PSI - Issue 75

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Procedia Structural Integrity 75 (2025) 219–233

© 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of Dr Fabien Lefebvre with at least 2 reviewers per paper The purpose of this paper is to provide in a single document a comprehensive collection of the most relevant fatigue crack growth laws and models allowing a more effective review and comparison of the available tools and facilitating decision making. © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Fatigue Design 2025 organizers Keywords: Fatigue crack growth; Crack Retardation Models; Looping Algorithms Fatigue Design 2025 (FatDes 2025) A comprehensive review of Fatigue Crack Growth laws and models Andrew Halfpenny a , Cristian Bagni a, *, Amaury Chabod b , Stephan Vervoort c a Hottinger Bruel & Kjaer UK Ltd, Advanced Manufacturing Park Technology Centre, Brunel Way, Rotherham, S60 5WG, United Kingdom b Hottinger Bruel & Kjaer France SAS, 2-4 rue Benjamin Franklin, 94370 Sucy-en-brie, France c Hottinger Bruel & Kjaer GmbH, Im Tiefen See 45, 64293 Darmstadt, Germany Abstract Fatigue is the most common cause of failure in structures subject to cyclic loading. Fatigue failure is a two-stage phenomenon consisting of a crack initiation stage followed by a propagation stage. Cracks can initiate due to fatigue but also as a result of manufacturing processes and material features such as inclusions or voids. Once a crack is present, if it is subject to a sufficiently high cyclic stress, it will propagate until failure. However, in some instances a crack can propagate into a low-stressed region causing the crack to stall and never propagate to failure. In this case the crack may be acceptable in-service as it might not compromise the durability of the component (damage tolerant approach). Fatigue crack growth refers to the propagation (or non-propagation) of cracks in structures subject to cyclic loading. Fatigue crack growth and damage tolerant analyses use fracture mechanics principles and therefore, the knowledge of pre-existing cracks is necessary. These pre-existing cracks can be detected in components using non-destructive techniques or assumed. From the second half of the 20th century, significant research effort was put in understanding and describing how cracks propagate under cyclic loading, including how to characterise the threshold, propagation and fast fracture regions, both from an experimental and numerical point of view, as well as how to account for mean stress effect and crack retardation. Unfortunately, this research effort is scattered in a multitude of scientific publications.

* Corresponding author. Tel.: +44-7768-091-654. E-mail address: cristian.bagni@hbkworld.com

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Fatigue Design 2025 organizers

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of Dr Fabien Lefebvre with at least 2 reviewers per paper 10.1016/j.prostr.2025.11.024

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1. Introduction Fatigue is a progressive damaging phenomenon in a material caused by variable loads, causing a component to fail even if the resulting stresses it experiences are well below the static strength of the material. Fatigue failure represents the main failure mechanism in components subject to cyclic loading and typically involves two main stages: • Crack initiation: one or more cracks nucleate in the material. • Crack propagation: after initiation, the cracks propagate until failure of the component, provided they are subject to sufficiently high cyclic stress. The duration of these two stages depends on several factors such as material properties, structural design, and application. It is also worth highlighting that there are instances where a crack, after entering a low-stress region, stops propagating before failure of the component. In such cases, the crack does not compromise the component's durability and structural integrity and may be considered acceptable in-service (damage tolerance approach). Historically, crack initiation and propagation have been analysed using different physical models. While crack initiation is typically studied using strain-life (EN) models, crack propagation is generally studied using principles of fracture mechanics. Fracture mechanics, born from the early works of Griffith (1921) and Irwin (1958), is based on the assumption that a crack already exists in a component and that, if subject to a favourable stress and strain state at its tip, it progressively propagates until complete failure of the component or to unacceptable deformation in the crack region. The crack propagation takes place through damage mechanisms at an atomic scale in the so-called process zone ahead of the crack tip. In the second half of the 20 th century, a significant research effort was dedicated to understanding and characterising crack propagation under cyclic loading, including the definition of the threshold, propagation, and fast fracture regions both experimentally and computationally, as well as taking into consideration mean stress effects and crack retardation. This produced a multitude of crack growth laws and crack retardation models that, unfortunately, scattered across numerous scientific publications. This paper aims to be a collection of the most relevant fatigue crack growth laws and crack retardation models, allowing engineers to efficiently review these laws and models and make informed decisions. 2. Crack Growth Laws An effective way of describing how a crack propagates in a given material is by plotting how the crack growth rate, , changes with the stress intensity range, ∆ . This type of plot is well-known as crack growth curve and, as shown in Fig. 1a, it is characterised by three distinct regions: • Region I describes the slow crack growth rate for small cracks close to a threshold value of stress intensity range, ∆ ℎ , below which it is assumed that a crack will not propagate. In this region, the microstructure has a significant effect. • Region II describes the propagation behaviour of a crack for most of its life. In this region, the crack growth curve follows a power-law behaviour and is controlled by Linear Elastic Fracture Mechanics (LEFM) with little influence of microstructure. • Region III describes the fast fracture behaviour before final failure. In this region, the influence of microstructure becomes significant again, together with the thickness of the component.

