Unsymmetric Cross-ply Laminates Research Articles

Bistable characteristics of unsymmetric cross-ply composite laminates considering different boundary shapes

The aim of this paper is to study the bistable characteristics of unsymmetric cross-ply composite laminates induced by thermal stress during the curing process when considering different boundary shapes. The changes in geometry for five types of planform shapes, have been focused, which are named rectangular planform (RP), elliptical planform (EP), trapezoidal planform (TP), diamond planform (DP) and cruciform planform (CP) respectively. The parameters that characterize the bistable performance of the laminates, including the principal curvature and snap load, were measured. The principal curvatures in different stable configurations were determined with analytical, numerical, and experimental methods, and the load-displacement curves in the snapping process were successfully obtained. Moreover, the boundary effect caused by the difference in planform shapes, which cause the local curvature of the bistable composite laminate to change, were also discussed. Predictions from an analytical shell model are verified through comparisons with nonlinear finite element simulations and experimental test results.

Buckling of shear-deformable unsymmetrically laminated plates

The couplings of unsymmetric laminated structures and the comparatively low transverse shear stiffnesses significantly influence their stability behaviour. The aim of this work is to improve the purely analytical stability analysis of unsymmetric laminated structures. This problem can be reduced to one single plate with the use of the discrete plate theory. In this study, laminates under uniaxial compressive load are considered that are simply supported at all edges and have rotational restraints on the unloaded edges (SRSR) or have a free edge (SFSR). Therefore, the present study provides explicit solutions for the buckling load in two ways. Firstly, the first explicit solution for the buckling load of unsymmetric laminates with the mentioned boundary conditions is developed, which takes the bending–extension couplings into account in a direct way. The method is an energy-based method in the form of the Rayleigh-quotient for classical laminated plate theory (CLPT), first order shear deformation theory (FSDT) and third order shear deformation theory (TSDT). Secondly, the reduced bending (RBS) method is investigated for the first time in the framework of TSDT. The comparison of unsymmetric cross-ply and angle-ply laminates shows a good agreement with finite element analysis (FEA). Due to the high computational efficiency of the presented methods, they are excellently suited for pre-design and optimization of composite structures, with simultaneous high usability in practical application.

An efficient semi-analytical framework to tailor snap-through loads in bistable variable stiffness laminates

Multistable laminates are potential candidates for adaptive structures due to the existence of multiple stable states. Commonly, such bistable shapes are generated from the cool-down process of the unsymmetric laminates from the curing temperature. In this work, we exploit unsymmetric variable stiffness laminates with curvilinear fiber paths to generate similar bistable shapes as unsymmetric cross-ply laminates, but with the possibility to tailor the snap-through loads. Snap-through is a complex phenomenon in that is difficult to characterize using simple analytical models. An accurate yet computationally efficient semi-analytical model is proposed to compute the snap-through forces of bistable variable stiffness (VS) laminates. The differential equations resulting from the compatibility and the in-plane equilibrium equations are solved with negligible numerical error using the Differential Quadrature Method (DQM). As a result, the in-plane stress resultants and the total potential energy is written in terms of curvatures. The out-of-plane displacements are expressed in the form of Legendre polynomials where the unknown coefficients of the displacement function are found using the Rayleigh-Ritz formulation. The calculated snap-through loads are then compared with the Finite Element (FE) results. A parametric study is conducted to explore the tailoring capabilities of VS laminates for snap-through loads.

Free-edge stress fields in cylindrically curved symmetric and unsymmetric cross-ply laminates under bending load

