A probabilistic damage accumulation solution based on crack closure model

International Journal of Fatigue 21 (1999) 531–547 www.elsevier.com/locate/ijfatigue

A probabilistic damage accumulation solution based on crack closure model G.S. Wang

*

The Structures Division, The Aeronautical Research Institute of Sweden, P.O. Box 11021, SE-161 11 Bromma, Sweden Received 3 November 1998; received in revised form 29 January 1999; accepted 8 February 1999

Abstract A total life probabilistic model has been developed based on both short and long crack growth data to analyse the fatigue crack growth under general loading conditions. The analysis is based on Elber’s crack closure model. An effective stress intensity factor is used as the major crack growth driving force in the crack growth analyses. Statistical fatigue crack baseline data from the constant amplitude loading tests are used as a basic governing function in the probabilistic crack growth analyses. A strip yield fatigue crack closure model is used together with a modified Miner–Palmgren’s damage accumulation model to account for uncertainties in the non stochastic crack growth process from the crack initiation to the final failure. In this model, effects of material inhomogeneity, initial flaws, and variations in geometry and applied load etc., have been considered for their individual effects. This model provides a probabilistic estimation of the fatigue crack growth for both small and large crack sizes. Several examples have been provided to illustrate a variety of problems that the model is capable of dealing with, and to clarify physical mechanisms behind some of the controversial experimental results.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Short crack; Spectrum loading; Crack closure; Probability; Damage accumulation

1. Introduction The fracture mechanics based damage tolerance method provides information only about the mean crack growth behaviour. Such a method doesn’t account for uncertainties in the fatigue crack growth for practical structural problems. A large initial crack size is often assumed in the damage tolerance analyses, and an empirical safety factor is adopted to cover the uncertainty and variability in the fatigue crack growth process. When there are not enough experimental or historical data to support a selection of the initial flaw size and the safety factor, conservative values are usually preferred which may lead to weight penalty or rejection of qualified structures and components. In addition, the unrealistic safety factor and the large assumed initial crack sizes may lead to a poorly defined maintenance program, resulting in uneconomical inspection intervals. Fatigue crack growth is subjected to various uncer-

* Tel.: +46-8-555-49000; fax: +46-8-25-8919. E-mail address: [email protected] (G.S. Wang)

tainties. Even when differences due to the type of products (plates, extrusion, forging), orientation with respect to grain direction, and thickness etc., are excluded by using different groups of base-line material data, the crack growth is still affected by uncertainties due to: 1) initial flaws, 2) metallurgical inhomogeneity (microstructures, batches, extent of cold work, and producers), 3) component stress variations due to production tolerances and methods (rivet, weld, etc. connections, joints), 4) residual stresses due to welding, cold working, heat treatment, and surface manufacturing processes etc., 5) structural assemble stress variations and corresponding residual stresses, secondary stresses, parasite stresses caused by indeterministic structures design (multiple-stress path design etc.), mismatching etc., 6) surface quality at stress concentrations where cracks may be initiated, and 7) external or thermal load variations (e.g. change of spectrum severity in service), etc. So far, most of the probabilistic investigations have been concentrated on the statistical features of material inhomogeneity under the laboratory condition. Good reviews in this area are given by Kozin and Bogdanoff [1], and Sobczyk and Spencer [2]. Many stochastic mod-

0142-1123/99/$ - see front matter.  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 9 9 ) 0 0 0 1 5 - 8

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G.S. Wang / International Journal of Fatigue 21 (1999) 531–547

els [3–17] have been proposed for the analysis of fatigue crack growth under the constant amplitude loading in the intermediate range of crack growth (Paris range or Stage II). For the fatigue crack growth under the spectrum loading, rather limited statistical investigations have been published [18–23]. Established models from these investigations are usually valid for a number of limited loading conditions like the constant amplitude loading or some spectra with the constant feature. Due to the complicated nature of the crack growth when the crack is small, there is a rather limited number of investigations concerning the scatter feature of fatigue life based on the propagation behaviour of small cracks [24,25]. One of the basic problems in many of the statistical crack growth analyses is the governing fatigue crack growth function. The stochastic models proposed for both the constant and random loading were often various forms of randomised versions of the Paris–Erdogan compatible equations [26] which are invalid even for the constant amplitude loading with different stress ratios. Significant error may be introduced when these models are used to solve practical problems involving the variable amplitude loading, small flaws, and slow crack growths etc. It is often doubtful to extend such models to analyse uncertainties due to the change of stress or load, and to analyse the effect of initial flaws since the Paris–Erdogan compatible equations are incapable of dealing with the fatigue crack growth behaviour under the general conditions. Novel models like those based on the reaction rate theory [27,28], though attractive being as a micro physical based model, still require much refinement for the variable amplitude loading. Recent investigations based on Elber’s [29,30] fatigue crack growth closure model provide a new way to analyse the probabilistic fatigue crack growth process [31– 34]. The crack closure model accounts for conditions not only ahead of the crack tip, but also on the crack surface [35–37]. According to this model, the stress density at the crack tip, which is a direct driving force for the dislocation movement at the crack tip, can be reasonably accounted for under the fatigue loading condition [38]. The crack closure model is not only successful in accounting for the stress ratio effect. It has been revealed that various spectrum and stress state related crack growth behaviours are closely related to the crack closure mechanism [39,40]. In addition, the crack closure model accounts even for the small crack behaviour of fatigue crack growth under both the constant amplitude and spectrum loading [41,42]. According to the crack closure model, the basic material data can be simplified by using an effective stress intensity factor range ⌬Keff that is calculated from a load level at which the crack tip begins to open to the maximum load level. It has been found that there is an almost same crack growth rate for the same ⌬Keff independent the load spectrum

or the crack size (even in the small crack growth region [43]). In this paper, a new probabilistic solution will be presented based on the crack closure model for the fatigue crack growth analysis of the crack initiated from natural initial flaws under general loading conditions. In this method, the experimental data for the fatigue crack growth rate against the effective stress intensity factor for the constant amplitude loading for both small and large cracks are used as the intrinsic baseline material data to analyse the fatigue crack propagation. Together with a strip yield crack closure solution [44], statistical uncertainties and variations in the fatigue life due to material inhomogeneity, initial flaw sizes, the surface quality, the spectrum severity, and the geometry etc., are to be accounted for. This model makes it possible for a statistical evaluation of the fatigue crack growth under practical structural and load conditions.

