Variable Stiffness Composites Research Articles

Buckling optimization of variable-stiffness composite plates with two circular holes using discrete Ritz method and potential flow

Potential flow around two equal-radius cylinders is derived analytically and applied to generate the curvilinear fiber path of variable-stiffness composite (VSC) plates with two circular holes. As complex variable theory and conformal mapping are used to generate the potential flow around two equal-radius cylinders, the location and size of the two equal-radius circular holes are arbitrary. By changing the angle of incoming flow, the global fiber angle of a variable-stiffness lamina can be simulated and the local fiber orientation angle at any point is determined by the global potential flow field. Buckling performance of variously-shaped VSC plates with two circular holes are investigated via a novel numerical method − discrete Ritz method (DRM). DRM combines the extended interval integral, Gauss quadrature, and variable stiffness within a rectangular domain and builds a discrete energy system to simulate the plate, which allows the geometric boundaries of the plate to vary. The strain energy of a plate is modeled in the rectangular domain, discretized using Gauss points, and is characterized by variable stiffness, which characterizes both the material distribution and plate geometry simultaneously. After that, a three-dimensional sampling optimization (3DSO) method is adopted to optimize the curvilinear fiber configurations of VSC plates, and its buckling performances are compared with those of constant stiffness composites (CSC) with optimal straight fibers. Significant improvements on load-carrying capacity can be achieved compared to straight ones, demonstrating that using potential flow is one of the most efficient way to generate curvilinear optimal fiber path with maximum load-carrying capacity for VSC plates with material discontinuity. Moreover, DRM exhibits good precision and stability for buckling analysis of VSC plates with complex geometries.

Fundamental Frequency Layer-Wise Optimization of Tow-Steered Composites Considering Gaps and Overlaps

AbstractThe advent of Automated Fiber Placement (AFP) in aerospace composites lay-up and manufacturing has allowed orientations to vary along pre-defined curved directions rather than being forced to remain constant within the lamina. These composites are called Variable Angle Tow (VAT) or Variable Stiffness Composites (VSC). Despite the enhancements in mechanical performance offered by VAT, constraints from the manufacturing process hinder their full potential. This paper explores the effect of primary defects, i.e., gaps and overlaps, on optimal design and fundamental frequency optimization. For doing so, the Carrera Unified Formulation (CUF) and the Defect Layer Method (DLM) are integrated directly into the optimization process to provide an efficient and cost-effective framework for modeling the structural behavior and manufacturing process of VSCs. Particular attention is given to manufacturing and tow-steering simulation to quantify and map defects for each laminate layer. This research serves a dual purpose: (i) examining the impact of process-induced defects on achieving an optimal design and (ii) exploring how the choice of structural theory may affect the optimal solution.

Machine learning-based modelling, feature importance and Shapley additive explanations analysis of variable-stiffness composite beam structures

In the current work, the displacement and inner stress state of orthotropic, composite, variable-stiffness beam structures is investigated, combining the extended Kantorovich method (EKM) with machine learning (ML) modeling and explainable artificial intelligence (AI) techniques. The complete displacement and inner stress fields of variable stiffness (VS) composite beams with a through-thickness variation of their material properties are computed for different orthotropy, material gradation and slenderness attributes. Both deep learning and tree-based machine learning architectures are considered, identifying high accuracy and low computational cost modeling architectures. Thereupon, the significance of underlying fundamental design features is assessed, classifying their importance for the first time. In particular, material orthotropy, stiffness gradation and slenderness are analyzed for different boundary conditions, elucidating their interdependence and relative significance. It is shown that reduced end kinematic constraints overall favor the importance of stiffness gradation. Differences in the importance of each design feature for the transverse displacement and inner normal and shear stress fields are identified and quantified using Shapley additive explanations (SHAP) analysis. It is found that material orthotropy is more important than stiffness gradation for the transverse displacement rather than for the normal or shear stress fields developed, for which their relative importance is inverted. What is more, evidence is provided that increased gradation and orthotropy values have opposite signed contributions to the stress fields developed at critical positions of the VS composite, while they uniquely yield equal-signed contributions to the transverse displacement field.

