Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates

: The present study focuses on the static and dynamic response of functionally graded carbon nanotube (FG-CNT)-reinforced rhombic laminates. The cubic variation of thickness coordinate in the displacement ﬁeld is considered in terms of Taylor’s series expansion, which represents the higher-order transverse cross-sectional deformation modes. The condition of zero-transverse shear strain at upper and lower surface of FG-CNT-reinforced rhombic laminates is imposed in the present formulation. The present two-dimensional model is formulated in a ﬁnite element, with the C 0 element consisting of seven nodal unknowns per node. The ﬁnal material properties of FG-CNT-reinforced rhombic laminates are estimated using the rule of mixture. The obtained numerical are compared with the results available in the literature to verify the reliability of the present model. The present study investigates the effect of CNT distribution, loading pattern, volume fraction, and various combinations of boundary constraints by developing a ﬁnite element code in FORTRAN.


Introduction
Recently, composite plates reinforced by carbon nanotubes (CNTs) have gained significant attention in civil, aeronautical, mechanical, and marine engineering due to their high strength/weight ratio and low density. The CNTs discovered by Iijima [1] are made up of the molecular-scale tube-like structure of carbon allotropes having fine materialistic properties. The CNTs are generally used in composites to improve their elastomechanical and thermal properties by dispersing in the matrix [2,3]. Various plate theories have been developed by the researchers to analyze the plates. The classical plate theory (CPT) model is based on the Kirchhoff-Love hypothesis that straight lines remain straight and perpendicular to the midplane after deformation. The transverse shear stress components are neglected in the CPT, where it is included in the shear deformation theory by Reissner [4]. In Reissner's shear deformation theory, a shear correction factor is required for strain equations. Zhu et al. [5] discussed the effect of single-walled CNTs on the bending and vibration analysis of a CNT-reinforced The volume fraction of uniform distribution and functionally graded distributions of the CNTs along the thickness direction of the CNT-reinforced rhombic plates shown in Figure 1 was assumed to be as follows: The volume fraction of uniform distribution and functionally graded distributions of the CNTs along the thickness direction of the CNT-reinforced rhombic plates shown in Figure 1 was assumed to be as follows: where w CNT is the mass fraction of the CNTs in the CNT-reinforced rhombic plates, whereas ρ m and ρ CNT are the densities of the polymer matrix and carbon nanotubes, respectively. In line with the rule of mixture, the effective material properties of FG-CNT-reinforced plates were employed by introducing the CNT efficiency parameters; thus, the final properties can be written as follows [38]: where E 11 CNT and E 22 CNT are Young's moduli and G 12 CNT is the shear modulus of singly walled CNTs, respectively. E m and G m are known as Young's modulus and shear modulus of the isotropic matrix. ν 12 CNT and ν m represent the Poisson's ratio of CNTs and matrix respectively. V m and V CNT are the volume fractions of the matrix and carbon nanotubes, respectively, and the sum of both volume fractions equals to unity. η 1 , η 2 , η 3 are the scale-dependent material properties and they can be calculated by matching the effective properties of CNT-reinforced composite obtained from the MD simulations with those from the rule of mixture.

Displacement Fields and Strains
The displacement field for the FG-CNT-reinforced rhombic plate is considered to derive the mathematical model based on the third-order shear deformation theory [39]: u(x, y, z) = u 0 (x, y) + zθ x (x, y) + z 2 ξ x (x, y) + z 3 ζ x (x, y) v(x, y, z) = v 0 (x, y) + zθ y (x, y) + z 2 ξ y (x, y) + z 3 ζ y (x, y) w(x, y, z) = w 0 (x, y) , (7) where u, v and w are the displacements of any generic point in the plate geometry, (u 0 , v 0 and w 0 ) are displacements at the mid-plane and θ x , θ y are the bending rotations defined at the midplane about the y and x-axes respectively. ξ x , ξ y , ζ x and ζ y are higher order terms of Taylor's series expansion. The function ξ x , ξ y , ζ x and ζ y will be calculated by vanishing shear stress at top and bottom of the plate. By applying the boundary conditions γ xz (x, y, ± h/2) = γ yz (x, y, ± h/2) = 0 at the upper layer and lower layer of the plate in Equation (7) and rearranging the terms that appear in the displacement field (u and v), we obtained By substituting Equations (8) and (9) into Equation (7), we obtain If the displacement field represented in Equation (10) is implemented in the strain part, the problem of C 1 continuity requirement in the higher-order theory may arise due to the existence of first-order derivatives of transverse displacement. By applying C 0 continuity to the present problem, the out of plane derivatives are exchanged by the following relations in Equation (10): The final form of higher order theory possessing C 0 continuity may be presented in the following manner: Hence, the basic field variables interpreted in the present investigation with the assumption of constant transverse displacement component are u 0 , ν 0 , w 0 , θ x , θ y , ψ x , and ψ y for each node. Mathematically, the nodal displacement vector {δ} corresponding to displacement field in Equation (12) may be represented as From the displacement field presented above in Equation (12), the strain can be written as Furthermore, the expression of strain vector {ε} can be correlated with the displacement vector {δ} by means of the following relationship: where [B] is known as a strain-displacement matrix and involves the derivatives of shape function terms. Since the plane stress problem is considered in the analysis, the components of strain vector may be represented as The strain relationships can be written as

