The design of a coated composite corrugated skin for the camber morphing airfoil

Abstract

One of the critical components of a morphing wing is the anisotropic skin, which has to be stiff to withstand the aerodynamic loads and flexible to enable the morphing deformations. This work presents the design of an elastomer coated composite corrugated skin for the camber morphing airfoil. The good in-plane strain capability and highly anisotropic behaviour of composite corrugated panels make them very effective in morphing wing applications. The behaviour of these corrugated skins must be investigated comprehensively and optimized in terms of aero-elastic effects and the boundary conditions arising from the internal wing structure. In this article, the geometric parameters of the coated composite corrugated panels are optimized to minimize the in-plane stiffness and the weight of the skin and to maximize the flexural out-of-plane stiffness of the corrugated skin. A finite element code for thin beam elements is used with the aggregate Newton’s method to optimize the geometric parameters of the coated corrugated panel. The advantages of the corrugated skin over the elastomer skin for the camber morphing structure are discussed. Moreover, a finite element simulation of the internal structure with the corrugated skin is performed under typical aerodynamic and structural loadings to check the design approach.

Keywords

Morphing skin design elastomer corrugated core optimization

Introduction

Actively varying an airfoil’s camber is an effective way to change the aerodynamic forces and moments generated by a wing. This allows for control of the vehicle’s flight path and optimization of the aerodynamic performance over different flight regimes. Traditionally, camber variation has been accomplished through the use of discrete trailing edge flaps, and indeed, this is the solution employed by nearly all aircraft currently flying. However, there has long been an interest in the aerospace industry in technologies, which would allow for a more smooth and continuous change in camber than that of a flap. The many different concepts explored over the last several decades have been summarized in several review articles (Barbarino et al., 2011; Chopra, 2000; Giurgiutiu, 2000). These systems are being pursued for the promise of a reduction in the significant drag penalty associated with flap deflections. In order to be effective, the control authority of the morphing mechanism must be substantial. The primary motivation for the use of a given morphing technology must be that it can radically alter the performance of the wing. Without a significant impact on the net aerodynamic performance, it is very hard to justify the added weight, complexity and cost of morphing systems, and they are therefore not likely to come to fruition. Moreover, it is of necessity that the morphing design be as simple as possible. The simplicity of the design would result in rapid reduction of the cost, minimization of the use of mechanical elements and potentially a reduction in the weight of the mechanism. Avoiding design complexity decreases the maintenance requirements as well.

The Fish Bone Active Camber (FishBAC) concept (Woods and Friswell, 2012) has been designed in light of these design criteria. It combines several different structural aspects into a single concise design concept. The biologically inspired compliant structure consists of a thin chord wise bending beam spine with stringers branching off to connect it to the coated corrugated skin surface. Actuators mounted in the non-morphing leading edge induce bending moments on the spine through an antagonistic pair of tendons. Figure 1 shows the baseline FishBAC concept built around an anisotropic compliant structural core. The baseline design uses a pre-tensioned elastomeric matrix composite (EMC) skin. Continuous bending deflections are driven by a high stiffness, non-backdrivable, antagonistic tendon system. This article considers the same morphing concept with the EMC replaced with an elastomer coated composite corrugated skin.

Corrugated cores are stiff along the corrugation direction, but flexible in the transverse direction and hence they have exceedingly anisotropic behaviour. For this reason, coated composite corrugated panels have been proposed as a candidate for application in morphing wings (Yokozeki et al., 2006). This is due to the fact that wing structures must be stiff so as to withstand bending due to aerodynamic forces and flexible so they can deform efficiently in flight. Another advantage of using sandwich structures (made of metals or composites) with corrugated cores is that they have high fatigue resistance. Moreover, composite corrugated panels decrease the number of parts used in a wing structure, which increases the speed of assembly and reduces the manufacturing costs (Roosen and Juhl, 2005).

Numerous investigations have been carried out on the mechanical behaviour of corrugated cores for general applications, such as studying the effect of different shapes of the corrugation on the bending stiffness of the panel (Luo et al.,1992), studying their geometric and material nonlinearities with different loading configurations through finite element (FE) analysis (Gilchrist et al., 1998), representing the panels by homogenized-based analytical models (Xia et al., 2012) and investigating the failure mechanism maps for the different failure modes of the corrugated panels (Kazemahvazi et al., 2009; Kazemahvazi and Zenkert, 2009).

However, in terms of morphing skin applications, recently a number of investigations have been performed by Dayyani et al. (2012, 2013b) on the mechanical behaviour of composite corrugated cores with and without elastomeric coating. They showed that the optimal design of these structures required high-fidelity models of the panels that would be incorporated into multi-disciplinary system models. They considered the nonlinear effects due to the material properties and mechanism of deformation and studied the mechanical behaviour of composite corrugated laminates with and without elastomeric coatings by means of experimental, numerical and analytical investigations. The importance of their work was that it provided detailed experimental and numerical models of the panel that could be used for further static and dynamic homogenization and optimization studies. The authors (Dayyani et al., 2013a) continued on this path and proposed equivalent structural models that retain the dependence on the geometric parameters of the coated corrugated panels. They presented two analytical solutions to calculate the equivalent tensile and bending flexural properties of a coated composite corrugated core in the longitudinal and transverse directions and verified the accuracy and efficiency of the presented equivalent model by investigating different experimental and numerical models. Figure 2 shows a schematic of the application of the coated corrugated core on the FishBAC internal wing structure.

