ON THE FATIGUE CRACK GROWTH PREDICTION UNDER VARIABLE AMPLITUDE LOADING S. U. KHAN1, R.C. ALDERLIESTEN, J. SCHIJVE AND R. BENEDICTUS Department of Aerospace Materials and Structures, Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, 2600 GB Delft, The Netherlands.
Abstract During the last decades, numerous papers have been published on fatigue life and fatigue crack growth prediction under variable amplitude loading. The fatigue crack growth prediction models are fracture mechanics based models that have been developed to support the damage tolerance concepts in metallic structures. After a thorough review of the literature, this paper attempts to provide a complete overview of prediction models for crack growth under variable amplitude loading. The prediction models are categorized and the concept of each model is described. The evaluation of the models as presented in this paper is based on the research and experimental data published by many authors. The scope of this paper is limited to aerospace problems and materials (aluminium alloy). To conclude, this paper will discuss the considerations for selecting crack growth prediction models by highlighting the main advantages and disadvantages of all models.
1. Introduction The phenomenon of fatigue has been discovered in the post incident findings of the Versailles incident in 1842. Since then, engineers and scientists have developed models to predict the fatigue life of components and to incorporate the fatigue analysis in the design. However, with the Comet aircraft accidents in 1954 it became evident that the fatigue phenomena were not fully understood. With the introduction of the damage tolerance approach in aircraft design, the fatigue life analysis has been extended with fracture mechanics based models to describe the fatigue crack growth phenomena under constant and variable amplitude loading. These prediction models for crack growth under variable amplitude loading vary from simple modifications on the constant amplitude baseline up to complex models with detailed description of the relevant fracture mechanisms. Some models for instance calculate the crack growth by averaging over the applied load spectrum, while many other models tend to calculate the crack growth with cycle-by-cycle analysis. Despite the ongoing development of prediction models towards more accurate description of phenomena, there seems to be no general agreement about which mathematical description is the most useful. Even the simple prediction models are still used by many engineers. Within this context, a thorough literature review has been performed to identify all the prediction models available today, to investigate the underlying concepts and to compare the accuracy of all the models. Based on this literature study, this paper tries to sketch a complete
1
Corresponding Author, Tel.: +31 15 2788672; fax:+31 15 2781151; Email address:
[email protected], (Sharif Ullah Khan)
overview of available crack growth prediction models and to provide a comparison, which the reader can use to select the appropriate model for his analysis. In this paper, the prediction models are categorized to structure the discussion. The underlying concept of each model will be presented, evaluated and compared with each other. It should be noted that this paper considers aerospace applications and aluminium alloys only.
2. Crack growth prediction models The fracture mechanics based crack growth models have been developed to support the economical fail-safe and damage tolerance concepts. In general, these models can be divided in global analysis and cycle-by-cycle analysis, see Figure 1. The global analysis concept predicts the fatigue crack growth considering the average of the applied loading cycles. The cycle-by-cycle analysis evaluates the crack growth for each load cycle and determines the crack growth life by accumulation. The cycle-by-cycle analysis can be performed with or without involving the interaction effects, i.e. the effect of a load cycle on crack growth in later cycles. A well-known interaction effect is caused by an overload on crack growth in subsequent load cycle. The models that take interaction effects into account can be divided into three main categories: yield zone models, crack closure models and strip yield models. Fatigue Crack Growth Concepts Global Analysis
Cycle-by-cycle Analysis
Linear Damage Accumulation
Considering Interaction Effects
Yield Zone Models
Root Mean Square Model
Crack Closure Models
Wheeler Model
Onera Model
Willenborg Model
Corpus Model
Strip Yield Models
Preffas Model
Figure 1
Classification of Fatigue Crack growth Concepts
3. Global analysis model The root mean square model is a global analysis model that has been introduced by Barsom [1]. This model averages the stress intensity factor ranges following from the load spectrum using the root mean square technique to obtain one value for ∆K that can be used in for instance the Paris relation for crack growth. This approach can be represented by the relation
∆K rms =
1 N
N
∑ ∆K i2
(1)
i =1
where N is the number of cycles and i is used to designate the current cycle number. Various authors have proposed modifications to this concept, by using either the Forman relation -2-
(Tadjiv [2]), by modifying equation (1) with maximum and minimum K value in order to put more emphasis on the load sequence (Hudson [3]), or by applying the stress intensity factor solution of Newman [4] which includes some shape factors. For uniformly stochastically distributed load spectra the RMS model correlates well with experimental data [1]. However, for other spectra including severe interaction effects, the model becomes inaccurate [2, 5, 6]. Bignonnet [7] compared the predictions with the experiments. In his observation the experimental crack growth was more than the predicted one. According to him, for materials having Paris relation exponent m ≤ 2 , ∆KRMS can be appropriate but for m > 2 , ∆KRMS is underestimated.
