1 Introduction

Due to technical difficulties and cost issues on full-scale testing of engineering components, laboratory experiments on smooth and small specimens are usually performed to obtain basic fatigue properties, which are then utilized for fatigue performance assessment of real structures. As a result, it is vital to study the size effect in metal fatigue and establish a valid transfer from smooth and small specimens data to full-scale component behavior (Sonsino and Fischer 2005; Wormsen et al. 2007; He et al. 2020; Zhu et al. 2020).

As is well-known, a key factor affecting size effect is raised by defects formed during manufacturing, processing or mechanical damage produced in service (Rafsanjani and Sørensen 2015; Ai et al. 2019). The presence of defects is one of the major causes for fatigue crack initiation, especially for those located on or near the surface. From this point of view, fatigue crack propagation caused by manufacturing defects is usually assumed to dominate the fatigue failure of engineering components, which correlates the size and number of defects with fatigue life, like Kitagawa–Takahashi diagram-based methods (Murakami and Endo 1994).

In general, modeling of the size effect originating from defects is inseparable from the study of these defects. In the past, scaling law parameter method (Carpinteri and Spagnoli 2004) based on experimental data fitting as well as monofractal (Carpinteri et al. 2009, 2010) and multifractal methods (Bažant and Yavari 2005; Carpinteri et al. 2002) based on fractal dimensions had been introduced to describe size effect. Recently, with the development of inspection technology [e.g. ultrasonic, radiographic, eddy-current and optical microscopy (Atkinson and Shi 2003)] and finite element (FE) technique, more attentions have been paid on the inherent defects. Based on this, various studies on the phenomenon of weakening fatigue strength under defects have been conducted to explain the differences between the tested and real properties (Goto 1991). For evaluating fatigue strength/life, a flow chart for general modeling procedure under defects is shown in Fig. 1.

Fig. 1
figure 1

Flow chart for fatigue strength/life assessment under defects

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Frost and Greenan (1964) conducted a series of studies, which initially considered the effects of geometrical discontinuities or cracks generation from fatigue, while the influence of defects or non-metallic inclusions in the material was ignored. The development of fracture mechanics theory shows a great significance during strength evaluation considering size effect under critical defects. However, most researches on small fatigue crack growth cannot be well explained by theory of Linear Elastic Fracture Mechanics (Goto 1991) and usually yield large scatter in fatigue data (Song et al. 2019; Fei et al. 2014). According to the material variability (including defects distribution) (Zhu et al. 2013, 2015, 2016, 2018a), to ensure the structural integrity of engineering components, the safety factor concept is normally included during deterministic analysis. However, these safety factor-based approaches cannot explain the origin of scatter well, which limit the optimal design of interest and the study of size effect (Zhu et al. 2012, 2017, 2018b, c; Qian et al. 2018; Hu et al. 2018; Li et al. 2018). Therefore, the key to conduct fatigue strength evaluation under size effect is to model the statistical behavior of defects. The idea—modeling fracture phenomenon based on a statistical model was originally introduced by Griffith (1921), i.e. the differences between the ideal and actual performances are mainly raised by the presence of defects weakening the structure. Specifically, the defect distribution determines the statistical distributions of fatigue properties, which constitutes the grounds of the weakest link theory studied by Pierce (1926).

Recently, a large amount of approaches based on statistical methods and fracture mechanics under defects have illustrated the significance of defects in fatigue size effect modeling (Tiryakioglu and Campbell 2010; Shirani and Härkegård 2014). Furthermore, fractography using scanning electron microscope (SEM) indicated that surface or near-surface defects are normally the major cause of fatigue failures (Yi et al. 2003; Besel and Brueckner-Foit 2008; Buffière et al. 2001; Murakami 2012). According to this, various studies have been conducted using defects relevant information (including size and position) for fatigue life estimation (Toft et al. 2011; Shirani and Härkegård 2012; Al-Owaisi et al. 2016), such as the competing failure modes described by Poisson defects statistics (Ravi Chandran and Jha 2005; Ravi Chandran 2005; Ravi Chandran et al. 2010) and the numerical simulation method with FE analysis (Fjeldstad et al. 2008).

In this research work, in terms of inherent defects of materials, recent advances on the statistical physics and probabilistic methods utilized for size effects modeling when transferring the fatigue properties of specimens with different geometries/scales are briefly reviewed. Firstly, available studies on the link between defects and size effect are introduced in Sect. 2. Then, three types of size effects are individually elaborated. In particular, the main concerns are their effects on fatigue behaviors and the quantification methods based on defect statistics. Based on recent advances in metal fatigue under defects, discussions are presented in Sect. 3. Finally, conclusions are drawn in Sect. 4.

2 Size effect in metal fatigue

In metal fatigue analysis under size effect, damage modeling under defects has been a hot spot in fatigue evaluation and integrity assessment of engineering structures (Ktari et al. 2016; Saffar et al. 2014; Åman et al. 2017; Abroug et al. 2018a). During the periods when casting technology was in its infancy, structural defects, such as shrinkage cavities and gas pores, were commonly presented in materials, which attracted much attentions in early fatigue studies (Murakami and Endo 1994). Fractography reveals that structural defects are normally the primary reason for fatigue crack initiation (Yi et al. 2003). In particular, defects influence the fatigue lives of casting alloys not only by triggering premature failure (Tiryakioglu 2009), but also cause considerable scatters by increasing the variability of mechanical properties (Yi et al. 2003). Therefore, the investigations on defects induced failure mechanism promote a better understanding of size effect in metal fatigue.

