Fatigue Characteristics Analysis of Carbon Fiber Laminates with Multiple Initial Cracks

Abstract

In the entire wind turbine system, the blade acts as the central load-bearing element, with its stability and reliability being essential for the safe and effective operation of the wind power unit. Carbon fiber, known for its high strength-to-weight ratio, high modulus, and lightweight characteristics, is extensively utilized in blade manufacturing due to its superior attributes. Despite these advantages, carbon fiber composites are frequently subjected to cyclic loading, which often results in fatigue issues. The presence of internal manufacturing defects further intensifies these fatigue challenges. Considering this, the current study focuses on carbon fiber composites with multiple pre-existing cracks, conducting both static and fatigue experiments by varying the crack length, the angle between cracks, and the distance among them to understand their influence on the fatigue life under various conditions. Furthermore, this study leverages the advantages of Paris theory combined with the Extended Finite Element Method (XFEM) to simulate cracks of arbitrary shapes, introducing a fatigue simulation method for carbon fiber composite laminates with multiple cracks to analyze their fatigue characteristics. Concurrently, the Particle Swarm Optimization (PSO) algorithm is employed to determine the optimal weight configuration, and the Backpropagation neural network (BP) is used to train and adjust the weights and thresholds to minimize network errors. Building on this foundation, a surrogate model for predicting the fatigue life of carbon fiber composite laminates with multiple cracks under conditions of physical parameter uncertainty has been constructed, achieving modeling and assessment of fatigue reliability. This research offers theoretical insights and methodological guidance for the utilization of carbon fiber-reinforced composites in wind turbine blade applications.

1. Introduction

As the pivotal component of a wind power generation system, wind turbine blades’ performance and reliability are essential for the safe and efficient operation of the entire setup [1]. Carbon fiber, with its exceptional specific strength, high modulus, and lightweight characteristics, has emerged as the premier material of choice in the wind energy sector for enhancing blade capabilities [2]. The expanded use of carbon fiber composites in the wind energy sector is contingent upon elements like dependability, cost, and upkeep requirements [3]. As depicted in Figure 1, the manufacturing process of wind turbine blades is subject to certain variability in their physical attributes due to technological constraints. Additionally, uneven resin infusion, inadequate vacuum sealing, and technical mishaps during the carbon fiber layering process can introduce defects. These imperfections, under the strain of complex environmental conditions and fluctuating loads, can expand into cracks, leading to blade fissures and potentially catastrophic breakage, which poses a severe risk to wind turbines’ secure functioning [4]. The presence of such defects not only escalates the likelihood of malfunctions but also results in substantial economic losses [5]. Therefore, investigating wind turbine blades, especially the impact of indeterminate variables on the fatigue performance of carbon fiber composites riddled with multiple fractures, is of utmost significance.

Drawing from the existing literature, in the realm of fatigue simulation methodologies, Gholamid et al. [6] utilized an advanced Hashin failure criterion coupled with a stiffness degradation model to perform FEA simulations of composite laminates under fatigue, predicting fatigue life across varying layup sequences. Maio et al. [7] developed a numerical model combining cohesive zone and ALE techniques, effectively simulating fracture processes in quasi-brittle materials, overcoming mesh dependency issues, and verifying the model’s accuracy in predicting crack propagation through experiments. Liu et al. [8] developed a novel CCZM-based fatigue analysis for wind turbine blade adhesive joints, factoring in uncertainties like material attributes and load fluctuations. Programmatic simulations yielded substantial fatigue life data, culminating in a reliability assessment model. Qian et al. [9], after obtaining stress data through experiments or simulations and applying the rain flow counting method and Miner’s rule combined with Palmgren’s linear accumulation principle to the estimation of fatigue lifespan, considered uncertainty factors and developed a fatigue reliability model, which was approximated and adaptively updated through the Kriging model to enhance computational efficiency. Ziane et al. [10] used PSO-ANN to forecast wind turbine composites’ fatigue strength as assessed with MSE. Resin infusion molding was used, and PSO-ANN accurately predicted the fatigue properties. In the field of reliability assessment, Li et al. [11] have developed a model for reliability assessment incorporating Shared Cause Failures by enhancing the theories of multi-state Bayesian Networks and fuzzy probabilities, improving the accuracy of the assessment through a process of defuzzification. Qian et al. [12] introduced a novel single-loop strategy combining Multi-Response Gaussian Processes (MRGPs) with Kriging models for the reliability analysis of systems subject to change over time. This approach eliminates the need for complex inner-loop extremum optimization by decoupling the extremum optimization process and directly employs MRGP to construct the response surface of extremum values. Additionally, they incorporated a Kriging surrogate model initialized with seed points to identify subsequent sampling locations, enhancing the precision of the strategy. Later, Qian et al. [13], focusing on issues related to low failure probabilities, proposed a reliability analysis method for time-variant systems that integrates MRGP with Subset Simulation (SS). By using a new single-loop decoupling strategy to avoid inner-loop extremum optimization, they utilized MRGPs to build a proxy model of the extremum response surface and combined it with the Kriging model to identify new sample points. Finally, the SS method was applied to address low failure probabilities, and the effectiveness of this method was confirmed through case studies. Zhang et al. [14] developed a reliability evaluation technique combining transfer learning and the Direct Probability Integral method to gauge offshore wind turbine blade dependability amidst complex physical interactions. By using adaptive sampling strategies and surrogate models, they were able to efficiently predict the performance of the blades, offering valuable insights for blade design and safety considerations.

Besides contemplating the effects of multiple initial cracks on the fatigue characteristics of carbon fiber-reinforced plastics, the interplay of various uncertainties during manufacturing and assembly processes cannot be overlooked; since they significantly affect the accuracy of fatigue life forecasting and trustworthiness evaluation, the examination of uncertainties deserves consideration. Wang et al. [15] observed that the majority of research on the simultaneous optimization of composite material layout and fiber direction is carried out under fixed conditions, neglecting the effect of variable factors on the structural dependability of fiber-reinforced composites. Balokas et al. [16] utilized experimental data on effective stiffness and strength to identify critical random parameters and updated their probabilistic models through Bayesian inference, markedly enhancing the accuracy of uncertainty in the micro-parameters of carbon fiber unidirectional composites. Joannès et al. [17], through the analysis of T700 carbon fiber single-fiber test data, discovered that the influence of sampling randomness on the uncertainty of fiber strength exceeds that of measurement-induced uncertainty. The research highlights the significance of employing stochastic uncertainty in the reliability of fiber composite material strength models.