Figure 2
Comparison between test results and predictions on fatigue crack growth life under flight simulation loading [18]
Numerous models have been developed using the theory proposed by Barsom [7, 8], but remained unable to increase the accuracy of basic model [9]. Figure 2 clearly shows the dissimilarity between the prediction made by non interaction models and test results.
4. Linear Damage Accumulation The linear damage accumulation model is simply a summation of calculated crack growth increments. As a result, it is the simplest model to predict the crack growth rate under VA loading. In general, this rule can be presented mathematically as:
(
a = a 0 + ∑ da
dN
)= a
n
0
+ ∑ f (∆K , r ,..)
(2)
i =1
The majority of the authors [10-17] reported the presence of interaction effects under VA loading and hence mentioned that the linear damage accumulation rule is physically unrealistic.
5. Interaction models The presence of interaction effects is evident from experiments and fracture mechanics dealing with fatigue under VA loading. These effects always alter the crack growth rate under
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the application of VA loading. For correctly predicting the crack growth under VA loading it is necessary to involve the interaction effects while developing the prediction models. The models developed to predict the crack growth in the presence of interaction effects are discussed hereafter, starting with simple yield zone models, followed by the more complex crack closure models, and finally the most advance strip yield models.
5.1.
Yield zone models
According to Gallagher [10] and Schijve [18], the models that try to explain the interaction effect by considering the condition in front of crack tip (plastic zone) are labelled as Yield Zone Models. Wheeler [12] and Willenborg [13] started this generation of prediction models involving interaction effects in the prediction of crack growth.
5.1.1. Wheeler The Wheeler [12] prediction model is based on the damage accumulation relation but modified by a simple retardation parameter CP (Equation 3). n
a = a 0 + ∑ C P f (∆K , r ,..)
(3)
i =1
where CP varies from 0 to 1 depending on the location of the crack tip in a previously created larger zone (rP in figure 3) and the plastic zone size of the current load cycle ri. With r p = (a OL + rOL ) − a i the vale of CP is ap calculated by: rOL
aO
r CP = i rP
m
and C P = 1
ri
ai
when
ri < rP
when
ri ≥ rP
∆a
KOL
where ri is current plastic zone size, rOL is overload plastic zone size, aOL is crack length at overloading and m is the experimentally calculated exponent. m depends on the stress level, the crack shape as well as the load spectrum. Wheeler used the Irwin plane stress relation [19] for the plastic zone size computation.
rp Kmax,r Kr
Kmax ∆K Kmin Kr
Kmin,i
Kmax,i Kmax,eff ∆Keff Kmin,eff
Figure 3 Relative sizes of plastic zones in the yield zone models.
Wheeler assumed that m once calibrated can be used for other spectra. But later it was shown that the accuracy of predictions will suffer if different loading spectra are used with the same m value [15, 16]. The Wheeler model is unable to predict the phenomenon of crack arrest after a high overload, because the predicted retardation factor immediately after the overload will not be zero [17]. Secondly, the Wheeler model did not recognize the occurrence of delayed retardation. Actually, the model assumes a very simple crack growth behaviour, whereas immediately after application of peak loads the phenomenon is very complex.