Kloos (1976) classified size effects into three types, namely the statistical size effect, geometrical size effect and technological (production and surface) size effect, which gives definite direction for size effect studies. Unfortunately, in general metal fatigue cases, different kinds of size effects work together, which creates barriers for independent analysis. In addition, different types of size effects exert different influences on structural integrity. In engineering practice, structural discontinuities are common in mechanical components, statistical and geometrical size effects always appear in pairs. When transforming the fatigue properties of laboratory smooth specimens to fatigue performances of engineering components, the impact of these two factors cannot be overlooked. Specifically, statistical and geometrical size effects normally yield more significant influence on fatigue strength than technological one (Kloos 1976). However, as this work focuses on describing and modeling the influence of different types of size effect, we only introduce them one by one. Certainly, the joint modeling of statistical and geometrical size effects will be an interesting work in the future analyses. This section reviews the existed studies on such three types of size effects, with more emphasis on statistical and geometrical size effects.

2.1 Statistical size effect

It is generally known that fatigue limit of structures normally decreases with its size/volume (Liao et al. 2020; Furuya 2011, 2008; Beretta and Regazzi 2016). In other words, the greater the highly stressed region volume of the studied object, the higher the probability of more critical microcracks or flaws inside it, which normally leads to eventual failure. This phenomenon is called statistical size effect (Bažant 1999; Kloos et al. 1981; Alava et al. 2009). Statistical size effect is vital for studying the fatigue behavior of real components/structures considering the influence of defects on fatigue strength (Muniz-Calvente et al. 2017; Zhu et al. 2018d). For specimens with different sizes, Fig. 2a and b show respectively the comparison of fatigue lives for AlMgSi1 6082-T6 aluminum alloy specimens (\(V_{\emptyset 3}=3.2\times 10\hbox {mm}^{3}\), \(V_{\emptyset 8}=2.21\times 10^{3}\hbox {mm}^{\mathrm {3}}\), \(V_{\emptyset 22}=1.13\times 10^{5}\hbox {mm}^{3})\) and EN-GJS-400-18-LT ductile cast iron specimens (\(V_{\emptyset 21}=2.45\times 10^{{4}}\hbox {mm}^{{3}}\), \(V_{\emptyset 50}=3.50\times 10^{{5}}\hbox {mm}^{\mathrm {3}}\), \(V_{120\times 120\times 140}=6.40\times 10^{{6}}\hbox {mm}^{{3}})\). Figure 2c shows the influence of size effect on fatigue lives of the TA19 titanium alloy specimens under the same nominal net section stress loading, which supports the above description of statistical size effect. Thus, statistical size effect modeling is the key to establish a sound relationship between fatigue performances of small-scale specimens and full-scale engineering components.

Fig. 2
figure 2

Fatigue behavior of specimens with different sizes

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The statistical study on size effect appears after Peirce’s work in 1926 (Pierce 1926), which introduced the extreme value statistics theory and formulated the weakest link model like a chain. Based on heuristics and experiments, the Weibull distribution was introduced into fatigue failure analysis of metals and ceramics, and a power law for the statistical size effect was then obtained, which lays down the capstone of the statistical theory (Weibull 1951). The theoretical justification was offered with means of a statistical distribution of microcracks or microscopic flaws in Jayatilaka and Trustrum (1977).

The size, position, shape, orientation and physical properties of inhomogeneities (like de-bonded zones or pores) within a microstructure are factors that influence the fatigue strength (Murakami 2012). Regarding this, fatigue lives of cast aluminium alloy strongly depends on the detect size (Ting and Lawrence 1993). Fatigue life is sensitive to the density of defects, with decreasing the number of pores in termite rail welds, the number of cycles forming dominant fatigue crack increased (Fry et al. 1996). Couper (1993) reported that the defect type has shown critical influence on the fatigue life, in which pores in termite rail welds are much more damaging to the fatigue life than the alumina inclusions. In another example, with the same size, sharp crack-like defects are more likely to triggering fracture compared to blunt crack-like defects (Todinov 2005). Therefore, spatial distribution, size distribution and shape distribution of defect population within a microstructure all are influencing factors for fatigue strength and size effect of metals.

The defect size is an exact value in the crack propagation analysis, which represents the effect of the original crack length on the fatigue life. However, in the size effect probability model, it is a statistical feature, which represents the coupling effects of microstructures (including defect size distribution, pore density, etc.) on fatigue behavior of metals. In the sufficiently small defect scale (like the only defects expected in the material are micro-shrinkages of size < 0.2 mm), the shape distribution of defects can be ignored and the spatial distribution of defects can be regarded as homogeneous distribution. The assumption of homogeneous distribution means that the small probability events with concentrated defects are ignored, which well completes the statistical feature represented by defect size. Furthermore, Murakami (2012) indicates that size distribution of the defect initiating dominant fatigue cracks determines the real fatigue limit, because once the crack initiates from defects or other inhomogeneities, the size of the crack determines the stress intensification at the crack tip and accordingly the threshold condition. Thus, the defect size distribution as a main statistical feature is normally used for describing macroscopically the effect of defect population in size effect modeling.

To model the influence of material defects on structural fatigue behavior, two main approaches are widely used for statistical size effect analysis (Fjeldstad et al. 2008; Tomaszewski et al. 2014), i.e. critical defect method and weakest link method.