Any morphing solution must be optimized at the system level. The FishBAC structure with a corrugated skin has many geometric and material parameters that could be varied. For example, the spacing between adjacent stringers may be varied, and indeed need not be uniform. A non-uniform stringer spacing would be justified because the aerodynamic pressure is not uniform over the airfoil, and higher stringer spacing could be used in regions of low aerodynamic pressure. Similarly, the corrugation geometry could be non-uniform. In this article, both the stringer spacing and the corrugation geometry are assumed to be uniform, so that the analysis can concentrate on the performance of the skin. This understanding of the skin may then be incorporated into a system-level model of the FishBAC by relying on the optimized equivalent skin properties developed here. Directly optimizing a structure with many unknown parameters using high-fidelity models is computationally expensive and often does not lead to a full understanding of the design drivers of the component parts.

In this article, a suitable corrugation configuration for the morphing skin application is identified and then optimized in terms of three objective functions: the mass of the skin, the tensile in-plane stiffness and the flexural out-of-plane stiffness. To calculate the equivalent tensile and flexural stiffness of the corrugated panels, a FE code for beam elements was written in MATLAB. The aggregate Newton’s method was used to perform the multi-objective optimization for different configurations of FishBAC stringers and corrugated unit cells. The results show the optimal design space of the FishBAC with a corrugated skin. The advantages of the corrugated skin over the elastomer skin for the FishBAC morphing structure are discussed comprehensively based on FE comparison studies. Furthermore, a FE simulation of the skin and internal structure under typical aerodynamic and structural loadings is performed to verify the design approach.

Corrugated skin optimization problem

Considering the application of a corrugated panel for the morphing skin, the first question to be answered is which corrugation shape has more out-of-plane stiffness and less in-plane stiffness. These parameters are important because higher out-of-plane stiffness of the skin results in smaller bending deformation under aerodynamic loading. Moreover, this increases the buckling stiffness of the skin, when the skin is subjected to compression due to morphing actuation. In other words, the higher out-of-plane stiffness of the skin results in a smoother surface of the wing during flight. In contrast to the out-of-plane stiffness, minimizing the in-plane stiffness results in less resistance of the skin to actuation of the morphing FishBAC deformations, thereby reducing the force and energy requirements. Investigating the effect of the various corrugation shapes on the balance of these two stiffnesses and selecting the proper corrugation shape are necessary steps before starting the optimization.

Mechanical properties of different corrugation shapes

To find the optimum corrugation shape that has high out-of-plane stiffness and low in-plane stiffness, three typical configurations are selected, as shown in Figure 3. The sinusoidal corrugation shape is not included since the presence of a single line of contact between the elastomer coating and corrugated core, as opposed to the rectangular contact surfaces of the other configurations, provides insufficient area for bonding. In this initial study, the portion of the elastomer coating, which overlaps the corrugation, is neglected, since the ratio of elastomer Young’s modulus to composite core material Young’s modulus is very small (Dayyani et al., 2013b). The composite corrugated core and elastomeric coating are labelled in Figure 3.

Based on Figure 3, six corrugated cores with and without elastomeric coating were modelled in ABAQUS. Each model consisted of 10 unit cells. In order to make a true comparison between these configurations, the values of parameters in each configuration are selected to keep the consistency of the length and the height of the corrugated unit cell in all cases. Table 1 presents the values corresponding to the parameters of the three corrugation configurations of Figure 3. The thickness of the corrugated core and elastomeric coatings and the width of the panel for all cases were 1, 0.8 and 25 mm, respectively. Glass fibre and elastomer with Young’s modulus of 4.5 GPa and 13 MPa were considered for the corrugated core and coatings, respectively. More details are presented in the literature (Dayyani et al., 2013b).

Table 1.

Parameters of the three corrugation configurations of Figure 3.

Parameters	a ₁	A ₂	A ₃	h
Reentrant	10	5	10	10
Rectangular	7.5	0	7.5	10
Trapezoidal	5	5	5	10

All values are given in millimetre.

Two sets of boundary conditions were applied to each model. In the first set, all degrees of freedom of one end of the panel were fixed, whereas a displacement boundary condition of 200 mm in the out-of-plane direction was applied to the other end of the panel, to simulate cantilever bending. In the second set, all degrees of freedom of one end of the panel were fixed, and a displacement boundary condition of 100 mm for the in-plane displacement was applied to the other end of the panel, to give a simple tensile test. In all cases, a fine mesh of beam elements was used.

Figure 4 shows the force displacement curves for reentrant, rectangular and trapezoidal corrugated cores with and without elastomeric coating. The interesting point in Figure 4 is that although the uncoated trapezoidal corrugated core has maximum tensile and bending stiffness, adding the elastomeric coating to the corrugated core results in minimum bending and tensile stiffness in contrast to other configurations. This story is reversed for the reentrant corrugation shape.

Considering the mechanism of deformation in the corrugated core, applying a tensile boundary condition to the reentrant corrugated core results in deformations that change the reentrant corrugation pattern to a rectangular and then to a trapezoidal configuration. Moreover, the presented analytical solutions for tensile and bending stiffness of the uncoated corrugated cores (Dayyani et al., 2012) revealed that with a fixed length of unit cell, the unit cell with smaller $θ$ , as illustrated in Figure 3(c), has a more flattened shape and therefore has increased tensile and bending stiffness. Therefore, the uncoated trapezoidal corrugation has more stiffness compared to the uncoated rectangular and reentrant corrugations. But adding the elastomeric coating to the corrugated core reverses the order of each configuration stiffness. This is due to the fact that the gap between two adjacent corners of a unit cell of a reentrant corrugated core is smaller than it is for the rectangular and trapezoidal corrugations, and hence, there is less available elastomer length to strain, resulting in higher elastomer strains to achieve a given global skin strain and an increase in stiffness.

Figure 5 shows a comparison of the bending flexibility and tensile stiffness for different corrugation types. The different configurations of the uncoated corrugated cores have 6.7 times more out-of-plane flexibility than the coated corrugated configurations on average. In contrast, although the coated corrugated cores have better aerodynamic surfaces and smaller out-of-plane flexibility, they have 5.5 times more in-plane stiffness than the uncoated corrugated configurations on average.

