1 Introduction

Fibre-reinforced plastic (FRP) composites offer high specific stiffness and strength compared to metallic materials. Hence, FRP composites are often used in mass-sensitive applications such as the aerospace, marine and automotive sectors. The advanced manufacturing methods used for FRP production, such as automated fibre placement, enable the manufacture of complex geometries with minimal waste, thus moving away from conventional subtractive manufacturing. Structures with tailored mechanical properties can also be formed [1, 2] to resist the applied load and customisable mechanical behaviour in operation. This production process, however, requires the machining of a bespoke mould, incurring substantial capital costs and restricting design flexibility. These factors limit the adoption of FRP composites to cost-insensitive production scenarios such as in aerospace.

Recent advances in the 3D printing of carbon fibre composites have opened the possibility of “mould-free” production [3, 4]. Zeng, et al. [5] manufactured carbon FRP horseshoe scaffolds using a co-fibre-matrix extrusion technique on a flat tool plate in a form not achievable with typical composite processes. Pervaiz, et al. [6] discussed the prospects of fused deposition modelling 3D printing of thermoplastic FRP composites for structural applications. The resultant components from these 3D printing processes typically exhibit poor mechanical performance due to high void content, low compaction force, and reduced interfacial adhesion. Hence, these see limited adoption for structural applications [6]. Other mould-free methodologies include adaptive mould composites [7] and coreless filament winding [8]. However, adaptive moulds consist of several adjustable parts and require machining to a high precision to reduce gaps in the mould. This is especially critical when manufacturing processes such as vacuum-assisted resin transfer and autoclave curing are used. Coreless filament winding is limited to continuous filaments and cylindrical shapes. Moreover, both instances require advanced automated equipment, increasing their upfront cost, hence adding barriers to their adoption.

Hoa et al. [9] first coined “4D composite printing” (4DCP) as a term to describe a process that leveraged automated layup and thermal stress to manufacture curved composite components off a flat mould. The anisotropic properties of the unidirectional carbon fibre prepregs in an asymmetric, unbalanced stacking sequence created a direction mismatch in thermal expansion during autoclave cure. The thermal stresses induced during the curing cycle led to a controllable curvature response. As the fibres were continuous and typical autoclaved curing process was applied, this 4DCP approach allowed the manufacture of load-bearing structures with good static and fatigue properties [10,11,12]. However, 4DCP sacrifices efficient fibre orientation to produce deformed geometries. Fibre orientations are firstly selected based on laminate imbalance, not the resultant stiffness or strength. Therefore, the laminate is a product of the desired shape and not driven by the structural requirements, potentially leading to heavier components than necessary.

Eccentric fibre prestress (EFP) has been demonstrated to produce a similar warping effect to 4DCP [13]. Benefiting also from continuous reinforcement fibres and proper consolidation, other authors have investigated the use of EFP to reduce process warpage resulting from unsymmetric laminates [14, 15]. Applications also include inducing bistability into FRP laminates to produce tuneable toggle energy configurations [16,17,18,19,20,21]. In these instances, EFP did not rely on excess material and hence produced lighter structures than 4DCP. Moreover, EFP is agnostic of the cure mechanism and can be performed at ambient temperatures to reduce cost. While EFP has been demonstrated in select studies to create composite beams with a single curvature, no work has considered EFP for compound curve surfaces.

The current investigation presents EFP as a novel method for producing dimensionally accurate single-plane compound curvatures from a flat mould by way of longitudinal stiffness tailoring. It aims to demonstrate experimentally that EFP can produce simple curved profiles, and then extend the experimental program to show for the first time, quasi-sine wave and compound curvatures for two-ply unidirectional EFP laminates of glass fibre (GF) and/or carbon fibre (CF). The study also seeks to show a novel EFP optimisation approach through mixed reinforcement material laminates. Finally, the work proposes a unique inverse design methodology to predict the required laminate to achieve the upper surface of a standard NACA 4412 aerofoil using a genetic algorithm (GA).

