Review of the mechanical performance of variable stiffness design fiber-reinforced composites

2.1 Design for maximum strength

For the necessity of the assembly and processing, numerous joints and cutouts were manufactured inevitably in the composite components. These regions of joints and cutouts were stress concentrated resulting a complicated stress state. Tailored fiber orientation displayed such tremendous potential to prevent stresses from concentrating and ensure the continuity of fibers [24], [25]. Tosh and Kelly [20] aligned the fiber orientation with principal stress vectors and load path in laminates, respectively, which improved the strength for a component with an open hole and a pin-loaded hole under tension. IJsselmuiden et al. [26] derived a series of equations for the conservative strength envelope that can be applied to formulate the strength- and stiffness-based designs of variable stiffness lamination. Based on this design method, panels under the binding force between axial and shear loads were designed for maximum strength and stiffness.

Khani et al. [27] investigated the design of variable stiffness panels to get the maximum strength. The lamination parameters [28] were considered as the design variable in process to ensure the largest design space and high-efficiency convergence. The maximum strength was defined as the minimization of critical failure index proposed as the reciprocal of safety factor. Finally, this design method was applied to a supported square panel with a center hole under tension and the simulations were performed to confirm the efficiency of the method. The numerical results show that this VSD method exhibited about three times improvement in strength about the quasi-isotropic design. Falcó et al. [29] used finite element analyses to simulate the first-ply failure of variable stiffness panels under tensile load. This analysis approach can be applied directly to the optimal design of fiber orientation to avoid tensile failure.

Legrand et al. [30] mentioned a design procedure to reduce concentration, which uses genetic algorithm to define the fiber orientation angles directly in each finite element of composite laminate. The quantity of meshed element decides the improvement extent of the accuracy and the continuity of the fiber trajectory. Therefore, this method has low efficiency because of the large number of analyses required to carry out. To achieve the maximum load-carrying capacity of the composite laminates with a central hole, Huang and Haftka [31] described a procedure for steering the fiber orientations around the hole in a single layer of a multilayer. The fiber angles are directly designated as the design variable. To avoid local convergence, the authors alternated the conjugate gradient Fletcher-Reeves method from DOT [32] and the genetic algorithm iteratively. The intention of this formulation is to use the genetic algorithm to move the optimization to a better region, and the conjugate gradient method is used to quickly converge to the local optimum in this region.

2.2 Design for buckling resistance

The buckling performance is often the primary design consideration of composite structures in aerospace applications and other fields. The VSD method provides the designer more tailoring options to scheme fiber orientations with desired buckling resistance. For example, Tatting and Gürdal [33] showed that up to 60% enhancement can be gained by the idea of curved fiber paths to design variable stiffness panels with a center hole for maximum buckling loads. Through defined stiffness variation over a structure domain, IJsselmuiden et al. [34] obtained the significant improvement in excess of 100% in buckling load-carrying capability of variable stiffness lamination under the simulation experiment condition.

Lund et al. [35] applied the discrete material optimization approach, which evolved from multiphase topology optimization [36], to design variable stiffness plates for maximum buckling load. The stacking sequence and fiber orientation are computed simultaneously using the gradient-based method. Based on numerical simulations, Lopes et al. [37] demonstrated the advantage of variable angle laminates with compressive buckling and first-ply failure. During their analysis, a smooth reference curve, as proposed by Tatting and Gürdal [33], is defined as the fiber path orientation. Wu et al. [22], [38] and Khani et al. [39] also presented the optimal design methods of buckling capacity of laminated composite. However, compared to the conventional composite design, the VSD method is more complicated because of the large number of variables with respect to design continuity and manufacturability. This is the reason why many literatures used a numerical simulation tool to verify their conclusions. Combining the manufacturing constraints [40], [41], Weaver et al. [42] used an embroidery tow-placed machine to manufacture the variable stiffness panels and test such plates. The test results show the postbuckling performance seriously improved relative to quasi-isotropic lamination.

To achieve the VSD of maximum buckling load, Setoodeh et al. [43] presented a generalized reciprocal approximation method. The fiber orientation angle is selected as the design parameter. The buckling load is expanded according to the inverse of the stiffness tensor. However, the design issue is changed into a nonconvex design problem because of using fiber angles as the design parameter. Hence, the outcome of the optimization design process is highly dependent on the starting fiber orientation angles. To solve this problem, IJsselmuiden et al. [34] attempted to change the design parameter to the lamination parameter in the following research. Thus, a conservative reciprocal approximation was created. The approximation scheme reproduces the homogeneity properties of the buckling factor and is convex. The analysis results showed that the yielding buckling load is significantly improved with respect to quasi-isotropic laminates.

