Validation of Finite Element-Based Crack-Tip Driving Force Solutions Using Fractal Analysis of Crack-Path Microfeatures

Abstract

Accurate quantification of the crack-tip driving force $\left(∆K\right)$ is fundamental to predicting the fatigue life of engineering structures. Analytical formulations of $∆K$ are rarely available for components with complex geometries. In such cases, finite element (FE) analysis has become a widely accepted approach for determining $∆K$. In this study, an FE-based solution for the crack-tip driving force of a fatigue crack in an asymmetric L-shaped bell crank geometry, a representative complex structure, is established. The structure is fabricated from AISI 410 martensitic stainless steel. The FE-predicted $∆K_I$ for crack growth in the Paris regime has been independently validated using the fractal crack-tip driving force model. Results show that the fatigue crack in the bell crank structure is driven by a combined Mode-I (opening) and Mode-II (shearing) crack-tip loading along a curved crack-path trajectory, as dictated by the asymmetric stress distribution. The fatigue crack edge exhibits fractality with fractal dimensions ranging from 1.00 (Euclidean) to 1.18 along the crack length ($a-a_0$) up to 9.947 mm. The FE-calculated crack-tip driving forces of the bell crank structure are comparable with those computed based on the corrected crack edge fractal dimensions, thus validating the FE simulation outcomes. The resulting fatigue crack growth rates, determined from crack-tip driving forces based on validated FE-computed contour integrals, are comparable to those obtained from the ASTM standard tests.

1. Introduction

Fatigue fracture in engineering structures is driven by the initiation and propagation of cracks under cyclic loading [1]. The cracks weaken structural components [2] and grow under the influence of fatigue. Accurate prediction of crack growth rate is fundamental to ensuring structural integrity and developing cost-effective maintenance schedules in safety-critical industries, such as aerospace, automotive, and nuclear. Central to fatigue crack growth analysis is the estimation of the crack-tip driving force [3], typically characterized in terms of stress intensity factor range $\left(∆K\right)$, which characterizes the local stress field at the crack front. While catastrophic failure occurs when $∆K$ reaches the material’s fracture toughness $\left(K_C\right)$, subcritical crack growth takes place when the operating $∆K$ exceeds a threshold value $\left(∆K_{TH}\right)$, even if it remains below $K_C.$ Within the exponential crack growth stage (described by the Paris law), the crack growth rate curve can be integrated to estimate the remaining life to catastrophic fracture of the component. This phenomenon forms the basis for damage tolerance and time-to-failure estimations in practical applications [2,4].

The stress intensity factor range ΔK is expressed in terms of the operating stress range, stress ratio, and the crack-geometry factor. Closed-form solutions of $∆K$ are well documented for standard geometries [5,6,7], and are derived through rigorous mathematical formulations and idealized boundary conditions. However, real-world structures often incorporate design features with geometric discontinuities, such as branched connections, asymmetric linkages, and load-bearing webs, which render analytical solutions for $∆K$ impractical [8]. In such cases, finite element (FE) analysis has become the preferred approach for evaluating $∆K$ in cracked bodies. Finite element methods rely exclusively on accurate modeling of the singular asymptotic response at the crack tip. Numerous recent works employ commonly used FE methods, such as the contour integral, X-FEM, and VCCT, to estimate $∆K$ [9,10]. Despite their analytical sophistication and widespread use, FE-based $∆K$ estimates are sometimes prone to significant uncertainty, particularly when an inadequate understanding of boundary conditions and/or an uncertain operational history impedes accurate modeling of the fracture process. Phase-field modeling approaches, which represent cracks as diffusive interfaces, offer a unified treatment of crack initiation and propagation [8,11], yet remain computationally demanding and sensitive to parameter calibration [12], thus limiting their industrial applicability. These computational models call for independent, credible strategies to validate FE-based $∆K$ predictions for complex crack geometries and/or loading conditions.

