Geometrically nonlinear free vibration analysis of shear deformable laminated composite elliptic plates using the p -version FEM

Abstract

This study examines the large amplitude free vibration behavior of symmetric and antisymmetric constant stiffness composite laminated (CSCL) and variable stiffness composite laminated (VSCL) elliptic plates, addressing a gap in nonlinear vibration research on elliptic composite structures. A curved rectangular $p$ -element, based on the first-order shear deformation theory (FSDT) and Von Kármán nonlinearity, accurately model their dynamic responses. The nonlinear equations of motion are transformed into the frequency domain using the harmonic balance method and solved iteratively via the linearized updated mode method. The model’s efficiency and accuracy are validated through convergence and comparison studies. Parametric studies reveal that aspect ratio, thickness ratio, stacking sequence, boundary conditions, modulus ratio, and fiber orientation angles significantly influence the nonlinear frequency response, hardening behavior, and mode shapes. VSCL plates, in particular, demonstrate greater potential for tailoring nonlinear vibrations through fiber path customization. Simply supported plates exhibit stronger hardening effects, while clamped plates yield higher nonlinear fundamental frequencies. Additionally, symmetric configurations exhibit greater hardening behavior than antisymmetric ones. These findings enhance nonlinear vibration modeling and provide key insights for designing composite structures under large amplitude vibrations, serving as benchmark data for future studies on nonlinear dynamics and optimization.

Keywords

Geometric nonlinearity large amplitude free vibration constant stiffness variable stiffness laminated composite elliptic plate degree of hardening

Introduction

Laminated fiber-reinforced composite structures have revolutionized industries like aerospace, automotive, and naval engineering due to their exceptional combination of high stiffness, lightweight design, and superior resistance to fatigue and corrosion. Traditional constant stiffness composite laminates (CSCL) have seen extensive use, but the emergence of variable stiffness composite laminates (VSCL) with curvilinear fiber paths offers new possibilities for optimized structural performance.¹ Advanced manufacturing techniques, such as Automated Fiber Placement (AFP), enable precise control over fiber orientation in VSCLs, resulting in tailored mechanical properties, reduced material usage, and improved energy efficiency.

VSCL structures with curvilinear fibers are well-known for their advantages, and their mechanical behavior has been thoroughly investigated for various composite structures, including beams,^2–7 plates,^8–17 and shells.^18–22 For a comprehensive review of the mechanical behavior and modeling of VSCL structures, readers may refer to Ribeiro’s review.²³ Generally, the findings indicate that curvilinear fiber paths outperform traditional straight fibers in terms of vibrational and buckling performances due to their tailored stiffness and improved load-bearing efficiency they offer. While the linear vibration behavior of VSCL laminates has been extensively studied, their nonlinear vibration characteristics remain less explored. Ribeiro and Akhavan²⁴ conducted the first study in this area, investigating the nonlinear forced vibrations of VSCL plates using a $p$ -version of the finite element method ( $p$ -FEM). By integrating hierarchical basis functions, FSDT, and Von Kármán’s nonlinear terms, they demonstrated that adjusting fiber orientation can significantly impact the nonlinear vibrational response. Ribeiro²⁵ extended the $p$ -FEM approach to investigate the nonlinear periodic free vibration of conservative symmetric VSCL rectangular plates. Using linear vibration modes, the equations were reduced in the time domain and solved in the frequency domain via the harmonic balance method and arc-length continuation. Houmat²⁶ further employed the $p$ -FEM along with Von Kármán’s nonlinear classical thin plate theory (CPT) to investigate the geometrically nonlinear free vibration of clamped symmetric and antisymmetric VSCL rectangular plates. Results showed that while antisymmetric laminates had slightly lower fundamental frequencies, they exhibited the most extreme hardening behavior. Houmat²⁷ also analyzed the nonlinear free vibration of VSCL symmetric skew plates by introducing a skew $p$ -element in conjunction with CPT and Von Kármán’s nonlinear model. Akhavan and Ribeiro^28,29 studied the large amplitude forced vibration and free oscillations perfect and imperfect clamped VSCL rectangular plates. Their study employed $p$ -FEM with Von Kármán’s third-order shear deformation theory (TSDT) and the shooting method. Ribeiro³⁰ and Ribeiro and Stoykov³¹ explored the geometrically nonlinear free vibrations of thin VSCL shallow shells and nonlinear forced vibrations of cylindrical shells using $p$ -FEM. Their analysis was based on an original formulation incorporating Kirchhoff-Love’s hypothesis and Von Kármán’s strain displacement relations. The nonlinear system was solved by an arc-length continuation method. These studies demonstrated that curvilinear fibers significantly influence the hardening and softening behaviors of VSCL shells. Akhavan and Ribeiro³² studied the periodic vibrations of VSCL rectangular plates with geometric imperfections under harmonic excitation, incorporating damage assessment. They modeled the vibrational problem using $p$ -FEM combined with TSDT and Von Kármán’s nonlinear relations. The results showed that increasing the fiber angle at the plate center enhances stiffness, while larger edge fiber angles reduce it.

Ganapathi et al.³³ examined the large amplitude free flexural vibrations of thick and thin symmetric and antisymmetric laminates with curvilinear fibers. Their methodology employed a higher-order theory incorporating geometric nonlinearity through Green’s strain vectors. The problem was solved using a direct iterative procedure. The results highlighted that VSCLs demonstrate a significantly higher degree of hardening behavior than CSCLs. Akhavan and Ribeiro³⁴ investigated the nonlinear flutter of VSCL rectangular plates in supersonic airflow using $p$ -FEM based on TSDT. The equations of motion were formulated using the principle of virtual work and solved via a specialized method for determining the amplitudes of limit-cycle oscillations. Farsadi et al.³⁵ optimized VSCL skew and tapered cylindrical panels to maximize their fundamental natural frequencies using a genetic algorithm. The amplitude-dependent nonlinear frequency behavior was analyzed through the Generalized Differential Quadrature (GDQ) method, combined with FSDT and Green’s strain-based nonlinearity. Results showed that the radius ratios, fiber paths, and boundary conditions significantly impact natural and nonlinear frequencies, mode shapes, and maximum achievable frequency. Yan et al.³⁶ developed a semi-analytical method for the nonlinear vibration analysis of thin VSCL plates. The authors used a mixed variational principle and Von Kármán’s relations. The Ritz method was employed for spatial analysis, while four temporal methods, including the harmonic balance method, perturbation, averaging, and direct integration were applied. Their findings indicated that the perturbation method allows for rapid analysis. Ribeiro et al.³⁷ examined the influence of fiber orientation and boundary stiffness on the nonlinear forced vibrations of thin VSCL rectangular plates. The $p$ -FEM based on CPT and Von Karman’s formulations was applied. Gupta and Pradyumna^38,39 analyzed the nonlinear free and forced vibrations of VSCL and sandwich shell panels featuring VSCL face-sheets and an auxetic honeycomb core. Using the FEM combined with TSDT, Von Kármán nonlinearities, and the Murakami zig-zag model, they demonstrated that edge fiber angles had a significant impact on forced vibration responses compared to centered fiber angles.

Barathan et al.⁴⁰ studied the nonlinear free vibration behavior of VSCL beams, using a shear-flexible model that incorporates Poisson’s effect. The problem was formulated via FEM and solved iteratively through an eigenvalue approach. The results showed that increasing the fiber edge angle reduces the degree of hardening, while increasing the center angle enhances hardening characteristics. Anilkumar et al.⁴¹ developed a semi-analytical model based on Von Kármán’s relations to explore the nonlinear dynamics of bi-stable, square cross-ply VSCL plates. Their findings demonstrated that curvilinear fibers significantly influence the nonlinear frequencies and mode shapes of these plates. Pagani et al.⁴² developed a numerical approach using the two-dimensional Carrera Unified Formulation (CUF) and the layerwise method to study the nonlinear vibration and buckling behaviors of various VSCL plate and shell structures. Andérez González and Vescovini⁴³ presented a mixed formulation based on TSDT for analyzing the nonlinear vibrations of thick VSCL plates. They combined the Ritz, differential quadrature, and harmonic balance methods to solve this problem.

The mechanical behavior of composite structures represents a complex and evolving field, focused on understanding how these advanced materials respond to diverse loading conditions. Over the years, this area has attracted significant research interest, with various methodologies and theoretical frameworks developed to capture its complexities. For instance, Wang et al.⁴⁴ addressed the free vibration response of composite materials under complex damping by proposing an equivalent viscous damping approach, enabling stable and efficient time-domain computations. Wang et al.⁴⁵ investigated the hydrostatic buckling behavior and design optimization of composite cylindrical shells. Collectively, these works offer valuable insights into both the dynamic and stability performance of composite structures. Moreover, Cong et al.⁴⁶ extended the scope by considering the impact of multi-source uncertainties, thereby enhancing the reliability and safety of composite systems, especially in thermal environments.

Nonlinear vibration analysis is particularly essential for understanding the dynamic behavior of composite structures subjected to large deformations and high-frequency excitations. Unlike linear models, nonlinear approaches account for phenomena such as stiffness variation, geometric nonlinearity, and amplitude-dependent frequency shifts, which are crucial for capturing the complex response of advanced laminated composites. Among the computational techniques employed to address these challenges, the FEM remains the most widely adopted. Numerous studies, including those cited in Refs.24–34 and 37–40 have successfully applied FEM to analyze the nonlinear vibrational behavior of composite structures, accurately capturing localized effects and complex stress distributions. Chen et al.⁴⁷ investigated the nonlinear resonance behaviors of bi-directional functionally graded microbeams using FEM, accounting for material gradation, size effects, and geometric nonlinearity. Ramteke et al.⁴⁸ explores the nonlinear eigenfrequencies of doubly-curved functionally graded panels, incorporating multi-directional grading and large geometric deformations. All these studies highlight the robustness of the FEM in capturing the nonlinear dynamic responses of advanced composite structures.

As an alternative to FEM, meshless methods, employed in studies such as^35,43 and,^49–51 offer greater flexibility for modeling large deformations and complex boundary conditions without relying on a fixed mesh. These methods complement FEM by enhancing adaptability in simulating nonlinear dynamics of composite plates. Additionally, semi-analytical approaches^36,41 combine the precision of analytical formulations with the computational efficiency of numerical techniques, offering a balanced framework for analyzing complex structural behaviors. These methods are particularly effective in capturing nonlinear dynamics while reducing computational costs compared to fully numerical approaches. Each of these computational strategies contributes distinct advantages, enabling researchers to address wide range of challenges and to further advance the understanding of composite material dynamics.

The $p$ -version of FEM has been successfully used to analyze the nonlinear vibrational behavior of VSCL structures.^24–32 This method offers higher accuracy by increasing the polynomial order of the shape functions. It significantly improves the representation of complex geometric nonlinearities, limit-cycle oscillations, and mode coupling effects, which are commonly observed in VSCL structures. Unlike the $h$ -version, it does not require mesh refinement to improve accuracy, making it particularly advantageous for nonlinear analysis, where iterative solutions necessitate the reconstruction of nonlinear stiffness matrices. Additionally, the $p$ -FEM allows for a detailed representation of stress and strain distributions, particularly in regions of high nonlinearity and stiffness gradients induced by variable fiber orientations. Moreover, a structure can be modeled using a single element, further enhancing computational efficiency. These advantages make $p$ -FEM a powerful tool for capturing both global dynamic responses and localized nonlinearities in VSCL structures, especially under large amplitude excitations.

Elliptic plates are widely used in various engineering fields, such as aerospace and nuclear engineering, where their shape offers advantages such as improved aerodynamics and reduced stress concentrations. Understanding their vibrational behavior is crucial for improving the performance and reliability of engineered systems. Although numerous studies have investigated the nonlinear vibrations of isotropic^52–58 and orthotropic^59–61 elliptic plates, as well as the linear vibration of CSCL^62–64 and VSCL circular and elliptic plates,^10,65 however, solutions for geometrically nonlinear free vibrations of composite laminated elliptic structures remain unavailable in the literature. Building on prior studies, this work extends large amplitude nonlinear free vibration modeling to symmetric and antisymmetric CSCL and VSCL elliptic plates, incorporating complex stiffness distributions and geometric nonlinearities. A high-accuracy computational approach is introduced to analyze their nonlinear dynamic behavior under various design parameters. The analysis employs a curved hierarchical finite element based on FSDT and Von Kármán nonlinear equations. This approach ensures exponential convergence and an efficient representation of complex displacement fields with a single curved rectangular element, making it particularly suitable for vibration analysis. However, while the $p$ -FEM offers high accuracy and computational efficiency, it may introduce geometric approximation errors for curved elements at very high polynomial orders. To mitigate this issue, the blending function method is employed to enhance the accuracy of the elliptic structure’s representation. The nonlinear equations of free motion are mapped from the time domain into the frequency domain using the harmonic balance method and solved iteratively via the linearized updated mode method. The accuracy of the proposed method is verified through convergence tests by varying the polynomial order, and through comparisons with existing results for isotropic and orthotropic elliptic plates. The nonlinear fundamental frequencies of elliptic plates reinforced with straight and curvilinear fibers presented here, for the first time, may serve as benchmark results for future studies. Further, a parametric study investigates the effect of fiber orientation angles, boundary conditions, number of layers, aspect ratio, thickness ratio, modulus ratio, and stacking sequence on the nonlinear frequency response, hardening behavior, and mode shapes. These findings offer insights into the dynamic behavior of CSCL and VSCL elliptic plates, contributing to the advancements in nonlinear vibration modeling for aerospace, mechanical, and materials engineering applications.

