Laminated Plate Structures Research Articles

An efficient neural network approach for laminated composite plates using refined zigzag theory

This paper presents an innovative methodology employing the One-dimensional Convolutional Gated Recurrent Unit neural network (1D-CGRU) algorithm for the analysis of laminated composites using the Refined Zigzag theory (RZT). The RZT methodology is utilized to assess laminated plate structures and generate essential data, forming the basis for training the 1D-CGRU model. The synergistic application of RZT and 1D-CGRU demonstrates exceptional global–local accuracy in predicting the mechanical behavior of laminated composite plates. For efficient data generation, RZT not only provides high precision, but also exhibits computational efficiency, making it suitable for finite element simulations with a C0-continuous kinematic approximation. Additionally, the 1D-CGRU model integrates the strengths of a One-dimensional Convolutional Neural Network (1D-CNN) for spatial feature extraction and dimensionality reduction, coupled with a Gated Recurrent Unit (GRU) network for discerning temporal relationships and mapping them to the target domain. Furthermore, quantitative accuracy measurements, including Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE), are used to validate the superior performance of the 1D-CGRU model compared to other surrogate models. For angle-ply and cross-ply composite laminates, the 1D-CGRU model achieves remarkable accuracy (98.15% and 99.45%, respectively) with low RMSE values. These results highlight the potential of the proposed framework to enhance predictive analysis for laminated composite structures, offering valuable insights for engineering applications and design optimizations.

A mechanical analysis of variable angle-tow composite plates through variable kinematics models based on Carrera’s unified formulation

Variable Angle-Tow (VAT) laminates offer a promising alternative to straight fiber composites. By varying fibers orientation within the structure plane, ambitious design and performance goals can be achieved. However, the wider design space results in a more complex problem with more parameters to consider. Carrera’s Unified Formulation (CUF) has been used in previous works performing buckling, vibrational and stress analyses of VAT plates. Usually, one-dimensional (1D) CUF beam models are used, while two-dimensional (2D) plate models are obtained as a particular case of shells by considering a null curvature. In most cases, a linear law is considered to describe the variation of fibers orientation in the main plane of the structure. The purpose of this article is to extend the CUF 2D plate finite elements family to the mechanical analysis of composite laminated plate structures with curvilinear fibers. The main contribute consists in the development of a CUF FE model within the Reissner’s Mixed Variational Theorem (RMVT) context for an improved calculation of the out-of-plane stress components. Results show that RMVT can predict in-plane stresses and satisfy the though-the-thickness transverse stresses continuity due to inter-layer equilibrium. The accuracy of RMVT-based models is also investigated using two different approximation points distributions along the plates thickness.

A FEM Free Vibration Analysis of Variable Stiffness Composite Plates through Hierarchical Modeling

Variable Angle Tow (VAT) laminates offer a promising alternative to classical straight-fiber composites in terms of design and performance. However, analyzing these structures can be more complex due to the introduction of new design variables. Carrera’s unified formulation (CUF) has been successful in previous works for buckling, vibrational, and stress analysis of VAT plates. Typically, one-dimensional (1D) and two-dimensional (2D) CUF models are used, with a linear law describing the fiber orientation variation in the main plane of the structure. The objective of this article is to expand the CUF 2D plate finite elements family to perform free vibration analysis of composite laminated plate structures with curvilinear fibers. The primary contribution is the application of Reissner’s mixed variational theorem (RMVT) to a CUF finite element model. The principle of virtual displacements (PVD) and RMVT are both used as variational statements for the study of monolayer and multilayer VAT plate dynamic behavior. The proposed approach is compared to Abaqus three-dimensional (3D) reference solutions, classical theories and literature results to investigate the effectiveness of the developed models. The results demonstrate that mixed theories provide the best approximation of the reference solution in all cases.

S0 Lamb Mode Scattering Studies in Laminated Composite Plate Structures with Surface Breaking Cracks: Insights into Crack Opening Behavior

Ultrasonic-guided waves are attractive for rapid inspection of laminated composite structures, where cracks developed transverse to the loading direction are the severe type of damage. This paper presents studies of the interaction of fundamental symmetric S0 Lamb mode with vertical surface-breaking cracks in laminated composite plate structures. Finite element simulations and experimental investigations are used to study the effect of crack depth on S0 wave reflection behavior. Results show a monotonic rise of the reflection coefficient for different crack depths in a manner that is strongly dependent on the orientation of the plies and transverse ply location in the vicinity of the crack. Scattered wave packets in the reflection regime are captured using an in-plane laser. The S0 Lamb mode's sensitivity is numerically presented for the different crack depths in the long wavelength limit. We also observed that the reflected wave mode depicts the information of the corresponding broken interfaces. An attempt was made to show that this behavior relates to the crack-opening behavior in response to in-plane excitation. The reflection coefficient as a characteristic polynomial is proposed for various orientations. It was observed that the dispersion at receiver nodes makes the analysis challenging for distinguishing the signal from crack faces due to the smaller dimension. The study outcomes show its prospect as a promising NDE tool for crack damage detection in thin laminated plate structures.

