Validation Study on the Statistical Size Effect in Cast Aluminium

Abstract

Imperfections due to the manufacturing process can significantly affect the local fatigue strength of the bulk material in cast aluminium alloys. Most components possess several sections of varying microstructure, whereat each of them may inherit a different highly-stressed volume (HSV). Even in cases of homogeneous local casting conditions, the statistical distribution parameters of failure causing defect sizes change significantly, since for a larger highly-stressed volume the probability for enlarged critical defects gets elevated. This impact of differing highly-stressed volume is commonly referred as statistical size effect. In this paper, the study of the statistical size effect on cast material considering partial highly-stressed volumes is based on the comparison of a reference volume $V_0$ and an arbitrary enlarged, but disconnected volume $V_{\alpha }$ utilizing another specimen geometry. Thus, the behaviour of disconnected highly-stressed volumes within one component in terms of fatigue strength and resulting defect distributions can be assessed. The experimental results show that doubling of the highly-stressed volume leads to a decrease in fatigue strength of 5% and shifts the defect distribution towards larger defect sizes. The highly-stressed volume is numerically determined whereat the applicable element size is gained by a parametric study. Finally, the validation with a prior developed fatigue strength assessment model by R. Aigner et al. leads to a conservative fatigue design with a deviation of only about 0.3% for cast aluminium alloy.

1. Introduction

Complex cast aluminium parts possess a severely heterogeneous microstructure and therefore it is essential to consider its interaction with the highly stressed volume (HSV). The result of elevated highly stressed volumes in terms of cyclic loading is generally a reduced the fatigue strength. According to References [1,2], size effects can be classified into technological, geometrical, statistical and surface technology size effects. Larger components, respectively larger HSV, increase the probability of critical defect sizes, thus lessening the endurable fatigue strength. The aim of this work is the validation of the statistical size effect with consideration of the microstructural properties, as introduced as probabilistic design method for aluminium castings in References [3,4,5]. In general, the local fatigue strength correlates well with the dedicated microstructure because of the statistical distribution of the defects, apparent in preliminary studies [6,7,8,9,10,11]. Therefore, it is essential to consider the local pore size distribution in the fatigue design process. Fatigue initiating defects in cast parts can be described well with extreme value statistics, like the generalized extreme value distribution (GEV) or the Gumbel distribution [12,13,14]. Further methodologies to assess the statistical size effect with regard to volumetric dependencies and highly stressed surface models are given in References [15,16,17,18,19,20,21,22]. In these latter cases, a highly stressed volume, which is defined as the volume with a particular percentage of the maximum stress node, is taken into account. One of these approaches is the volumetric model of Sonsino [16], who invokes the 90% highly-stressed volume $V_{90}$ and the Weibull exponent $\kappa$ to assess the size effect related fatigue strength, represented in Equation (1).

$$\frac{{\sigma }_{\mathrm{LLF},0}}{{\sigma }_{\mathrm{LLF},1}}={\left(\frac{V_{90,1}}{V_{90,0}}\right)}^{\frac{1}{.}\kappa }$$

(1)

In this equation, ${\sigma }_{LLF,0}$ and ${\sigma }_{LLF,1}$ represent the long-life fatigue strength of the highly-stressed volumes $V_{90,0}$ and $V_{90,1}$. The material dependent Weibull exponent $\kappa$ specifies the slope in the double logarithmic ${\sigma }_{LLF}$-$V_{\alpha }$-plot and therefore the reduction of the fatigue strength against the highly-stressed volume. Its value and can be taken either by a common guideline [23], which defines the parameter as $\kappa =10$ for aluminium castings, or be calculated dependent on the probability distribution of the fatigue data, represented by $T_S$ [17], see Equation (2). In this equation, $T_S$ is the scatter index of the high cycle fatigue region at ten million load cycles, defined as the stress ratio between a 10% and 90% probability of survival. In Reference [24], Sonsino proposed a threshold volume $V_{\infty }=8000{\mathrm{mm}}^3$ for cast aluminium material, implying that no further noticeable decrease in fatigue strength may be observed.

$$\kappa =\frac{1.3151}{log\left(T_s\right)}.$$

(2)

The weakest link model of Weibull [25] as well as the discussed volumetric model are in good accordance to the experimental fatigue data [26]. Studies on artificial defects in References [27,28] exhibit that the highly-stressed volume approach is more suitable to investigate the statistical size effect. Both, the common engineering guideline [23] and short-crack growth findings in Reference [1] recommend a highly stressed surface model but refer additionally to highly-stressed volume models. Hence, the model of Sonsino [16] is used in this study for the validation of the statistical size effect. Kitagawa and Takahashi recommended in Reference [29] that the long life fatigue strength ${\sigma }_{LLF}$ can be related to a dedicated crack length a, respectively to equivalent defect size, which can be defined as equivalent circle diameter (ECD) or by the equivalent edge length of a square ($\sqrt{area}$), see Equation (3). The sound applicability of the model from Kitagawa and Takahashi has been proven in several studies, see References [3,4,7,30,31,32,33,34,35,36].

