Prediction Model for Low-Cycle Fatigue Life of Cast TiAl Alloys Based on Defect Stress Concentration Effects

Abstract

Internal defects cause significant fluctuations in the dispersion of low-cycle fatigue life of titanium–aluminum alloy specimens under fully reversed strain control ($R_{\epsilon }=-1$) at room temperature. To accurately analyze the relationship between internal defects and low-cycle fatigue behavior, this study adopts an energy-based approach to investigate the variation patterns of plastic strain energy density (PSE) during low-cycle fatigue testing of specimens. Research has revealed that the decline process of plastic strain energy dissipation is distinctly divided into two stages, and the low-cycle fatigue life exhibits a pronounced nonlinear relationship with the plastic strain energy dissipation rate (PSEDR) during the first stage. Based on internal defect characteristics obtained from X-ray scans, a defect intensity parameter $K_{t\_micro}$ was proposed to establish a life prediction interval under the influence of internal defects. By correcting the stable plastic strain energy using stress concentration factors, the prediction error of the stable plastic strain energy model was reduced from a 3× error band to a 1.5× error band. The maximum relative error decreased from 132% to 34.30%, significantly narrowing the overall prediction error. Compared with the Mason–Coffin (M-C) model and the stable plastic strain energy model, the prediction accuracy is significantly improved.

1. Introduction

With the growing demand for high-performance structural materials in aerospace, automotive, and energy sectors, lightweight properties, high strength, and high-temperature resistance have become key indicators for evaluating advanced materials’ performance [1,2]. Among numerous emerging materials, titanium–aluminum intermetallic compounds stand out for their exceptional comprehensive properties. These alloys exhibit a density (approximately 4–5.9 g/cm³) significantly lower than that of traditional nickel-based high-temperature alloys (approximately 8 g/cm³), while simultaneously offering extremely high specific strength and specific stiffness, outstanding high-temperature creep resistance, and excellent oxidation and corrosion resistance. Titanium–aluminum alloys, owing to their outstanding lightweight properties and high specific strength, have become the ideal material for manufacturing automotive turbocharger rotors. Currently, such complex structural components primarily rely on casting processes for production. Although significant progress has been made in numerical simulation techniques for casting processes [3,4,5], process-related defects such as micro-pores, shrinkage porosity, inclusions, and tearing remain difficult to completely avoid in actual manufacturing. These defects not only compromise the reliability of the material’s mechanical properties but also significantly reduce its fatigue life under cyclic loading [6,7]. This severely limits the durability and safety of components operating in high-temperature, high-stress environments [8,9,10,11,12]. Internal defects within materials induce significant stress concentrations during cyclic loading, serving as primary initiation points for fatigue crack growth. This drastically reduces fatigue life and may even lead to catastrophic structural failure, resulting in severe safety incidents [13,14]. Therefore, accurately predicting the fatigue life of defect-containing titanium–aluminum alloys is a core issue for ensuring their safe and reliable engineering applications.

In recent years, incorporating defect characteristics into fatigue life prediction models has become research hotspots. Scholars have discovered that the size, type, location, and morphology of volumetric defects are the four key parameters influencing material fatigue properties [15,16]. Murakami [17] first employed the square root of the internal defect area as an influencing factor to modify the fatigue limit of materials. Ryota et al. [18] described the characteristic variations of defects using probability density distribution functions, treating defects as initial cracks and applying the Pairs-law to predict the fatigue life of crack initiation. Chi et al. [19] discussed the competition mechanism between artificial surface defects and internal manufacturing defects based on defect size and stress intensity factor ranges, modeling the projected area of artificial surface defects. Zhu et al. [20] introduced the Z parameter in stress-controlled high-cycle fatigue analysis, incorporating micro-defect location and micro-defect area factor, revealing the micro-structural dependency of internal fatigue cracking. Hu et al. [21] proposed the P-parameter model based on defect characteristics, building upon the Mason–Coffin model. This model controls the prediction error for the lifetime of defective materials within a 1.5× error band. Leven [22] defined the stress concentration factor as the ratio of the maximum stress to the nominal stress through bending tests on notched flat steel, demonstrating that geometric discontinuities significantly amplify local stresses. Afazov et al. [23] proposed a defect factor to quantify local stress concentration in materials by combining defect location and size with a notch sensitivity coefficient. Serrano-Munoz et al. [24] investigated natural shrinkage and artificial defects, observing that shape is a factor to consider regardless of defect size, location, or nature and that the earliest crack nucleation occurred in regions with the highest $K_t$ values.