2.1.1 Critical defect method

Fatigue crack initiation is a local phenomenon, which is generally triggered by different kinds of defects statistically distributed within the material volume (Hertel and Vormwald 2012). Based on statistical theories, a common basic understanding is that, in a given material volume, fracture is usually triggered by a critical defect defined via its strength or size (Makkonen 2001; Todinov 2001). Based on statistical theories of extremes, the size distribution of defects is defined by a characterizing function of material (Hertel and Vormwald 2012; Nohut 2014). Then, the critical defect size leading to prospective fatigue failure can be estimated. Based on the defined critical defect, the lower bound of fatigue life or fatigue strength can be determined by an appropriate fracture criterion. Accordingly, by calculating the sizes of critical defects in different material volumes, the statistical size effect can be modeled. This strategy is named as the critical defect method (Fjeldstad et al. 2008), its procedure is shown in Fig. 3.

Fig. 3
figure 3

Flow chart of critical defect method

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Statistically, it implies that critical defect determines the fracture-related mechanical properties (Makkonen et al. 2014). Different from the small volume of testing materials in which defect information can be obtained by conventional inspection methods (Dai et al. 1993), the defect size distribution in real components can only be postulated based on statistical methods (Tiryakioglu and Campbell 2010).

In the following paragraphs, two methods for predicting fatigue critical defect size based on the statistics of extremes are presented (Fjeldstad et al. 2008), namely the block maximum method (which models the size distribution of the characteristic largest defect only (Casellas et al. 2005; Gruenberg et al. 1999; Zamber and Hillberry 1999; Beretta et al. 2005)) and peak over threshold method (which takes the distribution of the defects above a certain size into account (Shi et al. 1999; Smith 2009; Shirani and Härkegård 2009)).

(1) Block maximum method   Block maximum method was developed based on the generalized extreme value (GEV) distribution (Mohammad et al. 2014). In this method, only the size of the largest defect in each of k control areas is measured (Fjeldstad et al. 2008). Particularly, the total volume (polished area) is separated into k equally inspection volumes \(V_{0}\) (area \(A_{0})\). As stated in (Anderson and Coles 2002; Anderson et al. 2005), the size distribution of defects in three-dimensional studied volume can be determined from the size distribution of defects in two-dimensional area by stereological approximation. Therefore, the control region of volume \(V_{0}\) can be deduced from the control region of area \(A_{0}\).

The information of all defects over the detection size are measured in k control regions, and the largest defect in each of control regions can be decided, i.e. \(a_{max}=\mathrm {max?}a_{1},...,a_{i}\). Hence, the results of the block maximum method are a set of k observations of maximum defect sizes \(a_{max,1},...,a_{max,k}\).

(2) Peak over threshold method For the peak over threshold method, all defects with size larger than a defined threshold (sufficiently high) \(a_{th}\) are chosen. The results of the peak over threshold method are a set of iobservations, \(a_{1},...,a_{i}\). The statistical analysis is based on the defects with sizes over the threshold, i.e. the values in \(\left[ a_{i},a_{th} \right] \).

The common point of block maximum method and peak over threshold is that both methods utilize the defect data in small inspection volumes to predict the critical defect in larger volumes, which avoid difficulties in measurement, especially the perspective of resolution (Fjeldstad et al. 2008). However, in block maximum method, only the largest measured defect is used, while the rest of defects data are then abandoned, which may ignore other important information of these defects data. In contrast, the peak over threshold method allows more effective usage of inspected data to estimate the defect size distribution.

In the early stage of critical defect method, due to limited experimental conditions, more discussions focused on how defects information could be described more reasonably by statistical approach. Among them, Shi et al. (1999) firstly applied the peak over threshold method to defect size distribution in clean steels. Makkonen (2001) utilized the lognormal distribution to describe the parent population of initial cracks. Based on the Linear Elastic Fracture Mechanics theory, fatigue limit of the target specimens can be estimated with the largest initiated crack calculated from tested specimens. However, for better quantifying the statistical uncertainties, the Makkonen’s method still needs further improvement on scatter prediction. In addition, Makkonen et al. (2014) mentioned that the quantile of the largest defect may be more suitable for modeling size effect than the characteristic size of the largest defect. Note from Beretta et al. (2005) that the estimated characteristic size of maximum defect based on Gumbel extreme value distribution can be utilized in crack propagation model to analyze size effect and predict the fatigue limit of full-scale structures, although some additional parameters are required.

Moreover, Fjeldstad et al. (2008) compared the predicted results of fatigue life and fatigue strength under the assumption that the defect size follows Gumbel and Frechet distribution, respectively. As shown in Table 1, due to the heavier upper tail of Frechet distribution, for smooth specimens, the critical defect size based on Frechet assumption is larger than that by Gumbel assumption, thus its predicted fatigue life and fatigue strength are lower. Nevertheless, regarding their applications in notch fatigue analysis, there is no significant difference.

Table 1 Comparison of simulation results under Gumbel and Frechet distributions (Fjeldstad et al. 2008)
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Since Gumbel and Frechet distributions are similar in simulation results, Khoukhi (2019) chose Gumbel distribution comparing with Log-normal and Weibull distributions by statistically fitting the critical defects. Calculation results are shown in Table 2, Log-likelihood is a fit index and high Log-likelihood indicates that result is reliable.

Table 2 Comparison of fit goodness for the critical defects under different distributions (Khoukhi 2019)
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Table 2 shows that the distribution fitting of critical defects under Log-normal and Weibull distributions both produced better fitting results with higher ranking points (goodness of fitting) than that under Gumbel distribution, which indicates that these two distributions are both suitable for defect statistics.

