Variable Angle Tow Composite Plates Research Articles

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.

Aeroelastic control of delaminated variable angle tow laminated composite plate using piezoelectric patches

The free vibration and aeroelastic analysis of variable angle tow (VAT) laminated composite plate bonded with discrete piezoelectric patches are considered for investigation. A finite element (FE) model is developed using eight noded isoparametric elements to study the impact of delamination size and location on the dynamic response of smart VAT laminated plates. For aeroelastic analysis, the FE model is coupled with MSC. Nastran through Direct Matrix Abstraction Program (DMAP) and the structural parameters obtained with the FE model are supplied to MSC. Nastran solver to generate aerodynamic force matrix using Doublet Lattice Method (DLM). Various parametric studies are performed to find the most critical location of the delamination that severely affects vibration and flutter characteristics. The results showed that the delamination near the leading edge which faces free stream wind is found to be the most critical for the dynamic aeroelastic failure of structures. Further, the most effective location of the piezoelectric patches is identified to enhance the aeroelastic boundary of smart delaminated VAT composite plates using active displacement and velocity feedback gains.

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.

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.

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.

Multi-patch NURBS formulation for anisotropic variable angle tow composite plates

Conventional methods result in sharp variation of stiffness in analysis of variable angle tow (VAT) composite plates, which can be enhanced using high-order continuity of non-uniform rational B-spline (NURBS). To this end, a Bézier extraction-based isogeometric (IGA) model based on the higher-order shear deformation theory (HSDT) is developed for the parametric instability study of anisotropic VAT composite plates. Bézier extraction-based IGA approach expresses exactly the orientation function of VAT composites for multi-patch geometries, which provides accurate solution for general-shape plates. The accuracy and convergence of the developed framework is compared with recently formulated methods. Primary results shows that the developed concept solves VAT problems with much lower number of equations than variable kinematic models. Further, response characteristics of analyzed anisotropic VAT plates are presented and discussed considering various fiber orientation functions, boundary conditions and multi-patch geometry. Based on results, Bézier control mesh is more accurate than the NURBS-based IGA solution due to considering the continuity of strain field along coupled boundaries. Anisotropic VAT plates have severe deformation couplings and the current concept accurately captures this feature even with courser meshes. The interaction between boundary conditions and fiber orientations along edges affects mainly the dynamic instability regions (DIR) of modeled VAT plates, dislike to straight fiber ones. The dynamic instability opening (DIO) of VAT plates relies on the fiber orientation functions and boundaries. In anisotropic VAT plates, maximum frequency case and maximum buckling force case would not necessarily be the same. Modes shapes are variable of external force intensity and therefore the Rayleigh-Ritz based methods are not appropriate for dynamic stability analysis of VAT plates. It is essential to change the NURBS control mesh to the Bézier type to offer the required accuracy for the multi-patch domains.

Robust design optimization of variable angle tow composite plates for maximum buckling load in the presence of uncertainties

A robust design optimization algorithm is proposed for variable angle tow composite structures in the presence of uncertainties in the constituent material properties and applied loads. The proposed algorithm uses a stochastic perturbation method to propagate these uncertainties through to the simulated structural response, measured in terms of buckling load. The expected value plus a selected number of standard deviations of the response in the form of a bi-criteria problem. To describe the curvilinear fibres, two types of fibre path function, namely linear- and nonlinear-variation formulae, are adopted to illustrate the proposed methodology. A comparison between the resulting robust designs and deterministic designs is made, and changes to the final designs of fibre tow paths arising from the inclusion of uncertainty are discussed. It is shown that the robust designs out-perform the deterministic designs under real-world situations that include uncertainties.

Towards an understanding of variations in the buckling of tailored variable angle tow composite plates

In this paper, variable angle tow (VAT) composite plates tailored to enhance buckling performance are studied with the use of stochastic finite element method to quantify uncertainties in buckling measures arising from variations in material properties and fibre tow path. Detailed formulations for predicting buckling statistics in terms of mean value and standard deviation are derived to enable a perturbation-based stochastic finite element analysis. The derivations are built on a linear variation formula for fibre tow path and plate element based on the first order shear deformation theory. They are integrated with Taylor series expansion to propagate uncertainties from inputs to buckling performance measures, including buckling eigenvalues, critical buckling coefficients, etc. A twelve-layer VAT composite plate, with optimally designed fibre tow paths under various boundary conditions, has been investigated to illustrate the uncertainty quantification procedure. The performance of the perturbation-based stochastic finite element method has been validated using Monte Carlo simulation. Influences of variations in material properties and fibre tow path are thoroughly examined to understand the variability of buckling performance of VAT composites.

Buckling analysis of variable angle tow composite plates with a through-the-width or an embedded rectangular delamination

Variable angle tow (VAT) composite laminates, in which fibre orientation varies spatially in-plane in a continuous fashion yet is piecewise constant through-thickness, have been made possible by advanced automated fibre placement technology. Such designs have shown considerable potential to improve the performance of lightweight composite structures. In the present study, an analytical model is developed to study the buckling behaviour of VAT composite plates with a through-the-width or an embedded rectangular delamination under compression loadings. The proposed model can accurately capture the global, local and mixed buckling response of delaminated VAT composite plates. Both free and constrained modes are assumed in the delamination buckling analysis. A constrained point approach is employed to analyse the buckling response when contact occurs between delaminated layers. The accuracy and reliability of this proposed delamination buckling model is validated by finite element analysis and with prior results. The influence of delamination size, position and varying fibre orientation angles on the buckling response of delaminated VAT composite plates is studied by numerical examples. It is shown that the buckling loads decrease with an increase of the delamination size. The VAT laminates with an off-midplane delamination may lead to the delamination opening up, which further reduces the buckling loads. Finally, the mechanism of taking advantages of VAT laminates to improve the buckling performance of delaminated composite plates is thoroughly investigated in a parametric study. This study also shows that the residual buckling resistance of the delaminated composite plates can be significantly improved through using the VAT design concept.