Developing trends in recent research suggest generalizable validation frameworks that integrate deterministic mechanics analysis with data-rich observable parameters for predictive failure analysis [13,14,15,16,17]. Some alternative non-destructive measurement methods reported in the literature, such as Digital Image Correlation (DIC) [1,18,19], Acoustic Emission Monitoring (AEM) [20,21], and Thermoelastic Stress Analysis (TSA) [1,22,23], can be used to validate the computed $∆K$. These methods, while providing high measurement accuracy, often become impractical when applied to complex geometrical structures in a service environment. Additionally, these techniques are typically expensive and labor-intensive. Fractal and multifractal descriptors of fracture surfaces have emerged as physics-aware, data-driven metrics capable of linking morphology to the mechanics of fatigue cracks [24,25]. Multifractal features in the crack wake encode the stress–strain history of the fracture. In this respect, $∆K$ could be inferred directly from these features by treating the fracture zone as a fractal continuum, thereby providing an independent means of verifying FE-computed results. While FE models provide a deterministic prediction of fatigue crack growth based on first principles, the multifractal approach directly extracts the same information from experimental data in a non-destructive manner. The latter approach processes high-resolution binary crack images to estimate the fractal dimension $\left(d_F\right)$ using the box-counting algorithm or similar methods. The crack-tip driving force $∆K$ is then obtained from the fractal dimension paired with the material-specific coefficient of fractality $\left(C_F\right)$ [24].

The objective of the study is to introduce a fractal-based model for determining the crack-tip driving force $∆K_I$ and to demonstrate its application in validating the FE-predicted $∆K_I$ of a fatigue crack in a general structure where the crack-geometry factor $Y$ is not readily available. In this respect, this work describes the fatigue crack growth response of a general structural component, represented by an asymmetric bell crank plate with a straight-through edge crack. Because the crack-geometry factor is not readily available and the nominal stress distribution is relatively complex, the analytical determination of the crack-tip driving force $∆K$ is challenging. Alternatively, the $∆K$ value at discrete crack lengths is calculated using FE simulations. Fractal dimensions of the crack-path microfeatures are quantified at the respective crack lengths. These fractal descriptors, extracted from high-resolution crack images, provide a non-destructive, geometry-independent quantification of the crack-tip driving force, thus enabling direct validation of the FE-calculated $∆K$ of the propagating fatigue crack.

2. Theoretical Background

The crack-tip driving force under cyclic loading is commonly expressed through the stress intensity factor range $∆K$, which governs the amplitude of the singular stress field surrounding the crack front. Within the framework of linear elastic fracture mechanics (LEFM), the asymptotic crack-tip fields are uniquely characterized by the three stress intensity components $\left(K_I, K_{II}, K_{III}\right)$ corresponding to opening, sliding, and tearing modes, respectively. Fatigue crack growth is primarily controlled by Mode-I (opening), although geometric asymmetry and local constraints may induce significant mixed-mode contributions. In FE formulations, $∆K$ is typically obtained using the contour-integral approach, most notably the J-integral, which represents the energy release rate per unit crack extension under linear elastic conditions [26]:

$$J={\int }_{\Gamma }\left(W{\delta }_{1j}-{\sigma }_{ij}{\displaystyle \frac{\mathsf{\partial }{u}_i}{\mathsf{\partial }{x}_1}} \right)n_{j}d\Gamma$$

(1)

where $W$ is the elastic strain energy density, ${\sigma }_{ij}$ the Cauchy stress tensor, $u_i$ is the displacement vector, and $n_j$ the outward normal to the virtual contour Γ, enclosing the crack tip (Figure 1a). For linear elastic materials, the J-integral is directly related to the stress intensity factor vector, $\left\{K\right\}={\lfloor K_I K_{II} K_{III}\rfloor }^T$, through the following equation:

$$K={\left[B\right]}^{-{\displaystyle \frac{1}{2}}}\sqrt{J}$$

(2)

