Stiffness Of Composite Laminates Research Articles

Conical-shaped variable stiffness composite laminates: Design and fiber path planning

This study presents an innovative design methodology for conical-shaped variable stiffness composite laminates. It employs the spectral Chebyshev method to solve dynamic equations, optimizing lamination parameters to maximize fundamental frequency. Subsequently, the fiber angles at the sampling points across the domain are retrieved. In the final step, a custom normalized cut segmentation approach is proposed to partition the domain into clusters and delineate fiber paths based on discrete fiber angles, while satisfying the manufacturing constraints such as curvature, gaps, and overlaps. The effectiveness of the proposed design methodology is demonstrated through several case studies, where maximum enhancement in fundamental frequency for the same part weight is compared to optimized constant stiffness composite laminates. The results show that an enhancement of 16% is achievable for a variable stiffness laminate disregarding manufacturing constraints for the fully clamped case study. However, the improvements reduce to 12.6% and 14% (deviating from the ideal optimal fundamental frequency by 4.5% and 3.3%), respectively, once the manufacturable constant and curvilinear fibers in each cluster are computed. This work can lead significant advances of engineering applications by enabling the design of variable stiffness laminates with complex geometries improving mechanical performance and/or reducing structural weight compared to conventional laminates.

A geometrically nonlinear Hellinger–Reissner shell element for the postbuckling analysis of variable stiffness composite laminate structures

Variable stiffness (VS) composite laminates provide larger freedom to design thin-walled structures than constant stiffness (CS) composite laminates. They showed to allow the redistributing of stresses, improving buckling and post-buckling performance and, therefore, reducing material weight and costs. This work extends a recently developed mixed shell element, MISS-4C, to the postbuckling analysis of VS composite laminate structures. MISS-4C has a linear elastic closed-form solution for the stress interpolation of symmetric composite materials. Its stress field interpolation is obtained by the minimum number of parameters, making it an isostatic element. Moreover, its kinematic is only assumed along its contour, leading to an efficient evaluation of all operators obtained through analytical integration along the element contour. MISS-4C uses a corotational approach within a fast multi-modal Koiter algorithm to efficiently obtain the initial post-buckling response of VS composite laminate structures.First, the element performance is investigated by analysing a carbon fibre VS composite laminate plate subjected to compressive stresses. Numerical results obtained with MISS-4C are compared with those obtained with the MISS-4 element, showing good accuracy and a high convergence rate. Subsequently, the structural response of a glass fibre VS composite laminate girder of a short-length bridge is optimised through a multi-objective optimisation that exploits the robustness of the MISS-4C element and the efficiency of the multi-modal Koiter algorithm.

Open loop flutter control of optimally oriented smart variable stiffness plates under hygrothermal environment

The present study investigates the impact of hygrothermal effects on the free vibration and flutter characteristics of a smart variable stiffness composite laminate (VSCL). This research seeks to underscore the inherent advantages of VSCLs in contrast to conventional straight fibre constant stiffness composite laminates (CSCLs). Utilizing curvilinear fibres, VSCLs offer a heightened level of customization, culminating in notably enhanced structural performance. The Genetic Algorithm is employed to obtain optimal fibre paths for the variable stiffness laminates. Introducing an active fibre composite layer as an actuator atop the optimally oriented VSCL laminate enhances its capabilities. A parametric inquiry into the influence of hygrothermal loading on the smart structure's free vibration and flutter traits is undertaken. While hygrothermal loading imparts vulnerability to the laminate, the introduction of voltage to the active fibre composite layer counteracts this weakening effect, effectively augmenting stiffness and concurrently enhancing the flutter parameters (velocity, frequency) intrinsic to the plate structure.

Bending performance analysis of variable stiffness composite laminates

AbstractFor fiber‐reinforced polymer structures, the out‐of‐plane stiffness is much lower than the in‐plane stiffness and is extremely prone to damage and instability. The potential of variable stiffness (VS) laminates was explored to enhance the stiffness in the thickness direction. Based on the finite strip method (FSM), a semi‐analytical solution for bending problems of laminates was derived, which is called the variable‐stiffness discrete strip (VSDS) method. By comparing with traditional constant stiffness (CS) laminates, the influence of design parameters of VS laminates on the bending shape and stress distribution of laminates under two load cases were studied. Combining the results of FE models, the accuracy of VSDS has been verified, and the strategy of VS layout configuration design to improve the bending resistance was proved.Highlights Variable stiffness discrete strips (VSDS) method was presented based on FSM. Bending performances of VS laminates under two load conditions were studied. The method has the ability to reduce the computation cost of bending problems. The solutions of VSDS agree well with the results of FE models. The improvement mechanism of VS laminates on bending resistance was explored.