Inverse differential quadrature solutions for free vibration of arbitrary shaped laminated plate structures

An essential aspect of design of laminated plate structures in many engineering applications is the analysis of free vibration behaviour in order to model the structure for random excitations. In this regard, numerical solutions to the systems of high-order partial differential equations governing free vibration response of the structure become important. Direct approximation of such high-order systems are prone to error arising from the sensitivity of high-order numerical differentiation to noise necessitating the demand for improved solution techniques. In this work, a novel generalised inverse differential quadrature method is developed to study the dynamic behaviour of first-order shear deformable arbitrary-shaped laminated plates. The ensuing underdetermined system is operated upon by Moore-Penrose pseudo-inverse preconditioning to form a squared eigenvalue system. Free vibration solutions of square, skew, circular, and annular sector plates for different boundary conditions are obtained and validated against exact and numerical solutions in the literature and ABAQUS. It is demonstrated with numerous examples that iDQM solutions are in excellent agreement with exact solutions for square plates and the results for arbitrary shaped plates are comparable with solutions in the literature while saving up to 96% degrees of freedom required for ABAQUS solution. Finally, refined parametric studies conducted reveal that, subject to varying geometric configurations, iDQM solutions are numerically stable and potentially converge faster than DQM.

On the free vibration behavior of nanocomposite laminated plates contained piece-wise functionally graded graphene-reinforced composite plies

The numerical answer to the free vibration behavior of laminated plate structure contained piece-wise carbon-based nanocomposites plies is investigated. Plies are made from a combination of an isotropic polymer matrix and a nanoscale carbon-based reinforcements namely graphene. The graphene reinforcements are dispersed across the plate thickness either in uniform or non-uniform fashion according to four different distribution patterns. To assess the effect of the grading graphene patterns five types of lay-up arrangements are comparatively investigated. The effective elastic moduli of the laminated nanocomposite including Young’s and shear moduli are obtained by using the so-called extended Halpin-Tsai model. Theoretical approaches are based on the first-order shear deformation theory (FSDT), while Lagrange's equation and a finite element formulation are employed to derive the natural frequencies. The accuracy of the results using the present framework model is examined with those available in previous attempts and an excellent agreement is achieved. A large range of parametric studies was carried out with a view to investigate the effect of several factors including lay-up arrangements, graphene distribution patterns, lamination sequence, plate geometric parameters, plate boundary conditions, and plate mode shapes on the free vibration behavior of functionally graded graphene-reinforced composite (FG-GRC) laminated plates. New suggestions are presented with the aim of providing useful remarks and new design criteria for future and innovative nanocomposite structures.

Vibration characteristics and power flow analysis of irregular composite coupling laminated plate structures

This article presents a predictive model to investigate the vibration and power flow characteristics of irregular composite coupling laminated plates based on the 2D Chebyshev–Ritz method. First, the simplified physical model of coupling plate with arbitrary coupling angle is specified and the coordinates are defined for better describing the displacement components of irregular structures using Chebyshev polynomials. The Lagrange energy functionals of coupling plates are derived by the first-order shear deformation theory and the employment of boundary and coupling springs. The displacement admissible functions of coupling plates are constructed based on the Chebyshev polynomials of the first kind. Then, the vibration solutions of coupling plate structure can be well evaluated by the Ritz procedure. The convergence characteristics and calculation accuracy of the model are verified by a series of numerical examples. The effects of relevant parameters on the inherent characteristics and steady-state response of the coupling plate are further investigated. The proposed vibration model, based on the 2D Chebyshev–Ritz method, is not only capable of physically revealing the complex vibration and power flow phenomena of irregular composite coupling laminated plates and similar structures, but also provides a basis for structural low-noise design engineering.