2. The crack closure model A reasonable probabilistic crack growth model should account for the material uncertainty, the manufacture effect, and the stress effect etc. Some of these effects are inherent like material inhomogeneity. Others are external like the load, the stress, the residual stress, the scratch and mishap etc. According to the fracture mechanics method, the crack growth rate as a function of the stress intensity factor range for the constant amplitude loading can be used to characterise the fatigue crack growth for a given material. The fatigue crack growth rate is usually determined using standard test procedures. An example is shown in Fig. 1(a) for 2024 T3 aluminium alloy for the crack growth rate as a function of the stress intensity factor range at different stress ratios. The crack growth data are obtained in different laboratories for a wide range of stress ratios and load levels for different types of specimens. Various methods have been used to process the experimental a–N raw data to the da/dN–⌬K data. The test results in Fig. 1(a) can be considered to be representative for this alloy. The experimental results in Fig. 1(a) show considerable scatter in the crack growth rate. Scatter up to three orders of magnitude is presented in some of the crack growth rate range. Such scatter, however, contains the material and test related uncertainty as well as the stress related uncertainty. A systematic shift in the crack growth rate may be observed in the crack growth rate data for different stress ratios. Generally, high stress ratio may lead to high crack growth rate for a given stress intensity factor range (see Fig. 1(a)). According to Elber’s crack closure model, the effect of a stress ratio in the intermediate range of crack growth rate is mainly due to the contact of crack surfaces so that the crack tip will not open for the whole range of load cycles especially at the low stress ratios. Fre-

G.S. Wang / International Journal of Fatigue 21 (1999) 531–547

Fig. 1.

533

Crack growth rate as a function of the stress intensity factor range (a), and the effective stress intensity factor range.

quently, the plastically deformed material, the debris, and oxidation on the crack surface will lead to an early contact of the crack surfaces and, in turn, a slow crack growth rate. At the high stress ratio, the crack surface may be kept open during the whole load range. The condition on the crack surface will have minimal effect on the crack tip during the fatigue cycle. To account for the crack closure effect, Elber has proposed an effective stress intensity factor range which is computed using the part of load cycle from the opening of crack to the peak load level. The quasi-static failure may affect the crack growth rate at the high crack growth rate when the maximum stress intensity factor approaches to its critical value. For a given critical maximum stress intensity factor, the critical stress intensity factor range will be different for different stress ratios. A high stress ratio will lead to a low critical stress intensity factor range. Such an effect causes a systematic shift in the crack growth rate towards a high value for the high stress ratio in the high crack growth rate range. The quasi-critical crack growth should be considered together with the crack closure to understand the basic crack growth rate for a given material. The quasi-critical crack growth acceleration can be empirically accounted for by using the critical stress intensity to modify the crack growth rate. A weighed crack growth rate against Elbers effective stress intensity factor range can be defined by excluding the systematic effect of the crack closure and the quasistatic failure on the crack growth rate data. Another crack growth rate relation can be established [45] as shown in Fig. 1(b). Compared to the original data, the scatter in the crack growth rate has been reduced from the threshold up to the static failure for a wide range of stress ratios, load levels, and specimen types. No systematic shift can be further observed in the crack growth rate for different stress ratios. The crack growth rate can be now approximated as a single function of the effective

stress intensity factor range and the maximum stress intensity factor. The weighed crack growth rate against the effective stress intensity factor range may be used as an “intrinsic” relation in evaluating the fatigue crack growth since external effects of the stress ratio and maximum stress intensity factor have been excluded. Scatter in the crack growth rate in the intrinsic relation as shown in Fig. 1(b) may be considered to be due to material inhomogeneity since the control of load, stress at the cracking location, and the measurement of the crack size can be considered to be accurate in laboratories where the experimental data have been obtained. The scatter in the experimental crack growth rate is hardly an intrinsic material property. The measurement intervals, the crack length measurement accuracy, and the method to transfer a–N results into da/dN results etc., have all their signified effect on the crack growth rate. Before these effects have been excluded, the scatter in the crack growth rate as shown in Fig. 1 can not be used reasonably to analyse the statistical crack growth behaviour despite that many stochastic crack growth models use such a scatter as an approximation to the first order distribution in the stochastic crack growth solutions. The comparison shown in Fig. 1 is used here mainly to illustrate the effect of crack closure due to the stress ratio. As shown in the figure, the discrepancy in the crack growth rate due to the mean crack closure is much less than the discrepancy due to the scatter in the crack growth rate. The the stress ratio effect appears not to be due to the material inhomogeneity, but due to the external effect of load. It is more rational to consider directly the crack propagation when the effect of material homogeneity on the fatigue crack growth is to be considered. An example is the experimental statistical fatigue crack growth data obtained by Virkler et al. [46] as shown in Fig. 2(a). Virkler et al’s experimental data were performed using specimens cut from the same sheet in the same orien-

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Fig. 2.

Probabilistic fatigue crack growth of long cracks.

tation for the same alloy. Tests were performed at the same machine, in room temperature, and at a fixed stress ratio. All the cracks were started from the same initial size. The measurement of the crack size were made in a fixed crack increment. 68 tests have been performed. In the Virkler et al’s a–N results, scatter in the crack growth can be considered mainly due to the material inhomogeneity. As shown in the a–N results in Fig. 1(a), the fatigue crack growth has much less scatter than that which could been expected from the scatter in the crack growth rate as shown in Fig. 1(a) and (b) for the same alloy. The crack growth results as shown in Fig. 2(a) can be converted into the crack growth rate. By using the secant method which gives largest scatter in the crack growth rate, the crack growth rate against the effective stress intensity factor can be computed. The effective stress intensity factor can be computed using the empirical functions given be Newman [47]. The results are shown in Fig. 2(b) compared to the 3s-envelope obtained from the results as shown Fig. 1(b). The scatter from Virkler et al’s results shows a remarkable agreement with the general crack growth rate as shown in Fig. 1(b). The scatter in the crack growth rate is large and arbitrary. Much less scatter may be obtained in the crack growth rate if some smoothing methods have been used for the raw a–N data when the crack growth rate is computed. The scatter in the crack growth rate as shown in Fig. 2(b) can therefore not be considered as intrinsic. Even when the measurement interval and accuracy are assumed to be accurate, the processing methods to transfer a–N into da/dN may give different scatters. Virkler et al’s data, together with Ghonem and Dore’s [48] data etc., are the basic data for the development of many stochastic fatigue crack growth models. Although most of the developed models are capable of dealing with the stochastic feature of these data, these models

