRP materials, the frequency parameters show unified behaviors with the change of Young's modulus in the fiber direction.It was demonstrated that it is feasible to derive approximate formulas to simultaneously predict the frequency parameters for laminated plates composed of different CFRP materials.The coefficients in the linear formula were tabulated and used to estimate the frequency parameters for limited cases, and it is hoped that this formula will be extended and used in many design situations.