# A Literature Review of Incorporating Crack Tip Plasticity into Fatigue Crack Growth Models

^{*}

## Abstract

**:**

## 1. Introduction

_{eff}in the cyclic stress–strain hysteresis curves characteristic of fatigue processes (Figure 1a,b), the authors demonstrated that crack closure also occurs in high cycle fatigue (HCF) processes and affects ΔJ and ΔJ

_{eff}, consequently affecting the FCGR.

## 2. Cumulative Damage Theories Based on Stress and/or Strain History

_{n}, Equation (1), which is a candidate for a more robust model. In fact, after numerical correlations, the model is found to be good at all scales.

_{r}is addressed (in Equations (33) and (34) in the original publication) and all other necessary intensity factors are listed (in Equations (35)–(40) in the original publication). With all the above, the fatigue crack driving force ΔK and the fatigue crack growth expression da/dN − ΔK are derived analytically (in Equations (41)–(60) in the original publication). Figure 2 shows a high correlation of the model results with the experimental ones. The authors point out that the fact that being able to relate the Smith–Watson–Topper damage parameter and the stress-strain history of the crack allowed introducing residual stresses into the model has improved its goodness of fit.

_{m}) and one cyclic (r

_{c}), this work relates this second zone r

_{c}to the FCGR, which is produced by the local compression generated at the crack closure of each cycle. Several models can be used to relate the plastic zone to crack propagation. In this study, a simplified model based on the effect of damage accumulated in front of the crack tip da/dN = f(r

_{c}) is used. Taking the expression of the stress value around the linear elastic fracture mechanics (LEFM) crack, it implies that a plastic deformation zone is formed around the crack which has an extension relative to the value of the applied load and the crack length. Furthermore, if the plastic zone is large, a large amount of energy is absorbed during crack propagation, and if the zone is small, crack propagation requires less energy; thus, the size of the plastic zone is directly proportional to the hardness of the material. Therefore, the mechanical properties of the material and the stress state govern the size and shape of the plastic zone. It is concluded that the plastic zone depends on the stress field, the applied stress, the specimen thickness, and the crack length, and that the crack size is an ideal fatigue crack propagation parameter to determine crack growth, better than any other elasto–plastic fracture mechanics (EPFM) parameter. The proposed model is based on da/dN = B(r

_{c})

^{2}, where B = 3.2 × 10

^{−6}(R

_{e})

^{1.3}, R

_{e}being the yield stress. Experimental tests were carried out on 12NC6 steel, yielding consistent and conservative results.

_{max}− Δε/2. The driving force of crack growth can be expressed as a function of two parameters: the residual cyclic plastic stresses σ

_{res}(χ) and the residual intensity factor K

_{r}, which are expressed in different equations. The study shows the correlation of the experimental results and those obtained with the model, validating the use of the model which is based on modelling the crack as a succession of blunt notches of radius r

_{p}, and not as a sharp crack, therefore with realistic stresses and strains around a rounded crack tip.

## 3. Damage Theories Based on Crack Growth Concepts—CTOD, Plastic Zone Size, r_{p}, ΔK

_{pc}is used as a key parameter. Starting from the equation of r

_{pc}based on the Dugdale model, and applying plane strain conditions, Equation (3) is obtained for r

_{pc}, and with plane stress conditions, Equation (4) is obtained. That is, a specimen of zero thickness would be ideal for plane strain and one of infinite thickness for plane strain conditions. The FCGR is expressed as da/dN = C

_{e}(ΔK

_{eff})m

_{c}, where C

_{e}and m

_{c}are material constants. Figure 4a,b show the correlation of the experimental FCGR with that obtained in the model for different specimen thicknesses, as a function of ΔK and ΔK

_{eff}. Figure 4c,d show the FCGR correlations assuming plane strain and plane stress respectively; it is observed that plane strain conditions favour the FCGR, so it increases with specimen thickness. Following the emphasis on the plastic zone size in [58], the study in [22] extends the discussion by measuring crack length and fatigue crack closure to understand the variation of crack growth rate with SIF under different loading conditions.

_{max}or ΔK

_{eff}for different values of R (stress ratio) and B (specimen thickness), to obtain the da/dN − ΔK

_{eff}curves, and to propose an empirical model that correlates a new normalised loading parameter U with R, B, and ΔK. The results were obtained for constant load amplitude in tension with three stress ratios: R = 0, 0.2, and 0.4, and three specimen thicknesses: B = 6, 12, and 24 mm. To measure the crack opening values, gauges were used, and with them ΔK

_{eff}

_{.}could be calculated. The authors started from a set of equations (in Equations (1)–(3) in the original publication) to relate, in the first instance, U, K

_{op}, and ΔK

_{eff}. In view of the experimental results (shown in Figure 4, Figure 5 and Figure 6 in the original publication), an experimental relation between da/dN and ΔK

_{eff}and a readjustment of U was proposed. As a final model, the relation between da/dN and ΔK

_{eff}in Equation (5) remains. An important conclusion obtained in the work is that, despite some scatter in obtaining the da/dN − ΔK

_{eff}curves, the results of the stress ratios and thicknesses were presented using a simple curve for two parameters of the crack growth rate relationship applied to CK45 steel. A model of U as a function of R, B, and ΔK was proposed based on the experimental data of the crack closure. This model is limited to 0 < R < 0.4 and 6 < B < 24 mm.

