Variable Angle Tow Composites Research Articles

Uncertainty quantification for natural frequency and mode of variable angle tow composite plates with random and interval uncertainties

Variable angle tow (VAT) composite laminated structures are produced by automated tow-placement technology which can take advantage of the designability of composites to meet the aerospace industry’s need for dynamic properties. Complex production techniques inevitably lead to uncertainties in material properties due to manufacturing defects. The aim of this work is to present an efficient computational method for estimating the dynamical properties of VAT composite laminated plates due to multiscale hybrid uncertainties. This method consists of perturbation stochastic finite element method and particle swarm optimization combined with Mori-Tanaka homogenization. The degree of influence of the multiscale parameters on the dynamic properties of VAT composites with different structural forms has been obtained by sensitivity analyses of the frequency and mode. The method is applied to investigate variations in natural frequency and mode of a variety of VAT composite laminated plates with different fiber tow paths, layers, and boundary conditions. The results indicate that variations in natural frequencies and mode shapes are strongly linked to fiber tow paths and boundary conditions. The frequency and modes of the VAT show a nonlinear variation with angle. This method can quickly obtain the probability distribution intervals of the frequency, which is of great significance for assessing the reliability of VAT composites laminated plates dynamic designs.

Evolution of variable angle tow composite structures: Data analysis and relevance of the theme

Variable angle tow composites were developed to increase stiffness in notched laminates. Scientific advancements in manufacturing and fiber path optimization have since improved variable angle tow composites. This study uses bibliometric analysis to track the evolution of variable angle tow composites in the literature. Bibliometric analysis is a useful tool for measuring paper contributions and accurately assessing performance, aiding authors in the research process. This article presents a co-occurrence map of keyword frequency in research papers. Furthermore, an experimental section highlights manufacturing defects found in variable angle tow composites produced by filament winding, showing that manufacturing real structures in variable angle tow composites poses a significant challenge. Lastly, the perspectives of variable angle tow composite applications are discussed.

Variable angle tow composites in fibre-reinforced polymer bridges

Fibre reinforced polymers are increasingly used in bridge structures for reducing their environmental impact, extending the service life and saving costs. In this work, we propose the use of composite laminates called Variable Angle Tow (VAT) to realise bridge structures. In VAT composite laminates, the fibre orientation changes pointwise over the structure, enhancing stiffness tailoring capabilities with respect to traditional straight fibre (SF) laminates. In aerospace engineering, VAT composites have been successfully used to tailor structures’ stiffness to enhance linear elastic response, reduce buckling phenomena, and optimise the postbuckling behaviour. However, VAT laminates have never been applied in civil structures. This work shows how using VAT composite laminates in a bridge girder improves its buckling and postbuckling behaviours and increases its overall stiffness. A numerical investigation is conducted to assess the benefits given by VAT composite laminates to the structural performance of bridge girders. For different spans, results of a multi-objective optimisation considering both traditional SF and VAT laminates are presented. It is found that VAT laminates exhibit a buckling load up to 80% over SF laminates, with equal or lower deflections and without adding extra material, thereby opening new design scenarios for bridge structures.

An adaptive Ritz formulation for progressive damage modelling in variable angle tow composite plates

In this work, an adaptive Ritz model for the analysis of variable angle tow composite plates featuring damage initiation and evolution under progressive loading is proposed, developed, implemented and tested. The plate kinematics is represented employing a first-order shear deformation theory, while the plate equilibrium equations at a given load step are obtained by minimizing the structure potential energy. The constitutive behaviour is modelled within the framework of continuum damage mechanics. In particular the initiation and evolution of damage, up to failure, are tracked by defining irreversible damage indices related to both fibres and matrix, both in tensile or compression loading. The discrete equations are then obtained by assuming a polynomial Ritz approximation of the primary kinematic variables in the energy minimization. Preliminary tests show how the application of the method as a single-domain approach induces the emergence of problematic spurious effects, related to Gibbs artefacts due to the inability of the selected polynomial basis to represent damage localization. An adaptive multi-domain technique is thus proposed to circumvent such issues, which has been successfully validated by benchmark tests. Eventually, original results about variable angle tow plates featuring damage evolution under progressive loading are presented.