Incorporating parameters associated with material properties, sampling variability, measurement uncertainties, and stochastic elements into the analysis of fatigue life and reliability is important. Considering this, the present study selects carbon fiber composite laminates with multiple initial cracks as the research subject, taking into account the impact of cracks in different combinations and the uncertainty of physical parameters on the fatigue performance of carbon fiber composite laminates. A new method for assessing their fatigue characteristics and fatigue reliability under conditions of physical parameter uncertainty is proposed in this study. By combining static and fatigue tests, a simulation method has been developed to predict the fatigue life of the laminates. In response to the challenges of data requirements and time consumption associated with single fatigue life simulations of carbon fiber laminates, this study utilizes a Particle Swarm Optimization–Backpropagation (PSO–BP) neural network as a surrogate model for predicting fatigue life. This model is then applied for the modeling and assessment of fatigue reliability of carbon fiber laminates with multiple cracks, as shown in Figure 2.

2. Methods

2.1. Paris-XFEM Method

This paper utilizes the Paris-XFEM technique for fatigue modeling, forecasting the service life of carbon fiber composite laminates riddled with multiple cracks. By incorporating the Paris law, which delineates the velocity of crack propagation due to fatigue, and using the Extended Finite Element Method (XFEM) to model crack propagation that is independent of the underlying mesh [18], this method offers a robust solution for predicting fatigue behavior. The Paris formula [19] is articulated in detail as follows:

$$\mathrm{d}\mathrm{a}/\mathrm{d}\mathrm{N}=\mathrm{C}{(\mathrm{\Delta }\mathrm{K})}^{\mathrm{m}}$$

(1)

where a denotes the crack depth; N indicates the cycle count of fatigue stress; da/dN is the velocity of crack extension; ΔK represents the amplitude of the stress intensity factor due to fatigue stress; and C and m are material-specific parameters that characterize the fatigue crack growth behavior.

The fatigue life is expressible through the cumulative count of fatigue cycles. Integration of the Paris equation yields the correlation between crack length and fatigue cycle count, thereby determining the lifespan of fatigue crack propagation.

$$\mathrm{N}={\displaystyle {\int }_{{\mathrm{a}}_0}^{{\mathrm{a}}_{\mathrm{c}}}\frac{\mathrm{d}\mathrm{a}}{\mathrm{C}{\left(\mathrm{\Delta }\mathrm{K}\right)}^{\mathrm{m}}}}$$

(2)

where $a_0$ represents the initial size of the crack, and $a_c$ indicates the critical size at which the crack becomes critical.

In pursuit of enhanced applicability, ABAQUS has adopted an energy release rate-based fatigue crack propagation methodology, which meticulously delineates the thresholds for energy release rates. Illustrated in Figure 3, the crack maintains a state of dormancy and refrains from expanding when the rate of energy liberation at the crack’s edge is beneath the minimum critical value $G_{thresh}$. Upon surpassing this minimum threshold $G_{thresh}$, the crack initiates growth, with its rate of expansion incrementally escalating. When the energy discharge rate at the fracture tip goes over the highest threshold $G_{pl}$, the rate of crack growth precipitously increases, culminating in the eventual failure of the structure.

In the realm of numerical simulation for expansion of cracks under cyclic stress with ABAQUS, the progression of a crack is not exclusively dictated by the comparison of the velocity of energy release at the fracture’s tip to the predetermined critical threshold, as delineated in Equation (3). It is also subject to the impact of the relative release rate [20]. Consequently, the initiation expansion of fatigue-induced cracks can be delineated with Equation (4) as the foundational criterion.

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{G}}_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}}}{{\mathrm{G}}_{\mathrm{C}}}}=0.01\\ {\displaystyle \frac{{\mathrm{G}}_{\mathrm{p}\mathrm{l}}}{{\mathrm{G}}_{\mathrm{C}}}}=0.85\end{array}\right.$$

(3)

$$\mathrm{f}={\displaystyle \frac{\mathrm{N}}{{\mathrm{c}}_1\mathrm{\Delta }{\mathrm{G}}^{{\mathrm{c}}_2}}}\ge 1.0$$

(4)

where the pivotal energy liberation thresholds $G_c$ in mixed-mode fracture scenarios are characterized by specific values, with $c_1$ and $c_2$ serving as material parameters, conventionally assigned the values of 0.5 and −0.1 [21], respectively. The term ΔG denotes the disparity between the peak $G_{max}$ and the trough $G_{min}$ of the rate of energy liberation during the entirety of cyclic loading. Accordingly, the formula governing the crack growth rate within ABAQUS, employing the Paris-XFEM approach, is hereby presented.

$$\mathrm{d}\mathrm{a}/\mathrm{d}\mathrm{N}={\mathrm{c}}_3{(\mathrm{\Delta }\mathrm{G})}^{{\mathrm{c}}_4}$$

(5)

where the parameters $c_3$ and $c_4$ are material-specific constants that are ascertained through experimental means. The interplay between the stress concentration factor and the rate of energy liberation, a concept deeply rooted in the principles of fracture mechanics, unfolds as follows:

$${\mathrm{G}}_{\mathrm{C}}={\displaystyle \frac{{\mathrm{K}}_{\mathrm{C}}^2}{\mathrm{E}\prime }}, \mathrm{E}\prime =\left\{\begin{array}{l}\mathrm{E}, \mathrm{Plane} \mathrm{stress} \mathrm{state}\\ {\displaystyle \frac{\mathrm{E}}{1-{\mathrm{\mu }}^2}}, \mathrm{Plane} \mathrm{strain} \mathrm{state}\end{array}\right.$$

(6)

where the term E signifies the elastic modulus, while µ is indicative of the Poisson’s ratio. By applying the principles encapsulated in Equation (5) in conjunction with Equation (6), the subsequent relationships and outcomes are elucidated:

$$\mathrm{d}\mathrm{a}/\mathrm{d}\mathrm{N}={\mathrm{c}}_3{({\displaystyle \frac{\mathrm{\Delta }{\mathrm{K}}^2}{\mathrm{E}\prime }})}^{{\mathrm{c}}_4}$$

