Recent developments in manufacturing,mechanics,and design optimization of variable stiffness composites

Automated fiber placement

The AFP devices are the improved form of conventional automated tape laying (ATL) and filament-winding machines. The ATL process, developed during the 1970s, advances from the expensive, time-consuming, and imprecise hand-laying processes through the modification of computer numeric control (CNC) machines. Grimshaw et al. ³⁷ discussed the developments and worldwide availability of tape-laying machines in an early report. Through EADS CASA (merged with present-day Airbus Defense and Space) data, they have reported that the ATL process is ten times more productive and has a material scrap rate of 5% compared to 25–30% in case of hand-layup. In general, any ATL setup is supported on large gantry structures with cross rails for precise movement over the tool. The delivery head handles the prepreg tape, which is typically 75-, 150-, or 300-mm wide and is supported on a stiff and protective backing paper. ³⁸

The ATL process cannot properly form curved profiles and specific geometries with a non-flat layup. To improve the tape-laying process, the AFP device adopts smaller tows generally 3.2-, 6.4-, or 12.7-mm wide. These smaller tows are provided over the tool through the delivery head, supported on either the gantry system or using the robotic arm. Most of the AFP machines can handle up to 32 tows or slit tapes simultaneously with individually controlled, delivered, and cut-out processes for each such tow or slit tape. This flexibility can accommodate even complex geometries, cut out sections, and curvilinear fiber placement in manufacturing. Overall, the AFP delivery head has similar arrangements like ATL delivery heads with tape spools passing to the compaction rollers through robotic movement or an offline program. The heat and pressure treatments are provided to eliminate the vacuum voids using infrared, laser heater–like processes before finally pressing the ply layup on the mould. Figure 2(a) shows robotic arm–supported AFP delivery head functioning over a curved tool.

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Figure 2.

Different methods of manufacturing for angle tow composites. (a) Placement of fibers on a curved tool using an automatic fiber placement device, courtesy: Coriolis Composites ³⁹ ; (b) continuous tow-shearing approach^33,40; (c) tailored fiber placement approach ⁴¹ ; (d) integrated AFP delivery head with a mounted 3D printer ³⁵ ; and (e) textile fiber steering method. ⁴²

The narrow tape placement introduces some challenges like layup accuracy, productivity, gap and overlap defects, etc. Tool path optimization^25,43,44 and downtime reduction are also possible obstacles which need attention. Lukaszewicz et al. ²⁷ have presented a systematic and more dedicated review on the development of AFP processes. The constraints and defects associated with the AFP process for manufacturing of curvilinear fiber composites are given in the Manufacturing Limitations and Defects in Fiber Steering section. Various researchers have utilized AFP machines from different manufacturers in studying VSCs. Tatting and Gurdal ¹⁷ used the VIPER^TM45 series AFP device from Cincinnati machines (acquired by present-day Fives group). Ingersoll machine tools, Forest-Liné, ⁴⁶ Automated dynamics, MTORRES, and Coriolis ³⁹ are some of the leading manufacturers of ATL and AFP machines.

Tailored fiber placement technique

Researchers at Leibniz Institut fur Polymerforschung Dresden have developed the TFP process for placing curvilinear fiber profiles ³¹ with greater flexibility as shown in Figure 2(c). This approach works with the flexible placing of the fiber roving on a two-dimensional (2D) plane and zig-zag stitching by another yarn. A schematic showing the fundamental principle of the TFP process is given in Figure 3. TFP process is an embroidery-based process which uses dry tow. The dry tow is impregnated after stitching by resin using resin transfer molding (RTM). Some studies^31,32,47 have tried to investigate the utility of TFP-based composites for stress distribution in advanced composites and optimization of fiber paths. TFP composites have more waviness due to zig-zag stitching, and therefore, the estimation of stiffness reduction due to waviness becomes essential. ⁴⁸

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Figure 3.

Fundamentals of TFP process. ⁴⁷

On the other hand, the defects related to overlap and gaps are less prone in the TFP process due to convergence of individual tows of smaller width. A representative volume element (RVE)–based finite element (FE) study is conducted by Uhlig et al. ⁴⁷ to study the effect of stitching on effective material properties of TFP composites. The sizes of roving, stitching width, and resin volumes are evaluated from the micro sections and provided as input parameters for the RVE model. The mean stiffness of the experimental sample was 1.7% lower than the reference, whereas the RVE sample predicted it to be 1.9% lower. Through the experimental tests, the authors argued that for the same volume content, the reduced stiffness/strength of TFP specimen is still around 13% greater than conventional unidirectional CFRP composite under tensile load. The application of the TFP process is mostly limited to 2D structures, and minimal studies to large-scale structural applications are available. While most of the studies use thermoset composites in TFP, applications of thermoplastics are also investigated.^49,50 The TFP process has a relatively faster rate of tow deposition, but smaller width of the tows takes longer to manufacture complex geometries.

Some works also present on the embroidery-based approach with electrodeposition resin molding^51,52 for uniform impregnation of resin. However, these applications are still limited to small-scale test samples. The typical resin impregnation stage in the TFP-fabricated sample can be energy-intensive and full of harmful chemical substances. Further, the use of vacuum-assisted resin transfer molding (VaRTM) can disturb the placement of non-crimp fabrics and lead to void regions. This can be efficiently handled by dipping the sample in electrodeposition solution which impregnates the resin cover around the fiber by maintaining specific voltage difference in the solution. This process does not require any autoclave curing or vacuum packing but impregnation in thicker laminates may leave void spaces due to insufficient resin precipitation.

Continuous tow shearing method

There have been regular efforts to reduce the manufacturing defects of the AFP process. In this direction, Weaver et al. ⁵³ first used dry tow for the flexible placement of curvilinear fibers. The authors pointed out that wrinkling and straightening defects and fiber curvature limitations cannot be subdued using the in-plane bending deformation-based fiber placement approach. In-plane shear deformation of the tows was proposed by Kim et al. ³⁴ to overcome these limitations. Figure 4(a) shows that the regular AFP process induces buckling (compression) of the inner fibers and straightening (tension) of the outer fibers due to difference in length of the inner and outer fibers. Figure 4(b) shows the arrangement of in-plane sheared tows in a continuous manner which avoids the possibility of fiber wrinkling. The tow gaps and overlaps can be avoided by combining the CTS approach with the conventional tow shifting method.

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Figure 4.