Thermal stress analysis of variable angle tow composite plates through high-order structural models

This work focuses on thermo-elastic analysis of Variable Angle Tow (VAT) composite plates. Emphasis is given to the through-the-thickness 3D stress component distributions as a consequence of constant temperature profiles. High-order structural models are used, in order to take into account the volumetric and the distortion contributions. The last contribute arise from boundary conditions or asymmetric staking sequences. Governing equations of the uncoupled thermo-elastic problem are derived in the domain of the Carrera Unified Formulation (CUF), which in turn is coupled with a classical finite element method to obtain opportune numerical solutions. The advantage of using CUF lies in the fact that different theories of structures can be implemented automatically and a critical study on the use of standard to high-order plate finite elements can be performed with ease. In this manner we classify models based on their efficiency and depending on the degree of accuracy needed, both in terms of displacements and stresses. Different numerical problems are considered and it is demonstrated that layerwise approximations are needed whenever shear stresses trends are of interest. Boundary Conditions influence more the choice of the adequate expansion theory otherwise.

Buckling analysis and optimization of variable angle tow composite plates via Ritz method and variable stiffness optimization

A novel variable stiffness optimization (VSO) algorithm is proposed accounting for the inherent mechanical characteristics of variable angle tow (VAT) plates. Through sampling in the design space of lamination parameters (LPs), a good initial point is identified in stacking sequence design domain. The concept of tailoring stiffness point by point through-thickness is then introduced. By using the linear fiber path function, design variables for VAT plates are divided into three groups, and then a layerwise optimization approach (LOA) is applied to optimize the angles from the outermost to the innermost stacking position within the three groups, sequentially and iteratively. Through a point-by-point optimization of fiber paths, the stiffness of VAT plates is enhanced part by part. Ritz-type buckling solutions are presented for VAT composite plates with arbitrary boundary conditions. Using first-order shear deformation theory (FSDT), the translational and rotational degrees of freedom of the plate are divided into in-plane and out-of-plane components to solve the prebuckling and buckling problems independently. Buckling loads of VAT plates are optimized under multiple boundary conditions and loadings. Results indicate that VSO algorithm is efficient and robust, and the buckling resistance capacity of optimized VAT plates can be improved approximately twice over constant-stiffness laminates.

Buckling and post-buckling of variable stiffness plates with cutouts by a single-domain Ritz method

Structural components with variable stiffness can provide better performances with respect to classical ones and offer an enlarged design space for their optimization. They are attractive candidates for advanced lightweight structural applications whose functionalities often impose the presence of cutouts that requires accurate and effective analysis for their design. In the present work, a single-domain Ritz formulation is proposed, implemented and validated for the analysis of buckling and post-buckling behaviour of variable stiffness plates with cutouts. The plate model is based on the first-order shear deformation theory with nonlinear von Karman strain–displacement relationships. The plate generalized displacements are approximated with trial functions built as products of one-dimensional Legendre orthogonal polynomials. The non linear governing equations system is then deduced from the stationarity of the energy functional; the involved matrices are numerically computed by a special integration algorithm based on the implicit description of the cutout via suitable level-set functions. The formulation has been implemented in a computer code which has been used to validate the method through comparison with literature solutions for variable angle tow laminates with circular cutouts. Several investigations on buckling and post-buckling behaviour of variable angle tow composite plates with cutouts having different shapes and dimensions are then presented to illustrate the approach capabilities, provide benchmark results and point out features and design opportunities of the variable stiffness concept for the buckling and post-buckling design of advanced lightweight structures.