where $\left[B\right]$ denotes the pre-logarithmic energy factor matrix that incorporates material elastic constants. In practice, FE analysis software (Abaqus 2025) computes the J-integral using domain or interaction integrals, and numerical stability is ensured through convergence of $J$ across successive contours. Under mixed-mode loading, fatigue cracks often deviate from a straight path. Numerous criteria [27,28,29,30,31,32,33] have been developed to predict potential propagation direction, denoted by the kink angle ${\theta }_k$, as illustrated in Figure 1b. This work considers the Maximum Tangential Stress (MTS) criterion, which stipulates that crack growth occurs in the direction of $K_{II}$ = 0, thus providing the local propagation angle as follows [27]:

$${\theta }_k={\mathit{cos}}^{-1}\left({\displaystyle \frac{3K_{II}^2+\sqrt{K_I^4+8K_I^2{K}_{II}^2}}{K_I^2+9K_{II}^2}}\right)$$

(3)

The effective Mode-I crack-tip driving force in the kinked direction is then expressed as $K_I^{\left(k\right)},$ which forms the basis for calculating $∆K$ in subsequent FE analyses.

Complementary to continuum mechanics, the fractal-based fracture model provides an alternative, morphology-based approach for estimating $∆K$. Fracture surfaces generated under fatigue exhibit self-similar and multifractal features that encode the interaction of the crack front with the local stress field. The fractal crack-tip driving force model expresses the stress intensity factor range $∆K_I$ of the crack under Mode-I (opening) as follows [25]:

$$∆K_I=∆K_{TH}+C_F(d_F-d_E)K_{IC}$$

(4)

where $∆K_{TH}$ is the threshold stress intensity factor range for crack propagation, $K_{IC}$ is the plane strain fracture toughness, $d_E$ is the Euclidean dimension of the crack (equal to one for a line crack), $d_F$ is the fractal dimension extracted from the crack morphology, and $C_F$ is a material-specific coefficient of fractality. The value of $C_{\mathrm{F}}$ and other fracture properties of AISI 410 martensitic steel used in this study are listed in

Table 1. Mechanical properties and fatigue crack growth parameters of AISI 410 steel.

Properties and Parameters	Values
$\mathrm{Tensile}\mathrm{strength}, S_U$	657 MPa
$\mathrm{Yield}\mathrm{strength}, S_Y$	620 MPa
Young’s $\mathrm{modulus}, E$	200 GPa
Poisson’s ratio, ν	0.29
$\mathrm{Hardness}, H_V$	268
$\mathrm{Fracture} \mathrm{toughness}, K_{IC}$	$55.0 \mathrm{MPa}\sqrt{\mathrm{m}}$
$\mathrm{Threshold} \mathrm{stress} \mathrm{intensity} \mathrm{factor} \mathrm{range}, {∆K}_{TH}$	$15.1 \mathrm{MPa}\sqrt{\mathrm{m}}$
Paris crack growth law	$6.53\times {10}^{-9}∆K_I^{3.04}$ mm/cycle
$\mathrm{Coefficient} \mathrm{of} \mathrm{fractality}, C_F$	2.50

. Because the fractal dimension quantifies the fractal nature of crack edges at the microscale, it reflects characteristics of microstructural parameters such as grain size and grain orientation, as well as the operating stress range and the localized stress field. The fractal dimension is obtained by applying the box-counting algorithm to high-resolution crack images, thereby quantifying the geometric complexity of cracks and the driving force at their tips.

3. Materials and Methodology

3.1. Material and Bell Crank Geometry

The material used in this study is AISI 410 martensitic stainless steel, commonly used for compressor blades in land-based power generation turbines. These blades and compressor components are designed to safely tolerate fatigue cracks up to the next inspection interval. The material is supplied in the annealed and hot-finished condition. The chemical composition (in wt.%) of the steel is as follows: 0.15 C, 12.40 Cr, 0.35 Si, 0.495 Mn, 0.027 P, 0.002 S, 0.08 Mo, the balance being Fe. The steel exhibits randomly oriented plates and a needle-like microstructure. A series of fatigue crack growth tests was performed on compact tension C(T) specimens fabricated from the same AISI 410 steel plate. The resulting reference mechanical properties and fatigue crack growth characteristics of the steel are listed in Table 1.