Formulation

Curvilinear fiber

A variable stiffness laminated composite elliptic plate with dimensions $2 a \times 2 b$ , is constructed using a reference curvilinear fiber path. As shown in Figure 1, the fiber orientation angle $θ^{k}$ varies linearly along the $x$ -axis between two angles $T_{0}$ and $T_{1}$ . Here, $T_{0}$ is the fiber orientation angle at the center, while $T_{1}$ is the fiber orientation angle at a distance $a / \sqrt{2}$ from the center. The reference fiber path orientation is defined by $〈 T_{0} | T_{1} 〉$ . The elliptic laminate consists of $N C$ layers of uniform thickness. The shifted fiber path method¹ is employed to construct the remaining fibers as shown in Figure 2. Notably, $T_{0} = T_{1}$ represents the case of constant stiffness laminated composite elliptic plate. It is worth mentioning that the shifted fiber method may introduce gaps and overlaps during manufacturing. However, as this study focuses on the theoretical nonlinear vibration analysis of VSCL elliptic plates, addressing manufacturing-related challenges such as these, falls beyond its scope. Such issues are typically managed through experimental validation and process optimization techniques in AFP systems.

Figure 1.

Elliptic layer with curvilinear reference fiber path and orientation angles ( $T_{0} and T_{1}$ ).

Figure 2.

Shifted fiber paths of an elliptic layer $〈 45 ° | 15 ° 〉$ with aspect ratio $a / b = 2$ .

The reference fiber path equation $y (x)$ and the fiber orientation angle $θ (x)$ can be expressed as functions of $T_{0}$ , $T_{1}$ , and $x$ as follows:

\begin{array}{l} y (x) = & {\begin{cases} \frac{a}{\sqrt{2} (T_{1} - T_{0})} {- \ln [\cos (T_{0})] + 2 \ln [\cos (T_{1})] - \ln [\cos ({- T}_{0} + 2 T_{1} + \frac{\sqrt{2} (T_{1} - T_{0})}{a} x)]} \\ (\frac{- 2 a}{\sqrt{2}} \leq x \leq \frac{- a}{\sqrt{2}}) \\ \frac{a}{\sqrt{2} (T_{1} - T_{0})} {- \ln [\cos (T_{0})] + \ln [\cos (T_{0} - \frac{\sqrt{2} (T_{1} - T_{0})}{a} x)]} \\ (\frac{- a}{\sqrt{2}} \leq x \leq 0) \\ \frac{a}{\sqrt{2} (T_{1} - T_{0})} {\ln [\cos (T_{0})] - \ln [\cos (T_{0} + \frac{\sqrt{2} (T_{1} - T_{0})}{a} x)]} \\ (0 \leq x \leq \frac{a}{\sqrt{2}}) \\ \frac{a}{\sqrt{2} (T_{1} - T_{0})} {\ln [\cos (T_{0})] - 2 \ln [\cos (T_{1})] + \ln [\cos ({- T}_{0} + 2 T_{1} - \frac{\sqrt{2} (T_{1} - T_{0})}{a} x)]} \\ (\frac{a}{\sqrt{2}} \leq x \leq \frac{2 a}{\sqrt{2}}) \end{cases} \end{array}

(1a)

\begin{array}{l} θ (x) = & {\begin{cases} \frac{\sqrt{2}}{a} (T_{1} - T_{0}) x + T_{0} + 2 (T_{1} - T_{0}) (\frac{- 2 a}{\sqrt{2}} \leq x \leq \frac{- a}{\sqrt{2}}) \\ - \frac{\sqrt{2}}{a} (T_{1} - T_{0}) x + T_{0} (\frac{- a}{\sqrt{2}} \leq x \leq 0) \\ \frac{\sqrt{2}}{a} (T_{1} - T_{0}) x + T_{0} (0 \leq x \leq \frac{a}{\sqrt{2}}) \\ - \frac{\sqrt{2}}{a} (T_{1} - T_{0}) x + T_{0} + 2 (T_{1} - T_{0}) (\frac{a}{\sqrt{2}} \leq x \leq \frac{2 a}{\sqrt{2}}) \end{cases} \end{array}

(1b)

The fiber orientation angles $T_{0}$ and $T_{1}$ must satisfy the fiber curvature constraint $κ$ of the reference fiber path throughout each layer to avoid fiber kinking and ensure manufacturability of the laminate. It is important that the curvature $κ$ does not exceed the prescribed value of $3.28 m^{- 1}$ ¹. The curvature $κ$ of the reference fiber path is given by:

κ = \frac{\sqrt{2}}{a} (T_{1} - T_{0}) \cos [T_{0} + \frac{\sqrt{2}}{a} (T_{1} - T_{0}) x]

(2)

Figure 3 illustrates curvilinear fiber paths for four different cases ( $〈 45 ° | 0 ° 〉$ , $〈 45 ° | 15 ° 〉$ , $〈 0 ° | 45 ° 〉, 〈 15 ° | 60 ° 〉$ ) demonstrating the variation in curvature along the elliptic lamina with $a = 0.5 m$ . The maximum curvature values are explicitly indicated in the figure, with their respective locations. This visualization clarifies how different fiber orientations influence the curvature distribution, which is critical for manufacturability and structural performance.

Figure 3.

Curvilinear fiber paths and maximum curvature locations in a VSCL elliptic layer with $a = 0.5 m$ .

Curved hierarchical finite element model

The stress( $σ$ )–strain ( $ε$ ) relations for an orthotropic $k^{t h}$ layer in the local coordinate are expressed as follows:

{\begin{cases} σ_{11} \\ σ_{22} \\ τ_{12} \\ τ_{13} \\ τ_{23} \end{cases}}^{(k)} = {[\begin{array}{l} \begin{array}{l} Q_{11} \\ \begin{array}{l} Q_{12} \\ \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} Q_{12} \\ \begin{array}{l} Q_{22} \\ \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} Q_{66} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} Q_{44} \\ 0 \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ Q_{55} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array}]}^{(k)} {\begin{cases} ε_{11} \\ ε_{22} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{cases}}^{(k)}

(3)

where

Q_{i j}

are the stiffness coefficients, expressed in terms of the longitudinal and transverse modulus of elasticity (

E_{1}, E_{2})

, Poisson’s ratios

(ν_{12}, ν_{21})

, and shear modulus (

G_{12}, G_{13}, G_{23})

as:

Q_{11} = \frac{E_{1}}{1 - ν_{12} ν_{21}}

(4a)

Q_{22} = \frac{E_{2}}{1 - ν_{12} ν_{21}}

(4b)

Q_{12} = \frac{ν_{12} E_{2}}{1 - ν_{12} ν_{21}}

(4c)

Q_{66} = G_{12}

(4d)

Q_{44} = G_{13}

(4e)

Q_{55} = G_{23}

(4f)

where

ν_{21} = \frac{ν_{12} E_{2}}{E_{1}}

(4g)

the subscripts 1 and 2 refer to the fiber direction and the transverse direction to the fiber, respectively.

The stresses and strains in the global coordinates are related by:

{\begin{cases} σ_{x x} \\ σ_{y y} \\ τ_{x y} \\ τ_{x z} \\ τ_{y z} \end{cases}}^{k} = {[\begin{array}{l} \begin{array}{l} {\bar{Q}}_{11} \\ \begin{array}{l} {\bar{Q}}_{12} \\ \begin{array}{l} {\bar{Q}}_{16} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} {\bar{Q}}_{12} \\ \begin{array}{l} {\bar{Q}}_{22} \\ \begin{array}{l} {\bar{Q}}_{26} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} {\bar{Q}}_{16} \\ \begin{array}{l} {\bar{Q}}_{26} \\ \begin{array}{l} {\bar{Q}}_{66} \\ \begin{array}{l} 0 \\ 0 \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} {\bar{Q}}_{44} \\ {\bar{Q}}_{45} \end{array} \end{array} \end{array} \end{array} & \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} 0 \\ \begin{array}{l} {\bar{Q}}_{45} \\ {\bar{Q}}_{55} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array}]}^{k} {\begin{cases} ε_{x x} \\ ε_{y y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{cases}}^{k}

(5)

The stiffness coefficients ${\bar{Q}}_{i j}$ can be expressed as:

{\bar{Q}}_{11} = Q_{11} m^{4} + 2 (Q_{12} + 2 Q_{66}) m^{2} n^{2} + Q_{22} n^{4}

(6a)

{\bar{Q}}_{12} = (Q_{11} + Q_{22} - 4 Q_{66}) m^{2} n^{2} + Q_{12} (m^{4} + n^{4})

(6b)

{\bar{Q}}_{16} = Q_{11} m^{3} n - Q_{22} {m n}^{3} - 2 (Q_{12} + 2 Q_{66}) m n (m^{2} - n^{2})

(6c)

{\bar{Q}}_{22} = Q_{11} n^{4} + 2 (Q_{12} + 2 Q_{66}) m^{2} n^{2} + Q_{22} m^{4}

(6d)

{\bar{Q}}_{26} = (Q_{11} - Q_{12} - 2 Q_{66}) n^{3} m + (Q_{12} - Q_{22} + 2 Q_{66}) m^{3} n

(6e)

{\bar{Q}}_{66} = (Q_{11} + Q_{22} - 2 Q_{12}) m^{2} n^{2} + Q_{66} {(m^{2} - n^{2})}^{2}

(6f)

{\bar{Q}}_{44} = Q_{44} m^{2} + Q_{55} n^{2}

(6g)

{\bar{Q}}_{45} = (Q_{55} - Q_{44}) m n

(6h)

{\bar{Q}}_{55} = Q_{44} n^{2} + Q_{55} m^{2}

(6i)

where

m = \cos (θ)

and

n = \sin (θ)

A curved rectangular p-element is used to discretize the laminated composite elliptic plate, as illustrated in Figure 4. This element is formulated based on FSDT and incorporates high-order polynomial shape functions to improve accuracy in capturing both linear and nonlinear free vibration characteristics of CSCL and VSCL elliptic plates. 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. The blending function method is employed to accurately represent the curved sides of the element.

Figure 4.

Curved rectangular $p$ -element used for modeling the elliptic laminate.

The local and global coordinates are related by:

ξ = \frac{\sqrt{2} x}{a}

(7)

η = \frac{\sqrt{2} y}{b}

(8)

Based on the FSDT, the displacement fields $u$ , $v$ and $w$ at a point $(x, y, z)$ can be defined as

u (x, y, z, t) = u_{0} + z ϕ_{y} (x, y, t)

(9)

v (x, y, z, t) = v_{0} - z ϕ_{x} (x, y, t)

(10)

w (x, y, z, t) = w (x, y, t)

(11)

where

u_{0}

v_{0}

and

w

represent the displacements of mid-surface, while

ϕ_{x}

and

ϕ_{y}

denote the rotation of transverse normal to the mid-surface about the

x

and

y

axes, respectively.

According to Von Kármán’s assumptions, the strain-displacement relations are expressed as

ε_{x x} = \frac{\partial u}{\partial x} + z \frac{\partial ϕ_{y}}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}

(12)

ε_{y y} = \frac{\partial v}{\partial y} - z \frac{\partial ϕ_{x}}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2}

(13)

γ_{x y} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} + z (\frac{\partial ϕ_{y}}{\partial y} - \frac{\partial ϕ_{x}}{\partial x})

(14)

γ_{x z} = \frac{\partial w}{\partial x} + ϕ_{y}

(15)

γ_{y z} = \frac{\partial w}{\partial y} - ϕ_{x}

(16)

Using the curved rectangular p-element, the transverse displacements and rotations of cross-sections about the $x$ and $y$ axes are expressed as:

u = \sum_{β = 1}^{{(p + 1)}^{2}} q_{2 β - 1} (t) N_{β} (ξ, η)

(17)

v = \sum_{β = 1}^{{(p + 1)}^{2}} q_{2 β} (t) N_{β} (ξ, η)

(18)

w = \sum_{β = 1}^{{(p + 1)}^{2}} q_{3 β - 2} (t) N_{β} (ξ, η)

(19)

ϕ_{x} = \sum_{β = 1}^{{(p + 1)}^{2}} q_{3 β - 1} (t) N_{β} (ξ, η)

(20)

ϕ_{y} = \sum_{β = 1}^{{(p + 1)}^{2}} q_{3 β} (t) N_{β} (ξ, η)

(21)

where

t

is time and

p

is the polynomial order.