$$\Delta {\sigma }_{LLF}=\frac{\Delta K_{th,lc}}{Y\sqrt{\pi a}}.$$

(3)

In this equation, $\Delta K_{th,lc}$ is the long crack threshold and Y a geometry factor depending on the geometrical shape and location of the defects, as discussed in preliminary studies such as that in Reference [37]. The design strength is limited on the one hand by the long life fatigue strength of the near defect free material $\Delta {\sigma }_0$, evaluated at specimens with hot isostatic pressed (HIP) condition with T6 heat treatment (HIP + T6). Otherwise, the fracture mechanical approach takes into account the long crack threshold value $\Delta K_{th,lc}$ and the effective crack threshold value $\Delta K_{th,eff}$. These crack threshold values come into effect for flaw sizes becoming larger than a intrinsic crack length $a_{0,eff}$, respectively $a_{0,lc}$. Further improvements of the Kitagawa Takahashi diagram by El Haddad [38,39] and Chapetti [40] are considering the crack resistance curve. A schematic representation of the Kitagawa Takahaschi diagram (KTD) and its modifications are given in Figure 1.

The crack extension from the intrinsic threshold $\Delta K_{th,eff}$ to the long crack threshold $\Delta K_{th,lc}$ can be represented by applying the cyclic crack resistance curve (R-cureve), as introduced by Reference [41]. The build-up of the crack resistance from the intrinsic $\Delta K_{th,eff}$ to the long crack threshold $\Delta K_{th,lc}$ with elevating crack length is caused by crack closure effects [42,43], whereat a premature contact of the crack faces generally leads to a minor real effective load $\Delta K_{eff}$ for further crack propagation, see Equation (4).

$$\Delta K_{eff}=K_{max}-K_{op}.$$

(4)

Crack closure effects can be classified into plasticity-, roughness-, and oxide-induced crack closure fractions as the most pronounced ones, whereat an explicit separation of these effects is not possible [44,45,46,47,48,49]. Maierhofer recommended in Reference [41] a procedure to describe the R-curve in a unified manner as given in Equation (5).

$$\Delta K_{th,\Delta a}=\Delta K_{th,eff}+\left(\Delta K_{th,lc}-\Delta K_{th,eff}\right)\left[1-\sum _{i=0}^n{\nu }_i·\exp \left(-\frac{\Delta a}{l_i}\right)\right],$$

(5)

with

$$\sum _{i=0}^n{\nu }_i\equiv 1.$$

In this equation, the crack closure effects are considered using the parameters ${\nu }_i$ and $l_i$, implying that if the crack reaches length $l_i$ the corresponding closure effect ${\nu }_i$ is completely developed. By inserting Equation (5) in Equation (3), the cyclic R-curve can be implemented in the KTD whereby this extension of the KTD is useful to assess both, physically short and long, cracks. Therein, the crack length a is substituted by the equivalent defect size $\sqrt{area}$. Murakami introduced in Reference [50] the $\sqrt{area}$-parameter, which is the cross section of a defect in respect to the load direction. According to a study in Reference [51], the stress field surrounding the defect correlates well with the $\sqrt{area}$-parameter. Hence, this parameter is used to assess the crack initiating defects. Preliminary studies [5,52,53] contributed to the measurement methods of defects in cast aluminium alloys, as also applied within this study.

In References [12,13,14,54] it was shown that the statistical distribution of defect sizes follow an extreme value distribution. The Generalized Extreme Value (GEV) distribution includes the Frechet, Gumbel and Weibull distribution [55] and is applicable for characterizing crack initiating defect sizes [12]. Its cumulative distribution function (CDF), see Equation (6), is defined by three parameters, named as location $\mu$, scale $\delta$ and shape $\xi$ parameter which can be estimated by using the maximum likelihood method, applied in the studies [12,53,56,57]. The shape parameter $\xi$ determines the type of extreme value distribution, differentiating between three cases: $\xi \to 0$ indicates a Gumbel, $\xi <0$ a Weibull and $\xi >0$ a Fréchet distribution, see Reference [13].