2.3 Design for postbuckling capacity

Various experimental results demonstrated that bearing capacity of the thin-shell component for composite materials beyond initial buckling under axial compression load. They could sustain load in a postbuckling state without being broken down, which offers the potential of even lighter structures than are currently used on the new generation of composite passenger aircraft [44], [45]. The VSD method can significantly increase the postbuckling performance, which was early confirmed by Wu et al. [46]. To more accurately predict the postbuckling capacity of variable stiffness panels, Lopes et al. [47] considered the effect of the residual thermal stresses that arise from the curing process in the analysis process. The predicted results compared well to the experimental results. In addition, the tow-steered design embodied more tolerance to traditional design according to the experimental data.

Based on stress function and transverse displacement, Raju et al. [48] presented a comparatively new method that is known as the differential quadrature method (DQM). This method was proposed to solve the postbuckling analysis of variable stiffness plates under axial compression. Afterwards, Raju et al. [49] presented an asymptotic numerical method that transforms the nonlinear problem into a set of well-posed recursive linear problems. This approach was then successfully implemented using a generalized differential-integral quadrature method to solve the nonlinear postbuckling problem of variable stiffness laminates. A similar approach was adopted by White et al. [50], who made a successful application of this method into the postbuckling analysis of variable stiffness cylindrical shell. White et al. [51] further investigated the postbuckling performance of two variable stiffness cylindrical shells with the tow overlapping and the dropped to avoid overlapping under axial compression. The specimen models were analyzed by finite element method in both linear and nonlinear ways. Moreover, the results were compared to experimental results.

Using Koiter’s asymptotic theory and the methods of generalized differential and integral quadrature, White and Weaver [52] constructed an efficient model of panel’s postbuckling. The tangent stiffness of the variable stiffness panel’s postbuckling model was maximized with a genetic algorithm. Besides, they also said that multiobjective VSD, in which the performance such as stiffness and strength were taken into consideration, would be realized while the external forces exist. Wu et al. [53], [54] presented an optimization strategy using a genetic algorithm for the design of postbuckling performance of variable stiffness plates. To illustrate the superiority of the variable stiffness plates, they compared the simulation results to an optimal straight fiber plate.

2.4 Design for maximum fundamental frequency

Fundamental frequency is one of the key factors that affect the dynamic performance of composite structure and has a direct influence on the performance and service life of the designed parts. One of the earliest applications of a variable stiffness concept to improve the vibration performance of composite plates was reported by Leissa and Martin [55]. Moreover, the numerical results showed that using nonuniformly spaced fibers could improve the fundamental frequency by as much as 21%.

To achieve the maximum fundamental frequency, Honda et al. [56] introduced a new vibration design method of variable stiffness plates based on the Ritz energy methodology. The Ritz energy method was applied to determine the frequency parameters of the plates. The results of design were compared to the traditional plates with parallel fibers to show its superiority. Honda and Narita [57] also suggested steering the fibers to the direction of the projections of contour lines for the cubic surfaces. For determining fiber orientation, the parameters of the cubic polynomial surfaces were defined as design variables.

The research on the design of composite conical shell for maximum fundamental frequency was carried out by Blom et al. [58]. They designed conical shells using curvilinear fibers for maximizing fundamental frequencies, and manufacturing constraints were taken into account in the process. Luersen et al. [59] presented a design method of variable stiffness cylindrical shell for maximum fundamental frequency using a surrogate method to optimize the fiber path parameters. The maximization of the fundamental frequency of the shell was studied for three different boundary conditions including clamped-free, clamped-clamped, and pinned-pinned edges.

Akhavan and Ribeiro [60] proposed a new p-version finite element model that can ensure that the fiber curvature does not exceed the maximum value and the fiber orientation changes linearly based on the third-order shear deformation theory. It is also found that the mode shape can be changed using the VSD method, which can increase or decrease the natural frequency significantly. In a later research [61], with the same model and method, the relationship between fiber angle and deflections and normal and transverse stresses was studied. The effect of fiber angle changes on the mode shape and natural frequency were investigated by Ribeiro and Akhavan [62] based on same element model. It was found that the effect of the variation of fiber orientation on the higher-order mode was more severe than the first-order mode. Linear and nonlinear vibration and the variation of fiber orientation may lead to different dynamic behaviors and may behave in a more rigid pattern in a few cases.

2.5 Minimized weight or thickness

To lower the structural weight, composites are widely used in aerospace structures. Therefore, some researchers selected the weight and thickness as the design target directly. Blom et al. [63] used a series of polynomial and trigonometric functions to present a concept of streamline analogy, which can be successfully used to show the thickness distribution in variable fiber angle design. It realized the fiber orientation angle design for minimizing the maximum ply thickness, maximizing surface smoothness or combining these objectives.