2 Analytical modelling

Classical laminate theory (CLT) is used to determine the residual stresses arising from thermal loads in an unbalanced stacking sequence and the subsequent deformation upon cooling [14, 15, 22]. In this work, a two-dimensional plane strain model using the Euler–Bernoulli beam theory (EBBT) is presented to predict the internal residual stress of thin, eccentrically prestressed unidirectional fibre laminates [23, 24]. Here, the x-axis (width) orientation is neglected, and so the analysis can be simplified into two-dimensional plane strain in the y–z plane. While this predictive model is highly simplified, it is an efficient method of analysing an EFP laminate, shown in Fig. 1a.

Fig. 1
figure 1

Free body diagram of EFP side view orientation, a prestressing of fibres, b deflected beam geometry. Here, superscript \(p\) refers to prestressed material whilst superscript \(n\) the neutral material. Terms \(t\) refer to thickness parameters and \(\overline{y}\) the neutral axis location. The eccentricity, \(e\), is taken as the distance between the prestress material centroid and the neutral axis, and \(r\) is the radius at the neutral axis. All other terms are as defined in the text

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To perform EFP, a selection of fibres during the cure cycle are loaded to \({P}_{fibre}\), at an offset distance, \(e\), from the neutral axis of the beam. On release of the pretension, after laminate cure, the internal moment, \({M}_{int}\), for an unloaded EFP laminate is presented in Eq. (1):

$$\sum {M}_{int}=-{P}_{fibre}e$$
(1)

Therefore, the internal moment is linearly proportional to the pre-stress load and eccentricity offset distance. Assuming flaws within the fibres are negligible, the tension load will evenly distribute across the length of the prestressed specimen. Hence, fibre pre-tension stress is assumed a constant along the fibre orientation. Upon release of the pre-tension, the internal bending moment is immediately redistributed and balanced by the non-stressed layers, resulting in a circular profile as shown in Fig. 1b. Hence, the curvature of the beam, \(k\), is the inverse of the radius, \(r\), and can be calculated using Eq. (2) [25]:

$$k=\frac{{M}_{int}}{{E}_{11}I}$$
(2)

where \({E}_{11}\) is the in-plane modulus of the laminate, and \(I\) is the moment of inertia. For a constant rectangular cross-sectional area, the moment of inertia, \(I\), is given as \(w{t}^{3}/12\), where \(w\) and \(t\) are the beam’s width and thickness, respectively. A function can then be derived, correlating the laminate thickness to the radius of curvature for a given pretension applied to the bottom ply of the ply sequence.

Hybridisation can be defined as the inclusion of two or more reinforcement materials in a laminate. Hybridisation of the neutral reinforcing fibres allows for the location of the neutral axis, and consequently eccentricity to be manipulated based on their relative stiffness. The characteristics of hybrid laminates containing two reinforcement materials are accounted for by the cross-sectional transformation of the neutral fibres with respect to the ratio of the tensile moduli. The ratio of modulus between the prestressed layers \({E}_{11}^{p}\) and neutral layers \({E}_{11}^{n}\) is given as \(n\), defined in Eq. (3):

$$n=\frac{{E}_{11}^{n}}{{E}_{11}^{p}}$$
(3)

Considering \(n\), the analytical width value \(w\) of the neutral material may be modified to \({w}{\prime}\) as shown in Eq. (4) such that the entire laminate may be analysed as if it were composed of the prestressed material only:

$${w}{\prime}=nw$$
(4)

Defining the bending compliance as \(e/I,\) a load-independent value is created which can be used to assess the influence of the thickness ratio \({t}^{n}/{t}^{p}\) on the magnitude of EFP curvature. In this case, a higher bending compliance denotes a tighter radius of curvature for a given prestress load. This relationship is controlled by both prestress eccentricity and stiffness. Eccentricity increases with laminate thickness which increases the bending moment. However, the moment of inertia, given as \(I=w{\prime}{({t}^{p}+{t}^{n})}^{3}/12\), increases as well, resulting in a reduction in beam bending compliance. Figure 2 presents this dynamic relation for different values of \(n\) where an initial region, denoted by “Eccentricity,” is dominated by the geometrically controlled bending moment. This then gives way to stiffness effects, which increase at a cubic rate with thickness. The dotted line outlining the boarder of the “Stiffness” Eccentricity zones represents the optimal thickness ratio configuration for a given modulus ratio, \(n\).

