Fatigue Life Prediction in SAE-AMS 7475-T7351 Alloy: a Comparative Analysis of Retardation Model Implementation in Commercial Software

Abstract

Accurate fatigue-life prediction under variable-amplitude loading (VAL) is critical for damage-tolerance assessment. This study presents a systematic comparative analysis of experimental data and numerical predictions of crack propagation in SAE-AMS 7475-T7351 alloy. NASGRO, AFGROW, and CRACK 2000 were employed to simulate standardized spectra (TWIST, FALSTAFF) and an actual commercial aeronautical load history, focusing on the calibration of Wheeler, GW, and MGW retardation models. The results demonstrate that calibration parameters (e.g., R_so) are highly spectrum-dependent. Physical analysis revealed that underloads attenuate retardation in the TWIST spectrum but are ineffective in the FALSTAFF spectrum. Discrepancies between the software packages, even with identical parameters, were attributed to varying ΔK_th implementations.

Keywords:
Fatigue; Crack propagation; Variable amplitude loading; SAE-AMS 7475-T7351 aluminum alloy; Retardation models

1. Introduction

High-strength aluminum alloys, notably SAE-AMS 7475-T7351, are crucial materials in the manufacturing of aeronautical structural components, such as spars and ribs, owing to their superior combination of low density and high strength¹,². However, fatigue (a progressive damage phenomenon characterized by crack nucleation, a microstructure-influenced process, and subsequent propagation under cyclic loads) fundamentally limits the service life and operational safety of these components³,⁴.

In actual service conditions, these critical components are subjected to variable-amplitude loading (VAL) spectra, the primary cause of fatigue damage⁵,⁶. Analyzing crack propagation under such conditions is essential for residual-life assessment and serves as a pillar of the damage-tolerance philosophy⁷, which applies linear-elastic fracture mechanics (LEFM) to ensure structural integrity⁸,⁹.

Predicting crack propagation under VAL, however, is inherently complex¹⁰,¹¹. Load interaction effects, rendering it dependent on the applied loading history¹²,¹³, intrinsically govern the growth rate (da/dN). Among these phenomena, the propagation retardation induced by overloads (OL) is prominent¹⁴,¹⁵, an effect frequently modeled via compressive residual stresses at the crack tip, as postulated in the Willenborg model¹⁶,¹⁷.

Failure to accurately account for these transient interactions can result in imprecise and dangerously non-conservative life predictions¹⁸. This sensitivity is exacerbated by the fact that errors in determining stress intensity factors (SIF) are magnified by exponential crack growth laws¹⁹,²⁰.

The need for accurate fatigue crack growth (FCG) predictions is vital to damage tolerance, driving the development of reliable numerical models that reduce reliance on costly full-scale testing²¹,²². Although LEFM provides the framework for SIF calculations, and models such as Paris, Forman, and NASGRO are widely utilized²³,²⁴, their accuracy depends on the calibration of empirical material constants²⁵,²⁶. The development of robust probabilistic models and their application to complex structural components remains a central challenge in the field²⁷,²⁸.

While the SAE-AMS 7475-T7351 alloy is extensively characterized, a specific methodological gap persists: quantifying prediction discrepancies among leading commercial FCG software suites (NASGRO, AFGROW, CRACK 2000). Internal implementation of models (e.g., Wheeler, GW) and thresholds (e.g., ΔK_th) differs between codes, generating uncertainty. This work utilizes validated experimental data as a benchmark to isolate and critically analyze these implementation discrepancies.

The central objective is not to perform a new material characterization, but rather to evaluate the robustness and limitations of these software packages in predicting life under VAL. To this end, a systematic comparative analysis is performed between experimental results²⁹ and numerical predictions implemented in NASGRO 4.0, AFGROW, and CRACK 2000.

2. Literature Review

The prediction of fatigue crack growth (FCG) under variable amplitude loading (VAL) remains a cornerstone of damage tolerance philosophy in aeronautical structures. This section provides the theoretical framework for the mathematical models employed to describe crack propagation rates and the subsequent retardation effects induced by load interactions. By establishing the fundamental relationships between stress intensity and material response, this review facilitates a deeper understanding of the numerical methodologies used to assess the structural integrity of the SAE-AMS 7475-T7351 aluminum alloy.

