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Abstract

Designing superior braided thermoplastic composites that account for uncertainties in internal structural parameters (SPs) and material properties (MPs) can enhance the mechanical properties of integrated structural components. This study proposes a novel approach to design high-performance Braided Carbon Fiber Reinforced Poly Ether Ether Ketone (BCF/PEEK) using computational analysis to assess variability in SPs and MPs, and determine the sensitivity of fiber and Poly Ether Ether Ketone (PEEK). Initially, a large dataset of parameter samples for sensitivity analysis is acquired using a command stream-driven finite element (FE) method, employing the scale-span mechanical property characterization model of BCF/PEEK. Subsequently, Grey Relation Analysis (GRA) of SPs for BCF/PEEK is performed to establish an optimal prediction model. Then, a global sensitivity analysis model for MPs of the carbon fiber and PEEK is developed and solved using MATLAB software, along with input and output sample sets. An innovative approach is proposed, employing variable parameter amplitudes to validate the sensitivity coefficients of MPs. Various BCF/PEEK configurations that can withstand different types of loads are designed using the developed sensitivity analysis model. Finally, the strong correlation of MPs in BCF/PEEK is confirmed using the scale-span FE characterization model, validating the accuracy and efficiency of the proposed design approach.

Keywords

BCF/PEEK scale-span FE high-performance design material properties sensitivity analysis

Introduction

Poly Ether Ether Ketone (PEEK), a thermoplastic resin, is renowned for its exceptional performance across diverse metrics, encompassing high-temperature, corrosion, wear, and radiation resistance.^1,2 Meanwhile, Braided Carbon Fiber Reinforced Poly Ether Ether Ketone (BCF/PEEK) is increasingly preferred in aviation applications over conventional metallic materials, such as aircraft rotor hub cores, turbine blades, and engine inlet noise reduction liners. This shift is attributed to its excellent overall mechanical properties, high strength-to-weight ratio, corrosion resistance, enhanced design flexibility, ease of transportation, and unique capability to sustain high-temperature loads.^3,4 Research on the mechanical properties of BCF/PEEK, an inhomogeneous material, has garnered considerable scholarly attention in recent years.^5,6 The macroscopic mechanical properties of BCF/PEEK demonstrate comprehensive performance arising from the integration of multiple carbon fiber bundles within a high-performance PEEK matrix.^7,8 Variations in structural parameters (SPs) and material properties (MPs) across different types of fibers and PEEK directly influence the mechanical properties of BCF/PEEK. Importantly, these mechanical properties serve as critical indicators influencing the entire lifecycle of BCF/PEEK, encompassing optimization of machining processes,⁹ determination of braided structure parameter designs,^10,11 and assessment of service life in operational environments.¹²

BCF/PEEK is a complex composite material characterized by numerous uncertain SPs, which must be prioritized during its design to achieve an optimal structure. Equally critical are the MPs of the carbon fiber and PEEK, which are essential for the composition design of BCF/PEEK. The most convenient and straightforward approach involves analyzing the impact of each parameter, a method known as sensitivity analysis. Concurrently, investigating the sensitivity of SPs and MPs of the carbon fiber and PEEK is crucial in the structural design of BCF/PEEK.¹³ Based on sensitivity analysis results, referred to as the sensitivity coefficients, internal SPs and MPs suitable for varying operational conditions are chosen. Parameters showing high sensitivity and strong correlation are given priority, while those with low sensitivity and weak correlation are either disregarded or addressed using traditional empirical methods. Targeted improvements in the material performance of BCF/PEEK are crucial to ensure that its mechanical properties meet the service requirements under complex operating conditions during the material design process.

The sensitivity analysis method has been applied in engineering to assess mechanical properties and structural parameters of the conventional composite, such as Carbon Fiber Reinforced Plastics (CFRP).^14–16 For instance, regarding SPs, Huang et al.¹⁷ investigated sensitivity metrics for three-dimensional woven composites, analyzing how uncertainties in weave structure and angle influence vibrational frequencies of internal structures. Sakata et al.¹⁸ used local sensitivity analysis, discussed multi-scale stochastic stress analysis at the micro-scale for unidirectional CFRP, considering random fiber position variations. Tao et al.¹⁹ employed a direct derivative-based method to compute sensitivity coefficients for optimizing structural designs of unidirectional CFRP components. Zhang et al.²⁰ applied an efficient stochastic sensitivity analysis strategy based on Sobol’s sequence algorithm to quantify contributions of different sensitivity parameters to load sharing, stiffness, and peak loads. Andrés et al.²¹ conducted the analysis of variance to explore contributions of warp and weft directions in composite materials to preparation, molding, and mechanical properties. Zhu et al²² developed a high-precision model and conducted global sensitivity analysis via high-dimensional model representation to examine the micro-level parameters affect elastic responses of unidirectional CFRP.

Within the domain of MPs research, Liu et al²³ conducted a comprehensive investigation utilizing weighted indicators derived from mechanical properties and process parameters to explore their effects on thrust forces during CFRP drilling operations. This methodology facilitates rapid prediction of thrust force in drill bits. Kushari et al²⁴ proposed an innovative matrix-based stochastic sensitivity analysis method independent of external factors, specifically tailored for laminated composite plate. This approach primarily addresses uncertainties in MPs and geometric parameters by using them as input variables, utilizing them as input variables, thereby establishing a unified metric to measure the impact of multi-objective parameters. Ilyani et al²⁵ employed global sensitivity analysis techniques, encompassing regression-based and variance-based methodologies, to assess the sensitivity of four intermediate-scale uncertain input parameters on macro-scale responses. Additionally, they employed the Pearson coefficient to evaluate the precision of influence coefficients associated with each uncertain input variable in structural response assessments. Prior research underscores the extensive scholarly investigations into uncertainties pertaining to composite SPs and MPs. These investigations have prioritized key influencing factors crucial for material design, thereby fostering advancements in high-performance composite development. Nevertheless, these studies have not comprehensively evaluated the collective impact of uncertainties in SPs and MPs on the corresponding mechanical properties. Furthermore, the quantification of the influence of each factor has been insufficient, rendering each weight coefficient unverifiable. Moreover, conducting sensitivity analyses necessitates a substantial sample size, typically obtained through labor-intensive experimental procedures. Lastly, the lack of effective methods for sensitivity verification has frequently resulted in significant deviations between the designed and actual mechanical properties of composites.

