Stiffness Of Composite Laminates Research Articles

Curvature-controlled trajectory planning for variable stiffness composite laminates

Variable-stiffness laminates can redistribute the applied load and increase critical buckling loads compared to traditional straight fiber laminates. To take full advantage of fiber reinforced composite materials, a practical trajectory planning method is generated based on the maximum principle stress vector field. A reference path is represented as a blend curve of a sequence of uniform cubic B-spline segments passing through some given maxi-stress points. Based on the local-support property of each B-spline segment, subsequent paths within single lamina can be easily obtained by shifting the reference path along a specific direction. A fast localized curvature-correction algorithm is proposed to control the curvatures of the reference path and strictly constrain the void gap or overlap in a variable stiffness lamina. This trajectory planning method takes the requirement of automated fiber placement machines into account, and improves the mechanical properties of the variable stiffness composite laminates by decreasing the occurrence of gap-errors, such as buckling and wrinkling between adjacent paths. A practical case of variable stiffness trajectory planning is provided to demonstrate the feasibility and efficiency of the proposed method. In this practical case, the gap-error rate has decreased from 45.8% to 4.2%.

Aeroelastic panel flutter optimization of tow-steered variable stiffness composite laminated plates using isogeometric analysis

The optimization of linear flutter behavior of the thin plates made from tow-steered fiber-reinforced laminates is investigated. In the tow-steered variable stiffness composite laminates, the fiber reinforcements are assumed to follow prescribed curvilinear path orientations according to a linear function of geometry longitude. The effect of location-dependent fiber reinforcements is taken into account in the integration procedures. A stream is considered to travel over the panel area in different heading directions. The flutter response of the laminated panel is then improved by adjusting the fiber placement properties through applying a genetic algorithm optimization procedure. The isogeometric analysis method based on the cubic nonuniform rational B-splines is developed. The accuracy and reliability of the optimization process is shown through some representative comparisons. The effect of changes in the flow direction, panel edge constraints, and layup characteristics are inspected.

Stability of variable stiffness composite laminates under compressive and shearing follower forces

Dynamic stability characteristics of variable stiffness composite rectangular panels subjected to non-conservative compressive or shear follower loads are investigated here using the shear-deformable finite element method. At the beginning, the veering of modal frequencies are studied by the eigenvalue analysis to classify the divergence and flutter type of instabilities and evaluate the critical loads for isotropic and variable stiffness composite plates under both types of follower forces. Thereafter, the governing finite element equations are reduced into a set of Mathieu-Hill equations employing the modal transformation technique and the method of multiple scales is used to determine the regions of instability of composite laminates under pulsating follower forces. Limited numerical examples are presented to demonstrate the simple and combination (additive/difference) type of parametric instabilities of edge supported and cantilever panels with curvilinear fibers under compressive or shear pulsating follower forces.

Analysis of variable angle tow composites structures using variable kinematic models

This work presents a refined beam model based on the Carrera Unified Formulation for the free-vibration analyses of variable stiffness composite laminate characterized by layers with curvilinear fibers. These models introduce a refined kinematic description over the cross-section to obtain a 3D displacement field. Taylor and Lagrange polynomials have been used to describe the cross-sectional variables that is, equivalent single layer and layer-wise approaches have been considered. Variable stiffness composite materials have been introduced considering a continuous variation of the lamination angle thanks to an ad hoc integration scheme. Extensive validation of the models has been performed including convergence analyses and comparisons with commercial codes. The results obtained using the present models have been compared with those from open literature considering composites with different values of thickness, lamination, and boundary conditions. The results show that the use of refined models is mandatory for the analysis of such structures where complex laminations may create strong mechanical couplings.