With the advances of defect studies, how to use defect statistics effectively for size effect modeling becomes a new hotspot in the field. Khoukhi et al. (2018) presented a numerical approach based on Monte–Carlo simulation for linking the extreme value distribution of the critical defects generated by parent defect distribution and size effect, which can be applied for local fatigue strength estimation. Shirani and Härkegård (2011a, 2011b, 2009) performed a series of studies in fatigue crack growth analysis, in which both the size distribution and the location of defects were utilized for predicting fatigue life distribution and size effect modeling. Based on Shirani’s researches, Rafsanjani and Sørensen (2015), Wan et al. (2014) and Ai et al. (2019) concluded that not only defect location, but also the damage mechanism of surface initial crack (surface correction) should be considered for deducing P-S-N curves and fatigue failure analysis of engineering components with different sizes.

In Ai et al. research (Ai et al. 2019), when performing fatigue failure analysis, a model with two sub-regions, namely an internal region and a surface rim region, was introduced and the volume with highest failure probability was defined as the critical volume for failure probability estimation of studied object. Based on this, a surface position correction coefficient according to damage mechanism of surface initial crack was proposed to characterize the influence of surface defect on fatigue life. As shown in Table 3, for specimens with surface defects, prediction results with defect location and surface correction show better agreement with experimental results than others, which directly affects size effect modeling.

Table 3 Comparison of predicted results with/without defect location and surface correction (Ai et al. 2019)
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It is worth noting that the major issue of critical defect method is that, due to the limited experimental condition, it is difficult to know the size and number distributions of initial cracks in advance (Alava et al. 2009). Moreover, there are still a lot of obstacles in the transfer among defect distribution of different materials (Fjeldstad et al. 2008), like the prediction error caused by diverse material microstructures. Therefore, the choice of block maximum method and peak over threshold method depend on the inspection conditions, and critical defect method were commonly used for statistical size effect modeling (Smith 2009). Note that critical defect method mainly deals with the failure cases caused by the critical defect or when the size and number distributions of initial cracks can be obtained in advance; while for other failure cases, the weakest link method works can be utilized.

2.1.2 Weakest link method

As one of the most commonly used theories for statistical size effect description, there are two kinds of the general idea of weakest link argument for the weakest link theory under the Weibull model (Tomaszewski et al. 2014). Firstly, the basic assumption for the weakest link theory is the material heterogeneity, i.e. defects randomly distribute with a certain density per unit volume (Wormsen and Haerkegaard 2015). Trantina and Johnson (2009) presumed that the dispersity of fatigue life or strength depends both on the size distribution of critical initial defects and the specimen size effect on the average defect size. When observing the defect distribution (size and number) of the material, if only one specimen or a small amount of material is tested, the probability of sampling a critical defect leading to fracture is very small (Todinov 2001; Brückner-Foit et al. 1990; Okeyoyin and Owolabi 2013). However, with increasing stressed volume and/or region, the size and number of the severe defects also increase while fatigue strength of the material decreases (Cova et al. 2014; Song 2013; Tiryakioğlu 2015).

The other general idea is that, to investigate the influence of size effect on fatigue behavior, fatigue strength of an investigated object is controlled by its weakest local sub-volume (Alava et al. 2009; Wang et al. 2012b). For a disordered system, the increased specimen volume leads to higher probability with weak regions (Trujillo et al. 2014). The main difference between weakest link method and critical defect method is that fatigue life is not determined according to the information of detected critical defect only, each defect can evolve to be the ‘weakest link’ (Abroug et al. 2018a, b; Xu et al. 2019). Compared with the latter, the weakest link method within the weakest region does not involve complex defect information, and is therefore widely used in engineering practice (Wang and Shao 2014; Muñiz-Calvente et al. 2015; Zok 2017; Zhang et al. 2016; Sun and Song 2018). It’s worth noting that these two methods are complementary in reality.

As aforementioned, weakest link method is widely applied to reliability evaluation under statistical size effect, in which its typical form based on the Weibull distribution is written as:

$$\begin{aligned} P_{N\mathrm {,}chain}\left( \sigma \right) =1-\hbox {exp}\left\{ -{n\left( \frac{\sigma }{\sigma _{\mathrm {0}}} \right) }^{b} \right\} \end{aligned}$$
(1)

where \(\sigma _{0}\) is the reference stress and b is the Weibull modulus. \(P_{N,chain}\left( \sigma \right) \) denotes the probability that the failure stress is smaller than \(\sigma \) for the chain composed of n links, which means that the average strength of interest decreases with the increasing of n.

In the early study of weakest link theory, its application range is one of the research highlights. Among them, Weibull (1951) developed the concepts of weakest link theory to model the probability distribution of tensile fracture behavior of brittle solids. Then, the weakest link theory was commonly used to explain the strength problem of brittle materials which fail in brittle fashion, then ductile metal and alloy materials (Kittl and Diaz 1988; Kittl and Díaz 1990; Carpinteri 1989; Zaitsev and Wittmann 1974; Zech and Wittmann 1977).