As published in a previous study [3], the CDF of an $\alpha$-times enlarged volume $V_{\alpha }$ of the defect distribution $P^{\alpha }$ can be derived based on the distribution of the reference volume $V_0$ according to Reference [58], expressed in Equations (6)–(12).

$$\begin{array}{c}{V}_0\sim P(\sqrt{area};\mu ,\delta ,\xi )=\exp \left\{-{\left[1+\xi \left(\frac{\sqrt{area}-\mu }{\delta }\right)\right]}^{-\frac{1}{\xi }}\right\}\end{array}$$

(6)

$$\begin{array}{c}{V}_{\alpha }\sim P^{\alpha },\end{array}$$

(7)

with

$${\xi }_{\alpha }=\xi$$

(8)

$$\begin{array}{c}\hfill {\delta }_{\alpha }=\delta ·{\alpha }^{\xi }\end{array}$$

(9)

$$\begin{array}{c}\hfill {\mu }_{\alpha }=\mu +\frac{\delta }{\xi }·\left({\alpha }^{\xi }-1\right),\end{array}$$

(10)

which leads to

$$P^{\alpha }=\exp \left\{-{\left[1+\xi \left(\frac{\sqrt{area}-(\mu +\frac{\delta }{\xi }({\alpha }^{\xi }-1))}{\delta {\alpha }^{\xi }}\right)\right]}^{-\frac{1}{\xi }}\right\}$$

(11)

$$\begin{array}{c}\hfill V_{\alpha }\sim P\left(\sqrt{area};\mu +\frac{\delta }{\xi }\left({\alpha }^{\xi }-1\right),\delta {\alpha }^{\xi },\xi \right).\end{array}$$

(12)

Detailed methodologies to calculate the maximum defect in geometries with enlarged HSV are given in References [59,60], whereat it is shown that the most extremal defects are commonly Gumbel distributed, applied in Equation (13), using the location parameter $\mu$ and scale parameter $\delta$.

$$P\left(\sqrt{area}\right)=exp\left\{-exp\left[-\frac{\sqrt{area}-\mu }{\delta }\right]\right\}.$$

(13)

Now the size of a critical defect in an enlarged control volume $V_{\alpha }$, which is considered by the ratio of the enlarged volume $V_{\alpha }$ divided by the reference volume $V_0$, can be calculated by Equation (14).

$$\begin{array}{c}\sqrt{area}\left(\alpha \right)=\mu -\delta ·ln\left[-ln\left(1-\frac{1}{\alpha }\right)\right]\end{array}$$

(14)

with the return perid $\alpha$ denoted as:

$$\begin{array}{c}\hfill \alpha =\frac{V_{\alpha }}{V_0}.\end{array}$$

(15)

Complex components exhibit various HSVs whereas, mostly, each of them features differences in microstructure due to dependency on local casting process conditions. Thus, a local fatigue assessment considering the microstructural characteristics is beneficial. Even in case of the same microstructure, respectively basic defect distribution, the HSV depends on the component geometry and load condition, which lead to the question if single HSVs may be added together, resulting in a HSV of the whole part and fatigue strength design according to Equation (1), or if each HSV has to be considered individually for all unconnected ones. This paper clarifies this task regarding size effect based fatigue strength design in cast aluminium. Summing up, this paper scientifically contributes to the following points:

• The influence of disconnected highly-stressed volumes as statistical size effect based on accumulated highly-stressed volumes.
• The impact of the highly-stressed volume on the defect distribution and its associated parameters is verified for samples with not-yet investigated casting process conditions. This enhances the existing database and strengthens the a priori established model framework of probabilistic fatigue strength design.
• The effect of the element size during numerical evaluation of the highly-stressed volume is studied and supports recommendations for engineering applicability.
• The work validates the prior developed statistical size effect approach which depends not only on the return period of the highly-stressed volume but takes also the defect distribution of the fractograpic analysis and the material resistance as probabilistic values into account.

2. Investigated Alloy

The material is taken out of a gravity cast automotive part, manufactured using the core package system casting process [61,62]. The components are made of EN AC-46200 with T6 heat treatment [63], whose nominal chemical composition is given in

Table 1. Nominal chemical composition of the investigated cast alloy in weight percent [63].