Thermoelastic geometrically nonlinear analysis and optimization of variable stiffness composite plates in presence of buckling

Thermoelastic geometrically nonlinear analysis and optimization of variable stiffness composite plates in presence of buckling

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

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

Frequency veering and nonlinear coupled vibration analysis of variable stiffness composite plates with curvilinear fiber paths

The coupling forced vibration behaviors of a variable stiffness plate with curvilinear fiber paths at mode veering regions caused by variations of fiber angle have been rarely found in previous papers, especially in the presence of considering the complex material nonlinearity of composite materials. In this paper, the variable stiffness plate of carbon fiber reinforced composites (CFRC) with curvilinear fiber paths is taken as the research object, and the influence of curvilinear fiber paths on its natural characteristics and vibration responses are studied in detail on the basis of considering the material nonlinearity of CFRC. Firstly, it is found that the variations of curvilinear fiber paths can cause complex frequency veering behavior of the variable stiffness plate, and lead to the appearance of irregular modal shapes formed by modal interaction. Then, the forced vibration response of the variable stiffness plate in the frequency veering region is analyzed methodically, and the variation rules of the vibration response under modal interaction is obtained. After analyzing the nonlinear vibration response of the variable stiffness plate under the combined influence of modal interaction and CFRC material nonlinearity, the mechanism of complex nonlinear dynamic behavior of variable stiffness plate in frequency veering region is further investigated.

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

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

Structural optimization of composite corrugated cores with variable stiffness

Outstanding progress in manufacturing technology of fiber-reinforced composite materials, allows designers to go beyond classical design rules. Composites with variable stiffness are an innovative division of composite materials that provide more efficient designs for engineers due to their diverse design parameters. In this study, a straightforward methodology is presented for the design of sandwich panels with corrugated variable stiffness core. Based on the homogenization method, a unit-cell of the sandwich panel is used as a representative volume element for the sandwich panel with corrugated core. Due to multiple design variables, conventional numerical and analytical methods cannot be applied to variable stiffness composites, thus PYTHON/ABAQUS framework is developed to determine the extensional and bending behavior of unit-cells. Details of this framework including PYTHON scripting and ABAQUS finite element solver are given along with their corresponding functions. Furthermore, the elitist genetic algorithm is incorporated into a framework for the optimum design of variable stiffness cores. Two optimization problems, namely composites with variable thickness and variable angle are considered for unit-cells, and associated constraints, boundary conditions, and fitness functions are discussed in detail. In addition to the benefits of the proposed framework, the obtained results reveal notable enhancement in the extensional and bending behavior of unit-cell. Results reveal that corresponding displacements are significantly improved (up to 80%) when using the optimum variable thickness case. For variable angle cases, the improvement of displacements is up to about 28%.

Equivalent single layer and layerwise models for flutter and buckling analysis of supersonic variable stiffness laminated composite plates

This work provides an assessment of Equivalent Single Layer (ESL) and Layerwise finite element models, aimed for panel flutter and buckling analysis of supersonic variable stiffness composites, exploring variable-order shear deformation theories and Lagrange z-expansions. Numerical applications focus on simply supported panels with curvilinear fibre composite layers, including a cross-ply configuration for comparison purposes, and various side-to-thickness ratios. The accurate prediction of transverse shear and bending–twisting coupling is highlighted, as rather necessary for proper tailoring and analysis of supersonic curvilinear composites, especially when including in-plane loads. This is attainable through high-order ESL descriptions, in thin panels, even without thickness stretching.

Energy absorption of variable stiffness composite thin-walled tubes on axial impacting

In order to realize the engineering application of automated fiber placement for composite laminates, a method of the variable angle fiber placement was proposed to design the tube structures of variable stiffness composites based on the quadratic Bezier curve. The axial crushing responses were simulated to investigate the energy absorption characteristics of composite tubes. The results showed that the method of the variable angle fiber placement contributed to the improvement of the energy absorption effects. The maximum crushing force efficiency of the variable stiffness composite thin-walled tubes designed by the method was 49.04% which was 106.48% higher than the constant stiffness composite tube. The results could be helpful for the process of automated fiber placement and the design of the energy absorption for composite thin-walled structures.