_{eff}was established based on the ACM. Subsequently, the CWI was used to understand the intrinsic limitations of the ACM, and by applying adjustments to account for the CWI and its mechanisms, develop an improved model. Starting from the expression of ΔK

_{eff}= K − K

_{maxop}

_{,}corrections were applied to arrive at Equation (6) where the parameters exhibit a high degree of similarity (ΔK

_{eff}, K

_{max}and K

_{op}), but with different nuances.

_{eff}= ACR − ΔK

_{app}

_{,}where ACR is a coefficient. The CWI method was modelled by subtracting K

_{cwi}from K to obtain K

_{eff}—K

_{cwi}is related to V

_{cwi}, which is the midline shift in a series of equations—to obtain the final expression of K

_{eff}in Equation (7). Two further modifications were proposed based on the adjusted compliance ratio method and the adjusted compliance ratio/opening load blend method. From this point on, the study focuses on representing the FCGR versus the ΔK, both experimental and those obtained in the models, an example of which is shown in Figure 5. The FCGR is the da/dN, expressed in mm/Cycle, and the ΔK, the diverse intensity factors proposed in MPa(m)

^{0.5}. In total, seven different methods of estimating the effective intensity factor were compared. The ACR method is easy to apply but does not reflect crack closure well. The CWI is mathematically more complex to apply, limiting its applicability. A second version, ACRn2, improves the problems of ACR without adding mathematical difficulty, and finally, the empirical approach for AOP (simply an empirical combination of the ACR and OP methods) makes it a simpler model.

_{eff}= K − K

_{maxop}, establishing an effective intensity factor that, in turn, depends on the intensity factor at crack opening. The model proposed in this paper is developed (in Equations (3)–(10) in the original publication), resulting in Equation (8) for ΔK

_{effM}, which adequately correlates the effects of R-ratio on crack growth at low and intermediate load levels in aluminium alloys. Until the publication of this research, the effect of the R-ratio had been systematically neglected. It is important to emphasize that the R at the crack tip does not necessarily coincide with that applied to the specimen. It is noteworthy that the key parameter in this model, ΔK

_{effM}, is related to the crack opening profile, which obviously depends on the plastic deformation at the tip. Experimental tests show that the proposed model shows improvements to the model in the Elber growth threshold region.

**Figure 6.**The comparison between the predicted K

_{max}vs. ΔK curve and the experimentally obtained data for (

**a**) 2324 aluminium alloy and (

**b**) 6013 aluminium alloy. From [61].

_{I}– iK

_{II}can be calculated with the above data using Equation (11). Figure 7 plots the intensity factor K, the effective intensity factor K

_{eff}, and the real and imaginary components of the intensity factors K

_{I}and K

_{II}. A difference in values obtained with the model and those obtained experimentally exists as the model lacks the crack closure effect, while experimentally there were clear indications of its existence and possible effects.

_{max}and σ

_{max-comp}(maximum applied compressive stress) are the two external loading parameters that determine the above three variables. As an inner parameter, r

_{pc}, the reverse plastic zone size is appropriate to correlate the fatigue crack propagation rate; see also Figure 8. Finally, he proposes an equation for the fatigue crack growth rate under tension–compression loading, with special emphasis that it has nothing to do with the concept of crack closure, but with plasticity at the crack tip and based on the plastic damage theory [61].

_{max}, K

_{eff}, K

_{cl, rs,}K

_{cl, rl}, and K

_{cl, p}

_{.}The subscript cl stands for closure, rs for range—short, rl for range—long, and p for plasticity. Once these characteristic parameters of RICC and PICC are calculated, they can be related to crack growth as deemed best. Finally, ΔK

_{eff}= K

_{max}− K

_{cl}; K

_{max}= ΔK/(1 − R); the expression is given in Equation (12).

_{P}were measured. A relationship between da/dN and CTOD

_{p}was established for the AA6082-T6 aluminium alloy, which is a quantifiable relationship between plasticity and crack growth rate. In addition, the relationship between CTOD

_{P}and other parameters of interest such as ΔK was quantified. When modelling without contact (visible in Figure 4), both plastic and elastic CTODs are higher, i.e., without contact, without the effect of crack closure, the effect of loading is higher. Without contact, without crack closure, the crack propagation only depends on ΔK. The CTOD

_{p}is plotted with respect to the deformation, linear relationship, and with respect to the energy, characterised by a quadratic relationship. In [26], the main objective was to verify the effectiveness of the crack closure concept. Therefore, the crack flank contact was modelled and the following non-linear parameters in the crack tip environment were calculated: cyclic plastic strain range; both elastic and plastic CTODs; size of the plastic inverted zone, r

_{pc}in Figure 8; and the plastic energy dissipated per cycle. Direct relationships of these four parameters with the crack growth rate and ΔK are shown throughout the paper, which can be said to form a plasticity-based growth model. The paper also demonstrates the effect of crack closure on the linear parameters, as well as the effect of mesh size on the linear parameters. In another paper [68], the same numerical finite element model is essentially used as in the two previous works, which will be now described briefly: the numerical modelling is assumed to be elasto–plastic with: 1-Isotropy and 2-the Von Mises criterion is followed by the Voce isotropic hardening law [69] and the Lemaitre–Chaboche’s kinematic law of hardening [47]. The numerical model was implemented with DD3IMP in-house code. Only 1/8 of the specimen is modelled by symmetry with the consequent computational savings. The main result of the work is the elucidation of the relationship between the COTD

_{P}with da/dN for 7050-T6 aluminium. Specifically, there is a linear relationship, with multiplication by 0.5245. Following the basis of the numerical modelling of the previous publications, in [70], the relationship between plasticity and crack growth is further explored and the relationship between CTOD