2.1. Fatigue crack growth models

Fatigue crack propagation is characterized by a sigmoidal relationship between the growth rate (da/dN) and the stress intensity factor range (ΔK) spanning the near-threshold regime (Region I), stable propagation (Region II), and the transition to instability (Region III)³⁰,³¹. While foundational power-law approaches describe Region II behavior, they often fail to account for stress ratio effects and near-threshold sensitivities³²,³³. To achieve a precise representation of the entire fatigue life under complex loading, advanced numerical formulations such as the NASGRO equation are required.

The NASGRO model (Forman-Mettu equation) incorporates load ratio (R) effects and plasticity-induced crack closure:

$$\frac{da}{dN}=C{\left[\left(\frac{1-f}{1-R}\right)\Delta \text{K}\right]}^n\frac{{\left(1-\frac{\Delta {\text{K}}_0}{\Delta \text{K}}\right)}^p}{{\left(1-\frac{K_{máx}}{K_c}\right)}^q}$$ $$ (1)

Where C and n are material constants, and p and q are empirical exponents. Notably, (ΔK₀) represents the intrinsic fatigue threshold (for R = 0), which differs from the effective threshold (ΔK_th) by accounting for the crack-opening function (f). This formulation effectively models the asymptotic behavior in both the near-threshold and fracture toughness (K_c) regimes³⁴.

The empirical constants C and n, along with the exponents p and q, are derived systematically by curve-fitting experimental da/dN data across the entire range of stress intensity factors. The exponent p is adjusted to characterize the curvature in the near-threshold regime (Region I), while q is tuned to model the rapid acceleration toward instability and critical fracture toughness (Region III). Accurate calibration of these parameters is essential to ensure the mathematical model aligns with the material's physical behavior throughout the fatigue life³⁵.

The crack-opening function f is a critical component of the NASGRO algorithm, typically using the model proposed by Newman (1984) to quantify plasticity-induced closure. This function calculates the ratio of the crack-opening stress to the maximum applied stress, thereby enabling determination of the effective stress-intensity-factor range (ΔKeff).

$\Delta {\text{K}}_{eff}=K_{máx}-K_{op}$ $$ (2)

Consequently, it captures the influence of the stress ratio (R) and crack-closure phenomena, making the model robust for analyzing variable-amplitude loading conditions³⁶.

2.2. Retardation models and interaction loadings

The implementation of the aforementioned crack growth equations assumes a constant relationship between the driving force and the material response; however, under variable amplitude loading, the sequence of cycles significantly alters the plastic zone at the crack tip. These load interactions, particularly overloads, induce retardation effects that deviate from the predictions of linear accumulation models. To accurately simulate these phenomena, it is essential to incorporate retardation models that account for the transient changes in propagation rates, shifting the focus from steady-state growth to the competitive mechanisms of plasticity-induced closure and residual stress fields.

2.2.1. Wheeler model

The Wheeler model is a yield zone interaction formalism designed to predict FCG retardation by analyzing the geometric relationship between competing plastic zones. The model posits that retardation persists as long as the current cyclic plastic zone (r_i) remains encapsulated within the larger plastic zone (r_ol) generated by a prior overload event³⁷.

To quantify this interaction, the model introduces a scalar retardation factor, Cp (0 ≤ C_p ≤ 1), which serves as a correction factor to reduce the baseline crack growth rate³⁸:

$\frac{da}{dN}=C_p\left(\frac{da}{dN}\right)$ $$ (3)

The factor Cp is formally defined based on the proximity of the crack tip to the boundary of the overload-induced plastic zone:

$C_p={\left(\frac{r_i}{a_{ol}+r_{ol}-a_i}\right)}^m$ $$ (4)

Where a_i is the current crack length and a_ol is the crack length at the moment of the overload. The Wheeler exponent (m), an empirical shaping parameter that must be calibrated to the specific load spectrum and material state, governs the decay of retardation³⁹.

2.2.2. Generalized Willenborg model (GW)

The Generalized Willenborg (GW) model, formulated by Gallagher (1974), distinguishes itself from yield zone interaction theories by employing a residual stress-based methodology to predict retardation⁴⁰. This model postulates that overload events induce compressive residual stresses at the crack tip, characterized by a residual stress intensity factor (K_r) that effectively reduces the local maximum stress intensity factor (K_{max, eff}). Consequently, the retardation of crack propagation is governed by the suppression of the effective stress ratio (R_eff), which captures the interaction between the applied load and the residual stress field⁴¹.

$K_R^W=\Phi K_r=\left(\frac{1-\frac{\Delta K_{th}}{\Delta K}}{\left(R_{so}-1\right)}\right)K_r$ $$ (5)

To enhance the predictive capability of the original formulation, the GW model incorporates the Shut-off Overload Ratio (R_so) as a critical empirical parameter.