Research has revealed that plastic strain energy represents the energy dissipated by a material during each cycle due to plastic deformation. It is directly linked to irreversible damage mechanisms at the microscopic level, such as dislocation motion and slip zone formation, and is considered a parameter that more closely reflects the physical essence of damage [25]. Unlike conventional damage mechanics, which typically rely on macroscopic phenomenological models and equivalent assumptions, the energy approach evaluates damage based on cyclic plastic energy dissipation. This method more directly reflects the actual energy evolution and damage-driving mechanisms around defects. Therefore, the energy approach provides stronger physical justification and predictive accuracy in capturing local stress–strain concentrations and nonlinear damage accumulation caused by microscopic defects [26,27,28,29]. Although previous studies have investigated the stress concentration effects of defect characteristics within materials [22,23] and the application of stable plastic strain energy models in predicting low-cycle fatigue life [18,26,27,28,29,30]; however, research integrating defect-induced local plastic strain energy variations into stable plastic strain energy models, establishing models that explicitly reflect the impact of internal defects on fatigue life, and achieving accurate low-cycle fatigue life prediction for defect-containing materials remains relatively scarce [31].

The inherent performance variability of engineering materials and the uncertainty of service parameters significantly impact the accuracy of structural fatigue life predictions. Sarfarazi et al. [32] developed a physics-based information evaluation framework addressing uncertainties, proposing the use of physical constraints, uncertainty treatment, and evidence-based decision-making logic to effectively mitigate model prediction deviations, thereby providing a theoretical foundation for incorporating defect correction coefficients. Furthermore, addressing the challenges of mechanical modeling with limited samples and uncertain parameters, Sarfarazi et al. [33] reviewed the applications of artificial intelligence, physics-based information agent models, and interpretable algorithms in the dynamic characterization of composite structures, offering insights for quantifying and reducing parameter variability while enhancing the reliability of fatigue life models.

To achieve this, this paper aims to further elucidate the mechanism by which microstructural defect characteristics influence fatigue life dispersion. Based on internal defect data obtained through non-destructive testing techniques, a stress concentration factor reflecting defect effects was established and incorporated into the plastic strain energy life prediction model. This approach enables quantitative characterization of fatigue life prediction dispersion bands under different defect conditions, significantly enhancing the prediction accuracy and engineering applicability of fatigue life for materials containing internal defects.

2. Experiments and Methods

2.1. Test Specimen

To obtain the basic mechanical properties of titanium–aluminum materials, tensile tests at room temperature and low-cycle fatigue tests under symmetrical loads at room temperature ($R_{\epsilon }=-1$) were conducted in accordance with ASTM E8M-04 and ASTM E606 standards. The parent material of the test specimens is shown in Figure 1, and the processing drawings for the tensile test specimens and low-cycle fatigue test specimens are shown in Figure 2. The chemical composition of titanium–aluminum alloy is shown in

Table 1. Chemical composition of titanium–aluminum alloy.

Element	Ti	Al	V	Cr	Zr
Nominal composition	Balance	33.8%	3.0%	1.0%	0.5%

.

2.2. Experimental Equipment

To obtain the stress–strain curve of the specimen, static tensile tests were conducted using an electronic tensile testing machine, as shown in Figure 3a. LCF tests were conducted using uniaxial symmetric load fatigue tests under strain control. The test equipment used was the LFV-100HH thermomechanical fatigue testing machine, as shown in Figure 3b. Fatigue tests were conducted in strain control with triangular waveforms at a strain rate of 0.01/s. Failure was defined as a 25% drop in tensile stress or complete fracture, using strain amplitudes of 0.5%, 0.45%, and 0.4%.

2.3. Experimental Results

The yield strength of the material is defined as the engineering stress corresponding to ${\epsilon }_p=0.2\%$. The mechanical properties, including ultimate tensile strength and elastic modulus are listed in

Table 2. Tensile properties of titanium–aluminum alloys at room temperature.

Temperature (°C)	Elastic Modulus ($\mathbf{G}\mathbf{P}\mathbf{a}$)	Yield Strength (MPa)	Tensile Strength (MPa)	Elongation Ratio (%)
25	160.56	308.9	424.5	25.02

.

Table 3. Low-cycle fatigue properties of titanium–aluminum alloys at room temperature.