Constitutive Relationship
The stress-strain relationship for the CNTRC rhombic plate can be written as where the constitutive matrix is where

Element Description
A nine-node C 0 isoparametric Lagrangian element was utilized in the present investigation. The element has a total of 63 degrees of freedom and each node has seven degrees of freedom. The element has inconsistent rectangular geometry in the x-y coordinate system. In order to ensure a consistent rectangular geometry, the element was plotted to the ξ-η plane. For the assumed nine-node element, the expressions for shape functions N i are described below.
For corner nodes: For middle nodes: For the center node:

Governing Equation for Bending Analysis
The expression of strain energy may be given as By utilizing the relationship of Equation (18), the above equation can be written as where [H] is the matrix that contains the terms involving z and h.
The change in strain vector may be written as {δε} = [B]{δX}. By using Equations (15) and (25), the stiffness matrix [K] can be written in the following form:

Governing Equation for Free Vibration Analysis
The time derivative of velocity at any given point within the element may be expressed in terms of the mid-surface displacement parameters (u 0 , v 0 and w 0 ) as where the vector {f } represents the nodal unknowns, which is of the 7 × 1 order and contains the terms of Equation (7). The nodal unknowns {f } are decoupled into a matrix [C] that involves the shape functions (N i ) and global displacement vector {X}: where the matrix {X} contains the nodal unknowns of the nine nodes. By utilizing the Equations (27) and (28), the mass matrix of an element can be written as where the expression of the matrix [L] can be expressed as where ρ is termed as the density of the CNT-reinforced rhombic plate. The derivation of element stiffness matrix and the mass matrix is given in the Appendix A. Hence, the governing equation for free vibration analysis of rhombic plate becomes where [K] and [M] are the linear stiffness matrix and mass matrix, respectively.

Skew Boundary Transformation
For the rhombic plate shown in Figure 2, it is important to alter the element matrices from global axes (x, y) to local axes (x', y') because the skew boundary of the laminate is not parallel to the global axes of the rhombic laminate. Hence, the transformation matrix [T] is required at the element level to transform the element matrices from global to local axes.
where c = cosα, s = sinα and α is the skew angle of the plate.

Numerical Results and Discussion
The static and free vibration analyses were performed for FG-CNT-reinforced rhombic plate under different combination of end support, volume fraction, and several geometric parameters. The above-discussed formulation has been incorporated into a computer code. The nine-noded isoparametric elements with seven degrees of freedom per node were chosen for the present model for discretizing the FG-CNT-reinforced rhombic plate. Poly{(m-phenylenevinylene)-co-[(2,5-dioctox y-p-phenylene) vinylene]}, basically known as PmPV [40], was chosen as the matrix and the armchair (10, 10) SWCNTs were considered as the reinforcing material. The material properties of the matrix were taken as E m = 2.1 GPa, ρ m = 1150 kg/m 3 and ν m = 0.34 at room temperature (300 K). The material properties of (10,10) SWCNTs at 300K are tabulated in Table 1. Three types volume fraction were used in present study. In the case of V*CNT = 0.11, η1 = 0.934 and η2 = 0.149, in the case of V*CNT = 0.14, η1 = 0.150 and η2 = 0.941, and for V*CNT = 0.17, η1 = 0.149 and η2 = 1.381. We assume that η2 = η3 and G12 = G13 = G23. The abovementioned values are used for the following numerical results.