Variable-stiffness composites optimisation under multiple design requirements and loads

The aim of this paper is twofold. On the one hand, it presents a methodology for the deterministic optimisation of a general class of variable-stiffness composite (VSC) structures, including a solution obtained by using laminæ with a curvilinear fibres-path and variable-thickness, by considering different design requirements under multiple load cases. The considered framework is the multi-level design methodology based on the polar parameters (PPs) to describe the macroscopic behaviour of the VSC structure. Particularly, only the first-level problem is addressed in this work: the design variables are, thus, the PPs and the thickness of the VSC laminate, whose spatial distribution is described via basis spline (B-spline) surfaces. The goal is to minimise the mass of the VSC structure subject to design requirements on feasibility, strength, first buckling load and maximum curvature of the fibres-path. This latter is formulated as an equivalent (conservative) constraint in the PPs space, regardless of the fibres-path within each lamina. Moreover, a general formulation of the gradient of the requirements related to buckling load and strength is proposed, which takes advantage from the main properties of B-spline entities and PPs. On the other hand, this paper aims to propose a new benchmark problem that is representative of a panel belonging to the fuselage of a standard civil aircraft subjected to multiple loading conditions. To this end, a wide campaign of numerical tests has been performed by considering a sensitivity analysis of the optimised solution to: (a) the integer parameters involved in the definition of the B-spline entities describing the distribution of the PPs and, possibly, of the thickness, (b) the type of VSC structure, (c) the type of deterministic optimisation algorithm. The results can be used as a database to assess the effectiveness of different design strategies against the optimised solutions presented in this paper.

Uncertainty quantification of bistable variable stiffness laminate using machine learning assisted perturbation approach

Morphing structures have received growing interest in aerospace structures and wind turbines due to their rapid shape-changing ability in response to the change in operating conditions. Bistable laminates using variable stiffness composites are considered potential candidates in morphing structures for their ability to tailor the design space with a plethora of multiple stable configurations and satisfy the conflicting requirements of load-carrying capacity and deformability. Even though extensive works have been reported on the analysis and design of the cured shape of unsymmetrical variable stiffness laminates for morphing application, the effect of uncertainty in design variables on the behavior of bistable laminates is not profoundly assessed in the literature. In particular, uncertainty propagation through a highly non-linear map can lead to a significant discrepancy between the numerically predicted and experimental observations. Therefore, for adaptability in practical application, it is imperative to quantify the uncertainty as well as to characterize the non-linearity present near the design point of interest. In this work, a general purpose machine learning assisted uncertainty quantification (MLAUQ) framework is developed and demonstrated on unsymmetrical bistable laminate. The study considers three different variants of the approach based on the order of approximation (O(hk)) used for the training purpose. It is found that the MLAUQ−3 approach performs better than other approaches, and a theoretical justification for the same is provided. The method relies on the fact that expensive computation of the Hessian required for standard perturbation approaches can be bypassed by training a neural network while retaining accurate gradient information near the design point of interest. In the case of bistable laminate, a network trained with a few training samples can capture the local model non-linearity. Further, numerical investigations reveal that the proposed approach is computationally efficient and accurate compared to traditional uncertainty quantification (UQ) approaches.