(7)

Using the method of integration, the fatigue life is calculated, yielding the following expression:

$$\mathrm{N}={\displaystyle {\int }_{{\mathrm{a}}_0}^{{\mathrm{a}}_{\mathrm{c}}}\frac{\mathrm{d}\mathrm{a}}{{\mathrm{c}}_3{\left(\frac{\mathrm{\Delta }{\mathrm{K}}^2}{\mathrm{E}\prime }\right)}^{{\mathrm{c}}_4}}}$$

(8)

Upon juxtaposition with Equation (2), the following insights are revealed:

$$\left\{\begin{array}{c}{\mathrm{c}}_3={\mathrm{c}}_0{\left(\mathrm{E}\prime \right)}^{{\mathrm{c}}_4}\\ {\mathrm{c}}_4={\displaystyle \frac{\mathrm{m}}{2}}\end{array}\right.$$

(9)

During the fatigue analysis conducted with ABAQUS software, upon encountering a defined sequence of cyclic load cycles, the crack is projected to propagate from its original length $a_0$ to a subsequent length $a_c$. Within this context, ‘unit extension cycles’ denote the minimal cycle count necessary for the crack to advance one grid cell along its path of expansion through the carbon fiber composite laminate. Following this, ABAQUS software autonomously employs Equation (5) to forecast the cyclic iterations required for the crack to progress to the subsequent cell.

2.2. An PSO–BP Life Prediction Model

In the quest to perform uncertainty analysis on carbon fiber laminates that exhibit multiple initial cracks, one is typically faced with the demand for a significant volume of experimental simulation data. To elevate the analytical process’s efficiency, the present study employs a Particle Swarm Optimization–Backpropagation (PSO–BP) neural network as a proxy model. This approach is leveraged to forecast the lifespan of carbon fiber laminates riddled with multiple cracks, thus paving the way for further investigation.

2.2.1. BP Neural Network

The Backpropagation (BP) neural network operates on an algorithmic principle that refines the precision of predictive or classificatory tasks. It achieves this by dynamically adjusting the synaptic weights of network neurons in response to error signals conveyed backward during training [22]. The network’s framework, shown in Figure 4, is made up of an initial input layer, a chain of hidden layers, and a terminal output layer. As forward propagation takes place, the input data flows through the input layer, is altered by the hidden layers, and is then delivered to the output layer. Discrepancies between the output layer’s predictions and the target values trigger an error feedback loop through the backpropagation mechanism. This feedback loop catalyzes the network to fine-tune its weights and thresholds, striving to minimize the sum of squared errors to a pre-established threshold. The detailed model is delineated as follows:

$$\mathrm{Y}={\mathrm{f}}_2({\mathrm{V}}^{\mathrm{T}}{\mathrm{f}}_1({\mathrm{W}}^{\mathrm{T}}\mathrm{X}+{\mathrm{b}}_1)+{\mathrm{b}}_2)$$

(10)

where Y represents the predicted output value. The connection weights between the input and hidden layers are denoted as $W^T$, with bias terms $b_1$, and the activation function $f_1$ is linear function. The synaptic weights linking the hidden layer with the output layer are $V^T$, with bias terms $b_2$, and the activation function $f_2$ is a sigmoid function.

2.2.2. PSO–BP Neural Network

The quintessential mechanism of the Backpropagation (BP) neural network hinges on the utilization of gradient descent to mitigate the aggregate of squared prediction errors, effectively narrowing the gap between the anticipated and the actual outcomes. Nonetheless, this methodology is susceptible to the pitfall of local optima during optimization, which can impede the network’s capacity to realize its peak theoretical performance and compromise its forecasting prowess on novel datasets. Integrating Particle Swarm Optimization (PSO) amplifies the global search proficiency within the neural network’s parameter optimization regimen, steering the network clear of local minima while it endeavors to identify the optimal configuration of weights and thresholds, thus augmenting the network’s predictive capabilities. The foundational ideology of this technique draws inspiration from the synergistic interactions and information dissemination observed within biological communities, harnessing swarm intelligence to navigate and uncover the most efficacious solution [23]. The detailed implementation is delineated as follows:

$${\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}+1}=\mathrm{w}{\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{c}}_1{\mathrm{r}}_1({\mathrm{p}}_{\mathrm{best},\mathrm{i}}^{\mathrm{k}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}})+{\mathrm{c}}_2{\mathrm{r}}_2\mathrm{(}{\mathrm{g}}_{\mathrm{best},\mathrm{i}}^{\mathrm{k}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}})$$

(11)

$${\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}+1}={\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}+1}$$

(12)

Within this framework, $V_i$ symbolizes the kinetic velocity of particle i, which mirrors its rate of advancement through the parameter space. The term $x_i$ signifies the position coordinates of particle i, marking its precise location within the expanse of potential solutions. The iteration index, denoted by k, serves to monitor the algorithm’s progression. The learning coefficients $c_1$ and $c_2$ dictate the particle’s sensitivity to its individual best performance and the collective best performance of the swarm, respectively. The inertia weight, represented by ω, alongside the stochastic elements $r_1$ and $r_2$ that reside within the [0,1] range, contribute to the dynamics of the system. The individual best position for particle i is encapsulated by $p_{best,i}$, while $g_{best,i}$ denotes the optimal position that the swarm converges upon as a collective achievement.

