Antisymmetric Cross-ply Laminates Research Articles

Cross-well dynamics for vibration energy harvesting of base-excited antisymmetric bi-stable laminate with a point being fixed

The cross-well dynamic of the bi-stable laminates triggered by ambient vibration is a typical nonlinear dynamic, the larger deformation makes the bi-stable laminates attract much attention in vibration energy harvesting. To obtain the performance of cross-well dynamics and vibration energy harvesting of the bi-stable laminate theoretically, a 4-layer harmonic base-excited antisymmetric cross-ply bi-stable laminates with d31 type Marco fiber composite (MFC) for vibration energy harvesting is proposed. The bi-stable laminate is fixed at a chosen point where a base excitation is applied as displacement excitation. The first order shear deformation laminated plate theory (FSDT), geometric nonlinear, and the energy principle are employed to obtain the governing equation of the system in terms of the polynomial coefficients of configuration functions. Two stable configurations are calculated by the principle of minimum potential energy, which is used as the initial state for the energy harvesting analysis. The stable configurations are predicted, and evaluated by comparing them with ABAQUS. The cross-well nonlinear dynamic behaviors and the performance of energy harvesting are analyzed. The effects of excitation level, excitation frequency, and the position of the fixed point on the cross-well dynamics and energy harvest characteristics of the antisymmetric bi-stable laminates are investigated in-depth, and some new aspects are revealed. It demonstrates that the base displacement harmonic excitation can induce chaotic cross-well vibration in a high-frequency band when the fixed point is far away from the center point, the various cross-well vibration of the system is more likely to occur in the dominate-frequency resonant band.

Three-dimensional scaled boundary finite element method to study interfacial imperfections in thick laminated composite plates undergoing bi-axial bending

In this work, the scaled boundary finite element method (SBFEM) is employed to model weakly bonded thick laminated composite plates subjected to bi-axial bending. A three-dimensional modelling technique incorporated in the SBFEM framework is used for thick laminated plates, where two-dimensional plate models cease to provide accurate results. Weakly bonded interfaces are modelled using spring layer models with the incorporation of zero-thickness interface elements between the layers. The interface element is subjected to behave in a linearly elastic manner capturing the slippage between the layers. Using SBFEM, it is possible to have fewer discretisation in the through thickness direction of the plate (or individual lamina) without compromising the accuracy of the results. Additionally, the size of interface elements can be kept small without altering the size of the adjoining bulk element. Validation examples covering symmetric, anti-symmetric cross ply and angle ply laminates have been solved using the proposed approach. Results from the proposed method using SBFEM are compared with well-established methods for modelling thick laminated composite under bi-axial bending.

The bending behaviour of antisymmetric cross-ply laminates using high-order shear deformation theories and a Radial Point Interpolation Method

The analysis of composite laminates is often performed using plate models suitable to study, with accuracy, the distribution of the shear stresses across the laminate’s thickness. Several High-Order Shear Deformation Theories (HSDTs) are proposed in the literature since they fulfil the insufficiencies of the First-Order Shear Deformation Theory (FSDT) without the computational cost of approaches such as Layerwise (LW) theories. In this work, five of the most popular HSDTs existing in the literature were used to study the bending of antisymmetric cross-ply laminates. The plate theories were computationally implemented within a meshless method algorithm – the Radial Point Interpolation Method (RPIM). The numerical analysis of engineering problems is commonly performed using the Finite Element Method (FEM), but in problems dealing with transitory geometry, the FEM may not be the most efficient numerical method. Additionally, in meshless methods, the shape functions have virtually a higher order which results in a higher continuity and reproducibility. Meshless methods only require an unstructured nodal distribution to discretize the problem domain, so there is no previous relationship between nodes, which makes the refinement procedure easier than in the FEM.Thus, using the RPIM and HSDTs, different laminates were analysed, which marks the first time in the literature that the RPIM is used to study the bending of antisymmetric cross-ply laminates through HSDTs. In the end, the meshless solutions are compared with analytical and FEM solutions available in the literature, and the accuracy and robustness of the numerical approach are proved.

New shape function in the free-vibration analysis of antisymmetric angleply composite laminates

The paper analyzes the problem of free vibrations in antisymmetric crossply laminates. A new shape function which is used in higher order shear deformation theories has been introduced. The comparative analysis was performed with the known shape functions. The procedure for obtaining dynamic equations of motion in the Matlab software package has been developed. For theoretical considerations, the module with the symbolic variable has been used. The paper shows which of the existing shape functions are applicable in the free vibration analysis of antisymmetric angleply laminates. The advantages and disadvantages of the newly developed shape function are clearly highlighted. Analytical procedures have been used to obtain the results of partial differential equations, based on Navier's solutions. Numerical integration procedures were used as an integral part of the developed Matlab codes for those shape functions where it was necessary. The results are presented in a table and figures. The procedure itself has been verified by comparison with the reference results from the literature.

