Homogenisation of Double-Double (DD) laminates: Warpage mitigation and new stacking sequence approaches

Abstract

Homogenisation of Double-Double (DD) laminates is investigated herein. DD laminates are a new layup design approach composed of repeated sub-laminates or Building Blocks (BB), comprising two sets of bi-angled plies, $\pm Φ °$ and $\pm Ψ °$ . The DD laminate approach simplifies the design of composite structures, potentially reducing manufacturing complexity, and allows for rapid thickness tapering, presenting substantial prospects for reducing structural mass. Sufficient through-thickness homogenisation of the laminate eliminates undesirable out-of-plane distortions, commonly observed in asymmetric composite laminates. DD homogenisation criteria previously proposed in literature are evaluated through laminate manufacture using Carbon Fibre Reinforced Polymer (CFRP) prepreg material. Based on these evaluations, revised homogenisation criteria are proposed, which are found to effectively suppress warpage in DD laminates. Two new DD stacking sequence approaches, termed Symmetry-Enhanced Double-Double (SEDD) laminates, are introduced. These approaches are based on a foundation symmetric DD sub-laminate, ${[B B]}_{s}$ , and involve either the continuous addition of asymmetric BBs or the mirroring of the BB configuration at every alternate repeat. SEDD laminates are shown to have the capability to produce warpage-free panels using a reduced number of sub-laminate repeats, while maintaining the benefits of the traditional DD method.

Keywords

Double-Double DD laminates homogenisation laminate warpage warpage mitigation

Introduction

Quad or Legacy Quad Laminate (LQL)s, are traditional composite laminates where the orientation of plies is restricted to a set of $0 °$ , $\pm 45 °$ , and $90 °$ angles. Designing with these conventional LQLs requires numerous design rules and guidelines, summarised by Bailie et al.¹ These guidelines have introduced vast complexity to the design process, limiting innovation and presenting design challenges regarding layup optimisation. Laminate thickness is often optimised throughout a structure to minimise structural weight for the given load cases.^2,3 Symmetry must be maintained in LQLs when tapering, creating complex blending issues,^4,5 internal discontinuities⁶ and changes in laminate properties.

Double-Double (DD) laminates are a new layup design approach which seeks to address the limitations of LQLs. First proposed by Tsai et al.⁷ in 2017, DD simplifies the design of composite structures and allows for simplified and rapid thickness tapering with the potential for structural mass reduction^8,9 and increased manufacturing speeds,¹⁰ particularly with the use of Non-crimp Fabric (NCF)s¹¹ and automated tape laying, while eliminating the need for legacy design rules. DD laminates are typically made up of repeated building blocks or sub-laminates, composed of two sets of bi-angled plies, e.g. ${[Φ / - Ψ / Ψ / - Φ]}_{r}$ , where $r$ represents the number of sub-laminate repeats.

Through-thickness homogenisation of the laminate is key to the DD layup methodology.^11,12 Sufficient homogenisation of the laminate suppresses coupling behaviour typically present in asymmetric laminates, thereby preventing warpage during cool-down from cure temperature. Similar thickness normalised in-plane and out-of-plane stiffness is a characteristic of homogenised laminates, simplifying the design process by streamlining the selection of laminate stacking sequences. Homogenisation of the laminate allows ply drops to be carried out on the external surface of the laminate.

Warpage in composite laminates

The Coefficient of Thermal Expansion (CTE) of a Carbon Fibre Reinforced Polymer (CFRP) ply can vary significantly along the fibre and transverse directions, since the CTE of carbon fibres is lower than the matrix material.¹³ During cool down in a cure cycle, plies will expand or contract according to their individual CTEs and the CTE of the constituents. Since plies at different angles are bonded together in the laminate, they constrain each other, inducing internal residual stresses.^14,15

Warpage or out-of-plane displacement of an initially flat laminate, observed post-cure, is caused by out-of-plane material property gradients and stresses.¹⁶ These unbalanced stresses can be attributed to the asymmetry of the laminate or/and induced through the curing process due to tool-part interaction, pressure or the cure cycle itself.^17,18 This study will focus on the stacking sequence induced warpage of laminates, with emphasis on the ability to produce asymmetric flat laminates with the DD layup methodology.

Through-thickness laminate homogenisation

To understand through-thickness homogenisation of a composite laminate, several stiffness terms based on Classical Laminate Theory (CLT) must be defined. Terms denoted with “*” indicate thickness normalisation, as shown in equation (1).

[A^{*}] = \frac{1}{h} [A] [B^{*}] = \frac{2}{h^{2}} [B] [D^{*}] = \frac{12}{h^{3}} [D]

(1)

where

h

is the thickness of the laminate.

Tsai’s Modulus (TM) is an engineering constant that characterises the stiffness of a fibre-reinforced composite.^19–21 It is calculated using the trace of the reduced stiffness matrix [Q] or the thickness-normalised [A] matrix, as shown in equation (2). Terms with the superscript “ ^Tr ” indicate normalisation by TM, as shown in equation (3), or equivalently, their values computed using master ply values^19,22,23, which represent the average proportions of stiffness in each direction of typical CFRP materials.