Alloy	Si [%]	Cu [%]	Fe [%]	Mn [%]	Mg [%]	Ti [%]	Al [-]
EN AC-46200	7.5–8.5	2.0–3.5	0.8	0.15–0.65	0.05–0.55	0.25	balance

. In general, applied steps for T6 heat treatment at aluminium alloys are solution treatment, quenching and age hardening, following defined temperature and time conditions [61,64,65]. First, solution treatment is conducted at high temperatures of approximately 490 °C to 510 °C for about 0.5 h to 8 h to dissolve Cu-rich particles [64,65,66,67,68]. The following quenching in water at ambient temperature, or at 60 °C, leads to a over-saturated solid solution [64,69]. In the third step, the age hardening process is conducted at temperatures from 160 °C to 210 °C for about 4 h to 18 h, whereat in case of higher temperatures a reduced time span is needed to reach the peak hardness, which is the overall aim of T6 treatment [64,67,70,71,72,73,74]. Furthermore, the peak hardness decreases with increasing age hardening temperature [64]. The specimens are manufactured from two different sampling positions, denoted as A and B, where A possesses a highly-stressed volume $V_0$ and B an increased highly-stressed volume $V_1$. Further information about these positions and its local microstructural and mechanical properties are given in detail in References [3,53,75,76]. Within these preliminary studies, the fundamental KTD was built up.

The secondary dendrite arm spacing (SDAS) in position A and B is almost identical and differ by only five percent, resulting in a negligible technological size effect between these two positions. The SDAS was evaluated through an automated procedure described in Reference [77] for linking the microstructural properties to quasi-static [78,79,80] and fatigue properties [7,73,81,82]. Thus, the chosen positions A and B feature specimens of varying geometric sizes but with almost identical microstructural and mechanical properties. The investigated samples possess the same basic circular cross section, but their total length differ. For clarification, specimen A is taken from position A and specimen B is manufactured out of position B. Subsequently, only the specimens are denoted as A and B. To reduce the stress concentration factor within the cross section transition region, the specimens have been numerically shape optimized resulting in a stress concentration factor of only 1.04.

The difference between specimen A and specimen B is, that in case of specimen B, the basic geometry of sample A has been invoked two times in a row. Thus, it is the same as two specimens of type A. Figure 2 and Figure 3 depict the two specimen geometries for high cycle fatigue testing under uniaxial tension load.

The highly-stressed volume of specimen geometry B is roughly doubled in comparison to geometry A. Thus, considering the sum of both sections, a noticeable statistical size effect is expected. To determine the return period of the highly-stressed volumes more accurately, a numerical study regarding the applicable element seed is conducted. A linear elastic finite element analysis has been set-up featuring an uni-axial tension load with couplings to match the experimental clamping conditions.

The element types employed are 20-node quadratic brick C3D20R and 10-node quadratic tetrahedron C3D10 elements with 8 up to 116 elements on each circumference. Additionally, axisymmetric CAX8R elements are used with the same element dimensions to significantly reduce the simulation time. This results in an average element size of approximately $0.24$ mm to $3.5$ mm in the HSV-region, see Figure 4. Another possibility to define the element seed, respectively number of elements per unit length, is the deviation factor, which is defined as the ratio between height h of the segment and the chord length L with n as element number on the circumference, see Equation (16).

$$\frac{h}{L}=\frac{1}{2}·tan\left(\frac{\pi }{2·n}\right).$$

(16)

Thus, a number of about 32 elements on circumference, or a deviation factor of $0.03$, leads to a sound compromise between simulation time and accuracy. The numerically evaluated volume results in a value of $V_{0,90\%}=647$ ${\mathrm{mm}}^3$ for specimen A and $V_{1,90\%}=1284$ ${\mathrm{mm}}^3$ for specimen B, see Figure 5. Concluding, a 90 % highly-stressed volume ratio of $\alpha =1.98$ is obtained for specimen A and B.

In the first phase of the testing procedure the experiment is carried out by clamping part 1 and 3 of the entire specimen with subsequent high cycle fatigue testing until rupture, either at the upper (section 2–3) or the lower (section 1–2) specimen fraction, as depicted in Figure 6. Next, the fractured part is removed (shorter specimen part of section 1 or section 3). Subsequently, the specimen is clamped at the middle part (section 2), see secondary clamping in Figure 6, and the test is continued at the same load level until rupture of the remaining short specimen. It should be highlighted that this shortened specimen possess a HSV which is equivalent to specimen geometry A. The result of this testing procedure are two points in the S/N diagram, which will be discussed in more detail in Section 3.

In order to ensure a homogeneously distributed surface quality with prevention of human influence by polishing, the specimens are polished by a vibratory finishing process. After the CNC machining process, the components are placed in an oscillating bowl containing polishing media. Thereby, the specimens are precision grinded and polished with different abrasive media for several hours until the required surface quality is obtained.