${\text{R}}_{eff}=\frac{K_{eff,mín}}{K_{eff,máx}}=\frac{K_{mín}-K_R^w}{K_{máx}-K_R^w}$ $$ (6)

This variable defines the specific magnitude of the overload ratio required to induce complete crack arrest, effectively reducing the growth rate (da/dN) to zero. The utilization of R_so enables the model to be rigorously calibrated to distinct material properties and stress states, ensuring accurate correlation with experimental data across different loading spectra⁴².

2.2.3. Modified Generalized Willenborg model (MGW)

The Modified Generalized Willenborg (MGW) model represents a critical evolution of the standard Generalized Willenborg (GW) formulation, specifically addressing the latter's inability to account for the deleterious effects of compressive underloads on fatigue life. While the standard GW model effectively predicts retardation through a residual stress intensity factor, it often yields unconservative life predictions under variable amplitude spectra by neglecting the reduction of retardation caused by subsequent compressive excursions. This modification, attributed to Brussat, incorporates the physical phenomenon where compressive loads effectively "wash out" or diminish the beneficial compressive residual stresses formed at the crack tip following an overload⁴³,⁴⁴.

$K_{eff,máx}=K_{máx}-K_r$ $$ (7)

$K_{eff,min}=\left\{\max \left(K_{máx}-K_r\right),0\right\}{K}_{min}>0$ $$ (8)

$K_{eff,min}=K_{min}{K}_{min}\le 0$ (9)

To quantify this reduction in retardation, the MGW model introduces a specific correction factor, denoted as Φ_MGW, which utilizes an empirical parameter Φ₀ to adjust the residual stress intensity factor.

$${\Phi }_{MGW}=min\left(\mathrm{1,}\frac{\mathrm{2,523}{\Phi }_0}{1+\mathrm{3,5}{\left(\text{0}\text{,25}-R_{ul}\right)}^{\mathrm{0,6}}}\right)R_{ul}<\mathrm{0,25}$$ (10)

${\Phi }_{MGW}=1R_{ul}\ge \mathrm{0,25}$ (11)

This function calibrates the decay of the retardation effect based on the magnitude of the underload ratio (R_ul), defined as the ratio of the compressive underload stress to the maximum stress of the preceding overload. Consequently, the proper determination of Φ₀ is essential for accurately simulating complex loading sequences, as it ensures the model captures the interaction effects where underloads significantly accelerate crack growth relative to the baseline retardation predictions⁴⁵.

2.3. Standardized flight load spectra in aeronautics

The practical application of the aforementioned retardation models requires their integration with realistic load sequences that replicate the operational environment of aircraft structures. In the aeronautical industry, this is achieved through standardized flight-by-flight spectra, which organize thousands of individual load cycles into representative service units known as flights. These spectra, such as TWIST and FALSTAFF, incorporate complex combinations of maneuver, gust, and taxiing loads, including both overloads and compressive underloads. Understanding the statistical distribution and the sequential nature of these loads is fundamental, as the efficacy of numerical life predictions depends on the model's ability to capture the cumulative damage and the interaction effects unique to each mission profile.

Standardized load spectra such as TWIST and FALSTAFF serve as normative benchmarks for fatigue testing, representing the distinct stress histories of transport and fighter aircraft wings, respectively. TWIST characterizes commercial transport operations dominated by gust loads, whereas FALSTAFF simulates the severe maneuver-induced loading environment of tactical fighters. These spectra consist of variable amplitude loading sequences containing randomly distributed overloads and compressive ground-air-ground cycles, which are critical for evaluating interaction effects during numerical simulation⁴⁶.

The "flight" unit methodology involves grouping individual load cycles into representative blocks to simulate a complete mission profile, often referred to as flight-by-flight loading. A single flight unit encapsulates the full sequence of operational events, utilizing specific markers such as taxi loads to delimit the transition between missions as established. Consequently, this grouping establishes the standard metric for analyzing fatigue life, where the crack length (a) is plotted as a function of the "number of flights" rather than simple cumulative cycles to reflect the operational service life⁴⁶,⁴⁷.