Parnas et al. [64] presented an approximate methodology for the design of laminated composites with curvilinear tow to minimize the weight. They used the cubic Bezier curves and bicubic Bezier surfaces to formulate fiber angles and layer thicknesses, respectively. The total weight of the composite structure was selected to be optimal objective. The Tsai-Hill criterion was used for the failure analysis on the first ply. The result testified that the removal of straight-fiber restriction would provide increased flexibility to the designer and yield a lighter design, as the design space was enlarged.

2.6 Multiple-property simultaneous design

Some researchers tried to consider multiple properties simultaneously in the scheme of VSD and achieved the scheme using the multiobjective optimization theory. Honda et al. [65] used a multiobjective optimization approach to define the curvilinear fiber shapes in variable stiffness plates. Two mechanical properties were considered as the objective function. They included the in-plane strength represented by the Tsai-Wu failure index and the natural frequency of the panel. The numerical simulation showed that the two mechanical properties of the designed plate were higher than the plates with straight fibers. Alhajahmad et al. [66] applied the variable stiffness concept to the design of fuselage panel. They optimized the fiber orientation for maximum buckling loads and maximum failure loads.

Nik et al. [67] maximized simultaneously the in-plane stiffness and buckling resistance using a hybrid approach combing the polynomial regression and the nondominated sorting genetic algorithm-II (NSGA-II) [68] for the VSD of composite laminate. The approach of polynomial regression was chosen because it can simplify the analysis model for constructing a surrogate model. NSGA-II was a multiobjective evolutionary algorithm that is used to quickly find a set of much better distributed solutions and better convergence to a multiobjective optimization problem [69]. The results of numerical simulation demonstrate that this hybrid algorithm converges faster than the traditional genetic algorithm and the curvilinear fiber trajectory can increase both buckling load and stiffness simultaneously higher than the quasi-isotropic. In the poststudy [70], the buckling load of four different sizes of variable stiffness laminate was researched. The final result shows that the buckling capability of variable stiffness laminate could be reduced, as the laminate sizes are smaller than the minimum turning radius of fiber placement.

2.7 Design for other performance

Besides the above-mentioned mechanical performance, there were other performances that had been considered in VSD. For instance, Setoodeh and Gürdal [71] used the strain energy criterion to establish the design conditions for fiber orientation. They thought that an optimum solution was achieved when the strain energy was at a minimum. In the literature [72], [73], they presented the minimum compliance design of variable stiffness panels using fiber orientation as continuous spatial design variables. Later, to improve the efficient computation of laminate design, they [74] used lamination parameters as the design variables instead of the fiber orientation to perform VSD under in-plane and out-of-plane loadings.

Stodieck et al. [75] analyzed the influence of different layer orientations on the aeroelastic behavior using an idealized composite wing model as the research object. The results proved that the variable angle design could bring positive impact on such performance as the divergence speed, flutter speed, and gust response. The maximum shear, bending, and torsion loads were reduced by 18%, 29%, and 16%, respectively, using a variable stiffness laminate rather than a standard traditional laminate.

The multistable composite has attracted more people’s attention in recent years because it can be used to make simple, reliable, and lightweight deployable structure and widely used in aerospace vehicle design and manufacturing. Applying the concept of variable angle to the design of multistable structure to ensure the continuity of the fiber can impart additional structural strength meanwhile. Panesar et al. [76], [77] presented an approach to develop a bistable finite element model that can be used for VSD. The model can accurately predict the cured shape of variable stiffness laminates. In a later research [78], thermally induced laminates were studied to further promote blended bistable laminates for morphing flap application. Sousa et al. [79] also carried out a research on the bistable variable stiffness composite laminates. They analyzed the geometry of the curing shapes obtained with morphing laminates as well as its stability properties. The laminate was divided into two zones with different lay-ups from the view of plan form during the design process. Combined with the finite element model produced by ABAQUS, the variation of displacements and curvature of the multistable variable stiffness laminate were analyzed during the cool-down stage.

3.1 General numerical optimization design methods

The numerical optimization techniques mentioned here mainly include the gradient-based optimization methods and its derivate techniques using the convexity of the objective function or converting the nonconvex function into convex to solve the optimization. All these types of approaches have a wide application scope, and it has been applied in every optimization field at a very early time. To solve the issue of VSD, the design parameters of the composite material, such as tow angle, are viewed as directly or indirectly variable. Stodieck et al. [75] defined the fiber angle function as a linear variation of the fiber orientation along the coordinate. The aeroelastic behavior was formulated based on the energy balance equation for a conservative system. Meanwhile, the impacts of the fiber angle variation on free vibration, flexural axis, flutter and divergence speeds, and gust loads were investigated. In a further work [82], using the modified wing structural model, a multiobjective optimization method was presented. This method can make the simultaneous minimization of correlated gust loads at two different design airspeeds realizable. However, the author also pointed out that this method is appropriate only for the simple rectangular plate wind model. For wing with different geometries or further complications, more study is required.