Moreover, considering the corrugation geometries and the corresponding values of their parameters represented in Figure 3 and Table 1 and the density of the composite core and elastomeric coating presented in Table 3, the mass of the skin for different corrugation types were calculated precisely by including the elastomeric coating in the overlapped regions with the corrugated core. The calculation reveals that the coated reentrant and coated rectangular corrugated cores are 30% and 11% heavier than the coated trapezoidal corrugated core.

In terms of the morphing skin application, the preferred skin must have the minimum tensile stiffness and bending flexibility as well as a smooth aerodynamic surface. Therefore, as a starting point of the multi-objective optimization of the corrugated skin, the coated trapezoidal configuration is selected as the preferred configuration for the morphing skin.

FE analysis

After selecting the configuration of the corrugation, a FE code for beam elements was generated in MATLAB that calculates the equivalent tensile, flexural stiffness and the mass of a coated trapezoidal corrugated core with four unit cells. The parameters $a_{1}$ , $a_{2}$ , $a_{3}$ and $h$ defining the geometry of a trapezoidal corrugated unit cell, and $t_{c}$ and $t_{e}$ representing the thickness of the corrugated core and elastomeric coating, were the inputs to the FE code. The width of the panel was set to 25 mm. Furthermore, the material properties were defined by means of $ρ_{c}$ and $ρ_{e}$ representing the density of the composite core and elastomeric coating, and $E_{c}$ and $E_{e}$ representing Young’s modulus of the composite core and elastomeric coating. Since the ratio of elastomer Young’s modulus to glass fibre Young’s modulus is very small, the elastomer coatings were neglected in the areas overlapped with the composite corrugated core. This assumption is valid because these two materials are assumed to be well adhered together. Consequently, they would have the same displacement, and hence, the strain energy of the coating would be very small compared to the strain energy of the corrugated core in the overlapped areas (Dayyani et al., 2013b).

Three degrees of freedom $U_{x}$ , $U_{y}$ and $θ_{z}$ in the global coordinate system were considered at each node. The coated corrugated structure was discretized by 26 elements. The global stiffness matrix of each element with rotation angle of $θ$ is

K_{global} = [\begin{matrix} \begin{matrix} (\frac{C^{2} EA}{l} + \frac{12 S^{2} EI}{l^{3}}) & (\frac{CSEA}{l} - \frac{12 CSEI}{l^{3}}) & (\frac{- 6 SEI}{l^{2}}) \\ (\frac{CSEA}{l} - \frac{12 CSEI}{l^{3}}) & (\frac{12 C^{2} EI}{l^{3}} + \frac{S^{2} EA}{l}) & (\frac{6 CEI}{l^{2}}) \\ (\frac{- 6 SEI}{l^{2}}) & (\frac{6 CEI}{l^{2}}) & (\frac{4 EI}{l}) \end{matrix} & \begin{matrix} (\frac{- C^{2} EA}{l} - \frac{12 S^{2} EI}{l^{3}}) & (\frac{- CSEA}{l} + \frac{12 CSEI}{l^{3}}) & (\frac{- 6 SEI}{l^{2}}) \\ (\frac{12 CSEI}{l^{3}} - \frac{CSEA}{l}) & (\frac{- 12 C^{2} EI}{l^{3}} - \frac{S^{2} EA}{l}) & (\frac{6 CEI}{l^{2}}) \\ (\frac{6 SEI}{l^{2}}) & (\frac{- 6 CEI}{l^{2}}) & (\frac{2 EI}{l}) \end{matrix} \\ \begin{matrix} (\frac{- C^{2} EA}{l} - \frac{12 S^{2} EI}{l^{3}}) & (\frac{12 CSEI}{l^{3}} - \frac{CSEA}{l}) & (\frac{6 SEI}{l^{2}}) \\ (\frac{- CSEA}{l} + \frac{12 CSEI}{l^{3}}) & (\frac{- 12 C^{2} EI}{l^{3}} - \frac{S^{2} EA}{l}) & (\frac{- 6 CEI}{l^{2}}) \\ (\frac{- 6 SEI}{l^{2}}) & (\frac{6 CEI}{l^{2}}) & (\frac{2 EI}{l}) \end{matrix} & \begin{matrix} (\frac{C^{2} EA}{l} + \frac{12 S^{2} EI}{l^{3}}) & (\frac{- 12 CSEI}{l^{3}} + \frac{CSEA}{l}) & (\frac{6 SEI}{l^{2}}) \\ (\frac{- 12 CSEI}{l^{3}} + \frac{CSEA}{l}) & (\frac{12 C^{2} EI}{l^{3}} + \frac{S^{2} EA}{l}) & (\frac{- 6 CEI}{l^{2}}) \\ (\frac{6 SEI}{l^{2}}) & (\frac{- 6 CEI}{l^{2}}) & (\frac{4 EI}{l}) \end{matrix} \end{matrix}]

(1)

where $C$ , $S$ , $E$ , $A$ and $I$ represent the $\cos (θ)$ , $\sin (θ)$ , Young’s modulus, the cross-sectional area and the second moment of area of the element, respectively. Enlarging and assembling the stiffness matrix of all the elements resulted in the assembled stiffness matrix of the coated corrugated core as ${[K_{structure}]}_{60 \times 60}$ . While almost all degrees of freedom at one end of the panel were fixed, two prescribed displacements in the $x$ direction and $y$ direction were applied separately to one end of the panel for each model to calculate the tensile and bending stiffness of the panel. The boundary conditions applied to the coated corrugated core in tensile and bending models are presented in Table 2.

Table 2.