With the in-depth weakest link method, not only fatigue strength can be evaluated, but also more fatigue life information can be obtained from the statistical size effect modeling. Based on the weakest link theory, Karolczuk and Palin-Luc (2013) used a function of a local damage parameter to evaluate high cycle fatigue life considering notch effect. Castillo et al. (2006), Przybilla et al. (2011, 2013), Blasón et al. (2017) presented a three-parameters Weibull distribution to solve the problem of testing the hypothesis of length independence in fatigue life, and established the \(S{-}N\) diagrams using samples with different lengths. Shirani and Härkegård (2011a, 2011b) used the Weibull weakest link method to draw \(P{-}S{-}N\) diagrams under size effect, but the effectiveness of this method still remains to be further validated by specimens with diverse geometries.

Recently, more attentions have been paid on transferring the application of weakest link method from uniaxial fatigue to multiaxial one. Using the weakest link theory, Flaceliere and Morel (2004) tried to determine the scatter of strength properties in multiaxial fatigue, but further study is still expected. Based on the Weibull distribution, Muñiz-Calvente et al. (2017) presented a probabilistic model which enables the failure probability to be found for any selected plane orientation by considering the damage gradient as a parameter for both proportional and non-proportional loadings.

Overall, the behavior of statistical size effect where the crack initiation source emerges from the critical defect or the critical region (weakest link) of the material can be characterized by the weakest link theory (Lieb et al. 1981). Weakest link theory for probabilistic fatigue analysis has shown advantages like it does not require information concerning the size and number distribution of defects, even the crack growth rate behavior (Trantina and Johnson 2009) and shows great prediction ability in the size effect of brittle materials. However, note from Bažant and Yavari (2005) that the application of traditional weakest link method faces the following difficulties:

  1. (a)

    There are difficulties in explaining the energy release theory;

  2. (b)

    All structures are mathematically equivalent to uniaxial stress bar;

  3. (c)

    There are great limitations in applicable materials, it can only be applied to brittle material with similar geometries;

  4. (d)

    There are some statistically fundamental controversies on the Weibull scale parameter b (Zok 2017; Afferrante et al. 2006);

  5. (e)

    Assuming that local failures are considered independent, it ignores some of the facts since cracks induce multiple interactions that may correlate stresses in different regions of the material (Alava et al. 2009);

  6. (f)

    Between two- and three-dimensional similarity, there is an excessive prediction error on the size effect.

2.1.3 Comparison of critical defect method with weakest link method

Figures 4 and 5 show the calculating procedures of the Ai et al. model (Ai et al. 2019) based on critical defects information and the Weibull weakest link model, respectively.

Fig. 4
figure 4

Flow chart of the Ai et al. model

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Fig. 5
figure 5

Flow chart of the Weibull weakest link model

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As shown in Figs. 4 and 5, compared with the Weibull weakest link model, the Ai et al. model (Ai et al. 2019), which considers defect information, is more reasonable. While from another viewpoint, the Weibull weakest link model only requires fatigue test data, which conveniences its application in engineering practice.

For a quantitative comparison of these two methods, size effect of three materials (including AlMgSi1 6082-T6 aluminum alloy, EN-GJS-400-18-LT ductile cast iron and TA19 titanium alloy) were evaluated respectively by the Ai et al. model (Ai et al. 2019) based on critical defect method and the other three models [Wang’s model (Wang et al. 2017), Przybilla’s model (Przybilla et al. 2013), Shirani’s model (Shirani and Härkegård 2011a, b)] using the weakest link concept, which ignored both types and size distributions of defects. By comparing these methods, the prediction error is quantified by Eq. (2):

$$\begin{aligned} P_{error} =\hbox {lg}\left( N_{fp} \right) -\hbox {lg}\left( N_{ft} \right) \end{aligned}$$
(2)

where \(N_{\textit{fp}}\) is the predicted life and \(N_{\textit{ft}}\) is the experimental life. Exactly, a positive mean value of \(P_{\textit{error}}\) means overestimated predicted life, and negative for conservative predictions.

Figure 6 presents the comparison of model prediction errors on the size effect of three materials.

Fig. 6
figure 6

Comparison of prediction errors (Ai et al. 2019)

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As shown in Fig. 6, for statistical size effect of the three materials, the prediction results under both critical defect method and weakest link methods provide acceptable results, with mean values closer to 0 and smaller standard deviations. In particular, the prediction of the Ai et al. model based on critical defect characterization is slightly better than other three models using the weakest link concept. However, critical defect method, including the Ai et al. model, generally require an overall detection of the defect information inside the investigated body. Defect information and two sets of fatigue data were applied for the Ai et al. model, whereas only one set of fatigue data was used for weakest link model fitting based on Weibull distribution. Actually, under limited defect detection conditions, the weakest link method is more effective for statistical size effect modeling.

2.2 Geometrical size effect

Change of the stress gradient is responsible for decreasing fatigue limit and fatigue life with different geometries, like notches, which is named as the geometrical size effect (Kloos et al. 1981). As shown in Figs. 7 and 8, three sets of experimental data illustrate that different geometries correspond to diverse fatigue behaviors. In particular, the notch effect in fatigue (generally characterized by the fatigue stress concentration factor \(K_{t})\) has been known to be dependent with geometrical size effect. Under the same stress concentration factor, the notch effect is more significant with decreasing notch size. When notch size is below a certain condition, the notch effect shows small impact on the fatigue limit (Lukás et al. 1989). Thus, traditional method for notch effect cannot be directly used for the study of geometrical size effect.

Fig. 7
figure 7

Configurations of notched specimens of three materials. a TC4 alloy bar specimens (Editorial Committee China Aeronautical Materials Handbook 2002), b TC11 alloy bar specimens (Xin et al. 2012), c Al 7075-T6 alloy sheet specimens (2.286 mm thick) (Illg 1956).