Tsai ’ s Modulus (TM) = T r [Q] = Q_{11} + Q_{22} + 2 Q_{66} = T r [A] = A_{11}^{*} + A_{22}^{*} + 2 A_{66}^{*}

(2)

[Q^{T r}] = \frac{[Q]}{T M} [A^{* T r}] = \frac{[A^{*}]}{T M} [B^{* T r}] = \frac{[B^{*}]}{T M} [D^{* T r}] = \frac{[D^{*}]}{T M}

(3)

Homogenisation of the laminate is achieved by repeatedly stacking sub-laminate building blocks, shown graphically in Figure 1. Laminate homogenisation is defined by two conditions, based on the thickness-normalised matrix terms ${[A B D]}^{*}$ , as shown in equation (4).^24,25 Since a homogenised laminate suppresses coupling behaviour, it has been suggested that mid-plane symmetry is no longer necessary.²⁶

Figure 1.

Visualisation of heterogeneous versus homogeneous laminate.

[A^{*}] = [D^{*}] a n d [B^{*}] = 0

(4)

A homogenised laminate exhibits the following advantages:

• Similar thickness normalised in-plane and out-of-plane stiffness,

• No thermal warpage,

• No tension-bending, tension-shear or bend-twist coupling.

Tsai¹² proposed criteria to determine when a laminate could be deemed sufficiently homogenised. The criteria used were 2% of $[B^{* T r}]$ and $| [D^{* T r}] - [A^{* T r}] | .$ It was proposed that a DD stacking sequence of ${[+ ϕ, - ψ, - ϕ, + ψ]}_{r}$ , termed Stagger 1, was best for homogenisation. These homogenisation criteria were formulated “arbitrarily”, as stated by Vermes and Tsai in ref. 11. Miravete furthered this work by formulating a homogenisation grid, concluding that there were two optimal permutations, depending on the combination of DD angles in the building block, to achieve homogenisation.²⁵

The arbitrary 2% homogenisation criteria were incorporated into the DD design space, where the optimal permutations were identified within the feasible stiffness domain of a Lamination Parameter Plot (LPP).^27,28 This alternative method of representing homogenisation requirements through a LPP offers the advantage of a continuous region representation. However, it is less intuitive to interpret compared to the homogenisation grid approach and requires a comprehensive understanding of lamination parameters.

Tsai et al. demonstrated that warpage could be effectively eliminated in an unbalanced and asymmetric laminate, ${[0 / 25]}_{16 T}$ , through homogenisation.²⁹ Recent studies by Kappel also demonstrated that warpage can be effectively eliminated in asymmetric laminates through repeated sub-laminate stacking, thereby achieving homogenisation.^21,30 A study by Vermes et al.³¹ analytically and experimentally demonstrated that warpage in asymmetric cross-ply laminates could be effectively eliminated through homogenisation.

Summary

Literature has shown that DD laminates have various advantages over LQLs, including potential weight savings and tapering improvements. One manufacturing consideration with the production of any laminate is the potential for warpage. In LQLs, warpage is generally controlled by mid-plane symmetry. However, this cannot necessarily be accommodated in DD laminates. Therefore, other approaches have been proposed. While previous studies have shown that warpage in asymmetric DD laminates can be prevented through sub-laminate repetition, no robust, experimentally validated criteria for homogenisation of DD laminates have been established.

Therefore, this study introduces a novel framework for determining the optimal stacking sequence for homogenisation, for any combination of DD angles and defined homogenisation criteria, before considering the warpage of specimens post-manufacture and cure.

Initially, existing homogenisation criteria found in the literature^11,12,25 are used in the framework to generate a corresponding homogenisation grid. DD laminates were then manufactured and analysed to quantify the resulting panel warpage. Following this, updated homogenisation criteria were proposed to mitigate the warpage observed in the manufactured laminates. These updated homogenisation criteria were also experimentally validated through the manufacture of a range of DD laminates from prepreg materials.

The final part of this study proposes alternative DD stacking sequences, termed Symmetry-Enhanced Double-Double (SEDD), which introduce various levels of symmetry into the DD laminate, to minimise the required number of sub-laminate repeats to satisfy the homogenisation criteria.

Methods

This section introduces a Python-based implementation of CLT for determining thickness-normalised [ABD] matrix terms calculated using master ply CFRP values, essential for evaluating laminate homogenisation. Additionally, a Python-driven framework is presented for generating homogenisation grids, which convey the best DD stacking sequence to minimise laminate coupling effects. Subsequent sections detail validation steps including laminate manufacturing processes, surface scanning, and warpage quantification criteria.

Classical laminate theory Python implementation

To compute the thickness normalised [ABD] matrix terms necessary to evaluate the homogenisation of a composite laminate, CLT was implemented in a Python script. Further details on CLT are given in 7,16,32–34.