Stacking sequence optimization of laminated plate structures using the boundary element method

In the present paper, we implement a pure boundary element method for the static analysis of 2D and 3D structures composed of thin, unsymmetric laminated composite plates. The domain integrals related to distributed loads are transformed into boundary integrals using the radial integration method. Furthermore, we apply an equilibrium-based procedure to directly recover the out-of-plane stress state from the classical laminated plate theory. Some benchmark examples verify the accuracy and efficiency of the present formulation, highlighting its steady convergence in regions of stress concentration near holes with sharp corners. To exploit the computational performance of the resulting boundary element code, we use the resulting code as the analysis step of an optimization scheme, taking fiber orientation angles as design variables.

Study on Dynamic Response Characteristics of Bird Impact Glass Fiber Laminate

Composite structures are often subjected to bird impact loads during service, which causes visually invisible damage inside the structure. Such damage often weakens the mechanical properties of the laminated plate structure, resulting in a significant reduction in the bearing capacity and service life of the structure. Therefore, it is important to evaluate the impact resistance of composite structures and predict the corresponding impact damage characteristics for safety. Experimental research using real birds is costly and unrepeatable, so gelatin bird projectile was often used instead of real birds. In this study, gelatin projectile was prepared to meet the requirements of the experiment and impacted the glass fiber laminated at different speeds. The strain variation trend at each point was recorded and analyzed, Simultaneously capturing the test process with a high-speed camera. In order to understand the bird strike damage characteristics of fiberglass laminates, ultrasonic C-scans were performed on the test specimens after the experiment. The results show that the strain at the impact point is the largest during the impact process. When the impact speed is doubled, the strain at each point does not increase exponentially, but only slightly. In the test, point C is inner concave due to the affected of the impact, and the nearby area is outer convex, and then spread to the surrounding area in the form of "water wave". The gelatine pellets of 2bl did not cause lamination damage at the impact velocities of 44.4m/s, 66.6m/s, 88.8m/s and 111m/s, but plastic deformation will occur to varying degrees.

Analytical Solution for Thermal Flutter of Laminates in Supersonic Speeds

As a basic component of engineering fields such as aeronautics, astronautics and shipbuilding, panel structure has been widely used in engineering and scientific research. It is of great theoretical and practical significance to study the vibration of panels. The panel flutter problem has caused widely concerned by researchers at home and abroad during to the emergence of high-speed aircrafts. With regard to the eigenvalue problem of rectangular panels, it is generally believed that it is difficult to obtain a closed form eigen solution in the case of an adjacent boundaries clamped-supported or a free boundary that cannot be decoupled. Aiming at the problem, this paper studies the two-dimensional symmetric orthogonal laminated plate structure in the hypersonic flow in the thermal environment, and combines the first-order piston aerodynamic theory to study a high-precision separation variable method. Through this method, analytical solution to the closed form of the thermal flutter problem of rectangular panels can be obtained under any homogeneous boundary conditions.

Numerical solutions for magneto–electro–elastic laminated plates resting on Winkler foundation or elastic half-space

This article investigates the bending response performances of the magneto–electro–elastic (MEE) laminated plates resting on the Winkler foundation or the elastic half-space subjected to a transverse mechanical loading .By assuming that the foundation is not electrically and magnetically conductive, the scaled boundary finite element method (SBFEM) based on the three-dimensional (3D) theory of elasticity is applied for both the simulation of the MEE laminated plate and the elastic half-space. The SBFEM model considers the generalized displacement involving the elastic displacement, electric potential and magnetic potential as the nodal degree of freedom for the MEE laminated plates, and only the in-plane of the MEE laminated plate or the boundary of the elastic half-space needs to be discretized leading to reduce the spatial dimension by one. Furthermore, in the SBFEM, the governing equations can be solved by using an analytical approach in the radial direction of the scaled coordinate system, so that it is particularly suitable for the simulation of the elastic half-space. For the Winkler foundation–plate system, the global stiffness coupling governing equation that includes the interaction between the MEE laminated plate and the Winkler foundation is derived directly from the 3D elasticity equations of the MEE laminated plate by assuming that the foundation reactions are proportional to the transverse displacements of the plate structure. While for the MEE laminated plate-half-space system, the whole domain is divided into three sub-domains including the MEE laminated plate structure, the near and semi-infinite far foundation systems based on the sub-structure method, and then the stiffness matrix of each sub-domain can be determined by means of the SBFEM. As a result, the global stiffness equation of the plate-half-space system can be assembled according to the principle of the degree of freedom matching at the same nodes. The numerical results obtained for limiting cases by using the proposed method were compared with the published works and showed excellent agreements with the solutions based on the analytical and numerical approaches, so that the accuracy and applicability of the proposed formulations for the analysis of the interaction problems between the MEE laminated plate and the Winkler foundation or elastic half-space can be verified. Moreover, several numerical examples with various material properties, geometries, stacking sequences, aspect ratios, and supported boundary conditions were presented to show the effects of which on the responses of the plate–foundation system.