are limited by the range of the experimental data based on which the models were developed as shown in Fig. 2(b). They are limited by the individual method as in processing of raw material data. The experimental data cover a very limited range of the crack growth as the example shown in Fig. 2(b), typically the intermediate crack growth rate from 10⫺5 mm/cycle to 10⫺3 mm/cycle. Such a range may be a concern when the constant amplitude loading and large crack sizes are considered, and the fatigue loading is at the intermediate or high level. For a general application, the crack growth at both the low and high ends should be considered differently. The low end, especially the near threshold region, is very important since more than 90% fatigue life in many applications may be determined by the crack growth in this region. The use of crack growth data for large cracks in the near threshold region to evaluate the crack growth behaviour is often not successful for structural problems. Long fatigue lives may be predicted due to an often unrealistic high threshold value and a resulting low crack growth rate especially for the spectrum loading conditions. Cracks initiated from natural small flaws behave differently. From the part-through small crack growth investigations [42], one group of the small crack test results are presented in Fig. 3(a). Compared to the long crack growth data as shown in Fig. 2(a), the small crack growth data have much larger scatter. The scatter also includes the effect of the initial flaw sizes since no initial cracks have been introduced in the small crack growth tests. Meanwhile, the test results show that the small crack may have a rapid initial crack growth acceleration. The crack growth may be retarded before the second acceleration. The crack growth may also be arrested. The raw test results as shown in Fig. 3(a) contain not only the effect of material inhomogeneity, but also the effect of the initial flaw size. Unlike the results for the long

G.S. Wang / International Journal of Fatigue 21 (1999) 531–547

Fig. 3.

535

Probabilistic fatigue crack growth of small cracks.

crack as shown in Fig. 2(a) which can be used directly to characterise the effect of material inhomogeneity, the small crack growth results can be used only when the effect of the initial flaw is excluded. To consider the crack growth rate which may be less dependent on the initial flaw size, a modified secant method is used to compute the crack growth rate. The method excludes the negative crack growth in the raw data. Together with a stress intensity factor solution by Newman [42], the crack growth rate are determined and presented in Fig. 3(b) as symbols for the fatigue crack growth rate as a function of the nominal stress intensity factor range. The small crack growth rates have a very large scatter compared to the long crack growth data as shown in Figs. 1(b) and 3(b). The scatter increases with the decrease of stress intensity factor range. The crack grows even below the threshold of the long cracks. Scatter in the small crack growth tests are due to many factors. Some of them are intrinsic. Some of them are external. For example, the relative measurement error may increase as the crack becomes small. For a given accuracy, the error in the crack size measurement will significantly increase for very small crack sizes. When the measured crack sizes are used to compute the crack growth rate, such error will be magnified. This effect is in agreement with the trend as shown in Fig. 3(b). The effect is apparently external. In the same way, the metallurgical irregularities also increases their effects on the small crack growth rate when the crack growth rate approaches to the scale of the metallurgical irregularities. This effect is also in the same trend in the crack growth rate results as shown in Fig. 3(b). This effect is intrinsic. The initial crack size has less effect for the crack growth rate since only the increment of the growth is considered when computing the crack growth rate. In general, the scatter in the small crack range seems to be rather different from that of the long crack growth data as compared to the results as shown in Fig. 1 even when the nominal stress intensity factor range may be

the same. The mean crack growth trend for the small crack data is different from that of the long crack growth as the comparison in Fig. 3(b) shows. The mean crack growth rate seems to be higher for the small crack than for the long crack. By using the strip yield crack closure model as given in ref. [44], the deterministic small crack growth can be analysed based on the intrinsic mean crack growth data as shown in Fig. 1(b). Together with the consideration of gross yield at the crack location, a deterministic crack growth is analysed based on the mean metallurgical defects in the material matrix as the initial flaw sizes. The crack closure analytical results are shown in Fig. 3(a) as a symbol curve. The crack closure analysis seems to be satisfactory. There is no major physical mechanism to be violated in the crack closure analysis, and the long crack growth data is used directly to analyse the small crack growth data. The analytical crack growth results are used to derive the crack growth rate. The analytical crack growth rate for the small crack growth is shown in Fig. 3(b) as diamond symbols. There is a good agreement between the analytical results and the mean small crack growth data. Both the analytical and test results show that the crack growth rate is higher for small cracks than for long cracks. The long crack growth rate is shown in the same figure as a solid curve. The agreement indicates that the crack closure can be a major reason for the particular small crack growth behaviour. The crack closure model may be used to analyse the mean crack growth behaviour for the small crack growth problems. Details of the crack closure model can be found elsewhere [31–34,40,44]. The small crack growth tests indicate that it may be necessary to consider the scatter in the low stress intensity factor region in a different way as that for the long cracks in the same region. It should be very cautious in the low crack growth rate region since in this region, either the initial flaw size is very small or the structure is subjected to the spectrum loading which may contain many small load cycles with very low crack growth driv-

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ing force. Such cases should be considered for the long endurance structures as well as for the limit life structures. For the long endurance structures, the near threshold region needs to be considered since the fatigue crack growth in this region may occupy a significant part of fatigue life. For the limit life structures subjected to the spectrum loading, a large part of load cycles may have a crack growth driving force close to the threshold region. It is as important for the limit life structures subjected to the spectrum loading as for the long endurance structures that the near threshold fatigue crack growth should be correctly modelled. It is reasonable to consider a full range intrinsic fatigue crack growth rate as shown schematically in Fig. 4(a) instead of the long crack growth relation as shown in Fig. 1(b). In this figure, the mean crack growth rate is shown as a solid curve which is in agreement with both long and small crack growth data. The difference is in the scatter of the crack growth rate. The scatter in the near threshold region is expected to be large and should be determined by the small crack growth data. A mathematical simplification can be made to create a basic crack growth model to characterise the effect of material inhomogeneity on the fatigue crack growth rate using a piece-wise linear approximation in the log-log co-ordinates as shown in Fig. 4(b) for the crack growth rate being divided into different regions. In each region, Elber’s relation may be used and the effective stress intensity factor is used as the governing parameter. The intrinsic crack growth rate may be approximated by da w(Kmax)⫽XiCi(⌬Keff)bi. dN

(1)

where Ci and bi are determined by the mean crack growth relation. In this model, w(Kmax) is a function to account for the effect of the maximum stress intensity factor in the quasi-static growth range or in the near threshold range, and Xi is a probabilistic function used to characterise the uncertainty in the crack growth rate

due to material inhomogeneity which includes both the metallurgical inhomogeneity and the stochastic crack closure. The intensive investigations of the stochastic fatigue crack growth behaviour show that the stochastic fatigue crack growth may be reasonably approximated using a simple random variable solution. The stochastic process Xi given in Eq. (1) and shown in Fig. 4(b) can be approximated as random variables. Xi can be reasonably assumed to have a log-normal distribution without time correlation so long as the number of load cycles in each region is large enough. Since many of the external effects on the mean fatigue crack growth like the stress ratio effect, the load interaction effect, the stress state effect, the post yield effect, the residual stress effect, and the small crack effect can be reasonably accounted using the crack closure solution (typically the strip yield model), reasonable probabilistic analyses may be realised based on the same model. In the probabilistic solution, the stochastic function Xi in Eq. (1) as well as the stress and load variation should be solved. In the following sections, a solution will be presented.