5.1.2. Willenborg Willenborg [13] assumed crack growth retardation as a function of the stress intensity factor required to cancel the effect of plastic zone created by an overload. Contrary to Wheeler,
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Willenborg used a stress state variable in the plastic zone size relation similar to the Newman stress state constraint factor αg [20]. In the model, a required maximum stress intensity factor, Kmax,req is defined, which will produce a plastic zone just large enough to touch the border of the plastic zone created by an overload (figure 3). Willenborg assumed the occurrence of a residual compressive stress (Sr) due to the overload which reduces the effective stress at the crack tip. The values of Kmax,i and Kmin,i of the current cycle are reduced by Kred to obtain the effective values. Finally, the effective stress intensity factor and effective load ratio, were substituted in the Forman equation [21] in order to determine the crack growth rate for cycle i. If ai + ri ≥ ap, then the residual stress as well as the retardation effect are ignored (Figure 3). The Willenborg model is preferred over the Wheeler model; since it manages to omit the curve fitting parameter m (tune up parameter according to Wheeler). The drawback in the Willenborg model, however, is concerning the crack arrest prediction. The crack arrest is predicted at an overload ratio rOL= Kmax,OL /Kmax,r =2. If the stress intensity factor Kmax,r equals KOL then ∆Keff theoretically becomes negative for rOL>2, but Willenborg assumed the effective cyclic stress intensity factor to be zero. However, the shutoff ratio at which crack arrest is achieved depends on the material and the loading, making this concept to be unrealistic [22]. Some scientists [20] have doubts regarding the method Willenborg had adopted for residual compressive stress estimation.
5.1.3. Generalized Willenborg Gallagher [10] formulated the generalized Willenborg model. Like the original version it deals with retardation effects only. Gallagher tried to improve the residual stress and crack arrest assumption by stating a relation between the Willenborg residual stress intensity factor (Kr) and a new stress intensity factor (KR). ∆K th 1− ∆K ⋅ K KR = φ ⋅ Kr = ( R SO − 1) r
(4)
where RSO is the shut-off value of the stress ratio. When this value is exceeded, Kmax,eff is set equal to ∆Kth/(1-R) and crack growth is arrested. The shut-off ratio RSO of the generalized Willenborg model has only a small influence on the simulation results in case the overload ratio does not exceed this limit. The shut-off ratio for Aluminium 7075-T651 is assumed to be 3.0. Only 3% difference occurs in prediction of crack growth rate if RSO is varied from 2.5 to 3.5 in simulations with the standardized load spectrum FELIX/28 [22] for helicopter load spectrum. In other simulations even smaller differences have been observed. Multiple overloads still remain alien to the prediction models and their effects are treated somewhat similar to the single overload. Like the original Willenborg model, interaction effects induced by underloads are not considered in this generalized model. Brussat [23] proposed a modification to the generalized Willenborg Model. The modification is merely concerning the retardation due to the underload. It was assumed that an underload i.e., a compressive or tensile load that is lower than the previous minimum load subsequent to the last overload cycle can reduce retardation. The stress ratio RUL is used to adjust the factorφ of equation 4. -5-
φ=
2.523φ0
(1.0 + 3.5(0.25 − R
UL
φ = 1. 0
) 0 .6
)
,
RUL < 0.25
Single Overload FELIX/28
,
RUL ≥ 0.25
WISPER
where φ0 is a material dependent parameter that can be calculated by typical aircraft spectrum tests. The value of φ0 ranges from 0.2 to 0.8.
Figure 4 shows the influence of the factor φ0 in the modified generalized Willenborg model. It is surprising that the simulations of a single overload Figure.4 Influence of the material experiment always lead to non-conservative results parameter φO on the accuracy of predicted crack growth life, (CT specimen for all values of the factor φ0, while the simulations of 7075-T651 T-L) [22]. of full spectra like FELIX/28 and WISPER (Standardised fatigue loading sequences for wind-turbine blades) [22], yield good results. This behaviour is related to lack of interdependency of different load variations in the model, which resulted in erroneous predications in case of single overload. In figure 5, fitted curves using different parameters are compared to experimental data for the aluminium alloy 7075-T651 L-T for R = 0.1. It is obvious that the crack growth rate is overor underestimated using different parameters depending on the cyclic baseline level loading. In short, the reliability of prediction with this model depends not only on the concept but also on the appropriate fitting of the crack growth curve. [22]. Chang and Engle [23] suggested a modification to the generalized Willenborg model regarding the acceleration due to underload. This model employs the Walker equation for positive stress ratios and an equation developed by Chang for negative stress ratios while the retardation effects are modelled similar to the generalized Willenborg. The modification proposed to the basic two models was meant to address the issues of crack growth acceleration and delayed retardation. The modifications proved to be doubtful as the modified models are based on the same basic assumptions as the original models; they still can not predict accurately interaction effects other than retardation.