Comparison of (a) AFP approach tow deformation and (b) CTS approach tow deformation. ³⁴

The CTS process adopts a delivery head (Figure 2(b)) which is led by the compaction shoe at the front and roller pinching device beside it. The dry tows are passed through the gap between the roller head and gripping shoe along the defined paths and to restrict the movement parallel to the roller axis, they are pressed at the roller head by the gripping shoe. The resin is impregnated at the bottom of the tows after the removal of the backing paper. The compaction shoe presses the semi-impregnated tows while the roller pinching head moves along the shifting direction to shear the tows and provides the curved path. The rate of shifting is synchronized with the backing paper take-up speed during this in-plane shear deformation. The CTS process can reduce the gap and overlap defects of the conventional AFP process in a larger way but still has limitations on the range of shear deformation, speed of the in-situ impregnation, and uniformity in resin delivery with few rich resin pockets existing. Kim et al. ³³ have used a tow deposition rate of around 5 mm/s in the CTS process which is much lower than the rate of 50 mm/s used in the AFP process for controlled tow placement.

Kim et al. ³³ investigated the manufacturing characteristics of the CTS technique experimentally. The fiber placement technique was improved by introducing one extra encoder in the previous set up which synchronized between backing paper take-up speed and tow feed speed. The fiber angle variation in between takes place such that the thickness changes smoothly. Some errors, that is, layup inaccuracy, slightly uneven thickness, and shift width variation may be seen if the shear angle is greater than 50°, and little amount of wrinkle and overlap can be seen in the straight region of the tow path. The same group ⁵⁴ then proposed computer-aided modeling software, VATMESH, to model VAT composites made by the CTS method. The software could be directly used in the manufacturing process as it integrated design and manufacturing. Finite element modeling in ABAQUS was integrated with this CAD tool to visualize fiber orientation and thickness distribution properly. The thickness variability of CTS samples can be used as a design variable too. Lincoln et al. ⁵⁵ discussed about the reduction of geometric imperfection sensitivity on thin-walled cylinders under axial compression. The CTS process is used to produce angle tow composites, which includes fiber angle and thickness coupling as a design variable. Thicker parts of the cylinder can act as stiffeners which inevitably increase the buckling load capacity of the section. It is observed that geometric imperfection is drastically reduced for VSC, and as a consequence, buckling load capacity and knockdown factor (KDF) is enhanced. Further, this study is extended for experimental validation by Lincoln et al., ⁵⁶ which uses the rapid tow sheering technique (RTS) to produce VSC. The RTS is an advanced version of the CTS technique, with higher deposition rate, proposed by iCOMAT. ⁵⁷ Zympeloudis ⁵⁸ provided a detailed insight into the wide-tape CTS fiber placement process in his thesis. The effect of process parameters and material parameters is studied on the placement of dry unidirectional tape and prepreg tapes. It is also shown that with the CTS process, the steering radius can be decreased up to 50 mm for 100-mm–wide tow with a layup speed of 6 mm/s, which is highly beneficial in eliminating manufacturing defects.

Additive manufacturing (3D printing) technology

3D printing, also referred to as additive manufacturing (AM), rapid prototyping (RP), or solid freeform (SFF), was first introduced by Charles Hull in 1986. ⁵⁹ 3D printing is gaining interest due to its versatility in making complex geometries, reduction in a multi-step process, and freeform fabrication. Some studies have recently investigated 3D-printed curvilinear fiber composites.^60–63 Wang et al. ⁵⁹ have discussed various 3D-printing procedures for manufacturing fiber composites as shown in Figure 5. To form a continuous fiber path using 3D-printing procedures, the fused deposition modeling (FDM)–based co-extrusion method is mostly utilized.^59,64 FDM includes the melting of the filament inside the nozzle followed by extrusion using the rear end push. This approach can manufacture smaller radius fiber paths (up to 4 mm) with better control using thermoplastics. PLA, ABS, and PET type thermoplastic matrix are used for regular applications, whereas PEI, PEKK, and PEEK matrix are used for high-performance utilities.

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Figure 5.

(a) Fused deposition modeling (FDM) is a commonly used method because of its low cost, high speed, and simplicity in fiber placement. It is based on the working principle of thermoplastic filament extrusion controlling. The filament turns to semi-liquid state in the nozzle by heat application and placed in a layer-wise manner. The principal drawback of this method is the viscosity limitation of the thermoplastic polymer used; it should be capable of providing enough structural support along with enabling extrusion at the same time. (b) The powder bed and inject head 3D printing (3DP) is another approach in which polymer materials (in powder form) are spread on the platform and a binder coming from the inject print head binds it in the desired form. The print head can move in the x–y direction forming a 2D pattern, and a new layer is formed by lowering the platform once previous layer is complete. (c) The stereolithography (SLA) approach uses a UV laser to cure the photopolymers and orient them in the desired shape with high resolution. This procedure is costly but due to absence of the nozzle, the clogging problem can be avoided. (d) Selective laser sintering (SLS) is similar to previous 3DP processes. In this approach, instead of liquid binder, laser is used for fusing neighboring powders through molecular diffusion. (e) 3D plotting or the direct write method uses a syringe to place the material on a print bed under pressure. The syringe can move in three dimensions, and the bed remains stationary. Material flexibility is the key advantage of this method; it may need some support if low-viscous material needs to be placed (figure reproduced from Ref. 59).

Still, curvature-induced minor wrinkling and folding defects, voids and micro cracks, and non-uniform resin distribution are some of the defects present in 3D-printed continuous fiber composites.^62,65 To improve the fiber deposition and control in the 3D-printing procedure, several modifications in the manufacturing process are proposed. One of the main challenges in placing curvilinear fibers is the point-wise distance irregularities between adjacent fibers; hence, for proper printing, it is essential to change filament feed along the nozzle motion to achieve uniformly distributed polymeric materials. Akhoundi et al. ⁶⁶ calculated filament feed in the FDM process to appropriately model continuous curvilinear fiber without any fiber-deficient or fiber-congested areas. This detailed procedure of calculating the filament feed results in very accurate curvilinear path distribution without gaps and overlaps which affects the mechanical property of the composite greatly. Sugiyama et al. ⁶¹ considered the variable distance between the 3D-printed curvilinear fibers in the modeling to improve the estimates of numerical analysis with built samples. Matsujaki et al. ⁶⁴ demonstrated an in-nozzle impregnation method of 3D printing for continuous fiber plotting. Here, the thermoplastic resin filament and the fibers were supplied separately to the printer head. The reinforcing fibers are preheated before entering the nozzle which enhance the mixing of fibers with resin and the defused heat reduces the viscosity of the mixture resulting in smooth fiber placement. This process obtained higher strength and elastic modulus than commercially available 3D printers.