Fig. 2
figure 2

Bending compliance versus thickness ratio for a series of mixed modulus laminates

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In practice, composite structures’ geometries, such as an aerofoil, can be quite complex and consist of multiple planes and curvatures. The EBBT model has highlighted that the single-plane curvature of a beam can be modified by the magnitude of the prestress, thickness ratio, and modulus ratio. Hence, these design parameters can be employed to manufacture beam geometries with compound curvature using the EFP process. As the pretension load is constant, tailoring of the thickness and modulus ratios can result in a multi-curve response. In a simplified 2D example, Fig. 3 (a) shows thickness ratio tailoring along the laminate length. The result is a segmented deformation response, presented in Fig. 3 (b), with each segment characterised by the radii of curvature, \({r}_{1}, {r}_{2}, {r}_{3}\), in the EBBT model.

Fig. 3
figure 3

Laminate variation across length (a). Resulting 2D beam geometry (b). Prestressed fibre shown in orange and neutral fibre in grey

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Assuming that the profile is continuous along the neutral axis, and differentiable across segments, the compound curvature of a 2D beam can be mathematically defined as a piecewise discrete function. Considering \(s\) equal-length segments, where each segment has \(m\) possible radii based on the number of neutral layers, a continuous profile can be constructed. Dividing each segment into \(dy\) equispaced points along the arc allows for complete discretisation into the \(y\) and \(z\) coordinates as outlined in Eqs. (5) to (7):

$$ii=\mathrm{1,2},3,\dots ,s$$
(5)
$$i=\mathrm{1,2},3,\dots ,dy$$
(6)
$$r={r}_{1}, {r}_{2},{r}_{3},\dots {r}_{m}$$
(7)

Here, \(ii\) counts the number of arc segments, whilst \(i\) counts the number of data points in each segment. The term \(id{x}\) is presented in Eq. (8) as an integer index of each radius from the group, \(r\), such that the angle at the end of the first arc segment, \({\theta }_{1}\), and origin location \({O}_{\mathrm{1,1}},{O}_{\mathrm{1,2}}\), may then be defined using Eqs. (9) to (11) for any EFP radius \({r}_{idx}\):

$$id{x}_{ii}=\left[1,m\right] \in Z$$
(8)
$${\theta }_{1}=\left[\mathrm{0,2}\pi \right]\in R$$
(9)
$${O}_{\mathrm{1,1}}= {r}_{{\mathit{idx}}_{1}}\mathit{cos}\left({\theta }_{1}+\pi \right)$$
(10)
$${O}_{\mathrm{1,2}}={r}_{id{x}_{1}}\mathit{sin}\left({\theta }_{1}+\pi \right)$$
(11)

\(d\theta\), defined in Eq. (12), refers to the angular step change such that each arc segment of length \(l\) may be discretised into \(dy\) points:

$$d{\theta }_{ii}=\frac{2}{dy}{\mathit{sin}}^{-1}\left(\frac{{l}_{id{x}_{ii}}}{2{r}_{id{x}_{ii}}}\right)$$
(12)

Equation (13) to (15) presented can then be used to define \(\uptheta\) and \({\text{O}}\) for the remaining \({\text{s}}-1\) segments.

$${\theta }_{ii+1}=\left(i-1\right)d{\theta }_{id{x}_{ii}}+{\theta }_{ii} \left(ii\ne 1\right)$$
(13)
$${O}_{ii+\mathrm{1,1}}=\mathit{cos}\left({\theta }_{ii+1}\right){r}_{{idx}_{ii}}-{r}_{{idx}_{ii+1}}+{O}_{ii,1} \left(ii\ne 1\right)$$
(14)
$${O}_{ii+\mathrm{1,2}}=\mathit{sin}\left({\theta }_{ii+1}\right){r}_{{idx}_{ii}}-{r}_{{idx}_{ii+1}}+{O}_{ii,2} \left(ii\ne 1\right)$$
(15)