Analysis of antisymmetric cross-ply laminates using high-order shear deformation theories: a meshless approach

For many years finite element method (FEM) was the chosen numerical method for the analysis of composite structures. However, in the last 20 years, the scientific community has witnessed the birth and development of several meshless methods, which are more flexible and equally accurate numerical methods. The meshless method used in this work is the natural neighbour radial point interpolation method (NNRPIM). In order to discretize the problem domain, the NNRPIM only requires an unstructured nodal distribution. Then, using the Voronoi mathematical concept, it enforces the nodal connectivity and constructs the background integration mesh. The NNRPIM shape functions are constructed using the radial point interpolation technique. In this work, the displacement field of composite laminated plates is defined by high-order shear deformation theories. In the end, several antisymmetric cross-ply laminates were analysed and the NNRPIM solutions were compared with the literature. The obtained results show the efficiency and accuracy of the NNRPIM formulation.

Thermomechanical analysis of antisymmetric laminated reinforced composite plates using a new four variable trigonometric refined plate theory

The thermo-mechanical bending behavior of the antisymmetric cross-ply laminates is examined using a new simple four variable trigonometric plate theory. The proposed theory utilizes a novel displacement field which introduces undetermined integral terms and needs only four variables. The validity of the present model is proved by comparison with solutions available in the literature.

Modelling of laminated composite plates with weakly bonded interfaces using scaled boundary finite element method

Scaled Boundary Finite Element Method (SBFEM), a relatively new approach, is investigated for the analysis of the weakly bonded laminated composite plates, under cylindrical bending. A plane strain formulation, which does not require any assumptions on the displacement or stress field, is developed. The interface between two plies is simulated using a linear spring layer model and incorporated in the numerical modelling with the help of zero thickness interface elements. The model is coded and postprocessed in MATLAB to capture general delamination, i.e. separation of the plies at interfaces in both tangential and normal direction. Due to the semi-analytical solution procedure of SBFEM, the model provides highly accurate results. Use of SBFEM reduces the meshing burden and complexities in the discretisation. Moreover, the method can handle hanging nodes, so one can use small interface elements without decreasing the size of adjoining domains. These qualities make the proposed numerical model computationally cheaper as well as very accurate. The presented formulation is demonstrated for a variety of examples in laminated composites including unidirectional laminates, symmetric cross ply laminates and antisymmetric cross ply laminates. The proposed model's close agreement as well as its superiority over the other methods is established for future reference.

Modeling and analysis of cross-ply and angle-ply laminated plates under patch loads using RBF based meshfree method and new HSDT

Angle-ply laminated plates have been examined for flexure analysis with multiquadric radial basis function (MQRBF) base meshfree method. A new higher order shear deformation theory (HSDT) representing the transverse shear strain function is proposed in the displacement field. Proposed theory satisfies the condition of continuity and differentiability and is accurate enough to predict the results of flexure. Governing differential equations (GDEs) are derived using energy principle. A MATLAB code is developed to obtain the results for deflection and stresses of the angle-ply laminated plate. The flexure results for laminated plates under patch loads commonly used in general practice is obtained with proposed HSDTs and meshfree method. Present results are validated with published results for the cross-ply laminated plate, and some new results are produced for different loading conditions for angle-ply laminated plates. In the present study, eight layered symmetric cross-ply laminate of equal thickness, an antisymmetric cross-ply laminate of equal thickness and a general angle-ply laminate of equal thickness is considered. The effect of the span to thickness ratio, number of layers and shifting patch loads is presented.

A Comparison of Closed-Form and Finite-Element Solutions for the Free Vibration of Hybrid Cross-Ply Laminated Plates

The natural frequencies of hybrid cross-ply laminated plates are predicted using a high-order shear deformation theory and the three-dimensional finite-element analysis. The equations of motion for simply supported laminated hybrid rectangular plates are derived using the Hamilton principle. Closed-form solutions for antisymmetric cross-ply and angle-ply laminates are found employing the Navier solution. In the finite-element method, eight-node linear interpolation brick elements are used to model the composite plates. First, the analytical and numerical results are validated for an antisymmetric cross-ply square laminate by results available in the literature. Then, the effects of side-to-thickness ratio, aspect ratio, lamination schemes, and material properties on the fundamental frequencies for simply supported carbon/glass hybrid composite plates are investigated. Since no data are available in the literature for hybrid composite plates, the finite-element solution is used for comparison purposes. A comparison of the analytical solution with the corresponding 3D finite-element simulations shows a good accuracy of the proposed analytical solution in predicting the fundamental frequencies of hybrid cross-ply laminated plates.