Over-estimated prediction results Under-estimated prediction results
Figure.5 Experimental data of crack growth for R=0.1 compared to fitted curves [22]
Although good results are sometimes achieved with the generalized Willemborg model, the fundamental idea is in disagreement with experimental data. If the yield zone concept is correct the calculated plastic zone size must be equal to the influenced crack length. But the experimental results for aluminium 7075-T651 [20] show that the influenced crack length increment is always larger that the plastic zone size independent of the overload ratio or the underlying equation for the calculation of the plastic zone size. Also similar results for other
-6-
materials show that influenced crack length increment after an overload is much larger than the calculated plastic zone sizes.
5.2. Crack closure models The occurrence of crack closure of a fatigue at a positive tensile stress level after removing the load on the specimen is a physical reality [24]. In order to be accurate, this phenomenon should be an essential element of a crack growth prediction model. During crack growth, the plastic zone is moving with the tip of the crack as well as increasing in size, figure 6. The same will be true for the reversed plastic zone. Reversed Plastic Deformation This deformation involves elongation in the ydirection. As a result of this elongation the crack will close (at least partly) during unloading, and after full unloading (P=0) compressive residual stresses will be present in the wake of the crack. Monotonic Plastic Deformation As the fractured surfaces are pressed together by Figure 6 Plastic Zones plastic deformation left in the wake of the crack, the residual compressive stresses are transmitted through the crack. This phenomenon in literature is referred to as “Crack Closure”. It was first observed by Elber [24] and it is sometimes referred to as the Elber Mechanism. The presence of this phenomenon can be justified either by stiffness measurement [25], which is not an accurate way of measurement, or by the effect on fatigue crack growth. Elber suggested that only that part of the load cycle will contribute to crack extension where the crack is fully open until the crack tip, because crack tip singularity does not exist during the part of the load cycle when the crack tip is closed. This leads to the definition of an effective stress range and stress intensity factor. 1
Elber Relation
0.8
ONERA Model γ = Sop/Smax
0.6
Schijve Relation 0.4
CORPUS Model 0.2
PREFFAS Model
0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
R
Figure 7 The crack opening stress level according to different relations
-7-
0.8
1
∆S eff = S max − S op
∆K eff = K max − K op
(5) Elber devised the famous crack closure relation involving the stress intensity factor and stress ratio. U=
∆K eff ∆K
=
∆S eff ∆S
;
= 0.5 + 0.4 R
(6)
Figure 7 compares the different crack closure relations as a function of R. Elber’s relation indicate that Sop is increasing again for a negative R-value which is physically unrealistic. Analytical work of Newman [26] has shown that it should be a decreasing function for R → −1 . For this reason, Schijve [25] proposed a new relation between U and R based on the trends as predicted by Newman. U = 0.55 + 0.35 R + 0.1R 2
(7)
This relation shows a continuously decreasing Sop for a decreasing R-value. This trend should be expected because for a certain Smax value, a lower R-value implied a lower Smin value. The weakness of this approach is the implicit and unproven assumptions that crack closure is responsible for all load ratio effects and that these can be correlated by an equation. But this relation is proved to be the only crack closure estimation method due to unavailability of accurate direct crack closure measuring techniques [27]. After the introduction of the crack closure concept by Elber [24], a lot of effort was put in understanding the phenomenon to predict crack growth. These efforts include the early phase work which were mainly numerical techniques (finite element analysis) as detailed by Newman [28] and Ohiji et al. [29]. Numerical techniques were shown to be very accurate but are more complicated for modelling and meshing aspects. The major issues were the calculation costs and time, which made scientists developing simple analytical crack closure models. [30-32]. Crack closure models for VA-loading require cycle-by-cycle calculation of the crack opening stress, Sop and the corresponding Kop. The three main models which are based on Elber’s crack closure assumption were primary developed to predict fatigue crack growth under flight simulation loading [65]. These three models are discussed hereafter.