With the segment radius, angle and origin known, the discretised arc profile may then be converted into its \(y\) and \(z\) coordinates using Eqs. (16) and (17):

$${y}_{i,ii}={O}_{ii,1}+{{r}_{idx}}_{ii}\mathit{cos}\left(\left(i-1\right)d{{\theta }_{idx}}_{ii}+{\theta }_{ii}\right)$$
(16)
$${z}_{i,ii}={O}_{ii,2}+{{r}_{idx}}_{ii}\mathit{sin}\left(\left(i-1\right)d{{\theta }_{idx}}_{ii}+{\theta }_{ii}\right)$$
(17)

In practical applications of EFP, the objective is to attain a specific laminate with a compound curve from a flat mould. The solution is non-trivial as the designer must control the pretension force and laminate configuration, as opposed to the radius. Hence, a method of inverse design is required. A minimisation problem capable of performing inverse design exists when the desired compound profile is compared to a computational estimation using the analytical model. This is described in Fig. 4 and calculated as a root mean square error (RMSE).

Fig. 4
figure 4

Fitness function of minimisation problem between two curves

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For complex minimisation problems, evolutionary GAs have been demonstrated to be successful in heuristic searches [26]. When used to control the segment stiffness, a low computational effort inverse design workflow is produced and presented in Fig. 5.

Fig. 5
figure 5

Inverse design workflow

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The prestress ratio is chosen as the EFP input while the magnitude of prestress and modulus ratio is held at a constant load. This is achieved by controlling the ply count for each segment, in particular the ratio of prestressed to neutral layers. By defining a maximum number of plies, the analytical model can develop a radius dictionary for each configuration. From this, the GA can pick a radius from this known set for each segment and compare the modelled curve to the target curvature through the RMSE fitness function in Eq. 18:

$$RMSE=\frac1{s\cdot dy}\left[\sum_{ii=1}^s\sum_{i=1}^{dy}\sqrt{\left({yy}_{i,ii},-y_{i,ii}\right)^2+\left({zz}_{i,ii}-z_{i,ii}\right)^2}\right]$$
(18)

Over successive iterations, the GA learns favourable radii for each segment, and converges to a stable best penalty value (fitness function score) for the optimised solution. Notably, the computation of this dictionary and building of the 2D geometry through the MATLAB model negates the need for complex and time-consuming FEM iterations.

3 Experimental development

3.1 Specimen manufacture

A bespoke jig is presented to impart fibre pretension. The fixture, shown schematically in Fig. 6a and pictured in Fig. 6b, is composed of a mild steel frame and 6061 aluminium clamping hardware and machined parts. Instron eyelets measuring 25.4 mm enable the assembly to mount into an Instron 34TM machine for accurate application of the pre-tension load to within ± 0.5% load cell accuracy. Guide rods constraining the fibre clamps to axial translation only prevent axial twist of the laminate during manufacturing. The locking nuts on the pictured threaded rods allow for the removal of the jig from the Instron machine whilst retaining tensile loading in the fibres.

Fig. 6
figure 6

Schematic of the bespoke prestress jig (a) and jig pictured with fibre blank installed (b)

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The selected reinforcement fibres in this study are Toray T300 unidirectional carbon fibre and Saertex unidirectional glass fibre fabrics. The polymer matrix is Allnex R180 epoxy resin, mixed to a 5:1 ratio with H180 slow hardener. The composite specimens are manufactured under controlled laboratory conditions using the hand lamination process. The material properties of the laminates, manufactured and their constituents are presented in Table 1.

Table 1 Fibre, matrix and laminate properties
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The manufacturing steps taken are depicted in Fig. 7. Firstly, metallic end tabs are adhesively bonded to the dry fabric, with an actual blank pictured in Fig. 7e. The bonded end tabs prevented the fibre from splitting and aid in spreading the load evenly across a larger area. Care is taken to ensure at this stage that the fibre is evenly taut across the width of the blank before curing to minimise twist in the final specimen. With the blank installed, the prestress jig is loaded into an Instron universal testing machine. Here, tensile load is applied to the fibres by a displacement control program at a rate of 2 mm/min. Once the desired tensile load is achieved, the tensioned fibres are secured in place by the locking stud nuts, and the jig is removed from the Instron. Epoxy is then applied to the tensioned fibre using the hand lamination technique. The dry, neutral fibre is placed and wetted similarly. As the neutral fibre is added after the application of prestress, it remains effectively stress-free during polymerisation. Peel ply and breather are used to absorb excess resin and ensure an even surface finish. The composite specimen is further consolidated in a vacuum bag and cured for 12 h under ambient laboratory conditions.