In order to assess the efficacy of the Particle Swarm Optimization–Backpropagation (PSO–BP) approach, the present study adopts a suite of metrics that are delineated as follows:

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{Y}}_{\mathrm{i}}-\widehat{{\mathrm{Y}}_{\mathrm{i}}}\right|}$$

(13)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{Y}}_{\mathrm{i}}-\widehat{{\mathrm{Y}}_{\mathrm{i}}}\right)}^2}}$$

(14)

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\left|\frac{{\mathrm{Y}}_{\mathrm{i}}-\widehat{{\mathrm{Y}}_{\mathrm{i}}}}{\widehat{{\mathrm{Y}}_{\mathrm{i}}}}\right|}\times 100\%$$

(15)

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{Y}}_{\mathrm{i}}-\widehat{{\mathrm{Y}}_{\mathrm{i}}}\right)}^2}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{Y}}_{\mathrm{i}}-\overline{{\mathrm{Y}}_{\mathrm{i}}}\right)}^2}}}$$

(16)

where MAE stands for Mean Absolute Error, RMSE stands for Root Mean Square Error, MAPE stands for Mean Absolute Percentage Error, R² stands for the coefficient of determination, n stands for the number of samples, $Y_i$ stands for the predicted value, $\widehat{Y_i}$ stands for the actual value, and $\overline{Y_i}$ stands for the mean value.

The PSO–BP neural network leverages the synergy within the particle swarm to iteratively refine particle parameters, thereby identifying the ideal weights and decision boundaries for the neural network. The in-depth strategy for estimating the lifespan of carbon fiber laminates via the PSO–BP algorithm, as outlined in this paper, is depicted in Figure 5.

2.3. Sobol Sensitivity Analysis

Sobol sensitivity analysis hinges on variance-based decomposition, dissecting model intricacies into a series of functions that correspond to different combinations of input variables. Executed through comprehensive sampling, this method quantifies the influence of variable interactions on the model’s output [24]. Central to Sobol’s approach are the first-order sensitivity indices, which gauge the effect of individual variable fluctuations; second order and subsequent sensitivity indices, which evaluate the compounded impact of variable interactions; and the total sensitivity index, which amalgamates both primary and higher-order influences, normalized to encapsulate the overall contribution of each variable to the model’s outcome. The formulas [25] pertinent to this analysis are as follows:

The first-order sensitivity index:

$${\mathrm{S}}_{\mathrm{i}}={\displaystyle \frac{\mathrm{V}\mathrm{a}\mathrm{r}\left(\mathrm{E}\left(\mathrm{Y}\left|{\mathrm{X}}_{\mathrm{i}}\right.\right)\right)}{\mathrm{V}\mathrm{a}\mathrm{r}\left(\mathrm{Y}\right)}}$$

(17)

The total sensitivity index expression is given by:

$${\mathrm{S}}_{\mathrm{T}\mathrm{i}}=1-{\displaystyle \frac{\mathrm{V}\mathrm{a}\mathrm{r}\left(\mathrm{E}\left(\mathrm{Y}\left|{\mathrm{X}}_{\sim \mathrm{i}}\right.\right)\right)}{\mathrm{V}\mathrm{a}\mathrm{r}\left(\mathrm{Y}\right)}}$$

(18)

where $S_i$ reflects the standalone sensitivity of the input parameter $X_i$. The vector $S_{Ti}$ consists of all input variables excluding $X_i$. The notation Var is used for variance, and E stands for the expectation mean. $X_{~i}=\left(X_1,X_2,\dots ,X_{i-1},X_{i+1},X_n\right)$.

3. Experimental Tests

This paper prepares specimens based on the GB/T3354-2014 standard, with specific manufacturing conditions detailed in

Table 1. Carbon fiber laminate manufacturing conditions.

Category	Typical Value or Range
Resin type	Epoxy resin
Enhancer type	T700 Carbon fiber
Type of knitting	3K Twill preparation
Carbon fiber volume ratio	60%
Prepreg preparation conditions	Indoor temperature
Resin content	35–40%
Curing temperature	120 °C–150 °C
Curing pressure/MPa	0.5–1.0
Curing time/hours	2–4
Lamination direction	[0°/90°]

. Carbon fiber laminates with multiple initial cracks are used as standard tensile test specimens. During the experiment, the prefabricated internal crack length L₂, the angle θ, and the relative distance S between cracks in the specimens are varied, while the length of the edge crack L₁ is kept constant. The depth of all cracks is set to 1.5 mm, and the relevant experimental setup and results are shown in Figure 6. Furthermore, to more accurately assess the impact of the inter-crack distance S on the performance of carbon fiber laminates under different crack length and angle conditions, this study measures S as the distance between the geometric centers of the internal crack L₂ and the edge crack L₁. This method not only facilitates experimental measurements and numerical simulations but also provides a more conservative estimate by considering the distance between crack centers, thereby more effectively ensuring the safety and reliability of carbon fiber laminates. The test was conducted through an electro-hydraulic servo testing machine model QBS-50 whose maximum static force is ±50 KN, with a maximum dynamic amplitude of ±50 KN, and an accuracy level of 1. The corresponding test-specific process is shown in Figure 7.

3.1. Static Test

Following the control variable method, a static testing protocol was devised for the edge crack L₁ and internal crack L₂ as well as the internal crack angle θ in carbon fiber laminates. A series of five specimen sets, each with distinct parameter combinations, were fabricated. The detailed parameter configurations and corresponding outcomes are presented in

Table 2. Sample parameters and results of static tests.

Serial Number	S Inter-Crack Distance (mm)	θ Crack Angle (°)	${\mathbf{L}}_1$ Crack Length (mm)	${\mathbf{L}}_2$ Crack Length (mm)	Maximum Load (KN)	Average Value (KN)	CV (%)
A1-1-1	0	30	4	6	14.8, 13.93, 16.18, 14.75, 14.15	14.762	5.94
A1-2-1	0	60	4	6	13.63, 13.12, 13.98, 13.43, 14.32	13.696	3.42
A1-3-1	0	90	4	6	13.17, 13.04, 13.16, 13.71, 13.86	13.388	2.76
B1-1-1	0	90	4	4	14.28, 13.35, 14.77, 13.95, 15.21	14.312	5.03
B1-2-1	0	90	4	8	11.47, 11.5, 10.86, 11.33, 11.2	11.272	2.30

.