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(a)

(b)

Fig. 1. Typical crack growth curve (a) and mean stress effect on crack growth rate (b). Mean stress also has a strong influence on the crack growth rate as graphically shown in Fig. 1b. This is usually quantified for a particular cycle using the ratio, , of the minimum stress to the maximum stress in the cycle: = min max = min max (1) The most relevant laws proposed to describe the crack growth curve (or portions of it) are detailed below. 2.1. Paris Crack Growth Law Paris and Erdogan (1963) proposed a simple crack growth law to describe the crack propagation region (Region II) of the crack growth curve. The Paris law does not describe the threshold region (Region I), while the onset of fast fracture (Region III) is usually modelled as a discrete termination point corresponding to the plane strain or plane stress fracture toughness, and it does not consider mean stress effects. The Paris law is expressed as: = ∙ ∆ (2) where is the Paris coefficient and is the Paris exponent, that are obtained from tests and constitute material dependent constants. According to the hypothesis of crack closure, compressive stresses in the cycle do not contribute to crack propagation and therefore, can be ignored. As a consequence, when ≤0 , ∆ should be replaced with max (assuming max >0 as a necessary condition for crack propagation), leading to the following formulation: = { ∙ ∆ for > 0 ∙ max for ≤0 (3)

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2.2. Forman Crack Growth Law Forman et al. (1967) proposed an extension of the Paris law to allow also the description of the fast fracture region (Region III). Forman’s crack growth law is described by the following equation: = ∙ ∆ (1 − ) ∙ −∆ (4) where and are the Forman coefficients, different from the Paris coefficients due to the units in the denominator of Eq. 4, is the plane stress fracture toughness and the effective ∆ is expressed as: ∆ = max ∙ (1− ) (5) By substituting Eq. 5 into Eq. 4, the Forman law takes the following expression: = ∙ m ax (1− ) −1 − max (6) The ability of the Forman crack growth law to implicitly model mean stress effects is often criticised because it misses an independent controlling parameter allowing to ‘fine - tune’ the fit to experimental data. Finally, it is worth highlighting that the Forman law is valid for stress ratios ∈ [ min , max ] , where min and max are the minimum and maximum stress ratios used in the tests, respectively. 2.3. Walker Crack Growth Law Walker (1970) also proposed an extension of the Paris law to account for mean stress effects. Walker achieved this by introducing the stress ratio, , into Paris formula (Eq. 2) as follows: = ∙ [∆ ∙ (1 − ) −1 ] (7) where the exponent controls the degree of influence of the mean stress on the crack growth allowing to ‘fine tune’ the fit to experimental data. Since Eq. 7 is a power law, similarly to the Paris law, it is not suitable for describing the threshold (Region I) and the fast fracture (Region III) regions, but only the crack propagation region (Region II). Walker, considering the hypothesis of crack closure, proposed to neglect compressive stresses in the cycle, leading to the following formulation (with max >0 necessary condition for crack propagation): = { ∙ [∆ ∙ (1 − ) −1 ] for > 0 ∙ [ max ∙ (1− ) −1 ] for ≤0 (8) Similarly to the Forman law, the Walker law can be considered valid for stress ratios ∈ [ min , max ] . 2.4. Austen Crack Growth Law Austen proposed another modification of the Paris law to implicitly consider both threshold and onset of fast fracture (nCode (2003)) . Austen’s crack growth law is given by:

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= ∙ (∆ )

(9) where the effective stress intensity range, ∆ = max − min , accounts for both threshold and onset of fast fracture, with max = max + and min =max( min , ) . Austen accounted for the onset of fast fracture through the expression (modification for static failure): = max 1 − max (10) where 1 is the plane strain fracture toughness. Threshold and short cracks, instead, were considered through a crack closure stress intensity factor, , expressed as: = max − max ∙ √ + 0 + ∆ ℎ 1− (11) where is the crack length and 0 = 1 ∙ ( ℎ ∆ 0 ) 2 is the smallest crack size that will propagate as defined by Kitagawa and Takahashi (1976), with ℎ the threshold stress intensity factor and ∆ 0 the unnotched fatigue strength. The threshold stress intensity range, ∆ ℎ , is defined as a bi-linear function of (or mean stress) as: ∆ ℎ ={ 1 if ≥ 0 −( 0 − 1 ) ∙ if < (12) where 0 is the threshold stress intensity at =0 , 1 is the threshold stress intensity at > and the stress ratio at the threshold knee. Austen also made the assumption, supported by test data, that compressive stresses do not contribute to damage. Finally, it is worth highlighting that the Austen law does not explicitly correct for mean stress effects, but Austen argued that it was irrelevant and attributed it to crack closure and retardation. NASGRO3 was proposed by NASA (1999) and represents the most comprehensive crack growth law. It considers mean stress effects, threshold values, the onset of fast fracture and crack closure. Another benefit is that NASA made available a large quantity of material data expressed in this format. The only real downside of the NASGRO3 law is the complex curve fitting of the experimental data. The NASGRO3 law is expressed as: = ∙ (1 − 1 − ∙ ∆ ) ∙ (1− ∆ ℎ ∆ ) (1− max ) (13) where , , and are empirical coefficients derived from test data. The crack tip opening function, , is given by: 2.5. NASGRO3 Crack Growth Law