Through leveraging master ply properties, which encapsulate the average stiffness proportions in each direction of typical CFRP materials,^19,22,23 this approach offers a material-independent framework for homogenisation. The master ply values used in this study were calculated by Tsai et al.³⁵ to be $Q_{11}^{T r} = 0.8849$ , $Q_{22}^{T r} = 0.0525$ , $Q_{66}^{T r} = 0.0313$ and $ν_{12}^{T r} = 0.3$ .

Homogenisation framework

A novel framework is introduced to determine the minimum number of repeats required across the DD domain to satisfy the homogenisation criteria. The framework was implemented in Python as outlined in Figure 2. In the first stage, unique permutations of the sub-laminate were obtained by eliminating those that are effectively the same in terms of a DD sub-laminate, i.e. mirrored permutations, permutations where the $Φ$ ’s and $Ψ$ ’s are interchanged and permutations where the signs of the $Φ$ and $Ψ$ angles are inverse. These unique permutations were then evaluated at specified increments between $0 °$ and $90 °$ to determine the minimum number of sub-laminate repetitions to meet the homogenisation criteria.

Figure 2.

Workflow for determining the minimum number of repeats and the best DD sub-laminate permutation for homogenisation of the laminate.

If a half-grid representation is used, it is necessary to define the angles in the grid as DD Angle 1 and DD Angle 2, then assign the angles to $Φ$ and $Ψ$ in subsequent steps. For each combination of angles, the ${[A B D]}^{* T r}$ matrix is calculated for both configurations, corresponding to the two possible orders of angle assignment within the sub-laminate (i.e., DD Angle 1 = $Φ$ , DD Angle 2 = $Ψ$ and vice versa). The angle order that minimises both the required repeats and the highest value in the $[B^{* T r}]$ coupling matrix is selected. If both metrics are equal, no preferential order is specified. Although consideration was not given to the angle order in the initial work of Miravete,²⁵ angle orders were specified in later work with homogenisation and LPPs.²⁸ The novel consideration of angle order, to minimise the $[B^{* T r}]$ coupling matrix, was not considered in homogenisation grids developed previously.

After evaluating all permutations for a specific sub-laminate, those that require the fewest repeats to satisfy the criteria are identified. In cases where multiple permutations remain, the permutation that minimises the highest value in the $[B^{* T r}]$ coupling matrix is selected. A hypothetical DD homogenisation grid is depicted in Figure 3. Cell values at the intersection of DD Angle 1 and DD Angle 2 represent the minimum sub-laminate repetitions to meet the homogenisation criteria, using the optimal stacking sequence permutation, indicated by the cell style. Homogenisation grids can be generated with any required angle resolution. In the main text, grids with a 5° resolution are presented, with higher resolution, 2° increment grids provided in Supplementa1 Information A. The following sections will cover the use of this framework and present homogenisation grids based on various homogenisation criteria.

Figure 3.

Hypothetical Double-Double homogenisation grid.

Laminate manufacture

Laminates were manufactured from two different carbon-epoxy prepregs, either IM7 977-2 or IM7 8552. The particular DD angles selected for laminate manufacture and warpage evaluation were chosen to capture the behaviour of a range of different stiffnesses and angle combinations. To create angled plies, a template was orientated using a digital angle rule to enable accurate and reliable alignment. Two panel sizes were used in this study, 390 × 220 mm and 400 × 300 mm. During layup, plies were stacked on an aluminium build plate and the laminate underwent debulking approximately every five plies to enhance stack consolidation. Laminates manufactured using IM7 977-2 were cured at 180°C under 6 bar pressure for 3 h. A two-stage cure cycle was used for IM7 8552 laminates. Under 7 bar pressure, the temperature was held at 110°C for 1 h, then increased to 180°C for a further 2 h. Ramp and cool-down rates were set to 3°C/min for both cure cycles. Post-cure, excess resin bleed was removed from all sides of the panel.

Surface scanning & point cloud data processing

To establish the amount of warpage post cure, panels were photographed and subsequently surface scanned using a Hexagon Romer Absolute Arm 7325SI with RS6 laser scanner, to generate a point cloud. The raw point cloud data was processed using a custom MATLAB script to enable comparison between laminates. Processing involved data downsampling, removal of spurious points through density-based clustering and automated translation and rotation operations to correct for skewness and uneven levelling during scanning. A surface of the form $z = p 00 + p 10 . x + p 01 . y + p 20 . x^{2} + p 11 . x y + p 02 . y^{2}$ was mapped to the processed point cloud data and the maximum deflection of the panel was reported. The maximum deflection is defined as the vertical distance along the z-axis between the lowest point on the panel, constrained to the z = 0 plane, and the highest point. The z-axis was scaled by 5x, compared to the x and y-axis, for all point cloud plots to aid visualisation of the distortion. Further details of the point cloud data processing including a flow diagram of the steps are given in Supplemental Information B.