Design of manufacturable variable stiffness composite laminates using spectral Chebyshev and normalized cut segmentation methods

The in-plane fiber orientations of variable stiffness (VS) laminates can be tailored to achieve enhanced structural properties compared to conventional constant stiffness (CS) laminates. However, VS laminate manufacturing faces challenges such as wrinkles, gaps, and overlaps. To address these challenges, we present a novel three-step design methodology. First, the laminate is modeled using lamination parameters (LPs) and the spectral Chebyshev method, and the optimal LPs are determined to maximize the fundamental frequency. Then, the discrete fiber angles are retrieved using the optimal LP distribution. Lastly, a normalized-cut segmentation method is applied to divide the domain into clusters and to generate manufacturable curvilinear fiber paths. Case studies focusing on designing clusters containing both straight and curvilinear fiber paths demonstrate that the designed VS composites can significantly enhance the dynamic performance with up to 20% enhancement in the fundamental frequency compared to CS laminates, under fully clamped boundary conditions with manufacturing constraints.

Buckling optimization of variable-stiffness composite plates via variable stiffness optimization algorithm

A variable stiffness optimization (VSO) algorithm is presented for optimizing variable-stiffness composite (VSC) plates with linear fiber path functions. A new definition of lamination parameters characterizing the distances between stacking configurations of VSC plates is presented. Using the inherent sensitivity of bending stiffness of composite laminates, the concept of “through thickness design” is introduced. The first step is to determine a good initial point for the VSC plates using three-dimensional sampling optimization (3DSO). The fiber angle design variables are then categorized into three groups. By optimizing the three groups' design variables sequentially and iteratively, the stiffness of the VSC plates is stiffened part by part until an optimum is reached. Lastly, the obtained optimum is redesigned accounting for the curvature constraint. The finite element method (FEM) is developed for the buckling analysis of VSC plates, and both Q4 and Q8 elements are employed to verify the accuracy and convergence of the FEM. Under a variety of boundary conditions and loading cases, FEM and VSO algorithm are used to maximize the buckling load of square and rectangular symmetrical VSC plates. Optimal results are compared with those available in the literature, demonstrating the effectiveness, robustness, and efficiency of the VSO algorithm.

Collaborative optimization for variable stiffness composite laminates using a fiber angle description method based on Archimedean spiral function

When optimizing the fiber orientation of multilayer composite laminates, with the increasing number of layers, the number of design variables increases sharply, resulting in a large amount of computational cost. To address this challenge, this paper proposes a novel discrete fiber angle optimization method based on the Archimedean spiral function and applies it to a collaborative design framework of topology and fiber orientation. The proposed method uses the normal distribution function as the angle selection function in every layer. To prevent convergence issues in optimizing the fiber angle, a new candidate angle weighting formula is proposed. Further, a discrete fiber angle parameterization method based on the Archimedean spiral function for the thickness direction of laminates is presented, which requires only one variable to represent the fiber angle of any two layers in the element. To circumvent the issue of local optima and expand the design space, a collaborative optimization strategy is employed to improve the discrete fiber angle optimization results. Finally, the numerical examples indicate that in comparison to conventional approaches, the correlation of the proposed method with initial values is significantly reduced (σ2=0.00188). Under identical initial conditions, this method can improve the structural stiffness by over 20% at maximum.

Maximisation of Bending and Membrane Frequencies of Vibration of Variable Stiffness Composite Laminated Plates by a Genetic Algorithm