_{P}and da/dN for aluminium 2050-T8 is also obtained. As a novelty in this work, the effect of the number of load cycles that elapse until another node is released from the crack tip with the consequent crack growth is studied. The higher the number of load cycles between node openings, the higher the CTOD

_{P}. The da/dN is plotted against the numerical CTOD

_{P}, and the strong influence of the numerical parameters can be seen. In plane strain, the higher the number of cycles, the lower the CTOD

_{P}, unlike in plane deformation. In [67], the methodology for the calculation of the CTOD

_{P}in numerical tests is established in detail, as well as the parameters that will affect this calculation, such as the crack propagation distance, the size of the crack, the distance of the points with respect to the edge, the size of the elements, and the number of previous cycles. The mesh size and the software used are those mentioned earlier in Antunes’ work. The distance of the first released node after the crack edge is the most sensitive parameter together with the mesh size.

_{s}of 0.1 and 0.6 on compact tension (CT) specimens made of commercial pure titanium. A sensitivity analysis was performed to explore the effect of the selected position behind the crack tip on the CTOD measurement. Analysis of a full load cycle allowed the elastic and plastic components of the CTOD to be identified. For the plastic, CTOD was found to be directly related to the plastic deformation at the crack. Furthermore, a linear relationship between da/dN and plastic CTOD was observed in both tests (Figure 10). The results show that CTOD can be used as a viable alternative method to ΔK in characterising fatigue crack propagation because the parameter considers the fatigue threshold. This work aims to contribute to a better understanding of the different mechanisms driving fatigue crack growth and the direction of fatigue crack growth, a controversy associated with plasticity-induced fatigue crack closure.

_{I}and K

_{II}in Equation (5) in the original publication). To apply the method, it is necessary to adjust the experimental displacement field obtained with the series described by William [78]. The parameter was determined to facilitate the difference between the SIF result of the (nominal) model and the experimental one. It was observed that, with increasing amplitude in cyclic loading, the difference between nominal and experimental K increased. This may be because of crack tip plasticity, which was not considered in the nominal evaluations. To account for plasticity, Irwin’s approximation was used in the analytical model. The plastic term is added to the crack length, a

_{corr}= a + r

_{y}, where r

_{y}is the plastic radius given by the Irwin approximation, ${r}_{y}=1/2\pi \xb7{\left(\frac{{K}_{I}}{{\sigma}_{ys}}\right)}^{2}$. The results showed a better agreement in the evolution of K under cyclic conditions, increasing the load up to the sharp fracture of the specimen.

## 4. Energy-Based Theories of Damage

_{L}, in Equations (1)–(14) in the original publication). The applied loads are cyclic in both stress and strain control. The author presents an approach based on the treatment of elasto–plastic deformations in the net section, from the material resistance point of view. The physical driving parameters of the crack are calculated as the difference of the energy of the net section and the energy before the crack appeared. ΔC

_{pσ}, Equation (19), for stress-controlled fatigue and ΔC

_{pε}, Equation (20), for strain-controlled fatigue. Both parameters are expressed in terms of material hardening and deformation parameters (k, n), crack size (a), and specimen width (W). It achieves excellent correlations between the crack growth ratios as a function of the strain energy parameters in the net section as seen in Figure 12. Furthermore, the correlations were obtained without any consideration of the concept of crack closure. As a conclusion, it was also added that this approach can be used in many inelastic deformation situations, such as creep crack growth and crack growth in viscoelastic materials.

_{pl}/dN), the increase in stored strain energy (dU

_{e}/dN) and the energy dissipated in the formation of new crack surface (dU

_{a}/dN). Experimental measurements of fatigue crack growth were performed to obtain the relationship between fatigue crack growth rate (da/dN) and energy variables. The result shows that dU

_{pl}/dN and dU

_{e}/dN are not directly related to da/dN. The dU

_{a}/dN, whose value cannot be obtained experimentally with sufficient regression, may be the variable directly related to da/dN within the test range.

## 5. Hybrid Damage Theories or Parameter Definition

_{u}, A, and ΔCTOD (S

_{u}is the maximum force) is presented. Seemingly uncomplicated, the authors attempt to outline all the possibilities and steps (Equation (1) in the original publication). For example, the number of cycles is expressed as N

_{T}= N

_{INC}+ N

_{MSC}+ N

_{PSC}+ N

_{LC}, where INC is incubation, MSC is propagation of a microstructurally small crack, PSC occurs during the transition from MSC status to that of a dominant long crack, and LC is long crack. These cycle numbers are related to the porosity size using various coefficients. For the MSC/PSC regimes, the cycle numbers are also expressed as a function of the most characteristic parameters of the process according to the authors. The whole process detailed so far is repeated for five types of inclusion models: (a) distributed microporosity and Si particles—no significant pores or oxides; (b) high levels of microporosity—shrinkage or gas pores with maximum diameter D

_{p}< 3DCS but greater than the maximum Si particle diameter; (c) large pores near the free surface (D

_{p}> 3DCS); (d) large pores near the free surface (D

_{p}> 3DCS); (e) large oxide films. The model results are correlated with the experimental A356-T6 plate results (in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 of the original publication). The work ends with some interesting graphs (Figure 13 and Figure 14 in the original publication), where the size of the inclusions is related to the crack growth mechanisms, as well as to the fatigue life of the materials. Transitioning from the microstructural emphasis in [15], the subsequent study by Khelil et al. [83] introduces an energetic perspective, illustrating how different approaches can converge to enhance the understanding of fatigue crack growth.