Numerical Results and Discussion
The static and free vibration analyses were performed for FG-CNT-reinforced rhombic plate under different combination of end support, volume fraction, and several geometric parameters.
The above-discussed formulation has been incorporated into a computer code.
The nine-noded isoparametric elements with seven degrees of freedom per node were chosen for the present model for discretizing the FG-CNT-reinforced rhombic plate. Poly{(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]}, basically known as PmPV [40], was chosen as the matrix and the armchair (10, 10) SWCNTs were considered as the reinforcing material. The material properties of the matrix were taken as E m = 2.1 GPa, ρ m = 1150 kg/m 3 and ν m = 0.34 at room temperature (300 K). The material properties of (10,10) SWCNTs at 300K are tabulated in Table 1. Three types volume fraction were used in present study. In the case of V* CNT = 0.11, η 1 = 0.934 and η 2 = 0.149, in the case of V* CNT = 0.14, η 1 = 0.150 and η 2 = 0.941, and for V* CNT = 0.17, η 1 = 0.149 and η 2 = 1.381. We assume that η 2 = η 3 and G 12 = G 13 = G 23 . The abovementioned values are used for the following numerical results. The non-dimensional quantities used are: • For the bending analysis • For the free vibration analysis The loading patterns used for the bending analysis are: For uniform loading q = q 0 ; For sin-sin loading q = q 0 sin πx a sin πy b ; For cos-cos loading q = q 0 cos πx a cos πy b .
The boundary conditions taken in the present analysis are as mentioned below:

Free Vibration Analysis
Example 1. The convergence study for the non-dimensional frequency parameter was carried out for UD and FG-CNT-reinforced rhombic plate shown in Table 2. The dimensionless frequency parameter of the UD, FG-V, FG-O, and FG-X type distributed CNTRC rhombic plate was computed for different mesh sizes and clamped boundary conditions. The results were computed for V* CNT = 0.11 and a skew angle of 15 • . The convergence study indicated that 16 × 16 mesh is satisfactory for the free vibration analysis of functionally graded CNTRC rhombic plate using current nine-noded isoparametric elements. Hence, 16 × 16 mesh size was adopted for all the cases of free vibration analysis of a functionally graded CNT-reinforced rhombic plate.  Tables 3 and 4 show the results of the bending and free vibration analyses for an isotropic square plate (ν = 0.3), respectively. The maximum deflection and axial stress were compared with the results provided by Reddy [41] and the frequency parameter of the isotropic plate was compared with an exact solution [42] and HSDT results for a moderately thick plate [43].  Tables 5 and 6. The first six non-dimensional frequencies were compared with Zhu et al. [5]. Three volume fractions (V* CNT = 0.11 and 0.14) and three side-to-thickness ratios (a/h = 10, 20 and 50) were taken for comparison. The frequency parameter of simply supported and clamped boundary condition was found to be closer to Zhu et al. [5].   Table 7 depicts the convergence study for dimensionless maximum deflection for UD and FG-CNT-reinforced functionally graded rhombic plate. The dimensionless maximum deflection of the UD, FG-V, FG-O and FG-X CNTRC rhombic plate was computed for different mesh size and clamped boundary condition. The results were computed for V* CNT = 0.11. The convergence study showed that 16 × 16 mesh size is acceptable for the present model using the discussed nine-noded isoparametric elements. Hence, 16 × 16 mesh size was chosen for all the parametric studies of the bending analysis of functionally graded CNTRC rhombic plate.  Tables 8  and 9. The non-dimensional central deflection was compared with Zhu et al. [5]; V* CNT = 0.11, 0.14 and a/h = 10, 20, 50 were used for the comparison study. The dimensionless central deflection of different types of boundary condition was found to be in decent agreement with Zhu et al. [5].

Results and Discussion
The comparison study indicates that the present mathematical model and its finite element implementation results are in agreement with the previously published results. The present study has been conducted to investigate the effect of loading pattern, side-to-thickness ratios (a/h), aspect ratio (a/b), skew angle (α), volume fraction of CNT (V* CNT ), and different boundary conditions (SSSS, CCCC, CCSS, CSCS, CCFF, and CFCF) on the bending and free vibration behavior of functionally graded CNT-reinforced composite rhombic plate.