Flatness quantification

A dedicated standard does not exist to define acceptable limits for the flatness of a CFRP composite panel. Previous publications have referred to general tolerances in ISO 2768 to classify a composite panel as sufficiently flat.¹⁴ Considering the dimensions of the panels in this study and the influence of ISO 2768, laminate warpage has been classified into four distinct categories, tabulated in Table 1.

Table 1.

Classification of laminate warpage within the scope of this study.

Classification	Maximum deflection, X (mm)
Negligible warpage	X $<$ 0.6
Minor warpage	0.6 $\leq$ X $<$ 1.2
Moderate warpage	1.2 $\leq$ X $<$ 2.4
Severe warpage	X $>$ 2.4

As previously mentioned, warpage can occur even in laminates with a symmetric layup. Therefore, to assess the warpage of asymmetric DD laminates, it is necessary to establish reference laminates to quantify the level of process-induced warpage. For this study, two symmetric quad laminates were manufactured from IM7 977-2, ${[45_{2} / 0_{4} / - 45_{2} / 90_{2}]}_{s}$ and ${[0 / 45 / 0 / 90 / 0 / - 45 / 0 / 45 / 0 / - 45]}_{s}$ . Panel dimensions corresponded to that of the DD laminates shown in subsequent sections, 390 × 220 mm and 400 × 300 mm, respectively. Both baseline laminates exhibited negligible warpage, confirming that the processed-induced distortion was insignificant. Supplemental Information C contains images and the corresponding point cloud data of these panels.

Homogenisation based on original criteria

As previously mentioned, homogenisation criteria have been proposed in literature²⁵ and are summarised in equation (5). The following sections detail the homogenisation grid generation for the standard DD sub-laminate $[\pm Φ °, \pm Ψ °]$ based on this established homogenisation criteria. A selection of laminates based on this homogenisation grid were manufactured and the results are presented.

| [D^{* Tr}] - [A^{* Tr}] | < 2 % an d | [B^{* T r}] | < 2 %

(5)

Homogenisation grid - original criteria

For the standard 4-ply DD sub-laminate, there are six unique permutations, confirmed in ref. 25 and shown for reference in Supplemental Information D - Table 1. Only permutations 5- $[Φ / - Ψ / - Φ / Ψ]$ and 6- $[Φ / - Ψ / Ψ / - Φ]$ are needed to maintain the minimum number of repeats across the domain. Using the aforementioned framework, with the original criteria, a homogenisation grid was generated and is shown in Figure 4. The permutations recommended here align with those proposed by Miravete.²⁵ However, this approach represents an advancement due to the specified angle orders minimising both the required repeats and the highest value in the $[B^{* T r}]$ coupling matrix.

Figure 4.

Grid showing the required repeats for homogenisation with a 4-ply sub-laminate and the recommended permutation for each region based on the original homogenisation criteria in 25.

Manufacture & point cloud data - original criteria

Multiple test laminates were manufactured to assess the previously proposed criteria. Details of the manufactured laminates are given in Table 2. Although the laminates

{[+ 61.5 / - 9 / - 61.5 / + 9]}_{5}

{[+ 47 / - 9 / - 47 / + 9]}_{5}

and

{[+ 22.5 / - 67.5 / + 67.5 / - 22.5]}_{5}

do not use the recommended stacking sequences according to the homogenisation grid, these laminates meet the original homogenisation criteria set out in equation (5) and are valid in assessing their efficacy. The

| {[A B D]}^{* T r} |

matrix terms for these laminates are shown in Supplemental Information E − Table 1. Images of the manufactured laminates are shown in Figure 5 with corresponding processed surface scans shown in Figure 6.

Table 2.

Laminates manufactured to assess original homogenisation criteria.

Permutation	Stacking sequence	Material	Warpage
5	${[+ 22.5 / - 67.5 / - 22.5 / + 67.5]}_{4}$	IM7 977-2	Severe
5	${[+ 61.5 / - 9 / - 61.5 / + 9]}_{5}$	IM7 977-2	Severe
5	${[+ 47 / - 9 / - 47 / + 9]}_{5}$	IM7 977-2	Severe
6-2	${[+ 22.5 / - 67.5 / + 67.5 / - 22.5]}_{5}$	IM7 977-2	Severe

Figure 5.

Laminates manufactured based on original homogenisation criteria.

Figure 6.

Processed point cloud data and fitted surface for manufactured laminates based on the original homogenisation criteria.

The results show that all laminates suffered severe warpage, with the worst case exceeding the permissible deflection for minor warpage classification by more than 4 times. These results suggest that the original arbitrary DD homogenisation criteria are insufficient to prevent laminate warpage.

