2.1 Reference fiber path

A laminated composite circular plate with curvilinear fibers requires an appropriate method for describing the fiber orientation at any point in a single layer. The plate has radius a. The fiber orientation angle θ is defined as a linear function of x between two angles T₀ and T₁ along the x-axis, where T₀ denotes the fiber orientation angle at the center and T₁ denotes the fiber orientation angle at a distance a/2 from the center. The orientation of a single curvilinear fiber path is symbolized by 〈T₀/T₁〉. The reference fiber path and orientation are shown in Figure 1. The reference fiber path equation y(x) and orientation θ(x) are as follows:

Figure 1:

Reference fiber path and orientation.

The rest of fiber paths are defined using the method of shifted paths [13]. In this method, the reference fiber path is used to create a layer. Because tow paths are identical, all that is necessary to model the layer is to duplicate the reference tow path and shift the duplicate tow paths the correct distance along the y-axis. A tow placement machine can be used to spatially change the fiber orientation within a single layer.

2.2 Fiber curvature constraint

A laminated composite circular plate with curvilinear fibers is created by curving the tow paths. To prevent a tow from kinking, the magnitude of the largest curvature in any layer must not exceed a specified maximum value. The fiber paths of a layer made of shifted fibers are identical to the reference fiber path. It is therefore sufficient to apply the fiber curvature constraint only to the reference fiber path. The curvature κ of a function y(x) is

where y′(x) and y″(x) denote the first and second derivatives of y(x) with respect to x.

Equation (3) yields

At each location along the reference fiber path, the curvature κ is required not to exceed the maximum allowable curvature of 3.28 m^-1 [13]. The design space for a circular layer made of shifted fibers and of radius a=0.5 m is shown in Figure 2. The fiber curvature constraint is satisfied everywhere in the gray region. The orientation angles T₀ and T₁ are equal along the dashed line.

Figure 2:

[∓〈30°/60°〉]_s layer with shifted curvilinear fibers.

3.1 Elastic parameters

The stresses and strains of the kth layer in the principal material directions under plane stress conditions are related by

The material stiffnesses Q_ij are defined as

where

The stresses of the kth layer in local and global coordinates are related by

The stresses and strains of the kth layer in global coordinates are related by

where the material stiffnesses Q¯ij are written in terms of m=cos(θ) and n=sin(θ) as

3.2 Shape functions

The laminated composite circular plate is modeled as one curved hierarchical square finite element as shown in Figure 3. The coordinates of a vertex i are symbolized by X_i, _,Y_i. The sides 1-2, 2-3, 3-4, and 4-1 are numbered 1, 2, 3, and 4, respectively.

Figure 3:

The curved hierarchical square finite element.

Using the hierarchical square finite element, the transverse displacement and rotations of cross-sections about the x and y axes will be expressed as

where t is time and p is the polynomial order.

The shape functions N_β are

where the subscript β is defined as

The shape functions g_i(ξ) (i=1, 2, …, p+1) are

where P_j-1 is the Legendre orthogonal polynomial of order j-1.

3.3 Stiffness and mass matrices

Using the blending function method, the curved sides of the hierarchical square finite element will be expressed in the parametric forms

where the subscript i denotes the side number.

The mapping functions of the curved hierarchical square finite element are

The derivatives with respect to local and global coordinates are related by

The Jacobian matrix is

The coefficients of J are

Equation (36) gives

where

The determinant of J is

The strains, transverse displacement, and rotations in global coordinates are related by

The strain energy U of the curved hierarchical square finite element is

where k_s is the transverse shear correction factor and

The kinetic energy T of the curved hierarchical square finite element is

The constants of inertia are

where h and ρ are the thickness and density, respectively.

Using Lagrange’s method, the equations of motion will be obtained as

where

The coefficients of the block stiffness and mass matrices K_α,β and M_α,β are given in Appendix 1.

Applying boundary conditions, Equation (56) will take the form

The symmetric generalized eigenvalue problem in Equation (59) can be solved for the frequencies using known routines.

4.1 Convergence test and comparison with published data

Because of the unavailability of results for laminated composite circular plates with curvilinear fibers, a comparison is made with published data for isotropic and symmetrically laminated composite circular plates with rectilinear fibers.

Table 1 shows the first five frequency parameters Ω of an isotropic circular plate with v=0.3. A comparison is made with values from [4]. A fast convergence is seen to occur as the polynomial order p is increased. The comparison of the results obtained with p=10 with values from [4] shows excellent agreement.

Table 2 shows the first five frequency parameters Ω of the clamped [30°, -30°, 30°] composite circular plate with h/a=0.02. The composite material has the mechanical parameters E₁=15.4E₂, G₁₂=G₁₃=G₂₃=0.792E₂, v₁₂=0.3. A comparison is made with the values from [4]. As can be seen in the table, a fast convergence occurs as the polynomial order p is increased. The results obtained with p=10 compare very well with the values from [4]. Accordingly, subsequent calculations are performed using p=10.