The complex structural geometry is represented by an L-shaped bell crank plate structure with asymmetric limbs, as shown in Figure 2. The structure is machined from a 20 mm-thick AISI 410 martensitic stainless-steel plate. The through-hole ahead of the machined notch is intended to induce mixed-mode crack-tip loading and a non-linear crack path, which are commonly encountered in structural applications. Fatigue pre-cracking is performed to establish a fatigue pre-crack of length $a_0=$30 mm (measured from the load line).

3.2. Fatigue Crack Growth Test of Bell Crank Structure

The fatigue crack growth test of the bell crank structure is intended to illustrate typical crack growth behavior in a general structure with complex geometry and/or loading. A 100 kN servo-hydraulic fatigue testing machine operates in load-controlled mode. A sinusoidal load waveform with a load range $∆P$ = 5.4 kN, load ratio $R$ = 0.1, and frequency $f$ = 10 Hz is applied. Crack propagation is tracked using a microscopic camera mounted on a mobile platform. Crack lengths are measured from the load line and along the crack path. Data pairs of crack length and the corresponding accumulated load cycles are recorded throughout the test.

It is of practical interest to determine the crack growth rate at any stage of the fatigue life. However, the associated crack-geometry correction factor for the bell crank structure is not readily available for calculating the stress intensity factor range ${∆K}_I.$ Alternatively, ${∆K}_I$ is determined based on the FE-calculated stress field in the crack-tip region.

3.3. FE Simulation of Bell Crack Structure for Crack-Tip Driving Force, ${∆K}_I$

A series of FE simulations of the bell crank structure, each with a different crack length, is performed to establish the crack-tip stress field for use in calculating the corresponding ${∆K}_I$. The digitized crack trajectory, based on experimental observations, is explicitly modeled to ensure a realistic representation of the observed crack path. Each FE model of the bell crank geometry is discretized using an adjusted element mesh topology to capture crack-tip parameters and local stress gradients accurately. A mesh convergence study is performed using $K_I$ as the monitoring variable. The mesh convergence analysis result for the crack-tip zone is shown in the inset of Figure 3. In this zone, the maximum element size has an edge length of 1.4 mm. The resulting solid element mesh is non-uniform, with finer mesh applied at geometric discontinuities and traction-free edges to better capture localized deformation and stress gradients, as shown in Figure 3. An ultrafine mesh is used in the immediate vicinity of the crack front where steep stress and strain gradients are anticipated. A coarser mesh is employed in the bulk of the material, away from critical locations, to reduce computational effort while preserving numerical accuracy. Fatigue load cycle is simulated with the maximum and minimum forces of 0.6 and 6.0 kN, respectively (i.e., $∆P$ = 5.4 kN, $R$ = 0.1) applied at the bottom loading hole, as illustrated in the figure. All translational and rotational degrees of freedom (DOF) of the top hole are constrained (U_X = U_Y = U_Z = 0) and (UR_X = UR_Y = 0) except the Z-rotational DOF. These loading and constraint conditions closely represent the testing setup.

Figure 4 depicts the methodology for mapping the actual crack length in the bell crank structure. The crinkled nature of the fatigue crack path necessitates the use of FE to predict the crack growth direction. The experimentally obtained crack trajectory (including pre-crack length) is digitized at 102 discrete points, as illustrated in Figure 4A. These points are input into commercial Abaqus 2025 FE analysis software to accurately reconstruct the irregular crack path for the analysis. Then, to determine the crack-tip driving force at a given crack length, a local coordinate system is prescribed at the crack front, as shown in Figure 4B. The origin is located at the midpoint of the crack front where the maximum value of $K_I^{\left(k\right)}$ is expected. The local Cartesian coordinates (x₁, x₂, x₃) are oriented such that the x₁ and x₂ directions are normal and parallel to the load line, respectively, while x₃ is tangential to the crack front. FE simulations are iteratively conducted at distinct crack front locations to evaluate provisional values of $K_I, K_{II}, K_{III}$, and ${\theta }_k$, assuming x₁ as the virtual crack extension direction. Once the initial estimates are acquired, $K_I^{\left(k\right)}$ is computed in the direction of ${\theta }_k$, in which $K_{II}$ = 0. The stress intensity factor range ${∆K}_I^{\left(k\right)}$ can be computed as the difference between K_I^(k) evaluated at $P_{max}$ and $P_{min}$. The stress intensity factor range is calculated as ${∆K}_I^{\left(k\right)}=K_{I,max}^{\left(k\right)}-K_{I,min}^{\left(k\right)}$.