Vibration Analysis of Fiber Reinforced Composite Laminated Plates with Arbitrary Boundary Conditions

A Fourier series method based on Mindlin theory and Hamilton variation principle has been proposed for the vibration modeling and analysis of composite laminated plates with arbitrary boundary conditions, in which the vibration displacements are sought as the linear combination of a double Fourier cosine series and auxiliary series functions. Three types of constrained springs are introduced to establish a general structural model of composite laminates, and the vibration model is established by combining the Hamilton energy principle. The accuracy, efciency and validity of the two solutions presented are demonstrated via comparison with published results. The influence of boundary conditions, laying angle and laying layer on vibration characteristics is analyzed. And it can be seen that the frequency of the structure increases with the increasing of the spring stiffness and the number of laying layers of the boundary of the laminated plate structure.

Modeling on Actuation Behavior of Macro-Fiber Composite Laminated Structures Based on Sinusoidal Shear Deformation Theory

A new piezoelectric composite, macro fiber composite (MFC) is recombined with piezoceramic fibers, an epoxy resin basal body, and an interdigitated electrode. It has been widely applied in vibration reduction and deformation control of thin-walled structures, due to its great deformability and flexibility. Research on its actuation performance is mostly concentrated on the MFC actuating force calculation based on classical plate theory (CPT), and the overall modeling of MFC and its structure. However, they have some deficiencies in the tedious calculating process and neglect of shear deformation, respectively. To obtain a precise MFC actuating force, the sinusoidal shear deformation theory (SSDT) is adopted to deduce the MFC actuating force formula, and global–local displacement distribution functions are introduced to help the MFC laminated plate structure satisfy the deformation compatibility and stress balance. For instance, in the end displacement calculation of the MFC laminated beam structure. The experimental result of the MFC laminated beam is compared with those of the MFC actuating force based on SSDT and on CPT, which indicates that the MFC actuating force formula based on SSDT can reach higher computational accuracy.

Mechanical analysis of photovoltaic panels with various boundary condition

The photovoltaic (PV) panels currently existed on market are a kind of laminated plate structure, which is composed of two stiff glass skins and a soft interlayer. Some of those panels are installed on the buildings and integrated as the components of the structures, such as wall and roof. In different locations, the installations of the panels are different and the boundary conditions are not simply supported any more. The generating electricity requirement of the panels needs the mechanic behaviour of them is stable with any boundary condition. In this paper the bending behaviour of PV panels with various boundary conditions is analysed and the influence of boundary condition is studied carefully. The Kirchhoff theory which is one of the classical lamination theory (CLT) is adopted to build governing equations of photovoltaic panels under static force. A Rayleigh-Rita method is modified to solve the governing equations and calculate the static deformation and stress. Different boundary condition requires different assumptions of the deflection function and a modified general function is developed to solve that problem. A theoretical solution is derived out and used to do the numerical calculation. A bending experiment of PV panel with two opposite edges simply supported and the other two free is used to verify the correctness and accuracy of the proposed solution. Finally, the influence of different boundary condition is stated by comparing the numerical results and some guides for the PV panel installation are proposed.

Uncertainty propagation in modal analysis of viscoelastic sandwich structures using a stochastic collocation method

This paper presents an uncertainty propagation analysis of viscoelastic sandwich laminated structures using a stochastic collocation method based on Smolyak’s sparse grids approach. Shear deformation in the viscoelastic layers is accurately captured by means of layer-wise plate finite elements. Modal properties are determined by solving nonlinear eigenvalue problems due to the frequency-dependent nature of the considered viscoelastic materials. Variations in material properties and layer thicknesses are both treated in a probabilistic way. The variabilities of natural frequencies and modal loss factors of two different viscoelastic laminated plate structures (plane and cylindrical) are studied. Results show that the present method converges quickly with small sample sizes when compared to the classical Monte-Carlo method. The influence of the variability of several parameters on viscoelastic structure modal properties is commented.

Mechanical analysis of photovoltaic panels with various boundary condition

The photovoltaic (PV) panels currently existed on market are laminated plate structures, which are composed of two stiff glass skins and a soft interlayer. Some panels are installed on the buildings and integrated as the components of the structures, such as wall and roof. In different locations, the installations of PV panels are different and the boundary conditions are not always simply supported. In this paper, the bending behaviour of PV panels with various boundary conditions is analysed and the influence of boundary condition is studied carefully. The Kirchhoff theory is adopted to build governing equations of PV panels under static force. A Rayleigh-Rita method is modified to solve the governing equations and calculate the static deformation and stress. Different boundary conditions usually require different assumptions of the deflection function, but a modified general function is developed in here to solve that problem. A theoretical solution is derived out and used to do the numerical calculation. The bending experiments of PV panels with two boundary conditions are used to verify the accuracy of the proposed solutions. Finally, the influence of different boundary condition is stated by comparing the numerical results and some guides for the PV panel installation are proposed.