5.2.1. The ONERA Model The ONERA model [34] is a simple model based on the Elber mechanism as it is a stress history dependent concept of crack growth threshold. It was assumed that the crack opening level is limited by two extreme conditions i.e. CA and the single overload, but in case of VA it was assumed to be in-between depending on the load spectrum parameter α. The ONERA model is utilizing the comparison of current plastic zone size and stress intensity factor to the equivalent plastic zone size rp,eq and stress intensity factor. In case of an overload, a new monotonic plastic zone will be created, having a size greater then the equivalent size of the last plastic zone. The equivalent stress intensity Kmax,eq factor is calculated using the Irwin [19] relation for plane stress. In case of absence of monotonic plasticity, Kmax,eq becomes smaller when the crack is growing, because rp,eq becomes smaller. In the ONERA model, the crack opening K-level can be expressed as,
[
K op ,i = K max, eq ,i ⋅ α ⋅ f 1 ( Req ,i + (1 − α ) ⋅ f 2 ( Req ,i )
]
(8)
-8-
where Req is the equivalent stress intensity factor ratio, α is loading parameter and varies from 0 to 1 depending on the type of loading amplitude and f1 and f2 are material dependent empirical function. The loading parameter α varies from 0 for CA loading to 1 for the single overload cycle. In [35], α was adjusted to match predictions to crack propagation test results obtained under FALSTAFF (Flight simulation load sequence for fighter aircraft) and MINITWIST (Sing load spectrum for civil transport aircraft) spectra (α=0.65 and 0.5 respectively), which led to the determination of a simple relation accounting for the type of load spectrum. In order to calculate the crack growth rate da/dN, it is necessary to determine the material data i.e. f1, f2 and Paris relation constants C and m. For the validation of ONERA model, Baudin and Robert [34] determined the value of f1 experimentally for 2mm thick 2024-T3. The crack growth relation used in the ONERA model utilizes a modified version of the Paris equation, written by Ellyin and Li [36]. da m 1 − f 2 ( R ) = C O ⋅ K max dN 1 − f 2 ( 0)
m
(9)
Elber’s U(R) defined in the ONERA model is similar to the relation proposed by Kurihara et. al [37]: U ( R) =
∆K eff ∆K
=
0.9 1.9 − 0.9 R
(10)
Padmandinata [38] evaluated the ONERA model thoroughly by application of the model to numerous results of crack growth data obtained in flight-simulation fatigue tests. He observed a few shortcomings and suggested modification. According to Padmandinata: 1. The ONERA model was not able to fully predict the interaction effects as well as transient phenomena (initial fast crack growth). Padmandinata [38] mentioned underestimation of transition effects and overestimation of retardation effects in some predictions of the ONERA model. 2. For simple VA load sequences, a significant difference was predicted by the model between periodic underload-overload and overload-underload sequences, which was not observed in the tests. This error in prediction is depending on the exponent β, a thickness parameter, used to calculate ∆K min,eq ,i . Exponent β is apparently an important factor in the model, but physically it is difficult to understand the result of β in the thickness effect and in the relaxation of Kmin,eq. 3. The state of stress (plane strain or plane stress) influences the predictions. Plane strain leads to a shorter predicted life, whereas plane stress leads to a longer predicted life. 4. The accuracy of input data and empirical relations is important to give accurate prediction. A different R value adopted in CA test results in a different material parameter. 5. Selection of a better U(R) function is required in order to reduce the scatter. In a double linear representation of (da/dN – ∆K) the scatter is less then for a single linear relation. 6. The multiple overload effect is still not predicted in the ONERA model.