Based on the aforementioned researches, this study proposes a novel design framework for BCF/PEEK to address diverse loading conditions, leveraging a global sensitivity analysis model to characterize its uncertain structural features. Firstly, a comprehensive scale-span mechanical property characterization model will be developed for BCF/PEEK to accurately characterize its structural uncertainties. Subsequently, a large dataset of parameter samples necessary for sensitivity analysis will be generated using a command stream-driven FE method. Next, Grey Relation Analysis (GRA) will be employed to determine optimal SPs for BCF/PEEK. Lastly, sensitivity analysis of MPs will identify the most influential factors of the carbon fiber and PEEK, thereby enhancing the systematic design of BCF/PEEK.

Characterization and Validation of the Mechanical Properties of BCF/PEEK Based on the Scale-Span Method

Scale-Span Characterization Modeling and Analysis of BCF/PEEK

The macroscopic mechanical properties of BCF/PEEK are fundamentally dictated by its intricate microscopic architecture. At the microscopic level, the mechanical characteristics of BCF/PEEK are intricately governed by the disposition of individual filaments or bundles thereof. During molding, BCF/PEEK undergoes high temperatures and pressures, which can cause uncertainties in its structural dimensions. The prepreg material employed is T800S-6 K/PEEK, incorporating 6K Toray T800S carbon fiber monofilament sourced from Toray Industries (Japan) and high-performance PEEK resin.²⁶ In such cases, an idealized structure fails to accurately represent the true cross-section of BCF/PEEK. Therefore, the establishment of a microstructure Representative Volume Element (RVE) model, capable of faithfully and effectively mirroring the morphological intricacies of BCF/PEEK, emerges as a paramount prerequisite for optimizing its material performance.

The establishment of a scale-span RVE model for characterizing the mechanical attributes of BCF/PEEK encompasses two pivotal stages: (i) Preparing BCF/PEEK specimens, constructing a micro-scale Unidirectional Carbon Fiber Reinforced Poly Ether Ether Ketone (UCF/PEEK) RVE model based on observed fiber bundle structures in BCF/PEEK, and delineating its mechanical attributes, illustrated in Figure 1(a) and (c). (ii) Formulating a meso-scale RVE model founded upon optimal braided architectures, depicted in Figure 1(b)–(d), amalgamating mechanical property characterization findings from UCF/PEEK and subsequently quadratic characterizing the holistic mechanical traits of BCF/PEEK. Finally, corresponding experimental specimens are prepared to validate the characterization of mechanical properties, depicted in Figure 1(e)–(g).

Figure 1.

Scale-span RVE model and experimental specimens.

Various mechanical properties of BCF/PEEK are characterized via a scale-span finite element (FE) method that integrates micro-scale mechanical theories with FE analysis according to author’s prior work.²⁷ The proposed method considers the diverse geometric structural parameters of BCF/PEEK at the microscale. Initially, a plugin is developed to simulate the random distribution of fibers within PEEK, incorporating dimensions and positions based on periodic boundary conditions. This leads to a detailed representative volume element (RVE) model with randomly distributed diameters and positions for UCF/PEEK. Then, a Python script is employed to efficiently characterize the macroscopic mechanical properties of UCF/PEEK, considering structural heterogeneity through homogenization theory.

The mechanical properties of unidirectional fiber bundles derived from a micro-scale homogenization model are characterized for BCF/PEEK adopting a quadratic homogenization method. Initially, three-dimensional structural information is extracted from experimental specimens, and mathematical expressions accurately matching the basic shape of the actual yarn cross-section and path are determined using a trust-region algorithm.²⁸ This process facilitates the construction of a micro-scale RVE model that includes random structural dimensions of fiber bundles and characterizes their mechanical properties accordingly. Subsequently, predictions are made regarding the effects of variations in fiber fluctuation amplitude, fiber volume fraction, and changes in the angle between fiber bundles on the mechanical properties of BCF/PEEK, based on the established RVE model of UCF/PEEK. Finally, standard tensile and shear experiments are conducted to validate the accuracy of the established scale-span characterization model, as depicted in Figure 1(f) and (g).

The laboratory-prepared BCF/PEEK, depicted in Figure 1(e), serves as the focal point of investigation. The structure of BCF/PEEK is assumed to remain intact, characterized by impeccable fiber-matrix bonding. Voids and initial micro-cracks formed during preparation are neglected. The fibers display transverse isotropy, whereas the PEEK matrix exhibits isotropy. The elastic modulus and Poisson’s ratio of PEEK are 3.8 GPa and 0.4, correspondingly. The prepreg has a mass per unit area of 200 g/m². Subsequent to consolidation, the final thickness measures 0.22 mm, the fiber volume fraction of the UCF/PEEK is approximately 85%, and the fiber volume fraction of BCF/PEEK is approximately 59% post-molding,^1,26 determined via combustion analysis using a muffle furnace. The mechanical property parameters of PEEK and T800S/6K carbon fiber monofilament are delineated in Tabel 1.

Table 1.

Mechanical property parameters of PEEK and T800S/6K carbon fiber monofilament.

Elastic parameter (GPa)	Value	Poisson’s ratio and coefficient of thermal expansion	Value
E _p	3.8	v _p	0.4
G _p	1.4	α_p, (1/°C)	4.5*10⁻⁵
E _f1	294	v _f12 /v _f13	0.2
E_f2/E_f3	15	v _f23	0.25
G _f12 /G _f13	24	α_f11, (1/°C)	5.3*10⁻⁶
G _f23	5.03	α_f22/α_f33, (1/°C)	−1.01*10⁻⁵

Based on the established UCF/PEEK and BCF/PEEK RVE models, normal and shear stresses of 1 MPa are applied in three directions using ABAQUS/Standard. The C3D8R element type is selected to ensure accurate calculations under deformation. Subsequently, stress contours under each loading condition are acquired, with two examples of characterization outcomes illustrated in Figure 1(d). The mechanical properties corresponding to each type of imposed load are deduced from the data in reference.^29,30 Table 2 displays the characterized mechanical properties of BCF/PEEK in various orientations, as determined utilizing the scale-span FE model, which accounts for the magnitude of external loading and resulting strain responses.