$\mathbf{I}\mathbf{D}$	${\mathit{\epsilon }}_{\mathit{a}}$/2(%)	${\mathit{\epsilon }}_{\mathit{e}\mathit{a}}$/2(%)	${\mathit{\epsilon }}_{\mathit{p}\mathit{a}}$/2(%)	$\Delta {\mathit{\sigma }}_{\mathit{a}}$ ($\mathbf{M}\mathbf{P}\mathbf{a}$)	${\Delta \mathit{W}}_{\mathit{p}\_0.5{\mathit{N}}_{\mathit{f}}}$ ($\mathbf{M}\mathbf{J}/{\mathbf{m}}^3·\mathbf{C}\mathbf{y}\mathbf{c}\mathbf{l}\mathbf{e}$)	$\mathit{E}$ ($\mathbf{G}\mathbf{P}\mathbf{a}$)	${\mathit{N}}_{\mathit{f}}$
LCF-01	0.500	4.02 × 10−3	9.36 × 10−4	512	2.3069	127	33
LCF-03	0.450	4.14 × 10−3	3.29 × 10−4	458	1.1373	110	202
LCF-04	0.450	4.30 × 10−3	1.67 × 10−4	471	0.9412	109	390
LCF-05	0.450	4.04 × 10−3	4.23 × 10−4	445	1.2166	110	85
LCF-06	0.400	3.65 × 10−3	2.98 × 10−4	440	0.6517	120	288
LCF-07	0.400	3.91 × 10−3	6.52 × 10−4	450	0.5931	115	1001
LCF-08	0.400	3.93 × 10−3	5.31 × 10−4	479	0.4735	122	1478

presents the low-cycle fatigue test results obtained under multiple strain amplitude conditions. Post-fracture macroscopic observation confirmed that fatigue cracks of the tested specimens initiated and fractured exactly within the scanned central gauge section, rather than at the transition region or gauge-length edges. Key parameters listed include total strain amplitude ${\Delta \epsilon }_a/2$, elastic strain amplitude $\Delta {\epsilon }_{ea}/2$, plastic strain amplitude $\Delta {\epsilon }_{pa}/2$, stress amplitude ${\sigma }_a$, stable plastic strain energy $\Delta W_{p-0.5N_f}$, cyclic elastic modulus $E$, and ultimate fatigue life $N_f$. As shown in Table 3, even under identical total strain amplitude conditions, the low-cycle fatigue life of each specimen still exhibits significant dispersion characteristics. This phenomenon is presumed to be related to factors such as micro-structural heterogeneity within the material, initial defect distribution, or localized stress concentration. It reflects the sensitivity of fatigue damage evolution to micro-structural features and highlights the uncertainty challenges encountered in fatigue life prediction. In addition, the elastic response measured in monotonic tension typically differs from the effective cyclic stiffness obtained in low-cycle fatigue, because cyclic loading facilitates damage initiation and evolution, which reduces the effective modulus under fatigue conditions. Furthermore, the fatigue failure of specimen LCF-02 occurred in the transition section rather than the designed effective working section. Thus, the corresponding test data are invalid, and this data point is eliminated.

3. Lifespan Dispersion Analysis and CT Detection

3.1. Energy Dissipation Analysis

The energy conversion mechanisms during low-cycle fatigue testing are illustrated in Figure 4. Under the external load applied by the testing machine, mechanical work is continuously input into the specimen. According to the thermodynamic principles of inelastic deformation, this total plastic work is partitioned into two primary components [31]: a large fraction is dissipated as thermal energy (manifested through heat conduction and convection with the surrounding environment), while the remaining fraction is converted into stored internal energy within the material. The quantitative energy balance between these components is described by Equation (1). The accumulated stored internal energy acts as the fundamental driving force for microstructural evolution, specifically manifested as the nucleation, motion, and continuous proliferation of dislocations that gradually form slip bands. Although only a portion of the mechanical work is stored, the low strain rate ($0.01 {\mathrm{s}}^{-1}$) applied in this study ensures that the generated heat is effectively dissipated, maintaining an essentially isothermal condition. Consequently, rather than isolating the exact stored energy, the macroscopic total plastic strain energy can be robustly utilized as a phenomenological metric to represent the overall fatigue damage driving force. As the cyclic loading progresses, when the accumulated plastic strain energy reaches the material’s critical energy dissipation capacity (threshold), macroscopic failure occurs, as formulated in Equations (2) and (3).

During low-cycle fatigue testing, specimens are subjected to alternating stresses exceeding their yield strength, leading to a significant increase in internal dislocation density accompanied by the formation and evolution of numerous slip bands. Under these conditions, energy dissipation is primarily dominated by energy storage, with a relatively minor proportion dissipated through thermal means. Furthermore, since the temperature difference between the specimen surface and the ambient environment during testing is insignificant, the heat loss effect can be neglected in energy analysis. It can be approximated that the mechanical energy input from the external environment is entirely converted into the specimen’s internal energy [31]. Based on this assumption, Equation (1) can be simplified to the energy conversion relationship expressed in Equation (4), thereby providing a theoretical basis for subsequent fatigue damage analysis.

$$W=Q_{CV}+Q_{TR}+Q_{CD}+{\Delta U}_S$$

(1)

$$W_f={\displaystyle \sum _{i=1}^N}{\Delta W}_i$$

(2)

$$D_m={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N}{\Delta W}_i}{W_f}}$$

(3)

$$W={\Delta U}_S$$

(4)

Among these, $W$ represents mechanical energy input from outside, $Q_{CV}$ represents heat dissipation, $Q_{TR}$ represents heat radiation, $Q_{CD}$ represents heat conduction, and ${\Delta U}_S$ represents stored internal energy. $D_m$ represents the cumulative damage to the material. $W_f$ is the energy dissipation threshold. $\Delta W_i$ represents the plastic strain energy absorbed by the specimen per cycle.