Free vibration analysis of variable stiffness composite laminate (VSCL) plates coupled with fluid

In this article, natural frequencies of variable stiffness composite laminate (VSCL) plates with curvilinear fibers submerged in fluid with different depths of submersion are studied. In each layer of VSCL plate, the fiber-orientation angle changes linearly with respect to the horizontal coordinate. The problem is solved by using the hierarchical finite element method (HFEM) (p-version), with seven degrees of freedom per node and the higher-order shear deformation theory (HSDT) for C0 continuity. The plate theory ensures a zero shear–stress condition at the top and bottom surfaces of plate and does not require a shear correction factor. The fluid/structure interaction is examined in this study, in which the effect of fluid is assumed as pressure acting on the plate. The mathematical model of fluid is given by employing velocity potential and Bernoulli equation. The obtained frequency parameters shown an excellent agreement when compared with experimental and analytical results of available solutions in literature. Accurate results are obtained by increasing the degree of polynomial shape function of HFEM. In order to examine the effect of fluid on VSCL plate, different cases are studied to find out the effect of plate thickness, curvilinear fiber angle and depth of submerged plate.

Modeling crack propagation in variable stiffness composite laminates using the phase field method

The fracture processes of the constant/variable stiffness composite laminate (CSCL/VSCL) subjected to uni-axial tension is investigated using the phase-field method. The main objectives of this work are: (a) to study the crack propagation path and (b) to investigate the influence of the fiber-matrix interface on the crack path. In particular, the emphasis is to explore the role played by the orientation of the fiber and the inter-fiber spacing on the fracture pattern in the composite. From the numerical study, it is inferred that: (a) the cracking pattern of the composite is significantly influenced by the orientation of the fiber and (b) the load carrying capacity of the composite enhances with the increase in the orientation of the fiber. For a larger inter-fiber spacing, the load-displacement curve does not alter significantly for different orientations of the fiber. Furthermore, a novel approach to predict the load-displacement curve of the composite using an equivalent homogenized orthotropic material is presented, showing a very good agreement between them. It is seen that the location of the crack in the VSCL influences the failure load.

Cost analysis of variable stiffness composite structures with application to a wind turbine blade

This paper presents a design framework for cost analysis of a wind turbine blade made of variable stiffness composite laminates. The framework consists of a design optimization, time-variant reliability analysis, structural performance analysis, and life-cycle cost evaluation phases. The objective of the optimization phase is the maximization of the stiffness by searching the optimum values of the design variables. The design variables are piecewise patch orientations and material properties of the fiber reinforced composites. Different volume constraints of carbon fiber reinforced polymer (CFRP) are imposed on composite laminates in the load-carrying component. Next, the structural performance and the service lifetime of the blade designs are estimated based on the time-variant reliability assessment. The time-variant reliability is evaluated using an outcrossing asymptotic method. The wind speed and the material properties are considered, respectively as the random process and the random parameters during the reliability assessment. The maintenance cost of the blade designs is determined by the combination of the estimated structural performance and an analytical method. Finally, the final designs are selected according to the cost-effectiveness values of the designs under the different discount rates and the undiscounted costs.

Aeroelasticity of composite plates with curvilinear fibres in supersonic flow

The target of this investigation is to understand the aeroelastic (dynamic or static) instability of variable stiffness composite laminates (VSCLs) in the presence of supersonic airflow. The studied VSCLs have curvilinear fibre paths. Two different types of VSCLs are considered: the first type (already adopted in several previous researches) includes fibre path angles changing linearly from T1 at right and left edges to T0 at the centre, and the second type (not introduced before) includes fibre paths where their angles change linearly from T0 at the left edge to T1 at the right edge. Displacements and rotations in the plate are modelled by a Third-order Shear Deformation Theory (TSDT) and are discretised by a p-version finite element model. Aerodynamic forces due to supersonic airflow, in the steady or unsteady regime, are modelled using linear Piston theory and the equations of motion of the self-excited vibration system are formed using the principle of virtual work. Dynamic (flutter) and static (divergence) instabilities are identified using the eigenvalues of the linear system. Effects of different boundary conditions and various fibre angles as well as the influence of airflow direction on the flutter and divergence occurrence in VSCL plates are studied.