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= { max{ , 0 + 1 ∙ + 2 ∙ 2 + 3 ∙ 3 } if ≥0 0 + 1 ∙ 0 −2 ∙ 1 if <−2 where 0 = (0.825 − 0.34 + 0.05 2 ) ∙ [cos ( 2 ∙ )] 1 if −2≤ <0

(14)

(15) 1 = (0.415 − 0.071 ) ∙ 2 =1− 0 − 1 − 3 3 =2 0 + 1 −1 where and are parameters obtained through tests that represent the plane stress/strain constraint factor and the ratio of maximum applied stress to the flow of stress, respectively. The threshold stress intensity range, ∆ ℎ , is defined as: ∆ ℎ =∆ 0 ∙ √ + 0 [ 1− (1− 0 ) ∙ (1 − )] 1+ ℎ ∙ (16) where ∆ 0 is the threshold stress intensity range at =0 obtained from tests, is the crack length, 0 = 0.0381mm is the intrinsic crack length and ℎ is the threshold coefficient determined from tests. The critical stress intensity, , is function of the thickness and is given by: = 1 ∙ [1+ ∙ −( ∙ 0 ) 2 ] (17) where 1 is the plane strain fracture toughness, and are fitting parameters obtained experimentally, is the plate thickness and 0 is the plane strain reference thickness defined as: 0 =2.5( 1 ) 2 (18) with the yield stress. For values of stress ratio < it was proposed to adjust the effective stress intensity range, ∆ , as: ∆ = { max ∙ (1− min ) if < min max − min otherwise (19) For values of stress ratio > , instead, it was proposed to adjust the maximum stress intensity as:

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max ={ ∆ (1− max ) if > max max otherwise

(20)

3. Retardation Models Elber (1971) observed that, following a tensile overload, the surfaces of a fatigue crack remained in contact even after the application of smaller tensile loads, with consequent reduction of the crack growth rate. This phenomenon is known as crack retardation. This phenomenon was attributed to the large plastic deformation region produced by the tensile overload around the crack tip causing the material to expand in this region due to plastic strain. Consequently, when the tensile overload ceases, the expanded material, due to the containment induced by the surrounding elastic material, applies a residual compressive stress to the surfaces of the crack pushing them close together (Fig. 2a). Crack propagation resumes if and when the stress at the crack tip overcomes the residual compressive stress caused by the tensile overload. The pre-compression remains effective until the crack has overgrown the plastic zone created by the tensile overload. According to the Paris law (Eq. 2), the crack growth rate

can be expressed as: = ∙ (∆ )

(21)

with the effective stress intensity range defined as: ∆ = max − (22) where is the stress intensity needed to overcome the residual compressive stress caused by the tensile overload.

(a)

(b)

Fig. 2 . Plasticity in the crack wake after Elber (a) and illustration of the plastic zone in Willenborg’s retardation model (b).

Other phenomena can also be observed:

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• Delayed retardation: the crack has to propagate into the plastic zone created by the tensile overload of a certain amount before crack growth retardation takes place. • Overshoot: this is a crack growth rate increase occurring when the crack reaches the end of the plastic zone generated by the tensile overload. • Crack acceleration: following a significant compressive underload, a residual tensile stress region is generated around the crack tip causing the crack growth rate to increase. This is the opposite of crack retardation; however, this effect is significantly smaller than the crack retardation caused by a tensile overload and usually leads to a reduction of the effects of a previous crack retardation. At the basis of any retardation model is the definition of the size of the plastic zone ahead of the crack tip caused by a tensile overload. Irwin (1960) proposed to estimate the size of the plastic zone, , ahead of a crack tip as a function of the stress intensity factor, , and the yield stress, : = 1∙ ∙ ( ) 2 (23) where is a parameter representative of the stress/strain state (Bannantine et al. (1990), NASGRO3 (1999), Harter (2000), nCode (2003)). 3.1. Wheeler Model Wheeler (1972), based on empirical observations, proposed to modify the crack growth rate through a retardation coefficient, , defined as: =( + − ) (24) where is the size of the plastic zone generated by the current cycle, is the crack length at the current cycle, is the size of the plastic zone generated by the tensile overload, is the crack length when the overload occurred and is an empirically derived fitting factor. The retardation coefficient, , ranges between 0 for no crack growth (crack arrest) and 1 when the crack reaches the boundary of the plastic zone created by the tensile overload. The fitting factor can be evaluated only through tests and this is often rather difficult, making this model not very appealing. In addition, the Wheeler model contradicts the observed effect of delayed retardation, as it predicts maximum retardation immediately following an overload. Finally, it is worth highlighting that this model does not consider crack acceleration due to compressive underload. Willenborg Model The retardation model proposed by Willenborg et al. (1971) assumes that, following a tensile overload, a plastic zone characterised by a residual compressive stress forms around the crack tip and that the crack propagates only if the stress at the crack tip overcomes this residual stress. The maximum and minimum effective stress intensity factors at the crack tip are respectively defined as: max = max − (25) min = min − (26) 3.2.