Review of Laminated Composite Plate Theories, with Emphasis on Variational Asymptotic Method

Most of the engineering structures are built using beams, plates, and shells as the structural elements, and in recent times using laminated composite structures is increasing drastically. The accurate analysis of laminated plate structures is an important responsibility; to this end, there are different ordered theories developed by the research community. The variational asymptotic method developed in a mathematically rigorous method practiced by Prof. Hodges and group to analyze laminated structures promises the accurate analysis. A review of the variational asymptotic method employed in the analysis of laminated plate structures is presented in this paper, along with the brief review on other plate theories. The solution strategies employed by the different plate theories are briefly reviewed. A flow chart or tree structure on different plate theories and different solution strategies are presented as a quick reference for the researchers. The possible research areas or problems to be addressed using the variational asymptotic method in conjunction with different solution strategies are briefly mentioned.

Bending of FGM plates under thermal load: Classical thermoelasticity analysis by a meshless method

Abstract Thermal stresses, especially at the interface between two different materials, often play an important role in the failure of laminated composite structures. Thus, it is a strong motivation to replace laminated plate structures by FGM ones if possible. The functional gradation of material coefficients, however, yields new coupling effects between the in-plane deformation and bending modes. Therefore the study of behaviour of FGM plates under thermal loadings has become important. In this paper, the bending of thin and/or thick FGM plates under thermal load is considered within the classical theory of thermoelasticity. The variable material properties of plate (such as the Young's modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The governing equations which are given by the 4th order partial differential equations (PDE) are decomposed into the 2nd order PDEs in order to overcome the inaccuracy of approximation of high order derivatives of field variables. The strong form meshless formulations for solution of thin plate bending problem is developed in combination with Moving Least Squares (MLS) approximation scheme. The attention is paid to the study of the influence of various parameters of gradations of material coefficients on bending of plates.

On the formulation of a high-order discontinuous finite element method based on orthogonal polynomials for laminated plate structures

Laminated plate structures are analyzed by a discontinuous finite element method with emphasis on determining the transverse shear and normal stress components at the interface of adjacent layers accurately. A Consistent Orthogonal Basis Function Space is used for the interpolation of the displacement field and the traction field between two adjacent layers. The mass matrix of the laminated plate becomes diagonal. Moreover, it is observed that the basis functions are very similar to the vibration mode shapes, even through we do not solve any eigenvalue problem in their generation. These basis functions are uniquely determined by the structure's configuration and associated boundary conditions. The stress field between the layers can be accurately calculated, even for the region near the boundaries that might have sharp stress gradients. Several numerical examples are studied with different boundary conditions. The results for both the deformation and the stress components are compared with the traditional finite element method, especially in terms of the number of degrees-of-freedom (DOF) used in the proposed method and the classic FEM. It is observed that the proposed method is able to use a much fewer number of DOF than that of commercial FEM software (ANSYS etc) to obtain accurate solutions to both the deformation of the plate and the stress field between adjacent layers.

Composite laminate oriented reliability analysis for fatigue life under non-probabilistic time-dependent method

The problem of composite laminate fatigue assessment under small samples and less probability information is always concentrated, and the mechanical properties of composite structure are deteriorated due to damage accumulation effect of fatigue issue. On the basis of analyzing the current reliability modeling method, this paper developed the reliability evaluation strategy based on the non-probability theory, and considering composite structure response to the external cycle load influence, time-varying reliability analysis model is established, which is different from the quasi-static analysis method, and finally the application for composite structure fatigue problem based on the non-probability of the time-varying reliability analysis is put forward. In the proposed method, every single layer reliability is analyzed combining the non-probabilistic theory with Tsai–Wu failure criterion model, comparing with the traditional probabilistic reliability theory, the method of non-probability is not constrained by probability distribution information, in view of that, the approach has stronger practicability; Introducing for the first-passage conceptions, the crossing rate is evaluated through the non-probabilistic reliability method, finally the time-varying reliability index of composite material structure is established. After analysis steps are given in detail, laminated plate structure is presented to demonstrate the validity and reasonability of the developed methodology.