Dynamic analysis for anti-symmetric cross-ply and angle-ply laminates for simply supported thick hybrid rectangular plates

In this paper, static and vibration analysis for anti-symmetric cross-ply and angle- ply carbon/glass hybrid laminates rectangular composite plate are presented. In this analysis, the equations of motion for simply supported thick laminated hybrid rectangular plates are derived and obtained through the use of Hamilton\'s principle. The closed-form solutions of anti-symmetric cross-ply and angle- ply laminates are obtained using Navier solution. The effects of side-to-thickness ratio, aspect ratio, and lamination schemes on the fundamental frequencies loads are investigated. The study concludes that shear deformation laminate theories accurately predict the behavior of composite laminates, whereas the classical laminate theory over predicts natural frequencies. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. It can be concluded that the proposed theory is accurate and simple in solving the free vibration behaviors of anti-symmetric cross-ply and angle- ply hybrid laminated composite plates.

Buckling analysis of anti-symmetric cross and angle-ply laminated composite plates

This article has for objective to analyze the buckling behavior of the unidirectional laminated plates. In this purpose, we propose an analytically method, based on the theory of classical, orthotropic plate theory. The governing equations are solved using Navier solution for uniform uniaxial loading in longitudinal direction. We were interested to identify the critical buckling load for simply supported antisymmetric cross-ply and antisymmetric angle-ply laminates of rectangular shape. Some important progress has been made on these relatively complicated buckling problems, involving coupling between bending and midplane stretching during a buckling deformation. Effects of different parameters such as fiber orientation angles, aspect ratio, modular ratio and number of layers were examined. Results are presented in the form of plots showing the variation in non-dimensional buckling load.

Free vibrations of laminated composite plates using a novel four variable refined plate theory

In this research, the free vibration response of laminated composite plates is investigated using a novel and simple higher order shear deformation plate theory. The model considers a non-linear distribution of the transverse shear strains, and verifies the zero traction boundary conditions on the surfaces of the plate without introducing shear correction coefficient. The developed kinematic uses undetermined integral terms with only four unknowns. Equations of motion are obtained from the Hamilton's principle and the Navier method is used to determine the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical examples studied using the present formulation is compared with three-dimensional elasticity solutions and those calculated using the first-order and the other higher-order theories. It can be concluded that the present model is not only accurate but also efficient and simple in studying the free vibration response of laminated composite plates.

Bending and free vibration analyses of antisymmetrically laminated carbon nanotube-reinforced functionally graded plates

In this paper, a simple four-variable first-order shear deformation theory is further applied to solve the bending and free vibration problems of antisymmetrically laminated functionally graded carbon nanotube (FG-CNT)-reinforced composite plates. The adopted four-variable theory contains only four unknowns in its displacement field which is less than the Reddy’s first-order theory. The equations of motion are derived from the Hamilton’s principle with the help of specific boundary conditions. Laminated FG-CNT-reinforced plates with different distribution types of carbon nanotube through the thickness are considered. The material properties of individual layer are estimated by using the extended rule of mixture. Analytical solutions of various simply supported antisymmetric cross-ply and angle-ply laminates are given for case study. The effects of carbon nanotube volume fraction, length to width ratio and thickness to width ratio on the non-dimensional fundamental frequency and the central deflection are investigated for antisymmetrically laminated FG-CNT-reinforced plates.

A novel four variable refined plate theory for laminated composite plates

A novel four variable refined plate theory is proposed in this work for laminated composite plates. The theory considers a parabolic distribution of the transverse shear strains, and respects the zero traction boundary conditions on the surfaces of the plate without employing shear correction coefficient. The displacement field is based on a novel kinematic in which the undetermined integral terms are used, and only four unknowns are involved. The analytical solutions of antisymmetric cross-ply and angle-ply laminates are determined via Navier technique. The obtained results from the present model are compared with three-dimensional elasticity solutions and results of the first-order and the other higher-order theories reported in the literature. It can be concluded that the developed theory is accurate and simple in investigating the bending and buckling responses of laminated composite plates.