3.4. Fractal Dimensions Determination of Fatigue Crack in Bell Crank Structure

Post-test, high-resolution digital images of the fatigue crack edges are obtained using an Olympus BX51M (Olympus Corporation, Tokyo, Japan) optical microscope at 100× magnification and a spatial resolution of 1090 pixels/mm to capture the crack-edge morphology. The grayscale crack image is converted to a binary image using OTSU’s global thresholding, yielding an optimal threshold for segmenting crack-edge features. Fractal dimension $d_F$, is determined from binary images using the box-counting algorithm scripted in Python version v3.10.7. Parameters for the current measurement process used to determine the fractal dimension have been optimized as described in [24]. This approach superimposes a grid of box size ε and records the number of intersecting boxes $N\left(\epsilon \right)$ by successively decreasing $\epsilon$. A typical log₂ $N\left(\epsilon \right)$ versus log₂ $\epsilon$ is shown in Figure 5. The data show a prominent straight line, indicating that the crack edge exhibits self-similarity or fractal behavior. The (negative) slope of the line defines the fractal dimension corresponding to the crack-tip position (crack length). Complete information needed to analyze the image and calculate the fractal dimension $d_F$ is given in [24]. Fractal dimensions of the fatigue crack were determined at selected positions along the fatigue crack path. Once the fractal dimension value at a given crack length has been established, the fractal crack-tip driving force model (Equation (4)) can be used to determine the corresponding crack-tip driving force, and subsequently the fatigue crack growth rate.

4. Results and Discussion

4.1. Fatigue Crack Growth Response of Bell Crank Structure

The torturous fatigue crack path of the bell crank structure from the notch tip is shown in Figure 6. The crack trajectory deviates from the reference horizontal line at the start of the test and gradually curves as it propagates through the structure. This suggests that the crack front is driven by mixed-mode loading, with contributions from Mode-I (opening) and Mode-II (shearing). Crack growth tends toward the relatively slender limb due to higher stress magnitude and lower fracture resistance. It is expected that the crack growth path aligns itself perpendicular to the maximum (tensile) principal stress direction. The crack-tip stress field predicted by the FE simulation is discussed in Section 4.2. The fatigue crack growth test is terminated with the final crack length extension of ($a-a_0$) = 9.947 mm, measured from the notch tip along the curved crack trajectory. The final crack-tip position is located at 8.453 and 4.409 mm in the X1 and X2 directions, respectively, as illustrated in Figure 6.

The measured fatigue crack growth curve of the bell crank structure is shown in Figure 7. A typical exponentially increasing trend of the crack length, ($a-a_0$) with the accumulated load cycles $N,$ is noted. The initial fatigue crack growth rate is relatively high, as reflected in the curve’s initial slope. This is likely due to the long initial crack length, thereby resulting in a relatively high crack-tip driving force. Post-processing this curve enables the determination of the structure’s fatigue crack growth rate at different crack lengths. However, for an engineering structure in field operation, the crack growth curve is unavailable a priori. Additionally, the crack-geometry correction factor and the operating stress range are not readily available for determining the crack-tip driving force, $∆K_I.$ Otherwise, the current crack growth rate of the structure could be established from the characteristic ${\displaystyle \frac{da}{dN}}-∆K$ curve of the material. In this respect, FE simulation of the cracked structure can be performed to calculate $∆K_I$, as deliberated in this work. The FE-calculated results are discussed in the next section.