3. Methodology

The experimental database for the numerical simulations is derived from tests previously conducted by Ruchert²⁹, who investigated the behavior of the aeronautical aluminum alloy SAE-AMS 7475-T7351 using M(T) (Middle-Cracked Tension) specimens, widely used in variable-amplitude testing⁴⁸,⁴⁹. The geometry and dimensions of these specimens, extracted in the transverse (T-L) orientation, feature a nominal thickness of 3 mm, a width of 100 mm, and a central hole with a 3 mm diameter. From this hole, two 0.2 mm lateral notches (EDM) were machined, resulting in a total initial crack length (2 a₀) of 10 mm, in accordance with ASTM E647-00, as detailed in Figure 1⁵⁰.

Figure 1
Geometry and dimensions (in mm) of the M(T) specimen used for variable amplitude fatigue tests on SAE-AMS 7475-T7351 alloy. Detail A shows the central hole and notches. Reprinted from Ruchert²⁹.

The variable amplitude loading (VAL) utilized included standardized flight histories, such as TWIST and its reduced version Mini-TWIST, as well as FALSTAFF and Mini-FALSTAFF, and a proprietary industrial flight spectrum provided by real aeronautical commercial history load. These loading sequences were obtained through the GENESIS (Generator for Standardised load Sequences for Fatigue) program, with TWIST being characterized by 4000 flights with a mean stress of 80 MPa and FALSTAFF by 200 flights with a maximum stress of 200 MPa. Table 1 presents a summary of the main standardized flight histories generated by GENESIS that were employed.

Table 1
Specifications of the standardized load histories employed.

Flight History	No. Reversed load points	Total of flights	Savg (a), Smax (b)
TWIST	805,331	4000	80 (a)
Mini-TWIST	124,885	4000	80 (a)
FALSTAFF	35,966	200	200 (b)
Mini-FALSTAFF	18,012	200	200 (b)

• Source: The authors, based on the GENESIS software.

GENESIS generated the load spectra in the Plain ASCII file text format, which stores only the sequence of reversed loads. However, to ensure compatibility with the simulation programs used (AFGROW and CRACK 2000), format conversion was indispensable⁵¹. For this purpose, small C-language programs were created, which processed the Plain ASCII file into the required input formats, ensuring maximum accuracy. The simulation was conducted on a cycle-by-cycle basis.

To ensure a precise correlation between numerical simulations and experimental benchmarks, a conversion methodology was implemented to translate the cycle-based output from NASGRO 4.0 into the operational 'flight' unit. This procedure is fundamental because, while the software executes calculations on a cycle-by-cycle basis, the aeronautical spectra are physically defined by discrete mission profiles. The conversion is based on the total number of reverse loading points (P_total) generated by the GENESIS software, where a single fatigue cycle is defined by the transition between two consecutive points, namely a peak and a valley. By establishing a linear proportionality between the simulated cycles and the spectral density, the fatigue life in flights (N_f) is mathematically determined by Equation 12:

$$N_f=\left[\frac{N_{sim}·F_{total}}{\raisebox{1ex}{$P_{total}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]$$ (12)

Where N_sim represents the cumulative cycles calculated by NASGRO, F_total is the total number of flights defined in the GENESIS spectrum, and P_total denotes the total number of reverse loading points, also established during the spectrum generation in GENESIS. The use of the floor function ensures a conservative structural integrity approach, as it considers only fully completed mission profiles, disregarding decimal fractions that lack physical meaning in an operational context.

Fatigue crack propagation simulations under Variable Amplitude Loading (VAL) were conducted using three Fatigue Crack Growth (FCG) analysis software packages widely recognized in the scientific community and the aerospace industry. The framework included NASGRO 4.0 (Fracture Analysis Software), AFGROW (Air Force Grow), and CRACK 2000 (Structural Integrity Program). These dedicated FCG packages utilize built-in libraries of Stress Intensity Factors (K) for idealized geometries⁵².

The NASGRO equation, also known as the Forman equation (full version), was adopted as the fundamental model for calculating the crack growth rate (da/dN) across all simulation platforms⁵³. Although the general expression, which accounts for the three regions of the propagation curve, remains the same, notable differences exist in implementation across the software, particularly in the calculation of the crack propagation threshold (ΔK_th), which varies among NASGRO 4.0, AFGROW, and CRACK 2000.

To ensure a methodologically valid comparison, the material properties and the empirical constants of the propagation equation (such as C, n, p, q, K_c) were rigorously harmonized. It is important to distinguish between the intrinsic material constants of the NASGRO equation (C, n, p, q, ΔK₀, K_c), which remained fixed across all simulations to ensure consistency, and the interaction model parameters (m for Wheeler, R_so for GW, and Φ₀ for MGW). The latter are not direct software defaults but empirical variables that require calibration against experimental data to account for the plasticity-induced effects specific to each loading history.