Prescribed boundary conditions applied to the coated corrugated core in tensile and bending models.

Node labels (see Figure 6)	Prescribed DOFs
	Tensile modelling			Bending modelling
	$U_{x}$	$U_{y}$	$θ_{z}$	$U_{x}$	$U_{y}$	$θ_{z}$
1	0	0	0	0	0	0
18	3 (mm)	0	0	Free	−1 (mm)	Free
19	3 (mm)	Free	0	Free	−1 (mm)	Free
20	0	Free	0	0	0	0

DOF: degree of freedom.

Figure 6 shows the deformed and undeformed configurations of the coated corrugated core subjected to the tensile and bending displacements. The obtained results were verified against ABAQUS results with a fine mesh and the difference between them was smaller than 1%. This high accuracy is because the cubic shape functions of the beam element interpolate the deformations precisely. Although discretizing the corrugated structure with more beam elements would modify the results slightly, it would increase the computation time, especially since FE code is called more than10,000 times in the optimization loops for each configuration of the FishBAC stringers and corrugation unit cells. Moreover, it must be mentioned that the difference between the equivalent properties of a long corrugated panel and a corrugated panel with four unit cells was examined in ABAQUS and was smaller than 1%.

Multi-objective optimization

The multi-objective optimization was performed using a weighted-sum method to minimize the following three objectives: the equivalent tensile stiffness, ${EA}_{eq}$ , the reciprocal of the equivalent flexural stiffness, $1 / {EI}_{eq}$ , and the mass of the skin. These objectives are directly related to the equivalent skin properties that have system-level implications. The equivalent tensile stiffness is directly related to the actuator force (and therefore energy) required and should be minimized. The equivalent flexural stiffness is directly related to the out-of-plane deformation of the skin due to aerodynamic pressure loading and should be maximized. And the mass of the skin will contribute to the aircraft mass and must be minimized. Of course, the system-level objectives will be to maximize aerodynamic efficiency (lift to drag ratio) and to minimize aircraft mass. The aerodynamic properties of the airfoil are defined by the global FishBAC parameters, but rely on the stringers and skin to maintain the required airfoil thickness profile and to ensure the aerodynamic surface is smooth. Hence, it is reasonable to decouple the skin design from the airfoil design, at least at the conceptual design stage.

The multiple objectives are combined into one single-objective scalar function in this method. In more detail, the weighted-sum method minimizes a positively weighted convex sum of the three objectives

min \sum_{i = 1}^{3} w_{i} f_{i} (x)

(2)

where $w_{i} \geq 0$ are the weighting coefficients representing the relative importance of the objective functions $f_{i} (x)$ and $x$ is the vector of the design variables in the optimization problem. It is assumed that the sum of the weights is equal to 1, that is

\sum_{i = 1}^{3} w_{i} = 1

(3)

This approach represents a new optimization problem with a unique objective function. In this article, this optimization problem with a single-objective function is solved using Newton’s method. Since the solutions of this optimization problem can vary significantly as the weighting coefficients change, and because very little is usually known about choosing these coefficients, it is necessary to solve the same problem for many different values of these weights. Hence, the weights were considered to vary from 0.01 to 0.99 with increments of 0.01.

Based on the geometry of the FishBAC, which is illustrated in Figure 7, the morphing length $l_{morph}$ , between the rigid leading edge and rigid trailing edge, was fixed as 160 mm, which is 52% of the chord length. The number of FishBAC stringers, $n$ , was allowed to vary between 3 and 15, and hence, the stringer spacing (and thus the unsupported length of the skin) is optimized. Considering manufacturing limits, the minimum allowable length of a corrugation unit cell, $L_{uc_a}$ , was set to 5 mm. Based on the distance between two adjacent stringers, that is, $d_{s}$ , and manufacturing constraints, the number of corrugation unit cells $N$ was allowed to vary from 1 to $N_{final}$ , which is calculated as $d_{s} / L_{uc_a}$ . Figure 7 also shows $l_{rt}$ , which is the height of the rigid part at the end of trailing edge. The value for this parameter is 5.21 mm in the FishBAC configuration, as shown in Figure 7, and is used as the upper bound for the height of the corrugated unit cell. The materials for the core and the elastomer are selected from the literature with the density and Young’s modulus values presented in Table 3, which correspond to glass fibre and elastomer.

Table 3.

Fixed material properties of the corrugated skin.

Variables	Description	Values
$ρ_{c}$	Density of composite corrugated core	0.001799 g/mm³
$ρ_{e}$	Density of elastomer coating	0.000878 g/mm³
$E_{c}$	Young’s modulus of composite corrugated core	34473 MPa
$E_{e}$	Young’s modulus of elastomer coating	10 MPa

Table 4 shows the geometrical parameters of a coated corrugated core and their corresponding upper and lower bounds. The parameter $L_{hu}$ represents the length of half of the unit cell and may be calculated as

L_{hu} = \frac{d_{s}}{2 N}

(4)

Table 4.

Variables of the optimization problem and their upper and lower bounds.

Variables	Description	Lower bound	Upper bound
$t_{c}$	Thickness of composite corrugated core	$L_{hu} / 30$ (mm)	$2 \times L_{hu} / 30$ (mm)
$t_{e}$	Thickness of elastomer coating	$L_{hu} / 30$ (mm)	$4 \times L_{hu} / 30$ (mm)
$a_{1}$	Corrugation unit cell geometry	$L_{hu} / 6$ (mm)	$2 \times L_{hu} / 3$ (mm)
$a_{2}$		$L_{hu} / 6$ (mm)	$2 \times L_{hu} / 3$ (mm)
$a_{3}$		$L_{hu} / 6$ (mm)	$2 \times L_{hu} / 3$ (mm)
$h$		$L_{rt} / 2$ (mm)	$9 \times L_{rt} / 10$ (mm)

The focus of this article in this section is the optimization of the geometrical parameters of the structure. However, it may be necessary to perform a full optimization of both geometry and material properties of the coated corrugated panel for the design of these structures. The full optimization solution requires an expression to relate Young’s modulus and density of materials. Usually materials with higher Young’s modulus have higher density and a database could be used to select a material from a number of candidates. On the other hand, considering the development of technology in material engineering, it is probable to have new materials with higher Young’s modulus and lower density, which is beyond the scope of this article.