Dynamic Performance of Hygrothermal Mechanically Preloaded Variable-Stiffness Composite Fairing Structures

Since the introduction of variable-stiffness composites, the design philosophy for high-performance lightweight composite structures has broadened greatly. Indeed, variable-stiffness composites have been shown to increase buckling performance and dynamic stability, as well as to modify the dynamic response by tailoring stiffness distributions. Thus, efficient linear analysis tools play a significant role in the early design of variable-stiffness structures, allowing designers to identify many viable solutions when considering preloaded dynamically excited aerospace components. To address this need, a Ritz-based method for eigenfrequency and dynamic instability analysis of hygrothermal and mechanically prestressed variable-stiffness laminated doubly curved payload fairing structures is presented. Flexibility in modeling and design is achieved using Sanders–Koiter-based shell kinematics that allow general orthogonal surfaces to be modeled without further assumptions on the shallowness or on the thinness of the structure. The efficiency of the proposed Ritz method is enabled by using Legendre orthogonal polynomials as displacement trial functions. By comparing the present approach with finite element solutions for variable-curvature, variable-angle tow fairing shell geometries, excellent accuracy is shown, accompanied by an order-of-magnitude reduction in variables by the present method. Original solutions are presented comparing the dynamic behavior of prestressed constant and variable-stiffness composite shell structures, showcasing the viability of the variable-stiffness concept to significantly increase the structural performance of critical doubly curved variable-curvature components such as launch vehicle payload fairings.

Isogeometric-based mapping modeling and buckling analysis of stiffened panels

Modeling and analysis of stiffened panels are two key technologies in the design of aerospace thin-walled structures. For the stiffened panels with complex geometry, classical finite element analysis (FEA) and conventional isogeometric analysis (IGA) based on explicit geometry usually require time-consuming and labor-intensive geometric processing, and additional coupling matrices to be ready for analysis. In this study, a new method for modeling and buckling analysis of stiffened panels is proposed, which provides a more efficient and simpler way. During the modeling process, the stiffeners are treated as curves on surfaces, which is not explicitly defined using the control-point-based representation of curves, but implicitly defined using parameter curves in the parametric space of the surface. Mapping modeling provides more accurate geometric description and transfer the complex modeling problems (three-dimensional space) of stiffeners on free-form surface into simple modeling problems in the regular parametric space (two-dimensional space). During the buckling analysis process, a new mapped stiffener element based on mapping modeling is proposed, which can model the section of the eccentric stiffener without changing the geometry. The precise normal information of the Non-Uniform Rational B-Splines (NURBS) surface can ensure that the stiffeners are perpendicular to the skin. In addition, the coupling of the stiffener and the skin is automatic, without any additional coupling matrix. This buckling analysis framework realizes the complete integration of modeling and analysis. Furthermore, for the stiffened panels with cutouts, the trimmed surface analysis (TSA) method is extended to be used for the numerical integration of the trimmed stiffeners, which means that no additional geometric process is required. Finally, four numerical examples of different types of stiffened panels are constructed, involving metal, trimmed surface, classical grid-stiffener, free-form surface, variable-stiffness composites, and curvilinear grid-stiffener. Several numerical examples of static and buckling analysis of stiffened panels with high fidelity demonstrate the effectiveness of the proposed framework.

Stress analysis of adhesive-bonded tubular-coupler joints with optimum-stiffness-composite adherend under axial tension, pressure, and thermal loading

With descriptions of fiber orientation variation, variable-stiffness composites (VSCs) have opened many new opportunities for structural designs because they offer tailorable material properties to increase the performance of composite structures. Using the concept of VSCs, this study aimed to determine the optimum adhesive-bonded tubular coupler joints subjected to tension, pressure, thermal load, and/or combinations thereof. The joint consists of three axisymmetric cylindrical adherends. The two inner adherends are made of a linear elastic isotropic material, whereas the outer adherend is composed of a symmetric balanced laminate. As the most complicated cases among others, owing to various stress components induced in the adherends and adhesive, the joints under these dominant in-plane normal force loadings are modeled to efficiently minimize stress concentrations in the thin adhesive layer. The unified theory of adhesive-bonded tubular joints was modified one step further to compute the optimum fiber angle variation using the constraint of linear distribution of the axial resultant forces in the adherends. From the methodological viewpoint, the results unveiled that the interfacial shear stress concentration in the adhesive layer was considerably diminished or even vanishes under all loading scenarios. The possibility of multiple optimum composite joints can be achieved, which otherwise are not straightforward to determine when using other analytical tubular joint models and optimization methods. Parametric studies of geometrical variables were also performed to provide a design guideline for optimum coupler joints with efficient mitigation of peak adhesive interfacial stresses.