In this paper, an analysis method for the determination of displacement, strain and stress fields in cylindrically curved cross-ply laminates under bending load is presented. The considered cross-ply laminates may be either symmetrically or unsymmetrically laminated and are clamped at one end while the other end is loaded by an evenly distributed bending moment. The analysis method employs a layerwise plane strain approach in the inner regions of the laminate in which the stresses in each layer are represented by adequate formulations for Airy’s stress function. In the regions of the free laminate edges where significant three-dimensional and possibly singular interlaminar stress fields are to be expected, the plane-strain approach is upgraded by a layerwise displacement-based formulation wherein the physical laminate layers are discretized into a number of mathematical layers with respect to the thickness direction. The governing differential equations for the unknown additional displacement functions with respect to the width coordinate in the form of the Euler-Lagrange equations stemming from the underlying variational statement can be solved exactly and eventually lead to an eigenvalue problem that needs to be solved numerically. Usage of continuity conditions between the individual laminate layers and formulation of adequate boundary conditions at the free edges in an integral sense then lead to complete representations for displacements, strains and stresses at every location in the considered laminate. While the analysis approach relies on a discretization of the laminate into a number of mathematical layers with respect to the thickness direction and further requires a numerical solution of a quadratic eigenvalue problem, it provides closed-form analytical solutions concerning the width direction and can thus be classified as being a semi-analytical solution. The presented analysis method is compared to the results of comparative finite element simulations and is shown to be in good agreement, however with only a fraction of the computational effort that is required for according finite element simulations.

Curvature of unsymmetric cross-ply laminates: Combined effect of thermal stresses, microcracking, viscoplastic and viscoelastic strains

Curvature of unsymmetrical [0/90] specimens caused by thermal stresses changes if the specimen is subjected to large axial strains introducing intralaminar cracks in the 90-layer. It is shown that the large curvature reduction can not be explained by cracking related stress release only. The large irreversible viscoplastic strains introduced during the axial tensile loading (with 5 min holding at high strain for crack counting) give the main contribution to the curvature change. The effect of transient viscoelasticity (VE) was found to be of minor significance. Simple approach based on effective damaged layer stiffness and constant irreversible strain is used in the framework of laminate theory to extract the viscoplastic and VE strains from experimental curvature data. The obtained fitting expressions for viscoplastic- and VE-strain development are successfully used to describe curvature change in [0/902] laminate subjected to the same test procedure. It is suggested that the used curved beam tests could be efficient to characterize the viscoplastic strain development in the thin 90-layers.

Exact vibration solutions for cross-ply laminated plates with two opposite edges simply supported using refined theories of variable order

Abstract This paper presents exact solutions for free vibration of rectangular cross-ply laminated plates with at least one pair of opposite edges simply supported using refined kinematic theories of variable order. Exact natural frequencies are obtained using an efficient and unified formulation where the solving set of second-order differential equations of motion and related boundary conditions are expressed at layer level in terms of so-called fundamental nuclei having invariant properties with respect to the order of the plate theory. The nuclei are then appropriately expanded according to the number of layers and the order of the theory and the resulting equations are transformed into a first-order model whose solution is obtained by using the state space concept. In this way, the mathematical effort needed to derive analytical solutions is highly reduced. Both higher-order equivalent single-layer and layer-wise theories are considered in this study. Comparisons with other exact solutions are presented and useful benchmark frequency results for symmetric and un-symmetric cross-ply laminates are provided.

The effect of moisture ingress on through-thickness residual stresses in unsymmetric composite laminates: A combined experimental–numerical analysis

Some unsymmetric polymer–matrix composite laminates show multi-stable behaviour, which can provide opportunities for morphing composite structures. It is known that through-thickness residual stresses are introduced in unsymmetric laminates during the manufacturing process. In this research work, a combined experimental–numerical analysis is proposed to study the effect of moisture ingress on unsymmetric composite laminates by analysing through-thickness residual stresses at dry and saturation states. A range of unsymmetric cross-ply laminates were manufactured using carbon–fibre epoxy pre-preg material. The laminates were subsequently immersed in water to full saturation. The curvatures of dry and saturated test coupons were measured and compared to quantify the effects of moisture ingress on the shapes of the unsymmetric laminates. In addition, a finite element model was developed to predict the experimentally measured curvatures of the dry laminates. By calibrating the numerical model, the through-thickness residual stresses were analysed for different unsymmetric cross-ply layup conditions. The calibrated dry model was then extended to develop a moisture-dependent numerical model in order to incorporate the effect of moisture ingress on the shapes of unsymmetric laminates. The moisture-dependent model was calibrated with the experimentally measured curvatures and then used to predict the through-thickness residual stresses in the fully saturated unsymmetric cross-ply laminates. It is shown that moisture ingress can considerably influence through-thickness residual stresses and thus the shapes of unsymmetric laminates.