3. Damage accumulation model Generally, the probabilistic solution is not straight forward for the spectrum loading conditions since one load cycle in a stationary region may be followed by a load cycle in another region. When only the fatigue life is considered, one of the most used methods is Miner– Palmgren’s [49] theory of linear accumulative damage for random loading (see Miles [50]). Miner–Palmgren’s linear accumulative solution is an empirical S–N curve method which is based on significant number of experimental data for the same test items under the constant amplitude loading for different load levels. In addition to being an empirical S–N curve method, the inadequacy of Miner–Palmgren’s theory has been generally recog-

Fig. 4. Comparison of crack growth rate for long and short cracks.

G.S. Wang / International Journal of Fatigue 21 (1999) 531–547

nised [51] to be overconservative for the probabilistic fatigue life analyses for spectrum loading histories due to too much weight put on the large scatter of low amplitude load cycles. Large part of the load cycles which have no effect on either the crack growth or the scatter are treated the same as those which do have effect on the crack growth. This is probably one of the main reasons that the model often provides unacceptable large scatter for the crack growth under the spectrum loading. This shortcoming may be overcome by considering the crack closure mechanism. An example is shown in Fig. 5 for the crack closure compared to a spectrum load sequence. Under the spectrum loading, the crack closure is mainly determined by the large load cycles. When there is a high crack closure, many of the small load cycles may be below the crack closure level. They may not contribute to the crack propagation. Significant development has been made during the past decades on the crack growth analytical model based on the crack closure model. It has been made possible and affordable nowadays to analyse the crack closure whenever the crack grows into several percent of the plastic zone. Cycle-by-cycle crack growth computations can be performed based on the crack closure model. The difficulty in the crack growth analyses under the spectrum loading is very much relieved based on this method. The fatigue cycles or the part of fatigue cycles can be separated between those which affect the crack growth and those which do not contribute to the crack growth. For the example as shown in Fig. 5, the crack closure evaluation reveals that most part of fatigue cycles will not contribute to the fatigue crack growth. For most of the load cycles, only a small portion of the load cycle is effective to cause the crack to grow. The cycle-by-cycle crack growth analysis based on the crack closure model and the random variable approximation of the stochastic fatigue crack growth makes it

Fig. 5.

Crack closure compared to the spectrum loading.

537

possible to extend the basic principle of Miner– Palmgren’s linear accumulative damage model into a non linear fracture mechanics crack growth analytical solution. The non linear fatigue damage accumulation can be analysed by using the effective part of load cycles as a damage parameter instead of the whole load range. A solution can be developed based on the crack closure model for the probabilistic evaluation of the fatigue crack growth according to this modified damage accumulation method. A schematic of the probabilistic damage accumulation solution based on the crack closure model is shown in Fig. 6. In this solution, a deterministic crack closure analysis is performed to compute a concurrent crack opening stress. According to the crack opening stress, the effective part of load cycle can be determined in the way as shown as the thick lines in Fig. 6(a). The effective stress range is determined by

冦

Smax−Smin if SminⱖSop

⌬Seff⫽ Smax−Sop if Smin⬍Sop

(2)

if SmaxⱕSop

0

The effective stress range is a mean value. It can be used together with the mean crack size to determine an effective stress intensity factor ⌬Keff as shown in Fig. 6(b) according to

冑

⌬Keff⫽⌬Seffy(a) pa

(3)

where a is the crack size and y(a) is a geometry function which is determined by both the geometry and the crack size. The mean crack growth rate can be determined according to the intrinsic crack growth rate and the quasi-static acceleration function as shown in Fig. 6(c)

Fig. 6. Schematic of the probabilistic damage accumulation solution for the fatigue crack growth analysis.

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when ⌬Keff is determined for each load cycle. The effective stress intensity factor determines not only the mean crack growth rate, but also the stochastic crack growth region for each load cycle. According to this model, the crack propagation can be determined by the accumulation of the crack growth rate. The crack closure model determines in the same time the accumulation of load cycles in each stochastic region. As compared to the original Miner–Palmgren’s model, a new damage model for spectrum loading can be developed by using a more physically related damage parameter D which is defined by the part of load cycles which actually contributes to the crack growth. The damage D in the number of load cycles for a given spectrum loading can be computed from the cycle-by-cycle evaluation of the fatigue crack growth following the procedure as shown in Fig. 6(d). D is determined by a sum of

冘 M

D⫽

ni[(⌬Keff)i]⫹n0,

(4)

i⫽1

where n0 are the number of load cycles with stress ranges which do not contribute to the fatigue crack growth as shown in Fig. 6(b). The load cycles ni[(⌬Keff)i] are the accumulated load cycles within an effective stress intensity factor range determined by (⌬Keff)i for which the crack growth may be approximated as a stationary process. Notice that, instead of using the rain flow counting method for the stress range and the mean stress level, ni[(⌬Keff)i] in Eq. (4) are load cycles defined by the effective stress intensity factor range. Unlike the linear accumulation method for which the load cycles in each group are not necessarily correlated to the physical crack growth, ni[(⌬Keff)i] is a direct measurement of the part of load cycles which cause the crack growth under the spectrum loading for a given stochastic region. If the total life is used to divide the damage parameter D as given in Eq. (4), it will give the same dimension as the damage parameter in the original Miner–Palmgren’s model. The significant difference is that the damage parameter is now determined not by the nominal range of load cycles, but by the effective range of load cycles which is determined by the crack closure evaluation. For the constant amplitude loading, this model model gives the same solution as Miner–Palmgren’s model since the crack closure is nearly constant, but for the spectrum loading, the model is different since the damage accumulation is determined by the effective stress intensity factor instead of the load range. The model can therefore effectively account for the load interaction effect, the stress state effect, and the crack size effect. In such a solution, the advantage of Miner–Palmgren’s model has been significantly increased. An intrinsic crack growth rate data is used in the model instead of using a S–N curve for the constant amplitude loading. The intrinsic crack growth rate data is determined by the material pro-

perty, independent of geometry and loading conditions. They can be determined using the standard specimens subjected to the constant amplitude loading conditions.