Free Vibration Analysis
Tables 10 and 11 represent the first six dimensionless frequency parameter of UD-CNTand FG-CNT-reinforced rhombic plate for the three different types of CNT volume fraction V * CNT = 0.11, 0.14 and 0.17 and four different skew angles (α = 15 • , 30 • , 45 • and 60 • ). The results were tabulated for simply supported and clamped boundary condition, respectively. It was noticed that an increase in the skew angle results in an increase in dimensionless frequency parameter for all types of CNTs distribution and all considered CNT volume fractions. An approximately 6% increase was noticed in the dimensionless fundamental frequency parameter of functionally graded CNTRC rhombic plate when skew angle changes from 15 • to 30 • , 22% and 60% increase was noticed for 15 • to 45 • and for 15 • to 60 • . Table 12 shows the dimensionless frequency parameter of FG-CNTR-reinforced rhombic plate for CCSS-, CSCS-, CCFF-, and CFCF-type boundary support. For the all considered boundary conditions and skew angles, the FG-O distribution retains the minimum dimensionless frequency parameter while the FG-X distribution shows maximum values of dimensionless frequency parameter among other kinds of distribution. Additional distribution of CNTs should be provided at the top and the bottom section rather than at the mid-section for attaining maximum stiffness. Therefore, the FG-X and FG-O distributions yield maximum and minimum stiffness, respectively. Apart from this, the CCCC end support yields the highest dimensionless frequency parameter while the CFCF end support shows the lowest value among all considered boundary condition resulting from the fact that the higher constraints at support impart higher stiffness to the FG-CNTRC rhombic plate. The effect of the side-to-thickness ratio for various types of skew angles was presented in Table 13. The dimensionless fundamental frequency parameter for all type of CNT distribution was increased along with the a/h ratio. The dimensionless frequency parameter also increases with the aspect ratio of FG-CNTR-reinforced rhombic plate, as depicted in Figure 3. The results were calculated for skew angles of 15 • and 30 • . The first four mode shapes for FG-V CNT-reinforced rhombic square plate having simply supported boundary condition and 30 • skew angle are presented in Figure 4.

Static Analysis
The dimensionless maximum deflection of a FG-CNT-reinforced rhombic plate under uniform loading for simply supported and clamped boundary conditions is shown in Tables 14 and 15, respectively. The volume fraction of CNT was taken as 0.11, 0.14 and 0.17. The results were tabulated for UD and FG-CNT-reinforced rhombic plate with a/b = 1 and a/h = 10. It can be observed that an increase in the volume fraction of CNTs results in a decrease in the deflection of CNTRC rhombic plate because of the fact that the higher value of volume fraction has higher stiffness; thus, the deflection is reduced. It is anticipated that there is a nearly 36% decrease shown in maximum deflection for both clamped and simply supported boundary conditions as the value of V* CNT increases from 0.11 to 0.17 and approximately 6% decreases are noticed when V* CNT changes from 0.11 to 0.14. Maximum dimensionless deflection decreases with an increase in the skew angle because it reduces the length of the shorter diagonal leading to an enhancement in the stiffness of the rhombic plate. Thus, the deflection is reduced. Appl. Sci. 2018, 8