The shape functions $N_{β}$ are expressed as:

{N_{β} = g}_{m} (ξ) g_{l} (η) (m, l = 1, 2, \dots, p + 1)

(22)

where the subscript

β

is defined as:

β = l + (k - 1) (p + 1)

(23)

the functions

g_{m}

are given by:

g_{1} (ξ) = \frac{1 - ξ}{2}

(24)

g_{2} (ξ) = \frac{1 + ξ}{2}

(25)

g_{m + 1} (ξ) = \int_{- 1}^{ξ} P_{k - 1} (ζ) d ζ (m = 2, 3, \dots, p)

(26)

where

g_{m + 1}

(p = 2, 3, . . ., 10)

are internal shape functions, which are used to describe the displacement in the interior of the element.

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

{x = x}_{i} (ξ), {y = y}_{i} (ξ), (- 1 \leq ξ \leq 1) (i = 1, 3)

(27a)

{x = x}_{i} (η), {y = y}_{i} (η), (- 1 \leq η \leq 1) (i = 2, 4)

(27b)

where the subscript

i

denotes the side number.

The mapping functions for the elliptic shape are expressed as:

x (ξ, η) = \frac{1}{2} [(1 - η) x_{1} (ξ) + (1 + ξ) x_{2} (η) + (1 + η) x_{3} (ξ) + (1 - ξ) x_{4} (η)] - \frac{1}{4} [(1 - ξ) (1 - η) X_{1} + (1 + ξ) (1 - η) X_{2} + (1 + ξ) (1 + η) X_{3} + (1 - ξ) (1 + η) X_{4}]

(28a)

y (ξ, η) = \frac{1}{2} [(1 - η) y_{1} (ξ) + (1 + ξ) y_{2} (η) + (1 + η) y_{3} (ξ) + (1 - ξ) y_{4} (η)] - \frac{1}{4} [(1 - ξ) (1 - η) Y_{1} + (1 + ξ) (1 - η) Y_{2} + (1 + ξ) (1 + η) Y_{3} + (1 - ξ) (1 + η) Y_{4}]

(28b)

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

{\begin{cases} \frac{\partial}{\partial ξ} \\ \frac{\partial}{\partial η} \end{cases}} = J {\begin{cases} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{cases}}

(29)

The Jacobian matrix $J$ is given by

J = [\begin{array}{l} \begin{array}{l} J_{1, 1} (ξ, η) & J_{1, 2} (ξ, η) \end{array} \\ \begin{array}{l} J_{2, 1} (ξ, η) & J_{2, 2} (ξ, η) \end{array} \end{array}] = [\begin{array}{l} \begin{array}{l} \frac{\partial x (ξ, η)}{\partial ξ} & \frac{\partial y (ξ, η)}{\partial ξ} \end{array} \\ \begin{array}{l} \frac{\partial x (ξ, η)}{\partial η} & \frac{\partial y (ξ, η)}{\partial η} \end{array} \end{array}]

(30)

the components of

J

are

J_{1, 1} (ξ, η) = \frac{1}{2} [(1 - η) \frac{d x_{1} (ξ)}{d ξ} + x_{2} (η) + (1 + η) \frac{d x_{3} (ξ)}{d ξ} - x_{4} (η)] + \frac{1}{4} [(1 - η) X_{1} - (1 - η) X_{2} - (1 + η) X_{3} + (1 + η) X_{4}]

(31a)

J_{1, 2} (ξ, η) = \frac{1}{2} [(1 - η) \frac{d y_{1} (ξ)}{d ξ} + y_{2} (η) + (1 + η) \frac{d y_{3} (ξ)}{d ξ} - y_{4} (η)] + \frac{1}{4} [(1 - η) Y_{1} - (1 - η) Y_{2} - (1 + η) Y_{3} + (1 + η) Y_{4}]

(31b)

J_{2, 1} (ξ, η) = \frac{1}{2} [- x_{1} (ξ) + (1 + ξ) \frac{d x_{2} (η)}{d η} + x_{3} (ξ) + (1 - ξ) \frac{d x_{4} (η)}{d η}] + \frac{1}{4} [(1 - ξ) X_{1} + (1 + ξ) X_{2} - (1 + ξ) X_{3} - (1 - ξ) X_{4}]

(31c)

J_{2, 2} (ξ, η) = \frac{1}{2} [- y_{1} (ξ) + (1 + ξ) \frac{d y_{2} (η)}{d η} + y_{3} (ξ) + (1 - ξ) \frac{d y_{4} (η)}{d η}] + \frac{1}{4} [(1 - ξ) Y_{1} + (1 + ξ) Y_{2} - (1 + ξ) Y_{3} - (1 - ξ) Y_{4}]

(31d)

equation (29) gives

{\begin{cases} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{cases}} = J^{- 1} {\begin{cases} \frac{\partial}{\partial ξ} \\ \frac{\partial}{\partial η} \end{cases}}

(32)

where

J^{- 1} = \frac{1}{\det J} [\begin{array}{l} \begin{array}{l} J_{2, 2} (ξ, η) & - J_{1, 2} (ξ, η) \end{array} \\ \begin{array}{l} {- J}_{2, 1} (ξ, η) & J_{1, 1} (ξ, η) \end{array} \end{array}] = [\begin{array}{l} \begin{array}{l} \frac{\partial y (ξ, η)}{\partial η} & - \frac{\partial y (ξ, η)}{\partial ξ} \end{array} \\ \begin{array}{l} - \frac{\partial x (ξ, η)}{\partial η} & \frac{\partial x (ξ, η)}{\partial ξ} \end{array} \end{array}]

(33a)

and the determinant of

J

is evaluated as

\det J = J_{1, 1} (ξ, η) J_{2, 2} (ξ, η) - J_{1, 2} (ξ, η) J_{2, 1} (ξ, η)

(33b)

The strain energy $U$ of the curved p-element is given by

U = \frac{1}{2} \underset{v}{∭} [{\bar{Q}}_{11} {(\frac{\partial u}{\partial x} + z \frac{\partial ϕ_{y}}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2})}^{2} + {\bar{Q}}_{22} {(\frac{\partial v}{\partial y} - z \frac{\partial ϕ_{x}}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2})}^{2} + {\bar{Q}}_{66} {(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + z (\frac{\partial ϕ_{y}}{\partial y} - \frac{\partial ϕ_{x}}{\partial x}))}^{2} + 2 {\bar{Q}}_{12} (\frac{\partial u}{\partial x} + z \frac{\partial ϕ_{y}}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}) (\frac{\partial v}{\partial y} - z \frac{\partial ϕ_{x}}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2}) + 2 {\bar{Q}}_{16} (\frac{\partial u}{\partial x} + z \frac{\partial ϕ_{y}}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}) (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + z (\frac{\partial ϕ_{y}}{\partial y} - \frac{\partial ϕ_{x}}{\partial x})) + 2 {\bar{Q}}_{26} (\frac{\partial v}{\partial y} - z \frac{\partial ϕ_{x}}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2}) (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + z (\frac{\partial ϕ_{y}}{\partial y} - \frac{\partial ϕ_{x}}{\partial x})) + {\bar{Q}}_{44} {(\frac{\partial w}{\partial x} + ϕ_{y})}^{2} + {\bar{Q}}_{55} {(\frac{\partial w}{\partial y} - ϕ_{x})}^{2} + 2 {\bar{Q}}_{45} (\frac{\partial w}{\partial x} + ϕ_{y}) (\frac{\partial w}{\partial y} - ϕ_{x})] dV

(34)

The strain energy $U$ is expressed as $U = \bar{U} + \hat{U} + \tilde{U} + \overset{˘}{U} + \tilde{\tilde{U}} + U_{b}$ , with detailed expressions provided in Appendix A.

The kinetic energy $T$ of the curved p-element is given by

T = \frac{1}{2} \iint [ρ h ({{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} (\frac{\partial w}{\partial t})}^{2}) + \frac{{ρ h}^{3}}{12} ({(\frac{\partial ϕ_{x}}{\partial t})}^{2} + {(\frac{\partial ϕ_{y}}{\partial t})}^{2})] d x d y

(35)

where

ρ

is density and

h

represents the total thickness of the plate.

The Lagrange equations of free motion are given by: (35)

\frac{d}{d t} (\frac{\partial T}{\partial {{\dot{q}}_{i}^{*}}}) - \frac{\partial T}{\partial {{q_{i}}^{*}}} + \frac{\partial U}{\partial {q_{i}}} = 0 ({q_{i}}^{*} = \bar{q_{i}} or q_{i}; i = 1, 2, \dots, n)

(36)

By substituting equations (17)–(21) into equations (34) and (35), and subsequently into equation (36), yields the following equations of free motion:

{\bar{M}}_{α, β} \frac{d^{2} {\bar{q}}_{β}}{d t^{2}} + {\bar{K}}_{α, β} {\bar{q}}_{β} + [{\hat{K}}_{α, β} + {\overset{˘}{K}}_{α, β}] q_{β} = 0

(37)

M_{α, β} \frac{d^{2} {\bar{q}}_{β}}{d t^{2}} + [K_{α, β} + {\tilde{K}}_{α, β}] q_{β} + [2 {\hat{K}}_{α, β} + {\overset{˘}{K}}_{α, β}] {\bar{q}}_{β} = 0

(38)

Here, $\bar{K}$ , $\bar{M}$ , and $\bar{q}$ denote the membrane stiffness matrix, mass matrix, and displacement vector, respectively, while $K$ , $M$ , and $q$ represent the bending stiffness matrix, mass matrix, and displacement vector. Additionally, $\hat{K}$ and $\tilde{K}$ denote the nonlinear stiffness matrices. These matrices and displacement vectors are defined as follows:

{\bar{K}}_{α, β} = [\begin{array}{l} {\bar{K}}_{2 α - 1, 2 β - 1} & {\bar{K}}_{2 α - 1, 2 β} \\ {\bar{K}}_{2 α, 2 β - 1} & {\bar{K}}_{2 α, 2 β} \end{array}]

(39a)

K = [\begin{array}{l} \begin{array}{l} K_{3 α - 2, 3 β - 2} \\ \begin{array}{l} K_{3 α - 1, 3 β - 2} \\ K_{3 α, 3 β - 2} \end{array} \end{array} & \begin{array}{l} K_{3 α - 2, 3 β - 1} \\ \begin{array}{l} K_{3 α - 1, 3 β - 1} \\ K_{3 α, 3 β - 1} \end{array} \end{array} & \begin{array}{l} K_{3 α - 2, 3 β} \\ \begin{array}{l} K_{3 α - 1, 3 β} \\ K_{3 α, 3 β} \end{array} \end{array} \end{array}]

(39b)

{\hat{K}}_{α, β} = [\begin{array}{c} {\hat{K}}_{2 α - 1, 3 β - 2} & 0 & 0 \\ {\hat{K}}_{2 α, 3 β - 2} & 0 & 0 \end{array}]

(39c)

{\tilde{K}}_{α, β} = [\begin{array}{c} {\tilde{K}}_{3 α - 2, 3 β - 2} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]

(39d)

{\overset{˘}{K}}_{α, β} = [\begin{array}{l} 0 & {\overset{˘}{K}}_{2 α - 1, 3 β - 1} & {\overset{˘}{K}}_{2 α - 1, 3 β} \\ 0 & {\overset{˘}{K}}_{2 α, 3 β - 1} & {\overset{˘}{K}}_{2 α, 3 β} \end{array}]

(39e)

M = [\begin{array}{c} \begin{array}{c} M_{3 α - 2, 3 β - 2} \\ \begin{array}{c} 0 \\ 0 \end{array} \end{array} & \begin{array}{c} 0 \\ \begin{array}{c} M_{3 α - 1, 3 β - 1} \\ 0 \end{array} \end{array} & \begin{array}{c} 0 \\ \begin{array}{c} 0 \\ M_{3 α, 3 β} \end{array} \end{array} \end{array}]

(40)

{\bar{q}}_{β} = {\begin{array}{c} {\bar{q}}_{2 β - 1} \\ {\bar{q}}_{2 β} \end{array}}

(41a)

q_{β} = {\begin{array}{c} q_{3 β - 2} \\ q_{3 β - 1} \\ q_{3 β} \end{array}}

(41b)

α = i + (j - 1) (p + 1) (i, j = 1, 2, \dots \dots, p + 1)

(42)

The non-zero coefficients of the matrices $M, \bar{K}, \hat{K}, \overset{˘}{K}, K, \tilde{K}$ are given in Appendix B.

Applying boundary conditions by ignoring the lines and columns associated with restrained degrees of freedom, the equations of free motion are formulated as follows:

\bar{M} \frac{d^{2} \bar{q}}{d t^{2}} + \bar{K} \bar{q} + [\hat{K} + \overset{˘}{K}] q = 0

(43)

M \frac{d^{2} q}{d t^{2}} + [K + \tilde{K}] q + [2 {\hat{K}}^{T} + {\overset{˘}{K}}^{T}] \bar{q} = 0

(44)

Since the plate edges are fixed in the plane, the in-plane displacements are negligible compared to the out-of-plane displacement. Consequently, the in-plane inertia can be ignored, simplifying equation (43) to:

\bar{q} = - {\bar{K}}^{- 1} [\hat{K} + \overset{˘}{K}] q

(45)

Inserting equations (45) into (44) yields:

M \frac{d^{2} q}{d t^{2}} + [K - {\overset{˘}{K}}^{T} {\bar{K}}^{- 1} \overset{˘}{K} + \tilde{K} - 2 {\hat{K}}^{T} {\bar{K}}^{- 1} \hat{K} - 2 {\hat{K}}^{T} {\bar{K}}^{- 1} \overset{˘}{K} - {\overset{˘}{K}}^{T} {\bar{K}}^{- 1} \hat{K}] q = 0

(46)

The stiffness matrices ${\hat{K}}^{T} {\bar{K}}^{- 1} \overset{˘}{K}$ and ${\overset{˘}{K}}^{T} {\bar{K}}^{- 1} \hat{K}$ are linear functions of $q$ , while the matrices $\tilde{K}$ and ${\hat{K}}^{T} {\bar{K}}^{- 1} \hat{K}$ are quadratic functions of $q$ .