3. Experimental Results

3.1. Fatigue Strength

The fatigue strength of the material is determined at a resonant testing machine with a testing frequency of about 108 Hz with compression/tension loading at a stress ratio of R = −1. In order to focus on the long life fatigue region, the run-out number was set to ten million load cycles. Previous investigations [3] indicated that the transition knee point is close to about two million load cycles for such unnotched samples made of aluminium alloy. As proposed in Reference [83] and applied in preliminary studies [3,76,83,84], the slope of the S/N-curve in the long life region $k_2$ scales with the slope in the finite life region $k_1$ and therefore it is assigned with $k_2=5·k_1$. The S/N curve in the finite life region is evaluated by the statistical procedure given in the standard [85]. The long life region is assessed by the $arcsin\sqrt{P}$ methodology, as proposed in Reference [86]. In the following, the long life fatigue strength of specimen A, taken out of position A, at ten million load cycles and at a probability of survival $P_S=50$ % is used as unifying reference value. Figure 7 presents the statistically evaluated S/N curve of specimen A series including the 90% and 10% scatter band. Be aware that specimen A inherit the highly-stressed volume $V_0$.

Next, the evaluated fatigue data of specimen B at first failure is depicted in Figure 8, again with the 90% and 10% scatter band of the high cycle fatigue region. Thus, the mean long-life fatigue strength ${\sigma }_{LLF}$ of position B decreased by approximately five percent compared to specimen A. The doubling of the highly-stressed volume in position B reveals an evaluable decrease in fatigue strength contributed as statistical size effect.

The evaluated slope $k_1$ at position B in the finite life region is somewhat higher with respect to position A. Additionally, the number of load cycles $N_T$ of the transition knee-point is slightly enhanced. Comparing the scatter indices $T_S$ of the positions A and B, an increase at disconnected highly-stressed volumes is observed. The evaluated long life fatigue strength ${\sigma }_{LLF}$ is listed in

Table 2. Results of the fatigue tests of specimen A and B.

	Specimen	HT	Volume	${\mathit{k}}_1$ [-]	${\mathit{\sigma }}_{\mathbf{LLF},\mathbf{50}\%}$ [-]	${\mathit{N}}_{\mathit{T}}$ [-]	${\mathit{T}}_{\mathit{S}}$ [-]
	A	T6	$V_0$	7.84	1.00	1,100,000	1:1.08
	B	T6	$V_1$	10.73	0.96	1,900,000	1:1.23

, where all fatigue strengths are normalized by position A with a probability of survival $Ps=50$ %. Furthermore, the slope $k_1$ of the finite life region, the number of load cycles for the transition knee-point $N_T$ and the statistically evaluated fatigue scatter index $T_S$ are given in Table 2.

Thus, the statistical size effect may be clearly identified for such samples possessing an increased highly-stressed volume, even though this volume is not coherent as shown in Figure 5. On the other hand, if the highly-stressed volume would be considered separately, which means that no statistical size effect occurs in case of non-coherent highly-stressed volume, both S/N curves in Figure 7 and Figure 8 must coincide. Therefore, the experimental point ${\sigma }_{B,Ps50}$ should be congruent with the point ${\sigma }_{A,Ps50}$ for the same connected highly-stressed volume. But the experimental point of specimen B ($V_1$) with two separated highly-stressed volumes $V_0$ is below the fatigue strength of specimen A ($V_0$).

Therefore, as main finding based on the presented experiments, the entire highly-stressed volume has to be considered for the statistical size effect. Thereby, the entire highly-stressed volume $V_1$ is calculated by the sum of the separated, non-coherent highly-stressed volumes whereat the failure of one single highly-stressed volume leads to a collapse of the specimen. The working hypothesis for first, and second, fatigue failure of specimen B and a theoretical discussion is given in detail in Appendix A.

3.2. Fractography

The crack initiating defect sizes of the HCF specimens are evaluated subsequently to the fatigue testing utilizing a digital optical microscopy for macroscopic inspection and scanning electron microscopy respectively for magnification enhanced, local analysis. According to previous investigations [5,53], defect sizes are evaluated by their precise contour in contrast to the coarser method proposed by Murakami in Reference [59], where a smooth hull contour, which envelopes the original shape, is utilized. This measurement methodology leads to smaller, but more precisely evaluated defect sizes, and it minimizes the distortive effects of projected pore shape onto the statistical evaluation of defect sizes. Therefore, a spline is drawn manually at the contour of the defect using the software Fiji, allowing to calculate the enclosed area. The analysis of the initiating cracks of the specimen A and B revealed that in most cases the technical crack initiates at surface near defects, as depicted in Figure 9. Thus, the increased stress intensity of surface-intersecting defects and surface near defects lead to a lowered crack initiation phase compared to closed defects within the bulk volume [53]. With the existence of superior internal defects in a few samples, in the majority of them specimen B cases, origin of fracture is shifted into the centre of the specimen, exemplary see Figure 10.