The LFV-100HH thermomechanical fatigue testing machine records the hysteresis curve of the specimen during each cycle of low-cycle fatigue testing. The stress–strain hysteresis loop is a typical manifestation of the work performed by a material’s microstructure as it resists external cyclic loading during fatigue testing, as shown in Figure 5. According to the Ramberg–Osgood equation [34], total strain consists of elastic strain and plastic strain. The characteristics of the stress–strain response process are shown in Equation (5). Plastic strain energy is a physical parameter used to quantify the area enclosed by the stress–strain hysteresis loop [29,30]. The integral formula is given by Equation (6).

$${\displaystyle \frac{{\Delta \epsilon }_a}{2}}={\displaystyle \frac{{\Delta \epsilon }_e}{2}}+{\displaystyle \frac{{\Delta \epsilon }_p}{2}}={\displaystyle \frac{{\Delta \sigma }_a}{2E}}+{\left({\displaystyle \frac{{\Delta \sigma }_a}{2K^{\prime }}}\right)}^{1/n^{\prime }}$$

(5)

$${\Delta W}_i={\oint }_{Loop}\sigma d\epsilon$$

(6)

Integrating the area of each hysteresis loop yields the change in internal energy per cycle for the specimen, thereby revealing the overall energy trend throughout the life cycle. The energy variation trends of the specimen under different strain amplitudes are shown in Figure 6. By analyzing the results in Figure 6, the following key phenomena can be summarized: (i) Under identical strain amplitude conditions, the plastic strain energy accumulated by each specimen during the initial cycles was essentially consistent, demonstrating high repeatability; (ii) despite similar initial plastic strain energies, the fatigue lifetimes of same strain amplitude specimens exhibited significant variations, indicating high sensitivity of fatigue life to internal microstructural features or defects; (iii) a clear positive correlation was observed between strain amplitude and initial plastic strain energy, suggesting strong coupling between these parameters in cyclic response.

3.2. Normalization Analysis

To investigate the intrinsic mechanism underlying the dispersion of specimen fatigue life under identical strain amplitude conditions and to further explore the relationship between cumulative plastic strain energy and fatigue life, this study systematically processed experimental data obtained during cyclic deformation of specimens at different strain amplitudes. By normalizing the number of cycles relative to failure life and the plastic strain energy relative to the energy dissipation threshold, a normalized life-strain energy relationship curve is obtained, as shown in Figure 7. This normalization effectively eliminates the influence of absolute numerical scales, revealing more clearly the unified behavior characteristics of plastic strain energy evolution with life under different loading conditions.

As observed from the normalized results shown in Figure 7, the plastic strain energy per cycle of the specimens exhibits a pronounced decay trend with increasing fatigue cycle count. Furthermore, significant differences exist in the PSEDR among different specimens. Further analysis indicates that specimens with lower PSEDR typically demonstrate shorter fatigue lifetimes. This pattern remains consistent across different strain amplitude conditions, underscoring its universality and significance. This study finds that the divergence in plastic strain energy dissipation among various specimens mainly emerges before the curve slope drops to 0.3, which corresponds to the initial 30% of fatigue life. This interval is hereby defined as the rapid decline phase of plastic strain energy, as illustrated in Figure 7a. Thereafter, the attenuation rate of plastic strain energy gradually stabilizes and maintains a steady variation state, and this stage is defined as the steady attenuation stage of plastic strain energy. To further explore the correlation between plastic strain energy dissipation rate (PSEDR) and fatigue behavior, this study performs systematic fitting analysis on the PSEDR values in the rapid attenuation stage under various strain amplitudes. The fitting results are presented in Figure 7b–d, and the detailed calculated data are summarized in

Table 4. PSEDR at different strain magnitudes.

	${\mathit{\epsilon }}_{\mathit{a}} = 0.5\mathbf{\%}$	${\mathit{\epsilon }}_{\mathit{a}} = 0.45\mathbf{\%}$	${\mathit{\epsilon }}_{\mathit{a}} = 0.4\mathbf{\%}$
PSEDR	0.267	0.347	0.550
		0.4639	0.756
		0.610	1.019

.

Fitting the energy decay rate during the rapid decrease phase of plastic strain energy reveals a clear positive correlation between the decay rate and fatigue life. Specimens exhibiting a higher PSEDR during this phase demonstrate longer fatigue life, as shown in Figure 8. It can thus be concluded that the PSEDR during the strain energy decay phase of a specimen can predict its overall lifespan. The predictive model is shown in Equation (7), where $M_{ER}$ is the energy decay coefficient, and $N_{ER}$ is the energy decay exponent, and where $M_{ER}=1753.7$, $N_{ER}=2.891$, and ${dW}_i/d{N}_f$ represents the PSEDR.