PurposeIn comparison to traditional, constant stiffness laminates, variable stiffness composite laminates (VSCL) with curvilinear fibres represent an extra analysis effort. It is the purpose of this work to present and test a relatively simple optimisation procedure, in order to find the maximum fundamental frequency of vibration in bending and in in-plane vibrations. It is also intended to explain why certain fibre paths lead to higher frequencies.MethodsThe optimisation is performed using a genetic algorithm (GA), which is described in detail. The bending and the in-plane plate models are based on the p-version Finite Element Method. Each model requires a small number of degrees of freedom, an important feature because applying the GA involves the solution of a large number of eigenvalue problems. In order to support the physical interpretation of the optimal designs, mode shapes and stress fields corresponding to some optimal solutions are illustrated.ResultsSingle- and multi-layer plates with different boundary conditions and fibre path types are studied. Fibre paths that lead to maximum fundamental frequencies are found and justified. The consequences that maximising the first frequency has on the higher-order modes of vibration are studied.ConclusionThe proposed optimisation and modelling methods are effective. Curvilinear fibres with the characteristics considered led to the maximum first natural frequency of vibration in a few cases, but not all. Particularly in in-plane vibrations, curvilinear fibres can provide major gains in comparison to straight fibres. The increase in the vibration frequency is accompanied by, overall, larger stresses.

Supersonic flutter of variable stiffness circular cylindrical shells

Curvilinear reinforcement fibres allow to tailor the elastic properties of composite laminates in space, leading to Variable Stiffness Composite Laminates (VSCL). In this work, the aeroelastic stability of VSCL cylindrical shells under a supersonic airflow along the axial direction is analysed in order to verify if curvilinear fibres can be used to increase the flutter speed. A linear mathematical model for thin circular cylindrical VSCL shells is developed for that purpose. The aerodynamic pressure caused by the supersonic airflow is modelled recurring to the linearised piston theory. The p-version of the finite element method is used to transform the differential equations of motion from partial to ordinary. Using the latter, eigenvalue problems are defined to obtain the natural frequencies and mode shapes of vibration, and the critical flutter conditions. Various types of curvilinear fibre paths are explored, to find paths that result in higher values of critical flutter free stream pressure. The values are compared with the ones of conventional, constant stiffness, composite laminated shells.

Experimental study on the influence of lamination parameters on the vibration characteristics of variable stiffness composite laminates

In this study, the vibration characteristics of composite laminated plates with variable stiffness were studied using a modal analysis experiment based on the hammering method and a free vibration attenuation experiment. Twenty-five variable stiffness laminates fabricated from carbon fiber-reinforced resin matrix composite prepregs were manufactured via an automated fiber placement process. By varying the parameters, such as the fiber angle (<0°|15°>, <15°|30°>, <30°|45°>, <45°|60°>, <60°|75°>, <74°|89°>), curve ply ratio (0, 0.5, 1), and the number of layers (8, 12, 16), these laminates were tested in a dynamic signal analyzer to obtain their modal frequency and modal damping. These parameters were observed to significantly affect the modal parameters. As experimental investigations on the vibration characteristics of variable stiffness composite laminates are very rare, this study can be considered a reference for future work.

Analytical modeling of panel flutter and active control in supersonic variable stiffness composite laminates

This work focuses on panel flutter and active control in supersonic variable stiffness composite laminates, making progress on the analytical modeling and combined exploration of curvilinear fiber composites tailoring and piezoelectric sensors/actuators, as promising structural design technologies, for aeroelastic control. The Classical Laminated Plate Theory and the First-Order Piston Theory are used as structural and aerodynamic models, respectively. Flutter analyses are carried out for simply supported plates, either purely elastic laminates or piezoelectric composite laminates. The tailor-ability of curvilinear fiber composites and the effect of proportional control are discussed. Ultimately, the presented results provide a comprehensive benchmark for future assessments.

Multi-level intelligent design of variable angle tow laminates via image-driven method

Compared with the traditional design of constant stiffness composite laminates, the fiber variable angle-tow (VAT) laminates as variable stiffness composites have more flexible design space. In order to obtain continuous fiber paths and high-precision analysis model, the design of VAT requires more detail features, yet the large number of design variables presents a significant obstacle to conventional optimization methods. In this study, an image-driven intelligent optimization method is proposed. The VAT fiber angle vector field is accurately described by the B-spline surface with different control points, and is mapped into a gray image with fixed size. Through the isogeometric analysis (IGA) method, the buckling load is gained as the label of the images, and the dataset is obtained to train convolutional neural networks (CNNs) for deep learning models. Then, a multi-level refinement of the local key design space effectively optimizes the VAT laminates while taking into account manufacturing constraints. Due to the images providing as a single unit of input data, the CNNs model can be applied in variable design space without resampling. Three numerical examples demonstrate the effectiveness, and obtain higher buckling load of VAT laminates in different level design spaces.