Andrew Halfpenny et al. / Procedia Structural Integrity 75 (2025) 219–233 Author name / Structural Integrity Procedia (2025) 9 where = − max is the compressive stress intensity factor due to the confinement induced by the elastic body on the plastic zone, with representing the stress intensity factor required to extend the current plastic zone to the edge of the plastic zone created by the tensile overload, . This condition can be expressed as: + = + (27) where, as defined in Fig. 2b, is the current crack length, is the size of the current plastic zone, is the crack length when the tensile overload occurred and is the size of the plastic zone generated by the tensile overload. Using Irwin’s formula ( Eq. 23) to estimate the plastic zone size, Eq. 27 becomes: + 1∙ ∙ ( ) 2 = + 1∙ ∙ ( ) 2 (28) Solving for yields the expression: = ∙√1− − (29) Therefore, can be calculated as: = ∙√1− − − max (30) The Willenborg model, despite its popularity, does not include delayed retardation, overshooting and crack acceleration (compressive underload). However, it can be argued that is reasonable to neglect these phenomena as they are often second order effects. Generalised Willenborg Model Gallagher (1974) generalised the Willenborg model to improve its convergence with test results in the threshold region and near crack arrest. The generalised Willenborg model proposed by Gallagher is documented in NASA (1999) and is expressed as: ′ = ∙ (31) where = 1 − ∆ ℎ max −1 (32) with indicating the shut-off value of the stress ratio max , varied to optimise the fit between life prediction and test results. Reasonable values of are 3.0 for aluminium, 2.7 for titanium and 2.0 for steel (Harter (2000)). However, Harter stated that these values are not truly material dependent and can vary. This retardation model does not consider compressive underload, delayed retardation or overshoot. 227 3.3.

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3.4. Modified Generalised Willenborg Model Brussat (1974) proposed an extension of the Generalised Willenborg model proposed by Gallagher (1974) to consider the effects of compressive underload. This model is also documented in NASA (1999). Brussat modified the expression of the parameter as: = { 2.523 ∙ 0 1 + 3.5 ∙ (0.25 − ) 0.6 if <0.25 1 otherwise (33) where is the ratio of underload stress to overload stress and 0 is the value of for =0 . The parameter 0 is a material parameter determined experimentally, with typical values ranging between 0.2 and 0.8 as described in Mettu et al. (1999). Brussat also proposed an alternative definition of the minimum effective stress intensity factor, min : min = { max ( min − ′ ,0) if min >0 min otherwise (34) This retardation model does not account for delayed retardation or overshoot. B ased on Walker’s crack growth law , Chang and Engle (1984) also extended the Generalised Willenborg model proposed by Gallagher (1974) to consider the effects of compressive underload. The model is expressed as: = { ∙ [∆ ∙ (1 − ) −1 ] for ∆ >∆ ℎ and ≥ 0 ∙ [ max ∙ (1+ 2 ) ] for ∆ >∆ ℎ and < 0 0 for ∆ <∆ ℎ (35) 3.5. Walker-Chang-Willenborg Model

with

{ + if > + − if < − otherwise

=

(36)

where + and − are the cut-off values for positive and negative stress ratios, respectively. Also, the threshold stress intensity range, ∆ ℎ , is defined as: ∆ ℎ =∆ 0 ∙ (1 − ∙ ) where is a fitting parameter and ∆ 0 is the stress intensity range at =0 . This retardation model does not include delayed retardation or overshoot.