As shown in

Table 5. Statistical fitting reliability indexes.

Parameter	${\mathbf{R}}^2$	RMSE	MAE
Value	0.9137	154.0065	88.9289

, this paper conducts a systematic reliability evaluation on the fitted curve of plastic strain energy density rate versus fatigue life. The coefficient of determination ${\mathrm{R}}^2$, goodness-of-fit statistics, and residual analysis are used to quantitatively characterize the fitting accuracy and statistical reliability of the model.

$$N_f={M_{ER} \left({\displaystyle \frac{d{W}_i}{d{N}_f}}\right)}^{N_{ER}}$$

(7)

The index i represents the serial number of each test specimen. According to the research by Toda et al. [35], defects present within materials undergo significant stress concentration at their tips or edges under cyclic loading, resulting in local stress in these regions that is substantially higher than the macroscopic nominal stress. This stress concentration effect promotes plastic deformation to occur preferentially around defect regions, accumulating progressively with increasing cycle count. Consequently, it significantly alters the distribution and dissipation mechanisms of energy within the material. Specifically, when defects exist within the material, energy dissipation ceases to be uniformly distributed and instead becomes highly concentrated within a finite volume near the defect. The localized concentration of plastic strain and energy within this region further accelerates the evolution of micro-damage [36].

Therefore, it is inferred that the presence of defects induces localized stress concentrations within the specimens, leading to significant differences in the PSEDR among different specimens under identical strain amplitudes. PSEDR serves as a crucial indicator for characterizing the evolution of damage in regions near defects, and its changes directly influence the overall fatigue life of materials. Based on the preceding theory of energy localization and damage concentration mechanisms, it can be concluded that the random distribution of internal defects within materials and the resulting stress concentration behavior are the primary factors causing dispersion in low-cycle fatigue life.

3.3. X-Ray Non-Destructive Testing

3.3.1. Equipment and Test Samples

To obtain defect information within the low-cycle fatigue specimens used in this experiment, X-ray non-destructive testing was performed on each low-frequency specimen. The scanning system used in this study was the nanoVoxel-4000 system, as shown in Figure 9. Based on the actual scanning requirements and the scanning range of the specimens, the scanning resolution selected for this study was 10 μm. Select the 12 $\mathrm{m}\mathrm{m}$ section at the exact center of the sample during scanning. The scanning area of the specimen and the three-dimensional reconstruction results after scanning are shown in Figure 10.

3.3.2. Defect Statistical and Correlation Analysis

Through systematic analysis of three-dimensional reconstructed defects within the specimens, it is evident that during the casting process of titanium–aluminum alloys, a significant number of pore-type defects formed internally due to high cooling rates or non-uniform cooling conditions. From the overall distribution perspective, apart from a few large defects with complex morphologies, the more prevalent defects are smaller in size, nearly spherical in shape, and clustered together. According to existing research [17,18,19], when evaluating the impact of microdefects on material fatigue properties, the defect characteristic parameters—size, shape, and spatial distribution—exert a significant influence on fatigue life. Therefore, in investigating the relationship between internal defects and fatigue life, this paper focuses on the following four characteristic parameters: (1) square root of defect area—$\sqrt{area}$, defined as the square root of the projected area of the defect on a plane perpendicular to the loading direction; (2) aspect ratio—$A_t$, defined as the ratio of the longest axis to the shortest axis of the defect; (3) sphericity—$S_p$, representing the ratio of the surface area of a sphere of equivalent volume to the actual surface area of the defect; (4) shape factor—$S_f$, used to characterize the irregularity of the defect’s geometric shape.

To visually demonstrate the statistical properties of the above defect characteristics, this paper employs histograms to visualize the distribution of each parameter, with the results shown in Figure 11. The defect area, shape factor, aspect ratio, and sphericity all follow nearly normal distributions. The mean and extreme values of the defect feature parameters are shown in

Table 6. Mean and Extreme Values of Defect Feature Parameters.

Defect Characteristics	$\sqrt{\mathit{a}\mathit{r}\mathit{e}\mathit{a}}$	${\mathit{A}}_{\mathit{t}}$	${\mathit{S}}_{\mathit{p}}$	${\mathit{S}}_{\mathit{f}}$
mean value	0.153	2.089	0.742	3.328
standard deviation	0.134	0.507	0.140	3.471
minimum value	2.585	8.199	1.041	91.217
maximal value	0.037	1.159	0.222	0.886

. This statistical characteristic provides crucial evidence for subsequent refinement of the defect-lifetime correlation model.