These recent advancements in 3D printing of continuous fiber composites,^64,67 which are mostly limited to the thermoplastic matrix (comparatively lesser strength than thermosetting), have opened broader possibilities of manufacturing curvilinear fiber composites.^60,68

Hybrid manufacturing and textile fiber steering approaches

There have been regular efforts to improve the performance of variable angle tow composites by including additional techniques of forming variable stiffness concepts. The robotic movement of the delivery head and additive construction are two essential steps for future manufacturing. In this direction, an integrated 3D printer on the AFP delivery head is proposed to overcome the gap regions (Figure 2(d)). ³⁵ This hybrid manufacturing approach detects gaps by either using the edge detector mounted on a 3D printer head or the thermal camera installed on the AFP delivery head. The 3D print head places single filaments of carbon fiber–reinforced plastic composites directly in the gap regions. The process is utilized for thermoplastic composites as they do not require autoclave curing. Raspall et al. ⁶⁹ also proposed a combined AFP and AM process for fiber-reinforced composites. The fixed AM delivery head provides geometric-specific thermoplastic substrate/mold for the AFP step. Through a movable layup table held on the second robotic arm, the requirement of the AFP head to remain perpendicular to the working surface is maintained. While the independent functions of AFP and AM are possible in the proposed design, the compatible material requirement for prepreg and substrate and larger space requirement are the remaining few challenges.

Recently, a low-cost and fast fiber steering method has been proposed, which is suitable and productive for textile fabrics.^42,70 The approach is motivated by the shear deformation of fabric fibers during the draping process over the doubly curved surfaces (Figure 2(e)). The controlled deformation of the fabric perimeter and selected internal points using pin-jointed net kinematics lead to the fiber steering. This process has been numerically modeled in a MATLAB® tool SteerFab to visualize and analyze the fiber steering. The method needs control of the shear angle to avoid warping. Like the CTS process, the change in thickness of the fabric due to shear deformation may appear. The technique has been tested for simple 2D textile fabrics and still needs more development for industrial applications. The approach is also more suitable for thermoset composites than thermoplastics as they can be easily deformed with an uncured matrix.

To address the manufacturing defects of laser-assisted automated tape placement technique (LATP), Clancy et al. ⁷¹ introduced a tape-spreading device into the LATP mechanism. Integrating this mechanism, variable width tow is produced which can compensate the gaps and overlaps limitations. Depending on the layup temperature and pressure, the width can be increased up to 62%. It is also investigated that the physical properties remain unchanged after fiber spreading.

Closure

Unequivocally, there are no perfect ways of manufacturing curvilinear fiber composites. While the AFP process is still the most preferred; the stringent limitation of higher steering radius, slow deposition rate, gap and overlap defects, high initial investment, and limited use of thermoplastics raise few challenges. The CTS and TFP processes carry fewer defects, and their utility to large structural components is steadily catching up. On the other hand, the versatile 3D-printing setup is not capable yet of producing high-strength thermoset composites. In this situation, the hybrid approach, particularly the integrated AFP and 3D-printer device, and the RTS-based approach appear to be a few suitable alternatives.

Curvature constraint

As the radius of the fiber path reduces, the fibers buckle locally and wrinkles are formed in the fiber path. ⁸⁰ Due to this reason, the minimum turning radius is restricted for curvilinear fibers to avoid fiber buckling or wrinkling. This limitation is further dependent on the tow width, material, and manufacturing technique used. ⁸¹ In general, the minimum radius of curvature varies between 400–1000 mm ⁸² for the AFP process. The minimum radius restriction is more relaxed for TFP and AM^62,74 processes due to smaller width and dry tows. The TFP process can deliver fiber roving with a minimum radius of curvature of around 5 mm. The recent works with laser-assisted AFP devices have also shown better control on the minimum steering radius. ⁸³ The 3D-printed curvilinear fiber profiles have lower critical steering radius. Matsuzaki et al. ⁶² have shown that 5 mm radius of the curved fiber path can be obtained with a 1000–carbon-fiber (1K) filament using the FDM process. These 1K carbon fibers had a diameter of 0.4 mm. The larger diameter (4K or 8K) carbon fibers show increase in the minimum steering radius. It is also observed that the difference between the set steering radius and printed radius increases for more numbers of dry fibers. Table 1 summarizes some of the minimum steering radii adopted/prescribed in different studies.

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Table 1.

Curvature constraints in different manufacturing approaches.

Manufacturing process	Primarily used fiber material		Layup speed	Curvature constraint
	Thermoplastic	Thermoset		Tow width	Limiting radius
AFP	✓	✓	—	3.175 mm	635 mm [84,85]
			50–140 mm/s	6.35 mm	889 mm [78]
			50 mm/s	6 mm	400–800 mm [83]
TFP		✓	16–40 mm/s600 stitches/min	—	5 mm [86]
CTS	✓	✓	3 mm/s	7 mm	30 mm [34]
			2 mm/s	100 mm	50 mm [87]
			6 mm/s	100 mm	50 mm [58]
3D printing	✓	✓	1 mm/s	0.4 mm	5 mm [62]
			5 mm/s	2.5 mm	3.8 mm [68]

If f (x) defines the curvilinear fiber path for fiber angles varying only along x-axis, then the curvature constraint is generally provided as ⁷⁷

The maximum rate of tow shearing β ˙ and layup speed V are governing factors for the minimum steering radius R min .