Table 2.

Mechanical properties characterization results of BCF/PEEK.

Elastic parameter (GPa)	Value	Poisson’s ratio and coefficient of thermal expansion	Value
E ₁	71.388	v ₁₂	0.10
E₂/E₃	7.838	v ₁₃ /v ₂₃	0.03
G ₁₂	9.599	α₁₁/α₂₂, (1/°C)	−4.12*10⁻⁵
G₁₃/G₂₃	2.794	α₃₃, (1/°C)	−2.75*10⁻⁵

Experimental Results Analysis and Validation of Scale-Span FE Characterization Model

The mechanical properties of BCF/PEEK are determined via standardized experiments conducted in strict adherence to ASTM D3039/D3039M-17³¹ and ASTM D5379/D5379M-19³² protocols. An electro-hydraulic servo fatigue tester, PT-100DWG from Shenzhen Wan Test Equipment Co., LTD, is employed for the experiments. The test specimens are clamped at both ends within the chuck of the testing machine, using a clamping length of 56 mm. Additionally, an extensometer is attached to the center of both sides of each specimen along the tensile direction to record stress-strain data. The displacement loading rate for transverse tensile testing of BCF/PEEK is set at 2 mm/min until failure occurred. To observe surface strain and damage, a VIC-3D system from CSI (Correlated Solutions, Inc.) is utilized. A random speckle pattern is applied to the sample surface using white spray paint and speckle-making tools. Two charge coupled device (CCD) cameras and polarized lights are positioned in front of the specimen. The experimental procedure and setup details are illustrated in Figure 2.

Figure 2.

PT-100DWG BCF/PEEK experimental setup equipped with VIC-3D system.

Subsequent to the experimental phase, the elastic phase curve undergoes analysis to determine the tensile elastic modulus and in-plane shear elastic modulus of BCF/PEEK in accordance with ASTM D3039/D3039M-17 and ASTM D5379/D5379M-19 standards, as depicted in Figure 3(a). Finally, a comparative graph depicting the experimental results juxtaposed with the simulation results is presented in Figure 3(b).

Figure 3.

Comparison analysis of experimental results and characterization results.

As depicted in Figure 3(b), the minimum absolute deviation in the tensile modulus E₁₁ characterization results for each mechanical property is only 1.41%. In contrast, the maximum deviation found in the characterization results of the in-plane shear modulus G₁₂ for BCF/PEEK is 4.07%. This discrepancy primarily arises from the fact that the elastic modulus bears the primary load during stretching, while the shear modulus bears less load under tensile conditions compared to the elastic modulus. Consequently, this leads to a greater deviation in the shear modulus during the experiment. Thus, the validity of using scale-span FE model to characterize the mechanical properties of BCF/PEEK is confirmed, further validating the accuracy of the coefficient of thermal expansion.

Establishment and Solution of Sensitivity Model considering SPs and MPs Uncertainties

Generation and Acquisition of Sample Sets

Generation of Input Sample Sets

Conducting a thorough global sensitivity analysis of the diverse mechanical properties of BCF/PEEK necessitates a considerable sample size. Considering the requirement for sample uniformity and the complexity of interdependent material parameters of BCF/PEEK, the Sobol’s sequence³³ is well-suited to this scenario. The implementation of sensitivity analysis using the Sobol’s sequence method is pivotal in this study, as it profoundly impacts the accuracy of the results characterizing the mechanical properties of BCF/PEEK through sensitivity analysis.

MATLAB software is utilized to develop relevant programs based on the input sample set described in our previous study.^13,23 These programs used initial source data and predefined parameters to generate samples for each parameter within an appropriate range. The number and sets of samples to be generated are carefully defined. To enhance the accuracy of sensitivity coefficient calculations in the global sensitivity analysis model for respective parameters, while maintaining computational efficiency, an initial dataset comprising 50 sample sets is created for each mechanical property parameter of both fiber bundles and PEEK. Subsequently, a total of 650 training input sample sets are generated by exchanging sample data across parameters in the two collections. Since the material properties of carbon fiber bundles follow a transversely isotropic pattern in the scale-span mechanical property characterization model (ie., E_f2 = E_f3, G_f12 = G_f13, etc.), data sample sets for both carbon fiber and PEEK are listed as an example in Table 3. Similar input data samples are also created for a total of 643 groups.

Table 3.

Sample examples of main MPs and SPs data set of BCF/PEEK.

No.	MPs (GPa)						SPs (mm)
	T800S/6K				PEEK		SPs (mm)
	E _f1	E_f2/E_f3	G_f12/G_f13	G _f23	E _p	v _p	l _a	l _b
1	285.74	13.59	20.50	5.22	3.688	0.267	0.090	1.32
2	292.54	15.76	26.70	4.29	3.953	0.287	0.099	1.20
3	275.89	12.61	25.00	4.63	3.596	0.282	0.094	1.269
4	305.20	14.76	23.20	5.84	3.863	0.260	0.106	1.12
5	277.05	15.03	22.88	4.80	3.650	0.298	0.102	1.17
6	304.98	13.33	24.81	5.72	3.883	0.268	0.097	1.22
7	287.07	14.05	27.38	5.34	3.557	0.251	0.103	1.16

Acquisition of Output Sample Sets

To conduct sensitivity analysis of SPs and MPs influence on the mechanical property of BCF/PEEK, a substantial number of samples must be generated. As outlined in scale-span characterization modeling section and the previous studies,²⁷ the scale-span mechanical property characterization model elucidates that the mechanical properties of each input sample set are derived from importing 650 groups of input parameters, yielding the resultant sample set of outputs. Typically, sample generation entails conducting multiple simulations of mechanical property parameters to generate samples for analysis. Nevertheless, the complexity of the simulation process necessitates the utilization of ABAQUS software’s graphical user interface (GUI), resulting in substantial computational and temporal expenditures, wherein prolonged runtimes may conceivably culminate in errors. Moreover, owing to the considerable number of requisite samples, the manual alteration of numerous structural and material parameter data and subsequent job submissions proves to be time-intensive.