The NASGRO 4.0 database served as the baseline, with its values manually entered into AFGROW and CRACK 2000. This approach ensures that variations in the results arise primarily from differences in the implementations of the retardation models and the ΔK_th calculation, rather than from discrepancies in the default material properties of each software.

To quantify the complex effects of load interaction, multiple plasticity-induced retardation models were systematically evaluated and compared. The models employed included the Wheeler model, the Generalized Willenborg (GW) model, and the Modified Generalized Willenborg (MGW) Model. Additionally, a simulation without an interaction model (Non-Interaction or No Retardation) was included in all programs. This baseline case was essential for quantifying the magnitude of the propagation retardation effect, serving as a reference for analyzing the effectiveness of the interaction models. The implementation of retardation models across the three software packages followed the distribution summarized in Table 2.

Table 2
Load interaction models employed for each program.

Loading interaction Models	NASGRO 4.0	AFGROW	CRACK 2000
Non-interaction	X	X	X
Wheeler model		X	X
GW	X	X	X
MGW	X

• Source: The authors.

The calibration procedure for the load interaction models involved systematically adjusting empirical parameters to optimize life prediction⁵⁴. The retardation models used included Wheeler's, the Generalized Willenborg (GW), and the Modified Generalized Willenborg (MGW). The calibrated parameters were (m) for the Wheeler model, (R_so) for the GW, and (Φ₀) for the MGW. This adjustment was performed via multiple simulations for each loading spectrum type. For the retardation parameters, a precision of one decimal place was adopted, as is standard practice in the literature.

The calibration strategy was centralized to isolate the effect of each software's implementation⁵⁵. The GW and MGW models were calibrated primarily in NASGRO 4.0, and the Wheeler model in AFGROW. The resulting optimal parameters (e.g., R_so = 2.3) were then applied in the other programs. This approach allows comparison not of the 'best fit' of each software, but rather of how the different implementations (AFGROW, CRACK 2000) handle the same physical retardation parameter calibrated on a reference platform.

The criterion for a successful calibration focused on obtaining the simulated curve that most closely approximated the experimental data. However, maintaining a conservative approach was essential, adopting curves that predicted a residual life slightly inferior to the experimentally tested life. Drastically non-conservative predictions invalidate the model. Consequently, the calibration aimed to optimize the parameters to ensure accurate and safe results in correlation with the experiments⁵⁶.

For the analysis of the real aeronautical commercial history load, for which no experimental reference data are available, an exploratory approach was adopted. Due to operational similarities, the calibration parameters (Wheeler, GW, MGW) obtained for the TWIST spectrum were extrapolated to the industrial spectrum. Additionally, a numerical analogy was sought by comparing the real aeronautical commercial history load results with those of a TWIST spectrum modified to a 90 MPa mean stress in NASGRO 4.0. The calibration procedure involved a systematic iteration of empirical parameters to ensure the simulated curves followed a conservative approach relative to experimental data, as exemplified in Figure 2. This adjustment followed an iterative trial-and-error methodology for each loading spectrum. For a given spectrum (e.g., TWIST), multiple simulations were executed by varying the interaction parameter (e.g., incrementing R_so by 0.1) until the numerical crack growth curve converged toward the experimental benchmark. The load patterns themselves serve as the input for this optimization, as the interaction between cycles determines the required magnitude of the retardation parameter.

Figure 2
a) Example of the calibration process using (R_so 2.2) for the GW model and (Φ₀ = 0.9) for the MGW model - Curves with better final results, but not conservative. b) Example of the calibration process using (R_so 2.3) for the GW model and (Φ₀ = 0.8) for the MGW model - Conservative curves adopted.

4. Results and Discussion

Disregarding load interaction effects (Non-Interaction model) is fundamentally inadequate for damage tolerance analysis, as confirmed by the initial results. As presented in Table 3, the life predictions were drastically non-conservative for all analyzed spectra. This inadequacy was exacerbated in severe spectra. For FALSTAFF, the experimental life (Test 1: 4088 flights) contrasted sharply with the NASGRO 4.0 prediction (1425 flights), a 65.1% reduction. The most significant percentage disparity observed was in the Mini-FALSTAFF spectrum, where the same software simulated a life 74.4% shorter than the experimental test. These baseline results prove that retardation phenomena are not secondary effects but rather the governing mechanisms of propagation under VAL.