(37)

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3.6. Harter-Willenborg Model Harter (2000), proposed to use a modified overload plastic zone size, , in the Willenborg model to account for the effects of compressive underload. The modified overload plastic zone size reduces the retardation effect and is expressed as: = ∙ [1 − 0.9 ∙ | min |] (38) where min is the size of the residual tensile stress zone created by the compressive underload and is determined according to Irwin’s formula (Eq. 23). The Harter-Willenborg model is not able to predict lives shorter than the life with no retardation and does not consider delayed retardation or overshoot. Austen-Willenborg Model Austen modified the Willenborg model to consider delayed retardation, overshooting and compressive underload (nCode (2003)). The resulting Austen-Willenborg model calculates as follows: = ∙√1− − − −( − max ) ∙√1− − − max ∙√1− − (39) 3.7. 4. Looping algorithms Crack growth analysis is an iterative process and when implemented in commercial durability software, the crack growth laws and retardation models previously described are iteratively applied within a looping algorithm, until a termination condition is met. T here are two looping algorithms available called ‘Cycle -by- Cycle’ and ‘Rapid Integration’. ‘ Cycle-by-Cycle ’ looping algorithm The ‘Cycle -by- Cycle’ looping algorithm is the most basic but is also the most comprehensive. It calculates the incremental crack growth for each cycle in turn calculating the instantaneous stress intensity and retardation. The main disadvantage of this algorithm is the longer processing time required to carry out most analyses. The ‘Cycle by- Cycle’ algorithm is most useful where significant retardation is expected and the fatigue life is short, or where semi-elliptical crack growth analysis is required. The ‘Cycle -by- Cycle’ looping algorithm creates a ‘Cycle’ object used to store all the crack growth parameters and initialises the cycle pointer and the initial crack length . The main loop then commences, and iteration continues until some termination condition is met. The ‘Cycle’ object is read in turn by the following algorithms: ‘ Rainflow ’ (get current cycle), ‘ Stress Intensity Factor ’ (calculate SIF for current crack length), ‘ Retardation ’ (calculate ) and ‘ Crack Growth Law ’ (calculate ∆ ). The required output calculated by each algorithm are then 4.1. where = 121 ∙ ( − min ) 2 (40)

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sent back to the ‘Cycle’ object. The new crack length is then calculated followed by checks to determine if one of the termination conditions (e.g. target crack length achieved, crack arrest) is met. If a termination condition is established, the ‘Cycle’ object is passed to the reporting object and the calculation stopped. If a termination condition is not met, the cycle pointer is incremented and the process continued. The ‘Cycle -by- Cycle’ looping algorithm is summarised in Fig. 3.

Fig. 3. Flow chart of the ‘Cycle -by- Cycle’ looping algorithm .

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4.2. ‘ Rapid Integration ’ looping algorithm The ‘Rapid Integration’ looping algorithm is based on the method proposed by Brussat (1974). According to this method, the crack length is discretised over a number of small intervals and the crack growth rate is calculated at distinct integration points along the propagation path.

Fig. 4. Flow chart of the ‘Rapid Integration’ looping algorithm .

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Author name / Structural Integrity Procedia (2025) The crack growth rate at the ℎ integration point ( ) is determined using the standard ‘Cycle -by- Cycle’ algorithm except that the crack length is not incremented between cycles. This reduces the computational effort in determining the SIF and enables block loads to be dealt with using linear accumulation. The number of cycles to propagate the crack between limits and are then calculated using a standard numerical interpolation routine integrating the reciprocal value for ( ) . The ‘Rapid Integration’ looping algorithm is summarised in Fig. 4. This algorithm is compatible with all the retardation models presented in this paper. According to Brussat, where retardation is required, it is recommended to initialise the retardation model before each integration point calculation to establish the overload plastic zone size for the highest tensile load in the spectrum. Unlike the ‘Cycle -by- Cycle’ algorithm, the ‘Rapid Integration’ algorithm is not able to implicitly determine yield or the critical crack size, , and requires an explicit calculation. In particular, a check is necessary before the calculation process is started to ensure that the user does not try to set an integration point beyond . Three types of discretisation and numerical integration methods can be used within this algorithm (see Press et al. (2007)): • Equal spacing : this is where the integration points are equally spaced at constant crack length intervals. The numerical integration can then use a simple Trapezoidal rule or Simpson’s rule. These methods are ideal for estimating the intermediate points and are therefore ideal for plotting crack growth curves and calculating the estimated initial flaw size. This method is based on a prescribed discretisation and yields no indication of convergence. • Non-equal spacing : where, as the name suggests, the integration points are non-equally spaced. This effectively excludes Simpson’s rule as it requires three equally spaced integration points. This method is also based on a prescribed discretisation and yields no indication of convergence. • Adaptive integration : this is an iterative approach that starts using a fairly coarse discretisation and refines this comparing convergence along the way. The advantage of using this approach is that a tolerance for convergence can be entered, and the crack is discretised appropriately. This can yield very efficient calculations as there is generally no need for over conservative discretisation. The downside of this approach is that it does not provide intermediate points and so proves unsuitable for plotting crack growth curves. 5. Conclusions In the second half of the 20 th century, a significant number of crack growth laws and crack retardation models were proposed. Unfortunately, this invaluable inheritance is scattered across numerous scientific publications making the review and comparison difficult and time consuming. This paper collects the most relevant fatigue crack growth laws, crack retardation models and looping algorithms with the aim of helping engineers, dealing with fatigue crack growth applications, to efficiently review these laws and models and make informed decisions. In this paper, several well-known fatigue crack growth laws were described, starting from the simpler but groundbreaking Paris law, suitable to describe only the Region II of a crack growth curve, to subsequent laws proposed to consider mean stress effects, crack closure, threshold behaviour (Region I) or the onset of fast fracture (Region III). Crack retardation was also described, and a wide range of retardation models were reviewed, from the Wheeler and Willenborg models, that neglect delayed retardation, overshoot and crack acceleration, to the Austen-Willenborg model that includes all three phenomena. Finally, two looping algorithms allowing the implementation of the crack growth analysis in commercial durability software were also described. References