Fig. 7
figure 7

Specimen manufacture process: a tabs bonded to dry fibre tows, b fibre tows are prestressed in jig, c hand lamination with resin and d vacuum consolidation. Dry blank pictured (e)

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3.2 Specimen measurement

The final manufactured specimens are scanned using a Creaform HandySCAN 700 3D scanner to an accuracy of ± 0.03 mm. The 3D-captured image is exported as a series of discrete data points and projected onto a 2D plane such that a 2D scatterplot of the curved profile is visible. The data is then ordered sequentially along the longitudinal axis (y-axis as pictured), and a 200-point moving average is applied to produce a 2D representation of the average profile across the width, as depicted in Fig. 8. The window size of the moving average was found to have negligible impact on the average profile within the range used. By interpolating the experimental data such that it possessed the same y-axis vector as the analytical prediction, a RMSE can be calculated to compare the accuracy of the analytical model.

Fig. 8
figure 8

Raw data from 3D scan (a) filtered 2D-projected profile showing the outer surface radius (b)

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4 Results and discussion

The mechanical model is validated experimentally across three unique laminates. First, a single-curvature, single-material laminate demonstrate the model’s ability to predict curvatures. A second, quasi-sine wave laminate, which is composed of equal, alternating arcs and hybrid reinforcement fibres, is produced to demonstrate the geometric model’s performance in predicting beam with multiple curvatures and demonstrate compound curvature. Finally, a compound curve laminate is manufactured using the same conditions as the quasi-sine wave however using different arc lengths and a higher pretension to show a more complex and pronounced compound profile. In each case, the outer surface radii are compared to the calculated, and a single laminate of each geometry is produced.

4.1 Mechanical model validation—single-curvature beam

A constant cross-section [02] CF laminate with a single ply pretensioned to 4250 N is manufactured at 350-mm specimen length and 100-mm width according to the manufacturing procedure. The result after removal from the jig and end tabs trimmed is shown in Fig. 9 a and b. A single, consistent curvature is identified, confirming that the EFP method produced regular arc profiles.

Fig. 9
figure 9

Side profile (a) and perspective (b) view of manufactured [02] curved single-material laminate. Comparison of analytical and experimental profiles noting unequal axis scale (c)

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The scanned curve and analytical are compared and found to have good accuracy with a RMSE of 1.19 at a radius of 108.4 mm. The close agreement between the scanned and analytical profiles is depicted in Fig. 9c. The performance of the model demonstrated the accuracy for the prediction of single curvatures, which confirmed its appropriateness for use in compound curvatures produced from the same manufacturing method.

4.2 Mechanical model validation—quasi-sine wave beam

To create the quasi-sine wave specimen, as pictured in Fig. 10a, the surface of the reinforcement fibre is alternated relative to the prestressed ply. The alternating pattern is made clear in the perspective view in Fig. 10b. This specimen utilised the same specimen geometry and prestress as the single-curvature laminate. In addition, the laminate is hybridised such that \(n\ne 1\) where GF is used as the pretension medium whilst used as CF the neutral. An overlap of 5 mm between the alternating plies is employed to effectively transfer load between segments.

Fig. 10
figure 10

Side profile (a) and perspective (b) view of manufactured [02] quasi-sine wave hybrid laminate. Comparison of analytical and experimental profiles noting unequal axis scale (c)

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The scanned and analytical curves are compared in Fig. 10c. A RMSE of 2.18 is achieved with a radius of 101.7 mm. Notably, the radius is tighter than the laminate containing CF only with the same pretension load. This is attributed to a change in neutral axis location in the hybrid laminate resulting from the mixed moduli condition—effectively increasing the eccentricity of the prestress, without changing the stiffness significantly.