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Fig. 8
figure 8

Configurations of three materials

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Until now, large amounts of theories and approaches are available for notch and size effects estimation in fatigue analysis. Among them, to model geometrical size effect in engineering application, various empirical relationships have been established (McEvily et al. 2008), like the Kuhn’s empirical rules based on stress concentration factor (Kuhn and Hardraht 1952). Then, under the experimental data of 26 specimens with different geometries, Härkegrd and Halleraker (2010) attempted to rank the predictive capability of these methods, where coefficients of variation (COV) was utilized to characterize their prediction performances, see Table 4.

As mentioned above, studies on notch effect and geometrical size effect both handle the notch issue. While the study of notch effect focuses on stress and strain states at area near the notch, like the field intensity method (Yao 1993). For the study of geometrical size effect, the influence of notch size is explored based on the stress gradient. Due to the same concern in stress gradient, some simple and efficient methods of notch effect are generally applied for studying geometrical size effect, such as Neuber method (Neuber 1960) and critical distance theory (Susmel and Taylor 2007). As shown in Table 4, the notch effect methods can be directly used to model geometrical size effect, but it cannot get expected results.

Table 4 Rank of prediction quality in the assessment of geometrical and statistical size effect
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Specifically, Table 4 shows an apparent superiority of the highly stressed volume (HSV) method with minimum COV for the geometrical and statistical size effects. In addition, with the development of numerical techniques, the HSV method has shown broader prospect for development and engineering application, and has been widely used for geometrical and statistical size effects analysis. In next subsection, HSV method is discussed in detail for geometrical size effect modeling.

Fig. 9
figure 9

80% HSV from different specimens

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2.2.1 HSV method

As one of the most commonly used approaches for fatigue strength evaluation considering geometrical and statistical size effects (Khoukhi et al. 2018; Norberg and Olsson 2007; Van Hooreweder et al. 2012), the HSV method was originally developed by Kuguel (1961). In particular, the HSV is defined as the local volume subjected to more than \(n\mathrm {\% }\) of the maximum stress existing within the object of interest (Mohammad et al. 2014):

$$\begin{aligned} \sigma _{c,n\% }=n{\% \times }\sigma _{c,max} \end{aligned}$$
(3)

where \(\sigma _{c,max}\) is the maximum local stress in the critical region and \(n\mathrm {\% }\) is an empirical parameter for defining the scale of HSV. The terminology \(\sigma _{c,n\mathrm {\% }}\) is utilized to determine the volume of the material subjected to stress level higher than \(n\mathrm {\% }\) of the maximum stress. The HSV of specimens with different geometries are shown in Fig. 9.

The probability for crack initiation/propagation increases with the increasing critical volume, which leads to eventual failure. Consequently, there is no need to analyze what happens in the whole specimen, but only in its critical region where the stress levels are higher. Accordingly, Lin and Lee (1998) presented a relation between the maximum elastic stress of materials and the HSV:

$$\begin{aligned} \sigma _{max}={BV}^{-m} \hbox { or } \sigma _{A}={BV}^{-m} \end{aligned}$$
(4)

where \(\sigma _{max}=K_{t}\cdot S_{e}\) is the maximum elastic stress in a specimen which is equal to the elastic stress concentration factor times the nominal fatigue strength, \(\sigma _{A}\) is the fatigue strength and V is the highly stressed volume (surface). Both parameters B and m are determined experimentally. Particularly, their values depend primarily upon the type and strength of the material concerned. In addition, Eq. (4) can be transformed into Eq. (5):

$$\begin{aligned} \frac{\sigma _{max}}{\sigma _{max,o}}=\left( \frac{V_{0}}{V} \right) ^{m} \hbox { or } \frac{\sigma _{A}}{\sigma _{A0}}=\left( \frac{V_{0}}{V} \right) ^{m} \end{aligned}$$
(5)

where the subscript o represents the reference specimen. It indicates that a component can withstand higher stress amplitude than that with a larger HSV.

Based on the above-mentioned viewpoints, many researchers utilized Weibull weakest link model to study the influence of HSV change on the fatigue strength or life of interest (Kaffenberger and Vormwald 2012; Leitner et al. 2017). Notably, Sun and Song (2018) pointed out that Weibull weakest link method based on HSV gives better predictions than the peak stress and the critical distance methods of Neuber (1960) and Peterson (1959) during geometrical size effect modeling. The flow chart of HSV method is summarized in Fig. 10 (parameters and formulas of the Ai et al. model can be found in Ai et al. (2019)).

However, the partial HSV method still need to be intensively studied, as shown below:

Fig. 10
figure 10

Flow chart of the HSV method

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(1) Discussion of parameter m

Since Eq. (3) is originally proposed as an empirical equation (Kuguel 1961), the parameter m in Eqs. (4) and (5) has been considered as material constant determined by engineering experiences and is a measure of its sensitivity on the size effect. In the Weibull weakest link method, many studies assume that parameter m directly relates to the shape parameter of Weibull distribution (Wang et al. 2012a; Kaděrová and Miroslav 2013). However, note from Afferrante et al. (2006), it is perhaps overidealized to assume m as material constant, since it ignores the variability of the relationship in the HSV under different stress states. Therefore, the application of the HSV method is principally limited to the prediction of fatigue strength or life distribution in geometrically similar specimens (Bažant and Yavari 2005). In view of this, Ai et al. (2019) developed a dynamic model coefficient based on equivalent highly stressed volume to predict fatigue life distribution among specimens with different geometries, which provides great convenience for engineering applications.