Periodic motion is assumed, $q$ can be expressed as:

q = Q \cos (ω t)

(47)

where

Q

are unknown coefficients and

ω

is the natural frequency.

Inserting equations (47) into (45) yields an equation of the form:

F (ω, t) = [- ω^{2} M \cos (ω t) + K \cos (ω t) + [\tilde{K} - 2 {\hat{K}}^{T} {\bar{K}}^{- 1} \hat{K}] \cos^{3} (ω t)] Q = 0

(48)

By applying the harmonic balance method and integrating over the period, the following equation is obtained:

\frac{ω}{π} \int_{0}^{\frac{2 π}{ω}} F (ω, t) \cos (ω t) d t = [- ω^{2} M + K - {\overset{˘}{K}}^{T} {\bar{K}}^{- 1} \overset{˘}{K} + \frac{3}{4} \tilde{K} - \frac{3}{2} {\hat{K}}^{T} {\bar{K}}^{- 1} \hat{K}] Q = 0

(49)

The stiffness matrix $\tilde{K}$ is derived only from the nonlinear in-plane strains. The stiffness matrix ${\hat{K}}^{T} {\bar{K}}^{- 1} \hat{K}$ represents the coupling between linear and nonlinear in-plane strains, while ${\overset{˘}{K}}^{T} {\bar{K}}^{- 1} \overset{˘}{K}$ characterizes the coupling between linear in-plane strains and curvatures.

The nonlinear solution procedure for the free vibration analysis of composite elliptic laminates is clearly illustrated in the flowchart (Figure 5). The nonlinear generalized eigenvalue problem is solved iteratively using the linearized updated mode method. In the first iteration, the displacement vector is initialized as $Q = 0$ . In subsequent iterations, the nonlinear stiffness matrices $\hat{K}$ and $\tilde{K}$ are computed based on the eigenvectors corresponding to a specific amplitude at the point of maximum displacement. The dimensionless coordinates $ξ_{0}$ and $η_{0}$ of this point are determined using the Newton-Raphson method.

Figure 5.

Flowchart of the iterative solution procedure for geometrically nonlinear free vibration analysis of composite laminated elliptic plates.

The maximum amplitude $W_{\max}$ attained during the cycle is given by:

W_{\max} = \sum_{k = 1}^{p + 1} \sum_{l = 1}^{p + 1} Q_{3 β - 2} g_{k} (ξ_{0}) g_{l} (η_{0})

(50)

Convergence is assessed based on a predefined error criterion (e.g., $10^{- 5}$ ), ensuring the accuracy of the computed nonlinear frequencies and associated mode shapes. If the maximum number of iterations is reached without convergence, the procedure terminates without a solution.

Results and discussions

Numerical results are presented in terms of the non-dimensional frequency parameter

Ω_{i} = ω a \sqrt{ρ / E_{2}}

for both symmetric and antisymmetric CSCL and VSCL elliptic plates. In this context,

Ω_{L} = ω_{L} a \sqrt{ρ / E_{2}}

and

Ω_{N L} = ω_{N L} a \sqrt{ρ / E_{2}}

represent the linear and nonlinear frequencies, respectively. It is assumed that all layers of the laminate have constant thicknesses. The effects of various parameters such as the aspect ratio, thickness ratio, modulus ratio, number of layers, stacking sequence, boundary conditions, and fiber orientation angles on the nonlinear fundamental frequencies are examined. The mechanical properties of the composite materials used are summarized in Table 1. The boundary conditions (BCs) for clamped (CCCC) and simply supported (SSSS) laminates are

u_{0} = v_{0} = w {= ϕ_{x} = ϕ}_{y} = 0

and

u_{0} = v_{0} = w = 0

, respectively.

Table 1.

Mechanical properties of composite materials.

Material	$E_{1} / E_{2}$	$G_{12} / E_{2}$	$G_{13} / E_{2}$	$G_{23} / E_{2}$	$ν_{12}$
1	3	0.6	0.6	0.6	0.25
2	4.6674	0.5	0.5	0.5	0.26
3	15.40	0.7924	0.7924	0.7924	0.30
4	25	0.5	0.5	0.2	0.20
5	40	0.6	0.6	0.5	0.25

Validation of the curved rectangular $p$ -element for geometrically nonlinear vibration

To validate the accuracy and effectiveness of the current curved $p$ -element in analyzing the geometrically nonlinear free vibration of symmetric and antisymmetric CSCL and VSCL elliptic laminates, convergence and comparison studies were conducted. This $p$ -element has previously been successfully employed by the authors to analyze the linear free vibration of clamped and soft simply supported symmetric VSCL elliptic plates.¹⁵

The results obtained were compared with existing solutions for linear free vibration of symmetric and antisymmetric CSCL elliptic plates with clamped and simply supported boundary conditions. Additionally, comparisons were made with available solutions for the geometrically nonlinear free vibration of isotropic and orthotropic elliptic plates. Convergence tests were performed by incrementally increasing the degree $p$ of polynomial shape functions from 4 to 10.

Tables 2 and 3 present the first five linear modes for simply supported symmetric and clamped antisymmetric 4-layer composite laminated elliptic plates with $θ = 30 °$ and a thickness ratio $h / a = 0.01$ . The results are presented as a function of the order $p$ of shape functions and the aspect ratio $a / b$ . The frequencies were calculated using the mechanical parameters of Material 3.

• For symmetric 4-layer CSCL laminates, denoted as ${[θ °, - θ °]}_{s}$ , the stacking sequence is $[θ °, - θ °, - θ °, θ °]$ , while for antisymmetric laminates, denoted as ${[θ °, - θ °]}_{2}$ , the stacking sequence is $[θ °, - θ °, θ °, - θ °]$ .

• For symmetric 4-layer VSCL ( ${[\mp 〈 T_{0} | T_{1} 〉]}_{s}$ ) and antisymmetric ( ${[\mp 〈 T_{0} | T_{1} 〉]}_{2}$ ) laminates with curvilinear fibers, the stacking sequences are $[+ 〈 T_{0} | T_{1} 〉, - 〈 T_{0} | T_{1} 〉, - 〈 T_{0} | T_{1} 〉, + 〈 T_{0} | T_{1} 〉]$ and $[+ 〈 T_{0} | T_{1} 〉, - 〈 T_{0} | T_{1} 〉, + 〈 T_{0} | T_{1} 〉, - 〈 T_{0} | T_{1} 〉]$ , respectively.

Table 2.

Convergence and comparison of the linear frequency parameter $Ω_{i L}$ of simply supported ${[30 °, - 30 °]}_{s}$ elliptic plates ( $h / a = 0.01,$ Material 3).

		$Ω_{L}$
$a ⁄ b$	$p$	1	2	3	4	5
1	4	0.039	0.107	0.179	0.360	0.456
	6	0.036	0.072	0.128	0.148	0.198
	8	0.036	0.071	0.125	0.128	0.172
	10	0.036	0.070	0.124	0.124	0.165
	Nallim and Grossi⁶⁴	0.036	0.070	0.124	0.124	—
2	4	0.076	0.215	0.341	0.552	0.888
	6	0.072	0.148	0.250	0.278	0.393
	8	0.071	0.145	0.244	0.252	0.356
	10	0.071	0.144	0.241	0.249	0.347
	Nallim and Grossi⁶⁴	0.071	0.145	0.244	0.250	—

Table 3.

Convergence and comparison of the linear frequency parameter $Ω_{i L}$ of clamped antisymmetric ${[30 °, - 30 °]}_{2}$ elliptic plates ( $h / a = 0.01,$ Material 3).

$a / b$	$p$	$Ω_{L}$
$a / b$	$p$	1	2	3	4	5
1	4	0.097	1.770	1.772	3.584	4.102
	6	0.072	0.137	0.179	0.272	0.328
	8	0.071	0.127	0.167	0.203	0.252
	10	0.071	0.126	0.166	0.196	0.245
	Hosokawa et al.⁶²	0.071	0.126	0.166	0.195	—
1.5	Present	0.099	0.202	0.211	0.321	0.350
	Hosokawa et al.⁶²	0.099	0.202	0.211	0.321	—
2	Present	0.141	0.252	0.330	0.390	0.492
	Hosokawa et al.⁶²	0.141	0.252	0.332	0.391	—
3	Present	0.265	0.394	0.560	0.667	0.887
	Hosokawa et al.⁶²	0.265	0.393	0.552	0.672	—

The results demonstrate a clear pattern of rapid downward convergence as the order of shape functions increases. Frequencies calculated with an order of $p = 10$ showe excellent agreement with those reported by Nallim and Grossi⁶⁴ for symmetric plates and Hosokawa et al.⁶² for antisymmetric laminates.

It’s worth mentioning that the literature lacks investigations on the nonlinear free vibration of CSCL elliptic plates. Therefore, comparisons in this study are primarily focused on the nonlinear free vibrations of isotropic and orthotropic circular and elliptic plates.

Tables 4 and 5 display the nonlinear-to-linear fundamental frequency ratios

(Ω_{N L} / Ω_{L})

for isotropic circular and elliptic plates, respectively, with

ν = 0.3

as a function of the order

p

. These calculated frequencies are compared with results from various studies in the literature that employed the Von Kármán’s nonlinearity.

Table 4.

Convergence and comparison of the nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ for clamped isotropic circular plates ( $ν = 0.3$ ).

Method	$p$	$\| W_{\max} \| / h$
Method	$p$	0.2	0.4	0.6	0.8	1	1.5
Present	4	1.0058	1.0228	1.0506	1.1348	1.1893	1.2828
	6	1.0100	1.0390	1.0847	1.1436	1.2121	1.4055
	8	1.0081	1.0324	1.0718	1.1221	1.1807	1.3438
	10	1.0081	1.0324	1.0717	1.1219	1.1805	1.3436
Shooshtari and Asadi Dalir⁵⁹		1.0081	1.0322	1.0717	1.1234	1.1866	1.3711
Nowinski⁵²		1.0079	1.0313	1.0690	1.1194	1.1808	1.3711
Rao⁵⁷		1.0078	1.0307	1.0679	1.1179	1.1791	—
Allahverdizadeh et al.⁵⁶		1.0075	1.0296	1.0654	1.1135	1.1724	1.3567
Haterbouch and Benamar⁵⁴		1.0072	1.0284	1.0623	1.1073	1.1615	1.3255
Reddy and Huang⁵³		1.0073	1.0284	1.0624	1.1075	1.1620	—
Zhang and Zhou⁵⁸		1.007	1.030	1.065	1.113	1.172	1.357

Table 5.

Convergence and comparison of the nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ for isotropic elliptic plates ( $ν = 0.3$ , $h / a = 0.01$ ).

BCs	$a / b$	Method	$\| W_{\max} \| / h$
BCs	$a / b$	Method	$p$	0.2	0.4	0.6	0.8	1
CCCC	1.5	Present	4	1.0082	1.0326	1.0719	1.1246	1.1889
			6	1.0104	1.0405	1.0875	1.1476	1.2169
			8	1.0080	1.0310	1.0665	1.1195	1.1777
			10	1.0080	1.0310	1.0664	1.1192	1.1772
		Zhang and Zhou⁵⁸		1.007	1.030	1.065	1.113	1.172
	2	Present		1.0081	1.0316	1.0697	1.1208	1.1831
		Zhang and Zhou⁵⁸		1.007	1.030	1.065	1.113	1.173
SSSS	1.5	Present	4	1.0181	1.0661	1.1261	1.1669	1.2219
			6	1.0271	1.1000	1.1923	1.3375	1.4418
			8	1.0271	1.1016	1.2162	1.3725	1.5436
			10	1.0271	1.1016	1.2160	1.3720	1.5428
		Zhang and Zhou⁵⁸		1.027	1.103	1.220	1.369	1.541
	2	Present		1.0273	1.1031	1.2211	1.3792	1.5546
		Zhang and Zhou⁵⁸		1.027	1.103	1.220	1.371	1.545

A rapid convergence is observed as $p$ increases from 4 to 10. Stable convergence up to three significant figures is achieved with $p = 8$ . Therefore, $p = 8$ was used in subsequent studies. These tables reveal that the nonlinear frequency increases as the amplitude to thickness ratio $| W_{\max} | / h$ increases. Moreover, the results calculated for the clamped isotropic circular plates closely match those reported by Nowinski⁵² and Shooshtari and Asadi Dalir.⁵⁹ As for elliptic plates, the results demonstrate a close resemblance to those obtained by Zhang and Zhou,⁵⁸ who employed the Ritz method based on the CPT, despite varying aspect ratio $a / b$ and boundary conditions.