Homogenisation based on updated criteria

The previously proposed homogenisation criteria were found to be inadequate in preventing laminate warpage, necessitating the development of more stringent constraints. As it is not possible to completely eliminate all $[B^{* T r}]$ coupling matrix terms with the standard DD approach, it is necessary to establish the upper limit for which warpage is sufficiently suppressed. These updated criteria impose tighter constraints on the $[B^{* T r}]$ coupling terms, as detailed in equation (6). This modification was aimed at evaluating whether halving the $[B^{* T r}]$ criterion would effectively reduce laminate warpage to within acceptable limits. The $[D^{* T r}] - [A^{* T r}] < 2 %$ criterion imposes constraints on differences between in-plane and out-of-plane stiffness properties, intending to simplify structural design, but does not directly affect laminate warpage. Since the primary focus of this study was on mitigating warpage in DD laminates, this criterion was retained without modification.

| [D^{* T r}] - [A^{* T r}] | < 2 % a n d | [B^{* T r}] | < 1 %

(6)

Homogenisation grid based on updated homogenisation criteria

Figure 7 shows the resulting homogenisation grid for the standard 4-ply DD sub-laminate, based on the updated criteria. As with the grid based on the original criteria, only two unique DD sub-laminate stacking sequences are required to achieve the minimum number of sub-laminate repeats, [ $Φ / - Ψ / - Φ / Ψ]$ and $[Φ / - Ψ / Ψ / - Φ]$ . The tighter constraints imposed by the updated homogenisation criteria result in an increase in the number of sub-laminate repeats required for some DD angle combinations. The largest impacts are observed in the central region of the homogenisation grid, where $[B^{* T r}]$ terms are the critical constraint in meeting the criteria. For example, the quasi-isotropic DD sub-laminate case, with angles $\pm 22.5$ and $\pm 67.5$ , requires eight repeats, up from four, with the optimal stacking sequence remaining as permutation 5- $[Φ / - Ψ / - Φ / Ψ]$ .

Figure 7.

Grid showing the required repeats for homogenisation with a 4-ply sub-laminate and the recommended permutation for each region based on the updated homogenisation criteria.

Manufacture & point cloud data - updated criteria

As before, laminates were manufactured with various angles and prepreg materials to test and validate the updated criteria. Details of these laminates are given in Table 3. The

| {[A B D]}^{* T r} |

matrix terms for these laminates, demonstrating the homogenisation criteria are met, are given in Supplemental Information E − Table 2. The resulting manufactured laminates are shown in Figure 8 with corresponding processed surface scans for three of the panels shown in Figure 9.

Table 3.

Laminates manufactured to assess the updated homogenisation criteria.

Permutation	Stacking sequence	Material	Warpage
5	${[+ 22.5 / - 67.5 / - 22.5 / + 67.5]}_{8}$	IM7 977-2	Negligible
6-2	${[+ 17.0 / - 64.5 / + 64.5 / - 17.0]}_{8}$	IM7 977-2	Negligible
6-2	${[+ 9 / - 61.5 / + 61.5 / - 9]}_{4}$	IM7 8552	Moderate
6-2	${[+ 9 / - 61.5 / + 61.5 / - 9]}_{5}$	IM7 977-2	Minor
6-2	${[+ 9 / - 47 / + 47 / - 9]}_{5}$	IM7 977-2	Moderate

Figure 8.

Laminates manufactured based on updated homogenisation criteria.

Figure 9.

Processed point cloud data and fitted surface for manufactured laminates based on updated homogenisation criteria.

Table 3 shows the updated homogenisation criteria effectively eliminated severe warpage. The warpage classification for laminates ${[+ 22.5 / - 67.5 / - 22.5 / + 67.5]}_{8}$ and ${[+ 17.0 / - 64.5 / + 64.5 / - 17.0]}_{8}$ was given based on visual inspection. These panels were the flattest among those manufactured, as evident from the images, and were therefore considered to have negligible warpage. The two laminates with the largest maximum deflections, ${[+ 9 / - 61.5 / + 61.5 / - 9]}_{4}$ and ${[+ 9 / - 47 / + 47 / - 9]}_{5}$ , were classified as moderately warped, missing the classification of minor by less than 0.25 mm in both cases. Although possible to constrain the $[B^{* T r}]$ criteria even further, for example to <0.8%, the updated criteria of <1% is deemed sufficiently stringent to achieve flat laminates for most DD angle combinations.

New stacking sequence approaches for DD laminates

Despite the improvements achieved with the updated homogenisation criteria, potential issues arise for certain DD angle combinations. In cases such as the optimal quasi-isotropic DD sub-laminate $[\pm 22.5 / \pm 67.5]$ , a minimum of eight repeats were required to prevent laminate warpage. For many lightweight applications requiring relatively thin laminates, the use of thin plies would be necessary to employ the DD methodology, which is not without drawbacks.^36,37

To address this limitation, two new stacking sequence methods are proposed. While it is well known that fully symmetric laminates are ideal for preventing out-of-plane distortions, due to $[B] = 0$ , enforcing full symmetry in DD laminates would prohibit external tapering of the laminate, a key advantage of the DD approach. These new approaches aim to reduce the required sub-laminate repeats needed to meet the updated homogenisation criteria, while maintaining the advantages of the standard DD method. These methods introduce various levels of symmetry into the DD laminate and therefore have been termed Symmetry-Enhanced Double-Double (SEDD) laminates. To represent these new stacking sequences, it is efficient to define an individual sub-laminate as a Building Block (BB).