This study tackles these challenges by employing ABAQUS batch processing files, facilitating the characterization of mechanical properties in a non-GUI mode through the scale-span FE model. This approach effectively alleviates the aforementioned challenges. By leveraging a scale-span FE RVE model for BCF/PEEK with initial input parameters, a dataset consisting of 650 samples is generated. These samples encompass a variety of uncertain characteristics of MPs, such as fiber bundle length, thickness, and distinct MPs for both carbon fiber bundles and PEEK. The identical representation method is applied to each sample to characterize their individual mechanical properties.

Leveraging the command stream-driven FE method outlined in Figure 4, the resultant parameter sample set consolidates the representation outcomes of all respective parameters. The entire operation is GUI-independent from the ABAQUS software, facilitated by the adoption of the command stream-driven FE method, necessitating solely the invocation of the ABAQUS software kernel in the background. Moreover, the files generated during modeling and calculation do not consume the running resources of a computer workstation. Consequently, the entire modeling and calculation process is faster compared to conventional characterization methods.

Figure 4.

Flow chart of command stream-driven FE method.

Table 4 showcases multiple instances of output sample results achieved through the command stream-driven FE method, encompassing 643 sets of identical characterization outcomes. Each characterization model completes in approximately 15 minutes. The comprehensive computation of all input samples demands over a week, leveraging superior computers outfitted with dual 48-core 8160 Platinum processors and 128 GB of RAM.

Table 4.

Several examples of BCF/PEEK mechanical property characterization results considering SPs and MPs uncertainties.

No.	Mechanical properties of BCF/PEEK (GPa)
	MPs				SPs
	E ₁₁ /E ₂₂	E ₃₃	G ₁₂	G ₁₃ /G ₂₃	E ₁₁ /E ₂₂	E ₃₃	G ₁₂	G ₁₃ /G ₂₃
1	73.56	7.79	8.45	2.85	81.47	8.23	9.03	3.16
2	75.41	8.30	8.49	2.92	76.80	8.04	8.20	3.08
3	74.35	7.46	9.18	2.75	79.28	8.14	8.63	3.13
4	73.86	7.70	9.41	3.05	72.89	8.23	9.03	3.16
5	75.09	8.25	8.15	2.77	75.15	8.04	8.20	3.08
6	73.63	7.83	9.01	3.04	77.55	8.14	8.63	3.13
7	73.56	7.36	8.54	2.87	80.50	8.23	9.03	3.16

Establishment and Solution of Sensitivity Model

Global Grey Relational Analysis of SPs

The SPs of both the carbon fiber and PEEK significantly influence the mechanical property parameters of BCF/PEEK. Additionally, there is interdependence between the SPs of carbon fiber and PEEK. A grey correlation analysis method is employed to comprehensively investigate this relationship, directly assessing the post-cured SPs of carbon fiber and PEEK impact the mechanical properties of BCF/PEEK. Due to data with incompatible dimensions, it is necessary to normalize them to a common scale. The most common approaches for dimensionless processing are the extreme value method and the mean value method. Based on the data requirements, the mean value method is chosen as the basis for dimensionless calculation, as detailed in equation (1).

The performance indicators measured in experiments often have diverse dimensional units, which can complicate result analysis and lead to unsatisfactory outcomes. Therefore, before conducting correlation analysis, it is crucial to normalize all experimental data into dimensionless and comparable sequences. In this section, the characteristics of structural dimension errors encountered during the normalization process can be described as.³⁴

x_{m n} = \frac{\max_{n} (y_{m n}) - y_{m n}}{\max_{n} (y_{m n}) - \min_{n} (y_{m n})}; m = 1, 2, 3 . . . . . ., 27; n = 1, 2, 3

(1)

where, y_mn represents the mth experimental data from the nth group in the original dimensional units. x_mn represents the normalized data, x_mn∈[0,1] in this study,

Structural parameter deviations are the subject of analysis in this study, all deviation values are referenced to 0, which can be expressed as.

(x_{1}, x_{2}, x_{3}) = (0, 0, 0)

(2)

The formula for grey correlation coefficients can be expressed as.

γ (x_{0 n}, x_{m n}) = \frac{Δ_{\min} + 0.5 \times Δ_{\max}}{Δ_{m n} + 0.5 \times Δ_{\max}}; m = 1, 2, 3 . . . . . ., 27; n = 1, 2, 3

(3)

where, γ(x_0n, x_mn) represents the correlation coefficient corresponding to each comparative sequence and the reference sequence. x_mn represents the normalized data, with x_mn∈[0,1].

Δ

_max and

Δ

_min represent the minimum absolute difference and the maximum absolute difference between the reference data sequence and the comparative data sequence, respectively. And the subscripts max and min represent the maximum and minimum value, respectively.

Grey correlation measures the level of correlation between a reference sequence and a comparable sequence. After calculating the grey correlation coefficient, the grey correlation is determined using the following formula, and it can be written as.

Γ (x_{0}, x_{m}) = \sum_{n = 1}^{l} w_{q} γ (x_{0 n}, x_{m n}); m = 1, 2, 3 \dots \dots, 27; n = 1, 2, 3

(4)

where, Γ represents the grey correlation degree between X₀ and X_m, ω_q represents the weight of target q. Since the working conditions are uncorrelated with each other, ω_n = 0.5 in this study.

According to the specified evaluation criteria, (E₁₁, E₂₂, E₃₃, G₁₂, G₁₃, G₂₃, v₁₂, v₁₃, v₂₁, v₂₃, v₃₁, v₃₂, a₁₁, a₂₂, a₃₃) and a dataset comprising 650 entries, this study analyzes the correlation between these 15 evaluation criteria and the dimensions of fiber bundle length l_a and thickness l_b used as reference values, as shown in Figure 1(b). Based on this analysis, a related analysis is performed by incorporating the correlation coefficient calculation formula to determine the correlation coefficient values. Furthermore, these correlation coefficients formed the foundation for the respective evaluations, as illustrated in Table 5.

Table 5.

Fiber bundle length l_a and thickness l_b correlation results.