4. Effect of load and geometry So far, discussions are limited to the analyses of fatigue crack growth due to material inhomogeneity. The solution is under the assumption that the stress around the crack is deterministic. This condition can only be achieved in the laboratory condition where specimens are simple, their geometry and stresses can be well controlled, and the load is carefully monitored. For structural problems in practical applications, variation in stresses, typically due to load, manufacture tolerance, the surface conditions, aspect ratios of the crack, the assembly etc., have yet to be considered if a reasonable probabilistic analysis is to be made. When the maximum stress Smax, the crack opening stress Sop, and the geometry function y(a) are treated as random variables, representing variation in the applied load, the crack closure, the geometry, and the crack aspect ratio etc., the crack growth driving force ⌬Keff becomes a random variable for each load cycle. The statistical description of the fatigue crack growth driving force ⌬Keff can be approximated as:

冑

¯ (a)] pa. ⌬Keff⫽ZseffS¯eff[Zyy

(5)

The bars above variables represent mean values, and Zseff and Zy are random variables representing the randomness in stress and geometry including manufacture tolerance, surface condition, and aspect ratio. As the crack closure is directly affected by the applied load, a further simplification may be made by assuming a log normal variation in the applied load being in the same order as the log normal variations in the crack closure so that Eq. (5) can be replaced by ⌬Keff⫽ exp(zS⫹zy)⌬Keff,

(6)

where zS and zy are symmetrical random variables which can be approximated as normal distributions. They represent variations due to stress and geometry. When the piece-wise linear approximation is used for the mean crack growth data as shown in Fig. 4(b), the fatigue crack growth rate can be determined according to the crack growth model of Eq. (1), da w(Kmax)⫽XiCi[exp(zS⫹zy)⌬Keff]bi⫽YiCi[⌬Keff]bi dN

(7)

where Yi⫽Xiexp[bi(zS⫹zy)]

(8)

This model includes not only the variation in material

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property, but also variations in both the applied load and geometry.

5. Non stationary stochastic solution In general, the crack growth may cross several stochastic regions with different baseline material parameters Xi, Ci and bi especially for the variable amplitude fatigue loading. A solution should be found for the crack propagation which covers more than one region. As mentioned before, a random variable solution can be used to approximate stochastic fatigue crack growth in each region. The crack growth may be considered to be fully correlated between the neighbour regions since a highly loaded sample may have a greater chance to be kept in the same loading condition. It is at least conservative to assume a fully correlated solution for the crack propagation crossing several material regions. Yang and Manning [52] have proposed a solution for the fully correlated crack growth crossing several stochastic regions. Their method can be modified to account for the non stationary fatigue crack propagation in the damage accumulation model. As shown in previous sections, the fatigue crack growth analysis can be performed based on the mean fatigue loading sequence at the mean stress level. The number of load cycles within each stochastic region is recorded separately as given in Eq. (4) and Fig. 4(b). For each region, a random variable can be used to approximate the stochastic crack growth process according to the crack growth rate as given in Eqs. (7) and (8). For each region, the probabilistic solution for the number of load cycles can be approximately solved by using the mean load cycles and the random variable as ni⫽

冕

冘

w(Kmax)da 1 ⫽ YiCi[⌬Keff]bi Yi

w(Kmax)⌬a n¯ ⫽ YiCi[⌬Keff]bi Yi

(9)

with the bar above ni representing the mean value. Eq. (9) is determined by the sum of the mean load cycles in a stochastic region according to the crack growth relation given in Eq. (7). The probabilistic total number of fatigue cycles, the damage parameter D(a), can be solved by

冘 m

D(a)⫽

n¯i/Yi⫹n0/Y0

(10)

i⫽1

where n0 is the number of load cycles below the crack opening stress. n0 does not contribute to the fatigue crack growth. It can be treated separately as a random variable. It has been shown that the log normal distribution is a good approximation for the crack growth in the Paris range as the example in Fig. 2(b) shows. The log normal distribution assumption may also be accepted for the

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quasi-static crack growth range. For the small crack range, the distribution is often complicated when the crack growth threshold is involved since consideration should be made to judge whether or not the crack may continue to grow for each load cycle. According to the crack closure model, the crack growth in the small crack range is determined by a different consideration; the intrinsic crack growth rate. The threshold consideration is different in the crack closure analysis. An intrinsic threshold is used according to the crack closure model. This threshold is smaller than the long crack growth threshold especially for the low stress ratio. The crack closure analysis will exclude all the further evaluation of the crack growth when the effective stress intensity factor is less than the intrinsic threshold. In the crack closure analysis, only those cracks which can keep growing are considered. The extrapolation of the intrinsic crack growth rate as shown in Fig. 4(a) is used to evaluate the fatigue crack growth. In the crack closure analysis, the log normal distribution may still be assumed for the crack growth in the small crack range. Therefore, random variables can be assumed according to the crack closure model for all the crack growth ranges as log normal distributions with a distribution function of Yi as FYi(x)⫽⌽[ln(x/Mi)/si]⫽⌽[ln(x/Mi)1/si]⫽⌽[ln y]

(11)

Here, Mi is the 50% probability value for the random variable Yi. A substitution can be made according to the above distribution for Yi⫽Miysi

(12)

When the random variable y is consider to be the same for all the crack growth ranges, a complete correlated solution can be found for D(a). The fully correlated solution gives

冘 m

D(a)⫽

n¯i/(Miysi)⫹n0/(M0ys0)

(13)

i⫽1

where y is a single random variable. The distribution function of fatigue cycles can then be solved for the combined effect of the geometry, the load, and the stochastic crack growth by

冕 D

FD(D)⫽ flog normal[y(D)]d[y(D)]

(14)

0

where flog normal is the density function of a log-normal distribution, and y(D) is the reverse function determined by Eq. (13). The density function of D(a) can be determined according to Eq. (14) which gives

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fD(D)⫽flog normal[y(D)]兩∂[y(D)]/∂D兩

(15)

In this solution, numerical methods can be used to solve the reverse function y(D), and distribution and density functions of D(a). When the initial crack is considered to be a random variable, its effect can be included in the analysis according to the schematic as shown in Fig. 7. In this figure, IFSD is the initial flaw size distribution, TTCI is the time to crack initiation, e.g. the fatigue time for a given crack size. FSD is the flaw size distribution for a given time. TTCI is the damage parameter D(a). The damage parameter D(a) is conditional to the initial crack size. Therefore, the total life solution with variable initial crack sizes becomes

冕 ⬁

FD(D兩a)⫽ FD(D兩a,a0)fIFSD(a0)da0

(16)

0

Once the distribution of initial flaw sizes is determined, D(a) can be determined according to the solution given in Eq. (10) and the integral given in Eq. (16). Standard numerical procedures can be used to solve these functions. The initial flaws is usually approximated as a log normal distribution of f(a0)⫽

1

冑2ps a

再

exp ⫺

0 0

冎

[ln(a0/l0)]2 , 2s20

(17)

This distribution can be determined by a median (50 percent probability) of the initial flaw size l0 and the corresponding deviation of s0. They are computed from,

冑

l0⫽E{a0}/ 1+n (a0),

(18)

s20⫽ ln[1⫹n2(a0)],

(19)

2

where n(a0) is the coefficient of variance for the initial crack size which is defined by

Fig. 7.