A multi-parameter model for stiffness prediction of composite laminates with out-of-plane ply waviness

In this paper, an analytical multi-parameter model based on the classical lamination theory (CLT) is developed and the study of the stiffness response for unidirectional composite laminates with out-of-plane ply waviness is followed. A range of parameters associated with the waviness defect, including the waviness geometry, the fiber off-axis angle, the extent of wavy region and the waviness pattern, are incorporated into the analytical model. The fiber volume fraction and the type of reinforcement are also taken into account in the parametric analysis. The three-dimensional stiffness parameters of thick wavy laminates are investigated systematically for different cases. An observation of the negative in-plane Poisson’s ratios for wavy laminated composites reinforced with carbon fibers is presented under the given assumptions of CLT. This phenomenon provides a possible, new technique to design the auxetic materials. The modeling approach provides an effective procedure to quantify the effects of ply waviness on the stiffness of composite laminates.

Buckling and Post-Buckling Behaviors of a Variable Stiffness Composite Laminated Wing Box Structure

The buckling and post-buckling behaviors of variable stiffness composite laminates (VSCL) with curvilinear fibers were investigated and compared with constant stiffness composite laminates (CSCL) with straight fibers. A VSCL box structure was evaluated under a pure bending moment. The results of the comparative test showed that the critical buckling load of the VSCL box was approximately 3% higher than that of the CSCL box. However, the post-buckling load-bearing capacity was similar due to the layup angle and the immature status of the material processing technology. The properties of the VSCL and CSCL boxes under a pure bending moment were simulated using the Hashin criterion and cohesive interface elements. The simulation results are consistent with the experimental results in stiffness, critical buckling load and failure modes but not in post-buckling load capacity. The results of the experiment, the simulation and laminated plate theory show that VSCL greatly improves the critical buckling load but has little influence on the post-buckling load-bearing capacity.

Free vibration analysis of variable stiffness composite laminate plate with circular cutout

This paper presents the free vibration analysis of a variable stiffness composite laminate (VSCL) square plate with circular cutout. Although the h-version of the finite element method is quite attractive, it could not be used in the VSCL plate. This is due to the difficulties associated with the inter-element continuity for the curvilinear fibre and the fact that the circular cutout cannot be represented accurately. To overcome this problem, a new C0 enriched curved quadrilateral p-element is developed based on the first-order shear deformation theory, in which the transformed reduced stiffness elements are computed exactly within the stiffness matrix, and the curved edges are represented accurately using the blending function method. The convergence properties of the proposed p-version and the results of the numbers of test problems are in good agreement with the results of the literature. The natural frequencies are examined by considering the parabolic fibre orientation, radius of the circular cutout, the plate thickness and the boundaries conditions. It is found that the variation of the fibre orientation have significant effect on natural frequencies, especially for plates with large circular cutout.

Geometrically non-linear periodic forced vibrations of imperfect laminates with curved fibres by the shooting method

Abstract In this paper, the authors study periodic vibrations of variable stiffness composite laminates excited by a harmonic force. The plates have geometrical imperfection in the form of various sinusoidal out-of-plane initial deflections associated with zero stress. The angle of the curvilinear fibre path is introduced as a function of the horizontal Cartesian coordinate. The theory used to extract equations of motion for VSCLs is a third order shear deformation theory that retains rotary inertia. The relations of von Karman for elastic large deflection are used. A p -version finite element is employed and, to find the solution of the equations of motion, the shooting method is applied; frequency response curves are obtained. Static condensation and a modal summation method are applied to reduce the number of degrees of freedom. A damage analysis based on Tsai-Wu criterion is carried out during the studies on vibration. The effects of curvilinear fibres and the influence of modal interactions on the vibration of imperfect VSCLs are investigated. The stability of the periodic solutions is determined by applying Floquet's theory. The effect of geometric imperfections on the vibrational behaviour is studied.