Figure 6 presents the nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ for clamped orthotropic elliptic plates with an aspect ratio $a / b = 1.5$ and a thickness ratio $h / a = 0.01$ using the mechanical parameters of Material 1. A close agreement is observed between our results and those found by Sathyamoorthy and Chia⁶⁰ and Sathyamoorthy,⁶¹ demonstrating the accuracy of the current model in predicting the nonlinear vibration behavior of orthotropic elliptic plates.

Figure 6.

Comparison of the nonlinear-to-linear frequency ratio for clamped orthotropic elliptic plates.

Parametric study

Following the validation of the curved rectangular $p$ -element, a comprehensive parametric study is conducted to investigate the geometrically nonlinear free vibration behavior of both CSCL and VSCL elliptic plates. The analysis considers the influence of various geometrical and physical parameters, including: the thickness ratio, aspect ratio, modulus ratio, number of layers, boundary conditions, straight fiber orientation angle ( $θ °$ ), and curvilinear fiber orientation angles $T_{0} and T_{1}$ . This analysis focuses on how these parameters influence the nonlinear fundamental frequencies, hardening behavior, and mode shapes of the elliptic plates as a function of maximum amplitude to thickness ratio $| W_{\max} | / h$ . The following subsections provide a detailed exploration of each parameter’s effect on the dynamic behavior of the plates.

Nonlinear vibration behavior of CSCL elliptic plates

The influence of the aspect ratio

a / b

on the nonlinear behavior of CSCL is examined for both 4-layer symmetric and antisymmetric clamped elliptic laminates. The corresponding solutions are provided in Tables 6 and 7, respectively. Mechanical parameters for Material 3 are used, with a thickness ratio of

h / a = 0.01 .

The point of maximum amplitude is located at

ξ = η = 0

. Furthermore, Figure 7 presents a comparison of the fundamental backbone curves for clamped symmetric

{[θ_{1}, - θ_{2}]}_{s}

and antisymmetric

{[θ_{1}, - θ_{2}]}_{2}

elliptic laminates with

a / b = 2

. The results highlight the influence of laminate symmetry on the nonlinear frequency response, providing insight into the differences in stiffness characteristics and hardening behavior between symmetric and antisymmetric configurations.

Table 6.

Nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ of clamped ${[θ_{1}, - θ_{2}]}_{s}$ elliptic plates as a function of $a / b$ ( $h / a = 0.01$ , Material 3).

$a / b$	${[θ_{1}, - θ_{2}]}_{s}$	$\| W_{\max} \| / h$
$a / b$	${[θ_{1}, - θ_{2}]}_{s}$	0.2	0.4	0.6	0.8	1	$Ω_{L}$
1	${[0 °, 0 °]}_{s}$	1.0077	1.0303	1.0668	1.1158	1.1758	0.076
	${[30 °, - 30 °]}_{s}$	1.0080	1.0315	1.0694	1.1199	1.1808	0.076
	${[45 °, - 45 °]}_{s}$	1.0080	1.0315	1.0694	1.1197	1.1804	0.076
	${[90 °, - 90 °]}_{s}$	1.0076	1.0303	1.0668	1.1158	1.1758	0.076
1.5	${[0 °, 0 °]}_{s}$	1.0075	1.0298	1.0656	1.1137	1.1780	0.087
	${[30 °, - 30 °]}_{s}$	1.0080	1.0316	1.0695	1.1199	1.1804	0.105
	${[45 °, - 45 °]}_{s}$	1.0080	1.0315	1.0694	1.1198	1.1809	0.124
	${[90 °, - 90 °]}_{s}$	1.0079	1.0310	1.0668	1.1190	1.1807	0.157
2	${[0 °, 0 °]}_{s}$	1.0075	1.0297	1.0654	1.1131	1.1711	0.109
	${[30 °, - 30 °]}_{s}$	1.0080	1.0315	1.0694	1.1197	1.1805	0.150
	${[45 °, - 45 °]}_{s}$	1.0080	1.0315	1.0695	1.1203	1.1820	0.192
	${[60 °, - 60 °]}_{s}$	1.0081	1.0318	1.0701	1.1215	1.1842	0.234
	${[90 °, - 90 °]}_{s}$	1.0080	1.0315	1.0696	1.1209	1.1837	0.275
	${[90 °, - 45 °]}_{s}$	1.0064	1.0255	1.0565	1.0984	1.1500	0.267
	${[60 °, - 30 °]}_{s}$	1.0063	1.0284	1.0547	1.0951	1.1447	0.226
3	${[0 °, 0 °]}_{s}$	1.0076	1.0299	1.0660	1.1134	1.1732	0.187
	${[30 °, - 30 °]}_{s}$	1.0080	1.0315	1.0693	1.1200	1.1816	0.278
	${[45 °, - 45 °]}_{s}$	1.0081	1.0318	1.0702	1.1217	1.1846	0.385
	${[90 °, - 90 °]}_{s}$	1.0082	1.0323	1.0712	1.1237	1.1879	0.606

Table 7.

Nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ of clamped ${[θ_{1}, - θ_{2}]}_{2}$ elliptic plates as a function of $a / b$ ( $h / a = 0.01$ , Material 3).

$a / b$	${[θ_{1}, - θ_{2}]}_{2}$	$\| W_{\max} \| / h$
$a / b$	${[θ_{1}, - θ_{2}]}_{2}$	0.2	0.4	0.6	0.8	1	$Ω_{l}$
2	${[30 °, - 30 °]}_{2}$	1.0055	1.0217	1.0476	1.0820	1.1164	0.142
	${[45 °, - 45 °]}_{2}$	1.0040	1.0158	1.0340	1.0502	1.0668	0.180
	${[60 °, - 60 °]}_{2}$	1.0064	1.0253	1.0554	1.0960	1.1396	0.221
	${[90 °, - 45 °]}_{2}$	1.0085	1.0338	1.0744	1.1287	1.1946	0.225
	${[60 °, - 30 °]}_{2}$	1.0086	1.0338	1.0741	1.1268	1.1890	0.185
3	${[30 °, - 30 °]}_{2}$	1.0068	1.0269	1.0590	1.1012	1.1509	0.266
	${[45 °, - 45 °]}_{2}$	1.0068	1.0268	1.0587	1.1005	1.1508	0.363
	${[60 °, - 60 °]}_{2}$	1.0082	1.0324	1.0712	1.1230	1.1853	0.467
	${[90 °, - 45 °]}_{2}$	1.0089	1.0352	1.0774	1.1340	1.2029	0.482
	${[60 °, - 30 °]}_{2}$	1.0089	1.0349	1.0765	1.01320	1.1992	0.379

Figure 7.

Fundamental backbone curves for clamped symmetric and antisymmetric elliptic laminates.

The results show that nonlinear frequency parameters increase with an increasing of the aspect ratio, which is attributed to the reduced cross-sectional area of the plate as the aspect ratio increases. The stacking sequences also have an impact on the nonlinear frequency, as illustrated in the tables. The comparison revealed that symmetric laminates exhibit a higher degree of hardening behavior than antisymmetric laminates that have the same orientation angle $θ$ . However, for plates with orientation angles $θ_{1}$ and $θ_{2}$ , the opposite behavior is observed. As shown in Figure 7, symmetric laminates also display slightly higher fundamental nonlinear frequency values than their antisymmetric counterparts. The highest and lowest fundamental nonlinear-to-linear frequency ratios ( $Ω_{N L} / Ω_{L}$ ) for the symmetric CSCL , associated with the maximum amplitude of the order of the thickness $(| W_{\max} | = h)$ , are calculated as 1.1879 for ${[90 °, - 90 °]}_{s}$ with $a / b = 3$ and 1.1447 for ${[6 0 °, - 3 0 °]}_{s}$ with $a / b = 2$ , respectively. For the antisymmetric elliptic plates, the highest and lowest ratios are calculated as 1.2029 for ${[90 °, - 45 °]}_{2}$ with $a / b = 3$ and 1.0668 for ${[45 °, - 45 °]}_{s}$ with $a / b = 2$ , respectively. It can be concluded that symmetric elliptic laminates present a higher degree of hardening and slightly higher nonlinear frequencies compared to antisymmetric ones.

Table 8 and Figure 8 show the nonlinear-to-linear frequency ratios of clamped

{[θ, - θ]}_{s}

elliptic plates as a function of thickness ratio

h / a

, demonstrating how changes in thickness affect the nonlinear vibrational response of CSCL elliptic plates. The geometric parameters are set as

a / b = 2

, and use the properties of Material 3 listed in Table 1. A hardening type of nonlinearity is observed as both the fiber orientation angle and thickness ratio increase. However, for the thinnest plates (thickness ratios of

0.01

and

0.025

), variations in the fiber orientation angle had a negligible effect on

Ω_{N L} / Ω_{L}

. In contrast, for thicker plates, a notable change in the nonlinear-to-linear frequency ratio was observed. The highest and lowest fundamental nonlinear-to-linear frequency ratios for the elliptic CSCL , associated with the maximum amplitude

| W_{\max} | / h = 1

, are calculated as 1.2380 for

{[90 °, - 90 °]}_{s}

with

h / a = 0.1

and 1.1805 for

{[0 °, 0 °]}_{s}

with

h / a = 0.01

, respectively. This indicates that selecting the thickness ratio and the orientation angle is critical for enhancing nonlinear dynamic responses of CSCL elliptic plates in various applications.

Table 8.

Nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ of clamped CSCL elliptic plates as a function of thickness ratio $h / a$ ( $a / b = 2$ , Material 3).

$h / a$	${[θ, - θ]}_{s}$	$\| W_{\max} \| / h$
$h / a$	${[θ, - θ]}_{s}$	0.2	0.4	0.6	0.8	1	$Ω_{l}$
0.01	${[30 °, - 30 °]}_{s}$	1.0080	1.0315	1.0694	1.1197	1.1805	0.150
	${[45 °, - 45 °]}_{s}$	1.0080	1.0315	1.0695	1.1203	1.1820	0.192
	${[90 °, - 90 °]}_{s}$	1.0080	1.0315	1.0696	1.1209	1.1837	0.275
0.025	${[30 °, - 30 °]}_{s}$	1.0081	1.0321	1.0704	1.1212	1.1819	0.370
	${[45 °, - 45 °]}_{s}$	1.0082	1.0324	1.0713	1.1230	1.1854	0.473
	${[90 °, - 90 °]}_{s}$	1.0084	1.0333	1.0734	1.1271	1.1927	0.669
0.05	${[30 °, - 30 °]}_{s}$	1.0086	1.0339	1.0741	1.1266	1.1883	0.718
	${[45 °, - 45 °]}_{s}$	1.0090	1.0354	1.0774	1.1325	1.1978	0.903
	${[90 °, - 90 °]}_{s}$	1.0100	1.0394	1.0862	1.1479	1.2209	1.228
0.1	${[30 °, - 30 °]}_{s}$	1.0104	1.0406	1.0875	1.1465	1.2097	1.296
	${[45 °, - 45 °]}_{s}$	1.0119	1.0463	1.0993	1.1543	1.2132	1.562
	${[90 °, - 9 0 °]}_{s}$	1.0162	1.0617	1.1234	1.1856	1.2380	1.930

Figure 8.

Effect of thickness ratio on the nonlinear-to-linear frequency parameter of clamped ${[45 °, - 45 °]}_{s}$ elliptical plate.

Figure 9 shows the influence of the number of layers $N C$ on the nonlinear free vibration of clamped symmetric CSCL elliptic plates with $θ = 45 °$ . The mechanical parameters of Material 3 are used. The aspect and thickness ratios are taken as 2 and 0.01, respectively. The results provide insight into how layering influences the nonlinear frequency response and overall dynamic behavior of CSCL elliptic laminates. The figure shows that the number of layers does not significantly affect the nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ due to small thickness ratio. However, it is noteworthy that the 2-layer laminates exhibit the most pronounced hardening behavior. The highest and lowest fundamental nonlinear-to-linear frequency ratios $Ω_{N L} / Ω_{L}$ , associated with the maximum amplitude $| W_{\max} | / h = 1.2$ , are calculated as 1.2729 for $N C = 2$ , and 1.2485 for $N C = 10$ , respectively.

Figure 9.

Effect of the number of layers on the nonlinear-to linear frequency parameter of clamped symmetric elliptic plate with $θ = 45 °$ .

In this analysis, the number of layers is varied while maintaining a constant overall thickness. Since bending rigidity is directly proportional to the thickness of the laminate, increasing the thickness leads to higher natural frequencies. Therefore, changing the number of layers without changing the thickness has minimal impact on the natural frequencies. For this reason, the effect of the number of layers will not be considered in the analysis of VSCL elliptic plates.