A non-linear Ritz method for progressive failure analysis of variable angle tow composite laminates

A Ritz formulation for non-linear analysis of damage initiation and evolution in variable angle tow composite plates under progressive loading is presented. The model is built on a few key items. It assumes first order shear deformation theory kinematics and non-liner strains in the von Kármán sense. The constitutive relationships are formulated in the framework of continuum damage mechanics at the ply level, so that each laminate layer can experience in-plane damage initiation and evolution, then reflected in material softening and loss of local stiffness. A Ritz polynomial expansion of the primary variables and the minimization of the total potential energy provide the discrete solution equations, which are then solved with an incremental-iterative approach for capturing damage evolution. A few tests have been successfully performed to validate the approach. Eventually, some original results about variable angle tow plates under progressive in-plane compression loads highlight the effect of damage on the post-buckling response.

Peridynamics for failure prediction in variable angle tow composites

Peridynamics for failure prediction in variable angle tow composites

Geometrically nonlinear analysis and vibration of in-plane-loaded variable angle tow composite plates and shells

This work intends to present a novel numerical approach for studying the vibration behaviours of variable angle tow (VAT) composite structures in their quasi-static nonlinear equilibrium states. This methodology is able to predict the buckling load, to investigate the natural frequencies variation for progressively higher loads, and to provide a means for verifying experimental Vibration Correlation Technique results. The use of VAT composites, in which the fibre orientations are allowed to vary along with a curvilinear pattern within each lamina, dramatically increases the design space and provides a significant improvement in buckling performance and benefits in the postbuckling regime. This study has been performed using an innovative methodology based on the well-established Carrera Unified Formulation able to describe several kinematic models for two-dimensional structures. In detail, layerwise theories are employed to characterize the complex phenomena that may appear in VAT composite structures. All Green-Lagrange strain components are employed because far nonlinear regimes are investigated. Furthermore, the geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton–Raphson method along with a path-following approach based on the arc-length constraint. Different VAT composite structures have been analyzed to validate the proposed approach and provide some benchmark solutions. The computed equilibrium paths are compared with results obtained using the commercial code ABAQUS. The results document the good accuracy and reliability of the presented methodology and show this numerical tool’s potentialities.

Global buckling and wrinkling of variable angle tow composite sandwich plates by a modified extended high-order sandwich plate theory

The advanced Automated Fibre Placement (AFP) manufacturing technologies make the synthesis of Variable Angle Tow (VAT) composites, which enable the design of lightweight sandwich structures to possess variable stiffness facesheets. However, both global and local instability phenomena of VAT sandwich plates under in-plane compressive loads are rarely explored till now. The objective of this article is to fill this gap by developing a Rayleigh–Ritz procedure based on a modified extended high-order sandwich plate theory (EHSAPT). The original two-dimensional EHSAPT proposed by Phan et al. (2012) is extended to a three-dimensional case with a minor modification that the first-order shear deformation theory instead of the classical Kirchhoff–Love hypothesis is adopted for each facesheet, which brings several merits such as the conciseness in derivation process and the easiness in program implementation. The Rayleigh–Ritz approach combined with the principle of minimum potential energy is employed to derive the eigenvalue equation that governs the instability problem of VAT sandwich plates under in-plane compressive loads. Both global buckling and wrinkling patterns of VAT sandwich plates can be captured under this proposed analytical model framework. Before instability analysis, the nonuniform prebuckling stresses over the entire sandwich plate are determined under the assumption of membrane prebuckling state. The usage of Lagrange multiplier method in the prebuckling regime removes the restrictions inherent in conventional Rayleigh–Ritz formulation and thus provides a general way to model in-plane boundary conditions. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing with previously published results and FE solutions given by ABAQUS. Effects of core thickness, core orthotropy, and fibre orientation angle of the facesheets on the instability behaviour of VAT sandwich plates are investigated. Finally, the mechanism of steering the fibre trajectory over the facesheets to improve the buckling resistance for the sandwich plate is studied and discussed.