Effect of initial flaw distribution.

冑

n(a0)⫽ Var{a0}/E{a0}.

(20)

In applications, it is not enough to consider only the distribution of fatigue life. It is often required to know the crack distribution for a given time in order to make decisions on the status of fatigue damage and maintenance procedures. When the crack distribution is considered for a given time, the distribution function can be determined using the solution for D(a) since the even {xⱕa} is the same as the even {nⱖnf} (see the schematic in Fig. 7). The distribution function of crack size for a given time nf, the FFSD(a兩nf) can be determined by FFSD(a兩nf)⫽1⫺FD(nf兩a)

(21)

where FD(nf|a) is the probability of damage cycles nf for a given crack size a. The final solution for the fatigue life distribution or the crack size distribution for a given time is determined by the numerical solution which may not be fitted exactly to any known distribution since the bias of the initial crack is not in the same side as the bias of the stochastic crack growth process, and the fatigue crack growth is non linear which affect significantly the distribution of both the fatigue life and the crack size. To concentrate on the basic feature, above procedures can be used to determine central moments like the mean crack propagation and the coefficient of variation. A distribution function can be assumed afterwards for appropriate distributions for the damage parameter D(a) or for FSD. Empirical results showed that either the log normal distribution or the extreme distribution may be good approximations in these cases.

6. The mean crack growth When a deterministic analytical method is used for the probabilistic crack growth estimation, the accuracy of the method should be considered. Using a previously developed code by the author based on the modified strip yield model [39], the mean break-through fatigue lives for the small crack growth tests provided in ref. [42] have been analysed. The analytical and test results are compared in Fig. 8(a) for the constant amplitude loading with different stress ratios as well as two different types of spectrum loading; a fighter aircraft wing root spectrum FALSTAFF, and a symmetrical spectrum loading GAUSSIAN. The analytical results in Fig. 8(a) are shown as curves and experimental results are shown as symbols. In the analyses, the strip yield model is based only on the intrinsic mean crack growth rate data as the base line material data as shown in Fig. 2(b). The mean initial flaw size in the analysis is determined by the scanning electron microscope measurement of the metallurgical defects.

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Fig. 8.

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Comparison of analytical and experimental break-through fatigue lives for the small crack test results.

For a wide range of stress levels and load types, Fig. 8(a) shows that the analytical model consistently predicts the trend of the fatigue lives based only on the mean initial flaw sizes and the intrinsic crack growth rate. In the tests, specimens are carefully prepared at the crack initiation location so that scratches and residual stresses are minimised by using the chemical polish method to remove a layer of material. Therefore, the metallurgical defects are representative as the initial cracks. The analytical results show some systematic disagreements with the test results. However, such disagreements are expected when stresses at the notch are checked in detail. For example for the low stress ratio of R=⫺2, yield will occur at the notch due to the high compressive load so that a gross tensile residual stress will be created on the surface of the notch to promote the crack growth and hence the fatigue lives are reduced. On the other hand for the FALSTAFF spectrum and the high stress ratio of R=0.5, the maximum load will create a compressive residual stress so that the crack growth will be retarded and the fatigue lives will be prolonged. For the GAUSSIAN spectrum, there is a gross cyclic yielding on the surface of the notch which tends to reduce the fatigue life. To evaluate quantitatively the accuracy of fatigue life analyses according to the strip yield crack closure model, a probabilistic distribution can be found based on the comparison between the experimental data and the analytical results for the ratio of tested fatigue lives Ntest and the predicted fatigue lives Npredict. The distribution function Ntest/Npredict is shown in Fig. 8(b) as symbols. As shown in this figure, the distribution of Ntest/Npredict seems to follow a normal distribution but a log normal distribution can be used as well. The log normal distribution has an advantage that the unrealistic negative ratio of Ntest/Npredict can be prevented. In the fatigue life management of advanced structures like the aeroplanes, emphasis has been made nowadays towards analysis based methods implemented with lim-

ited test verifications. In such a methodology, the analytical accuracy should be considered so that the probability distribution in Fig. 8(b) should be used to characterise the accuracy of the analytical method when no experimental verification is available. In this distribution, we can see that the strip yield crack closure model is on average about six percent conservative with a large coefficient of variation of 0.65. Such a variation, as discussed previously, includes even the effect of gross compressive or tensile yield at the notch. The probability distribution shown Fig. 8(b) can be used together with the crack growth model to make a more realistic estimation of the crack growth behaviour. If the analysis can be justified by limited experimental tests, the scatter in the analytical mean crack growth can be reduced and the accuracy in the statistical analysis can be improved.

7. Examples and discussions According to the stochastic crack growth analysis for long cracks, the estimated coefficient of variation for the random variable Xi is about 0.08 for the Paris range for 2024-T3 aluminium alloy. Typically, the value is determined by the statistical test a–N data as those shown in Fig. 2(a) instead of the crack growth rate results. This value can not be determined from the crack growth rate data as those shown in Fig. 1(b) or Fig. 2(b) since the crack growth is a stochastic process and the distribution in the crack growth rate can not be related to the crack growth distribution when the correlation is not determined. There seems to be large scatter in the small crack growth range as the results from small crack growth tests indicated (see Fig. 3). The test a–N results are however not directly related to the random variable Xi since the small crack growth can usually not be performed from a predetermined initial crack size. The scatter in the test a–N results includes the effect of the random initial crack

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as well. In addition, the crack growth rate results from the small crack growth test is subjected not only to the uncertainty due to the stochastic crack growth, but also the systematic change due to the load level and geometry, the magnified measurement error, as well as the metallurgical uncertainties. For the small crack tests, the scanning electron microscope readings and inclusion image analyses according to Laz et al. [53] showed that a log normal distribution may be used to characterise the distribution of inclusions for 2024-T3 alloy in L-T orientation. The distribution of inclusions is in the whole material matrix. Their maximum width will not necessarily be on the surface of the notch where the crack is initiated. However, the scanning electron microscope readings showed that the average number of inclusions was about 316 particles/mm2 in the material matrix in the plane perpendicular to the load direction, resulting in about 900 inclusions in the 95% of the high stressed area. For such a inclusion population, it can be reasonably assumed that the probability of an inclusion with its largest width on the bore surface is close to 1 at the high stress region so that the distribution of inclusions can be used directly as the distribution of initial flaws. From SEM inspections, parameters in Eqs. (17)–(20) for the inclusion width distribution in the cross section are obtained as,

再

l0=3.4 mm

s0=0.79

.