Ranking	Fiber bundle length (l_a)		Fiber bundle thickness (l_b)
Ranking	Assessment items	Degree of association	Assessment items	Degree of association
1	G ₁₂	0.954	a ₃₃	0.910
2	E₁₁/E₂₂	0.895	v₁₂/v₂₁	0.876
3	a₁₁/a₂₂	0.844	v₃₁/v₃₂	0.843
4	G₁₃/G₂₃	0.802	v₁₃/v₂₃	0.745
5	E ₃₃	0.796	E ₃₃	0.676
6	v₁₃/v₂₃	0.725	G₁₃/G₂₃	0.672
7	v₃₁/v₃₂	0.665	a₁₁/a₂₂	0.643
8	a ₃₃	0.639	E₁₁/E₂₂	0.619
9	v₁₂/v₂₁	0.581	G ₁₂	0.572

The complete observation reveals that among the 15 structural parameter evaluation criteria, G₁₂ received the highest rating with a correlation coefficient of 0.954, closely followed by E₁₁/E₂₂ with a correlation coefficient of 0.895, and a₁₁/a₂₂ with a correlation coefficient of 0.844. Additionally, G₁₃/G₂₃ exhibited a correlation coefficient of 0.802, while E₃₃ demonstrated a correlation coefficient of 0.796. Analogously, among the 15 structural parameter evaluation criteria, a₃₃ attained the highest score with a correlation coefficient of 0.910, followed by Poisson’s ratio at 0.876. In contrast, E₃₃ showed the lowest correlation coefficient of 0.676, and G₁₃/G₂₃ had a correlation coefficient of 0.67.

To visually represent the correlation between fiber bundle length l_a and fiber bundle thickness l_b with BCF/PEEK, the correlation coefficients of l_a and l_b are depicted in a line chart, as illustrated in Figure 5. The geometric parameter l_a exhibits the strongest correlation with both the elastic modulus and shear modulus. Attributed to its greater load-bearing capacity and higher fiber volume fraction of fiber bundles when l_a is longer. Additionally, a negative correlation trend exists between the geometric parameters l_a and l_b and the representative results. Consequently, these uncertain SPs, l_a and l_b, are utilized as design variables for structural optimization. Subsequent work will focus on the integrated design of the BCF/PEEK structural part.

Figure 5.

Gray relational degree analysis diagram.

Based on the GRA algorithm, 650 sets of random SPs are computed for Grey Relational Degrees Γ, Γ_max = 0.667 and Γ_min = 0.496. The corresponding SPs for these extremes are presented in Table 6.

Table 6.

Corresponding SPs of Γ_max and Γ_min.

Type	l _a	l _b	Grey relational coefficient
Γ_max	1.197	0.096	0.667
Γ_min	1.013	0.109	0.496

According to the optimization results obtained through GRA, the highest Grey relational degree of 0.667 among various mechanical property parameters is achieved when l_a = 1.197 and l_b = 0.096. Conversely, the lowest Grey relational degree of 0.496 is observed when l_a = 1.013 and l_b = 0.109. Consequently, when designing SPs, careful consideration of the numerical values of fiber bundle parameters l_a and l_b is essential to achieve optimal design when designing SPs.

Global Sensitivity Analysis of MPs

Building upon the optimal SPs that characterize the mechanical properties of BCF/PEEK as delineated in global grey relation analysis section, subsequent global sensitivity analysis of mechanical property parameters will be undertaken. This approach is crucial for enabling superior design under varied loading conditions. Owing to the multitude of material property parameters embedded within the mechanical properties of the BCF/PEEK component structure, the contribution of each specific MPs to its characterization necessitates individual analysis. Accordingly, this study utilizes incremental variations in individual MPs of BCF/PEEK as input variables to discern their respective impacts on the comprehensive mechanical characterization results. Meanwhile, a slope method is proposed to denote the correlation between a single variable and the overall mechanical properties characterization results. The slope coefficient k can be written as.

k = \frac{y_{t} - y_{i}}{x_{t} - x_{i}}

(5)

where, k represents the slope. y and x represent the vertical axis variable and horizontal axis variable, respectively. Subscripts t and x represent the termination value and initial value, respectively.

To systematically structure the discussion on the influence of structural and material parameters on distinct mechanical properties, the analysis is organized into four primary sections: elastic modulus, shear modulus, Poisson’s ratio, and thermal expansion coefficients.

As illustrated in Figure 6, variations in E_f11, E_p and E_f22/E_f33 respectively show significant impacts on the elastic modulus, particularly for E₁₁/E₂₂, where the k-values reach maximums of 0.207, 0.147 and 0.116, respectively. Similarly, changes in E_p, E_f22/E_f33 and v_m also show notable sensitivity the elastic modulus E₃₃, with k-values peaking at 0.136, 0.089 and 0.056, respectively. This highlights strong dependencies of the elastic modulus on these parameters. Specifically, the elastic modulus E₃₃ is particularly affected by v_m, whereas E₁₁/E₂₂ is most responsive to E_f11 and variations in E_f22/E_f33.

Figure 6.

Mechanical properties of BCF/PEEK correlation analysis considering single variable.

Variations in G_f12/G_f13 and E_p exert a notable influence on shear modulus G₁₂ respectively. The k-value for G₁₂ reaches up to 0.298, indicating high sensitivity to changes in G_f12/G_f13, while the k-value for G₁₃/G₃₁ peaks at 0.057 under changes in G_f23. These findings underscore a robust correlation between shear modulus variations and these particular fiber orientation factors. The overall impact of these parameters on Poisson’s ratio seems relatively less pronounced compared to their effects on elastic and shear moduli, as indicated by the relatively smaller changes in corresponding k-values. Variations in thermal expansion coefficients a_f11, a_f22 and a_f33 lead to observable changes in the corresponding k-values for a₁₁/a₂₂ and a₃₃. This indicates a certain degree of sensitivity, although the discussion implies that other mechanical properties remain relatively unaffected by these variations. In summary, substantial changes in mechanical property parameters, such as elastic modulus and shear modulus, are primarily observed when significant material parameters (E_p, v_p, E_f11, E_f22/E_f33, G_f12/G_f13, G_f23) undergo variations. The alterations in other MPs have a comparatively minor impact on the various mechanical properties of BCF/PEEK.