4.2. FE Analysis Results of the Bell Crank Structure

Asymmetric limbs, the anterior hole, and the gradual crack propagation interact to cause stress redistribution in the bell crank structure as predicted in Figure 8 and Figure 9. The contour plots correspond to the peak load cycle. The evolution of the maximum principal stress field for different crack lengths ranging from 0.087 to 9.947 mm is illustrated in Figure 8. The (tensile) maximum principal stress direction is perpendicular to the crack trajectory and thus predominantly contributes to crack opening. A comparatively higher stress is distributed in the shorter (lower) limb of the asymmetric crank geometry, driving the crack to curve towards the shorter limb. The early asymmetric distribution of the crack-tip stress field intensifies as the crack grows along the curved trajectory. At the crack lengths of ($a-a_0$) = 7.656 mm (Figure 8c), the crack-tip stress field begins to acknowledge the presence of the hole. The crack-tip stress field of the advancing crack interacts with the hole edge, thereby amplifying the mode-mixity of the crack-tip driving force. The fatigue crack growth test is terminated when the crack reaches the length ($a-a_0$) = 9.947 mm. At this crack length, the crack-tip stress field strongly interacts with that of the hole, as illustrated in Figure 8d. It is noted that a relatively larger zone of high stress is confined to the crack tip for the mid-thickness plane of the bell crank structure (Figure 8e) when compared to the outer surface (Figure 8d). The reduced constraint on traction-free surface deformation (plane-stress condition) enables plastic flow, thereby reducing stresses, whereas the mid-plane experiences greater constraint in the thickness direction under the plane-strain condition.

The crack-tip stress field and the corresponding shape of the plastic zone under a pure Mode-I crack loading are often examined in terms of the opening stress component. Figure 9 illustrates the crack-tip stress field for different crack lengths, expressed in terms of the global Y-stress component ${\sigma }_{yy}$. Butterfly-like stress contours with asymmetrical lobes are predicted. The lobes increase in size as the crack propagates, and the stress intensity factor range $∆K_I,$ increases. Steeper stress gradients in the lower limb of the bell crank structure with smaller lobes drive the crack along the curved path, as represented in Figure 9d. The plastic zone shape would be similar to the stress lobe geometry, as it traces the elastic-plastic boundary of the material. The relatively small plastic zone size renders the LEFM analysis appropriate.

Figure 10 shows the measured crack trajectory of Figure 6, digitized at 102 data points. The x-axis is perpendicular to the loading direction. The origin (x = 0) defines the transition between the pre-crack and crack growth regions and corresponds to an initial crack length $\left(a-a_0\right)$ = 0. This coordinate system is specific to Figure 10 and distinct from the global frame described in Figure 4B. The predicted direction of crack propagation, based on the Maximum Tangential Stress criterion $\left(K_{II}=0\right)$, as stated in Equation (3), at the respective locations along the crack path is illustrated by the vectors. The magnitude of the vectors is scaled proportionally to the Mode-I stress intensity factor range ${∆K}_I^{\left(k\right)}$. Results show that the crack path predicted by the FE model is comparable to the observed crack trajectory. The prescribed crack growth criterion along the $K_{II}$ = 0 direction successfully predicts the instantaneous local crack advance under mixed-mode loading. Additionally, it partially verifies the implementation of the contour-integral approach and the FE modeling strategy used in Abaqus FEA, providing a solid basis for extending this methodology to extract components of the stress intensity factor vector, as discussed next.