Pressure loading, end- shortening and through- thickness shearing effects on geometrically nonlinear response of composite laminated plates using higher order finite strip method

A semi-analytical finite strip method is developed for analyzing the post-buckling behavior of rectangular composite laminated plates of arbitrary lay-up subjected to progressive end-shortening in their plane and to normal pressure loading. In this method, all the displacements are postulated by the appropriate harmonic shape functions in the longitudinal direction and polynomial interpolation functions in the transverse direction. Thin or thick plates are assumed and correspondingly the Classical Plate Theory (CPT) or Higher Order Plate Theory (HOPT) is applied. The in-plane transverse deflection is allowed at the loaded ends of the plate, whilst the same deflection at the unloaded edges is either allowed to occur or completely restrained. Geometric non-linearity is introduced in the strain–displacement equations in the manner of the von-Karman assumptions. The formulations of the finite strip methods are based on the concept of the principle of the minimum potential energy. The Newton–Raphson method is used to solve the non-linear equilibrium equations. A number of applications involving isotropic plates, symmetric and unsymmetric cross-ply laminates are described to investigate the through-thickness shearing effects as well as the effect of pressure loading, end-shortening and boundary conditions. The study of the results has revealed that the response of the composite laminated plates is particularly influenced by the application of the Higher Order Plate Theory (HOPT) and normal pressure loading. In the relatively thick plates, the HOPT results have more accuracy than CPT.

Curvature tailoring of unsymmetric laminates with an initial curvature

An unsymmetric laminated composite having bi-stability is interesting as a morphing structure because it can sustain large deformation without energy supply and the deformed state is stable. Especially, inducing different curvature values at each stable state is important for the practical application of unsymmetric laminates. In this study, we propose a method for tailoring the curvature of unsymmetric cross-ply laminates by curing the composite on a curved tool plate. The results show that the final normal curvature tensor of unsymmetric laminates is the summation of the tool plate curvature tensor and the normal curvature tensor of the laminates.

Morphing of Bistable Composite Laminates Using Distributed Piezoelectric Actuators

The use of bistable unsymmetric cross-ply laminates for morphing application has received growing attention in the last few years. So far, most studies use large rectangular piezoelectric Macro Fiber Composite (MFC) patches bonded at the center of the laminate to induce snap-through. However, the use of large rectangular MFC patches bonded in the center of the laminates significantly influences the shape of the laminate by greatly reducing the curvature at the midsection of the laminate where the MFC patches are bonded. This paper presents a study where narrow cocured MFC strips distributed over the entire surface are used to induce snap-through of unsymmetric cross-ply laminates. This MFC configuration allows having a more uniform curvature in the laminate. Since the strips are bonded on both sides, reverse snap-through should be obtained. The study was both theoretical and experimental. A finite element nonlinear analysis was used to predict the two stable cylindrical configurations and the snap-through induced by MFC actuation. For the experimental study, a laminate-MFC structure was manufactured and tested. The shapes were measured using a 3D image correlation system as a function of applied voltage. Good correlations for the cylindrical shape and displacement field were observed.

Natural vibration of unsymmetric cross-ply laminates

This study considers the linear vibration characteristics of square [ 0 n / 90 n ] T laminates relative to their room-temperature static equilibrium configurations. A Rayleigh–Ritz approach combined with Hamilton's principle is used to provide approximate solutions to this vibration problem. The vibration mode shapes are assumed to have the same spatial dependence as used in past investigations to study the room-temperature configurations of these laminates, and are thus assumed to be perturbations on the static equilibrium configurations. Hamilton's principle then results in the so-called zero- and first-order equations. The zero-order equations lead to the classic static equilibrium results of past investigations, presented here in nondimensional form with analytical solutions at the bifurcation point. The first-order equations, combined with zero-order results, lead to the vibration characteristics for each zero-order static configuration. Interest centers on the lowest natural frequency and the associated mode shape for laminates clamped at their midpoints, with special attention as to how these vibration characteristics depend on the laminate side-length-to-thickness ratio. With an imaginary-valued frequency, the static saddle configuration for side-length-to-thickness ratios larger than the critical value is correctly assessed as unstable. A finite element model is also used to study the vibration characteristics and to compare with the findings for the developed analysis. The qualitative comparisons between the developed analysis and the finite element model are generally good, and the quantitative comparisons are also satisfactory.