Figure 10 shows the nonlinear-to-linear frequency ratios for symmetric 4-layer CSCL elliptic plates with clamped and simply supported boundary conditions. In this study, we focus on clamped and simply supported boundary conditions because they are among the most commonly used constraints in engineering applications, particularly in aerospace, automotive, and structural design. These boundary conditions represent two extreme cases: clamped conditions, which provide maximum constraint, and simply supported conditions, which allow free edge rotation, enabling a comprehensive analysis of the structural response. The mechanical parameters of Material 3 were applied. The thickness ratio and aspect ratio are taken as $0.01$ and $2$ , respectively. The point of maximum amplitude is located at $ξ = η = 0$ for both cases. The change from clamped to simply supported condition appears to affect the degree of hardening nonlinearity. The highest and lowest fundamental nonlinear-to-linear frequency ratios, associated with the maximum amplitude of the order of the thickness ( $| W_{\max} | = h$ ), are calculated for the same ${[45 °, - 45 °]}_{s}$ laminates with simply supported boundary conditions as 1.4692, and 1.1820 with clamped boundary conditions, respectively. This indicates that simply supported CSCL elliptic plates exhibit a higher degree of hardening nonlinearity compared to clamped elliptic laminates.

Figure 10.

Effect of clamped and simply supported boundary conditions on the nonlinear-to linear frequency parameter of ${[θ, - θ]}_{s}$ elliptic laminates.

Figure 11 shows the effect of the modulus ratio $E_{1} / E_{2}$ on the nonlinear free vibration of clamped ${[θ, - θ °]}_{s}$ elliptic laminates with the fiber orientation $θ = 45 °$ . This analysis illustrates how variations in the stiffness ratio between the fiber and matrix materials influence the nonlinear dynamic behavior of the laminates. Understanding these effects is essential for optimizing stiffness properties in laminated composites, as changes in the modulus ratio significantly impact nonlinear frequency response, structural stiffness, and hardening behavior in large amplitude vibrations. The plate’s aspect and thickness ratios are taken as 2 and $0.01$ , respectively. It can be seen from the figure that the nonlinear vibrational response increases as the modulus ratio $E_{1} / E_{2}$ increases. This indicates that as the stiffness increases, the nonlinear dynamic behavior of the clamped elliptic laminates becomes more evident. The highest and lowest fundamental nonlinear frequencies associated with the maximum amplitude $| W_{\max} | / h = 1.2$ , are calculated as 0.362 for the laminates with Material 5 $(E_{1} / E_{2} = 40)$ , and 0.133 for the laminates with Material 1 $(E_{1} / E_{2} = 3)$ , respectively.

Figure 11.

Effect of the modulus ratio $E_{1} / E_{2}$ on the nonlinear fundamental frequencies of ${[θ, - θ]}_{s}$ elliptic laminates.

Nonlinear vibration behavior of VSCL elliptic plates

This section investigates the nonlinear vibrational behavior of VSCL elliptic plates under the influence of various parameters. Studying the nonlinear vibrational behavior of such structures provides important insights into their dynamic performance, demonstrating how factors such as aspect ratio, thickness ratio, fiber orientation angles, boundary conditions, and material properties affect their nonlinear dynamic response.

Figure 12(a) and (b) show the fundamental backbone curves of clamped ${[\mp 〈 T_{0} | T_{1} 〉]}_{s}$ elliptic laminates, as a function of the center fiber orientation angle $T_{0}$ and the edge fiber angle $T_{1}$ , respectively. In the first case, $T_{0}$ is fixed at $45 °$ , while $T_{1}$ is increases from $0 °$ to $90 °$ with an increment of $15 °$ . In the second case $T_{1}$ is fixed at $45 °$ , while $T_{0}$ varies from $0 °$ to $90 °$ in $15 °$ increment. All fiber orientation angles $〈 T_{0} | T_{1} 〉$ considered in this analysis have been systematically verified to ensure they remain within the maximum curvature constraint. This validation confirms their manufacturability and practical feasibility. The mechanical parameters of Material 3, a thickness ratio of $h / a = 0.01$ , and an aspect ratio of $a / b = 2$ are used in this analysis. The point of maximum amplitude is located at $ξ = η = 0$ , where the laminates exhibit a hardening behavior. The hardening effect is evident in the higher frequency required to achieve larger amplitudes when the laminate vibrates in the nonlinear regime. It is observed that increasing $T_{1}$ gives higher fundamental nonlinear frequencies. However, increasing $T_{0}$ results in larger nonlinear-to-linear frequency ratios $Ω_{N L} / Ω_{L}$ . Moreover, the highest and lowest fundamental frequencies are produced by VSCL elliptic laminates compared to CSCL laminates (where $T_{0} = T_{1}$ ). The largest and smallest ratios $Ω_{N L} / Ω_{L}$ , corresponding to a maximum amplitude of the order of the thickness $| W_{\max} | = h$ are calculated as 1.2022 for ${[\mp 〈 45 ° | 0 ° 〉]}_{s}$ and 1.1514 for ${[\mp 〈 45 ° | 90 ° 〉]}_{s}$ , respectively.

Figure 12.

Fundamental backbone curves showing the effect of fiber orientation angles (a) $T_{0}$ and (b) $T_{1}$ on the nonlinear fundamental frequencies of ${[\mp 〈 T_{0} | T_{1} 〉]}_{s}$ elliptic plates ( $a / b = 2$ , $h / a = 0.01$ , $E_{1} / E_{2} = 15.4$ ).

Tables 9 and 10 display the nonlinear-to-linear frequency ratios for clamped symmetric and antisymmetric VSCL elliptic plates, respectively. These tables illustrate the impact of varying the aspect ratio and stacking sequence on the nonlinear vibrational behavior of VSCL elliptic laminates. In this analysis,

T_{0}

is fixed at

45 °

, while

T_{1}

is varied. The analysis employs the mechanical parameters of Material 3, with a thickness ratio of

h / a = 0.01 .

The point of maximum amplitude is located at

ξ = η = 0

. Figure 13 provides a comparison of the nonlinear frequencies for clamped symmetric and antisymmetric 4-layer VSCL elliptic laminates, highlighting the influence of laminate configuration on the nonlinear vibrational response. These results provide insight into how symmetry and asymmetry affect frequency variation in nonlinear regimes.

Table 9.

Nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ of clamped ${[\mp 〈 T_{0} | T_{1} 〉]}_{s}$ elliptic plates as a function of aspect ratio $a / b$ ( $h / a = 0.01$ , Material 3).

$a / b$	${[\mp 〈 T_{0} \| T_{1} 〉]}_{s}$	$\| W_{\max} \| / h$
$a / b$	${[\mp 〈 T_{0} \| T_{1} 〉]}_{s}$	0.2	0.4	0.6	0.8	1	$Ω_{l}$
1	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0073	1.0287	1.0632	1.1090	1.1642	0.076
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0079	1.0311	1.0687	1.1188	1.1798	0.074
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0063	1.0250	1.0554	1.0966	1.1473	0.071
1.5	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0080	1.0314	1.0690	1.1190	1.1792	0.116
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0077	1.0304	1.0671	1.1163	1.1762	0.129
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0063	1.0248	1.0550	1.0959	1.1463	0.136
2	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0082	1.0325	1.0715	1.1234	1.1863	0.175
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0077	1.0303	1.0669	1.1159	1.1758	0.206
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0065	1.0258	1.0571	1.0993	1.1514	0.225
3	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0084	1.0330	1.0732	1.1270	1.1927	0.346
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0077	1.0305	1.0675	1.1168	1.1773	0.420
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0068	1.0271	1.0597	1.1042	1.1568	0.475

Table 10.

Nonlinear-to-linear frequency ratio of $Ω_{N L} / Ω_{L}$ of clamped ${[\mp 〈 T_{0} | T_{1} 〉]}_{2}$ elliptic plates as a function of $a / b$ ( $h / a = 0.01$ , Material 3).

		$\| W_{\max} \| / h$
$a / b$	${[\mp 〈 T_{0} \| T_{1} 〉]}_{2}$	0.2	0.4	0.6	0.8	1	$Ω_{l}$
2	${[\mp 〈 45 ° \| 15 ° 〉]}_{2}$	1.0084	1.0329	1.0714	1.1214	1.1792	0.150
	${[\mp 〈 45 ° \| 30 ° 〉]}_{2}$	1.0059	1.0233	1.0490	1.0814	1.1165	0.165
	${[\mp 〈 45 ° \| 45 ° 〉]}_{2}$	1.0040	1.0158	1.0340	1.0502	1.0668	0.180
	${[\mp 〈 45 ° \| 60 ° 〉]}_{2}$	1.0030	1.0120	1.0256	1.0364	1.0464	0.193
	${[\mp 〈 45 ° \| 75 ° 〉]}_{2}$	1.0027	1.0105	1.0217	1.0341	1.0441	0.204
	${[\mp 〈 45 ° \| 90 ° 〉]}_{2}$	1.0030	1.0115	1.0242	1.0389	1.0525	0.214
3	${[\mp 〈 45 ° \| 15 ° 〉]}_{2}$	1.0087	1.0342	1.0750	1.1287	1.1924	0.292
	${[\mp 〈 45 ° \| 30 ° 〉]}_{2}$	1.0076	1.0300	1.0655	1.1128	1.1687	0.327
	${[\mp 〈 45 ° \| 45 ° 〉]}_{2}$	1.0068	1.0268	1.0587	1.1005	1.1508	0.363
	${[\mp 〈 45 ° \| 60 ° 〉]}_{2}$	1.0064	1.0251	1.0549	1.0945	1.1421	0.396
	${[\mp 〈 45 ° \| 75 ° 〉]}_{2}$	1.0061	1.0239	1.0525	1.0903	1.1355	0.424
	${[\mp 〈 45 ° \| 90 ° 〉]}_{2}$	1.0057	1.0226	1.0497	1.0859	1.1293	0.450

Figure 13.

Fundamental backbone curves for clamped symmetric and antisymmetric VSCL elliptical laminates.

The aspect ratio of VSCL elliptic plates significantly influences their nonlinear vibrational behavior. As observed, the nonlinear frequency parameters increase with increasing aspect ratio, which can be attributed to the reduction in the plate’s cross-sectional area. Stacking sequences also affect the nonlinear frequency of VSCL elliptic plates. Comparison of their backbone curves reveals that symmetric laminates exhibit a higher degree of hardening than antisymmetric laminates, which show slightly lower fundamental nonlinear frequency parameters. This frequency reduction in antisymmetric laminates is attributed to a stiffness imbalance between layers, which reduces the resistance to deformation under vibrational loads. Consequently, this imbalance affects the dynamic behavior of the plates, resulting in a lower degree of hardening and slightly reduced nonlinear-to-linear frequency ratios. The antisymmetric configuration introduces more flexibility, diminishing stiffness and lowering the nonlinear frequency response compared to symmetric laminates.

The highest and lowest fundamental nonlinear-to-linear frequency ratios $Ω_{N L} / Ω_{L}$ for symmetric VSCL , associated with the maximum amplitude $| W_{\max} | / h = 1$ , are calculated as 1.1927 for ${[\mp 〈 45 ° | 30 ° 〉]}_{s}$ with $a / b = 3$ and 1.1463 for ${[\mp 〈 45 ° | 90 ° 〉]}_{s}$ with $a / b = 2$ , respectively. For the antisymmetric laminates, the highest and lowest frequency ratios associated with $| W_{\max} | / h = 1$ , are calculated as 1.1924 for ${[\mp 〈 45 ° | 15 ° 〉]}_{s}$ with $a / b = 3$ and 1.0441 for ${[\mp 〈 45 ° | 75 ° 〉]}_{s}$ with $a / b = 2$ , respectively. Overall, symmetric laminates present a slightly higher degree of hardening and higher fundamental nonlinear frequencies compared to antisymmetric ones.

The effect of thickness ratio on the nonlinear behavior of clamped 4-layer VSCL elliptic plates is investigated as a function of fiber orientation angles (

T_{0} = 45 °

T_{1} = 30 °

60 °

, and

90 °

), as shown in Table 11. The analysis used the mechanical parameters of Material 3 and the aspect ratio

a / b = 2

. The results show that the nonlinear-to-linear frequency ratio increases as the thickness ratio is increased, reflecting a greater degree of hardening in the plates. However, for the same thickness ratio, an opposite pattern is observed with an increase in

T_{1}

. Nevertheless, the nonlinear frequencies

Ω_{N L}

consistently increase with increasing thickness ratio and fiber orientation angle. This is due to the enhanced stiffness of the plate as the thickness ratio and fiber orientation angle increase, which improves resistance to deformation under vibration. As the plate becomes thicker or the fiber orientation shifts, the overall rigidity of the structure increases, resulting in higher nonlinear frequencies. This relationship indicates that both thickness ratio and fiber orientation play crucial roles in strengthening the dynamic response of VSCL elliptic plates. The highest and lowest fundamental nonlinear-to-linear frequency ratios

Ω_{N L} / Ω_{L}

for

{[\mp 〈 T_{0} | T_{1} 〉]}_{s}

, associated with the maximum amplitude of the order of the thickness

| W_{\max} | = h

, are calculated as 1.2290 for

{[\mp 〈 45 ° | 60 ° 〉]}_{s}

with

h / a = 0.1

and 1.1514 for

{[\mp 〈 45 ° | 90 ° 〉]}_{s}

with

h / a = 0.01

, respectively.