Nonlinear flutter of composite laminates with curvilinear fibres using a full linearized aerodynamic theory

This paper studies nonlinear flutter of rectangular variable stiffness composite laminates (VSCLs) subjected to a low supersonic airflow. The VSCLs are formed by different individual laminae where a polymeric matrix is reinforced by curvilinear carbon fibres. The plate’s displacement field follows a Third-order Shear Deformation Theory (TSDT); a p-version finite element is employed to discretize the displacement components. A full linearized (inviscid, potential flow) aerodynamic theory is used to approximate the aerodynamic pressure. The self-exciting vibrational model is formed using the principle of virtual work. The equations of motion representing the model are solved using Newmark method. Limit cycle oscillations (LCOs) of these plates are studied using time histories, phase-plane plots and FFT spectra. A damage onset criterion is utilized to control the structural health of the laminates during nonlinear flutter. Effects of curvilinear fibre path angles on linear and nonlinear flutter behaviour, as well as the damage onset in VSCLs are investigated.

Assessment of layerwise user-elements in Abaqus for static and free vibration analysis of variable stiffness composite laminates

In this work, user-elements (UEL) in Abaqus are taken a step forward into the high-order layerwise modelling of variable stiffness composite laminates with curvilinear fibre paths, extending the limited number of available literature on refined multilayered UEL models. Two layerwise UEL models with three discrete layers are here proposed, assigning to each layer the displacements of the first- and third-order shear deformation theories, thus named UEL1 and UEL3, respectively. A complete assessment of the models predictive capabilities is carried out by a comparison with available static and free vibration solutions in the literature – either for constant or variable stiffness laminates – considering various boundary and loading conditions, as well as thin and moderately thick plates. Numerical results demonstrate that the developed models are capable to render fairly accurate and computationally efficient results, with particular emphasis on the higher-order model for predicting the global–local response behaviour of moderately thick plates.

Variable-kinematic finite elements for the aero-thermo-elastic analysis of variable stiffness composite laminates

This paper explores the aero-thermo-elastic properties of variable stiffness composite laminates (VSCLs) with fibers steered along curved trajectories using a refined 1 D model. High-order solutions are generated by the p-version finite element method in conjunction with the Carrera Unified Formulation (CUF), which is applied to structural modeling by employing the improved hierarchical Legendre expansion (IHLE) in particular. The merit of using such a kind of expansion lies in that both the Equivalent Single Layer (ESL) and Layer-Wise (LW) theories can be formulated in a compact and straightforward manner. The linear piston theory is utilized to model the aerodynamic force, whereas the steady-state temperature field is applied during the flutter analysis. The weak-form governing equations are derived by taking a perturbation of the equilibrium configuration, and divergence and flutter instabilities are determined by solving the related eigenvalue problem. The accuracy of the proposed model has been verified through several test cases involving flutter, thermal buckling, and vibration of symmetric and antisymmetric VSCLs. The obtained results were compared to those obtained using the ABAQUS software and the available literature. Finally, the effect of the kinematic order, shear coefficient, and temperature variation on the critical dynamic pressure of VSCLs is thoroughly investigated.

A Three-Dimensional Equivalent Stiffness Model of Composite Laminates with Wrinkle Defects.

The stiffness of composite laminates is easily affected by wrinkle defects. In this paper, a new effective analytical model was proposed to predict the three-dimensional equivalent elastic properties of multidirectional composite laminates with wrinkle defects. Firstly, a geometric model was established according to the microscopic characteristics of wrinkle defects. Then, based on the classical laminate theory and homogenization method, the constitutive equation and flexibility matrix of the wrinkle region were established. Finally, the equivalent stiffness parameters of unidirectional and multidirectional laminates were derived, and the effects of different wrinkle parameters and ply-stacking sequences on the stiffness of unidirectional and multidirectional laminates were studied by using the analytical model. The results show that the mechanical properties of the lamina and laminates are affected by the out-of-plane angle and in-plane angle of the wrinkle defects. The accuracy of the analytical model has been verified by the numerical model and other theoretical models, and it has the characteristics of few parameters and a high efficiency. The analytical model can be used to predict the stiffness of composite structures with wrinkle defects simply, effectively, and quantitatively. It can also be used as a tool to provide the mechanical response information of laminates with wrinkle defects.