_{p}is formulated as the plastic energy of the hysteresis cycle characteristic of cyclic loads and U as the Specific Energy. U, Equation (22), is expressed as a function of da/dN and subdivided into the three phases of the kinetics fatigue failure diagrams (KFFD). On the other hand, taking as valid the value of the angular strain γ = ε(3)

^{0.5}near the crack tip, one can express R

_{N}(ϴ) as the singular dominant term approximating the elasto–plastic boundary. This allows further development of the ΔW

_{p}which in Equation (21) is left as an integral of stress and strain terms alongside r

_{p}(plastic radius), S

_{pz}(plasticised surface), and Q, Equation (23) (total energy dissipated).

_{R}) was determined using X-ray diffraction techniques with a synchrotron, which also served to map all the deformation in the vicinity of the crack tip. The modelling techniques used for FCGR prediction are based on three concepts: (i) Crack advancement is controlled by the damage processes occurring within the fracture zone located ahead of the crack tip. This zone is embedded within the plastic zone of the crack tip, which, in turn, is surrounded by the zone of dominance of the elastic mechanics stress field solution (K

_{field}). (ii) Prediction of crack advance should be possible by knowing the deformation of a small volume around the crack tip, including residual stress and damage accumulation. (iii) Strictly speaking, a distinction must be made between plasticity and damage: although both are dissipative mechanisms, ultimately it is the damage component that determines failure. However, in many metals, the two parameters are related, i.e., it can be assumed that the damage at each point in the material is a function of the plastic deformation. Therefore, FCGR must be correlated with the plastic deformation processes at the crack tip. To have a good growth model, it is essential to choose the parameters well. In this study, the relationship between crack tip blunting $\rho $ and crack length a is first obtained. The FCGR is related to the residual intensity factor, the ratio of cyclic loads R, and a series of adjustable constants. The adjustment and calculation of the parameters and constants leads to the equation where the FCGR is related to the dissipated plastic energy and a coefficient β; this equation implies the notion that a part of the dissipated plastic energy is converted into damage, causing propagation. Finally, the parameters ${\rho}_{p}$ (crack tip blunting) and ΔW

_{p}(plastic dissipate energy) are chosen as the constituents of the model whose results are compared with the experimental ones shown in Figure 14, offering a more than satisfactory correlation with better results for the blunting parameter.

_{max}and K

_{min}, or K

_{max}and ΔK, or other equivalent, as well as the indispensable variables that drive FCG directly without PICC mediation. One of the major problems of looking at the influence of PICC on crack growth is that under plane strain conditions PICC is not observable. Therefore, numerical modelling appears to be a good tool. The finite element models use a perfect elasto–plastic model with Von-Mises criterion associated with perfect plastic flow. No contact between the flanks was introduced, nor was meshing used. To maintain this criterion, the number of cycles was lowered, and different mesh models with very small element sizes were used. The most remarkable thing according to the authors is that the effect of PICC on FCG seems to be observable, even though PICC has not been modelled, so it is questionable whether PICC affects FCG or not.

_{eff}) reaches a percentage of the thickness. In the second, VCL, the tensile restraint factor α varies along the plastic zone according to a parabolic expression. The constraint decays from its value at the crack tip (α

_{tip}) to a flat stress value of 1.15 at the end in front of the plastic zone. The loss of restraint is also evaluated, but unlike the previous model, the value of α

_{tip}in both strain and plane strain is calculated from the ratio of the size of the plastic zone to the thickness of the specimen. The FCGR is formulated (Equation (24)), where C, n, p, and q are parameters that can be adjusted thanks to the NASMAT module. Three combinations of these parameters were considered, and the results compared with data from the NASGRO materials database and the literature. The ability of the models to estimate fatigue life and variability was analysed by comparing the simulated results with experimental fatigue crack growth data under different stationary Gaussian random loading processes on 2024-T351 aluminium CT specimens. The analysis showed that the two models with their three configurations provide good fatigue life predictions, with very similar results. The variability of the results due to the randomness of the loading was also analysed, and in this case the CCL model provides a better estimate than the VCL. Finally, it should be noted that the best correlation with the experimental results was achieved with one of the combinations of constants proposed by the authors. This combination implemented with the NASMAT module improved upon any combination using the NASGRO database.

## 6. Theory of Critical Distances (TCD)

_{CD}, the critical distance length, and the E

_{C}, the critical plastic energy. The determination of these parameters is of vital importance. For this purpose, the da/dN vs. ΔK curves were determined experimentally. Figure 15 plots the numerical results obtained for the da/dN vs.

**Δ**K curves for different values of L

_{CD}and E

_{C}. Finally, the parameters were fitted with the least-square method at L

_{CD}= 17.5 µm and E

_{C}= 0.55 J/mm

^{3}. Good correlations between the model and the experimental results were observed.

_{I}). Regarding the latter method, the relationship between the critical distance and the number of cycles to failure was calibrated using two different methodologies. When life estimations were performed based on the first calibration, the results were considered unsatisfactory. However, when the second calibration method was used, almost all predictions fell within scatter bands with respect to the experimental data used to calibrate the model.