To investigate whether correlations exist among different defect characteristics, scatter plots were used to visualize the relationships between these features parameters. The results are shown in Figure 12. As observed in Figure 12a, a clear positive correlation exists between defect area and shape factor. With increasing defect area, the shape factor exhibits an overall upward trend, accompanied by a gradual increase in the dispersion of data points. This indicates that large-area defects tend to exhibit more irregular geometric shapes, characterized by higher randomness and variability. As shown in Figure 12b, no significant correlation was observed between defect area and aspect ratio. The aspect ratios of the vast majority of defects were concentrated within the range of 1 to 3, with their morphology approximating ellipsoidal shapes. Only a very small number of small-area defects exhibited aspect ratios exceeding 4, appearing as strip-like structures. As shown in Figure 12c, a significant correlation exists between defect area and sphericity. As defect area increases, sphericity values generally decrease, indicating that larger defects exhibit a greater deviation from spherical shape due to their higher surface-to-volume ratio and more complex morphology. Conversely, smaller defects tend to be more spherical, suggesting that their formation process may be dominated by the mechanism of minimizing interfacial energy. Figure 12d illustrates the interrelationship between area, shape factor, and sphericity. When the defect area is less than 0.5, the three parameters exhibit a highly stable relative relationship. When the defect area exceeds 0.5, the sphericity remains relatively stable with changes in defect area, but the shape factor exhibits significant dispersion.

4. Low-Cycle Fatigue Model

4.1. Plastic Strain Energy Density Model

Traditional fatigue life prediction methods, such as the strain-based Coffin–Manson relationship, although widely applied, cannot accurately reflect the damage mechanisms of internal defects within materials or the complex damage accumulation process. In recent years, energy-based approaches, particularly those utilizing plastic strain energy as a damage parameter, have garnered increasing attention. As one of the core methods for structural damage assessment, the PSE model quantifies the influence of localized plastic deformation and energy concentration on regional damage evolution, revealing the energy-driven mechanism linking plastic deformation irreversibility to damage progression. By calculating the plastic strain energy during low-cycle fatigue, a relationship between stable plastic strain energy and fatigue life was established, as shown in Equation (8).

$${\Delta W}_{p\_0.5N_f}=a{N_{f\_t}}^b$$

(8)

where ${\Delta W}_{p\_0.5N_f}$ is the stable plastic strain energy, $\mathrm{a}$ is the static toughness coefficient, $\mathrm{b}$ is the static toughness exponent, and $N_f$ is the fatigue life. Therefore, the stable plastic strain energy is calculated by extracting the hysteresis curve corresponding to the median life of each test, and then the model parameters are fitted using the plastic strain energy-life data. After fitting, the model parameters are $a$ = 7.7197, $b$ = −0.38. The life prediction model curve is shown in Figure 13, and the predicted life and prediction accuracy of the model are shown in

Table 7. Model prediction accuracy under different strain amplitudes.

$\mathbf{I}\mathbf{D}$	${\mathit{\epsilon }}_{\mathit{a}}$ (%)	${\mathit{N}}_{\mathit{f}\_\mathit{t}}$	${\mathit{N}}_{\mathit{f}\_\mathit{p}}$	$\mathbf{R}\mathbf{e}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{v}\mathbf{e} \mathbf{E}\mathbf{r}\mathbf{r}\mathbf{o}\mathbf{r}$ (%)
LCF-01	0.500	33	24	−27.24
LCF-03	0.500	202	154	−23.56
LCF-04	0.450	390	254	−34.84
LCF-05	0.450	85	129	52.13
LCF-06	0.450	288	668	132.13
LCF-07	0.400	1001	856	−14.32
LCF-08	0.400	1478	1548	4.79

. $N_{f\_t}$ represents the actual lifespan, while $N_{f\_p}$ represents the predicted lifespan.

Predicting low-cycle fatigue life using the traditional plastic strain energy method yielded prediction errors within a 2× error band for most points, representing a significant improvement over the M-C models. However, substantial prediction errors were still observed for some data points. Based on the preceding analysis of life dispersion, micro-defects within the material can cause stress concentration in localized areas. This results in the actual stress in these regions exceeding the nominal stress, leading to the actual energy applied to these localized areas surpassing the energy integrated from the hysteresis curve. Consequently, this causes deviations in life prediction using the plastic strain energy method. Therefore, this paper modifies the stable plastic strain energy from the perspective of defect stress concentration to predict the fatigue life of specimens significantly affected by internal defects.