Post-dissection of the highest load endurance of the test pieces under a variety of conditions, as outlined in Figure 8, it is observed that the maximum load of the carbon fiber laminates diminishes with an increase in the angle of the internal crack L₂, initially starting at 14.762 KN and experiencing a 9.31% reduction. The overall trend of decline is moderate, as indicated by the regression line. With respect to the internal crack L₂ length, the load capacity of the laminates dropped significantly from 14.312 KN at 4 mm to 8 mm, marking a 21.24% decrease. This reduction indicates a notable decrease in load-bearing ability as L₂ lengthens, manifesting a swift downward trend. Additionally, the coefficient of variation reveals that the variability in load capacity narrows as both crack size and angle increase, suggesting that the uncertainty in maximum failure force, in the absence of initial crack length, is predominantly affected by the material’s inherent properties. The fracture details in Figure 7 illustrate that during the structural failure of the laminate, interlaminar delamination is initially observed, which is followed by progressive material fracturing until complete structural collapse. The fiber pull-out observed in the fracture zone demonstrates that the fibers and the resin matrix within the carbon fiber board significantly contribute to resisting tensile forces.

3.2. Fatigue Test

The fatigue testing parameters for carbon fiber laminates containing multiple cracks are detailed in

Table 3. Fatigue tensile test parameter settings.

Category	Fatigue Test Parameters
Stress amplitude	$0.8{\sigma }_s$
Loading frequency	30 HZ
Loading wave form	Sine wave
Stress ratios	0.1
Test environment	Temperature: 25 °C
	humidity level: 60%
Test equipment	Electro-hydraulic servo fatigue testing machine

, where the variables include edge crack lengths L₁ and internal crack lengths L₂, the internal crack angle θ, and the distance S between internal and edge cracks. The specific configurations of these parameters, along with the results of the experimental evaluations, are delineated in

Table 4. Fatigue test sample parameters and results.

Serial Number	S Inter-Crack Distance (mm)	θ Crack Angle (°)	${\mathbf{L}}_1$ Crack Length (mm)	${\mathbf{L}}_2$ Crack Length (mm)	Fatigue Life (Cycle)	Average Value (Cycle)	CV (%)
B2-3-1 (F)	1	90	4	6	33,404, 36,683, 35,673, 35,021, 36,041	35,359	3.53
C2-3-1 (F)	2	90	4	6	37,833, 37,586, 41,374, 39,241, 38,254	38,856	3.97
A2-1-1 (F)	0	30	4	6	54,505, 54,769, 50,922, 51,203, 48,759	52,032	4.92
A2-2-1 (F)	0	60	4	6	40,636, 37,579, 41,193, 38,542, 38,967	39,383	3.80
A2-3-1 (F)	0	90	4	6	32,754, 32,993, 31,993, 35,980, 33,691	33,482	4.55
A1-3-1 (F)	0	90	4	4	327,46, 49,530, 38,415, 45,107, 47,651	42,690	16.33
A3-3-1 (F)	0	90	4	8	21,464, 21,637, 18,470, 18,565, 19,547	19,937	7.69

. It can be observed that the fatigue life of the specimen can reach up to 52,032 cycles. This phenomenon can be attributed to the impact of changes in the internal crack angle on the stress field distribution. When the internal crack angle is tilted from 30° to 90°, the overlap of the cracks causes the stress field to redistribute. This redistribution alters the path and rate of crack propagation, thereby affecting the fatigue life. Specifically, a 30° crack angle may reduce the stress concentration at individual crack tips because the stress is more dispersed at the crack tips. In contrast, when the internal crack angle is 90°, the overlap of the cracks leads to more direct and continuous stress concentration. This concentrated stress accelerates the propagation of the crack, thereby reducing the fatigue life. In this case, the path of crack propagation is more direct, and the stress concentration at the crack tips is more severe, leading to a reduction in fatigue life. Additionally, the A2-1-1 (F) specimen has the shortest inner and outer crack lengths among the samples, which is also one of the reasons for it exhibiting the highest fatigue life. A shorter crack length means that more cycles are required for the crack to propagate to a critical size, thus extending the fatigue life under the same load conditions.

Figure 9 illustrates that the fatigue life of carbon fiber laminates diminishes progressively with increasing crack length. There is a notable 21.57% reduction in fatigue life as the internal crack L₂ extends from 4 mm to 6 mm and a more substantial 40.45% decrease as it extends further from 6 mm to 8 mm. This accentuates a marked escalation in the decline rate, highlighting that longer crack lengths precipitate a sharp decrease in the laminates’ fatigue life, with the trend intensifying as lengths increase. The fitting curve corroborates this with a sharp decline. Additionally, elevating the crack angle correlates with a reduction in the material’s fatigue resistance, with significant decreases of 24.3% from 30° to 60° and 14.99% from 60° to 90°. The fitting curve begins with a sharp plunge, which then transitions into a slower, more gentle decline. Conversely, an increase in the distance S between cracks correlates with an extended fatigue life, with increases of 5.6% from 0 mm to 1 mm and 9.89% from 1 mm to 2 mm. The above indicates that when the carbon fiber composite laminate has vertical cracks, the greater the spacing between the cracks S, the longer the fatigue life of the material. This means that under the same crack size and other conditions, the laminate can withstand longer cycles of loading without failure. However, if the length of the crack L₂ in the laminate increases, its fatigue life will decrease sharply, and this trend of decrease is more significant than the impact of changes in the internal crack angle θ on the fatigue life.

4. Fatigue Simulation Analysis

This section is based on the Paris-XFEM as a benchmark to establish a 3D finite element analysis model of carbon fiber laminates with multiple cracks, employing the ABAQUS software. The model dimensions are consistent with the sample sizes, and the fatigue simulation parameters are set at the same level as the experimental setup, with material parameters as shown in

Table 5. Carbon fiber-laminated material properties.

Elastic Modulus E1/GPa	Elastic Modulus E2/GPa	Elastic Modulus E3/GPa	Poisson’s Ratio ν1	Poisson’s Ratio ν2	Poisson’s Ratio ν3	Shear Modulus G1/MPa	Shear Modulus G2/MPa	Shear Modulus G3/MPa
105.5	7.2	7.2	0.34	0.34	0.378	3400	3400	2520

. Fatigue life data of the carbon fiber laminates were analyzed through fatigue tests to determine the material constants for the Paris formula C = 3.528 × 10⁻¹⁰ and m = 1.3096 m. These data were then input into the simulation software to predict the fatigue tensile fracture process and life under the action of multiple cracks. Figure 10 displays the fatigue crack propagation process for a model with a crack angle θ of 90°, L₁ of 4 mm, L₂ of 6 mm, and distance S of 2 mm.