Dynamic instability of variable angle tow composite plates with delamination

In this paper, the dynamic instability of variable angle tow (VAT) plates with a single rectangular delamination is studied using an analytical model. The analytical model is derived from the principle of potential energy based on the classical laminated plate theory. Both global and local behavior of delaminated VAT plates in the dynamic instability analysis are accurately captured by the use of multiple Legendre polynomial series. The equations for the motion in dynamic instability problem are derived using Hamilton’s principle. The dynamic instability regions are determined from the resulting Mathieu differential equations, which are solved using Bolotin’s approach. To validate the proposed analytical model, both critical buckling loads and natural frequencies of delaminated VAT plates are evaluated and compared with FEM results. The influence of delamination on the buckling load, natural frequency and dynamic instability region (DIR) of delaminated VAT plates is examined by numerical examples. A parametric study is subsequently carried out to analyze the effect of linearly varying fibre orientation angles on the dynamic instability response of delaminated VAT plates. Finally, the mechanism of applying variable angle tows to improve the dynamic stability performance of delaminated composite plates is studied.

Mixed shell element for static and buckling analysis of variable angle tow composite plates

A mixed quadrilateral 3D finite element, obtained from the Hellinger–Reissner functional, is presented for linear static and buckling analyses of variable-angle tow (VAT) composite plates. Variable-angle tows describe curvilinear fiber paths within composite laminae and are a promising technology for tailoring the buckling and post-buckling capability of plates. Due to the variable stiffness across the planform of the VAT plates, pre-buckling stresses can be tailored and redistributed towards supported edges, thereby greatly improving the buckling load. A linear mixed element called MISS-4 is used as starting point for this work. The element presents a self-equilibrated and isostatic state of stress. The kinematics lead to element compatibility matrix calculations based solely on the interpolation along element edges. The drilling rotations do not require penalty functions or non-symmetric formulations, thus avoiding spurious energy modes. The buckling analysis is reliably performed via a co-rotational formulation. In this work VAT plates with linear fiber angle variation in one direction, and constant stiffness properties in the orthogonal direction are studied. Numerical examples of VAT plates subjected to different loads and boundary conditions are investigated herein. The convergence of displacements, stress resultants and buckling loads are presented, and comparisons with numerical results, obtained using the S4R finite element of Abaqus and the pseudo-spectral Differential Quadrature Method, are shown.

Buckling and postbuckling of variable angle tow composite plates under in-plane shear loading

A geometrically nonlinear analysis of symmetric variable angle tow (VAT) composite plates under in-plane shear is investigated. The nonlinear von Karman governing differential equations are derived for postbuckling analysis of symmetric VAT plate structures which are subsequently solved using the differential quadrature method. The effect of in-plane extension-shear coupling on the buckling and postbuckling performance of VAT composite plates is investigated. The buckling and postbuckling behaviour of VAT plates under positive and negative shear is studied for different VAT fibre orientations, aspect ratios, combined axial compression and their performance is compared with that of straight fibre composites. It is shown that there can be enhanced shear buckling and postbuckling performance for both displacement-control and load-control and that the underpinning driving mechanics are different for each.

Postbuckling analysis of variable angle tow plates using differential quadrature method

Variable angle tow (VAT) placement has been shown to improve the buckling resistance of composite plates under compression loading. The problem of tailoring the in-plane tow path of VAT composite plates for enhanced postbuckling resistance is studied in this work. A pair of coupled geometrically nonlinear governing differential equations in terms of stress function and transverse displacement, based on classical laminated plate theory, is derived for postbuckling analysis of VAT plates. The differential quadrature method (DQM) is applied to solve the differential equations and the resulting nonlinear algebraic equations solved using a Newton–Raphson algorithm. The DQM was applied to study the postbuckling problem of VAT composite plates subjected to axial compression under different plate boundary conditions. The numerical results of DQM are compared with detailed finite element analysis and the relative accuracy and efficiency of the proposed DQM approach is studied.

Postbuckling optimisation of variable angle tow composite plates

The potential for enhanced postbuckling performance of flat plates using variable angle tow (VAT), in comparison with conventional laminated composites, has been shown previously. This paper presents an optimisation strategy for the design of postbuckling behaviour of VAT composite laminates under axial compression. The postbuckling performance of composite laminated plates for a given compression loading is assessed by studying both the maximum transverse displacement and the end-shortening strain. For the postbuckling analysis of VAT composite plates, an efficient tool based on the variational principle and the Rayleigh–Ritz method is developed. In the optimisation study, a mathematical definition based on Lagrangian polynomials, which requires few design parameters, is used to define a general fiber angle distribution of the VAT plate. A genetic algorithm is subsequently used to determine the optimal VAT configuration for maximum postbuckling performance. The optimisation of square VAT laminates under compression loading for different in-plane boundary conditions is studied and compared with straight fiber designs.