## 7. Conclusions

_{P}. The latter parameter has been proposed as replacing even the CTOD; the advantages of this are varied, from the unity of the parameter m, in the case of the CTOD, and above all the elasto–plastic nature of the CTOD as opposed to the elastic approach of ΔK. At present, it is a well-established hypothesis that the origin and first stage of the crack has a plastic and microstructural basis, while the second growth phase is of an elastic nature and much more influential on a macro level. Any numerical or analytical approach needs experimental evidence of this. In this direction, the works of Vaso-Olmo et al. [75] and Medhi et al. [77] give insights into the concept of CTOD and plasticity in the crack tip. Both works are able to explain the fatigue crack growth behaviour with an introduction to crack tip plasticity in different ways. It is important to highlight Antunes et al.’s latest studies [67,68,70] for their exceptional contributions to advancing the understanding of the implications of crack tip plasticity in fatigue crack growth predictions based on numerical models with validation with experimental results. In summary, this research successfully established a robust method for the numerical calculation of CTOD and CTOD

_{P}, emphasising their dependency on key parameters. The study initially focuses on the 7050-T6 aluminium alloy, where cyclic plastic deformation is meticulously determined through experimental tests and subsequently modelled analytically. The development of a 3D numerical model to predict CTOD

_{P}marks a significant advancement in this area. Experimental investigations provide a foundational understanding of the relationship between CTOD and FCGR. This is further complemented by numerical predictions of CTOD

_{P}, which are tailored for varying crack lengths and da/dN values, thereby enhancing the precision and applicability of the model in predicting fatigue crack growth.

_{p}[79,83,84], U

_{pl}[81], and Q [83]. All of these consider a threshold of the plastic energy that triggers the crack growth when surpassed. Nevertheless, there are some differences between them; while [79,83,84] refer to the same term ΔW

_{p}related to load cycle, other terms like [90,91], U

_{pl}[81], and Q [83] are related to crack or fracture cycle release. Quan and Alderliesten [81] also refer to the variation in elastic strain energy in one entire crack propagation cycle, which is included inside the elastic strain energy group. The strain energy gradient is mentioned in [90,91,93], with the difference that Zhu et al. reflects on stress and strain effects. Regarding the group where total strain energy is mentioned, terms like ΔWt [94], U

_{L}, ΔCpσ and ΔCpε [80], and Ua [81] are included. While ΔWt [94], U

_{L}, and ΔCpσ [80] are stress-related parameters, ΔCpε [80] is strain-related, and Ua [81] is surface-related. The last term is linked to specific energy, U [83], energy dissipated per unit volume during fatigue crack growth.

## 8. Future Directions

#### 8.1. Advancements in Multiscale Approaches

#### 8.2. Challenges in Experimental Measurement and Analytical Modelling

#### 8.3. Energetic Modelling Procedure

## Funding

## Conflicts of Interest

## Nomenclature

A_{0} | Parameter dependent of the stress intensity factor | r_{c} | Cyclic plastic zone size |

a_{0} | Initial crack length | RICC | Rugosity-induced crack closure |

a | Crack length | R_{int} | Interior radius for the area of interest |

a_{corr} | Corrected crack length | r_{m} | Monotonic plastic zone size |

ACR | Adjusted compliance ratio | R_{N} | Dominant singular term approximation to the elastic–plastic boundary |

B | Number of additions per column | R_{out} | Outer radius for the area of interest |

C | Matrix function of the polar coordinates in system of equations | r_{p} | Irwing plastic zone size for plane strain conditions |

CCP | Centre crack plate specimen | r_{pc} | Plastic zone size |

COD | Crack open displacement | R_{pr} | Reversed plastic zone size |

CT | Compact tension specimen | r_{y} | Plastic radius from Irwing |

CTOD | Crack tip open displacement | S | Shift applied to the crack tip coordinates |

CTOD_{BS} | CTOD in BS7448-1:1991 [96] | S_{PZ} | Plastified surface |

CTOD_{JWES} | CTOD proposed by the Japan Welding Engineering Society | SWT | Smith–Watson–Topper parameter |

CTOD_{p} | Plastic CTOD | T-Stress | Crack tip stress |

CWI | Crack wake Influence | U | Normalized load ratio parameter/fatigue crack energy |

d | Matrix including ε_{yy} information in system of equations | U_{a} | Surface energy dissipated through new crack surface formation |

D | Dissipated energy | U_{L} | Net section energy |

da/dN | Fatigue crack growth rate | U_{Pl} | Plastic energy dissipation |

dW/dN | Plastic work per cycle | UTS | Ultimate tensile strength |

E | Young’s modulus | V_{CWI} | E displacement at centreline due to concentrated force P on crack surface |

E_{c} | Critical plastic energy | V_{p} | Plastic component of crack mouth opening displacement |

F | Detection function for the symmetry axis | W | Specimen width |

f | Correction factors for plastic component of CTOD_{JWES} | WOL | Wedge opening loading specimen |

IFR | Influence ratio | X | Vector of unknowns in system of equations |

K | Stress intensity factor (SIF) | x, y | Crack growing coordinate and crack opening coordinate |

K_{ci,p} | Stress intensity factor when first contact between crack flanks occurs, plastic | x_{ct} | X coordinate of the crack tip |

K_{cl,rl} | Stress intensity factor when first contact between crack flanks occurs, range—long | x_{max} | X coordinate of the maximum ε_{yy} value |

K_{cl,rs} | Stress intensity factor when first contact between crack flanks occurs, range–short | x_{min} | X coordinate of the minimum ε_{yy} value |

K_{CWI} | Stress intensity factor due to concentrated force | YR | Yield-to-tensile ratio, rys/ruts |

K_{exp} | Experimental estimation of the stress intensity factor | δ | SIF relative error, % SIF |