4.2. Stress Concentration Factor

Zhu [20] et al. introduced the Z-parameter into stress-controlled fatigue analysis, accounting for the effects of volume defect location and volume defect area factor, thereby successfully enhancing the model’s predictive accuracy, as shown in Equation (9). Hu [21] proposed P-parameter model based on the M-C equation that accounts for defect location, size, and shape to modify fatigue life predictions from the M-C equation, as shown in Equation (10). Shen [37] introduced the concept of stress amplification factor, accounting for welding defects that occur in thin plates during the welding process, and validated the accuracy of this formula, as shown in Equation (11).

$$\mathrm{Z}={\mathsf{\sigma }}_{\mathrm{a}}{\left(\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\right)}^{1/12}{\mathrm{D}}^{\mathsf{\beta }}$$

(9)

$$\mathrm{P}={\mathrm{Y}\mathsf{\epsilon }}_{\mathrm{a}}{\left({\displaystyle \frac{\sqrt{\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}}{{\mathrm{R}}_{\mathrm{d}}}}\right)}^{\mathrm{m}}{\left({\displaystyle \frac{\mathrm{D}}{\mathrm{D}-{\mathrm{D}}_{\mathrm{d}}}}\right)}^{\mathrm{n}}$$

(10)

$$K_t={\sigma }_{notch}/{\sigma }_{nom}$$

(11)

Here, $\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}$ denotes the projected area of the defect on a plane perpendicular to the axial direction; $\mathrm{D}$ is the volume-defect region factor; and $\mathsf{\beta }$, $\mathrm{m}$, and $n$ are material constants.

To precisely characterize the influence mechanism of microscopic defect features on local plastic deformation and energy concentration behavior, this study introduces a defect intensity parameter based on defect morphology and distribution characteristics, as shown in Equation (12), building upon existing theoretical foundations. $L_s$ is the standard length, which is 1 μm. Under the influence of defect stress concentration factors, the traditional plastic strain energy model is modified to quantitatively describe the energy redistribution and localized concentration effects induced by internal defects. This enhances the accuracy of predicting specimen life under defect influence. The modified model is presented in Equation (13).

$$K_{t\_micro}=\gamma {\epsilon }_a{\left({\displaystyle \frac{\sqrt{area}}{L_s}}\right)}^p{\left(S_p\right)}^q$$

(12)

$${\Delta W}_{p\_0.5N_f}\times {\left(K_{t\_micro}\right)}^{\alpha }=a{N_f}^b$$

(13)

where $K_{t\_micro }$ is the defect intensity parameter, γ is the correction constant, α is the defect sensitivity exponent, $p,q$ is the material constant. Only the correction constant γ in the proposed formula is obtained by nonlinear fitting based on experimental data.

To explore the effect of defect selection interval on fitting accuracy, defects were randomly sampled via Latin hypercube sampling from the full set, top 50%, top 20%, and top 5% defects for formula fitting. The error band diagram of predicted life obtained by fitting with data from different intervals is presented in Figure 14.

Following the method proposed by Han et al. [38], internal defects inside materials are sorted in descending order according to projected area. Existing studies reveal that predicted lifetimes derived from the top 5% large defects show the optimal consistency with experimental measurements. Based on comparative results and previous research, the methodology adopted in this work is described as follows. All detected internal defects are ranked, and random sampling is conducted among the top 5% defects. The characteristic parameters of sampled defects are taken as input variables of the modified fatigue life prediction model based on plastic strain energy. Herein, ${\mathit{S}}_{\mathit{f}}$ is defined as the ratio of maximum Feret diameter to minimum Feret diameter.

Based on the original nonlinear relationship between plastic strain energy and fatigue life, and by assigning correction weighting factors to each data point according to the prediction accuracy of the original model, the differential evolution algorithm was employed to fit the parameters of the modified model. The parameters of the modified plastic strain energy model are shown in

Table 8. Fitting parameters for the modified plastic strain energy model.

Parameters	$\mathit{\alpha }$	$\mathit{p}$	$\mathit{q}$	$\mathit{a}$	$\mathit{b}$
value	0.1462	2	−0.0671	7.3330	−0.38

. Next, the defect characteristic parameters selected are substituted into Equation (12) to obtain the defect intensity parameters under various defect influences, as shown in

Table 9. Stress concentration factors for various defects.

$\mathbf{I}\mathbf{D}$	$\sqrt{\mathit{a}\mathit{r}\mathit{e}\mathit{a}}$	${\mathit{S}}_{\mathit{p}}$	$\mathit{\gamma }$	${\mathit{K}}_{\mathit{t}\_\mathit{m}\mathit{i}\mathit{c}\mathit{r}\mathit{o}}$
$K_{\_t\_1}$	1.8645	0.2705	0.4999	0.4410
$K_{\_t\_3}$	0.6236	0.4507	0.7957	0.7978
$K_{\_t\_4}$	1.2720	0.2757	0.5814	0.3535
$K_{\_t\_5}$	0.8026	0.3123	1.7700	2.1576
$K_{\_t\_6}$	1.5713	0.2513	1.9219	2.9177
$K_{\_t\_7}$	1.1146	0.2977	1.6071	0.2637
$K_{\_t\_8}$	1.4402	0.3440	2.2334	0.6222

.