As depicted in Figure 11, in a few cases another failure mechanism is recognizable. According to previous studies [5,87] large slip plane areas can operate as failure reason for load amplitudes within the finite life region. This is more likely to happen for increasing loads. It is stated in References [88,89], that in fine microstructures with a small SDAS, the dislocations are able to move across the cell boundaries of the dendrites, since there are no particles to block them. This is in contrast to larger SDAS values by means of coarse microstructures where the dendrite cell is isolated by a thick eutectic wall blocking the dislocations. Due to that, a critical defect size exists, below that the crack initiates at slip bands instead of interdendritic shrinkage pores. From preliminary studies [5,53] it can be assumed that this failure mechanism only occurs at specimens with lower SDAS and quite high load levels of the S/N-curve.

Summing up the experimental work, the fractographic analysis revealed that in most cases the crack initiation starts at interacting shrinkage porosity near the surface, see Figure 9. Therefore, defects are regarded as interacting if the distance between two defects is less than the size of the smaller defect, as proposed in Reference [90]. For the subsequent statistical evaluation of the critical defect sizes the generalized extreme value distribution is applied, following the proposal of Reference [13]. The associated cumulative distribution function (CDF) is given in Equation (6). Following Reference [91], a Kolmogorov-Smirnov (KS) test is conducted to evaluate the goodness of fit for the statistical assessment of the distribution. A perfect compliance for the fit is given with a value of $p_{KS}=1.00$ in the KS-test.

The evaluated probability of occurrence $P_{Occ}$ of casting defects for the reference volume $V_0$ as well as the two times enlarged volume $V_1$, reflecting specimen A and B, is drawn in Figure 12. Assuming that the failure of one section causes the failure of the whole component, only the first fracture and its associated flaw size are utilized for the evaluation of the distribution parameters. In addition, the parameters for the distributions in Figure 12 are statistically evaluated using the maximum likelihood estimation, as proposed in Reference [56]. The evaluated parameters of the distributions from specimen A and B, the result of the Kolmogorov-Smirnov test and the evaluated defect size with a probability of occurrence of 50% are listed for comparison in

Table 3. Statistically evaluated distribution parameters of the generalized extreme value distribution (GEV).

	Position	Volume	$\mathit{\mu }$ [$\mathsf{\mu }$m]	$\mathit{\delta }$ [$\mathsf{\mu }$m]	$\mathit{\xi }$ [-]	$\sqrt{\mathit{area}}\left({\mathit{P}}_{\mathbf{Occ}=0.5}\right)$ [$\mathsf{\mu }$m]	${\mathit{p}}_{\mathbf{ks}}$ [-]
	A	$V_0$	95.1	20.1	0.43	103	0.93
	B	$V_1$	118.4	28.1	0.36	129	0.69
	B (model)	$V_1$	111.3	27.1	0.43	122	0.58

.

As mentioned before, in this study only the first failures per specimen are considered for the validation of the statistical size effect. The probability of occurrence $P_{Occ}$ of a critical defect for an $\alpha$-times enlarged volume $V_{\alpha }$ can be estimated based on the statistical distribution of the reference volume as reasoned in Reference [3] and shown in Equation (17). The evaluation of the location ${\mu }_{\alpha }$, shape ${\xi }_{\alpha }$ and scale ${\delta }_{\alpha }$ parameter for the distribution $P^{\alpha }$ is given in Equations (8)–(10). The parameters of the distribution $P^{\alpha }$ are listed in Table 3.

$$P^{\alpha }=\exp \left\{-{\left[1+{\xi }_{\alpha }\left(\frac{\sqrt{area}-{\mu }_{\alpha }}{{\delta }_{\alpha }}\right)\right]}^{-\frac{1}{{\xi }_{\alpha }}}\right\}.$$

(17)

4. Verification of Size-Effect Related Fatigue Strength

In order to study the size effect as influence of the highly stressed volume, the probabilistic model of the preliminary work [3] has to be applied to evaluate the local Weibull factor $\kappa$. It depends on the return period $\alpha$ of the highly-stressed volume and the local defect population ${\mu }_0$. As the same aluminium alloy with T6 heat treatment was used also in the previous model development regarding fatigue strengths, the diagram can be easily rebuilt for the varying return period, respective defect population within the highly-stressed volume. The local Weibull factor $\kappa ({\mu }_0,\alpha )$ can be obtained by transforming Equation (1). This results in a value of ${\kappa }_{{\mu }_0,\alpha }=15.27$ based on the experimental results for the $\alpha$-times enlarged volume in case of specimen B, see Equation (18).