Linear modes of vibration of cylindrical shells in composite laminates reinforced by curvilinear fibres

Modes of vibration of thin cylindrical shells made up of layers with curvilinear fibres (variable stiffness composite laminates (VSCL) shells) are investigated in the linear regime. A p-version finite element type formulation is developed for that purpose; in the absence of data on vibrations of cylindrical VSCL shells, the formulation is verified by comparisons with published data on laminated shells reinforced by straight fibres and on VSCL plates. Parametric studies are performed, in order to investigate how curvilinear fibre paths can influence the modes of vibration. It is found that curvilinear fibre paths can have a very large effect, larger than on plates, on the natural frequencies and natural mode shapes of vibration of cylindrical shells. Factors that strongly influence the modes of vibration of VSCL shells are found; these include the fibre orientation at boundaries and in relation to principal normal sections.

Structural Strength Analysis and Measurement of Composite Laminate Using Finite Element Method and FBG Sensor

The aim of this study is to develop structural strength analysis technique and real-time measuring system of composite laminate using finite element method (FEM) and fiber bragg grating (FBG) sensor. A composite laminate of cantilever beam was designed and fabricated using glass fiber reinforced plastic (GFRP) for structural mechanics behavior research. Six design cases of different orientations composite laminate were considered for the better combinations by using FEM program. The bending test of a composite laminate of cantilever beam was performed by using FBG sensor to obtained relationship between strain and displacement. The study result shows that the higher stiffness of composite laminate of cantilever beam was obtained in the [0/90/0/90] orientation. The first natural frequency is 34.83 Hz and corresponding mode shape is bending mode in Z-direction. The FEM and FBG sensor have been successfully used in variety of composite laminate design with different layering sequences by this article.

Tensile properties of variable stiffness composite laminates with circular holes based on potential flow functions

In this study, a new type of variable stiffness laminate combining the curving path planning with the potential flow theory was experimentally investigated. By a reasonably discrete pair of mutual conjugate functions, contour curves in the flow function and potential fields were generated individually. According to these, the narrowband fiber tracks on the two variable stiffness plies were individually defined. On each intersection of the two plies, the tracks were mutually orthogonal and played a key role in improving the properties of the laminate. The tensile tests of the laminate with central holes show that the ultimate tensile load and elastic modulus of the variable stiffness laminate increased by \({\sim }129\) and 110 %, respectively. The curving paths were effective in delivering stress from the area around the hole to the edge regions of the specimen. The path planning method was based on the flow function, and all the tracks on the entire ply were defined as a whole and thus may significantly benefit the subsequent studies such as parameter optimization of scalar function and the structure of the variable stiffness laminate. The analysis of the experimental data and fracture morphology comparison indicates that the tensile property of the laminates is significantly enhanced.

Delamination growth in composite laminates of variable stiffness

SummaryThe production of new composite laminates with variable stiffness within the surface of plies was enabled by tow‐placement machines. Because of the variation of stiffness, these materials are called variable stiffness composite laminates (VSCL). Recently, many attempts were made to investigate their structural behaviour. In this contribution, a first‐order shear deformation theory is selected to model the multilayered composite laminates. The adopted theory is enhanced by the extended finite element method (XFEM) to describe discontinuities at element level of any interface of interest. To predict the location of the delamination onset, a traction–separation law is developed that is consistent with the XFEM topology. An exponential softening behaviour is implemented within the interface to model the delamination growth in a mixed‐mode direction. In order to solve the non‐linear equations of the delamination propagation, an arc‐length method is applied. The effect of the curvilinear fibre orientation on the location of the delamination onset is investigated. Subsequently, the structural response of the laminates is computed. According to the simplicity of the new approach using the XFEM; and based on the computational cost for calculating the stiffness of VSCL, the method is able to determine structural response of VSCL with less computational effort. Copyright © 2016 John Wiley & Sons, Ltd.