Whilst presenting a close correlation, discrepancies are evident in the second segment shown in Fig. 10c. Deviations are attributed to slight variations in fibre tension across the width of the tow, twisting the structure. Figure 11 a shows the 3D surface, segmented into a front and rear section of 15-mm width each. A 50-point moving average is taken across the y-axis for both and compared to the global average in Fig. 11b. Here, it is clear that there is a height discrepancy between the front and rear profiles, confirming the axial twist. This is further evidenced by Fig. 11c where an x–z section cut of the 3D scan data is taken at the point of maximum deviation, denoted by the purple markers in Fig. 11a.

Fig. 11
figure 11

Segmented 3D scanned curve (a), projected segment profiles noting unequal axis scale (b), section cut gradient (c)

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This across-width variation in fibre tension stems is attributed to the end tab gluing process. In dry form, the cross-stitching does little to prevent individual unidirectional fibres from sliding and can result in uneven fibre lengths across the tab width once glued. Future efforts to mitigate this may include the use of prepreg materials where the resin acts as a binder, preventing relative motion and so structural twist in the cured composite.

4.3 Compound curve

The manufacture of the compound curvature laminate followed a similar process to the quasi-sine wave except the segment length is varied across the total length. Specifically, each segment measures 72.5 mm, 122.5 mm and 155 mm long from left to right excluding the 5-mm overlap, as pictured in Fig. 12a. The overall width, length and moduli configuration of the laminate remain constant while the prestressing load is increased from 4250 to 7000 N. The resultant compound profile is shown in Fig. 12a whilst the laminate is viewed in Fig. 12b.

Fig. 12
figure 12

Side profile (a) and perspective view (b) of view of manufactured [02] compound curvature hybrid laminate. Comparison of analytical and experimental profiles noting unequal axis scale (c)

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The varied segment length method effectively induced compound curvature with different arc lengths, reinforcing the ability of the process to induce curved profiles into laminates manufactured off a flat tool surface. The comparison of the experimental to analytical profile, shown in Fig. 12c, results in a RMSE of 1.32 with a radius of 97.5 mm for each segment. Twist was less present in the compound curvature laminate, highlighting its occurrence as a user-induced, not systematic inaccuracy.

4.4 Inverse design of a NACA 4412 aerofoil profile

With the EFP model validated on the compound shaped laminates, the practical application of EFP is demonstrated by an analytical case study on an aerofoil section using the inverse design approach. Prismatic aerofoils exhibit single-plane, compound curvature with aero surfaces constructed from thin skins, making them well-suited to EFP. The National Advisory Committee for Aeronautics (NACA) defines a series of standardised aerofoil profiles for varied flow characteristics [30]. The NACA 4412, depicted in Fig. 13, is a popular four-digit configuration with applications in wind turbine blades due to favourable power generation performance [31]. In this case, the upper surface is selected as the target profile to demonstrate analytical skin design by EFP.

Fig. 13
figure 13

NACA 4412 section profile

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A 350-mm-long laminate is divided into twelve equal segments. The designated pretension load is set to 150 N/mm width and applied to a single CF ply. The maximum number of neutral plies is constrained to 14, composed of CF reinforcement only. The solver is constrained such that the maximum ply pick-up/drop-off between adjacent segments is three plies, minimising substantial changes in thickness and stress concentrations. A single capping ply is also applied to the outer surface of the neutral plies to mitigate delamination at the ply drop-off/pick-up locations. The solver is given the outer surface of the predicted laminate using the outer surface radius in Fig. 1b as the radius value in Eqs. (16) and (17).

Depicted in Fig. 14a is the convergence plot showing the best and average population’s performance against the fitness function over 300 generations. Three snapshots of the solution are taken during the solve, denoted by and shown in (b), (c), and (d) on the figure. Snapshots (b) and (c) demonstrated the solver’s ability to quickly find a close solution within the first 50 generations whilst (d) gave full convergence to a RMSE of 1.98 by 150 generations. Beyond 150 generations, improvements to the fitness score are negligible, indicating that an adequate solution could be obtained at this point. On an 8-core 3 GHz CPU with 16 GB RAM, this reduced the computational time by 41.9% to 10.112 s.