4.2 Free vibration analysis

The fundamental frequency parameter Ω is calculated for symmetrically laminated four-layer composite circular plates with E-glass, graphite, and boron fibers in epoxy matrices. The mechanical parameters of composite materials are given in Table 3. The material properties E_m, G_m, and v_m (for matrix) and E_f, G_f, and v_f (for fiber) are taken from [14]. The material properties E₁, E₂, G₁₂, and v₁₂ are calculated from E_m, G_m, v_m, E_f, G_f, and v_f using the rule of mixtures, as described in [15]. The average volume fraction of fiber is taken as 66.7%. It is well known that G₁₃=G₁₂, but G₂₃ is not necessarily equal to G₁₂; it can be different. In this work, it is assumed that G₁₂=G₁₃=G₂₃. The radius a and thickness h are taken as 0.5 m and 0.005 m, respectively. The symmetrically laminated composite circular plate is symbolized by [∓〈T₀/T₁〉]_s or equivalently by [+〈T₀/T₁〉, –〈T₀/T₁〉, –〈T₀/T₁〉, +〈T₀/T₁〉]. The angle T₀ is increased from 0° to 90° and the angle T₁ is increased from -90° to 90°. The increment is taken as 5°.

Figures 4–6 show the contour plots of the frequency parameter Ω for the clamped four-layer laminated composite circular plate. As can be seen in the figures, the dashed lines do not cross the largest and smallest frequency parameters. Thus, the largest and smallest frequency parameters are obtained with curvilinear fibers rather than with rectilinear ones. The largest frequency parameters are calculated for the [∓〈90°/25°〉]_s circular laminate with E-glass, graphite, and boron fibers as 0.050, 0.085, and 0.102, respectively. The smallest frequency parameters are calculated for the [∓〈20°/85°〉]_s circular laminate with E-glass, graphite, and boron fibers as 0.044, 0.070, and 0.083, respectively. The percentage changes are computed for E-glass, graphite, and boron fibers as 14%, 21%, and 23%, respectively.

Figure 4:

Contour plot of Ω for the clamped E-glass/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 5:

Contour plot of Ω for the clamped graphite/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 6:

Contour plot of Ω for the clamped boron/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figures 7–9 show the contour plots of the frequency parameter Ω of the soft simply supported laminated composite circular plate. In this case, the dashed lines cross the smallest frequency parameters. The dashed line crosses the largest frequency parameters for the sole case of boron/epoxy. This can be explained by the fact that boron/epoxy has the highest E₁/E₂. The largest frequency parameters are calculated for the [∓〈5°/70°〉]_s circular laminate with E-glass, graphite, and boron fibers as 0.016, 0.024, and 0.028, respectively. The smallest frequency parameter is calculated for the [∓〈0°/0°〉]_s circular laminate as 0.011 regardless of the type of composite material. As can be seen in the figures, the fiber material and boundary conditions influence the distribution of frequency throughout the design space.

Figure 7:

Contour plot of Ω for the soft simply supported E-glass/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 8:

Contour plot of Ω for the soft simply supported graphite/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 9:

Contour plot of Ω for the soft simply supported boron/epoxy [∓〈T₀/T₁〉]_s circular laminate.

To investigate the existence of mode crossing and mode veering, the variation of the frequency parameter Ω with the fiber orientation angle T₀ (T₁=30°) for the first five modes of clamped and soft simply supported E-glass/epoxy, graphite/epoxy, and boron/epoxy laminated composite circular plates is found. The frequency curves are represented in Figures 10–15. It can be seen that none of the modes is affected by crossing but modes 3 and 4 of clamped graphite/epoxy and boron/epoxy circular laminates are affected by veering.

Figure 10:

Frequency parameter Ω versus T₀ (T₁=30°) for the first five modes of the clamped E-glass/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 11:

Frequency parameter Ω versus T₀ (T₁=30°) for the first five modes of the clamped graphite/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 12:

Frequency parameter Ω versus T₀ (T₁=30°) for the first five modes of the clamped boron/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 13:

Frequency parameter Ω versus T₀ (T₁=30°) for the first five modes of the soft simply supported E-glass/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 14:

Frequency parameter Ω versus T₀ (T₁=30°) for the first five modes of the soft simply supported graphite/epoxy [∓〈T₀/T₁〉]_s circular laminate.

Figure 15:

Frequency parameter Ω versus T₀ (T₁=30°) for the first five modes of the soft simply supported boron/epoxy [∓〈T₀/T₁〉]_s circular laminate.