Bannantine, J. A., Comer, J. J., Handrock, J. L., 1990. Fundamentals of Metal Fatigue Analysis. Prentice Hall.

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Brussat, T. R., 1974. Rapid Calculation of Fatigue Crack Growth by Integration. Fracture Toughness and Slow-Stable Cracking, 298 – 311. Chang, J. B., Engle, R. M., 1984. Improved damage-tolerance analysis methodology. Journal of Aircraft 21(9), 722 – 730. Elber, W., 1971. The Significance of Fatigue Crack Closure. Damage Tolerance in Aircraft Structures, 230 – 242. Forman, R., Kearney, V. E., Engle, R., 1967. Numerical Analysis of Crack Propagation in Cyclic-Loaded Structures. ASME Journal of Basic Engineering 89(3), 459 – 463. Gallagher, J. P., 1974. A generalised development of yield zone models. Air Force Flight Dynamics Laboratory Technical Memorandum 74-28 FBR. Griffith, A. A., 1921. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character 221, 163 – 198. Harter, J. A., 2000. Afgrow Users Guide and Technical Manual. Air Vehicles Directorate, 2790 D Street, Ste 504, Air Force Research Laboratory, WPAFB OH 45433-7542, USA. Irwin, G. R., 1958. Fracture. In: Flügge, S. (eds) Elasticity and Plasticity / Elastizität und Plastizität. Handbuch der Physik / Encyclopedia of Physics 3/6, 551 – 590. Springer, Berlin, Heidelberg. Irwin, G. R., 1960. Plastic zone near a crack and fracture toughness. In: 7th Sagamore Ardance Materials Research Conference, Syracuse, Univ. Press. Kitagawa, H., Takahashi, S., 1976. Applicability of Fracture Mechanics to Very Small Cracks or the Cracks in the Early Stage. In: Proceedings of the 2nd International Conference on Mechanical Behaviour of Materials, 627 – 631. Mettu, S. R., Shivakumar, V., Beek, J. M., Yeh, F., Williams, L. C., Forman, R. G., McMahon, J. J., Newman Jr., J. C., 1999. NASGRO 3.0: A software for analyzing aging aircraft. NASA, 1999. Reference Manual for NASGRO version 3. Lyndon B. Johnson Space Center, 2101NASA Road 1, Houston, Texas 77058-3696, USA. nCode, 2003. ICE-Flow Crack Growth Quick Start Guide V2.0. Paris, P., Erdogan, F., 1963. A Critical Analysis of Crack Propagation Laws. Journal of Basic Engineering 85(4), 528 – 534. Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., 2007. Numerical Recipes in C: The Art of Scientific Computing (3rd ed.). Cambridge University Press. Walker, K., 1970. The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7075-T6 Aluminum. Effects of Environment and Complex Load History on Fatigue Life, ASTM STP 462. Wheeler, O. E., 1972. Spectrum loading and crack growth. ASME Journal of Basic Engineering 94(1), 181 – 186. Willenborg, J. F., Engle, R. M., Wood, H. A., 1971. A Crack Growth Retardation Model Using an Effective Stress Concept. Air Force Flight Dynamics Laboratory Technical Memorandum 71-1-FBR.