Partitioned parametrized variational procedure for the generation of theorems of structural analysis

Structural field modeling of displacements, stresses, and strains is conceptually linked with the generation of functionals. In particular, this work introduces for the first time the Partitioned Parametrized Variational Procedure (PPVP). In-plane stresses and strains are mathematically treated differently than the transverse counterparts. This allows one to generate all the classical variational theorems of elasticity (such as the Principles of Virtual Displacements and Complementary Energy, Hellinger-Reissner, and the Veubeke-Hu-Washizu theorems) and additional new types of variational theorems which can be exploited for the solution of structural problems in various areas, such as reinforced panels and variable angle tow composites. With the proposed variational procedure energetically consistent constraints naturally arise directly from the formulation. PPVP could also be used to formulate new enhancements of the finite element or meshless methods and to quantify functional errors not related to the numerical discretization.

GA Optimization of Variable Angle Tow Composites in Buckling and Free Vibration Analysis through Layerwise Theory

In the current research, variable angle tow composites are used to improve the buckling and free vibration behavior of a structure. A one-dimensional (1D) Carrera Unified Formulation (CUF) is employed to determine the buckling loads and natural frequencies in Variable Angle Tow (VAT) square plates by taking advantage of the layerwise theory (LW). Subsequently, the Genetic Algorithm (GA) optimization method is applied to maximize the first critical buckling load and first natural frequency using the definition of linear fiber orientation angles. To show the power of the genetic algorithm for the VAT structure, a surrogate model using Response Surface (RS) method was used to demonstrate the convergence of the GA approach. The results showed the cost reduction for optimized VAT performance through GA optimization in combination with the 1D CUF procedure. Additionally, a Latin hypercube sampling (LHS) method with RS was used for buckling analysis. The capability of LHS sampling confirmed that it could be employed for the next stages of research along with GA.

A complex Fourier series solution for buckling of VAT composite laminates with elastically restrained edges

Variable Angle Tow (VAT) composites have general anisotropic characteristics, and their fiber orientation angle varies continuously along an arbitrary direction. Therefore, the investigation of their buckling performance is quite challenging. Different from the commonly used numerical methods, a new method called complex Fourier series solution is proposed to study the buckling of VAT laminates with elastically restrained edges. First the prebuckling analysis is achieved by introducing Airy’s stress function and expanding it into a complex Fourier series form. And the corresponding mid-plane internal forces with non-uniform distribution are determined. Based on classical laminated plate theory, the buckling analysis of VAT laminate is carried out. The transverse displacement is also expressed as a complex Fourier series. And the critical buckling load and buckling mode shape of VAT laminate are obtained. The accuracy of the presented complex Fourier series solution is verified by comparing with the existing results. The influence of fiber orientation angles, boundary stiffness and boundary conditions on the critical buckling load and buckling mode shape is investigated in numerical examples. Results show that boundary stiffness as well as fiber path have great influence on the critical buckling load and buckling mode shape of VAT laminate. By adjusting the fiber orientation angles and the translational stiffness of the elastically restrained boundary can effectively improve the critical buckling load of VAT laminate. This paper provides a new method for the analysis of general anisotropic laminates with in-plane variable stiffness and the present complex Fourier series solution can also be used as a benchmark for the verification of other numerical results.

Nonlinear thermal flutter analysis of variable angle tow composite curved panels in supersonic airflow

The advanced Automated Fibre Placement (AFP) technologies make the synthesis of Variable Angle Tow (VAT) composites possible, which provides an extended space for the design of lightweight structures. At the same time, the variable stiffness feature inherent in tow-steered composite panels also poses a huge challenge to solve the mechanical problems including the panel flutter problem. In this article, the nonlinear thermal flutter characteristics of an infinitely long VAT composite curved panel in supersonic airflow is investigated for the first time. The proposed panel model is based on the Donnell’s shallow shell theory and accounts for the von Kármán’s geometrical nonlinearity. The aerodynamic load induced by the supersonic airflow passing over the panel surface is constructed from the first-order piston theory, whilst the thermal load imposed on the curved panel follows the quasi-steady thermal stress theory. The aerothermoelastic equations of motion that govern the nonlinear thermal flutter problem are determined via Hamilton’s variational principle. The Galerkin’s method is applied to convert the partial differential governing equation with variable coefficients into a set of ordinary differential equations, and the resulting nonlinear equations dependent upon time are solved through the fourth order Runge–Kutta method. Several qualitative tools, including the time history, phase portrait, Poincaré map and bifurcation diagram, are employed to study the dynamic behaviours, and a variety of attractors, both regular and chaotic, are observed. The accuracy and effectiveness of the proposed Galerkin procedure are validated by comparing with those available in literature. Effects of height-rise parameter, temperature parameter and fibre orientation angle on the nonlinear flutter responses of VAT composite curved panels in thermal environments are discussed in details.