The three-dimensional model is endowed with material and sectional attributes, and a geometric assembly is conducted to incorporate the multiple initial cracks. On the interacting plates, connections are forged between the carbon fiber laminate and the cracks, permitting the expansion of the cracks. Direct cyclic analysis steps are configured to specify the outputs of the field, and boundary conditions are set according to the amplitude curves that define the characteristics of the fatigue load (tensile fatigue). The meshing process is detailed as depicted in Figure 10, and the computational task is ultimately submitted for processing and resolution through the job module.

The stress distribution diagrams and the status of the enriched elements under various conditions are illustrated in Figure 11. S indicates the stress level, while STATUSXFEM indicates the condition of the enriched elements (an element with a status of 1.0 is fully cracked; one with a status of 0.0 is crack-free. Partially cracked elements have statuses between 1.0 and 0.0). It is evident from the figures that the edge cracks L₁ sustain higher stress levels compared to the internal ones when subjected to external forces. Fatigue damage initiates from the pre-existing edge cracks L₁, yet the internal cracks L₂ do not propagate due to the substantial separation between them, which does not reach the necessary stress for fatigue failure. With the accumulation of cyclic loading, the stress intensity factor eventually surpasses the threshold, causing the adjacent contact elements to fail as they cannot sustain the fatigue strength. Consequently, the cracks, now interconnected, swiftly propagate horizontally until the structure fails.

Table 6. Comparison of lifetime prediction results.

Specimen Number	Simulation Results	Experimental Mean	Relative Error
A1F	35,895	35,359	1.52%
A2F	40,214	38,856	3.49%
A3F	54,015	52,032	3.81%
A4F	41,156	39,383	4.5%
A5F	32,187	33,482	3.86%
A6F	44,895	42,690	5.17%
A7F	21,247	19,937	6.57%

and Figure 12 present a comprehensive review, illustrating the remarkable consistency between the Paris-XFEM simulation findings and the empirical data and revealing a maximum average absolute percentage error of 6.57%, complying with the industry standard of keeping errors within 10% [26]. Consequently, the fatigue simulation analysis approach introduced in this paper fulfills the requirements for the fatigue simulation analysis of carbon fiber plates with cracks.

5. Fatigue Reliability Modeling of Multi-Cracked Laminates

5.1. Uncertainty Analysis

During the analysis of multi-crack laminates, factors such as incomplete data and measurement inaccuracies can greatly affect the outcome. Introducing uncertainty analysis becomes essential to assess the cracks’ impact on service life, and this paper introduces a fatigue assessment methodology, aiming to establish a theoretical and methodological foundation for the investigation of fatigue characteristics in laminates.

The focus of this paper is on examining how physical uncertainties affect the fatigue life of laminates with multiple pre-set cracks. The uncertainties primarily arise from the material properties of the carbon fiber plate and the dimensions of the multiple cracks. Variations in the mechanical properties of carbon fiber plates during production and processing can cause fluctuations within an expected range. As such, we consider the mechanical parameters of the carbon fiber plate along three orthogonal axes—elastic modulus E, Poisson’s ratio ν, and shear modulus G—as sources of uncertainty. The precise measurement of initial crack dimensions, including length, angle, and the distance between cracks, is often challenging with conventional methods, leading to uncertainties that affect the progression of cracks and the endurance of the plate under fatigue conditions.

This study evaluates the combined effects of uncertain factors like crack dimensions, angles, and spacing on the projected fatigue life of carbon fiber plates containing multiple cracks. It presumes that the distribution of uncertain parameters adheres to a Gaussian normal distribution. As detailed in

Table 7. Uncertainty quantification for multi-cracked laminates.

Variable	Parameters	Average Value	CV	Distribution Type
Material properties	E1/Mpa	105,500	0.05	Normal distribution
	E2/Mpa	7200	0.05	Normal distribution
	E3/Mpa	7200	0.05	Normal distribution
	G1/Mpa	3400	0.05	Normal distribution
	G2/Mpa	3400	0.05	Normal distribution
	G3/Mpa	2520	0.05	Normal distribution
	ν1	0.34	0.05	Normal distribution
	ν2	0.34	0.05	Normal distribution
	ν3	0.378	0.05	Normal distribution
multi-cracked	L2/mm	6	0.05	Normal distribution
	θ/rad	60	0.05	Normal distribution
	L1/mm	4	0.05	Normal distribution
	S/mm	2	0.05	Normal distribution

, these random variables are not only normally distributed but also independent of each other, which is crucial for quantifying the potential influence of uncertainty analysis on fatigue life.

5.2. Fatigue Life and Reliability Modeling

5.2.1. Fatigue Life Prediction Method Based on Surrogate Model

Accurate determination of the fatigue life distribution of carbon fiber laminates is essential for conducting subsequent sensitivity analysis, which necessitates a robust dataset. However, obtaining such life data through experimental or simulation methods can be quite inefficient. To address this, a PSO–BP neural network is utilized in this section as a proxy model. The model takes the input parameters from finite element analysis and correlates them with the fatigue life of the laminates, establishing a link between uncertain factors such as material characteristics, crack spacing, length, and angles, with the fatigue life.

Utilizing the Latin hypercube sampling method [27] as indicated in Table 6, 100 sets of data were simulated to determine their fatigue lives, which are listed in

Table 8. Initial sample parameters and lifetime.