Wu et al. [83] developed an enhanced two-level optimization framework for the buckling optimization design of variable stiffness plates. This framework begins with establishing explicit stiffness matrices according to material invariants and lamination parameters. Subsequently, B-spline basis functions and Lagrangian polynomials were applied to determine the optimal fiber angle at each control point for the maximum buckling load. The advantages of this VSD framework are that only less design variables are required and the continuity of the fiber orientation is ensured. To reduce defects such as fiber wrinkling, the continuous Bezier curves are then replaced by piecewise quadratic Bezier curve, which is used to create the curved tow path [84], [85].

3.2 DQM

DQM is a numerical discretization technique that approximates derivatives using a linear weighted sum of all the functional values in the domain. For instance, as for 1D interval division, the kth derivative at the ith discretization point is

where x_i is the set of discretization points in the x-direction and Aij(k) is the weighting coefficients of the nth derivative. Recently, it has been proven that the solution of the variable coefficient, higher-order differential equations that govern the flexural behavior of variable stiffness plates can be obtained using this method [86].

Based on this method, Groh et al. [87] developed a reduced 2D equivalent single-layer formulation for the flexural behavior of variable stiffness plates incorporating transverse shear effects. In a further research, Groh and Weaver [88] realized variable stiffness plate buckling design by establishing and solving the equilibrium equations with DQM methods that use fiber orientation and lamination thickness as the variables. Raju et al. [89] and White et al. [50] realized the postbuckling design of variable stiffness plate and curved shell, respectively, by solving boundary equations with DQM methods. Tornabene et al. [90], [91] developed a strong-form finite element method based on DQM, which has been used to study the static and vibrational responses of doubly curved variable angle tow (VAT) laminates.

3.3 Intelligent optimization design methods

Intelligent optimization methods, also known as the modern heuristic algorithm, which possess the qualities of global optimization, high universality, are suitable for parallel processing. It is usually based on a strict theory rather than an expert experience and gets the precise or approximate optimal solution within a certain period of time theoretically.

For the purpose of reducing high computation time consumption, Bardy et al. [92] used an optimal genetic algorithm to detect the optimum fiber orientation and to improve the stiffness and strength of a square composite laminate. In the optimization process of genetic algorithm, the influences of variable parameter on computation time were analyzed. As a result, they discovered that faster convergence required a good balance of both elitism and mixing in the population. They also found that the use of a strain-based fitness criterion was better than the use of a stress-based criterion in terms of the efficiency of converge to an optimal solution. Panesar and Weaver [78] presented a design method of blended bistable variable stiffness laminates using ant colony optimization and reported a blended flap concept exhibiting multistability.

Rouhi et al. [93] developed a multistep optimization process of variable stiffness composite cylinders based on genetic algorithm. First, the orientation angles in the variable stiffness cylinder were defined to vary in six regions from the keel to the crown, meaning that the end angle of one region can be considered as the start angle of the next. Then, the genetic algorithms were performed to calculate the optimum angle of each regional boundary to get the maximum improvement in the performance of cylinder. To enhance significantly the computational efficiency, the radial basis functions introduced by Nik et al. [94] were used to simplify the finite element analyses. Finally, numerical simulation is implemented to verify that the aspect ratio has a significant impact on the buckling load but the radius of cylinder has not. Sliseris and Rocens [95] mentioned another multistep optimal design process of composite plates with discrete variable stiffness, which is also using the genetic algorithm.

3.4 Other methods

To reduce the computational expense, metamodel-based design optimization (MBDO) methods are used in which the costly high-fidelity functions are replaced by low-cost analytical approximation functions that are so-called surrogate models or metamodels. Luersen et al. [59] presented a design method of variable stiffness cylindrical shell for maximum fundamental frequency using a surrogate method to optimize the fiber path parameters. Rouhi et al. realized the VSD of cylindrical shells [96] and elliptical cylinders [97] using this model reduction method.

In contrast to the DQM mentioned before, Zucco et al. [98] presented a mixed finite element method for linear static and buckling analyses of variable stiffness composite lamination. This method exhibited strong performance in terms of convergence rate and numerical errors. Besides that, other higher-order finite elements have also been used to study the bending [99] and vibrational response [60], [100] of VAT plates.