Stochastic stress analysis and failure onset of variable angle tow laminates affected by spatial fibre variations

The usage of printed composites in the aerospace industry has been steadily increasing over the last years. Especially, 3D printers and automatic fibre placement machines have allowed the introduction of Variable Angle Tow (VAT) composites, which theoretically offer greater tailoring capabilities than classic composite laminates. Nevertheless, the steering of brittle fibres is not flaw-exempt and, in fact, is greatly affected by the printer signature. This manuscript aims to examine the influence of fibre misalignments on the stress and failure index distribution in laminated VAT composites. For doing so, the Carrera Unified Formulation (CUF) is employed to develop layerwise models with unprecedented accuracy. Flaws are introduced at the layer scale by means of stochastic fields and uncertainty analysis is carried out through a Monte Carlo analysis. The random variation (defects) is propagated through the scales and correlated with the stress and failure index distribution. The results show that misalignments greatly affect the in-plane normal stresses, which lately influence fibre tension and compression failure mechanisms.

Isogeometric static analysis of laminated plates with curvilinear fibers based on Refined Zigzag Theory

In this study, we propose a new IsoGeometric formulation based on Refined Zigzag Theory (RZT), abbreviated as IG-RZT, for static analysis of laminated plates and sandwich panels with curvilinear fiber paths for the first time in literature. The original RZT formulation defines constant, linear, and zigzag deformation contributions of thickness coordinate to the in-plane displacements. This kinematic relation can accurately predict in-plane displacement of composite with straight fibers. However, estimating a realistic variation of in-plane displacements in a variable angle tow (VAT) composite is a more challenging problem as compared to the straight fiber composites. This difficulty has been addressed herein by including a quartic (fourth-order) polynomial thickness expansion that includes the transverse normal strain effects to the kinematic displacement fields of RZT. Moreover, the modelling of VAT composites results in the RZT zigzag functions to depend not only on the thickness coordinate but also on the in-plane positions. The present IG-RZT methodology is free of shear-locking and shear correction factors due to the integration of RZT with Non-Uniform Rational B-Splines (NURBS) functions of isogeometric analysis. The use of NURBS functions also enable the exact geometry data to be taken directly from a computer aided design (CAD) software, e.g., Rhinoceros, into an in-house Mathematica code. The accuracy and efficiency of the present IG-RZT formulation is assessed and validated by solving various examples of curvilinear fiber laminated plates with different aspect and span-to-thickness ratios. Comparison of IG-RZT results with reference solutions available in literature and generated by a commercial software (ANSYS) using 3D finite elements have demonstrated the remarkable benefits of the proposed IG-RZT method for predicting highly accurate both displacement and stress distributions of VAT composite structures.

Application of Rayleigh-Ritz formulation to thermomechanical buckling of variable angle tow composite plates with general in-plane boundary constraint