A review on design for manufacture of variable stiffness composite laminates

The use of curvilinear fibre paths to develop variable stiffness laminates is now recognised as a promising technique offering great potential for performance improvements over conventional ‘straight fibre’ laminates. Its manufacture is feasible by fibre placement technologies, such as automated fibre placement. However, these technologies present a set of limitations that need to be included in the design to guarantee the manufacturability and quality of the composite laminates. Although this approach experiences an increasing interest from the specialised literature, most of the works completed overlook the manufacturing reality and, as a result, variable stiffness laminates are not used in industry. This work aims to provide a review of the State-of-the-Art on design for manufacture of variable stiffness in order to highlight the current gaps and research needs. As a conclusion, tools for analysis of the effect of manufacturing defects, manufacturing optimisation of gaps/overlaps or cycle time and the systematic integration of manufacturing constraints in design, are the main challenges that will be faced in the future to be able to exploit the potential of this advanced tailoring technique.

Generalized Unified Formulation shell element for functionally graded Variable-Stiffness Composite Laminates and aeroelastic applications

Composite materials have been extensively used in engineering thanks to their lightweight, superior mechanical performances and possibility to tailor the structural behavior, increasing the available design space. Variable Angle Tow (VAT) structures exploits this advantage by adopting a curvilinear patterns for the fibers constituting the lamina.This work, for the first time, extends the Generalized Unified Formulation (GUF) to the case of fourth-order triangular shell elements and VAT composites. Functionally graded material properties in both the thickness and in-plane directions are also possible. The finite element has been formulated with layers of variable thickness with respect to the in-plane coordinates.GUF is a very versatile tool for the analysis of Variable Stiffness Composite Laminates (VSCLs): it is possible to select generic element coordinate systems and define different types of axiomatic descriptions (Equivalent Single Layer, Layer Wise, and Zig-Zag enhanced formulations) and orders of the thickness expansions. Each displacement is independently treated from the others. All the infinite number of theories that can be generated with GUF are obtained by expanding six theory-invariant kernels (formally identical for all the elements), allowing a very general implementation. Finally, the possibility of tailoring the theory/order to increase the accuracy in desired directions makes the GUF VAT capability a very powerful tool for the design of aerospace structures.

A layerwise p-version finite element formulation for free vibration analysis of thick composite laminates with curvilinear fibres

Herein, modes of vibration of a novel type of composite laminates, manufactured by tow-placement machines and which are called Variable Stiffness Composite Laminates (VSCL), are of interest. Layerwise theory is chosen, because it leads to an accurate prediction of displacements through the thickness of laminates, even when these are thick. Furthermore, it naturally has the capability of modelling the bending-membrane coupling that occurs on free vibration of VSCL plates with unsymmetric stacking sequences. The continuity of displacements at layer interfaces is imposed and in-plane displacements vary in a zig-zag fashion along the thickness. The accuracy of the model is confirmed by testing Constant Stiffness Composite Laminates (CSCL) analysed in the literature by others, who employed either Equivalent Single Layer (ESL) or layerwise theories. Abaqus commercial finite element software is as well employed in order to test the present approach. Finally, published natural frequencies of VSCLs, which were studied by an ESL theory, are compared with frequencies resulting from the present layerwise theory. The verification of the proposed model is followed by an investigation on the effect of curvilinear fibre orientation parameters on natural frequencies of thin to thick composite plates, with symmetric and unsymmetric lay-ups.

Geometrically non-linear static analysis of unsymmetric composite plates with curvilinear fibres: p-version layerwise approach

This paper presents a layerwise theory, p-version finite element method for static analysis of unsymmetric laminates with curvilinear fibres. A displacement field that varies linearly along the thickness of each layer is considered and its continuity between the layers is imposed, in this way representing the zig-zag variation of displacements through the thickness. Von Kármán non-linear strain–displacement relationships are applied to take into account the effect of large displacements. Several test problems are examined to verify the accuracy of the proposed model for both symmetric and unsymmetric constant stiffness composite laminates (CSCL), in the linear and non-linear regimes. Deflections and stresses computed with the model proposed here are compared with results available in the literature and with results from Abaqus and Ansys commercial software. Then, static analysis of unsymmetric variable stiffness composite laminated plates (VSCL) are investigated in the geometrically non-linear regime and the effect of unsymmetric stacking sequences and fibre orientations are studied.