Symmetry-enhanced Double-Double - 1 (SEDD-1)

SEDD-1 introduces one symmetric BB to the laminate stack, creating a base laminate of ${[B B]}_{s}$ . Following the base laminate, additional sub-laminates or BBs are stacked asymmetrically on top, as with the original DD stacking sequence method. Therefore, SEDD-1 can be represented as ${[{(B B)}_{s} / {(B B)}_{r}]}_{T}$ . Two examples of SEDD-1 laminates are shown graphically in Figure 10.

Figure 10.

Visualisation of SEDD-1 Laminate stacking sequences.

For the base DD laminate, ${[B B]}_{s}$ , all $[B^{* T r}]$ terms will be zero due to symmetry of the laminate. As asymmetric layers are introduced, $[B^{* T r}]$ coupling terms will become present in the [ABD] laminate matrix. These $[B^{* T r}]$ terms reach their maximum value after the introduction of two additional BBs, ${[{(B B)}_{s} / {(B B)}_{2}]}_{T}$ , then decrease as subsequent BBs are added. This can be visualised in Supplemental Information F - Figures 1 and 3. Therefore, the optimal DD stacking sequence permutation can be determined for the laminate ${[{(B B)}_{s} / {(B B)}_{2}]}_{T}$ based on the updated homogenisation criteria, equation (6). ${[{(B B)}_{s} / {(B B)}_{2}]}_{T}$ meets the requirements of the new homogenisation criteria, for any angle combination. An additional computation can be performed to determine the minimum number of repeats for the SEDD-1 stacking sequence, ${[{(B B)}_{s} / {(B B)}_{r}]}_{T}$ . The minimum number of BBs is constrained to two, to maintain the ${[B B]}_{s}$ foundation. The homogenisation grid is shown in Figure 11.

Figure 11.

Grid showing the required repeats for homogenisation with a 4-ply sub-laminate and the recommended permutation for each region based on the updated homogenisation criteria with SEDD-1 ${[{(B B)}_{s} / {(B B)}_{r}]}_{T}$ .

The critical criterion driving the required number of repeats for ${[{(B B)}_{s} / {(B B)}_{r}]}_{T}$ is $[D^{* T r}] - [A^{* T r}]$ . This criterion constrains the differences between in-plane and out-of-plane stiffness properties, intending to simplify structural design, but does not affect laminate warpage. This difference is generally not a design consideration in LQLs, as minimising $[D^{* T r}] - [A^{* T r}]$ is difficult using this approach. If the primary focus of homogenisation is to prevent laminate warpage, omitting the $[D^{* T r}] - [A^{* T r}]$ criterion from the updated criteria, leaving only the requirement to meet $[B^{* T r}] < 1 %$ , can be useful to understand the minimum requirements to mitigate warpage. This results in the minimum number of BBs required being reduced to two, in the form ${[{(B B)}_{s}]}_{T}$ . Although additional repeats (r = 1,2…) will introduce $[B^{* T r}]$ coupling terms, the $[B^{* T r}] < 1 %$ criterion will not be breached for any DD angle combination or number of repeats.

Manufacture & point cloud data - SEDD-1

To assess the viability of SEDD-1, laminates were manufactured as shown in Table 4. The

| {[A B D]}^{* T r} |

matrix terms for these laminates are given in Supplemental Information E − Table 3. The resulting manufactured laminates are shown in Figure 12 with corresponding processed surface scans shown in Figure 13.

Table 4.

Laminates manufactured to assess SEDD-1.

Permutation	Stacking sequence	Material	Warpage
5	[(+22.5/−67.5/−22.5/+67.5)_s/(+22.5/−67.5/−22.5/+67.5)₂]_T	IM7 977-2	Minor
5	[(+22.5/−67.5/−22.5/+67.5)_s/(+22.5/−67.5/−22.5/+67.5)₂]_T	IM7 8552	Negligible
6-2	[(0/−53.5/+53.5/0)_s/(0/−53.5/+53.5/0)₁]_T	IM7 8552	Moderate

Figure 12.

Laminates manufactured with SEDD-1 and based on the updated homogenisation criteria.

Figure 13.

Processed point cloud data and fitted surface for manufactured laminates SEDD-1 based on the updated homogenisation criteria.

Both quasi-isotropic DD laminates, manufactured with IM7 977-2 and IM7 8552, exhibited negligible or minor warpage. The ${[{(B B)}_{s} / {(B B)}_{2}]}_{T}$ layup used for these laminates results in a maximum $[B^{* T r}]$ value of 0.92%, and hence is ideal for evaluating the efficacy of the updated criteria. The laminate ${[{(0 / - 53.5 / + 53.5 / 0)}_{s} / {(0 / - 53.5 / + 53.5 / 0)}_{1}]}_{T}$ was manufactured with three BBs, one less than that needed to meet the updated criteria fully. This laminate was classified as moderately warped, exceeding the classification of minor by 0.43 mm.