Variable Angle Tow (VAT) composites always exhibit in-plane variable stiffness property, which provides the designer with an extended freedom in stiffness tailoring to achieve higher structural performance for lightweight composite structures. In this paper, a methodology based on a generalised Rayleigh-Ritz formulation is developed to study the thermomechanical buckling response of symmetrical VAT composite plates with general in-plane boundary constraint. It is assumed that the material is of temperature-independent and the panel is exposed to an arbitrary in-plane temperature change. In the framework of thermoelastic theory, the principle of thermoelastic complementary energy in conjunction with Airy’s stress function formulation, for the first time, is applied to solve the in-plane thermoelastic problem of the tow-steered plate. The non-uniform distribution of in-plane force resultant over the entire plane is determined by utilizing the Rayleigh-Ritz formulation enhanced by Lagrangian multiplier method. The merit of the proposed modelling strategy lies in that the application of Lagrangian multiplier method removes the restrictions inherent in conventional Rayleigh-Ritz formulation and thus provides generality to model general in-plane boundary constraint against thermal expansion or contraction. During the buckling analysis, the governing equation of thermomechanical buckling problem of the tow-steered plate under a combination of both temperature change and general in-plane boundary constraint is derived based on the third-order shear deformation theory of Reddy’s type. The accuracy and robustness of the proposed Rayleigh-Ritz model is validated against finite element solutions and previously published results. Effects of boundary condition, fibre orientation angle and temperature change on the in-plane thermoelastic and thermomechanical buckling behaviours of VAT composite plates are studied by numerical examples. Finally, the mechanism of applying tow-steered technology to improve the thermomechanical buckling resistance of composite plates is explored.

Influence of fiber misalignments on buckling performance of variable stiffness composites using layerwise models and random fields

Additive manufacturing brought to the emergence of a new class of fiber-reinforced materials; namely, the Variable Angle Tow (VAT) composites. Automated fiber placement machines allow the fibers to be relaxed along curvilinear paths within the lamina. In theory, the designer can conceive VAT structures with unexplored capabilities and tailor materials with optimized stiffness-to-weight ratios. In practise, steering brittle fibers, generally made of glass or carbon, is not trivial and highly affected from the printer signature. This paper wants to explore the effect of fiber misalignment on the buckling response of laminated VAT composites. For doing so, we use the Carrera Unified Formulation (CUF), which allows to develop layerwise models with unprecedented accuracy in a straightforward and systematic manner. Variation patterns are generated at the layer scale by means of random fields through a Monte Carlo analysis. The stochastic variation (defects) is propagated through the scales and correlated with the global buckling response of VAT panels. The results show that layerwise models outperform equivalent single layer theories, since the former are able to foresee eventual switching between buckling modes, and thus making them fundamental in uncertainty analysis.

Prebuckling and buckling analysis of moderately thick variable angle tow composite plates considering the extension-shear coupling

In this paper, an analytical model based on a generalized Rayleigh-Ritz method is developed to deal with both the prebuckling and buckling problems of moderately thick VAT composite plates under a more general in-plane boundary condition. The non-uniform in-plane stress fields of VAT composite plates under various in-plane boundary conditions is firstly determined based on the minimum complementary energy principle combined with the Airy’s stress function. Afterwards, the governing equation of buckling problem of VAT composite plates under non-uniform in-plane stress fields is derived based on third-order shear deformation theory of Reddy’s type. The innovation of this paper lies in that a linear fitting method combined with Chebyshev-Gauss-Labotto point distribution is introduced to retrieve the boundary stress coefficients from the stress constraint conditions generated by the Lagrangian multiplier method, and meanwhile an integral expression of work done by the unknown boundary force along the prescribed in-plane displacement is adopted to satisfy the displacement constraint condition along the boundary line. In so doing, three kinds of in-plane boundary conditions are taken into account, that is, pure stress boundary condition, pure displacement boundary condition and mixed stress and displacement boundary condition. In addition, no assumption is made on the extension-shear coupling terms and thus the proposed analytical model is suitable to the unbalanced layup. The accuracy and robustness of the proposed analytical model is validated by finite element analysis and with prior results. Effects of the in-plane boundary condition, extension-shear coupling and varying fibre orientation angles on both in-plane stress and buckling responses of VAT composite plates are studied by numerical examples. It is shown that for the balanced panels under uniform compression, the benign load redistribution mechanism offered by the VAT layups remain in effect even if there exists the non-uniform shear constraint along the loading edges. However, the buckling load of the panel can be reduced by the shear constraint along the loading edges to a certain degree. Furthermore, it is found that for some displacement-controlled shear loading cases, the extension-shear coupling plays a positive role in the load redistribution, which results in a sight improvement in the longitudinal load of the panel when critical buckling occurs.