Planform effects upon the bistable response of cross-ply composite shells

This research aims at investigating the bistable characteristics of unsymmetric cross-ply laminates resulting from thermally induced stresses while curing. The bistable characteristics are examined with a focus on the change in geometry for a subset of configurations, namely, rectangular, trapezoidal and triangular. To this end, these geometries are systematically investigated leading to a set of non-dimensional parameters establishing the values required for bistable behavior. A new apparatus is designed to measure the magnitude of the force required for snap-through and snap-back. The influence of friction at the supports of the tested panels is minimized by utilizing air cushion for frictionless motion. Predictions from a nonlinear finite element based methodology using the ABAQUS™ code are validated through comparisons with test data.

Residual Stress Relaxation in CFRP Cross-ply Laminate

We measured residual stress relaxation in epoxy-based CFRP laminates and cyanate-based CFRP laminates. We estimated the residual stress in symmetric cross-ply laminates by measuring the curvature of unsymmetric cross-ply laminates and investigated the relaxation of the residual stress by measuring the curvature of unbalanced laminates in a nitrogen environment. Our experimental results demonstrated that the residual stress in both kinds of laminate can decay by approximately 15% after 500 hours at 40°C. The stress relaxation can be modeled by the power law model. Viscoelastic parameters were obtained by a quasi-static tensile test to predict stress relaxation. The validity of the prediction is discussed.

Anticlastic Stability Modeling for Cross-ply Composites

A viable aspect of fiber-reinforced composites is their elastic tailoring ability. Anticlastic curvature is one example of elastic tailoring that occurs with an unsymmetric cross-ply layup. Residual stresses resulting from differences in coefficients of thermal expansion and elastic properties in each lamina can cause large out-of-plane deformations. In the case of thin unsymmetric cross-ply laminates under thermal curing load, a cylindrical shape is observed because of this inherent geometrical nonlinearity as opposed to the saddle shape that classical lamination theory predicts. In this article, a finite element approach, using ABAQUS™, is implemented in order to predict the unsymmetric cross-ply laminate shapes under thermal curing stresses and understand the underlying limit point instability. Numerical results for curvatures of the predicted shapes are in agreement with published experimental and analytical data. The stability of the cylindrical laminates is also investigated. Depending on the aspect ratio of the rectangular laminate, a cylindrical shape may snap-through from its current stable configuration to another stable cylindrical shape with a different curvature. In particular, both the critical aspect ratio where snap-through will cease to occur and the buckling load are reported.

Distortion Simulation of Unsymmetrical Composite Laminates Using the Finite Element Analysis Method

Unsymmetric composite laminates were benefit to reducing the structure weight of some aircrafts. However, the cured unsymmetric laminates showed distortion at room temperature. Therefore, predicting the deformation before using the unsymmetrical composite is very important. In this study an attempt was made to predict the shapes of some unsymmetric cross-ply laminates using the finite element analysis (FEA). The bilinear shell-element was adopted in the process. Then the simulation results were compared with the experimental data. The studies we had performed showed that the theoretical calculation agreed well with the experimental results, the predicted shapes were similar to the real laminates, and the difference between the calculated maximum deflections and the experimental data were less than 5%. Hence the FEA method was suitable for predicting the warpage of unsymmetric laminates. The error analysis showed that the simulation results were very sensitive to the lamina thickness, 2 α and (T.

Theoretical study on shape control of thin cross-ply laminates using piezoelectric actuators

This paper investigates the application of a piezoelectric actuator to control deformation of thin unsymmetric cross-ply composite laminates . A theoretical model based on Rayleigh–Ritz principle is developed to predict laminate deformation and the effect of the piezoelectric actuator layer. Several stacking sequences are studied using the model. It is found that two stable shapes still exist for laminates with larger dimensions but the two shapes are not symmetric as in traditional cross-ply laminates. The bifurcation dimension increases when the thickness of piezoelectric layer increases. Laminate curvatures are in a perfectly linear relationship with an electric field applied to the piezoelectric actuator layer. Snap-through from one stable shape to the other can occur by applying an electric field. FEA results match model prediction fairly well. This model can be used as a guide to design piezocomposite laminates.

