MicrostructureBased Fatigue Modeling with Residual Stresses: Effect of Inclusion Shape on Very High Cycle Fatigue Life
Abstract
:1. Introduction
2. Modeling Strategy
2.1. Reconstruction of Microstructure
2.2. Digitalization of Inclusions
2.3. Residual Stress Simulation
2.4. Crystal Plasticity Model and Parameter Calibration
2.5. Fatigue Indicator Parameter and Fatigue Simulation
3. Results and Discussion
3.1. Residual Stress Distribution with Different Inclusions
3.2. Fatigue Crack Initiation Site Prediction
3.3. Fatigue Life Prediction
3.3.1. Results of Fatigue Life Prediction with Microstructure Modeling
3.3.2. Inclusion Parameters Unifying and the Formulation of Fatigue Life concerning Inclusion Parameters
3.3.3. Validation of the Formulation of Fatigue Life concerning Inclusion Parameters
4. Conclusions
 (1)
 In this study, the effect of the shape features of inclusions on fatigue life is systematically investigated via a microstructurebased modeling approach. Based on the findings, an analytical formulation to correlate the fatigue life with the size and shape features of inclusions is proposed. This formulation extends the former fatigue equation to shape parameters (aspect ratio, A, and tilting angle, θ).
 (2)
 To describe the inclusion shape, a new parameter, $\xi $, which unifies the aspect ratio and tilting angle of the inclusion, is introduced. This parameter, $\xi $, shows its representativeness in both the geometrical and physical sides. This new parameter and inclusion area jointly determine the equivalent inclusion area, which was adopted in the formulation of fatigue life as a single variable.
 (3)
 A critical θ is also offered. When θ > 53°, the effect of inclusion on the fatigue property will increase when A increases; when θ < 53°, the effect of inclusion on the fatigue property will decrease when A increases.
 (4)
 For smaller inclusions, the value of the maximum residual stress around the inclusion is affected by the microstructure of the steel matrix around the inclusion more seriously compared with the inclusion features itself; meanwhile, for larger inclusions, the inclusion shape is more important in residual stress analysis.
 (5)
 The predicted fatigue life in this study concerning the inclusion depends on the square root of the equivalent inclusion area to the −1/26th power.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Inclusion Shape  Circle  Ellipse1  Ellipse2  Ellipse3 

Geometry sketches of inclusions  
Parameters  A = 1 θ: unconstrained  A = 2 θ = 90°  A = 2 θ = 45°  A = 2 θ = 0° 
Material  Coefficient of Linear Expansion, α (10^{−6}·°C)  Young’s Modulus, E (GPa)  Poisson’s Ratio, v 

Calcium Aluminate  5.0  113  0.23 
Steel Matrix  23.0  206  0.30 
Constitutive Equation 
$${\dot{\gamma}}^{\alpha}={\dot{\gamma}}_{0}{\left\frac{{\tau}^{\alpha}{\chi}^{\alpha}}{{\tau}_{c}^{\alpha}}\right}^{\frac{1}{m}}sgn\left({\tau}^{\alpha}{\chi}^{\alpha}\right)$$

γ^{a}—slip rate along the slip system α; ${\dot{\gamma}}_{0}$—initial slip rate; ${\tau}^{\alpha}$—resolved shear stress along a slip system α; ${\chi}^{\alpha}$—backstress on slip system α; ${\tau}_{c}^{\alpha}$—critical resolved shear stress on slip system α; 1/m—strain rate sensitivity factor.  
Calculation of Shear Stress 
$${\tau}^{\alpha}=S\xb7\left({m}^{\alpha}\otimes {n}^{\alpha}\right)$$

n^{a}—normal to the slip plane; m^{a}—slip direction; S—second Piola–Kirchhoff stress tensor.  
Isotropic Hardening Law 
$${\tau}_{c}^{\alpha}={\tau}_{i}+{{\displaystyle \sum}}_{\beta =1}^{N}{q}^{\alpha \beta}\left[{h}_{0}{\left(1\frac{{\tau}_{c}^{\beta}}{{\tau}_{s}}\right)}^{a}\right]\left\mathsf{\Delta}{\gamma}^{\beta}\right$$

${\tau}_{i}$—initial resolved shear stress; ${q}^{\alpha \beta}$—latent hardening parameter; ${h}_{0}$, ${\tau}_{s}$,$a$—hardening parameter; $\mathsf{\Delta}{\gamma}^{\beta}$—plastic slip increment of each slip system β.  
Kinematic Hardening 
$$\dot{{\chi}^{\alpha}}={G}_{1}{\dot{\gamma}}^{\alpha}{G}_{2}\left{\dot{\gamma}}^{\alpha}\right{\chi}^{\alpha}$$

G_{1}, G_{2}—kinematic hardening constant; ${\chi}^{\alpha}$—backstress tensor of slip system α. 
C_{11}: 193.9 GPa  C_{12}: 94.6 GPa  C_{44}: 92.2 GPa  γ_{0}: 0.01 
1/m: 100  τ_{0}: 645 MPa  G_{1}: 100,000 MPa  G_{2}: 2000 
h_{0}: 1000 MPa  a: 1.1 
Circle  Ellipse1  Ellipse2  Ellipse3  

Geometry sketches of inclusions  
A  1  2  2  2 
θ  /  90°  45°  0° 
A − 1  0  1  1  1 
sinθ  /  1  $\sqrt{2}$/2  0 
sinθ × (A − 1)  0  1  $\sqrt{2}$/2  0 
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Gu, C.; Lian, J.; Lv, Z.; Bao, Y. MicrostructureBased Fatigue Modeling with Residual Stresses: Effect of Inclusion Shape on Very High Cycle Fatigue Life. Crystals 2022, 12, 200. https://doi.org/10.3390/cryst12020200
Gu C, Lian J, Lv Z, Bao Y. MicrostructureBased Fatigue Modeling with Residual Stresses: Effect of Inclusion Shape on Very High Cycle Fatigue Life. Crystals. 2022; 12(2):200. https://doi.org/10.3390/cryst12020200
Chicago/Turabian StyleGu, Chao, Junhe Lian, Ziyu Lv, and Yanping Bao. 2022. "MicrostructureBased Fatigue Modeling with Residual Stresses: Effect of Inclusion Shape on Very High Cycle Fatigue Life" Crystals 12, no. 2: 200. https://doi.org/10.3390/cryst12020200