Table 3
Comparison of experimental fatigue lives (flights) with numerical predictions neglecting retardation effects.

Load Spectra	Experimental life	AFGROW Non-Interaction	NASGRO 4.0 Non-Interaction	CRACK 2000 Non-Interaction
TWIST	2936	2565	2458	2541
Mini-TWIST	9234	4501	4239	4480
FALSTAFF	4088	1612	1425	1529
Mini-FALSTAFF	5451	1525	1391	1529
real history load	-	1075	1008	1057

• Source: The authors.

The TWIST-type spectra, representative of commercial transport aircraft, required retardation parameters of R_so = 3.9 (GW) and Φ₀ = 0.4 (MGW). The simulations in NASGRO 4.0, visualized in Figure 3c, demonstrated excellent correlation with the experimental data. The simulated curves adhered to the methodological criterion of conservatism, with predicted lives (GW: 2809 flights; MGW: 2748 flights) slightly lower than the test (2936 flights).

Figure 3
a) Representation of the TWIST Loading Spectrum, S_avg = 80 MPa Mean Stress. b) Comparison of Experimental Data with CRACK 2000 Predictions for TWIST Loading, using calibrated Wheeler (m = 0.1) and GW (R_so = 3.9) models. c) Comparison of Experimental Data with NASGRO 4.0 Predictions for TWIST Loading, using calibrated GW (R_so 3.9) and MGW (Φ₀ = 0.4) models. d) Comparison of Experimental Data with AFGROW Predictions for TWIST Loading, using calibrated Wheeler (m = 0.1) and GW (R_so = 3.9) models.

The comparison between the TWIST (Figure 3) and the Mini-Twist (Figure 4) revealed the reason for this behavior. The simple omission of low-amplitude cycles in the Mini-Twist resulted in an increase of over 300% in fatigue life (from 2936 to an average of 9234 flights). This drastic difference demonstrates that, for this type of spectrum, low-amplitude cycles—many of which manifest as compressive underloads—contribute significantly to the damage. The physical mechanism is the attenuation of retardation: these compressive excursions effectively 'wash out' or erode the beneficial residual stress fields generated by prior overloads⁵⁷. This explains why, for the complete TWIST spectrum, the MGW model (which explicitly accounts for the acceleration induced by underload excursions) correctly predicted a shorter life than the GW model, which neglects the interaction between low-amplitude cycles and the residual stress state.

Figure 4
a) Representation of the Mini-TWIST Loading Spectrum, S_avg = 80 MPa Mean Stress. b) Comparison of Experimental Data with CRACK 2000 Predictions for Mini-TWIST Loading, using calibrated Wheeler (m = 0.6) and GW (R_so = 2.3) models. c) Comparison of Experimental Data with NASGRO 4.0 Predictions for Mini-TWIST Loading, using calibrated GW (R_so 2.3) and MGW (Φ₀ = 0.8) models. d) Comparison of Experimental Data with AFGROW Predictions for Mini-TWIST Loading, using calibrated Wheeler (m = 0.6) and GW (R_so = 2.3) models. Source: The authors.

The FALSTAFF spectra (Figure 5a), which simulate the severe loadings of fighter aircraft, exhibited the most pronounced retardation effect. The calibrated parameters were R_so = 2.7 (GW) and Φ₀ = 0.7 (MGW). Figure 5c illustrates why the models are necessary: the "Non-Interaction" curve (1425 flights) catastrophically fails to predict the experimental life (Test 1: 4088 flights). This proves that the propagation is governed by the intense retardation induced by the severe and frequent overloads typical of this spectrum.

Figure 5
a) Representation of the FALSTAFF Loading Spectrum, S_max = 200 MPa Mean Stress. b) Comparison of Experimental Data with CRACK 2000 Predictions for FALSTAFF Loading, using calibrated Wheeler (m = 0.7) and GW (R_so = 2.7) models. c) Comparison of Experimental Data with NASGRO 4.0 Predictions for FALSTAFF Loading, using calibrated GW (R_so= 2.7) and MGW (Φ₀ = 0.7) models. d) Comparison of Experimental Data with AFGROW Predictions for FALSTAFF Loading, using calibrated Wheeler (m = 0.7) and GW (R_so = 2.7) models.