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Procedia Structural Integrity 75 (2025) 538–545 Structural Integrity Procedia 00 (2025) 000–000 Structural Integrity Procedia 00 (2025) 000–000

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© 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of Dr Fabien Lefebvre with at least 2 reviewers per paper © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2025 organizers. Keywords: fatigue assessment; spot-weld; structural stress; notch stress; multiaxial loading Abstract This paper proposes a multi-scale approach to account for additional Mode II / III loading using structural stresses. Two parametric models, one with shell elements and one with solid elements, are used to set up a high number of macro- and meso-scale models from which the correlation between forces and moments and local notch stresses at spot weld is evaluated. To reduce the computa tional e ff ort in practical application, a machine learning model is trained from the results of both models to estimates notch stresses (solid element) from nodal forces (shell element). This model can be used to reliably assess spot weld fatigue performance in full body analyses. © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2025 organizers. Keywords: fatigue assessment; spot-weld; structural stress; notch stress; multiaxial loading Fatigue Design 2025 (FatDes 2025) Advanced Machine Learning-Based Durability Assessment of Spot Welds Addressing Mode II and III Loading Jo¨rg Baumgartner a , Philipp Ro¨melt b a Fraunhofer LBF, Institute for Structural Durability and System Reliability, Bartningstr. 47, 64289 Darmstadt, Germany b Noa ffi liation Abstract This paper proposes a multi-scale approach to account for additional Mode II / III loading using structural stresses. Two parametric models, one with shell elements and one with solid elements, are used to set up a high number of macro- and meso-scale models from which the correlation between forces and moments and local notch stresses at spot weld is evaluated. To reduce the computa tional e ff ort in practical application, a machine learning model is trained from the results of both models to estimates notch stresses (solid element) from nodal forces (shell element). This model can be used to reliably assess spot weld fatigue performance in full body analyses. Fatigue Design 2025 (FatDes 2025) Advanced Machine Learning-Based Durability Assessment of Spot Welds Addressing Mode II and III Loading Jo¨rg Baumgartner a , Philipp Ro¨melt b a Fraunhofer LBF, Institute for Structural Durability and System Reliability, Bartningstr. 47, 64289 Darmstadt, Germany b Noa ffi liation

1. Introduction 1. Introduction

Spot welds are the main joining elements of the body-in-white. Depending on the complexity of the structure, around 2000 – 5000 spot welds are present [13]. The majority of these joints is cyclically loaded; therefore, these spot welds need to exhibit high reliability over the lifetime of the vehicle. For over 30 years, numerical assessment approaches exist to estimate the fatigue strength of spot weld. They can be subdivided in structural and notch stress approaches as well as approaches that use fracture mechanics. The most commonly used approaches have been developed to reliably predict crack initiation under mainly Mode I loading conditions, i.e., a ”peeling” combined with ”shear” is present. This is the case for standard overlap specimens under a global shear loading, since secondary bending moments are induced due to the eccentricity. This limitation can lead to Spot welds are the main joining elements of the body-in-white. Depending on the complexity of the structure, around 2000 – 5000 spot welds are present [13]. The majority of these joints is cyclically loaded; therefore, these spot welds need to exhibit high reliability over the lifetime of the vehicle. For over 30 years, numerical assessment approaches exist to estimate the fatigue strength of spot weld. They can be subdivided in structural and notch stress approaches as well as approaches that use fracture mechanics. The most commonly used approaches have been developed to reliably predict crack initiation under mainly Mode I loading conditions, i.e., a ”peeling” combined with ”shear” is present. This is the case for standard overlap specimens under a global shear loading, since secondary bending moments are induced due to the eccentricity. This limitation can lead to

∗ Corresponding author. Tel.: + 49-6151-705-474. E-mail address: joerg.baumgartner@lbf.fraunhofer.de ∗ Corresponding author. Tel.: + 49-6151-705-474. E-mail address: joerg.baumgartner@lbf.fraunhofer.de

2452-3216 © 2025 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of Dr Fabien Lefebvre with at least 2 reviewers per paper 10.1016/j.prostr.2025.11.054 2210-7843 © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2025 organizers. 2210-7843 © 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2025 organizers.

Jörg Baumgartner et al. / Procedia Structural Integrity 75 (2025) 538–545 Jo¨rg Baumgartner / Structural Integrity Procedia 00 (2025) 000–000

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a non-conservative dimensioning and subsequently to failure in service in cases where in-plane or out-of-plane shear dominates. The aim of this paper is to present a methodology that is easily applicable based on a standard simulation model but also able to capture the influence of all loading modes (I, II and III) on the stresses at the weld. With this approach, a higher assessment reliability is expected.

Nomenclature

t

thickness

w F

width of the sheets

d

diameter of spot weld

cutting forces M cutting moments

r

notch radius

2. Fatigue and assessment of spot-welded structures

Various di ff erent approaches exist for the fatigue assessment of spot-welded structures. The main approaches have been summarized almost 30 years ago as the structural stress approaches, the notch stress approaches, and fracture mechanics [17].