Assuming that the input variables governing the sensitivities of various MPs are denoted as M = (M₁, M₂, …, M_n), and M_n=(Y|0≤M_i ≤ 1, i = 1, 2, …, n). These MPs encompass the fundamental mechanical property characteristics of carbon fiber bundles and PEEK in the BCF/PEEK. Meanwhile, assuming a specific mechanical property of the BCF/PEEK compression-molded material can be represented as a function, denoted as Y(M), and it can be decomposed into the sum of 2ⁿ continuously increasing components.

k = \frac{y_{t} - y_{i}}{M}

(6)

Y (M) = Y_{0} + \sum_{i = 1}^{n} Y_{i} (M_{i}) + \sum_{i = 1}^{n} \sum_{j = i + 1}^{n} Y_{i, j} (M_{i}, M_{j}) + \dots Y_{1, 2, \dots n} (M_{1}, M_{2}, \dots, M_{n})

(7)

where, Y(M) represents the function of a relevant mechanical properties, such as elastic modulus, etc. M = (M₁, M₂, …, M_n) represents the variable input parameters. Y₀ represents the initial value of the mechanical properties of BCF/PEEK, which remains a constant.

Meanwhile, the value of the subsequent summation of integrals involving variables is 0.

\int_{0}^{1} Y_{i_{1}, i_{2}, \dots, i_{s}} (X_{i_{1}}, X_{i_{2}} \dots X_{i_{s}}) d X_{i_{k}} = 0

(8)

where k = 1, 2, …, n.

The specific expression meanings of equations (7) and (8) lead to the deduction that all the additive terms in equation (9) are orthogonal to each other, and it can be expressed as integrals of Y(M), which is written as.

Y_{0} = \int_{M_{n}} f (M) d M M = (M_{i}, M_{2}, \dots, M_{n})

(9)

Y_{i} (M_{i}) = \int_{0}^{1} \dots \int_{0}^{1} Y (M) \prod_{k \neq i} d X_{k} - Y_{0}

(10)

Y_{i, j} (M_{i}, M_{j}) = \int_{0}^{1} \dots \int_{0}^{1} Y (M) \prod_{k \neq i, j} d M_{k} - Y_{0} - Y_{i} (M_{i}) - Y_{j} (M_{j})

(11)

Combining the above formulas, it can be deduced sequentially.

\int_{X_{n}} Y^{2} (M) d M - Y_{0}^{2} = \sum_{s = 1}^{n} \sum_{i_{1} < i_{2} < \dots < i_{s}}^{n} Y_{i_{1}, i_{2} \dots, i_{n}}^{2} d M_{i_{1}} d M_{i_{2}} \dots d M_{i_{n}}

(12)

Then, the total variance of the function Y(X) is expressed as.

V (Y) = \int Y^{2} (M) d M - Y_{0}^{2}

(13)

Subsequently, the partial variance is given by.

V (E (Y | M_{i}) = \int Y_{i_{1}, i_{2}, \dots, i_{n}}^{2} d M_{i_{1}} d M_{i_{2}} \dots d M_{i_{n}}

(14)

The method for determining the sensitivity coefficient of the mechanical properties of BCF/PEEK, integrating the MPs of carbon fiber and PEEK, as well as the SPs of BCF/PEEK using the “Sobol” method, is described previously. When characterizing the relevant mechanical properties based on the MPs of carbon fiber and PEEK, this can be formulated as equation (15), where factors are defined to quantify the influence of various mechanical properties on the MPs of fiber bundles and PEEK.

S_{i} = \frac{V (E (Y | M_{i}))}{V (Y)}

(15)

where, S_i represents the respective sensitivity coefficient. V(Y) represents the total variance of relevant mechanical properties in samples following the compression molding of BCF/PEEK. And V (E (Y|X_i)) represents the conditional variance of the mechanical properties Y(X) with respect to the MPs of the carbon fiber bundle and PEEK X_i’s conditional expectation E (Y|X_i).

Equation (16) expresses that the value S_i reflects the degree to which the input variable X_i influences a specific parameter function Y(M), where 0≤S_i ≤ 1. As S_i approaches 1, it indicates that the corresponding MPs X_i has a significant impact on BCF/PEEK post-molding. Conversely, when S_i tends toward 0, it suggests that the influence of the MPs X_i is minimal or even negligible after the preparation of BCF/PEEK. Furthermore, it adheres to the condition that the sum of all sensitivity coefficients equals 1, which is written as.

\sum_{n = 1}^{n} \sum_{i_{1} < \dots < i_{n}} S_{i_{1} \dots i_{n}} = 1

(16)

A corresponding program was developed using Python software for modeling and analyzing the sensitivity model outlined previously. This method enables the determination of sensitivity coefficients for the mechanical properties of BCF/PEEK, as well as for the MPs of carbon fiber bundles and PEEK.

Calculating relevant sensitivities based on the acquired input-output dataset involves straightforwardly positioning input and output samples in the corresponding respective positions. This enables the characterization of various mechanical properties of BCF/PEEK based on the input sample set. Consequently, a specific parameter function Y(M) can be derived. The Y(M) function has been imported into a developed MATLAB code utilizing the “Sobol” method. This entire process ultimately results in the distribution of sensitivity coefficients for the mechanical properties of BCF/PEEK with respect to the MPs of the carbon fiber bundle and PEEK. Subsequent material characterization of BCF/PEEK reveals transverse isotropy, with E₂ = E₃ and G_f12 = G_f13, as illustrated in Figure 7.

Figure 7.

Verification results of sensitivity coefficients compared with “Sobol” method.

The sensitivity coefficient of the elastic modulus for mechanical properties in the X direction is 0.628, highlighting its pivotal role in influencing the mechanical properties of BCF/PEEK along the fiber direction X. Conversely, the mechanical elastic modulus in the Y/Z directions correlates primarily with the elastic moduli of carbon fiber and PEEK, with the highest correlation coefficient being 0.356. This suggests that the mechanical properties in the Y/Z directions may not demonstrate the same robustness in terms of load-bearing capacity as observed in the X direction. Sensitivity analysis shows that the shear modulus related to material parameters (MPs) in the XY/YZ directions reaches a peak value of 0.82646. Specifically, the sensitivity coefficient for shear modulus in the YZ plane is 0.678. Moreover, the shear modulus in the direction of the fiber bundle exerts a moderate influence, with sensitivity coefficients of 0.279. The Poisson’s ratios of the fiber bundle and PEEK significantly impact various mechanical properties of BCF/PEEK, with maximum sensitivity coefficients of 0.407. Additionally, the elastic modulus of PEEK can achieve a maximum sensitivity coefficient of 0.399, exerting a significant influence on various mechanical properties.