Selecting the upper and lower bounds for the thickness of both the corrugated core and the elastomeric coatings was based on practical considerations. The properties of the corrugated skin arise from localized bending with in the corrugations; if the ratio of the thickness of the corrugated core to the length of a unit cell is too high, then the mechanism of the deformation changes, resulting in a panel that is too stiff, especially when the size of the corrugation is very small. The lower bounds are set by the availability of suitable material and its robustness and handling properties. The upper bound and lower bound for $a_{1}$ , $a_{2}$ , and $a_{3}$ were selected to be consistent with the geometry of a trapezoidal shape. Considering the application of the corrugated panel for a morphing skin, the height of the corrugated unit cell is dependent on the external parameter $l_{rt}$ , which is the maximum size that the height of the panel can be. The only explicit constraint of the optimization problem was $a_{1} + a_{2} + a_{3} = L_{hu}$ , where $L_{hu}$ is implicitly fixed by the number of FishBAC stringers and corrugation unit cells. To ensure the best performance of the optimization scheme, all the parameters are normalized as follows

x_{n} (i) = \frac{x (i) - (\frac{x_{u} (i) + x_{l} (i)}{2})}{(\frac{x_{u} (i) - x_{l} (i)}{2})}, i = 1, 2, \dots, 6

(5)

where $x$ is the vector of parameters, $x_{u}$ and $x_{l}$ are the corresponding upper and lower bound vectors, and $x_{n}$ is the normalized vector of parameters. The objective functions are normalized as: $f_{n} (x) = f (x) / f (\bar{x})$ ; where the $\bar{x}$ is the average vector of upper bound and lower bound values, that is, $\bar{x} = (x_{u} + x_{l}) / 2$ .

Discussion and results

The results were obtained for different numbers of FishBAC stringers and corrugation unit cells between two adjacent stringers. These different configurations are tabulated in Table 5. Note that increasing the number of stringers reduces the range of the number of unit cells obtainable due to the 5 mm minimum length restriction of a unit cell.

Table 5.

Different configurations of stringers and unit cells in the optimization.

$n_{stringers}$	3	4	5	6	7	8	9	10	11	12	13	14	15
$N_{unitcells}$	[1–8]	[1–6]	[1–5]	[1–4]	[1–4]	[1–3]	[1–3]	[1–2]	[1–2]	[1–2]	[1–2]	[1–2]	[1–2]

Figure 8 shows the normalized optimum Pareto surface and its projection on three planes for the configuration of eight stringers and one coated corrugated unit cell between each two adjacent stringers, that is, $n_{stringer} = 8$ , $N_{Unitcell} = 1$ . In this Pareto surface, the relatively horizontal plane labelled as ‘3’ is obtained when the weight $w_{3}$ corresponding to the third objective, that is, the mass of skin, is dominant. In other words, this plane has mainly minimized the mass of skin but not the two other objectives: $1 / {EI}_{eq}$ and ${EA}_{eq}$ . The plane labelled as ‘1’ is obtained when the $w_{1}$ corresponding to the first objective, that is, $1 / {EI}_{eq}$ , is dominant since the two other objectives are not minimized significantly. Likewise, the surface titled as ‘2’ is obtained when the weight $w_{2}$ corresponding to the second objective is dominant.

Moreover, the best compromise point of this configuration is illustrated in Figure 8. The best compromise point was selected by first identifying an ideal reference point as the coordinates of minimum objective values, that is, $P_{ref} = ({(1 / {EI}_{eq})}_{min}, {({EA}_{eq})}_{min}, {({Mass}_{skin})}_{min})$ . Then the point at the minimum distance from the ideal reference point, in the normalized objective function space, was chosen as the best compromise point. Table 6 presents the corresponding weights of the optimization technique and the real values of the objectives. The range of the weights shows that all of the three objectives are involved efficiently in the process of optimization for the best compromise point.

Table 6.

Corresponding weights and real values of optimized objectives of the best compromise point in case 8.1 ( $n_{Stringer} = 8$ , $N_{Unitcells} = 1$ ).

Weights			Real values
$w_{1}$	$w_{2}$	$w_{3}$	$\frac{1}{{EI}_{eq}} {(N {mm}^{2})}^{- 1}$	${EA}_{eq} (N)$	Mass of skin (g)
0.35	0.33	0.32	1.2 × 10⁻⁴	1944.93	6.30

Figure 9 gives a good insight into the variation of the skin mass versus different combinations of number of stringers and unit cells for their corresponding best compromise point. The exact value for the minimum mass of skin for each configuration of stringers and unit cells is shown in Figure 9. For each number of stringers, the maximum mass of the skin occurs with the minimum number of unit cells.

Moreover, after collecting all the best compromise points corresponding to all the configurations of FishBAC stringers and corrugation unit cells, the design decision was made by repeating the process of finding the best compromise point among the collection in a normalized space of objectives. Figure 10 shows the trend and the projection of the best compromise points for the entire configuration of stringers and unit cells. The decision point of the best design is highlighted as red.