(2) Discussion of scale parameter n%

As the scale parameter of critical region, \(n\% \) is regarded as an empirical constant and not uniformed in literature. Therefore, there are still no generally acknowledged values. Kuguel (1961) firstly proposed the concept of 95% HSV, and numerous studies employed it later (Lin and Lee 1998; Grell et al. 2007). Recently, 90% HSV (i.e. ‘V90 criterion’) has been confirmed with higher prediction accuracy and is widely utilized in fatigue analyses of metallic components (Sonsino et al. 1997; Lipp et al. 2014). In addition, based on the experimental results, Khoukhi et al. (2018) found that V80 is more appropriate than V90 for short-fiber reinforced specimens and components. Thus, the selection of \(n\% \) directly relates to the studied materials, and further studies on this topic are still expected to choose \(n\% \) reasonably.

(3) Determination of the HSV

In early studies, due to technical limitations, a variety of empirical formulas based on mechanics analysis were used for HSV calculation with specific geometries and loadings (Grell et al. 2007). For example, the HSV of a notched specimen is originally estimated by Kuguel (1961) using a triangular approximation. Moreover, Przybilla et al. (2013), Shirani and Härkegård (2011b), Wang et al. (2017), Hertel and Vormwald (2012) developed a stress integral method considering the damage mechanism of structural surface, i.e. the highly stressed surface method.

Table 5 Comparison of average microstructural properties for T95 and T150 blocks (Shirani and Härkegård 2011a)
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Table 6 Comparison of mechanical properties for T95 and T150 blocks (Shirani and Härkegård 2011a)
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With the development of computing technology, the emergence of numerical FE software improves the accuracy of HSV method. Lin and Lee (1998) and Sonsino and Fischer (2005) combined mechanics analysis and stress data from FE analysis to calculate HSV, respectively. Van Hooreweder et al. (2012) put forward a method using the FE analysis of simplified 2D model to estimate the HSV directly. Similarly, based on numerical simulation, Leitner et al. (2017) extended a 2D model to a 3D model for the multiaxial loading assessment of HSV. However, Leitner’s method needs to be improved on the calculation accuracy. Later, based on the node stress of FE method, Ai et al. (2019) developed a highly stressed node method to directly obtain the HSV of 3D model with higher precision. However, it requires high density meshing near the highly stressed region, more attempts are desired to improve computing efficiency. In order to account for both notch and size effects, He et al. (2021) modified the Weibull distribution using critical distance theory and HSV method for probabilistic fatigue assessment of notched components.

2.3 Technological size effect

The production procedure changes with the size of structural components, which leads to attribute changes (such as microstructure, surface roughness or residual stresses) and shows different fatigue lives (Kloos 1976), i.e. technological (production or surface) size effect (Zamiri Akhlaghi et al. 2011).

(1) Microstructure

As mentioned above, the differences in fatigue strength are caused by diverse material microstructures, which stems from the different production techniques of materials with different sizes, i.e. production size effect (Shirani and Härkegård 2011a, b). The size effect of materials containing defects can be better understood in the perspective of microstructure, and then it can be described by statistical methods.

In order to study technological size effect under the metallurgical change within EN-GJS-400-18-LT ductile cast iron, two types of castings with 95 mm (T95) thickness and 150 mm (T150) thickness were processed (Shirani and Härkegård 2011a). For T95 and T150, the microstructural properties and mechanical behaviors are summarized in Tables 5 and 6, respectively. Particularly, Fig. 11 compares the fatigue test results for Ø21 specimens from T95 and T150 blocks.

Fig. 11
figure 11

Comparison of fatigue behavior for Ø21 specimens of T95 and T150 blocks Shirani and Härkegård (2011a)

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Note from the metallography analysis of ductile cast irons, by decreasing the cooling rate, the nodularity and nodule counts of the microstructure decreases (Shirani and Härkegård 2011a, b). In other word, when increasing the casting thickness, the cooling rate and the fatigue strength decrease, which confirms the technological size effect under metallurgical points of view. For microstructure change, its influence on fatigue strength/property is an overall phenomenon, this work focuses on quantifying macroscopically the relationship between them, the respective role of the microstructure and the defect populations are expected to be further studied in the future.

(2) Surface roughness

Similarly, surface roughness varying with sizes is another type of technological size effect, i.e. surface size effect in some studies (Kloos et al. 1981). As shown in Fig. 12, fatigue strength of material decreases with increased surface roughness.

Fig. 12
figure 12

Fatigue behavior of specimens with different surface roughness (Ra) Alang et al. (2011)

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From the perspective of fatigue analysis, both production and surface size effects (inhomogeneity) can be regarded as the influence of internal defect and surface defect respectively (Kim et al. 2007) and modeled by manufacturing defect analysis (Ai et al. 2019). In particular, production and surface size effects normally work together, which overlaps the statistical size effect in view of defects (Li et al. 2018; Pavlov et al. 2014). Therefore, the influence of these two factors is generally included into statistical size effect modeling (Abroug et al. 2018a, b; Ben Ahmed et al. 2017).

Furthermore, due to direct connection with specific materials and processes, the influence of these two factors is rarely studied separately for fatigue performance except for some special cases. For example, considering the influence of surface roughness on the micro sheet metal forming, Furushima et al. (2013) clarified the surface size effect of metal thin sheet on fatigue behavior by using experiments and inhomogeneous FE simulations.