Static Analysis
The dimensionless maximum deflection of a FG-CNT-reinforced rhombic plate under uniform loading for simply supported and clamped boundary conditions is shown in Tables 14 and 15, respectively. The volume fraction of CNT was taken as 0.11, 0.14 and 0.17. The results were tabulated for UD and FG-CNT-reinforced rhombic plate with a/b = 1 and a/h = 10. It can be observed that an increase in the volume fraction of CNTs results in a decrease in the deflection of CNTRC rhombic plate because of the fact that the higher value of volume fraction has higher stiffness; thus, the deflection is reduced. It is anticipated that there is a nearly 36% decrease shown in maximum deflection for both clamped and simply supported boundary conditions as the value of V * CNT increases from 0.11 to 0.17 and approximately 6% decreases are noticed when V * CNT changes from 0.11 to 0.14. Maximum dimensionless deflection decreases with an increase in the skew angle because it reduces the length of the shorter diagonal leading to an enhancement in the stiffness of the rhombic plate. Thus, the deflection is reduced.    Tables 16 and 17 represent the dimensionless maximum deflection of simply supported and clamped FG-CNT-reinforced rhombic plate under sin-sin loading, respectively. Here, an approximately 25% decrease in the maximum dimensionless deflection is noticed when the skew angle changes from 15 • to 30 • ; 40% decreases when the skew angle changes from 30 • to 45 • and 55% decreases when the skew angle changes from 45 • to 60 • for both uniform loading and sin-sin loading. The lowest and highest dimensionless deflection was found for FG-O-and FG-X-type CNT distribution, respectively.  Figure 5 shows the variation of dimensionless deflection of FG-V CNT-reinforced rhombic plate along the length (x/a) at y/b = 0.50 for four skew angles under sin-sin loading. It can be seen that all values of CNT volume fraction have the same nature of deflection along the length and for the skew angle 60 • , negative deflection is noticed for the farther end subjected to sin-sin loading.    The effect of loading type on the non-dimensional deflection of simply supported and clamped FG-V type CNT-reinforced rhombic plate with skew angle was shown in Figure 6. The effect of loading type on the non-dimensional deflection of simply supported and clamped FG-V type CNT-reinforced rhombic plate with skew angle was shown in Figure 6. The non-dimensional maximum deflection decreased with an increase in the skew angle for uniform and sin-sin loading, while under the cos-cos loading, the value of w increases first and then decreases as the skew angle grows. The effect of the skew angle on the maximum dimensionless deflection of CNT-reinforced rhombic plate subjected to sin-sin loading having various types of boundary condition was shown in Figure 7. For all considered boundary conditions except CFCF, the pattern of dimensionless deflection along the skew angle is linear. Figure 8 shows the variation in The non-dimensional maximum deflection decreased with an increase in the skew angle for uniform and sin-sin loading, while under the cos-cos loading, the value of w increases first and then decreases as the skew angle grows. The effect of the skew angle on the maximum dimensionless deflection of CNT-reinforced rhombic plate subjected to sin-sin loading having various types of boundary condition was shown in Figure 7. For all considered boundary conditions except CFCF, the pattern of dimensionless deflection along the skew angle is linear. Figure 8 shows the variation in dimensionless deflection of FG-CNT-reinforced rhombic plate along the length of the central line for four types of side-to-thickness ratios subjected to sin-sin load. The same nature of deflection along the length was noticed for all values of a/h. The results were calculated for the skew angle of 30 • and simply supported boundary condition. dimensionless deflection of FG-CNT-reinforced rhombic plate along the length of the central line for four types of side-to-thickness ratios subjected to sin-sin load. The same nature of deflection along the length was noticed for all values of a/h. The results were calculated for the skew angle of 30° and simply supported boundary condition.

Conclusions
The static and free vibration analyses of FG-CNT-reinforced rhombic plate under various types of load considering various combinations of end support using an efficient C 0 finite element model based on TSDT were presented. The actual material properties at any given section are calculated using the rule of mixture. The following conclusions written below were drawn from the obtained results for numerous values of side-to-thickness ratio, skew angle, and aspect ratio, and different types of end support.


The FG-O and FG-X type distributions inside the CNT rhombic plates have lower and higher non-dimensional frequency parameter as well as higher and lower dimensionless deflection, respectively.  The rise in the CNTs volume fraction results in a decrease in the deflection and an increase in the frequency parameter of the CNT-reinforced rhombic plate.  The dimensionless frequency parameter increases along with the skew angle, irrespective of the CNT distribution and boundary condition.  Maximum dimensionless deflection and dimensionless normal stresses decrease along with the skew angle.

Conclusions
The static and free vibration analyses of FG-CNT-reinforced rhombic plate under various types of load considering various combinations of end support using an efficient C 0 finite element model based on TSDT were presented. The actual material properties at any given section are calculated using the rule of mixture. The following conclusions written below were drawn from the obtained results for numerous values of side-to-thickness ratio, skew angle, and aspect ratio, and different types of end support.

•
The FG-O and FG-X type distributions inside the CNT rhombic plates have lower and higher non-dimensional frequency parameter as well as higher and lower dimensionless deflection, respectively.

•
The rise in the CNTs volume fraction results in a decrease in the deflection and an increase in the frequency parameter of the CNT-reinforced rhombic plate.

•
The dimensionless frequency parameter increases along with the skew angle, irrespective of the CNT distribution and boundary condition.

•
Maximum dimensionless deflection and dimensionless normal stresses decrease along with the skew angle.