	E1	E2	E3	G1	G2	G3	ν1	ν2	ν3	L2	L1	θ	S	Life
1	104,518.7	7001.72	7147.51	3098.22	3641.4	2260.58	0.33	0.34	0.38	5.49	4.19	90.85	1.98	37,380
2	103,392.4	7898.51	7288.83	3197.77	3130.45	2484.26	0.33	0.35	0.36	6.17	3.97	96.52	2	43,279
3	109,041	7241.36	6771.37	2928.85	3427.39	2685.95	0.33	0.32	0.39	5.94	4.16	81.32	1.88	35,264
4	103,013	7953.58	6994.15	3420.81	3280.89	2481.47	0.35	0.37	0.36	6.54	4.35	92.58	1.9	57,838
5	97,655.8	7264.51	6956.73	3347.17	3558.34	2640.01	0.35	0.35	0.38	5.99	3.7	89.75	1.99	31,361
6	106,243.1	7094.28	6848.09	3428.85	3481.3	2327.47	0.34	0.34	0.37	5.96	3.92	83.97	2	32,178
7	111,959.7	6824.78	7487.61	3378.8	3386.27	2634.78	0.37	0.32	0.36	5.25	3.93	86.83	1.97	30,045
8	104,983.1	7407.57	7222.26	3409.76	3490.12	2598.12	0.34	0.32	0.4	5.77	3.78	90.37	2.17	34,507
9	101,734	6981.44	7531.15	3311.62	3347.82	2661.69	0.35	0.33	0.39	6.09	3.95	100.2	2.08	44,050
10	88,483.18	6724.5	7188.01	3655.39	3254.09	2566.33	0.35	0.3	0.41	5.7	3.98	97.65	2.06	36,414
…	…	…	…	…	…	…	…	…	…	…	…	…	…	…
91	116,656.9	7286.38	7776.24	3383.33	3499.64	2668.98	0.33	0.34	0.38	6.1	4.02	83.19	2.04	41,155
92	116,213.5	7794.53	7026.55	3546.34	3323.9	2442.49	0.32	0.33	0.4	6.19	4.07	98.81	1.83	47,191
93	109,641.3	7171.53	7327.19	3255.83	3421.28	2351.54	0.37	0.36	0.38	5.83	4.47	91.98	2.11	49,794
94	110,219.3	7034.56	7716.96	3578.2	3455.32	2621.51	0.34	0.37	0.37	6.21	4.26	88.7	1.89	46,544
95	98,728.87	7477.29	6499.64	3353.18	3208.56	2742.31	0.35	0.33	0.36	6.47	4.04	91.4	1.98	42,851
96	105,406.3	6496.36	7856.82	3698.18	3606.51	2404.01	0.36	0.35	0.38	6.12	3.97	87.65	1.95	38,059
97	103,990.1	6916.35	7404.98	3271.24	3299.79	2452.75	0.32	0.34	0.39	5.92	4.17	88.25	1.82	37,859
98	110,591.1	6587.3	6576.46	3118.53	3196.18	2374.76	0.33	0.32	0.39	5.7	3.68	95.7	1.87	29,835
99	102,263.2	6304.57	7574.11	3283.91	3442.92	2496.36	0.35	0.35	0.35	6.29	3.81	94.42	2.01	38,366
100	116,656.9	7286.38	7776.24	3383.33	3499.64	2668.98	0.33	0.34	0.38	6.1	4.02	83.19	2.04	41,155

. From this dataset, 80 sets were allocated for training the model, while 20 sets served as the test samples. This division facilitates a direct comparison between the model’s predictions and the outcomes of the simulation, thereby enabling a rigorous assessment of the PSO–BP neural network’s forecasting accuracy. As a result, this method ensures the provision of reliable and trustworthy predictions

Table 9. Comparison of different evaluation indicators.

Evaluation Indicators	BP Training Set	BP Test Set	PSO–BP Training Set	PSO–BP Test Set
MAE (cycle)	915	1298	321	449
MAPE (%)	2.4081	3.2212	0.844	1.206
RMSE (cycle)	1115	1943	428	558
R2	0.95786	0.87818	0.99397	0.99051

presents a comparative evaluation of the PSO–BP neural network against the standard BP neural network, highlighting the superior predictive capabilities of the PSO–BP model. The performance metrics for the PSO–BP model, namely the MAE, RMSE, and MAPE, are reported to be 449 cycles, 558 cycles, and 1.206%, respectively. In contrast, the corresponding values for the BP neural network model were 1298 cycles, 1943 cycles, and 3.2212%. The PSO–BP model shows a significant reduction in MAE and RMSE, indicating a marked improvement in predictive accuracy. Additionally, the coefficient of determination (R²) for the PSO–BP model was 0.99051, compared to 0.87818 for the BP model. An elevated R² value denotes a stronger alignment between the forecasts generated with the PSO–BP model and the empirical data, signifying a greater degree of predictive accuracy in contrast to the conventional BP model.

Furthermore, as depicted in Figure 13, the vertical axis represents the predicted fatigue life N_p of the cracked carbon fiber plate via the neural network models, while the horizontal axis represents the finite element simulation values N_e. It is evident from the graph that the PSO–BP neural network model provides a more advantageous and valid estimation, with its fatigue life predictions for multi-cracked carbon fiber laminates closely aligning with the values calculated via traditional methods.

To enhance the efficiency of uncertainty analysis for multi-cracked carbon fiber plates, a PSO–BP neural network is employed as a surrogate model for predicting the fatigue life of multi-cracked laminates. The resulting 1000 predicted data points are illustrated in Figure 14.

5.2.2. Fatigue Reliability Modeling

To evaluate the distribution properties of the data, a statistical analysis is conducted on the fatigue life of carbon fiber laminates with multiple initial cracks, as forecasted via the PSO–BP neural network model. The data are fitted to the Lognormal, Weibull, and Gamma distributions, with the results of the fitting presented in Figure 15 and the specific parameters listed in

Table 10. Judgment coefficients of life fit curve.

Distribution Type	Normal Distribution	Gamma Distribution	Weibull Distribution
R2	0.99986	0.99794	0.99977

. An assessment of the goodness of fit using the determination coefficient R² for the fatigue life curves indicates that the Normal distribution aligns most closely with the fatigue life of the multi-cracked carbon fiber plate, rendering it the optimal choice for characterizing the fatigue reliability of such laminates.