Manufacturing defects

Tow wrinkling, waviness, and folding

The mismatch of the curved geometrical path and straight tow length introduces regions of tension, compression, and shear deformations in the steered tow. These deformations lead to the out-of-plane wrinkling, in-plane waviness, tow pull up (folding), and sheared fiber defects in VSCs. These defects are one of the ways of releasing the excessive compressive and tensile stresses of the tows. While the regions of different defects may exist along the tow path, most of the modeling studies consider them independently. Out-of-plane wrinkling is widely studied for steered tows,^{46,78,80,88,89} whereas limited studies are available for in-plane waviness. ⁹⁰ The compression in the inner fibers on a curved path leads to the local tow buckling which appears as repeating out-of-plane wrinkle patterns or in-plane waviness as shown in Figures 6(a) and (b). Several analytical and experimental works have studied the critical steering radius and frequency of wrinkle formation. Mostly, wrinkles are modeled as isolated patterns^78,89 due to consideration of uniform repeating frequency. The width of the tow and tackiness of the surface are the most vital parameters in deciding the extent of wrinkling and waviness defects. The process parameters, like temperature, compaction pressure, and layup speed, also influence the formation/control of such defect regions. Generally, below a critical steering radius, the possibility of these defects cannot be prevented. Most of the analytical models adopt the analogy of “plate resting on an elastic foundation” to model wrinkling and in-plane waviness defects of the AFP process. The identification of viscoelastic properties for substrate tacking and dependence on process parameters often become difficult to quantify in the absence of proper experimental results. Beakou et al. ⁴⁶ presented local wrinkling of steered prepreg tows considering the buckling of plate resting on elastic foundation. They experimentally evaluated adhesive normal stiffness (foundation) using a pressure-sensitive probe and investigated the critical steering radius, which will avoid wrinkling. Matveev et al. ⁸⁹ also adopted the similar model for dry tows with slightly different boundary conditions of the orthotropic plate model and investigated the relationship between critical steering radius, substrate tackiness, and process parameters. Figure 7(a) describes the variation of critical steering radius with substrate stiffness, and Figure 7(b) shows the dependence of substrate stiffness on process parameters. Bakshi and Hojjati ⁷⁸ adopted a global approach of wrinkle modeling based on the experimental evidences of non-uniform repeating patterns. The FE-based simulation is performed, and cohesive zone modeling is adopted for prepreg tack using bilinear traction separation law. They also conducted a number of experiments with different compaction pressure, delivery head speed, and temperature over a range of steering radii. The in-plane waviness, out-of-plane wrinkles, and mid-tow blisters are seen over the range of steering radius.

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Figure 6.

Manufacturing defects and process parameters in curvilinear fiber placement. (a) Wrinkle and blister formation in steered tows. ⁷⁸ (b) In-plane waviness in steered tows. ⁷⁸ (c) Fiber twisting and folding defects in the 3D-printing process. ⁶² (d) Waviness formation in the TFP process due to zig-zag stitching. ⁴⁸ (e) Regions of tow gap and overlap. ⁷⁹ (f) Variation of thickness during the CTS process. ³⁴

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Figure 7.

(a) The minimum permissible steering radius, defined for wrinkle-less tow placement, decreases with decreasing tow width and increase in substrate stiffness. Tackiness between tow and resin helps in effective control of wrinkling. ⁸⁹ (b) Tackiness is weakly affected by the process parameters, pressure, and layup speed, whereas the contact between tow and resin substantially improves at high temperatures due to decrease in resin viscosity. But this increase shows a peak value possibly associated with the strength reduction of the resin with decreasing viscosity. ⁸⁹

While the most of these defects are common in AFP manufacturing, some of similar deformities occur in other techniques as well. The fiber twisting and folding, shown in Figure 6(c), occur in 3D-printed curvilinear fiber composites due to non-rotation of delivery heads along a circular path. These defects emerge significantly with a smaller radius, and may degrade the mechanical properties of the composites. The zig-zag stitching in the TFP process (Figure 6 (d)) also introduces out-of-plane waviness and further reduces the stiffness.

Gaps and overlaps

These are the most common defects that are present in VSC manufacturing. The fiber placement is generally adopted in two ways. The tows are placed either using the shifting method or parallel to each other as shown in Figure 8. The gap and overlap regions, as shown in Figure 6(e), are resin-rich and thicker sections, respectively. ⁹¹ Gaps and overlaps can be denoted by the coverage rate also. 0% coverage represents a condition when tow is cut/dropped as soon as its edge reaches a limiting radius, whereas 100% coverage refers to the condition when both the edges of tow reach the limiting radius before tow drop. 0% coverage creates small triangular resin-rich regions (gap), and 100% coverage offers complete overlap. Most of the studies have shown that gaps are more prone to failure initiation, and overlap increases the buckling load capacity of the section. ⁹² Still, the increase in buckling capacity due to overlap can be attributed more to the stiffening effect of overlap regions rather than the action of the curvilinear fiber profiles; however, this improvement causes an adverse impact on its in-plane shear property. ⁷³ The geometry and extent of gaps and overlaps^93,94 control the mechanical properties of the laminates.

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Figure 8.

Methods of tow delivery. (a) Shifting approach shifts the reference fiber path by a distance along the normal direction to fiber angle variation. The approach is easy to manufacture using most of the AFP devices and mostly followed in practice. But due to the finite width and curvature of the tape, outer fiber of bottom tow does not align with the inner fiber of upper tow resulting in gap and overlap regions. (b) The parallel delivery builds the fiber path by defining each path as a group of points that are at the same distance from the reference curve. This process reduces the possibility of tow gaps and overlaps, but it may introduce pockets of large curvature in the tow placement path, which may wrinkle out of the plane. The parallel method of forming tow paths is more difficult to manufacture and therefore, is not used much.

Blom et al. ⁷² studied tow-drop effects in strength and stiffness of VSCs theoretically, and concluded that failure generally starts at tow-drop areas. Change in fiber angle orientation away from the mid-reference curve is obtained as a function of the tow width, and based on the coverage of tows at boundaries, tow-drop areas are identified. These tow-drop areas are modeled specifically along with the regular tow course with the damage model based on irreversible thermodynamics and solved using the commercial FE tool. The effect of tow width, laminate thickness, and staggering are studied, and it is shown that laminate strength decreases with increasing tow width. Staggering has a positive effect on strength, while thickness does not affect strength. It is understood that to model the tow drop correctly, minimum mesh size must be decided based on tow-drop areas which increases the computation time of the procedure. To improve the computational efficiency of such defect modeling algorithms, Fayazbakhsh et al. ⁹⁵ suggested a defect layer method which captures the location and extent of defect, irrespective of the element size. The method alters the material properties and thickness for elements in the percentage of gap or overlap area. The gap and overlaps are identified using vertical offset values of tow width in the shifting direction, which is coded as MATLAB subroutine. The FE analysis of 0% coverage shows around 15% decrease in buckling load capacity from defect-free VSC modeling, whereas 100% coverage shows maximum 71% increase from quasi-isotropic laminates. It is observed that gaps can reduce both the buckling strength and in-situ stiffness. Whereas, in case of overlap, in-situ stiffness decreases and buckling capacity of the structure increases. So, gaps are more vulnerable than overlaps. This model was further used to study the effect of defect variables on the optimization of buckling load and in-plane stiffness ⁹⁶ as well as on shear deformation-based kinematic models. ⁹⁷ Falco et al. ⁹⁸ have also tried to improve the computational efficiency using structured meshing for 0% coverage panels. Mishra et al. ⁸² presented the buckling strength and stiffness of VSCs with tow-drop defects in AFP process. Homogenization is used to establish the relationship between defect geometry and single-ply properties. The computationally efficient smeared layer approach provides accurate results for stiffness, but it overestimates the buckling capacity as stress concentration cannot be found accurately. The residual stress predictions using FE computational models are found to be more reliable for steered tow composites with an overlap region than non-overlap composites. ⁹⁹ These residual stresses appear due to the variable coefficients of thermal expansion with complex pattern of fiber orientations. Gap and overlap defects are less prominent in other manufacturing techniques of VSCs.