K_{F} | SIF opening mode | ΔC_{pε} | Change in net section strain energy for crack extension under plasticity in strain-controlled testing |

KFFD | Kinetics fatigue failure diagrams | ΔC_{pσ} | Change in net section strain energy for crack extension under plasticity in stress-controlled testing |

K_{field} | SIF field | Δ_{ij} | Kronecker delta |

K_{I} | Opening mode SIF | ΔJ | J-integral |

K_{Ic} | Mode I fracture toughness | ΔJ_{eff} | Effective J-integral |

K_{II} | SIF mode II | ΔK | SIF range |

K_{max} | Maximum SIF | ΔK_{eff} | Effective SIF range |

K_{nom} | Nominally applied SIF | δ_{p} | Crack tip blunting |

K_{op} | SIF open | ΔW_{e} | Elastic nominal strain energy density |

K_{R} | SIF retardation | ΔW_{e,eff} | Effective elastic nominal strain energy density |

K_{r} | Residual SIF | ΔW_{pl} | Plastic nominal strain energy density |

K_{S} | SIF shear | ΔW_{pl,eff} | Effective plastic nominal strain energy density |

K_{th} | Threshold stress intensity factor ranges | ε_{xx}, ε_{yy} | Elastic strain in x and y directions |

L_{CD} | Length critical distance | η | Statistical parameter |

n | Number of cycles | ν | Poisson’s ratio |

P | Pearson’s coefficient | $\rho $ | Plastic blunting |

PICC | Plasticity-induced crack closure | σ_{app} | Applied stress |

Q | Total energy dissipated | σ_{maxcomp} | Maximum stress at compression |

R | Stress ratio | σ_{xx}, σ_{yy} | Stress in x and y directions |

r, θ | Polar coordinates from the crack tip | σ_{y} | Yield stress |

R_{A} | Area roughness of crack flanks |

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**Figure 2.**Fatigue crack growth data for Al 2024 T351 aluminium alloy as a function of the two-parameter driving force, ΔK. From [39].

**Figure 3.**Opening stresses for an infinite plate with a circular hole and two radial symmetric cracks propagating under constant amplitude loading with and without residual stresses due to cold expansion (ur/r = 4%). (

**a**,

**b**) are for σ

_{0}/σ

_{max}= 0.4 and 0.6, respectively, with R = 0. From [44].

**Figure 4.**(

**a**,

**b**) Crack growth per cycle (da/dN) versus plastic CTOD range (ΔCTODp) for both tests for 304SS. (

**c**) Comparison of FCGR specimens with relation to cyclic plastic zone plane strain conditions for Inconel 718 and 304SS. (

**d**) Plane stress and strain for Inconel 718. From [58].

**Figure 5.**FCGR response of alloys (

**a**) 6061−T6 and (

**b**) 2024−T3 showing R = 0.7 data and constant K

_{max}data (K

_{max}= 22 MPa$\sqrt{m}$) as a function of K

_{app}. From [20].

**Figure 7.**(

**a**) Opening (K

_{I}) and (

**b**) shearing (K

_{II}) mode stress intensity factors measured during the loading portion of a fatigue cycle (R = 0) with θ = 30º on a 7010 aluminium alloy centre-cracked plate. The nominal values were calculated neglecting closure effects. The experimental values were determined relative to an image captured at zero load. From [62].

**Figure 8.**Interesting crack tip region/point, assuming that damage is a local issue—theory of critical distance (TCD). Region of reversed plastic zone r

_{pc}if fatigue damage accumulates. Region of forward plastic zone r

_{pm}.

**Figure 9.**Correlation between CTOD calculated from P-Vg curve and actual CTOD at mid-thickness experimentally measured by silicone rubber casting for specimen with thickness B = 30 mm. (

**a**) CTOD

_{JWES}. (

**b**) CTOD

_{BS}. From [35].

**Figure 11.**Evolution of fatigue crack propagation rate estimated by numerical simulation. (

**a**) Master plot for R = 0 and (

**b**) plastic dissipation per cycle vs. ΔK for a Titanium alloy. From [79].

**Figure 12.**Fatigue crack growth data under stress control for cracks growing from starter holes in round tension specimens of 0.45% C steel, plotted in terms of (

**a**) ΔK and (

**b**) the change in net-section strain energy, ΔC

_{pσ}. From [80].

**Figure 13.**Comparison of measured and estimated dissipated energy per cycle for (

**a**) 2024-T351 and (

**b**) 7075-T7351. Comparison of measured evolution of da/dN with Q (

**c**) for 2024-T351 (

**d**) and 7075-T7351. From [83].

**Figure 14.**(

**a**) The dependence of crack blunting parameter increment (per cycle) on the applied stress intensity factor, ΔK, that is described well by a power law relationship. (

**b**) The dependence of crack tip plastic energy dissipation (per cycle) on the applied stress intensity factor, ΔK, that is described well by a power law relationship. From [84].

Damage Parameter | Equation |
---|---|

Stress | σ_{a} = (σ′_{f} − σ_{m}) (2N_{f}) b |

Strain | ε_{a} = (σ′_{f} − σ_{m}) (1/E) (2N) fb + ε′f (2N_{f}) c |

Energy | ΔW_{t} = κ_{t} (2Nf) α^{t} + ΔW_{0t} |

SWT | ε_{a} σ_{max} = (σ′_{f})^{2} (1/E) (2N) f^{2b} + ε′_{f} (2N_{f}) c^{+b} |

**Table 2.**CTOD and/or CTOD

_{P}as a critical parameter used as a driving force in crack growth modelling. CTOD is Crack Tip Open Displacement, where

_{P}denotes Plastic.