Table 7 shows the corresponding objective values and parameters of the decision point of the design, corresponding to case 4.3 ( $n_{stringer} = 4$ , $N_{unitcells} = 3$ ) and case 14.1 ( $n_{stringer} = 14$ , $N_{unitcells} = 1$ ). Although these two cases have the same geometry of the skin, the number of the stringers of the FishBAC internal structure is different and this changes the weight of the assembled structure. However, it must be mentioned here that for the parameters $t_{c}$ and $a_{2}$ , the optimization algorithm has selected the corresponding values at the lower bound while for the parameter $h$ the upper bound is selected. This can be justified by considering the Young’s modulus of the composite (35 GPa) and the elastomer (10 MPa), as presented in Table 3. Hence, the optimization algorithm has selected the lower bound for the thickness of the corrugated core mainly to reduce the weight of the structure. The other interesting point is that for all cases, the height of corrugation is selected equal to the upper bound values and for the parameter $a_{2}$ close to its lower bound value. This implies that increasing the height of the corrugation gives a higher ratio of bending stiffness to tensile stiffness. In other words, the algorithm has tried to maximize the angle of corrugation, as shown in Figure 3(c). In this case, $θ$ , calculated as $\tan^{- 1} (h / a_{2})$ , is about 79.2°.

Table 7.

Corresponding optimized parameters of the decision point of the design.

Optimized objective functions			Optimized parameters (mm)
$\frac{1}{{EI}_{eq}} {(N {mm}^{2})}^{- 1}$	${EA}_{eq} (N)$	Mass of skin (g)	$t_{c}$	$t_{e}$	$a_{1}$	$a_{2}$	$a_{3}$	$h$
3.9 × 10⁻⁴	500.40	4.29	0.18	0.51	2.26	0.89	2.20	4.69

All parameters are given in millimetre.

Benefits over a simple elastomeric skin

Increasing the thickness of the skin of the airfoil allows the skin to resist more pressure caused by the airflow and increases the critical buckling load due to the morphing actuation. In other words, the thicker skin allows the FishBAC structure to have fewer stringers with smaller length. But the problem is that increasing the thickness of the skin increases the tensile stiffness of the skin, which requires more force to actuate the airfoil, and it increases the mass of skin, whereas the mass reduction of the stringers is negligible.

However, the coated corrugated skin has some benefits over the simple elastomeric skin to tackle these challenges. First, the possibility of using a compatible corrugated skin with more height (thicker corrugated skin) that increases the bending stiffness of the skin without increasing mass of skin significantly in comparison to a simple elastomeric skin. Second, considering the mechanism of deformation in the corrugated core, the corrugated skin with more height decreases the in-plane tensile stiffness of the panel that results in a smaller actuation force required for morphing deformation. This structural advantage of a corrugated skin provides the possibility of having fewer FishBAC stringers, which reduces the weight of the structure. Although more details about the interaction of the corrugated skin and the internal structure are presented in the next section, a comparison of the mechanical behaviour of the FishBAC with a coated corrugated skin and a simple elastomeric skin helps to understand the importance of the corrugated skin for morphing wing applications.

In this regard, the geometry of FishBAC as described in Table 9 was considered. First, the mass of the coated corrugated skin with the geometric parameters and material properties presented in Tables 3 and 7 was calculated without the assumption of neglecting the coating sections that overlap with the corrugated core. The mass of the corrugated skin consisting of two elastomeric coatings and a composite corrugated core was calculated as 35.12 g. This mass and the material properties of the elastomer coating were set fixed to compare both skins. Considering the length of the skin between the rigid leading edge and the rigid trailing edge as 161.08 mm, the thickness of the simple elastomeric skins was calculated as1.66 mm, more than nine times thicker than the elastomeric coating of the corrugated skin. Considering the density of the material used for the FishBAC stringers as 0.0011 g/mm³, the effective mass of the FishBAC with a corrugated skin, that is, the mass of non-common parts (skin and stringers), was 17.3% smaller than the FishBAC with a simple elastomeric skin. Moreover, comparing the mechanical behaviour of these two skins reveals that the corrugated skin is almost five times more flexible to stretch in the morphing actuation process and has almost 4.5 times more resistance to out-of-plane deformation due to aerodynamic loads and buckling deformations caused by actuation. Table 8 presents more details about the comparison of these two skins.

Table 8.

A comparison of mass, tensile in-plane and flexural out-of-plane stiffness of the coated corrugated skins and simple elastomeric skin for the FishBAC internal structure.

NACA 0012	Coated corrugated skin	Simple elastomeric skin
Mass of skin (fixed) (g)	35.12	35.12
Mass of skin and stringers (g)	100.82	118.34
EA (N)	500.4	2490
EI (N mm²)	2564	571.79

FishBAC: Fish Bone Active Camber.

Furthermore, in contrast to the simple elastomeric skin, the corrugated skin has the feature of the structural anisotropy, which helps the skin to withstand more aerodynamic loads in the spanwise direction. Considering the pressure distribution over a NACA 0012 airfoil shown in Figure 11, the corrugated skin would allow the internal structure to have variable distance between stringers; more distance between the stringers in regions exposed to lower pressure. In addition, the geometrical parameters of the corrugation provide the facility of having continuous variable out-of-plane and in-plane stiffness along the length of the skin, which leads to a further reduction of the mass of the skin. The mechanical behaviour of both of these skins can be improved more using more advanced materials such as a curvilinear fibre composite elastomeric skin (Murugan et al., 2012), or applying pre-stressed elastomeric coatings. However, in the case of pre-stressed skins, the corrugated core provides more regions to bond such pre-stressed coating and hence would have smaller shearing stresses between the elastomer and the corrugated core, compared to a simple elastomeric skin.