Fiber angle deviation and misalignments

For a curved fiber profile, the actual fiber angle continuously deviates from the reference fiber angle along the fiber course. Fiber course width is directly proportional to the angle deviation ⁹⁶ which leads to increase in gaps and overlaps simultaneously. The angle deviation can be restrained by decreasing the fiber course width, but it will make the process more time-consuming. The effect of fiber misalignment on buckling capacity of VSCs is evaluated in the stochastic framework using Monte Carlo analysis. ¹⁰⁰ The Gaussian distribution with 0.5 standard deviation and mean of 0 is stochastically investigated to evaluate the buckling load and mode shape distribution. The severe misalignments may induce mode switching in the VSCs. Fiber misalignment is also seen in 3D-printed curvilinear fiber composites. ⁷⁴ The weak fiber–matrix interface, porosity of the interface region, and uneven pressure of nozzle, other than the smaller steering radius, have been found to be the main contributors of such misalignments. Twisting of tows can also take place due to misalignment, which leads to a resin-rich area. Croft et al. ⁷³ studied the effect of twisted tow on strength. These twisting patterns are also observed in 3D-printed VSCs. ⁶²

Besides these limitations, there are few other defects present in the different processes. The variation in sample thickness occurs around the overlap regions in the AFP process. This thickness variation is also seen in the CTS process due to in-plane shear deformation which is shown in Figure 6(f). The modified thickness is given as t = t 0 / cos β , where t ₀ is initial thickness and β is the shear angle. Therefore, the large shear angle is generally avoided in the CTS process, and the 90° angle in fact becomes unfeasible. Further, fiber bridging may occur when curvilinear fibers are placed on a concave surface, as fibers are in high tension, and they can detach from the surface. A schematic in Figure 9 tries to list down major defects associated with different manufacturing techniques. The underlying mechanism of these possible defects is also listed. The AFP process is fast and versatile but at the same time contributes to maximum defects. As discussed, the occurrence of the defects can be controlled to some extent by the virtue of the process parameters, for example, temperature, speed, and compaction force, ⁷⁸ but with a lower steering radius some of these defects become unavoidable. The design philosophy shall accommodate these as the features rather than the defects at such smaller steering radii^80,101 for efficient design.

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Figure 9.

Schematic of various defects and underlying mechanisms associated with different manufacturing processes (¹The CTS process also has small amount of overlap in regions of higher shear angle, ²The AFP procedure also introduces patches of variable thickness around the overlap area).

Fiber profile (angle/path function) based optimization

Fiber profile can be optimized globally using predefined equations of fiber angle variation or path function to decrease the number of optimization variables. This curvilinear parameterization with a predefined set of fiber angle variation reduces the design space of the problem but maintains continuity of fibers. Several studies with linear^{12,17,107–109} and nonlinearly varying fiber angles^110–113 are available. Different parameterized curves are proposed to increase the design space and improve continuity and manufacturability. The local element-wise optimization of fiber angle orientation can be more effective with greater design space, but this increases the optimization variables, and fiber angle continuity may also suffer. To overcome these difficulties, few studies have adopted the fiber path using the linear combination of basis non-uniform rational B-spline (NURBS) curves, ¹¹⁰ potential flow field, ¹¹⁴ B-spline surfaces, ¹¹⁵ Shepard interpolation,^113,116 level set method, ¹¹⁷ etc. Parnas et al. ¹¹¹ used Bezier splines to maintain continuous non-uniform layer thickness and fiber angle variation. Some of the fiber angle profiles are discussed further.

The linear fiber angle variation is used in a number of studies^12,118 as

Due to the manufacturability requirements, fiber angle variations are also provided with constraint curvature conditions. To ease the implementation of curvature constraint on optimization parameters, a parameterized definition based on circular arc is proposed.^118–120 This gives a constant curvature profile which is more suitable for implementation in optimization algorithms. Equation (4) gives the fiber angle variation for the constant curvature path which is schematically shown in Figure 10. The floor function gives the nearest integer less than or equal to the real number

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Figure 10.

Schematic of the circular arc path having constant curvature. ¹¹⁹

Blom et al. ¹¹² defined the fiber angle for a conical surface as the angle between the longitudinal surface vector and tangent to the fiber path. They have adopted the following geodesic, constant angle, and constant curvature profiles to optimize the frequency response of the conical shells

Here, α is subtended cone angle, and r 0 defines smaller radius of truncated cone.

Nonlinear fiber angle variation is adopted in the form of Lagrangian polynomials as well^121,122

While the adaptation of the constant curvature path easily satisfies manufacturing constraints, ¹²⁰ the sharp curvatures of linear fiber angle profiles need strict curvature control. It is generally seen that parameterized curves reduce the computational task of optimization but may still need a large number of data points in Lagrangian polynomial form to avoid the local optima. These aspects generally govern the choice of optimization strategy for VSCs. Among the different approaches, the gradient search method is used by various authors.^{76,113,114,116,123} The approach shows fast convergence but may fall for the local optima as the convexity of objective function cannot be ensured. The complexity of composite profiles reduces the chances of analytical or closed form expression for objective function. The sequential breaking of the overall problem into approximate sub-problems has been utilized in such situations using cellular automata (CA),^124,125 quadratic sequential approach,^76,111 etc. These approaches become less reliable with increasing complexity and discretization due to its approximate nature but remain computationally efficient for possible implementation in parallel programming. The increasing number of variables in the multi-layered composite optimization problem requires assessment of global optima with enhanced sensitivity. The direct search algorithms are generally adopted for these cases. Most of the studies have used the stochastic direct search approach–based evolutionary algorithms like the genetic algorithm.^{107,120,123,126–128} The exploiting and exploratory nature of these evolutionary algorithms make them fast yet globally optimum. Still, the computation time of these algorithms is more due to large number of variables.