(22)

The mean initial flaw can be approximated by, a¯0⫽E{a0}⫽l0exp(s20/2)⫽4.6 mm.

(23)

This initial flaw size is with an average aspect ratio of 0.5 at the crack initiation locations. Assuming that the crack closure model can correctly predict the small crack growth behaviour, the random effect of the initial crack size can be evaluated analytically according to the deterministic crack closure analysis for different initial crack sizes. Trial computations can be performed according to the probabilistic solution. The analytical results are compared to all the available small crack growth test results for both the constant and spectrum loading conditions. Different values of Xi can then be tested for the small crack growth range. In this way, reasonably accurate values of Xi can be determined according to the small crack growth test results so long as the data are adequate to cover the useful application range. According to both long and short crack test results, COVs is estimated for 2024-T3 aluminium alloy for the random variable Xi for the whole crack growth rate range. The estimated Xi are given in Table 1. These

Table 1 Random variable parameters for the baseline crack growth data da/dN mm/cycle

√Var{da/dN)/E{da/dN}

⬍5E-9 ⬍1E-8 ⬍1E-7 ⬍1E-6 ⬍1E-5 ⬎1E-5

0.10 2.036 0.711 0.328 0.161 0.080

values are scaled estimations according to the trend in the small crack growth and the integrity requirement for the long crack growth data. The scatter in Table 1 is rather different from the data which are solely determined from the long crack tests as shown in Fig. 1(b). These parameters keep the integrity of the base line material data when the small crack growth is involved. They are different from the long crack growth data as shown in Fig. 1(b) especially for the low crack growth rate region. The non crack growth condition is defined in the base line data as shown in Fig. 4 for the crack growth data less than 5×10⫺9 mm/cycle. Such a low value is considered so that the fatigue crack growth may still be allowed for an effective stress intensity factor range well below the mean threshold value. For the crack growth rate higher than 10⫺5 mm/cycle, the parameters for the random variable are determined by the long crack growth data as shown in Fig. 1(b). Since the size of initial flaws directly affect the fatigue life, scatter in fatigue lives due to the initial crack sizes can be analysed by using various simulated initial flaw sizes without considering the stochastic crack growth process. The dashed curve shown in Fig. 9 gives a computed COV’s as a function of fatigue life for all the load cases as shown in Fig. 8(a). In comparison, symbols in Fig. 9 are experimental results. In several cases, the ana-

Fig. 9. Comparison of COVs between the experimental small crack growth results and the different analytical results.

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lytical results using variable initial crack sizes is acceptably conservative. But in general, the analytical results based only on the contribution of initial flaw sizes are non conservative since another major source of scatter, the scatter due to the stochastic fatigue crack growth, is omitted in the analyses. To achieve a better estimation of variations in the fatigue life, the scatter in the stochastic crack growth should be included. The computed COV’s according to the probabilistic model is shown in Fig. 9 as a solid curve. These results include effects of both initial crack sizes and the stochastic crack growth process. The analytical results seem to be more realistic. In the analyses, the effect of the multiple crack is not considered, the stress change due to geometry variation has not been considered, and the crack initiation stage is omitted. The evaluation is based only on the initial flaws and the stochastic crack growth process. The computed results still provide a good reference for the possible variation of crack growth for naturally initiated cracks at metallurgical inclusions. According to the experimental results, COV in the fatigue life seems to be a random variable. In the small crack growth tests, the specimens are simple, and the loading is carefully controlled. There is no significant effect of geometry and load on the fatigue crack propagation. When more complicated cases are considered, these effects should be included. Consider a structural detail in the aeronautical structures, the fastener joints. A common solution is shown in Fig. 10 for

Fig. 10.

Basic geometry and load for the fastener specimens.

543

a hole in a plate with a single part through crack at the edge of hole. Loads are considered for the remote tension, bending, and the load transfer through the fastener [54]. This configuration can be used to approximate various combinations of load transfer and secondary bending. The present of fastener in the hole introduces uncertainty in the stress along the crack front which may be characterised by variation in the geometrical function, y(a) in Eq. (3). An example is given for the test results performed by Noronha et al. [55] using different fastener dogbone specimens for the same alloy 2024-T3. Two groups of dogbone specimens have been tested; one with an unloaded fastener hole for 14 tests, the other with a fastener having about 15% load transfer at the hole for 21 tests. The specimens are subjected to a spectrum loading [56]. The fastener holes have a different extent of uncertainty in fitting conditions which is determined by the manufacture quality. The uncertainty will increase especially when the load transfer occurs through the fastener. Symbols in Fig. 11 show the test results of the log normal deviation as a function of flight hours for the test results of Noronha et al. [55]. The solid diamond symbols in the figure represent the test results for specimens without load transfer while the solid circles represent the test results for specimens with about 15% load transfer. Obviously, specimens with load transfer show larger scatter in the fatigue life. The scatter is initially high, and gradually stabilised. There seems to be an almost constant difference in the scatter between specimens with and without load transfer at the fasteners. Such a difference is reasonable since when the load transfer is involved at the fastener, the uncertainty due to manufacture, mishap, tolerance, and fitting will be signified, leading to a larger scatter in the fatigue crack propagation. Since the crack growth is initially located at the low

Fig. 11. Comparison of log-normal scatters for two different fastener specimens.

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end of the crack growth rate due to the low stress intensity factor for small crack sizes, the material inhomogeneity and initial crack size have a large effect on the crack growth. Therefore, the scatter is also large. Such an effect is “faded” out when the crack grows larger. As compared to specimens used for small crack growth tests, the surface of the fastener hole is the production standard. Large initial flaws with relatively smaller scatter are presented. In the crack growth analysis, trial-anderror tests are required to evaluate parameters in the model based on the final experimental data as shown in Fig. 11 since information about the surface and manufacture quality is not available. It has been found that a scatter in the stress for s{ln(Zy)}=0.04 gives a fairly good description to the scatter in the fatigue crack growth for specimens without load transfer. The analytical results with this value are shown in Fig. 11 as a solid curve compared to the experimental data shown as diamond symbols. This value is reasonable since fasteners are still installed in the specimens though there is no load transfer through the fastener. The tolerance between fastener and hole still affects the boundary at the hole and, in turn, the crack growth at the hole. The same base line material data given in Fig. 4 and Table 1 are used to analyse the scatter due to the material inhomogeneity. As can be found in the comparison in Fig. 11, both predictions and experimental results show that the scatter in the fatigue crack growth decreases with the increase of flight hours. The prediction is conservative for the long range of flight hours. The variation in manufacture and geometry will increase its effect on the fatigue crack growth when the load transfer occurs through fasteners. For specimens with 15% load transfer, a larger value of s{ln(Zy)}=0.08 is found to be a good approximation for the effect of fastener uncertainty on the fatigue crack growth. Fig. 11 shows the comparison between predictions and test results. The test results are shown as round symbols while the prediction is shown as a dashed curve. Very good agreement for scatters in the whole range of crack growth are achieved between prediction and test results. The same as for the non-load transfer tests, the fatigue crack growth is initially in the small range of stress intensity factor so that the scatter in the crack growth rate is large, leading to a large scatter in the fatigue crack growth. When the crack grows into intermediate range, the scatter in the fatigue crack growth is reduced. This seems to be the dominant factor affecting the scatter in the crack propagation. For even more complicated fastener joints involving different load types, a number of fatigue tests have been performed in the late 50’s for lap joints with two rows of eight rivets each [57], made of 2024-T3 aluminium alloy in order to investigate variations in fatigue lives for both the constant and the variable amplitude loading.