Morphing design considerations

The discussion in this article has thus far considered the skin in isolation and presented the advantages of the coated corrugated skin over the elastomer skin. The optimum design was obtained by identifying the best compromise on the Pareto surface between the in-plane stiffness, the out-of-plane stiffness and the mass of the skin. In practice, the skin would be optimized simultaneously with the internal structure; based on the environment, the airfoil would experience; and the optimum skin would be different if more importance is given to certain objective functions such as the flexibility of the whole structure, which reduces the required energy to morph and results in a lighter actuation system. In practice, the shape of the morphing airfoil and the FishBAC geometry would be optimized to achieve the highest possible lift-to-drag ratio. This would require an equivalent skin model to capture the actuator force required to deform the skin and the additional mass of the skin. The predicted aerodynamic pressure loads would allow the out-of-plane deformations of the skin to be estimated, and these deformations would be constrained so that their effect on the aerodynamics, particularly drag, is negligible. This step is not trivial and may require analysis of the fluid–structure interaction, particularly if unsteady effects are considered. Other constraints may be added, based on manufacturing requirements or structural integrity and fatigue, although these are rarely considered in current morphing aircraft design. Thus, some of the objectives may become constraints or additional constraints may be added. For example, a constraint on EI_eq is obtained from the maximum out-of-plane deformation due to the airflow or to prevent buckling due to the actuation of the internal structure. Such an optimization is beyond the scope of this article, although a FE simulation of the skin and internal structure under typical aerodynamic and structural loadings is performed to verify the design approach.

Prior to simulation of the internal structure with the corrugated skin, it is necessary to have a good insight of the aerodynamic loads on the morphed airfoil. In this regard, the airflow over the morphing trailing edge of the NACA 0012 airfoil was simulated using XFOIL for an air speed of 30 m/s. Figure 11 shows the estimated surface pressure distribution. This aerodynamic loading and its distribution over the trailing edge are important in terms of the deformation of the skin and the structural modelling of the FishBAC stringers.

Based on the optimized geometry of the corrugated skin presented in Table 7, different compatible geometries for the FishBAC internal structure can be considered as: case 14.1 ( $n_{stringer} = 14$ , $N_{unitcells} = 1$ ) or the case in which the distance between the stringers increases as the pressure distribution decreases, as shown in Figure 11. As mentioned earlier, this is one of the important benefits of the interaction of the corrugated skin and the internal structure, which leads to further reduction in the mass of the whole structure and must be considered in the full optimization. However, the geometry of the airfoil section for case 14.1 ( $n_{stringer} = 14$ , $N_{unitcells} = 1$ ) corresponds approximately by chance to those used in the original design of the FishBAC structure with elastomer skins. In this section, more attention is paid to the skin rather than to the internal structure and hence, the internal design is not optimized but assumed identical to the existing airfoil that has been studied in simulation and experiment (Woods and Friswell, 2012). The geometric parameters, taken from this article, are given in Table 9.This design currently has a simple elastomer skin and here this is replaced with a coated corrugated skin in this study. Young’s modulus and Poisson’s ratio of the FishBAC material were assumed to be 2.14 GPa and 0.3. These values are in range of the material properties of the previous manufactured FishBAC, which were printed from Acrylonitrile Butadiene Styrene (ABS) plastic using an HP Designjet 3D Fused Deposition Modelling printer.

Table 9.

FishBAC prototype geometric parameters.

Baseline airfoil	NACA 0012
Chord (c)	305 mm
Span (b)	150 mm
Start of morph (x_s)	0.35c = 107 mm
End of morph (x_e)	0.85c = 260 mm
Spine thickness (t_b)	2 mm
Number of stringer pairs	14
Stringer thickness (t_st)	0.8 mm

FishBAC: Fish Bone Active Camber.

Four different corrugated skins were modelled and assembled on the FishBAC internal structure in ABAQUS. These four skins, which were compatible with the geometry of the internal structure, correspond to the following points: best compromise, minimum $1 / {EI}_{eq}$ , minimum ${EA}_{eq}$ and minimum mass, as presented in Tables 7 and 10, respectively. Figure 3(c) shows a schematic of the corrugated core unit cell and the corresponding values of parameters presented in Tables 7 and 10. In terms of the material properties, the corrugated cores are assumed to be made of three plies of woven glass fibre with epoxy resin matrix (Aird, 1996; Clayton, 1988), with the material properties given in Table 3. The upper and lower skins of the sandwich panel are polyurethane elastomer (Bhowmick and Stephens, 2000; Callister, 1997; Harper, 1996) with Young’s modulus also given in Table 3.

Table 10.

Optimized parameters corresponding to the different dominant objectives of the design.

Dominant objectives	Optimized objective functions			Optimized parameters (mm)
Dominant objectives	$\frac{1}{{EI}_{eq}} {(N {mm}^{2})}^{- 1}$	${EA}_{eq} (N)$	Mass of skin (g)	$t_{c}$	$t_{e}$	$a_{1}$	$a_{2}$	$a_{3}$	$h$
$\frac{1}{{EI}_{eq}}$	0.0002	1162.32	6.85	0.36	0.71	2.81	0.89	1.64	4.69
${EA}_{eq}$	0.0011	206.01	2.68	0.18	0.18	1.46	2.43	1.44	4.69
Mass	0.0018	401.27	2.28	0.18	0.18	1.97	1.52	1.84	2.61

All parameters are given in millimetre.

The FishBAC and coated corrugated skin were modelled as described earlier. A fine mesh of cubic beam elements of approximate size of 0.5 mm was used for the assembled structures to simulate the mechanical behaviour of the morphing wing. Large deformation analysis was considered in case the deformations were outside the linear elastic range. The tendon actuation system was modelled as a moment on node ‘q’ in the rigid part at the trailing edge, as shown in Figure 7. The nodes at the leading edge section of the morphing section were fixed. Figure 12 shows the trailing edge displacement as an almost linear function of the actuation moment for the FishBAC internal structure with different corrugated skins. Figure 12 clearly shows that the optimum skin would be different if more importance is given to different objective functions. The skins with minimum in-plane stiffness and minimum mass require less actuation energy in contrast to the best compromise point for the morphing deformation.