Honda et al. ¹²⁷ gave optimized fiber profiles using meta-heuristic multi-objective (objective function on failure-index and fundamental frequency with conflicting argument for maximum curvature) non-dominated sorting genetic algorithm (NSGA-II) by improving their earlier model based on GA. ¹²⁹ The fiber path is assumed as a cubic polynomial and constant angle fiber; tangent to the surface are adopted for individual finite elements. Through optimized coefficients for cubic polynomial, they were able to show the fiber orientation tangential to the hole which agrees with the Hap (hydroxyapatite) crystal profiles around a naturally existing hole, foramina in bones. The similar contour lines, tangential to the surface with constant fiber angle for the individual finite elements, are used by Huang et al. ⁷⁷ In order to reduce the computational need and number of function evolutions to find the pareto-optimal solution in a multi-objective problem, Nik et al. ¹²⁸ used polynomial regression–based surrogate models coupled with GA. It is important to point out that though surrogate models increase the computational efficiency, the compromise on the accuracy of results may not give the best curvilinear profile for the fiber path. Rouhi et al.^130,131 also used a two-step multi-objective optimization procedure for optimum composite weight and buckling load criteria. The surrogate models are generated with radial basis function to define the buckling load as a function of fiber angles. The genetic algorithm optimization procedure is used to reduce the errors associated with the surrogate models.

A classical problem of the plate having holes (or cutouts) is studied by various authors.^{86,107,111,114,123,127,132–134} The optimization strategies with weight minimization, ¹¹¹ optimum strength,^123,127 and maximum buckling load^109,126,135 criteria are tested. The variable fiber angle profile is found to enhance the strength and buckling load capacity compared to the constant angle composites. Figure 11 gives a comparative estimate of increase in buckling load capacity of VSC plates with the center hole in comparison to the unidirectional plate with a hole. Though the estimates are not normalized for size of the plate, the diameter of the hole, fiber path, and boundary conditions, still, the utility of the curved fiber path can be observed. The adaptation of manufacturing limitations shows that buckling load capacity is influenced by the gap and overlaps. Jegley et al. ¹⁰⁷ conducted experiments on a number of samples with different hole sizes and tried to assess the measures of Rayleigh-Ritz–based FE models for possible deviations from experimental results. In order to consider the manufacturing defects, tow-drop and overlap approaches ¹⁷ are used. In comparison to straight fibers, the overlap method provides significantly higher buckling load capacity at a cost of an increased weight, whereas the tow-drop approach has a nominal increase in buckling load capacity. This observation is very similar to the results presented by Wu et al. ¹³⁶

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Figure 11.

Percent increase in buckling load capacity of a plate with a center hole in comparison to unidirectional composites reported by different studies.

To reduce the complexity of variable angle tow placement, Huang and Haftka ¹²³ showed that optimized fiber angles as piecewise bilinear interpolation functions in a small area around the hole can also increase the load-carrying capacity significantly. Such fiber distribution can avoid stress concentration without requiring rigorous fiber placement in large structure spaces. Recent work by Malakhov and Polilov ¹³⁷ considered non-constant distance between curvilinear fibers to obtain the stress concentration around a hole. It is shown that stress concentration reduces by a factor of 3.2 for curvilinear fiber composites compared to straight fiber composites.

Lamination parameter–based optimization

The conventional gradient search approaches may be computationally expensive for optimization of variable stiffness composites using finite elements due to a large number of design variables. Further, a number of layers enlarge the design space. The nonconvex nature of objective function for the fiber path optimization schemes also leads to the local optimum solution. In these situations, lamination parameter–based optimization techniques prove to be more effective. The lamination parameters (LP) are defined as the integral over thickness for trigonometric functions representing fiber angle orientations. Therefore, these parameters are independent of the number of layers in composite. In classical lamination theory for a regular composite, a total number of twelve lamination parameters can exist as given in equation (7)

In the case of orthotropic composites without coupling between bending and stretching terms, these parameters reduce to four. These parameters are defined in a feasible region governed by the inequality constraints which are given for the case of orthotropic laminates with curvilinear fibers. ¹²⁶ The feasible region defines the limitations on the values of LPs when another LP is defined. The identification of the outer boundary of the feasible region for a more generalized case of eight or twelve lamination parameters or for VSCs is still challenging. Therefore, often approximate feasible regions based on the convex hull approach are utilized. ⁷⁵

The typical VSC optimization procedure using LPs is a multi-level structural optimization. The first step expresses objective function (based on criteria of stiffness, strength, buckling, etc.) for the optimization process in terms of lamination parameters. The nonlinear objective function and constraints are expressed mostly in an approximate manner using the Taylor series form, reciprocal approximation approach, etc. The minimum compliance design which enforces the minima of strain energy is a widely taken optimality criterion for structural designs using lamination parameters. A number of works^75,138–140 have used this criterion for obtaining the maximum buckling load or optimum natural frequencies. Multi-objective optimization studies with maximum buckling and strength criterion are also present.^141,142 The reciprocal approximation approach expresses the objective function as the Taylor series form in terms of the design variables (compliance for this case). Conventionally, LP-based objective function in the feasible domain is considered to be convex based on the early proof of Grenestedt and Gudmundson. ¹⁴³ Therefore, gradient search–based approaches have been the preferred choice for first-level optimization. The recent results of Scardaoni and Montemurro, ¹⁴⁴ however, disprove this traditional claim and highlight that the actual feasible region in LP space may not be convex and shall be defined with dependence on the number of plies.

The optimized stiffness parameters are further incorporated into a second-level optimization strategy to retrieve the optimum stacking sequence and fiber angle profiles.^106,145 The second step optimization has been major source of functional deficiency in the process as retrieval of the fiber angle profile has to be accompanied with manufacturing constraints. The recovery of the optimum stacking sequence from optimized stiffness parameters is also difficult due to non-bijective nature of the relationship between ply-sequence and stiffness tensor. In fact, most of the feasible bounds in the LP space are obtained only for the symmetrical stacking sequence to maintain the null membrane-bending coupling tensor. This ignores other non-symmetrical stacking sequences (often termed as quasi-trivial stacking sequence) which may also give uncoupled stiffness tensor.^144,146 Further, in the FE framework, the retrieval step gives average values of fiber angle profiles at the node/element center for individual elements. The generation of continuous fiber angle profiles from these discrete optimization steps incurs losses on the efficiency of the optimization process. The fiber angle optimization at nodal points rather than element-based optimization reduces the discontinuity in the optimum fiber angle variation. To form the continuous fiber path, a number of studies have considered curvature control or adopted continuous functions to define the LPs. Khani et al. ¹⁴¹ have used quadratic functions to define the average curvature to construct the smooth fiber path in each layer. The maximum buckling load with realistic curvilinear fiber profiles shows an increase of 29.6% compared to constant angle fibers for linear analysis for VSC cylindrical shells. Wu et al. ¹²⁶ have employed B-spline functions to define the spatially varying LPs. The smooth and convex form of B-spline curve is determined using appropriate control points. Using a gradient search method, the optimum distribution of LPs, for maximum buckling load, is calculated. Finally, the genetic algorithm is used to recover fiber angle variation from optimum LPs. The approach works efficiently in case of similar mesh sizes for buckling analysis and optimization study. Hao et al. ¹⁴⁷ also used the same B-spline functions to define the lamination parameters. The isogeometric analysis for the buckling load calculation is adopted. The optimized LPs are converted into a layup sequence and fiber angle profile using an evolutionary algorithm. The conditions of curvature constraint and gap and overlap processing are also included to obtain the realistic fiber path.