Post-buckling behavior of composite laminated plates under end shortening and pressure loading, using two versions of finite strip method

Two different versions of finite strip method, namely spline and semi-analytical methods, are developed for analyzing the geometrically non-linear response of rectangular composite laminated plates of arbitrary lay-up to progressive end-shortening in their plane and to pressure loading. The plates are assumed to be thin so that the analysis can be carried out based on the classical plate theory. The in-plane lateral deflection υ is allowed at the loaded ends of the plate, whilst the lateral expansion of the unloaded edges is either free or completely prevented. Geometric non-linearity is introduced in the strain–displacement equations in the manner of the von Karman assumptions. The formulations of the finite strip methods are based on the concept of the principle of the minimum potential energy. A number of applications involving isotropic plates, symmetric and unsymmetric cross-ply laminates are described to investigate the effects of pressure loading. The comparison between the two sets of results obtained by different finite strip methods is very good. The study of the results revealed that the response of the laminates is significantly influenced by the application of the normal pressure loading. Particularly, the response of unsymmetric laminates is strongly affected by the sign of the normal pressure loading.

Cured Shape of Cross-Ply Composite Thin Shells

For a composite laminated plate, it has been found that classical laminate theory (CLT) can not always predict the final cured shape correctly and geometric nonlinearity must be considered. For composite laminated shells, experiments show that the cured shape depends on stacking sequence, radius, thickness, and size. This paper investigates the cured shape of several cross-ply composite shells. The cured shape of a cross-ply shell is generally cylindrical. A model is established to predict the cured shape. The model is based on the Rayleigh-Ritz energy method and considers geometric nonlinearity. Modeling and experiment show that, for certain stacking sequences, the generator of the cured shape may be orthogonal to the original generator, while other stacking sequences can lead to deeper or shallower cured shapes. The predictions of the model are very close to the results of FEM analysis and experiment. This model can be used as a guide in the manufacture of unsymmetric cross-ply laminates.

Non-linear analysis of the curved shapes of unsymmetric laminates accounting for slippage effects

The shape of unsymmetric laminates at room temperature has an out-of-plane deformation after autoclave curing processing. Unsymmetric cross-ply laminates show two stable curvature shapes because of snap-through when the side lengths of laminates are over a critical value. The shapes of the unsymmetric laminates at room temperature are influenced by various environmental factors such as temperature and moisture. Experiments show a significant effect of tool-plate, which cannot be ignored, on the cured shape of an unsymmetric laminate. We developed the simplified higher-order plate theory for laminates with linear transverse shear profile by applying the Rayleigh-Ritz approximation in the solution procedure. For a refined approximation, in-plane strains were expanded to complete fifth order polynomials. The slippage effects resulting from the interaction between the laminates and the tool-plate were considered. A dimensionless slippage coefficient was introduced, which was correlated to the asymptotic curvature value of the model corresponding experimental one was introduced, and the effect of processing parameters were studied. The previous models, which had been applied only to a square plate, were extended to predict curvatures of rectangular plate configurations with various aspect ratios. The slippage coefficient obtained from the square laminates can be used directly without modification to predict curvatures of rectangular plates with various aspect ratios.

Higher-Order Finite Strip Method for Postbuckling Analysis of Imperfect Composite Plates

Postbuckling analysis is essential to predict the capacity of composite plates carrying considerable additional load before the ultimate load is reached, and manufacturing-induced geometric imperfections often reduce the load-carrying capacity of composite structures. A higher-order finite strip method based on the higher-order shear deformation plate theory is developed for postbuckling analysis of laminated composite plates with initial geometric imperfection subjected to progressive end shortening. The arbitrary nature of initial geometric imperfection induced during manufacturing is accounted for in the analysis. Nonlinear equilibrium equations are solved by a Newton-Raphson procedure. Examples of postbuckling analyses of unsymmetric cross-ply, angle-ply, and arbitrary laminates are presented, and the accuracy and performance of the method are examined. The numerical higher-order finite strip method presented can be used as an accurate and efficient tool for postbuckling analysis of imperfect composite plates.