In direct contrast to TWIST, the omission of low-amplitude cycles (Mini-FALSTAFF, Figure 6) had an insignificant influence on the total life. The experimental lives of FALSTAFF (~4088 to 5572 flights) and Mini-FALSTAFF (~5451 flights) are of the same order of magnitude. This suggests that the overloads predominantly govern the damage, and the retardation they induce is so intense that the subsequent smaller cycles are largely ineffective in attenuating it. Corroborating this physical behavior, the MGW model predicted longer lives than the GW model, indicating that, in this high-stress regime, underloads have less influence on retardation attenuation⁵⁸.

Figure 6
a) Representation of the Mini-FALSTAFF Loading Spectrum, S_max = 200 MPa Mean Stress. b) Comparison of Experimental Data with CRACK 2000 Predictions for Mini-FALSTAFF Loading, using calibrated Wheeler (m = 1.0) and GW (R_so = 2.3) models. c) Comparison of Experimental Data with NASGRO 4.0 Predictions for Mini-FALSTAFF Loading, using calibrated GW (R_so 2.3) and MGW (Φ₀ = 0.9) models. d) Comparison of Experimental Data with AFGROW Predictions for Mini-FALSTAFF Loading, using calibrated Wheeler (m = 1.0) and GW (R_so = 2.3) models.

A central finding of this study is the direct comparison between the software packages using unified calibration parameters (Tables 4 -7). The observed divergences in life predictions, therefore, isolate the methodological cause: the distinct internal implementations of each program. The discrepancy is most evident in the Mini-Twist (Table 5): using the same physical parameter R_so = 2.3, the predicted life varied from 7976 flights (NASGRO 4.0) and 7360 flights (AFGROW) to only 5224 flights (CRACK 2000).

Table 4
Calibration Results for Retardation Models under TWIST Spectrum Loading (Life in Flights) across Different Software Platforms.

Retardation Models	AFGROW	NASGRO 4.0	CRACK 2000
Non-Interaction	2565	2458	2541
GW ( Rso = 3.9)	2825	2809	2589
MGW (Φ0 = 0.4)	-	2748	-
Wheeler (m = 0.1)	2761	-	2697

• Source: The authors.

Table 5
Calibration Results for Retardation Models under Mini-TWIST Spectrum Loading (Life in Flights) across Different Software Platforms.

Retardation Models	AFGROW	NASGRO 4.0	CRACK 2000
Non-Interaction	4501	4239	4480
GW ( Rso = 2.3)	7360	7976	5224
MGW (Φ0 = 0.8)	-	7529	-
Wheeler (m = 0.6)	7524	-	6787

• Source: The authors.

Table 6
Calibration Results for Retardation Models under FALSTAFF Spectrum Loading (Life in Flights) across Different Software Platforms.

Retardation Models	AFGROW	NASGRO 4.0	CRACK 2000
Non-Interaction	1612	1425	1636
GW ( Rso = 2.7)	3773	3597	2603
MGW (Φ0 = 0.7)	-	3744	-
Wheeler (m = 0.7)	3832	-	3112

• Source: The authors.

Table 7
Calibration Results for Retardation Models under Mini-FALSTAFF Spectrum Loading (Life in Flights) across Different Software Platforms.

Retardation Models	AFGROW	NASGRO 4.0	CRACK 2000
Non-Interaction	1525	1393	1539
GW ( Rso = 2.3)	4573	4773	3207
MGW (Φ0 = 0.9)	-	4913	-
Wheeler (m = 1.0)	4773	-	3472

• Source: The authors.

This variation is primarily attributed to the different formulations for calculating the crack propagation threshold (ΔK_th). CRACK 2000, which uses a specific formulation, consistently produced the most conservative results across all analyses. Conversely, NASGRO 4.0 demonstrated the most accurate predictions, aligned with experimental data in most scenarios. NASGRO 4.0, for example, employs a complex formulation based on Elber's closure function (parameter f), which adjusts ΔK_th based on the R-ratio and Kmax. In contrast, other codes may implement simpler ΔK_th models or different ΔK_eff (effective) formulations. The distinct calibration values obtained for different load patterns (e.g., R_so = 3.9 for TWIST vs. R_so = 2.7 for FALSTAFF) demonstrate that these retardation models are strictly phenomenological and highly spectrum-dependent. This implies that the optimal parameters are not universal material constants, but rather empirical fitting variables required to align the theoretical yield-zone interaction with the specific plasticity-induced retardation effects of each loading history. Consequently, predictive extrapolation is reliable only within the same class of operational spectra, as the physical interaction between overloads and the residual stress field is uniquely governed by the sequence of the applied cycles. This implies that the optimal parameters are not universal constants of the material, but rather fitting variables that adjust the theoretical yield zone to the specific interaction effects of each loading history. Consequently, predictive extrapolation is reliable only when applied within the same class of operational spectra (e.g., transport vs. transport), as direct transferability between disparate mission profiles (e.g., fighter vs. transport) is physically invalid.