Fig. 14
figure 14

Convergence plot for the GA solve (a) with snapshots of the best solution at different intermediate (b), (c) and final (d) generation intervals

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The leading-edge geometry of an aerofoil is important to the aerodynamic performance. Figure 15a shows the first 120 mm of the final GA surface against the target geometry. Here, the ply pick-ups are clear and present good surface alignment to the target geometry. Figure 15 b graphically shows the final laminate configuration, including the outer capping ply. As the front half of the NACA 4412 section is composed of tighter radii, the number of neutral plies is fewer, with the straighter segments at the trailing edge demanding more plies to reduce the curvature. Whilst visually significant in Fig. 14b, the abrupt thickness changes at the pick-up/drop-off locations will in practice be filled with resin, producing a smooth upper surface.

Fig. 15
figure 15

Cropped image of the final laminate at the leading edge (a) and graphical depiction of the total laminate where orange layers represent the prestressed lamina and blue, the neutral (b)

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Overall, the GA produced an acceptable solution to the upper surface of the NACA 4412 wing section, demonstrating the potential of EFP in design applications by an inverse design approach.

4.5 Aerofoil manufacturing and analysis

A 350 × 100-mm aerofoil demonstrator was manufactured using an adjusted methodology. As the aerofoil had a thicker and more complex layup, vacuum infusion was selected over wet lamination. Pre-cut dry fibres were stacked as depicted in Fig. 15b with small, randomly distributed drops of UHU Super Glue used to hold the preform together. After the peel-ply, flow mesh and vacuum bag were assembled, 15 kN of tension was applied as per the wet layup specimen procedure. The same R180 resin was degassed and infused under vacuum into the prestressed laminate. The remaining steps were identical to the wet layup method.

Figure 16 a presents the scanned top surface profile of the manufactured aerofoil and compares it to the profile designed by the GA. The experimental profile shows good agreement with a RMSE of 0.89. Moreover, ply steps were captured by the scanning process and shown to correspond closely to the analytical prediction. These steps are denoted by the abrupt changes in both the scanned and analytical profiles. Figure 16 b offers additional perspective to the ply steps where it is evident that the capping ply has blended but not concealed their transition.

Fig. 16
figure 16

Comparison of analytical (GA designed) and experimental profiles noting unequal axis scale (a), perspective view of manufactured aerofoil (b)

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4.6 Discussion

EFP has been shown as a promising method for producing mould-free compound curvatures in FRP laminates. Automated fibre placement offers the ability to accurately place fibre tows on segments where complex neutral layer configurations are required to achieve the desired curvature. Combining both manufacturing techniques will further advance the concept of 4DCP.

It is pertinent to address the limitations of the process to target its application and identify scope for future development. As the deformed curvature is a direct result of the internal bending moment, the maximum obtainable radius is limited by the tensile strength of the fibre. Hence, suitable structures for EFP should be composed of gentle, single-plane curvatures that exist within the possible remit of the material system. Structural applications that meet these criteria have potential. However, the influence of EFP on the mechanical properties of the constituent materials must be addressed and is the focus of a concurrent study.

Specific to the GA inverse design approach is the thickness of the laminate in flatter sections. As the prestress force is set to accommodate the tightest radius, mostly flat sections are manufactured by the addition of many neutral plies to increase bending stiffness. This increases the overall laminate weight and is a result of the built-in GA constraints which prevented the application of neutral lamina on the reverse side of the prestressed layer. In doing so, “flat” segments could be achieved with fewer plies, reducing the overall mass. This is however a concern of software only, and not inherent to EFP.

5 Conclusion

This investigation demonstrates that EFP can produce compound curvatures within FRP composites by tailoring the longitudinal stiffness of the laminate. Varying the location of the neutral plies relative to the prestressed results in a change in flexural rigidity and eccentricity which gives differing radii across the length. An analytical model founded on EBBT principles is demonstrated as accurate in predicting the resultant profile, with a close correlation to the experimental results. Finally, a simple inverse design model is proposed based on the GA to find the optimal laminate to achieve the upper surface of a NACA 4412 aerofoil. It is concluded that the GA is suitable for selecting an appropriate laminate to achieve the desired compound geometry.