2.1. Structural stress approach

The advantage of using structural stresses for fatigue assessment is primarily due to the simplicity of finite element models. The structural connection is modeled using a connector element, which can simply be a bar or beam element that exhibits the same diameter as the spot weld, Fig. 1. In case of a bar or beam element, two nodes need to be positioned exactly on the opposite sites of the sheets. Another commonly applied approach is the use of a hexahedron element as spot weld, which is connected with RBE3 connectors (spider) to the shell elements. This configuration e ff ectively distributes the loads from the shell elements to the joining element. With this approach, the nodes of the shell mesh do not need to be adapted to the connector. In addition, this approach allows a better mapping of the sti ff ness of the spot weld.

Shellmesh Spider

7

8

Beam

5

6

3

4

Hexahedron

1

2

Fig. 1. Representation of a spot-weld with a beam (left) and a hexahedron and spider element (right)

Di ff erent analytical formulae to derive all structural stress components at a spot weld are available [14, 20]. These are primarily determined by the three cutting forces F x , F y and F z as well as three cutting moments M x , M y and M z at the spot weld, which is represented as a beam element. These cutting forces and moments are then used for a fatigue assessment [18, 8]. Another approach is to use stresses measured at a certain distance from the spot weld on the sheet surface [22]. The most common approach to assess the fatigue strength of spot-welded structures is the so-called FESPOW (fatigue evaluation of spot welds) approach that was published in 1995 [18] as a result of a joint research project from the automotive industry in Germany [19]. The approach uses radial stresses determined at two bar elements that represent the spot weld as the main parameter for the fatigue assessment.

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Two assessments have to be performed: One for spot weld failure and one for failure in the sheet. The assessment of spot weld failure is, from a mechanical point of view, straightforward, as it uses section forces and moments in the bar element. However, these forces and moments cannot directly be used to determine the stresses in the sheet. A mechanical substitute model is used that consists of a metal sheet with the diameter of d a = 50 mm clamped on the outside. That diameter equals approx. 10 times the spot weld diameter d . Analytical formulae are used to calculate radial stresses for five cutting forces and moments: F x and F y , F z aswell as M x and M y . Whereas the approach is easily applicable, it has several drawbacks and a ”poor accuracy” [26]: • The assumption of d a = 10 × d of the mechanical substitute model is a simplification and does not represent the sti ff ness and deformation behavior in complex models. • Since only the radial stress is evaluated, the influence of all shear components is neglected. • Torque loading ( M z ) might also contribute to the damage, even though pure torque loading is not expected in structures with a huge number of spot welds. • The sti ff ness of the model is lower than that of 3D-model if sti ff ness or diameter of the bar element or neigh boring shell elements are not modified. The ideas of FESPOW have been develop further by adding beam elements in a radial pattern around the spot weld [21]. With this approach, the multiaxial stress states can be consider to a better extend; however, also the complexity of the model increases. The idea to model the fatigue critical notch at a spot weld with a reference radius was developed by Zhang [26]. The idea behind the approach is not connected to the reference radius of r = 1 mm with its FAT-class 225 [7]; furthermore, the derived notch stress can be interpreted as a value that is proportional to the stress intensity factor. Starting at spot welds, it was later used for the assessment of seam welds at thin sheets and a FAT-class of FAT630 was recommended [23]. This class is now included in the IIW-recommendations [3]. A benefit of notch stress is the possibility to assess multiaxial loading conditions by separating stresses normal to the weld and shear stresses [4, 1]. The interaction of normal and shear stress is assessed with the Gough-Pollard equation [6, 7]. Since the notches at a spot-weld have the characteristic of a crack, a lot of investigations have been conducted using fracture mechanics to assess the fatigue strength. The main di ffi culty in the assessment is the derivation of the stress intensity factor solution; i.e., an easy way to derive stress intensity factors (SIF). As input solutions, structural stress – in detail bending and membrane stress – is used to estimate the SIF [15, 9]. There also exists the possibility to calculate SIF directly with a Finite-Element model [10]; however, this approach cannot be used in industrial applications since it would consume too much time for setting up the models and too much resources to solve the models. As for the notch stresses, there exit various investigations that deal with the calculation of stress intensity factors for spot-welded structures under multiaxial loading. Here, not only the SIF under mode I loading, but also the SIF under mode II and mode III loading are evaluated [16, 27, 11]. As briefly introduced above, all three methods have in principle the possibility to assess multiaxial stress states. However, currently the structural stress based approaches are the most commonly used approaches to assess the fatigue strength of spot welds. The reason can be found in the simple models and fast calculation times. In addition, the main damage occurs in spot-welded structures due to mode I loading. Subsequently, in the majority of cases the assessment reliability is high. However, there are cases where shear, i.e., mode II loading dominates. In these cases, the assessment is non-conservative. So, in order to enhance the assessment reliability, a consideration of mode II and also mode III loading in the structural stress approaches would be beneficial. This can be achieved if notch stresses 2.3. Stress intensity factor for spot welds 2.4. Comparison of the approaches 2.2. Notch stress approach with radii smaller than 1 mm