By employing the Normal distribution to fit the data, we derive the probability density function of the carbon fiber plate.

$$\mathrm{f}(\mathrm{x})={\displaystyle \frac{1}{\sqrt{11684.42\mathrm{\pi }}}}{\mathrm{e}}^{\left(-{\displaystyle \frac{{(\mathrm{x}-39148.9)}^2}{2\times {5842.21}^2}}\right)}$$

(19)

Through the process of integration, we arrive at the cumulative fatigue failure distribution function for the carbon fiber plate, expressed with the following formula:

$$\mathrm{F}(\mathrm{t})={\displaystyle \frac{1}{5842.21\sqrt{2\mathrm{\pi }}}}{\displaystyle {\int }_0^{\mathrm{t}}{\mathrm{e}}^{\left(-\frac{{(\mathrm{x}-39148.9)}^2}{2\times {5842.21}^2}\right)}}\mathrm{d}\mathrm{x}$$

(20)

The formula for reliability is articulated as follows:

$$\mathrm{R}(\mathrm{t})=1-\mathrm{F}(\mathrm{t})$$

(21)

Figure 16 illustrates that in the early stages of the fatigue life for carbon fiber laminates with multiple cracks, the rate of crack growth is initially slow. However, the decline in reliability becomes precipitous when it falls below the threshold of 0.9. This behavior suggests that while the crack propagation is gradual at the onset of fatigue damage, it accelerates as the process advances, with the cumulative effects becoming pronounced. Eventually, this leads to a considerable surge in the incidence of cracks within the carbon fiber plates, which greatly compromises the reliability of these plates within the wind turbine blades.

5.3. Sensitivity Analysis

To ascertain the effects of parameter uncertainties on the fatigue life of carbon fiber laminates with multiple cracks, this section utilizes the Sobol method to conduct a sensitivity analysis on the fatigue life. The necessary matrix, derived from Latin hypercube sampling, is fed into the fatigue proxy model described earlier to obtain the fatigue life response. For each parameter, the first-order and cumulative sensitivities are evaluated, as indicated in

Table 11. Sensitivity analysis results.

Uncertainty Parameter	Si	STi
Edge crack length L1	0.44073	0.46249
Internal crack length L2	0.20386	0.23427
Internal crack angle θ	0.18209	0.18644
Distance between cracks	0.03724	0.04179
Elastic modulus E1	0.03387	0.03984
Elastic modulus E2	0.01586	0.01429
Elastic modulus E3	0.01064	0.0024
Shear modulus G1	0.01376	0.00941
Shear modulus G2	0.01127	0.00197
Shear modulus G3	0.01121	0.00227
Poisson’s ratio ν1	0.01454	0.00368
Poisson’s ratio ν2	0.01379	5.20072 × 10−4
Poisson’s ratio ν3	0.01113	6.22547 × 10−4

and visually represented in Figure 17.

The sensitivity analysis, as evidenced in Table 10 and Figure 17, reveals that the first-order Sobol sensitivity indices suggest the edge crack length L₁, internal crack length L₂, and angle θ have a pronounced influence on fatigue resistance of carbon fiber plates, with the edge crack length L₁ being the most influential. This finding aligns with the crack propagation trends depicted in Figure 11 for carbon fiber laminates. Additionally, the elastic modulus E₁, shear modulus G₁, and Poisson’s ratio ν₁ in the direction of the fibers is significantly more impactful than in other directions, which is related to the orientation of the applied loads and confirms the validity of the sensitivity analysis.

In the context of the Sobol total sensitivity analysis, the edge crack length L₁, internal crack length L₂, angle θ, and the distance between cracks S are identified as having substantial global influence, with a noticeable increase compared to the first-order analysis. In contrast, there is a decline in the impact of the elastic modulus E, shear modulus G, and Poisson’s ratio ν, with the shear moduli G₂ and G₃ and Poisson’s ratios ν₂ and ν₃ exerting a negligible effect on the life cycle in fatigue of the carbon fiber laminates.

In summary, the parameters exerting the most considerable influence on the fatigue life of carbon fiber plates are the edge crack length L₁, internal crack length L₂, and angle θ, which are identified as key factors contributing to material failure and fracture.

6. Conclusions

Wind turbine blades serve as essential structural elements in wind energy systems, with their efficacy and dependability paramount for the overall system’s security. As the trend towards larger and lighter blades continues to grow, carbon fiber composite materials, with their advantages of being lightweight, possessing high strength, and exhibiting corrosion resistance, have emerged as the material of choice for blade fabrication. Nonetheless, the manufacturing process of carbon fiber composites can easily introduce defects that, under adverse conditions and fluctuating loads, can cause cracks to propagate and interact, potentially leading to rapid blade fracture and jeopardizing operational safety. In light of this, the present study conducted static fatigue tests on carbon fiber laminates featuring multiple cracks of varying types and analyzed the fatigue life extension process of these multi-cracked plates through simulation. This research focused on the influence of material parameters, internal and edge cracks, crack angles, and the uncertainty associated with distances between cracks on fatigue life. Utilizing a PSO–BP neural network, a reliability model for carbon fiber plates was established, and sensitivity analysis of the physical uncertainty parameters was undertaken. The conclusions drawn are as follows:

• The results of static fatigue tests on carbon fiber laminates, which exhibit multiple cracks of varying configurations, reveal that variations in crack length are the predominant factor influencing the service life of carbon fiber laminates under fatigue conditions. Additionally, the dispersion of static fatigue data diminishes as the variables increase within the same level, whereas the dispersion in fatigue life data is significant and exhibits no clear pattern. At the specific condition where internal cracks measured L₁ = L₂ = 4 mm and the angle θ was 90°, the coefficient of variation peaked at 16.33%. This significant alteration highlights the considerable influence that uncertainties in physical parameters have on the service life of carbon fiber laminates when subjected to fatigue;
• To efficiently overcome the challenges of high costs, extended timelines, and data limitations in fatigue testing, the XFEM method is employed for simulating the fatigue behavior of carbon fiber laminates containing multiple cracks. The comparison with experimental data shows a relative error within an acceptable 10% margin, satisfying practical engineering standards. The simulation captures the progression of fatigue crack growth at various stages of the material’s lifespan, providing a means of ascertaining the physical parameters subject to uncertainty;
• Utilizing the PSO–BP neural network, a fatigue life prediction model for carbon fiber laminates with multiple cracks is established, and based on this, a reliability function is constructed to assess its fatigue reliability. The Sobol method is then applied for sensitivity analysis on the 13 parameters that influence fatigue life, as identified through simulation. The analysis concludes that the most significant parameters affecting the fatigue life of the carbon fiber plate are the lengths of the edge and internal cracks and the angle between them, with the length of the edge crack having the greatest impact.