Table 11.

Nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ of clamped VSCL elliptic plates as a function of thickness ratio $h / a$ ( $a / b = 2$ , Material 3).

$h / a$	${[\mp 〈 T_{0} \| T_{1} 〉]}_{s}$	$\| W_{\max} \| / h$
$h / a$	${[\mp 〈 T_{0} \| T_{1} 〉]}_{s}$	0.2	0.4	0.6	0.8	1	$Ω_{L}$
0.01	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0082	1.0325	1.0715	1.1234	1.1863	0.175
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0077	1.0303	1.0669	1.1159	1.1758	0.206
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0065	1.0258	1.0571	1.0993	1.1514	0.225
0.025	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0084	1.0332	1.0728	1.1256	1.1886	0.433
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0079	1.0313	1.0690	1.1193	1.1803	0.506
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0068	1.0268	1.0590	1.1033	1.1568	0.551
0.05	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0091	1.0357	1.0779	1.1331	1.1979	0.831
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0088	1.0346	1.0758	1.1303	1.1954	0.959
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0077	1.0305	1.0671	1.1156	1.1740	1.033
0.1	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0116	1.0451	1.0966	1.1617	1.2210	1.463
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0120	1.0467	1.1005	1.1656	1.2290	1.633
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0112	1.0433	1.0921	1.1454	1.2189	1.715

The boundary conditions play a crucial role in determining the nonlinear vibrational behavior of VSCL elliptic plates. Figure 14(a) and (b) present the fundamental backbone curves of clamped and simply supported ${[\mp 〈 T_{0} | T_{1} 〉]}_{s}$ laminates as a function of fiber orientation angle and aspect ratios $a / b = 2$ and $a / b = 3$ , respectively. It is known that simply supported boundary conditions lead to a greater degree of hardening behavior compared to clamped boundary conditions. This can be attributed to the increased flexibility allowed by the simply supported edges, which leads to larger deformations under the same vibrational conditions. Consequently, the nonlinear dynamic response is more significant, resulting in higher nonlinear-to-linear frequency ratios for simply supported plates. However, clamped plates produce higher fundamental nonlinear frequencies than simply supported plates. This is due to the increased edge constraints, which enhance the stiffness of the plate. The highest and lowest fundamental nonlinear-to-linear frequency ratios associated with the maximum amplitude of the order of thickness $| W_{\max} | = h$ , are calculated for simply supported ${[\mp 〈 45 ° | 30 ° 〉]}_{s}$ elliptic laminates with an aspect ratio $a / b = 3$ as 1.4716 and for clamped ${[\mp 〈 45 ° | 90 ° 〉]}_{s}$ as 1.1514, respectively.

Figure 14.

Backbone curves of clamped and simply supported ${[\mp 〈 T_{0} | T_{1} 〉]}_{s}$ elliptic plates for aspect ratios: (a) $a / b = 2$ and (b) $a / b = 3$ .

Table 12 presents the nonlinear-to-linear frequency ratio

Ω_{N L} / Ω_{L}

of clamped

{[\mp 〈 T_{0} | T_{1} 〉]}_{s}

elliptic plates as a function of the modulus ratio

E_{1} / E_{2}

and orientation angle

T_{1}

(

T_{0} = 45 °

T_{1} = 30 °

60 °

, and

90 °

). The aspect ratio and thickness ratio are taken as 2 and 0.01, respectively. Additionally, Figure 15 illustrates the variation of the nonlinear fundamental frequencies of

{[\mp 〈 45 ° | 30 ° 〉]}_{s}

as a function of the modulus ratio

E_{1} / E_{2}

. Studying the modulus ratio is particularly relevant for VSCL laminates, as their stiffness distribution is inherently non-uniform due to fiber orientation variations. Unlike CSCL plates, where the stiffness properties remain constant throughout, VSCL laminates can achieve enhanced vibration control by adjusting the modulus ratio in combination with tailored fiber paths. It can be seen that the nonlinear-to-linear frequency ratios and fundamental frequencies increase as the modulus ratio

E_{1} / E_{2}

is increased, indicating a greater degree of hardening behavior in the plates. The enhanced stiffness of the plate strengthens its overall dynamic response, leading to increased frequencies, as shown in Figure 15. However, increasing the edge angle reduces the hardening behavior of VSCL elliptic laminates. The highest and lowest fundamental nonlinear-to-linear frequency ratios, associated with the maximum amplitude of the order of thickness

| W_{m a x} | = h

are calculated for

{[\mp 〈 45 ° | 90 ° 〉]}_{s}

elliptic laminates with Material 5 as 1.1919 and 1.1233, respectively.

Table 12.

Nonlinear-to-linear frequency ratio $Ω_{N L} / Ω_{L}$ of clamped ${[\mp 〈 T_{0} | T_{1} 〉]}_{s}$ elliptic plates as a function of modulus ratio $E_{1} / E_{2}$ ( $a / b = 2$ , $h / a = 0.01$ ).

$Material$	$W_{\max} / h$
$Material$	${[\mp 〈 T_{0} \| T_{1} 〉]}_{s}$	0.2	0.4	0.6	0.8	1	$Ω_{l}$
1	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0076	1.0299	1.0660	1.1143	1.1731	0.117
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0075	1.0297	1.0656	1.1138	1.1724	0.111
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0074	1.0294	1.0650	1.1127	1.1711	0.103
3	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0082	1.0325	1.0715	1.1234	1.1863	0.175
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0077	1.0303	1.0669	1.1159	1.1758	0.206
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0065	1.0258	1.0571	1.0993	1.1514	0.225
4	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0085	1.0334	1.0734	1.1267	1.1910	0.208
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0076	1.0301	1.0664	1.1151	1.1747	0.249
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0056	1.0221	1.0490	1.0856	1.1305	0.274
5	${[\mp 〈 45 ° \| 30 ° 〉]}_{s}$	1.0085	1.0334	1.0738	1.1273	1.1919	0.258
	${[\mp 〈 45 ° \| 60 ° 〉]}_{s}$	1.0076	1.0299	1.0660	1.1145	1.1738	0.312
	${[\mp 〈 45 ° \| 90 ° 〉]}_{s}$	1.0052	1.0208	1.0462	1.0807	1.1233	0.344

Figure 15.

Fundamental backbone curves for clamped ${[\mp 〈 45 ° | 30 ° 〉]}_{s}$ elliptic plates as a function of the modulus ratio $E_{1} / E_{2}$ .

The normalized cross-section at $η = 0$ of the fundamental mode shape as a function of the maximum amplitude to thickness ratio $| W_{\max} | / h$ for clamped CSCL ${[45 °, - 45 °]}_{s}$ and ${[45 °, - 45 °]}_{2}$ as well as clamped VSCL ${[\mp 〈 45 ° | 15 ° 〉]}_{s}$ and ${[\mp 〈 45 ° | 15 ° 〉]}_{2}$ elliptic plates with $a / b = 2$ , are shown in Figures 16 and 17, respectively. As $| W_{\max} | / h$ increases, the fundamental mode shapes of elliptic laminated plates exhibit nonlinear changes due to the interaction between geometric nonlinearity and stiffness distribution. These changes are more pronounced in antisymmetric CSCL and VSCL compared to symmetric laminates. Antisymmetric laminates, characterized by non-uniform stiffness and enhanced bending-stretching coupling, show localized deformation patterns that evolve with increasing amplitude. In CSCL, sharper curvatures near the edges are observed, reflecting concentrated bending effects. However, for VSCL, the non-uniform stiffness leads to more complex deformation patterns, including significant flattening or reduced curvature near the edges at higher $| W_{\max} | / h$ values. These nonlinear effects emphasize the critical role of laminate asymmetry and stiffness tailoring in influencing the vibrational response.

Figure 16.

Normalized cross-section at $η = 0$ of the fundamental mode shape of clamped symmetric (a) and antisymmetric (b) CSCL elliptic plates as a function of $| W_{\max} | / h$ .

Figure 17.

Normalized cross-section at $η = 0$ of the fundamental mode shape of clamped symmetric (a) and antisymmetric (b) VSCL elliptic plates as a function of $| W_{\max} | / h$ .

Conclusion

This study presents a novel investigation into the geometrically nonlinear vibrational behavior of CSCL and VSCL elliptic plates using the $p$ -FEM based on FSDT and Von Kármán nonlinearity. The elliptic plate is exactly modeled using the blending function method. The nonlinear equations of free motion are derived using the harmonic balance method and solved iteratively via the linearized updated mode method. The curved $p$ -element demonstrates rapid convergence and high accuracy in capturing linear and nonlinear free vibration characteristics of elliptic laminates. Parametric studies reveal that aspect ratio, thickness ratio, stacking sequence, boundary conditions, modulus ratio, and fiber orientation angles significantly influence the nonlinear frequency response, hardening behavior, and mode shapes. VSCL plates, in particular, offer greater potential for tailoring nonlinear vibrations through fiber path customization. Compared to CSCL plates, which exhibit sharper and more uniform edge curvatures. Antisymmetric laminates exhibit greater sensitivity to geometric nonlinearity, leading to pronounced amplitude-dependent curvature changes. In contrast, symmetric laminates demonstrate a higher degree of hardening and slightly higher nonlinear fundamental frequencies. Additionally, simply supported plates show stronger hardening effects, while clamped plates yield higher nonlinear fundamental frequencies.

These findings provide valuable insights for optimizing the design and performance of composite laminated elliptic plates subjected to large amplitude vibrations in aerospace, automotive, and civil engineering applications. Future researches will focus on optimizing nonlinear vibration predictions, extending this model to investigate the buckling and post-buckling behavior of composite laminated elliptic plates, and incorporating higher-order shear deformation theories to improve the predictive accuracy of nonlinear vibration analysis.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Ahmed Guenanou

Mohammed Hachemi

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Appendix A

The strain energy $U$ can be expressed as: (A1)

U = \bar{U} + \hat{U} + \tilde{U} + \overset{˘}{U} + \tilde{\tilde{U}} + U_{b}

where

• $\bar{U}$ is the Extensional strain energy defined as:

(A2)

\bar{U} = \frac{1}{2} \iint [A_{11} {(\frac{\partial u}{\partial x})}^{2} + A_{22} {(\frac{\partial v}{\partial y})}^{2} + A_{66} {(\frac{\partial u}{\partial y})}^{2} + A_{66} {(\frac{\partial v}{\partial x})}^{2} + 2 A_{66} (\frac{\partial u}{\partial y}) (\frac{\partial v}{\partial x}) + 2 A_{12} (\frac{\partial u}{\partial x}) (\frac{\partial v}{\partial y}) + 2 A_{16} (\frac{\partial u}{\partial x}) (\frac{\partial u}{\partial y}) + 2 A_{16} (\frac{\partial u}{\partial x}) (\frac{\partial v}{\partial x}) + 2 A_{26} (\frac{\partial v}{\partial y}) (\frac{\partial u}{\partial y}) + 2 A_{26} (\frac{\partial v}{\partial y}) (\frac{\partial v}{\partial x})] d x d y

• $\hat{U}$ is the Extension-Bending coupling strain energy defined as:

(A3)

\hat{U} = \frac{1}{2} \iint [A_{11} (\frac{\partial u}{\partial x}) {(\frac{\partial w}{\partial x})}^{2} + A_{22} (\frac{\partial v}{\partial y}) {(\frac{\partial w}{\partial y})}^{2} + {2 A}_{66} (\frac{\partial u}{\partial x}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + 2 A_{66} (\frac{\partial v}{\partial x}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + A_{12} (\frac{\partial u}{\partial x}) {(\frac{\partial w}{\partial y})}^{2} + A_{12} (\frac{\partial v}{\partial y}) {(\frac{\partial w}{\partial x})}^{2} + {2 A}_{16} (\frac{\partial u}{\partial x}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + A_{16} (\frac{\partial u}{\partial y}) {(\frac{\partial w}{\partial x})}^{2} + A_{16} (\frac{\partial v}{\partial x}) {(\frac{\partial w}{\partial x})}^{2} + 2 A_{26} (\frac{\partial v}{\partial y}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + A_{26} (\frac{\partial u}{\partial y}) {(\frac{\partial w}{\partial y})}^{2} + A_{26} (\frac{\partial v}{\partial x}) {(\frac{\partial w}{\partial y})}^{2}] d x d y

• $\tilde{U}$ is the Bending strain energy defined as:

(A4)

\tilde{U} = \frac{1}{2} \iint [\frac{1}{4} A_{11} {(\frac{\partial w}{\partial x})}^{4} + \frac{1}{4} A_{22} {(\frac{\partial w}{\partial y})}^{4} + A_{66} {(\frac{\partial w}{\partial x})}^{2} {(\frac{\partial w}{\partial y})}^{2} + \frac{1}{2} {A_{12} (\frac{\partial w}{\partial x})}^{2} {(\frac{\partial w}{\partial y})}^{2} + A_{16} {(\frac{\partial w}{\partial x})}^{3} (\frac{\partial w}{\partial y}) + A_{26} {(\frac{\partial w}{\partial y})}^{3} (\frac{\partial w}{\partial x})] d x d y