Ref. | Authors and Date | Description/Main Contribution | Methods | Sug. |
---|---|---|---|---|

[26] | Antunes et al., 2015 | Analysis of remote compliance is the best numerical parameter to quantify the crack opening level. Establishment of an analytical relation between CTOD and da/dN. This relation was tested numerically. | Numerical | ++ |

[35] | Kawabata et al., 2016 | A new CTOD method is investigated considering the variation of crack tip blunting (strain hardening). The calculation formula is based on three-dimensional elasto–plastic FEM. | Numerical–Experimental | ++ |

[43] | Shih 1986 | Establish the relation between the J-integral and the crack opening displacement by exploiting the dominance of the Hutchinson–Rice–Rosengren singularity in the crack tip region. | Numerical–Experimental | ++ |

[66] | Pokluda 2011 | A discrete dislocation model of contact effects in small-scale yielding is presented. The model enables direct assessment of the magnitude of both plasticity and roughness-induced components of crack closure. | Analytical | ++ |

[67] | Antunes et al., 2018 | Establishment of a method of numerical calculation of the CTOD and CTOD^{P} and their dependence on certain parameters. | Numerical | +++ |

[68] | Antunes et al., 2017 | The 7050-T6 aluminium alloy cyclic plastic deformation was determined experimentally and modelled analytically. A 3D numerical model was developed to predict the CTOD^{P}. | Numerical–Experimental–Analytical | +++ |

[70] | Antunes et al., 2018 | First, experimental tests were conducted to obtain the relation between CTOD and FCGR. Then, numerical predictions of CTOD^{P} were obtained for different crack lengths and da/dN. | Numerical–Experimental | +++ |

[71] | Tagawa et al., 2014 | Numerical and experimental methods to determine a method to calculate CTOD and CTOD^{P}. | Numerical–Experimental | + |

[72] | Tagawa et al., 2009 | The effects of CTOD testing methodologies on CTOD values were investigated according to tests conducted by the Japan Welding Engineering Society (WES). | Numerical–Experimental | ++ |

[73] | Kayamori et al., 2010 | Experimental investigations and analytical developments into crack tip opening displacement (CTOD) were conducted to establish the relationship between BS7448-CTOD and ASTM E1290-CTOD. | Numerical–Experimental | + |

[74] | Kayamori et al., 2012 | Two new CTOD calculations were proposed: for deep-notched specimens, a displacement–conversion CTOD; and for shallow-notched specimens, a J-conversion CTOD was proposed. | Numerical–Experimental | ++ |

[75] | Vasco-Olmo et al., 2017 | A methodology is developed to measure and analyse the CTOD and CTODP from experimental data. | Experimental | +++ |

[78] | Yates et al., 2010 | The paper gives an overview of some DIC applications for crack tip characterisation such as CTOD and CTODP measures as well as data obtained. | Analytical–Experimental | ++ |

Ref. | Specific Parameter | Authors and Date | Description/Main Contribution | Methods | Sug. |
---|---|---|---|---|---|

[26] | R_{pr} | Antunes et al., 2015 | Reverse plastic zone size. The crack closure phenomenon has a great influence on crack tip parameters, decreasing their values. | Numerical | ++ |

[35] | CWI | Kawabata et al., 2016 | Crack wake influence. A new factor f is introduced to correct the plastic term. In this factor, the blunted crack tip shape is considered to depend on the strain-hardening exponent, and f is given as a function of the yield-to-tensile ratio (YR) of the material and the specimen thickness. | Numerical–Experimental | + |

[38] | $\rho $ | Pommier and Risbet 2005 | In the equations, special attention is paid to the elastic energy stored inside the crack tip plastic zone, sync. In practice, residual stresses at the crack tip are known to considerably influence fatigue crack growth. | Analytical | +++ |

[39] | Noroozi et al., 2005 | The results demonstrate the crack closure influence on LCF behaviour. The change of crack closure from LCF to high cycle fatigue and their consequences for lifetime prediction. | Analytical–Experimental | ++ | |

[40] | R_{pr} | Ould Chikh et al., 2008 | Plastic zone size. The cyclic plastic strain can be the principal parameter for fatigue crack growth under a cyclic loading. Generally, FCGR is a plastic zone size r_{c} function, and it increases as the plastic zone size increases. | Analytical | +++ |

[44] | De Matos et al., 2008 | This paper shows that the residual stress field due to cold expansion has a strong influence on closure behaviour and therefore on fatigue crack propagation. | Numerical–Analytical | ++ | |

[56] | εPA | Borges et al., 2020 | Fatigue crack growth (FCG) is simulated here by node release, which occurs when the accumulated plastic strain reaches a critical value. | Numerical | +++ |

[58] | R_{pc} | Park et al., 1996 | Plastic zone size experimental tests showed that plastic zone size was an important parameter in crack propagation. | Experimental | +++ |

[20] | r_{y} | Donald and Paris 1999 | The ACR and CWI methods measure the change in displacement at minimum load due to closure. That quantity is less subject to variability than the measurement of the opening load. | Analytical | +++ |

[61] | da/dS | Zhang et al., 2010 | Parameter da/dS defines the fatigue crack propagation rate with the change of the applied stress at any moment of a stress cycle. The relationship between this new parameter and the conventional da/dN is given. | Numerical | ++ |

[63] | R_{pc} | Zhang et al., 2010 | Plastic zone size. The results show that, near the crack tip, the reverse plastic zone size continues to change with a change in the applied compressive stress. | Numerical–Experimental | ++ |