Finally, the manufacturable fiber profile is defined in the third step. A schematic of this multi-step procedure is shown in Figure 12. A recent review summarizes many studies on the optimization of variable stiffness composites using lamination parameters. ¹⁰³ Other than the challenges of the fiber retrieval step, inability to consider the manufacturing constraints in the first-level optimization is also a major challenge for lamination parameter–based optimization. Peeters et al. ¹⁴⁹ used a multi-level optimization to define an optimum VSC stacking sequence with manufacturing constraint on the norm of the gradient of fiber angle distribution. The first assumed solution in terms of lamination parameters is updated based on fiber angles which are viable to manufacture. The constraint on the norm of gradient is implemented in average sense representing single global constraint per layer or local steering constraints for each element. The global constraints take control of gap and overlaps between tow but cannot maintain local steering curvature to avoid tow wrinkling. The local constraints therefore provide better manufacturable fiber angle optimization. Hong et al. ¹⁰⁴ expressed the norm of the gradient of lamination parameters in terms of the norm of fiber angle gradient using the chain rule and Cauchy–Schwarz inequality. This condition gives more conservative bounds but still reduces the computational effort in the optimization problem.

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Figure 12.

The multi-step optimization process of curvilinear fiber composites. ¹⁴⁸

Polar parameter–based optimization

Another method based on macro-scale stiffness modeling uses polar representation. Unlike the lamination parameters, the polar parameter (PP)–based approach uses invariants of a composite stiffness tensor. Importantly, polar representation of tensor components is also frame independent, and rotation of frame can be readily accommodated in the polar parameters. This avoids the cumbersome transformation exercise required for coefficients of tensors in Cartesian form. The polar representation of tensor invariants is based on the complex variable approach, and details of the formulation can be found in the early work of Verchery ¹⁵⁰ and review by Vannuci. ¹⁵¹ A second rank planer tensor T is given in terms of polar parameters with frame rotation of θ in equation (9)

Recently, a lot of interesting work on multiscale two-level (MS2L) modeling of VSCs using PPs has been taken up by Montemurro and other researchers.^153–155 The PP-based approach has clear advantage over the LP-based formulation. The first level optimizes macro-scale topology as well as stiffness/strength of the composite, and second level optimization identifies the optimum stacking sequence and fiber path continuity. The physically meaningful nature of polar parameters do not shrink the design space and also give the choice of non-symmetric, unbalanced stacking sequences in the second step. The earlier work by Montemurro and Catapano ¹⁵⁶ adopted manufacturing constraints in second-level optimization with B-spline formalization of fiber paths which may lead to the null optimum path fulfilling both the constrained condition and optimum distribution of PPs. This aspect is improved by Montemurro and Catapano^115,157 later by including the manufacturing constraint in the first-level optimization in the polar parameter space by using B-spline forms with constraints at control points only.

Taking a careful note of these studies, it is observed that mostly gradient search algorithms are utilized for first step optimization. The angle retrieval and path generation step mostly adopt evolutionary algorithms to optimize the fiber path function. The node-dependent optimization maintains better continuity control compared to the element-based optimization at the cost of greater computational effort.

Buckling and post-buckling analysis

The effectiveness of angle tow composites for enhanced buckling load capacity is a well-studied problem.^{30,53,75,114,121,158} Several studies have shown specific behavior of VSCs. Gurdal et al. ¹⁵⁹ used the Ritz method to study the prebuckling stresses and buckling load capacity of variable stiffness panels with a linear variation of fiber angles. The VSCs can obtain decoupling of the axial plane stiffness and buckling load, which otherwise is coupled for unidirectional composites. The fiber angle variation perpendicular to the loading direction provides better tailoring compared to fiber angle variation along the loading direction. Special differences in the localization of radial displacements are observed for the post-buckling analysis of VSC cylinders with and without overlap of fibers. ¹⁶⁰ Liang et al. ¹⁶¹ suggested a novel technique for pre- and post-buckling analysis for VSC using the Koiter–Newton method. It is an iterative process consisting of predictions and corrections to avoid fully nonlinear FE analysis, which is computationally expensive. Several other works have also studied buckling and dynamic stability characteristics of VSC composites.^162–164 The maximum buckling load capacity is affected by the manufacturing limitations and change in loading direction also. Rouhi et al. ¹³¹ have shown that increase of around 25% in maximum buckling load, for bending load along one axis, dramatically decreases by 57% when the bending direction is reversed.

The curvilinear stiffener profiles are shown to improve the maximum buckling load capacity of stiffened panels. These panels have significant applications as fuselage panels, ship hulls, and grid panels in large aerospace components, etc. The location, spacing, and curvilinear profiles of these stiffeners are shown to affect the maximum buckling load.^120,165 The curvilinear stiffeners in the form of NUBRS curves ¹⁶⁵ and B-spline curves ¹⁶⁶ are adopted. The stiffener geometry and angle ply arrangement of laminates are also shown to influence the buckling load capacity. In comparison to straight stiffeners, grid-stiffened panels with curved stiffeners improve maximum buckling load capacity by 35%, ¹⁶⁷ whereas weight can reduce by 30% for the same maximum buckling load capacity. ¹²⁰ The variable angle profile in both stiffener and laminate is also found to improve the buckling load capacity but has greater sensitivity to manufacturing limitations.^168,169 Structural stability, weight reduction, and damage tolerance can be enhanced by using non-uniform, curved, grid-stiffened composites (NCGCs).^19,20 Cui et al. ¹⁷⁰ discussed the optimization of the curved stiffener layout with a central hole in the structure for maximizing buckling load capacity. Wang et al. ¹⁷¹ has optimized buckling load, introducing global and local coupling effect using homogenization. Here, parametric and optimization studies are done to arrive at the optimal non-uniform stiffener configuration which has been used in post-buckling analysis as well.