In accordance with the sensitivity calculation rules defined in this study, the magnitude of the interrelationship between the original MPs sample set and the corresponding set of mechanical properties characterization results of BCF/PEEK is assessed. Similarly, variations in the amplitude of other mechanical property characterization results related to MPs align consistently with corresponding changes in MPs. Therefore, a parameter amplitude variation method is proposed to validate the sensitivity coefficients of the mechanical properties of BCF/PEEK in this study. This method defines the amplitude of characterization results as the product of the fluctuation range of parameters and the sensitivity coefficient of the MPs of carbon fiber bundle and PEEK. Based on the proposed sensitivity coefficient validation method, a corresponding mathematical formula is established, and the corresponding expression can be written as.

S_{i} = Δ Z / Δ C

(17)

where, S_i represents sensitivity coefficients. ΔZ represents the range of variation within the overall interval that characterizes the results. ΔC represents the range of variation within the overall interval of parameters.

Based on the mechanical properties of the typical BCF/PEEK model, the variation ranges of parameters and designs influencing mechanical properties within BCF/PEEK can be organized. These variations, derived from the initial mechanical property characterization model, have already been detailed and introduced in previously section. Furthermore, the scale-span FE characterization results are validated through standard experiments. These verification results serve as a foundation and guide for validating subsequent models. The parameter variation range is calculated by determining the absolute difference between the maximum and minimum parameter values, and dividing this result by the initial value.

Δ Z = | Z_{i \max} - Z_{i \min} | / Z

(18)

Δ C = | C_{i \max} - C_{i \min} | / C

(19)

where Z_imax and Z_imin represent the maximum and minimum values of the corresponding mechanical property characterization results, respectively. Z is the initial value for the mechanical property characterization results. Similarly, C_imax, C_imin and C represent the maximum, minimum, and initial values for MPs, respectively.

The sensitivity coefficient results obtained via the parameter amplitude variation method are compared and validated with results obtained using the “Sobol” method, as illustrated in Figure 7. The comparison indicates that the validation results of the sensitivity analysis model using the parameter amplitude method closely align with those obtained through the “Sobol” method. These findings affirm the accuracy and reliability of the parameter amplitude transformation method proposed in this study for verifying sensitivity coefficients.

High-Performance Design of BCF/PEEK Based on Sensitivity Analysis Models

Design Theoretical of High-Performance BCF/PEEK

In the intricate operational environments of BCF/PEEK, diverse structural components undergo a broad spectrum of loads during their operational lifecycle. For instance, these components necessitate a robust shear modulus to mitigate fracture and damage under torsional loading conditions. Consequently, an original design methodology for BCF/PEEK grounded in gray correlation and global sensitivity analysis models is proposed. By leveraging a preliminary mechanical property characterization model of BCF/PEEK, various superior variants capable of withstanding different load conditions are developed. The accuracy of the initial characterization model is verified through conventional experimental testing methodologies outlined in experimental results analysis and validation section, providing invaluable insights for the design of innovative BCF/PEEK materials tailored to diverse loading scenarios.

The sensitivity coefficients of mechanical properties in the BCF/PEEK component demonstrate correlations between their global macroscopic mechanical characteristics and the constituent composition, notably the fiber bundles and PEEK. Furthermore, the outcomes of the sensitivity coefficients have been substantiated. Hence, employing the suggested parameter range method, a methodology for devising BCF/PEEK that can withstand different types of loads is delineated. This approach primarily centers on crafting diverse mechanical properties of BCF/PEEK with optimal SPs at the meso-scale level. The underlying principle relies on the magnitude of variations in the MPs of the fiber bundles and PEEK, multiplied by their initial values and sensitivity coefficients. Consequently, when designing the overall mechanical properties of BCF/PEEK, the formula can be expressed as.

E_{D i} = \frac{S_{i} E_{i} (E_{C f i} - E_{f i})}{E_{f i}} + E_{i}

(20)

G_{D i j} = \frac{S_{i} G_{i j} (G_{C f i j} - G_{f i j})}{G_{f i j}} + G_{i j}

(21)

where, E_Di represents the elastic modulus of BCF/PEEK in the i direction after design optimization, i = 1,2,3. E_i represents the initial elastic modulus or Poisson’s ratio of the fiber or PEEK in the i direction. E_Cfi represents the elastic modulus of the carbon fiber in the i direction or the ratio of elastic modulus to the PEEK and Poisson’s ratio in the same direction. E_fi represents the initial elastic modulus of the carbon fiber and PEEK in the i direction. G_Dij represents the shear modulus of BCF/PEEK in the ij plane after design optimization, where i,j = 1,2,3. G_ij represents the initial shear modulus of BCF/PEEK. G_Cfij represents the shear modulus of the selected BCF/PEEK in the ij plane. G_fij represents the initial shear modulus or Poisson’s ratio of the carbon fiber or PEEK in the ij plane.

High-Performance Design Examples for Various Load Conditions

Based on the proposed BCF/PEEK design methodology, assuming nonzero correlation coefficients between the actual SPs of the fiber bundle and sensitivity coefficients of MPs between carbon fiber bundles and PEEK, samples for all relevant parameters are established across a variable range. For instance, the initial value of fiber bundle thickness l_b is set to 0.1 μm, with 10 samples collected at 0.01 μm intervals over the range 0.095–0.104 μm. Similarly, the initial value of E_f11 (carbon fiber axial elastic modulus) is set analogously to l_b, starting at 260 GPa. 10 corresponding samples are selected at 10 GPa intervals across the range 220–320 GPa. Correlation coefficients are computed using the GRA method, followed by the determination of sensitivity coefficients via the “Sobol” method. Lastly, equations (20) and (21) are employed to formulate BCF/PEEK variants, which exhibit distinct mechanical properties under diverse loading conditions, as illustrated in Figure 8.