Non-linear forced vibrations of variable stiffness plates on elastic supports

The effect of boundary conditions on the vibrations of variable stiffness composite laminates (VSCL) with curvilinear fibers is particularly interesting because fiber orientations at the plate edges and in the plate’s domain can differ. Since real boundaries are not the ideal limit cases usually assumed in the literature, a more correct model can be achieved by representing boundaries as elastically restrained edges. This approach is followed in this work, which investigates the combined effect of fiber orientation and boundary stiffness on the geometrically non-linear forced vibrations of rectangular VSCL plates. The boundary stiffness is adjusted using experimental data.

Finite strip method based on Carrera unified formulation for the free vibration analysis of variable stiffness composite laminates

AbstractThis work presents a refinement of the finite strip method (FSM) based on the Carrera unified formulation (CUF) and is applied to free vibration analysis of variable stiffness composite laminated (VSCL) plates. VSCL plates considered here are composed of layers with curvilinear fibers in which their orientation angle changes linearly with respect to one of the in‐plane coordinates. The FSM permits the plate to be divided into some finite strips connected along the so‐called nodal lines. The conventional finite strip method involves an approximation of the solution of a plate problem by using continuously harmonic series in the longitudinal direction along the nodal lines and simple polynomial shape functions (the usual beam shape functions) in the transverse direction of the plate. However, refined finite strip model that presented in this paper allows one to express the unknown variables as a set of thickness functions that only depend on the thickness coordinate and the corresponding variable that depends on the in‐plane coordinates which involve continuously harmonic series and polynomial shape functions. Thanks to this new finite strip model, the shape functions used in the transverse direction can be chosen as 1D bar shape functions, and also, all three components of the displacement field are considered in each nodal line. Moreover, based on this approach, three‐dimensional displacement field is approximated in a compact form as a generic N‐order expansion. Therefore, explicit expressions of fundamental nuclei of finite strip matrices are obtained in a compact form and presented here. The results obtained by the present formulation for VSCL plates are compared with those obtained from published data. The results show that this kind of strip is efficient and useful. Further, several numerical examples are presented, and the effect of the order of expansion, the number of strips, curvilinear fiber angle, and various boundary conditions are examined.

Simultaneous topology and fiber path optimization of composite structures with MAC constraints

The pursuit of stiffer lightweight structures has led to the development of new concepts such as the Variable Stiffness Composite Laminates (VSCL), which further enhance the design potential of these highly tailored materials. As the number of design variables rise, more robust topology optimization formulations are necessary to achieve optimal solutions. In this work, a design optimization framework is developed by assigning material and fiber orientation on discrete patches of the VSCL structures. A gradient-based optimization method is used in the framework by applying the Discrete Material Optimization (DMO) formulation. The main goal is to design structures to meet dynamic performance objectives, with particular emphasis on the synthesis of modes with a target design. Accordingly, the objective function depends on the eigenvalues (natural frequencies), while one set of constraints depends on the eigenvectors (mode shapes). The Modal Assurance Criterion (MAC) is simultaneously used to perform mode tracking between the different design iterations and to evaluate the correlation between the synthetized modes and the pre-defined objective shapes. In this study, the calculation of the weight derivatives is also revised for the DMO formulation to show the contribution of the missing terms in the final optimal results. The optimization framework is examined for some potential applications related to dynamic decoupling and dynamic synthesis. It is shown that the design approach tailors the dynamic response of the composite structure and it can effectively be used to design resonators based on a reference model.

Nonlinear dynamic analysis of sandwich shell panels with auxetic honeycomb core and curvilinear fibre reinforced facesheets

The nonlinear dynamic behaviour of sandwich shell panels having auxetic honeycomb core and facesheets composed of composite material with curvilinear fibres is investigated for the first time. A finite element study is performed considering a third-order shear deformation theory incorporating Murakami zig-zag effects for the analysis of the variable stiffness composite laminate (VSCL) auxetic honeycomb sandwich panels. A nine-noded C0 isoparametric quadratic shell element with eleven degrees of freedom per node is considered and von Kármán nonlinear strain-displacement equations are used to include the geometrical nonlinearity. The linear free vibration and nonlinear transient response of the panels are obtained using the eigenvalue solution and Newmark's time integration scheme, respectively. Most of the auxetic sandwich structures considered in the literature are made up of aluminium metal and therefore, the authors explored the viability of the VSCL based auxetic sandwich panels. A thorough numerical study including the effects of the geometrical parameters of the honeycomb unit cell, curvilinear fibre path angles and boundary conditions on the nonlinear dynamic response of auxetic sandwich panel with negative Poisson's ratio is presented.