Liguori et al. ¹⁷² tried to minimize the post-buckling displacement of the wing box made of VSCs to enhance the aerodynamic performance of the section. Multimodal Koiter’s asymptotic approach, which can consider the nonlinearity of the post-buckling regime, is used along with stochastic optimization strategies, that is, GA and Monte Carlo simulation. Manufacturing constraints are also considered in this study for practical implementation purposes. VSC sections show 81.31% displacement reduction in the post-buckling regime compared to quasi-isotropic sections. The asymptotic approach is further adopted by Zocco et al. ¹⁷³ to enhance computational efficiency in understanding the nonlinear modal interaction in the post-buckling regime of cylindrical VSC panels under compressive load. Initially, equilibrium equations are projected on its first thirty buckling modes. Depending on the percentage contribution of each buckling mode, it was concluded that for the orthotropic panel and the VSC panel, first 10 and 16 modes are significant, respectively. Subsequently, computational efficiency can be drastically improved by using the dominating modes only. In another study by Zocco et al., ¹⁷⁴ the importance of symmetry in post-buckling response analysis is discussed. Curie’s principle is used to exploit symmetric properties. This approach provides a computationally efficient and accurate post-buckling behavior analysis. A number of studies^175–177 have exploited the detailed analysis of post-buckling behavior of the VSC panel which considers the stress distribution property of VSC for optimized design purpose.

Vibration analysis

The vibration characteristics are also investigated by various researchers. Akhavan and Ribeiro ¹⁷⁸ studied the vibration behavior of curvilinear fiber composites using a third-order shear deformation theory. The linearly varying fiber angle distribution is adopted, and curvature constraint (curvature < 3.28 m⁻¹) is set up to consider the manufacturable fibers. It is shown that vibration mode shapes and natural frequencies can be influenced over a sufficient design space. They have also observed that vibration characteristics of thick plates are less influenced compared to thin plates in case of VSCs. It is argued that this behavior occurs possibly because of the alteration of only in-plane stiffness while translating from unidirectional to curvilinear fiber composites. Ribeiro and Akhavan ¹⁷⁹ further studied the nonlinear vibrations of VSC plates using first-order shear deformation theory and solving the governing equations using p-version finite elements. They observed that while only higher modes of vibration are mainly affected with change in fiber orientations, the oscillation amplitude for all modes under external excitations are strongly affected with fiber profiles. Vibration behavior and its optimization for laminated plates are also presented by Honda and Narita^129,180 using spline functions to define the fiber angle variation.

Yazdani and Ribeiro ¹⁸¹ implemented layer-wise formulation with constant transverse displacement for free vibration analysis of VSC plates. They have shown that symmetric layup can provide a greater natural frequency in comparison to unsymmetric composites. Though the layer-wise approach can give the realistic nature of transverse stresses in laminated composites, still, their application to VSC can generate a large number of primary variables in the FE procedure. A higher order theory in the ESL form is also implemented by Tornabene et al. ¹⁸² to study the free vibration response of variable stiffness general doubly curved sandwich shells. Zhao and Kapania ¹⁸³ studied the prestressed vibration response of a stiffened, angle tow plate under a uniform end shortening. Optimization studies show that nonlinearly varying (NLV) fibers improve buckling capacity significantly, but provide only a slight increase in the case of free or prestressed vibration fundamental frequencies. This improvement depends on boundary conditions, applied in-plane end shortening and presence of a stiffener.

Aeroelastic tailoring of angle tow composites

Aeroelastic tailoring has been one of the much-sought objectives of VSCs.^184–187 Several studies are devoted to this aspect. It is important to note that ideally, aeroelastic tailoring is a multi-objective phenomenon, often with a conflicting set of requirements. This criterion often leads to a small window in the design space for extensively varying large number of structural parameters. Murugan and Friswell ¹⁸⁸ studied the optimization of morphing wing skin. The reconfigurable nature of morphing wings requires skin material with small in-plane stiffness (flexible to deform with input actuation) and large out-of-plane stiffness (resistant of aerodynamic loads). Using a multi-objective genetic algorithm, the optimum fiber path is obtained for linearly varying fiber angles. It is observed that reasonable improvement in in-plane and out-of-plane stiffness can be gained using curvilinear fiber composites compared to unidirectional composites.

Stodieck et al. ¹⁸⁹ modeled the symmetric rectangular un-swept composite aircraft wing with linearly varying fiber angles across the length to study the aeroelastic tailoring of these composites. The eight-layered composite, with outer four layers made of variable angle fibers, is numerically tested for free vibration, flutter and divergence speed, and gust load response using the Ritz-based variational approach. The authors tried to show that though VSCs can overcome deficiencies of ply drop and other practices for aeroelastic tailoring, still their response could be both beneficial and detrimental. The optimum stacking sequence for different parameters, that is, flutter speed and peak gust load, is dependent on parameters like gust length and stead airspeed. They have concluded that VSC is not always superior to unidirectional (UD) composites. A similar set of conclusions are also made by Stanford et al. ¹⁹⁰ They have used a multi-objective genetic algorithm (NSGA-II) to optimize the flutter velocity and Tsai–Wu strength criteria. It is found that the two flutter modes show contradictory change with curved fibers, while the strength criterion is greatly benefitted due to stress redistribution in VSCs. Stanford and Jutte ¹⁹¹ further investigated the aeroelastic tailoring with curved fibers and curvilinear stiffeners. While independently, both the approaches (steered fibers and curvilinear stiffeners) improve the structural response equally, the advantage was lost with the simultaneous application. Wang et al. ¹⁹² optimized mass and local buckling load of a wing box under practical aerodynamic loading conditions. GA is used for optimization, taking into account the wing box skin thickness, geometry, and fiber path as design variables. It is observed from failure analysis that these structures are more prone to local buckling; hence, aeroelastic analysis and buckling analysis is coupled to produce a set of optimized design with different parameters. It is observed that fiber angle variation is more near the wing tip to produce greater torsional and bending rigidity. It is also observed that critical in-plane instability occurs at the mid span of the wing in general which can be minimized by utilizing the thickness variation of the CTS method. Along with that, using CTS, weight can be even further reduced as compared to constant thickness VSCs. Akhavan and Ribeiro ¹⁹³ studied aeroelastic stability and divergence behavior of VSC-laminated plates. The aeroelastic loads in supersonic flow are defined by linear piston theory. It is seen that for cantilevered VSC plates, the flutter speed can be increased by 86%, whereas for the clamped boundary condition, there is no significant change. It shall be added here that as long as the use of curved fibers does not deteriorate aeroelastic response for any function, improvements in other characteristics can still be motivating. There is always a trade-off between conflicting functionalities, and an optimum solution could be biased toward one of the functions/parameters.