Enhancing delamination resistance with intralaminar stiffness tailoring: A numerical study of variable stiffness composites

In general, strategies and techniques involving intralaminar stiffness tailoring are primarily used to improve compressive capacity (buckling and post-buckling) or tensile properties rather than interlaminate toughness for composites. However, in this work, we show the potential of utilizing intralaminar stiffness tailoring to enhance delamination resistance by specifically investigating the perforated variable stiffness composite plate with a different layout of curvilinear fibers. Firstly, a standard finite element model with the typical cohesive interface to evaluate the delamination performance of different composites is established and is verified against the averrable experimental data of intact constant stiffness composites. Then, two kinds of curvilinear fibers for the perforated composite plates under the Mode I and Mode II delamination test are investigated considering the different fiber layouts and the size of the central cutout. The reaction force–displacement curve and the R-curve of the composites with the curvilinear fibers are obtained to show the enhancement effect of the intralaminar stiffness tailoring on the delamination resistance. The interlaminate toughness of the curvilinear fibers laminate can overperform the straight fibers laminate up to 71% and 63% in the mode Ⅰ and mode Ⅱ cases (r = 8). The underlying mechanism is discussed in detail to inspire new ideas for intra-laminar stiffness tailoring to improve interlaminar toughness.

A partition of unity level set method with moving knot CS-RBFs for optimizing variable stiffness composites

A partition of unity level set method with moving knot Compactly Supported Radial Basis Functions (CS-RBFs) is proposed for optimizing variable stiffness composite structures. The iso-contours of a level set function are utilized to represent the curved fiber paths, and the tangent vector of the iso-contour defines the orientation of fiber. The level set function of the full design domain is constructed according to the Partition of Unity (POU) method by a set of local level set functions defined on an array of overlapping subdomains, and they are constructed by using the CS-RBFs. The positions of knots are iteratively changed during the optimization to improve the performance of composite structures. Several examples of compliance minimization are presented.

AFP tool path planning for manufacture of variable stiffness composites

Variable stiffness composites (VSC) demonstrate enhanced mechanical properties compared to conventional carbon fiber reinforced plastics (CFRP). Finite element analysis (FEA) is used to optimize layup directions, which need to be transformed into viable tool paths for automated fiber placement (AFP) processes. Manufacturing defects such as wrinkles, gaps, and overlaps should be handled. In this study, a novel path planning algorithm is developed for variable steering in AFP. The fiber directions are clustered and for each cluster a reference curve is fitted as B-Spline, where manufacturing constraints are considered. The proposed tool path planning approach is successfully demonstrated in a case study.

Variable stiffness optimization algorithm for vibration optimization of variable-stiffness composite plates

A novel variable stiffness optimization (VSO) algorithm is proposed for vibration optimization of variable-stiffness composite (VSC) plates uisng linear variation fiber path function. Firstly, an independent 2D-sampling optimization method (2DSOind) is employed to generate a good initial point. Then, optimization is conducted in multiple phases, within each phase, the problem is simplified to an approximately one-dimensional linear search problem. By dividing the fiber angles into two groups, VSO algorithm designs the laminate between two fiber angle groups sequentially and iteratively, making the stiffness of VSC plate stiffened part by part. Lastly, the VSC plate is redesigned accounting for the curvature constraint. The sequential permutation search (SPS) algorithm is coupled into VSO to optimize one group of fiber angles whilst the angles of another group are fixed during the iteration. Under a variety of boundary conditions, the free vibration frequencies of VSC plates are optimized using VSO. In comparison to conventional layerwise optimization (LO) and genetic algorithm (GA), the present results demonstrate much higher efficiency and similar robustness. Moreover, the maximum fundamental frequency of VSC plates can be improved by about 10% in comparison to straight fiber laminates.

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

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