The evolution of contributions from various stress intensity factor $\left\{K\right\}$ components, computed using FE analysis and plotted as a function of crack extension $\left(a-a_0\right),$ is illustrated in Figure 11. All components exhibit a consistent upward trend that accelerates as the crack grows. Although Mode-I (opening) remains the predominant fracture mechanism during crack propagation, significant mode-mixity is observed due to contributions from Mode-II (shearing) crack loading. The Mode-III (tearing) contribution is negligible at all calculation points and is therefore omitted from subsequent analyses. The Mode-I stress intensity factor, computed in the direction perpendicular to the load line, denoted as $K_I$, increases from approximately 20 MPa√m near initiation to over 40 MPa√m at 10 mm crack extension. The values of ${K_I, K}_{II},$ and $K_{III}$ shown in Figure 11 are computed at $P_{max}$ = 6.0 kN. However, the Mode-I stress intensity factor computed in the direction of crack propagation and at maximum load condition $\left(P_{max}\right),$ denoted as $K_{I,max}^{\left(k\right)}$, follows a different curve, as shown in Figure 11. Initially, it tracks K_I closely but gradually diverges as the crack length increases, reaching 48 MPa√m at 10 mm. This divergence arises from $K_{II}$ contributions, which, while secondary, are non-negligible at higher crack lengths. Starting near 2 MPa√m, $K_{II}$ increases to 12 MPa√m in the final stages of the test, thus revealing significant shear interaction and crack-path deviation potential. The stress intensity factor range in the crack propagation direction is computed as $∆K_I^{\left(k\right)}=K_{I, max}^{\left(k\right)}-K_{I,min}^{\left(k\right)}.$ It is used as a basis for comparison with the fractal model.

The much-needed crack-tip driving force $∆K_I$ of an engineering structure where the crack-geometry factor is unavailable can be established using FE simulation, as described above. However, validation of the FE-predicted results is required. In this study, the computed $∆K_I$ is validated using measured fractal dimensions along the fatigue crack edges.

4.3. Fractal Fracture Response of the Bell Crank Structure

The fractal dimension $d_F$ of the propagating fatigue crack in the bell crank structure is plotted against the number of load cycles $N$, as shown in Figure 12 (reproduced from [34]). The plot represents the fractal nature of growing fatigue crack edges. The scatter in the data is intrinsic to the material’s fatigue response, particularly at low stress amplitudes. Such scatter is attributed to the physical heterogeneity of the material, in which the distribution of near-tip stress and strain is influenced by microstructural discontinuities, inclusions, and local inhomogeneities, thereby leading to a stochastic fracture process [35]. The result shows that $d_F$ fluctuates between 1.00 and 1.05 during the initial loading up to ~75 × 10³ cycles, suggesting a Euclidean crack-like profile. This is followed by a gradual increase in the fractal dimension, reflecting enhanced branching and sub-branching patterns and a progressively rougher fracture-surface and crack-edge morphology. In the final fracture stage, $d_F$ rises sharply, peaking around 1.17 and 1.18 as the test terminates. The evolution trend of the fractal dimension is similar to that observed for the fatigue crack growth behavior, $a-N$ of the bell crank structure (Figure 7), and the standard specimen geometry [24,34]. Therefore, the fractal crack-tip driving force model described by Equation (4) can be used to calculate the corresponding crack-tip driving forces.