$$\kappa ({\mu }_0,\alpha )=\frac{\log \left(\alpha \right)}{\log (\Delta {\sigma }_{\mathrm{LLF},V_0})-\log (\Delta {\sigma }_{\mathrm{LLF},V_{\alpha }})}.$$

(18)

In the fundamental work of Reference [3], the Kitagawa-Takahashi diagram (KTD) was used to assess the fatigue strength $\Delta {\sigma }_{\mathrm{LLF},V_0}$ and $\Delta {\sigma }_{\mathrm{LLF},V_{\alpha }}$ depending on defects, respective microcracks. Therein, crack propagation tests have been conducted with specimens manufactured from the identical positions as used in this study to minimize microstructural deviations. To extend the KTD for physically short and long cracks, the crack-resistance curve was implemented [4]. A summary of the fracture mechanical variables, determined by crack propagation tests from previous studies [4], is given in

Table 4. Parameters resulting from crack propagation tests in position A and B for a probability of occurrence of $P_{Occ}=50$%.

	$\mathbf{\Delta }{\mathit{K}}_{\mathit{th},\mathit{lc}}\left[\mathbf{MPa}\sqrt{\mathbf{m}}\right]$	$\mathbf{\Delta }{\mathit{K}}_{\mathit{th},\mathit{eff}}\left[\mathbf{MPa}\sqrt{\mathbf{m}}\right]$	${\mathit{\nu }}_1$ [-]	${\mathit{\nu }}_2$ [-]	${\mathit{l}}_1$ [mm]	${\mathit{l}}_2$ [mm]
	3.95	1.06	0.4	0.6	0.03	0.75

for the investigated alloy. No statistically feasible difference in fracture mechanical material properties of position A and B has emerged.

The long life fatigue strength of the near defect free material $\Delta {\sigma }_0$, which defines the upper limit of the left side of the KTD, was evaluated with specimens in HIP treatment condition at the same position. In this model the fatigue strength $\Delta {\sigma }_{LLF,V_0}$ is determined using the R-curve extension [40] for a defect size represented by the size of an defect $a_m$ of the reference volume $V_0$ implying a probability of occurrence of $P_{Occ}=50$%. The critical defect size for an enhanced volume can be estimated by application of Equations (6)–(11). This results in a fatigue strength $\Delta {\sigma }_{LLF,V\alpha }$ for an enhanced volume $V_{\alpha }$ using the given defect distribution (${\mu }_{\alpha }$,${\delta }_{\alpha }$) with a specific defect size $a_{m,\alpha }$.

$$\Delta {\sigma }_{\mathrm{LLF},V_0}=\frac{\Delta K_{th,\Delta a}}{Y·\sqrt{\pi ·a_m}},$$

(19)

with

$$\Delta K_{th,\Delta a}=\Delta K_{th,eff}+\left(\Delta K_{th,lc}-\Delta K_{th,eff}\right)\left[1-\sum _{i=0}^n{\nu }_i·\exp \left(-\frac{\Delta a}{l_i}\right)\right],$$

(20)

$$a_m={\mu }_0+{\delta }_0\left(-log(-log(P\left)\right)\right),$$

(21)

$$\Delta a=a_m-a_{0,eff},$$

(22)

$$a_{0,eff}=\frac{\Delta K_{th,eff}}{{\left(Y·\Delta {\sigma }_0\right)}^2}·\frac{1}{\pi }.$$

(23)

Thus, the long life fatigue strength of the reference volume $V_0$ with a certain defect distribution can be calculated using Equations (19)–(23). Moreover, the fatigue strength of an enlarged volume $V_{\alpha }$ can be determined by means of Equation (24) to (27), (20) and (23), as exemplified in Reference [3].

$$\Delta {\sigma }_{\mathrm{LLF},V_{\alpha }}=\frac{\Delta K_{th,\Delta a}}{Y·\sqrt{\pi ·a_{m,\alpha }}},$$

(24)

with

$$a_{m,\alpha }={\mu }_{\alpha }+{\delta }_{\alpha }\left(-log(-log(P\left)\right)\right),$$

(25)

$$\Delta a=a_{m,\alpha }-a_{0,eff},$$

(26)

$${\mu }_{\alpha }={\mu }_0+log\left(\alpha \right)·{\delta }_0,$$

(27)

$${\delta }_{\alpha }={\delta }_0.$$

(28)

Now, the local Weibull factor $\kappa ({\mu }_0,\alpha )$ can be derived as a function of inhomogeneity population represented by its location parameter ${\mu }_0$ in a control volume $V_{\alpha }$. The course of the local Weibull factor $\kappa$ is plotted in dependence of $\alpha$ and ${\mu }_0$ in Figure 13. It is evident that $\kappa$ increases with rising return period $\alpha$ and defect population ${\mu }_0$. This relationship is a significant improvement compared to the common guideline [23], where the Weibull factor is determined with a constant value of ten.