Figure 13 shows the deformed configuration of the FishBAC, with both the aerodynamic pressure distribution shown in Figure 11 and the actuation moment required to give the tip deformation for the skin with the best compromise properties. The interaction of the structural behaviour between the corrugated skin and the FishBAC spine and stringers is important. In particular, Figure 13 shows that higher strains are present in the skin rather than the FishBAC spine.

As implied earlier, the out-of-plane deformation of the skin can occur because of buckling of the skin due to the actuation of the internal structure. There are two types of buckling modes for the corrugated skin: global corrugated core buckling and local buckling of the elastomer coating. This highlights the importance of simultaneously optimizing the skin with the internal structure. To demonstrate this phenomenon, three models of the FishBAC with a width of 150 mm and different numbers of stringers (n_s = 3, 7 and 15) were simulated. Two configurations for each number of stringers were modelled with different heights of corrugated skin, namely 2.5 and 5 mm. The parameters $a_{1}$ , $a_{2}$ and $a_{3}$ were set equal to 1.667 mm for both thin and thick skins. The material properties were selected as those given in Table 3. A large displacement of 54 mm transverse to the chord direction was applied as a boundary condition to the tip of the trailing edge of the FishBAC in each model. Table 11 presents a comparison of the stiffness resisting the actuation of the FishBAC with thin and thick corrugated skins for three different configurations of unit cells and stringers. Increasing the distance between the FishBAC stringers reduces the structural stiffness resisting the morphing actuation.

Table 11.

A comparison of stiffness resisting the actuation of FishBAC with thin and thick corrugated skins for three different configurations of trapezoidal unit cells and stringers.

Stiffness resisting the actuation of FishBAC	d_s = 10 mm (n_uc = 2, N_s = 15)	d_s = 20 mm (n_uc = 4, N_s = 7)	d_s = 40 mm (n_uc = 8, N_s = 3)
Thin skin (h = 2.5 mm)	0.1469 N/mm	0.1441 N/mm	0.1309 N/mm (buckled)
Thick skin (h = 5 mm)	0.1060 N/mm	0.1050 N/mm	0.1040 N/mm

FishBAC: Fish Bone Active Camber.

However, the global buckling of the corrugated core is observed for the case of the thin skin with the largest distance between the stringers, that is, d_s = 40 mm. This problem can be solved by increasing the height of the corrugation. This increase in height not only postpones the global buckling of the skin but also reduces the stiffness resisting the morphing actuation. However, the local buckling mode appears as wrinkling of the elastomer coating for all the simulated cases. This problem can be avoided by applying pre-stressed elastomeric coatings. This pre-stretching of the elastomer coating not only delays the local buckling of the elastomer coating but also decreases the out-of-plane deformations due to the pressure distribution over the airfoil. Figure 14 shows these two modes of buckling for both thin and thick corrugated skins for the case of d_s = 40 mm in Table 11.

Conclusion

In this article, the force displacement curves for reentrant, trapezoidal and rectangular corrugated cores with and without elastomeric coating were investigated in tensile and bending simulations. Comparing the results allowed the selection of a suitable corrugation configuration with regard to morphing skin applications. The geometric parameters of the coated trapezoidal composite corrugated panels were then optimized to minimize the in-plane stiffness and the mass of skin and to maximize the out-of-plane stiffness skin.

A FE code for beam elements was written in MATLAB to calculate the equivalent tensile and flexural stiffness of coated corrugated panels. The aggregate Newton’s method was used to perform the multi-objective optimization for different configurations of FishBAC stringers and corrugated unit cells. Moreover, the variation in the mass of the optimized skin for different configurations of stringers and unit cells was investigated. The normalized optimum Pareto surface and its projection on three planes of the objective function space were investigated. The weight distribution in the aggregate method optimization corresponding to the dominant objective functions was explained.

The best compromise point on the Pareto surface was selected by first identifying the ideal reference point with the minimum objective values and choosing the point that had minimum distance to the ideal reference point as the best compromise point. The variation of the skin mass versus different combinations of number of stringers and unit cells for their corresponding best compromise point was discussed, and it was shown that for each number of stringers, the maximum mass of skin occurs with the minimum number of unit cells. Collecting all of the best compromise points corresponding to all of the configurations of FishBAC stringers and corrugation unit cells, the design decision was made by repeating the process of finding the best compromise point among the collection in the normalized space of objective functions. The trend of parameters and their distance to their corresponding upper bounds and lower bounds showed that the algorithm maximized the angle of the corrugation, as shown in Figure 3(c). In other words, $θ$ was calculated to be about 79.2°.

The important advantages of using a corrugated skin rather than simple elastomeric coatings on the FishBAC internal structure were also discussed. Using a compatible corrugated skin with more height increases the bending stiffness of the skin to resist more pressure caused by the airflow and buckling forces due to morphing actuation. It was also shown that the corrugated skin with more height decreased the in-plane tensile stiffness of the panel that results in smaller actuation energy required for morphing deformation. The structural advantage of a corrugated skin provides the possibility of a smaller number of FishBAC stringers, which reduces the weight of the structure.

A FE simulation of the skin and internal structure under typical aerodynamic and structural loadings was performed to verify the design approach. The important interaction of the structural behaviour of the corrugated skin and the FishBAC structure highlighted the necessity of a full optimization, which considers the geometry and material properties of both FishBAC internal structure and corrugated skin.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

This study was funded by the European Research Council through grant no. 247045 entitled ‘Optimisation of Multiscale Structures with Applications to Morphing Aircraft’.

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