Figure 8.

Design of various mechanical properties of BCF/PEEK based on sensitivity analysis model.

Validation of the Mechanical Properties of Designed BCF/PEEK

The mechanical property characterization model for BCF/PEEK is formulated using the command stream-driven FE method, thereby substantiating the proposed approach to superior structural design. Through adjustments to the material property module code corresponding to various MPs, the mechanical property characterization of BCF/PEEK is conducted across diverse input MPs samples. Subsequent comparison with initial model data facilitates computation of the absolute deviation rate, thereby conclusively establishing the validation outcome.

Figure 9(a) illustrates that augmentation of the elastic modulus E_f11 of carbon fiber correlates with heightened elastic modulus E₁₁/E₂₂ in BCF/PEEK. In Figure 9(b), it is depicted that an elevation in the shear modulus G_f12/G_f13 of carbon fiber results in an amplified shear modulus G₁₂ of BCF/PEEK. The maximum deviation across all samples in shear modulus G₁₂ is merely 0.16%, and for elastic modulus E₁₁/E₂₂ it stands only 0.34%. These discrepancies primarily stem from the approximate nature of sensitivity coefficients associated with each material parameter.

Figure 9.

Prediction model and sensitivity model design verification result.

In light of the established sensitivity analysis model for MPs of carbon fiber and PEEK interface in BCF/PEEK, coupled with the validation method proposed for sensitivity coefficients based on variable parameter amplitudes, the prioritization of the following factors is recommended when employing sensitivity coefficients for designing BCF/PEEK structural parts capable of withstanding diverse loads.

To augment the load-bearing capacity of fiber bundles along the X/Y directions and mitigate potential deformation or damage in BCF/PEEK high-performance design, priority should be accorded to the X/Y directional elastic modulus E_f11 of carbon fiber. Likewise, for enhancing the XY directional load-bearing capacity of fiber bundles and preventing significant deformation or damage, the focus should be on the XY directional shear modulus G_f12/G_f13 of carbon fiber in BCF/PEEK structural design. Similarly, when BCF/PEEK is predominantly subjected to XZ directional loads, greater consideration should be given to the interlayer shear modulus G_f23 of carbon fiber and the elastic modulus E_p of PEEK, which exhibit sensitivity coefficients of approximately 0.279 and 0.504, respectively.

Extensive research and BCF/PEEK design schemes, coupled with comprehensive analysis results, underscore that the focal points in designing high-performance structures primarily encompass the elastic modulus and shear modulus. Other MPs exert relatively marginal influences on the mechanical properties of BCF/PEEK and thus are typically not the central focus in the design process.

Conclusion

This study proposed a novel approach to design BCF/PEEK by analyzing the variability of SPs and MPs computationally. Firstly, a comprehensive scale-span mechanical property characterization model for BCF/PEEK is established to accurately capture structural uncertainties. Subsequently, a sensitivity analysis model is developed to assess variations in characteristic parameters adopting sensitivity coefficients. These coefficients are verified via the proposed parameter amplitude variation method. Based on the calculation outcomes of each material parameter in the global sensitivity analysis model, various types of BCF/PEEK are designed and validated using the scale-span FE prediction model. This research reveals several critical considerations for designing the high-performance BCF/PEEK, as follows.

(1) The analysis showed that fiber bundle length l_a has the highest correlation 0.954 with shear modulus G₁₂, while fiber bundle thickness l_b has the lowest correlation 0.572 with G₁₂. Moreover, a negative correlation trend exists between fiber bundle length l_a and thickness l_b.

(2) The elastic modulus is mainly influenced by elasticity in direction X, with a high sensitivity coefficient of 0.628. Also, the shear modulus on planes XY and XZ showed high sensitivity coefficients of 0.826, mainly influenced by the elastic modulus of carbon fiber and PEEK.

(3) To enhance the load-bearing capacity of fiber bundles in the X/Y directions and prevent significant deformation or damage in BCF/PEEK structural design, prioritize the X/Y directional elastic modulus E_f11 of carbon fiber. Similarly, for improved XY directional load-bearing capacity and prevention of substantial deformation or damage, focus on the XY directional shear modulus G_f12/G_f13 of carbon fiber in BCF/PEEK design. Likewise, when BCF/PEEK faces XZ directional loads, emphasize the interlayer shear modulus G_f23 of carbon fiber and the elastic modulus E_p of PEEK, with sensitivity coefficients around 0.279 and 0.504, respectively.

Footnotes

Acknowledgements

The work reported herein is sponsored by the National Natural Science Foundation of China (52105450), the Key University Science Research Project of Jiangsu Province (No. 21KJB460016). Also, it was supported by marine and offshore equipment intelligent manufacturing technology team from Jiangsu University of Science and Technology, and Jiangsu Hengbo Composite Materials Co., Ltd, for their assistance in preparing BCF/PEEK specimens. The authors would like to acknowledge the editors and the anonymous referees for their insightful comments.

Declaration of conflicting interests

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

Funding

The authors received financial support for the research, authorship, and/or publication of this article: The work reported herein is sponsored by the National Natural Science Foundation of China (52105450), the Key University Science Research Project of Jiangsu Province (No. 21KJB460016).

ORCID iD

Yong Liu

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A global sensitivity analysis approach for designing high-performance braided thermoplastic composites: A case study of braided carbon fiber reinforced poly ether ether ketone

Abstract

Keywords

Introduction

Characterization and Validation of the Mechanical Properties of BCF/PEEK Based on the Scale-Span Method

Scale-Span Characterization Modeling and Analysis of BCF/PEEK

Experimental Results Analysis and Validation of Scale-Span FE Characterization Model

Establishment and Solution of Sensitivity Model considering SPs and MPs Uncertainties

Generation and Acquisition of Sample Sets

Generation of Input Sample Sets

Acquisition of Output Sample Sets

Establishment and Solution of Sensitivity Model

Global Grey Relational Analysis of SPs

Global Sensitivity Analysis of MPs

High-Performance Design of BCF/PEEK Based on Sensitivity Analysis Models

Design Theoretical of High-Performance BCF/PEEK

High-Performance Design Examples for Various Load Conditions

Validation of the Mechanical Properties of Designed BCF/PEEK

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References