(3) Residual stress

The fabrication procedure changes with the size of structural components, such as the weld size and number of welding passes (Gurney 1979; Zhao and Herion 2001), which shows different fatigue lives and is classified into technological size effect. In critical sections, residual stresses altering with welding procedure tend to keep the cracks open under fatigue loads, which is detrimental to structural fatigue strength. Note from Zamiri Akhlaghi et al. (2011), FE analysis is a versatile and powerful tool for assessing residual stress distribution with different processes. For example, the resulted residual stress field calculated by thermo-mechanical analysis of welding superposed into crack propagation analysis, which provides more realistic method for crack propagation analysis. Except for welding, other manufacturing processes also yield different residual stresses which significantly influences fatigue strength (Li et al. 2021; Liao et al. 2020). This have been demonstrated by De Jesus et al. (2020) when studying the AISI D2 tool steel processed by grinding and electro-discharge machining (EDM). Moreover, the surface finish is also a key influential parameter on its fatigue strength.

3 Discussions

As aforementioned, manufacturing defects inevitably appear in engineering components; consequently, fully understanding of manufacturing defects matters in size effect modeling and transforming fatigue performances from small (in most cases smooth) specimens to full-scale components. Based on fatigue and statistical behaviors of defects, three main types of size effects are individually reviewed in this study.

Statistical size effect, as a core element of size effect, research on this topic is relatively mature. Particularly, its relevant theories and engineering applications have been well developed. The inhomogeneities within a microstructure are normally influence factors for fatigue strength of metals. Increasing attentions have been paid on microstructure to identify influence factor and find the main influencing factors for fatigue strength assessment under size effect. As a main statistical feature but not the exact value, the defect size distribution represents the coupling effects of microstructures (including defect size distribution, pore density, etc.) on fatigue behavior of metals, which is normally used for describing macroscopically the effect of defect population in size effect modeling. Critical defect method and weakest link theory are two main strategies for statistical size effect modeling. Among them, critical defect method is closer to the reality, but its prediction accuracy relies heavily on defect detection technology and crack growth model. It is noteworthy that to model the actual crack growth process, quantifying the interaction of multiple defects should be intensively studied. In contrast, without the requirement of basic defect information, the weakest link method is simple and efficient in exploring the statistical size effect, and shows a better application prospect by combining it with the HSV method.

Geometrical size effect also plays an important role in fatigue design/assessment of engineering components. In previous studies, many researchers combined statistical size effect method with traditional notch effect method, which ignores the influence of notch size and generally leads to poor prediction accuracy. From the perspective of prediction precision and application, the HSV method has shown superior advantages in coupling statistical size effect with geometrical size effect. However, some empirical parts of the HSV method still deserve further improvements, like the selection of the scale parameter n%, which requires more experimental validations. Nowadays, most researches on size effects were conducted based on uniaxial fatigue data; however, in engineering practice, components normally operate under multiaxial loadings. In this regard, it would be a challenging and practical work to study the size effect in multiaxial fatigue.

Technological size effect is an important branch of size effect; however, it hasn’t been well investigated yet comparing with other branches. For technological size effect, the influences of microstructure and surface roughness are normally included in the calculation of statistical size effect, and the residual stresses influence involves specific production process. Therefore, the separate discussions of technological size effect, especially the impact of various processes with different sizes, are highly expected in future research.

The recent interests on metallic additive manufacturing for producing functional parts raises questions about structural integrity, with fatigue behavior at the forefront of the concerns. Defects, microstructural heterogeneities, high surface roughness and residual stresses are typical signatures of these manufacturing processes. Due to the high cost of full-scale testing and the limitation of equipment, testing small-scale specimens (e.g. miniature specimens) and extracting basic material parameters, then combining with FE method to check the fatigue strength of real structures is an inevitable general procedure. Therefore, accurate and comprehensive modeling of all types of size effects is vital to ensure the smooth transformation from mechanical properties of laboratory specimens to fatigue performances of engineering components.

In summary, in the past decades, the size effect modeling approaches have been well developed, especially statistical and geometrical size effects. However, due to the limited defect detection and FE techniques, there are still some difficulties on its application in fatigue design/assessment of engineering components.

4 Conclusions

The size effect is a problem of scaling, which is central to every physical theory. In this work, recent advances in size effect modeling based on inherent defect information and Fracture Mechanic or statistical theories are reviewed. The following conclusions are drawn:

  1. (1)

    Three kinds of size effects as well as their recent advances in metal fatigue based on defect information were summarized.

  2. (2)

    Two main approaches under defects (critical defect method and weakest link method) are elaborated to describe the influence of statistical size effect on fatigue strength. Then, the HSV method is introduced to describe the geometrical size effect, and the technological size effect caused with different factors are briefly discussed.

  3. (3)

    Further directions on the three size effects are explored, which includes their own shortcomings and possible solutions. For statistical size effect, the interaction of multiple defects should be studied for more real crack growth process. For geometrical size effect, in addition to further improving the empirical parts of HSV method, its application to complex loading conditions deserves further exploration. The study on technological size effect is rudimentary now comparing with the other two effects, various processes under different sizes should be systematically consolidated and summarized.

  4. (4)

    Size effect theories and methods need to be revisited in order to propose appropriate tools for fatigue design of reliable metallic additive manufactured parts (Naik and Fronk 2016).