4.3. Model Prediction Interval and Prediction Accuracy

For a set of defects of multiple types within the same batch of materials, differences in their geometric morphology and spatial distribution characteristics will result in varying degrees of stress concentration effects. The modified plastic strain energy model builds upon the parameter framework of the original model while further incorporating characteristic variables such as the square root of defect projection area, sphericity, and position parameters as inputs. By integrating the geometric and spatial properties of the defects into the system, this model generates multiple fatigue life prediction curves corresponding to different critical defect states. The life prediction interval is presented as a statistical dispersion band in the life prediction map. The maximum defect intensity parameter corresponds to the lower limit of the prediction interval, while the minimum defect intensity parameter corresponds to the upper limit of the prediction interval. The improved lifespan prediction range significantly enhances the model’s ability to quantitatively characterize defect uncertainties in actual materials and boosts the engineering reliability of its prediction results. Figure 15 results indicate that as $\sqrt{area}$ increases, the predicted curve progressively shifts downward. This demonstrates that the influence of defect area on stress concentration dominates, with its contribution far exceeding that of other defect characteristics.

The accuracy of the revised model’s life prediction is shown in

Table 10. Prediction accuracy of the corrected model.

$\mathbf{I}\mathbf{D}$	${\mathit{N}}_{\mathit{f}\_\mathit{t}}$	${\mathbf{\left(}{\mathit{K}}_{\mathit{t}\_\mathit{m}\mathit{i}\mathit{c}\mathit{r}\mathit{o}}\mathbf{\right)}}^{\mathit{\alpha }}$	${\mathit{N}}_{\mathit{f}\_\mathit{p}}$	$\mathbf{R}\mathbf{e}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{v}\mathbf{e} \mathbf{E}\mathbf{r}\mathbf{r}\mathbf{o}\mathbf{r}$ (%)
LCF-01	33	0.8872	29	−12.92
LCF-03	202	0.9675	147	−27.17
LCF-04	390	0.8590	331	−15.10
LCF-05	85	1.1189	84	−1.14
LCF-06	288	1.1694	387	34.30
LCF-07	1001	0.8229	1250	24.97
LCF-08	1478	0.9329	1624	9.87

, with its prediction error band and prediction-test life comparison results presented in Figure 16a and Figure 16b, respectively. As shown in Figure 16a, the energy-based method demonstrates superior overall prediction accuracy for defect-containing materials compared to the strain-based M-C model. The modified $K_{t\_micro}$ model reduces the overall prediction error range from 3× to 1.5× that of the original PSE model. As shown in the comparison results in Figure 16b, the predicted lifetimes from the modified $K_{t\_micro}$ model more closely aligned with the actual experimental lifetimes. Additionally, the maximum relative error of the revised model decreased from 132.13% to 34.30%, while the average relative error decreased from 43.7% to 26.3%. According to the mean absolute percentage error (MAPE) metric in Equation (14). The average MAPE of the revised model decreased from 41.29% to 17.92%, indicating an improvement in fitting accuracy. The above results consistently demonstrate that the proposed $K_{t\_micro}$ model significantly enhances the overall prediction accuracy and reliability for the low-cycle fatigue life of defect-containing materials.

$$MAPE=mean(abs({\displaystyle \frac{N_{f\_t}-N_{f\_p}}{N_{f\_t}}}))$$

(14)

5. Conclusions

In this study, stress–strain tests and low-cycle fatigue experiments were conducted on titanium–aluminum alloys. To analyze the causes of fatigue life dispersion in these materials, an energy-based approach was adopted in conjunction with internal defects identified through CT non-destructive testing. The following conclusions were drawn:

• The evolution of strain energy during low-cycle fatigue can be divided into two distinct stages. Material fatigue life exhibits a clear correlation with the rate of decrease in plastic strain energy during the first stage.
• Defect area exhibits a significant linear correlation with the shape factor and follows a power-law relationship with sphericity. Furthermore, it was found that when the defect area is below a certain threshold, the three parameters maintain a remarkably stable relative relationship. When the defect area exceeds a certain threshold, the sphericity remains relatively stable, but the shape factor exhibits significant dispersion.
• Compared to traditional plastic strain energy models, the modified $K_{t\_micro}$ model provides better prediction of fatigue life dispersion for specimens containing internal defects. It can generate prediction intervals, significantly improve prediction accuracy, and reduce overall error.
• This study has certain limitations that call for further research. First, the actual fatigue fracture origin of the specimens was not identified. As this paper argues that internal defects govern LCF life dispersion, fracture surface evidence is essential to confirm whether cracks initiated from the predicted critical defects. Due to experimental constraints, such characterization was not performed, leaving the link between critical defects and fatigue life unproven. This limitation is noted, and confirming fracture origins will be a focus of future research.