Buckling Analysis of Anti-Symmetric Cross-Ply Laminated Composite Plates Under Different Boundary Conditions

The buckling analysis of anti-symmetric cross-ply laminated composite plates under different boundary conditions is examined by using a refined higher order exponential shear deformation theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In this investigation, the equations of motion for simply supported thick laminated rectangular plates are derived and obtained through the use of Hamilton’s principle. The closed-form solutions of anti-symmetric cross-ply and angle-ply laminates are obtained using Navier solution. Numerical results for critical buckling loads anti-symmetric cross-ply laminated composite plates are presented. The validity of the present study is demonstrated by comparison with other higher-order solutions reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the buckling behaviors of anti-symmetric cross-ply laminated composite plates under different boundary conditions

Free vibration analysis of multilayered composite and soft core sandwich plates resting on Winkler–Pasternak foundations

Free vibration of laminated composite and soft core sandwich plates resting on Winkler–Pasternak foundations using four-variable refined plate theory are presented. The theory accounts for the hyperbolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the dynamic version of the principle of virtual work. Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply, angle-ply, and soft core laminates or soft core sandwich plates resting on elastic foundations. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate, but also efficient in predicting the natural frequencies of laminated composite and soft core sandwich plates resting on Winkler–Pasternak foundations.

Non-linear dynamic instability analysis of laminated composite cylindrical shells subjected to periodic axial loads

The dynamic instability of thin laminated composite cylindrical shells subjected to harmonic axial loading is investigated in the present work based on nonlinear analysis. The equations of motion are developed using Donnell’s shallow-shell theory and with von Karman-type of nonlinearity. The nonlinear large deflection shallow-shell equation of motions are solved by using Galerkin’s technique that leads to a system of nonlinear Mathieu–Hill equations. Both stable and unstable solutions amplitude of the steady-state vibrations are obtained by applying the Bolotin’s method. The nonlinear dynamic stability characteristics of both symmetric and antisymmetric cross-ply laminates with different lamination schemes are examined. A detailed parametric study is conducted to examine and compare the effects of the magnitude of both tensile and compressive axial loads, aspect ratios of the shell including length-to-radius and thickness-to-radius ratios, and different circumferential wave numbers as well on the parametric resonance particularly the steady-state vibrations amplitude. The present results show good agreement with that available in the literature.

Bounds on the natural frequencies of laminated rectangular plates with Extension–Bending coupling

Extension–Bending coupling behaviour is believed to arise uniquely in anti-symmetric Cross-ply laminates. The results in this article serve to dispel this misconception by presenting solutions for both Cross-ply laminates and Standard laminates, defined as containing combinations of angle plies and cross plies, for a range of different sub-sequence symmetries. Details of the algorithm used to develop the definitive list of laminate stacking sequences with up to 21 plies are presented. Finally, natural frequency factor envelopes are developed, which identify significant differences in the bounds, across a range of aspect-ratios, between Standard laminate configurations and the previously assumed Cross-ply designs.

Analytical solution to bending of stiffened and continuous antisymmetric laminates

Abstract Analytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

Localization of buckling modes in plates and laminates

We use the Mindlin plate theory and the finite element method to delineate the effect of fixing points on a transverse normal to the mid-surface of the plate on the localization of buckling modes in clamped–clamped rectangular plates made of linear elastic, homogeneous and either isotropic (monolithic) or orthotropic (fiber-reinforced composite) materials. The in-plane loads considered on the bounding edges are: (i) normal tractions on the length, (ii) normal tractions on the width, (iii) equal normal tractions on the length and the width (equal biaxial loading), (iv) shear (tangential), and (v) combined same normal and shear tractions on all sides. It is found that clamping points on a transverse normal passing through the mid-point of a line parallel to the short side increases the critical buckling load of plates of only low aspect ratios over that of the corresponding plates unconstrained at interior points. However, for plates of all aspect ratios (thickness/length) fixing points on a transverse normal divides it into two regions with negligible transverse deflections in only one of the two regions. Only for loads (i)–(iii) the dividing line is parallel to the short side of the plate. For both thin and thick isotropic plates the slope of the dividing line is found to monotonically increase with an increase in the aspect ratio of a plate until it reaches a saturation value. A parameter based on the modal strain energy is used to quantify the degree of localization of a buckling mode. For an isotropic plate the degree of localization is found to increase with the increase in the aspect ratio for load cases (i)–(iii) but is found to be moderate for load cases (iv) and (v). For an orthotropic layer the degree of localization with an increase in the aspect ratio of the plate increases more for the 90° lamina than that for the 0° and the 45° laminae. Also, the mode localization in the (45°,−45°) laminate is stronger than that in the (0°,90°) laminate for the five load cases. However, moderate degree of mode localization is found in symmetric and anti-symmetric cross-ply and angle-ply laminates.