A summary of the test scatters is shown in Fig. 12 as symbols for the log normal standard deviations as a function of the fatigue life. The test data for the constant amplitude loading are shown as round symbols and test data for the variable amplitude loading are shown as diamond symbols. An interesting outcome of the tests is that the scatter in the fatigue life for the constant amplitude loading increases with the increase of fatigue life while the scatter in fatigue life for the variable amplitude loading remains approximately constant for a wide range of fatigue lives. These tests indicated that the constant amplitude loading can not be used to substitute the variable amplitude loading for the statistical fatigue tests of fastener joints. The statistical model is used to analyse these test results based on the material parameters as shown in Fig. 4 and Table 1. The deviation due to variation in the manufacture is assumed to be s{ln(Zy)}=0.10. This value is larger than that used in the tests as shown in Fig. 11 due to the high fastener load transfer and the possible multiple site crack effect. The scatter in the initial flaw distribution is assumed to be the same as for the case as shown in Fig. 11. In this case, the mean crack growth initiation time is small and its variation has small effect on the solution [58]. The crack growth analysis is performed using the closure model based on the basic stress intensity factor solution as shown in Fig. 10. Here, only results for the scatter are presented. The predicted results for the constant amplitude loading is shown in Fig. 12 as a dashed curve, which compares favourably well to the experimental results. Both the analytical and test results show that scatter in the fatigue life tends to be larger for the longer life region. A detailed check to the analysis for the constant amplitude loading conditions shows that every load cycle contributes to the fatigue crack growth. For the long fatigue life, most of the load cycles are located in the small stress intensity factor range which has large scatter as shown in Figs. 3

Fig. 12. Comparison of log-normal scatters for the constant and variable amplitude loading.

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and 4. Therefore, the scatter in the fatigue life is large as well. For the variable amplitude loading, the predicted result is shown in Fig. 12 as a solid curve. The prediction agrees reasonably well with the test results. The analytical results show a trend of an approximately constant scatter over a large range of the fatigue life. The results even show that the scatter in the fatigue life is high in intermediate range while it is reduced both for the short and the long life region. Such a trend can also be found in the test data although it is less obvious. For the constant amplitude loading, the effective stress intensity factor at the crack tip increases smoothly with the increase of the fatigue crack size as the crack closure is stabilised. For the low load level, the effective stress intensity factor is mainly within the low end of fatigue crack growth rate, corresponding to a large scatter due to material inhomogeneity and the sensitivity to stress variation. Especially for the near threshold region, infinite fatigue life may appear for some cases. Therefore, the scatter in the fatigue life increases with the increase of fatigue life, approaching infinite somewhere around the threshold value. For the variable amplitude loading, both the load level and the crack opening level is a function of the load history. Only part of the load cycles have enough crack growth driving force to promote the crack growth. A large part of the load cycles may not contribute to the fatigue crack growth process. For high load levels when almost all the load cycles contribute to the fatigue crack growth, the statistical behaviour may become similar to the constant amplitude load, see the results as shown in Fig. 12. In the intermediate load range, the high loads may cause high crack closure, the crack driving force for small load cycles is reduced. Therefore, the scatter in fatigue life is increased. When the load level is reduced further, leaving a significant part of load cycles below the threshold, the scatter in the fatigue lives is again reduced since now only a small part of the load cycles contributes to both the crack growth and the scatter. Therefore, in contrary to the constant amplitude loading, the scatter in the fatigue lives is reduced for the variable amplitude loading when the fatigue life becomes longer. The tendency is clearly shown in Fig. 12 for both the test and analytical results. Comparison between tests and predictions shows that there is no conclusive trend in the constant and the variable amplitude loading. Under certain combinations, scatter in the fatigue life may be larger for the variable amplitude loading than for the constant amplitude loading while in other cases, the scatter may behave in another way. There is no direct correlation between the scatter for constant and variable amplitude loading unless the individual statistical effect of load cycles is analysed. Understanding such a difference in the fatigue crack growth analyses for the constant and variable

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amplitude loading is very important in translating the test results from the laboratory to applications.

8. Summary A simplified probabilistic fatigue crack growth solution is proposed based on the extension of Miner– Palmgren’s linear damage accumulation model and the crack closure model. Reasonable accuracy has been achieved based on the model for the probabilistic analysis of fatigue crack growth under both the constant and variable amplitude loading conditions involving uncertainties due to initial flaws, stochastic crack growth, geometry, and load. Statistical uncertainties in fatigue lives in this model are divided into material inhomogeneity related uncertainty, the geometry related uncertainty, and the applied load related uncertainty. As a result, their individual effects can be separately analysed. An intrinsic characteristic of material inhomogeneity in the fatigue crack growth rate is used in this model based on the assumption that the scatter of fatigue crack growth rate should be determined by the long as well as the small crack growth data. This assumption simplifies the statistical characterising of the fatigue crack growth rate. In addition, this assumption can reasonably account for the near threshold statistical fatigue crack growth behaviour which is very important in analysing the fatigue crack growth for the constant as well as variable amplitude loading when a large portion of small load cycles may fall in the near threshold region. Several test data for both the constant and variable amplitude loading have been used to demonstrate the model. The analysed results agree fairly well with the test results. Since the statistical model is based on the intrinsic description of the material data and the physical fatigue crack growth mechanism, it has not only the advantage of being simple to use, but it also reliable in nature.

Acknowledgements Financial support from FMV (the Swedish Defence Material Administration) and the Internal Founding of the Aeronautical Research Institute of Sweden is gratefully acknowledged.

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