The analysis of the real aeronautical commercial history load served as a preliminary methodological exercise. By applying the calibrated parameters from the TWIST spectrum R_so = 3.9; Φ₀ = 0.4), the predicted lives presented in Table 8 were obtained.

Table 8
Fatigue Life Predictions (Flights) for the real aeronautical commercial history load, with Comparative Results for the TWIST Spectrum (S_avg = 90 MPa).

Retardation Models	AFGROW	NASGRO 4.0	CRACK 2000
Non-Interaction	1075	1008	1057
GW ( Rso = 3.9)	1470	1666	1176
MGW (Φ0 = 0.4)	-	1762	-
Wheeler (m = 0.1)	1170	-	1144

• Source: The authors.

Figure 7c demonstrates a numerical approximation by comparing the industrial spectrum with a TWIST spectrum (S_avg = 90 MPa). This approach provides an initial estimate but lacks experimental validation. Therefore, the results for this spectrum must be treated as speculative until test data become available, serving here to illustrate the extrapolation methodology.

Figure 7
a) Representation of the real aeronautical commercial history load, S_avg = 90 MPa Mean Stress. b) Comparison of Experimental Data with CRACK 2000 Predictions for real aeronautical commercial history load and TWIST Loading, using calibrated Wheeler (m = 0.1) and GW (R_so = 3.9) models. c) Comparison of Experimental Data with NASGRO 4.0 Predictions for real aeronautical commercial history load and TWIST Loading, using calibrated GW (R_so = 3.9) and MGW (Φ₀ = 0.4) models. d) Comparison of Experimental Data with AFGROW Predictions for real aeronautical commercial history load and TWIST Loading, using calibrated Wheeler (m = 0.1) and GW (R_so = 3.9) models. Source: The authors.

5. Conclusion

The systematic study demonstrated that the physics of load interaction governs fatigue life and that its influence is fundamentally dependent on the loading spectrum history. The disregard for these effects leads to drastically non-conservative and invalid life predictions, reinforcing the criticality of including retardation models.

It was proven that the spectra exhibit distinct physical behaviors. In TWIST spectra, low-amplitude cycles are damaging because they frequently incorporate underload events that attenuate retardation; this is demonstrated by the Mini-TWIST spectrum (which omits these cycles), resulting in a life more than three times longer. In contrast, for FALSTAFF, the damage is governed by severe overloads, with the influence of low-amplitude cycles being insignificant to the fatigue life.

The work’s primary contribution is to demonstrate that the optimal calibration parameters (m, R_so, Φ₀) are not universal and depend critically on the loading spectrum class. This is evidenced by the specific values required for different mission profiles, such as R_so = 3.9 for transport spectra (TWIST) and R_so = 2.7 for fighter spectra (FALSTAFF). This finding confirms that the robust application of these models in the aerospace industry necessitates spectrum-specific calibration to ensure the accuracy and conservatism required for damage tolerance assessments.

In the inter-software comparison, NASGRO 4.0 provided the most accurate predictions, while CRACK 2000 consistently proved the most conservative. This divergence is primarily due to the distinct internal formulations used to calculate the crack propagation threshold (ΔK_th) in each program.

The study reinforces the importance of the adopted calibration strategy, which prioritized the conservatism criterion—essential for damage tolerance—over a 'best fit' that could be unsafe.

Finally, the methodology was extrapolated to the real aeronautical commercial history load. The analogy with the TWIST spectrum (S_avg = 90 MPa) permitted a preliminary life estimation. However, it is reiterated that this analysis is purely numerical. Experimental validation of this proprietary spectrum is recommended for future work to rigorously ratify the calibration parameters estimated herein.

• Data Availability
  All the data supporting the results of this study were published in the article itself.

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Edited by

• Associate Editor:
  Aloisio Klein.

• Editor-in-Chief:
  Luiz Antonio Pessan.

Data availability

All the data supporting the results of this study were published in the article itself.

Publication Dates

• Publication in this collection
  05 June 2026
• Date of issue
  2026

History

• Received
  31 Oct 2025
• Reviewed
  23 Apr 2026
• Accepted
  04 May 2026