4.4. Validation of FE-Calculated Crack-Tip Driving Force Using Fractal Measurements

The fractal dimensions of the fatigue crack in the bell crank structure, plotted as a function of the applied load cycles in Figure 12, are established using the morphology of the crack edges, which inherit the plane-stress condition. Consequently, the resulting fractal-based $∆K_I$, predicted using Equation (4), also represents the plane-stress crack-tip driving force. However, the FE-predicted $∆K_I$ is based on the computed value at the mid-thickness plane of the bell crank plate structure and represents the plane-strain condition. Therefore, the fractal-based $∆K_I$ values are corrected for the Poisson’s ratio effect, such that ${\left({∆K}_I\right)}_{plane strain}={\displaystyle \frac{{\left({∆K}_I\right)}_{plane stress}}{\sqrt{1-{\nu }^2}}}$. Taking the Poisson’s ratio value of 0.29, the plane-strain $∆K_I$ is only 4.5% higher than its plane-stress counterpart. The corrected $∆K_I$ is then compared with FE-calculated values, plotted as a function of the crack length $(a - a_0)$, shown in Figure 13. A similar trend of increasing ${∆K}_{I,max}$ with the crack length is observed. It is acknowledged that the deterministic FE-predicted data display a smooth trend, whereas the measured fractal-based data exhibit statistical scatter. Additionally, the data represent the crack-tip driving forces in the Paris crack growth regime, with the starting $∆K_I$ of about 18 MPa√m for the AISI 410 steel. Both the deterministic FE-predicted ${\Delta K}_{\mathrm{I}}$ and the measured fractal-based ${\Delta K}_{\mathrm{I}}$ data points are least squares fitted with a quadratic relationship to the crack length $(a - a_0)$, as shown in the figure. The coefficient of fit r², for the FE-data and the fractal-based data is 0.9872 and 0.8905, respectively. The relatively low r² value for the fractal-based data is attributed to inherent measurement scatter driven by variations in microstructurally related fractal features of the material. The maximum difference between the fitted lines is 9.56% at $\left(a-a_0\right)$= 5.01 mm. Improved corrected fractal dimension values could be obtained by considering the fractality of the fracture-surface morphology in the crack wake. The observed high degree of concordance between the two data sets validates the FE-predicted $∆K_I$ of the asymmetrical L-shaped bell crank structure.

The growth rates of the fatigue crack in the bell crank structure can now be established using the validated FE-computed contour integrals of $∆K_I$, as illustrated in Figure 14, for the Paris crack growth regime. The standard fatigue crack growth rate curve for AISI 410 steel using the compact tension (C(T)) specimen geometry is shown for comparison [24]. The Paris law for the exponential crack growth regime is given in Table 1. Good agreement between the validated FE-predicted $∆K_I$ and the standard test curve is demonstrated. The spread of the measured multifractal data used in the validation exercise is shown in Figure 14. The result indicates that most of the predicted fatigue crack growth data for the bell crank structure lie within the 99% confidence interval of the reference test data. This supports the validity of the crack-tip driving force model, described in Equation (4), for validating the FE-predicted values.

5. Conclusions

Finite element (FE)-based crack-tip driving force solutions for complex structural components have been established using an asymmetric L-shaped bell crank geometry as a representative structure. The bell crank structure is fabricated from AISI 410 martensitic stainless steel. The FE-predicted $∆K_I$ results for crack growth in the Paris regime have been independently validated using the fractal crack-tip driving force model. The following can be concluded:

• The fatigue crack in the bell crank structure is driven by a combined Mode-I (opening) and Mode-II (shearing) crack-tip loading along a curved crack-path trajectory, as dictated by the asymmetric stress distribution.
• The fatigue crack edge exhibits fractality with fractal dimensions ranging from 1.00 (Euclidean) to 1.18 along the crack length, $(a-a_0)$ up to 9.947 mm.
• The FE-calculated crack-tip driving forces of the bell crank structure compare well with those computed based on the corrected crack edge fractal dimensions, thus validating the simulation outcomes.
• Fatigue crack growth rates determined from crack-tip driving forces based on the validated FE-computed contour integrals are comparable to those obtained through ASTM standard tests.

It is important to characterize a crack and determine its growth rate under the operating fatigue loading. In a general engineering structure, such as the tubular joints of an offshore structure or the aluminum longeron of a military plane, neither the stress amplitude acting on the crack nor the crack-geometry modifying factor is readily available to determine the crack-tip driving force ${\Delta K}_{\mathrm{I}}$. However, if FE simulation is feasible, independent validation of the predicted results is indispensable. This work demonstrates the FE validation process using the measured fractal dimension of the fatigue crack tip. Once the crack tip ${\Delta K}_{\mathrm{I}}$ is quantified, the fatigue crack growth rate can be determined, and the Paris law can be integrated to estimate the fatigue crack growth life of the structure. In fact, since the fractal crack-tip driving force model (Equation (4)) establishes ${\Delta K}_{\mathrm{I}}$ from measurements of the fractal dimension at the crack-tip region and fracture properties of the material, FE simulation of the crack is optional for determining ${\Delta K}_{\mathrm{I}}$ in reliability assessment of the structure.