[66] | PICC | Pokluda 2011 | Plasticity induced in the crack closure. There is a good qualitative agreement between the plasticity-induced shielding terms employed in the dislocation-based model and the continuum-based multi-parameter model. | Analytical | + |

[67] | R_{pr} | Antunes et al., 2018 | Reverse plastic zone size. The increased crack plastic deformation also produces an increased crack closure phenomenon, which cancels the increased plastic deformation. | Numerical | ++ |

[84] | $\rho $ | Korsunsky et al., 2009 | Plastic blunting, crack tip blunting. Two approaches were considered in the present study: the approach based on the consideration of crack tip blunting due to Pommier and Risbet [38], and the presently proposed approach based on the analysis of local energy dissipation in the immediate vicinity of the crack tip. | Analytical–Experimental | ++ |

Ref. | Specific Parameter | Authors and Date | Description | Definition | Methodology |
---|---|---|---|---|---|

[67] | CTOD_{P} | Antunes et al., 2018 | Plastic energy dissipated per cycle | Plastic portion of the crack tip opening displacement. Plastic CTOD is obtained by subtracting the elastic CTOD from the total. | Numerical |

[87,88] | PM | Zheng et al., 2013, 2014 | Critical plastic energy, at the point close to the crack tip | The fatigue damage experienced by a point located within a specific distance from the crack tip can accurately represent the average damage condition at the crack tip area. | Numerical |

[90] | EC | Kujawsky and Ellyin 1984 | Critical plastic energy | Amount of plastic strain energy that a material can dissipate before experiencing fatigue failure. | Numerical |

[91] | EC | Chalant and Remy 1983 | Critical plastic energy | The strain gradient inside the grain at the crack tip. | Numerical |

[94] | ΔWt | Branco et al., 2021 | Critical plastic energy | Accounts for the mean stress effect, a measure of the energy dissipated per cycle. and is capable of unifying both the low-cycle and high-cycle fatigue regimes. | Numerical–Analytical |

[93] | χ_{W} | Zhu et al., 2018 | Critical plastic energy | Reflects the distribution of both stress and strain gradients within the actual structure. | Analytical |

[84] | ΔW_{p} | Konsunsky et al., 2009 | Equivalent deformation energy | Amount of energy dissipated due to plastic deformation at the crack tip during each loading cycle. | Analytical–Experimental |

[79] | ΔW_{p} | Klingbei 2003 | Strain energy gradient | Change in total plastic dissipation per unit width during a specific cycle. | Numerical–Analytical |

[80] | U_{L} | Ravi Chandran 2018 | Total dissipated plastic energy | Total net section strain energy in the crack plane of the ligament. Combination of elastic and plastic strain energies due to the increased stress. | Analytical–Experimental |

[80] | ΔC_{pσ} | Ravi Chandran 2018 | Total dissipated plastic energy | Change in net section strain energy parameter in stress-controlled fatigue. Equation (19) | Analytical–Experimental |

[80] | ΔC_{pε} | Ravi Chandran 2018 | Cumulative change in cyclic strain energy of the net section | Change in net section strain energy density in strain-controlled fatigue. Equation (20) | Analytical–Experimental |

[81] | U_{pl} | Quan and Alderliesten 2022 | Plastic energy difference in the net section | Energy consumed in the process of crack growth through plastic deformation of the material surrounding the crack. | Numerical–Experimental |

[81] | U_{e} | Quan and Alderliesten 2022 | Elastic energy difference in the net section | Variation in elastic strain energy stored throughout one full cycle of crack propagation. | Numerical–Experimental |

[81] | U_{a} | Quan and Alderliesten 2022 | Surface energy difference in the net section | Surface energy differential dissipated through new crack surface formation. | Numerical–Experimental |

[83] | ΔW_{p} | Kheli et al., 2013 | Stored deformation dissipation | Cyclic plastic strain energy, corresponding to one loading cycle. Equation (21) | Numerical–Experimental |

[83] | U | Kheli et al., 2013 | Dissipation new crack surface formation | Specific energy, energy dissipated per unit volume during fatigue crack growth. Equation (22) | Numerical–Experimental |

[83] | Q | Kheli et al., 2013 | Plastic energy of the hysteresis cycle characteristic of cyclic loads | Total dissipated energy in the specimen during fatigue crack growth. Equation (23) | Numerical–Experimental |

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**MDPI and ACS Style**

Garcia-Gonzalez, A.; Aguilera, J.A.; Cerezo, P.M.; Castro-Egler, C.; Lopez-Crespo, P.
A Literature Review of Incorporating Crack Tip Plasticity into Fatigue Crack Growth Models. *Materials* **2023**, *16*, 7603.
https://doi.org/10.3390/ma16247603

**AMA Style**

Garcia-Gonzalez A, Aguilera JA, Cerezo PM, Castro-Egler C, Lopez-Crespo P.
A Literature Review of Incorporating Crack Tip Plasticity into Fatigue Crack Growth Models. *Materials*. 2023; 16(24):7603.
https://doi.org/10.3390/ma16247603

**Chicago/Turabian Style**

Garcia-Gonzalez, Antonio, Jose A. Aguilera, Pablo M. Cerezo, Cristina Castro-Egler, and Pablo Lopez-Crespo.
2023. "A Literature Review of Incorporating Crack Tip Plasticity into Fatigue Crack Growth Models" *Materials* 16, no. 24: 7603.
https://doi.org/10.3390/ma16247603