Symmetry-Enhanced Double-Double - 2 (SEDD-2)

Incorporates a symmetric BB every other repetition. This stacking sequence can be represented as ${[{(B B)}_{s . r} / {(B B)}_{n}]}_{T}$ . Where $r \geq 1$ and n = 0 or n = 1. Two examples of SEDD-2 laminates are shown graphically in Figure 14. For an even number of BBs, the laminate will be symmetric, eliminating all $[B^{* T r}]$ coupling terms. Laminates with an odd number of BBs will have one asymmetric BB introducing $[B^{* T r}]$ coupling terms. When r = 1 and n = 1, this will lead to the laminate with the largest $[B^{* T r}]$ terms. This is a special case of SEDD-1, $[{(B B)}_{s} / {(B B)}_{1}]$ . As mentioned previously, this stacking sequence meets the criterion $[B^{* T r}] < 1 %$ for any angle combination, using the optimal permutation for that combination. Any subsequent asymmetric laminates with a larger number of BBs will have decreasing magnitudes of $[B^{* T r}]$ coupling terms.

Figure 14.

Visualisation of SEDD-2 Laminate stacking sequences.

The homogenisation grid for SEDD-2 based on the updated homogenisation criteria is shown in Figure 15. The optimal stacking sequence permutation and angle order for each angle combination were determined based on the $[B^{* T r}] < 1 %$ criterion for a laminate consisting of three BBs. Subsequently, the minimum number of BBs were computed based on the updated homogenisation criteria. The minimum number of BBs is constrained to two, to maintain the ${[B B]}_{s}$ foundation. In some instances, five BBs are required, in contrast to the maximum of four building blocks with SEDD-1. This is caused by the slower decrease in $| [D^{* T r}] - [A^{* T r}] |$ with this stacking sequence approach, demonstrated in Supplemental Information F - Figures 2 and 4. For some angle combinations with a minimum of three repeats, when an additional BB is added, leading to ${[B B]}_{s 2}$ , the $| [D^{* T r}] - [A^{* T r}] |$ criterion is breached. The criterion is met again when further BBs are added. As aforementioned, this criterion is not needed to prevent laminate warpage.

Figure 15.

Manufacture & point cloud data - SEDD-2

Details of the manufactured laminates based on SEDD-2 are shown in Table 5, with images shown in Figure 16. The

| {[A B D]}^{* T r} |

matrix terms for these laminates are given in Supplemental Information E − Table 4. As previously mentioned, when r = 1 and n = 1, this leads to a stacking sequence of

[{(B B)}_{s} / {(B B)}_{1}]

, a special case of SEDD-1. Therefore, the laminate

{[{(0 / - 53.5 / + 53.5 / 0)}_{s} / {(0 / - 53.5 / + 53.5 / 0)}_{1}]}_{T}

in Figure 12 can also be used to assess this layup approach.

Table 5.

Laminates manufactured to assess SEDD-2.

Permutation	Stacking sequence	Material	Warpage
5	[(+ 31.5/ − 58.5/−31.5/+58.5)_s2/( + 31.5/−58.5/−31.5/+58.5)₁]_T	IM7 977-2	Negligible
5	Angles $(+ 22.5 / - 67.5 / - 22.5 / + 67.5)$ , tapered with layers ${[B B]}_{S 2}$ , $[{(B B)}_{S} / {(B B)}_{1}], {[B B]}_{S}$	IM7 8552	Negligible

Figure 16.

Laminates manufactured with SEDD-2 and based on the updated homogenisation criteria.

Figure 16(a) shows the laminate ${[{(+ 31.5 / - 58.5 / - 31.5 / + 58.5)}_{s 2} / {(+ 31.5 / - 58.5 / - 31.5 / + 58.5)}_{1}]}_{T}$ with corresponding point cloud data shown in Figure 17. The maximum warpage for this laminate was the lowest of all manufactured DD laminates, and also in comparison to the reference quad laminate of the same panel dimensions. Figure 16(b) uses SEDD-2 in a tapered laminate, reducing the thickness from a four BB laminate (16 plies) to a two BB laminate (8 plies). Point cloud data was obtained, but no surface equation was mapped due to the thickness changes of the panel. No visible warpage was present and hence was deemed negligible.

Figure 17.

Processed point cloud data and fitted surface for ${[{(+ 31.5 / - 58.5 / - 31.5 / 58.5)}_{s 2} / {(+ 31.5 / - 58.5 / - 31.5 / 58.5)}_{1}]}_{T}$ with IM7 977-2 based on the updated homogenisation criteria.