Hence, this generalized model of Aigner et al. [3] can be used to check on the size effect of the return period $\alpha$, thereby validating the influence of disconnected highly-stressed volumes, as discussed in Section 3.2. Therefore the evaluated defect distribution for the reference volume $V_0$ in Section 3.2 and the return period of $\alpha =1.98$ are utilized and leading to a model-based local Weibull factor of $\kappa (\mu ,\alpha )=13.8$, as depicted in Figure 13 as red marked triangle.

By applying the common guideline [23], respectively, the volumetric model of Sonsino [16], (Equation (1)), the fatigue strength of an elevated HSV with return period $\alpha$, defined as $V_{\alpha }$, can be calculated as a function of a reference volume $V_0$ and its associated parameters. The calculation is done for the HSV of specimen B.

Aside from the discussed volumetric approaches, the estimation of the Weibull factor can be related to the scatter index of the experimental fatigue strength distribution only [17], see Equation (2). Substantiated by the high manufacturing quality of the samples and quite homogeneous manufacturing process conditions within the HSV, a comparably small fatigue scatter index $T_S$ in the long-life fatigue region is obtained. This approach leads to a value of $\kappa =39.3$, resulting in non-conservative fatigue data. This is depicted as dash-dotted line in Figure 14.

In Figure 14, all three different approaches [3,17,23] are compared, where each of them leads to different Weibull factors $\kappa$ resulting in differing fatigue strength values.

Table 5. Comparison of the normalized fatigue strength resulting from different Weibull parameters $\kappa$ using a return period of $\alpha =1.98$ (specimen B in this study).

${\mathit{\sigma }}_{\mathit{LLF},50\%}$ [-]	$\mathbf{\Delta }$	$\mathit{\kappa }$ [-]	Model
0.96	Reference	-	Experiment
0.93	−2.28%	10.0	[23]
0.98	+2.84%	39.3	[17]
0.95	−0.42%	13.8	[3]

lists the normalized fatigue strength results from the three different $\kappa$-values. The fatigue assessment model proposed in Reference [3] fits the experimental data with a value of $\kappa =13.8$ best, plotted as continuous line in Figure 14. The common guideline (dotted line in Figure 14) leads to a more conservative fatigue design compared to the experimental results, because a constant weibull factor $\kappa$ is defined for groups of materials. The model published by Reference [17] leads to an improper, non-conservative fatigue design due to the small scatter band of the fatigue data.

To summarize, the fatigue assessment model of Reference [3] is validated for EN AC-46200 in sand cast condition for volume ratios up to a value of $\alpha$ about two, leading to an enhanced fatigue assessment avoiding an over-conservative design. The updated size effect model, utilizing a highly stressed volume of 90%, now covers a return period of about two and up to six [3]. Nevertheless, other return periods shall be investigated to approve the statistical method even further.

5. Conclusions

This paper evaluates the size-effect based fatigue strength design of EN AC-46200 in T6 heat treatment condition. Therefore, a special specimen geometry was designed, which possess a non-coherent highly-stressed volume. Volumetric approaches are reviewed and their applicability for conservative fatigue designs is discussed. Overall, the following conclusions can be drawn:

• Based on a numerical parameter study, a deviation factor of about 0.03 is recommendable for numerical evaluation of the highly-stressed volume (HSV) in engineering applications.
• If several independent HSVs with the same microstructural properties are attached as one component and loaded simultaneously, the failure of each HSV leads to failure of the whole component. Hence, the aggregated sum of disconnected HSVs has to be considered as size effect in fatigue strength design. But in the case of varying microstructures between the individual highly-stressed volumes, the local microstructure has to be considered as well.
• The conducted validation of the aforesaid defect based probabilistic fatigue assessment model, originally published in Reference [3], is based on samples with a return period of about two. The results confirm that the model assesses the fatigue strength in terms of statistical size effect best by applying the local Weibull factor $\kappa$ depending on the return period $\alpha$ and defect population ${\mu }_0$. Thus, the verified probabilistic approach is recommendable for engineering design of complex parts, whereat the HSV has to be linked to the local microstructural properties for proper fatigue strength design.

Current work focuses on the design strength related interaction between HSV and associated microstructure in cast aluminium alloys, especially in case of service load cases which enforces locally varying HSV and subsequent feasible damage sum calculations. Moreover, the applicability of the design concept for notched components considering different load cases and local stress gradients will be investigated.