• $\overset{˘}{U}$ is the Extension-rotation coupling strain energy defined as:

(A5)

\overset{˘}{U} = \frac{1}{2} \iint [B_{11} (\frac{\partial u}{\partial x}) (\frac{\partial ϕ_{y}}{\partial x}) - B_{22} (\frac{\partial v}{\partial y}) (\frac{\partial ϕ_{x}}{\partial y}) + B_{66} (\frac{\partial u}{\partial y}) (\frac{\partial ϕ_{y}}{\partial y}) - B_{66} (\frac{\partial u}{\partial y}) (\frac{\partial ϕ_{x}}{\partial x}) + B_{66} (\frac{\partial v}{\partial x}) (\frac{\partial ϕ_{y}}{\partial y}) - B_{66} (\frac{\partial v}{\partial x}) (\frac{\partial ϕ_{x}}{\partial x}) - B_{12} (\frac{\partial u}{\partial x}) (\frac{\partial ϕ_{x}}{\partial y}) + B_{12} (\frac{\partial v}{\partial y}) (\frac{\partial ϕ_{y}}{\partial x}) + B_{16} (\frac{\partial u}{\partial x}) (\frac{\partial ϕ_{y}}{\partial y}) - B_{16} (\frac{\partial u}{\partial x}) (\frac{\partial ϕ_{x}}{\partial x}) + B_{16} (\frac{\partial u}{\partial y}) (\frac{\partial ϕ_{y}}{\partial x}) + B_{16} (\frac{\partial v}{\partial x}) (\frac{\partial ϕ_{y}}{\partial x}) + B_{26} (\frac{\partial v}{\partial y}) (\frac{\partial ϕ_{y}}{\partial y}) - B_{26} (\frac{\partial v}{\partial y}) (\frac{\partial ϕ_{x}}{\partial x}) - B_{26} (\frac{\partial u}{\partial y}) (\frac{\partial ϕ_{x}}{\partial y}) - B_{26} (\frac{\partial v}{\partial x}) (\frac{\partial ϕ_{x}}{\partial y})] d x d y

• $\tilde{\tilde{U}}$ is the bending strain coupling energy defined as:

(A6)

\tilde{\tilde{U}} = \frac{1}{2} \iint [B_{11} (\frac{\partial ϕ_{y}}{\partial x}) {(\frac{\partial w}{\partial x})}^{2} - B_{22} (\frac{\partial ϕ_{x}}{\partial y}) {(\frac{\partial w}{\partial y})}^{2} + B_{12} ((\frac{\partial ϕ_{y}}{\partial x}) {(\frac{\partial w}{\partial y})}^{2} - (\frac{\partial ϕ_{x}}{\partial y}) {(\frac{\partial w}{\partial x})}^{2}) + {2 B}_{66} ((\frac{\partial ϕ_{y}}{\partial y}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) - (\frac{\partial ϕ_{x}}{\partial x}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y})) + {2 B}_{16} (\frac{\partial ϕ_{y}}{\partial x}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + B_{16} ((\frac{\partial ϕ_{y}}{\partial y}) {(\frac{\partial w}{\partial x})}^{2} - (\frac{\partial ϕ_{x}}{\partial x}) {(\frac{\partial w}{\partial x})}^{2}) - {2 B}_{26} (\frac{\partial ϕ_{x}}{\partial y}) (\frac{\partial w}{\partial x}) (\frac{\partial w}{\partial y}) + B_{26} ((\frac{\partial ϕ_{y}}{\partial y}) {(\frac{\partial w}{\partial y})}^{2} - (\frac{\partial ϕ_{x}}{\partial x}) {(\frac{\partial w}{\partial y})}^{2})] d x d y

• $U_{b}$ is the bending strain energy defined as

(A7)

U_{b} = \frac{1}{2} \iint [{D_{11} (\frac{\partial ϕ_{y}}{\partial x})}^{2} + D_{22} {(\frac{\partial ϕ_{x}}{\partial y})}^{2} + D_{66} {(\frac{\partial ϕ_{y}}{\partial y} - \frac{\partial ϕ_{x}}{\partial x})}^{2} - 2 D_{12} (\frac{\partial ϕ_{y}}{\partial x}) (\frac{\partial ϕ_{x}}{\partial y}) + 2 D_{16} (\frac{\partial ϕ_{y}}{\partial x}) (\frac{\partial ϕ_{y}}{\partial y} - \frac{\partial ϕ_{x}}{\partial x}) - 2 D_{26} (\frac{\partial ϕ_{x}}{\partial y}) (\frac{\partial ϕ_{y}}{\partial y} - \frac{\partial ϕ_{x}}{\partial x}) + k_{s} A_{44} {(\frac{\partial w}{\partial x} + ϕ_{y})}^{2} + k_{s} A_{55} {(\frac{\partial w}{\partial y} - ϕ_{x})}^{2} + 2 k_{s} A_{45} (\frac{\partial w}{\partial x} + ϕ_{y}) (\frac{\partial w}{\partial y} - ϕ_{x})] d x d y

where (A8)

A_{i j} = \int_{- h / 2}^{h / 2} {\bar{Q}}_{i j} d z (i, j = 1, 2, 6)

(A9)

A_{i j} = \int_{- h / 2}^{h / 2} {\bar{Q}}_{i j} d z (i, j = 4, 5)

(A10)

D_{i j} = \int_{- h / 2}^{h / 2} {\bar{Q}}_{i j} z^{2} d z (i, j = 1, 2, 6)

(A11)

B_{i j} = \int_{- h / 2}^{h / 2} {\bar{Q}}_{i j} z d z (i, j = 1, 2, 6)

The shear correction factor $k_{s}$ , is taken as $5 / 6$ in this study.

Appendix B

The entries of the linear and nonlinear stiffness and mass matrices $\bar{K}, K, \overset{˘}{K}, \hat{K}, \tilde{K}, M$ are as follows: (B1)

{\bar{K}}_{2 α - 1, 2 β - 1} = \int_{- 1}^{1} \int_{- 1}^{1} (A_{11} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + A_{66} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + A_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + A_{16} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x}) | J | d ξ d η

(B2)

{\bar{K}}_{2 α - 1, 2 β} = \int_{- 1}^{1} \int_{- 1}^{1} (A_{66} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + A_{12} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + A_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y}) | J | d ξ d η

(B3)

{\bar{K}}_{2 α, 2 β - 1} = \int_{- 1}^{1} \int_{- 1}^{1} (A_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + A_{12} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + A_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y}) | J | d ξ d η

(B4)

{\bar{K}}_{2 α, 2 β} = \int_{- 1}^{1} \int_{- 1}^{1} (A_{22} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + A_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y}) | J | d ξ d η

(B5)

{\hat{K}}_{2 α - 1, 3 β - 2} = \int_{- 1}^{1} \int_{- 1}^{1} (A_{11} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial x} + A_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial y} + A_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial x} + A_{12} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial y} + A_{16} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial x} + A_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial y} + A_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial y}) | J | d ξ d η

(B6)

{\hat{K}}_{2 α, 3 β - 2} = \frac{1}{2} \int_{- 1}^{1} \int_{- 1}^{1} (A_{22} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial y} + A_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial y} + A_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial x} + A_{12} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial x} + A_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial y} + A_{26} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial y}) | J | d ξ d η

(B7)

{\tilde{K}}_{2 α, 3 β - 2} = \int_{- 1}^{1} \int_{- 1}^{1} (A_{11} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial x} \frac{\partial N_{ε}}{\partial x} + A_{22} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial y} \frac{\partial N_{ε}}{\partial y} + (A_{12} + 2 A_{66}) \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial y} \frac{\partial N_{ε}}{\partial y} + (A_{12} + 2 A_{66}) \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial x} \frac{\partial N_{ε}}{\partial x} + {3 A}_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial x} \frac{\partial N_{ε}}{\partial y} + A_{16} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} \frac{\partial N_{γ}}{\partial x} \frac{\partial N_{ε}}{\partial x} + 3 A_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial y} \frac{\partial N_{ε}}{\partial x} + A_{26} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} \frac{\partial N_{γ}}{\partial y} \frac{\partial N_{ε}}{\partial y}) | J | d ξ d η

(B8)

{\overset{˘}{K}}_{2 α - 1, 3 β - 1} = - \int_{- 1}^{1} \int_{- 1}^{1} (B_{66} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + B_{12} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + B_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + B_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y}) | J | d ξ d η

(B9)

{\overset{˘}{K}}_{2 α - 1, 3 β} = \int_{- 1}^{1} \int_{- 1}^{1} (B_{11} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + B_{66} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + B_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + B_{16} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x}) | J | d ξ d η

(B10)

{\overset{˘}{K}}_{2 α, 3 β - 1} = - \int_{- 1}^{1} \int_{- 1}^{1} (B_{22} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + B_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + B_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + B_{26} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y}) | J | d ξ d η

(B11)

{\overset{˘}{K}}_{2 α, 3 β} = \int_{- 1}^{1} \int_{- 1}^{1} (B_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + B_{12} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + B_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + B_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y}) | J | d ξ d η

(B12)

K_{3 α - 2, 3 β - 2} = \int_{- 1}^{1} \int_{- 1}^{1} k_{s} (A_{44} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + A_{55} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + A_{45} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + A_{45} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x}) | J | d ξ d η

(B13)

K_{3 α - 2, 3 β - 1} = - k_{s} \int_{- 1}^{1} \int_{- 1}^{1} (A_{55} \frac{\partial N_{α}}{\partial y} N_{β} + A_{45} \frac{\partial N_{α}}{\partial x} N_{β}) | J | d ξ d η

(B14)

K_{3 α - 2, 3 β} = k_{s} \int_{- 1}^{1} \int_{- 1}^{1} (A_{44} N_{β} \frac{\partial N_{α}}{\partial x} + A_{45} N_{β} \frac{\partial N_{α}}{\partial y}) | J | d ξ d η

(B15)

K_{3 α - 1, 3 β - 2} = - k_{s} \int_{- 1}^{1} \int_{- 1}^{1} (A_{55} N_{α} \frac{\partial N_{β}}{\partial y} + A_{45} N_{α} \frac{\partial N_{β}}{\partial x}) | J | d ξ d η

(B16)

K_{3 α - 1, 3 β - 1} = \int_{- 1}^{1} \int_{- 1}^{1} (D_{22} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + D_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + D_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + D_{26} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + k_{s} A_{55} N_{α} N_{β}) | J | d ξ d η

(B17)

K_{3 α - 1, 3 β} = - \int_{- 1}^{1} \int_{- 1}^{1} (D_{12} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + D_{66} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + D_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + D_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + k_{s} A_{45} N_{α} N_{β}) | J | d ξ d η

(B18)

K_{3 α, 3 β - 2} = k_{s} \int_{- 1}^{1} \int_{- 1}^{1} (A_{44} N_{α} \frac{\partial N_{β}}{\partial x} + A_{45} N_{α} \frac{\partial N_{β}}{\partial y}) | J | d ξ d η

(B19)

K_{3 α, 3 β - 1} = - \int_{- 1}^{1} \int_{- 1}^{1} (D_{12} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + D_{66} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + D_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + D_{26} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + k_{s} A_{45} N_{α} N_{β}) | J | d ξ d η

(B20)

K_{3 α, 3 β} = \int_{- 1}^{1} \int_{- 1}^{1} (D_{11} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial x} + D_{66} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial y} + D_{16} \frac{\partial N_{α}}{\partial x} \frac{\partial N_{β}}{\partial y} + D_{16} \frac{\partial N_{α}}{\partial y} \frac{\partial N_{β}}{\partial x} + k_{s} A_{44} N_{α} N_{β}) | J | d ξ d η

(B21)

M_{3 α - 2, 3 β - 2} = \int_{- 1}^{1} \int_{- 1}^{1} I_{1} N_{α} N_{β} | J | d ξ d η

(B22)

M_{3 α - 1, 3 β - 1} = \int_{- 1}^{1} \int_{- 1}^{1} I_{3} N_{α} N_{β} | J | d ξ d η

(B23)

M_{3 α, 3 β} = \int_{- 1}^{1} \int_{- 1}^{1} I_{3} N_{α} N_{β} | J | d ξ d η

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Geometrically nonlinear free vibration analysis of shear deformable laminated composite elliptic plates using the p -version FEM

Abstract

Keywords

Introduction

Formulation

Curvilinear fiber

Curved hierarchical finite element model

Results and discussions

Validation of the curved rectangular p -element for geometrically nonlinear vibration

Parametric study

Nonlinear vibration behavior of CSCL elliptic plates

Nonlinear vibration behavior of VSCL elliptic plates

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Data Availability Statement

Appendix A

Appendix B

References

Validation of the curved rectangular $p$ -element for geometrically nonlinear vibration