Symmetry-enhanced Double-Double summary

The introduction of SEDD laminates provides a practical solution to the high minimum sub-laminate repetition limitation of the traditional DD stacking sequences. SEDD-1 and SEDD-2 offer varying degrees of symmetry, enabling reduced sub-laminate repeats, while preserving external tapering capabilities. Among these, the SEDD-2 stacking sequence is recommended due to its ability to maintain lower levels of $[B^{* T r}]$ coupling when multiple BBs are added to the symmetric foundation. In scenarios where minimising the difference between in-plane and out-of-plane stiffness properties is desired, SEDD-1 can be more optimal.

In terms of manufacturing, no additional complexity was added to the manual prepreg layup procedure, when moving from the traditional DD to SEDD approach. Similarly, from an industrial manufacturing standpoint, using either automated tape laying or NCFs, the authors do not envision the SEDD approach presenting any added manufacturing challenges.

Conclusions

A novel framework for the through-thickness homogenisation of Double-Double (DD) laminates, addressing the challenges of warpage in these asymmetric laminates, has been introduced. Homogenisation grids serve as an effective design tool, presenting the optimal DD sub-laminate permutation and the required number of sub-laminate repetitions to achieve sufficient homogenisation for any given DD angle combination.

DD homogenisation criteria previously documented in the literature proved inadequate to prevent laminate warpage for the cases studied herein, with all manufactured laminates classified as severely warped (maximum deflection $>$ 2.4 mm). This inadequacy led to the formulation of updated criteria, which impose stricter constraints on the $[B^{* T r}]$ coupling terms to $<$ 1%. Laminates manufactured in accordance with these updated criteria showed significant improvements in flatness, with the majority having their warpage classified as negligible or minor (maximum deflection $<$ 0.6 mm & $<$ 1.2 mm, respectively). Of the two laminates that were classified as moderately warped, the maximum deflection exceeded the classification of minor by less than 0.25 mm, representing an order of magnitude improvement over the previous criteria. While further constraining the $[B^{* T r}]$ criterion to, for instance, $<$ 0.8% is feasible, the proposed updated criteria were deemed sufficiently stringent to achieve flat laminates for most DD angle combinations. Ultimately, the criteria required will be case-dependent, determined by the specific level of tolerance required for the application.

Two new DD stacking sequence approaches, of the form ${[{(B B)}_{s} / {(B B)}_{r}]}_{T}$ and ${[{(B B)}_{s . r} / {(B B)}_{n}]}_{T}$ , are also presented. These new stacking sequences, termed Symmetry-Enhanced Double-Double (SEDD)-1 and SEDD-2, respectively, introduce various levels of symmetry into the laminate. SEDD laminates reduce the required sub-laminate repeats needed to meet the homogenisation criteria. The manufactured laminates demonstrated negligible warpage using a reduced number of sub-laminate repeats. These layups retain the ability to easily and rapidly taper the laminate, as with the traditional DD layup approach, but have the advantage of meeting the criterion $[B^{* T r}]$ $<$ 1% for any angle combination with $\geq$ 8 plies, as demonstrated in this study.

These advancements in homogenisation provide a robust foundation for the design and manufacture of warpage-free DD laminates, paving the way for more efficient and lightweight composite structures. Future work could focus on a larger-scale validation program including the manufacture of laminates with a broader spectrum of DD angles, as well as expanding the study to encompass a wider array of materials, including thermoplastics.

Supplemental Material

Supplemental Material - Homogenisation of Double-Double (DD) laminates: Warpage mitigation and new stacking sequence approaches

Supplemental Material for Homogenisation of Double-Double (DD) laminates: Warpage mitigation and new stacking sequence approaches by Aidan Hawkins, Scott L. J. Millen and Ali Aravand in Journal of Composite Materials

Footnotes

Acknowledgements

This study was conducted in collaboration with Spirit AeroSystems Belfast. Surface scanning capabilities were provided by the Northern Ireland Advanced Manufacturing Innovation Centre (AMIC). The assistance of Jean-Aubin Thiebot is gratefully acknowledged. Significant knowledge transfer from Professor Stephen W. Tsai and the team at Stanford University is also gratefully acknowledged.

ORCID iDs

Aidan Hawkins

Scott L. J. Millen

Ali Aravand

Author contributions

Aidan Hawkins: Writing - Original Draft, Conceptualisation, Methodology, Software, Investigation, Validation, Formal analysis, Data curation, Visualisation. Scott L.J. Millen: Writing - Review & Editing, Conceptualisation, Methodology, Resources, Visualisation. M. Ali Aravand: Writing- Review & Editing, Conceptualisation, Methodology, Resources, Visualisation, Funding acquisition.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by an EPSRC Standard Research Studentship (2774253), co-funded by Spirit AeroSystems Belfast as the industrial partner through a collaborative studentship at Queen’s University Belfast.

Declaration of conflicting interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability Statement

Data not available within the main manuscript or supplementary information can be requested from the corresponding author.*

Supplemental Material

Supplemental material for this article is available online.

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