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5.2.2. The CORPUS Model The CORPUS model (Computation Of Retarded Propagation Under Spectrum loading) was proposed by De Koning [39] in 1981. This model developed used for crack growth prediction under flight simulation load sequences. The CORPUS model was based on the hump mechanism, i.e. crack closure is visualized by the hump formation (figure 8) on crack surfaces. There is no evidence supporting the formation of the humps on the crack surface presented in literature. However, only schematics are available to understand the hump creation and flattening. In case of an overload, a larger hump will be created and will be flattened by a later compressive load in the Figure 8 A hump created by an OL and flattened by an UL spectrum. In every cycle, a hump is created with associated Sop level, and for the estimation of Sop a cycle-by-cycle calculation is required, since Sop is an essential part of CORPUS model for crack extension. De Koning [39] was able to introduce a few new concepts in the crack growth models. These were related to the concept of primary and secondary plastic zones, the consideration of plane strain/plane stress for plastic zone estimation and the multiple overload effect. Although the concept behind the model is quite simple, the mathematical interpretation of the model appears to be fairly complex. Padmadinata [38] and Putra [40] explained the CORPUS model very systematically in their thesis. The description in this paper is mainly attributed to both authors and is based on their analyses. In order to describe the hump behaviour after application of an overload-underload combination, a form similar to Elber’s function was determined empirically for 7075-T6 and 2024-T3 material: U = (−0.4 R 4 + 0.9 R 3 − 0.15R 2 + 0.2 R + 0.45)
;R > 0
(11)
U = (−0.1R 2 + 0.2 R + 0.45)
;−0.5 ≤ R ≤ 0
(12)
Using Finite Element Analysis, Newman [41] demonstrated that Sop depends on Smax,n, Smin,n and on the level of σmax in comparison to the yield stress, which Newman assumed as an average yield stress. In order to incorporate the influence of high load levels, De Koning defined a correction factor h for the Sop values. The correction function was obtained by a curve fitting procedure to Newman’s results. An overload is playing a major rule in creating the hump while an underload will reduce the hump and hump opening stresses. A lower underload decreases the Sop level of the previous cycles while an overload higher than the previous overload cycles increases the Sop level. An important feature of the CORPUS model is that it also differentiates between a plastic zone developing into virgin (elastic) material and a plastic zone extending in already plastically deformed material. The first ones are called Primary Plastic Zone (PPZ) and the latter ones are called Secondary Plastic Zone (SPZ).
- 10 -
De Koning formulated a special equation by modifying the Irwin [19] equation as well as the Dixon finite width correction for centrally cracked specimen, in order to account for a large zone if Smax approaches the net section yield-limit. This resulted in a fairly complicated equation for calculating a PPZ involving a variable for the stress state assumption. The plastic zone size has an important role in the delay switch and the material memory consideration. Interactions between an overload with an overlapping PPZ causes an increase of the crack opening levels, which will give more crack growth retardation. This effect plays an important role in the CORPUS model. The hump opening stress given by the equations 11 and 12 is valid for a single overload Smax,n combined with an underload Smin,n. If a series of overloads is applied, de Koning assumes that Sop,n will reach an upper bound stationary level defined by 1 1 + m st , n − 1 U
(13)
Where mst,n is a stationary parameter which depends on the crack growth increment ∆a between the overloads and the plastic zone size Dn of the overload. For the CA case ∆a/Dn goes to zero and gives a value of mst=0.1. Finally, if the crack has grown through the overload plastic zone (∆a/Dn>1), the overload interaction is ignored and equation 13 is used to calculate the ∆Keff values for Paris relation. If a series of overloads is applied, the hump opening stress will reach the stationary value given by equation 13. After the application of the overload, the value of Sop,n is increased step by step. To compute the load interaction effect, a relaxation factor δ was taken into consideration (0.28 for 2024-T3). This value is valid for the interaction effects of overloads of the same level in plane stress condition. For a general case, where overload of different levels interact at different states of stress, two correction factors were introduced. The corrected relaxation factor is
δ = 0.28 ⋅ δ 1 ⋅ δ 2 .
(14)
δ1 accounts for interaction of different overload levels and δ2 accounts for the effect of reduced interaction in plane strain condition. It should be kept in mind that only interaction between the recent overload and the overload associated with dominant hump is considered. Concepts adopted in the CORPUS model are related to crack closure (Elber mechanism), plastic zone size, location of crack tip in plastic zone, hump and retardation mechanism. After comparing the predicted and tested results, Padmadinata stated the following conclusions: Crack growth in most severe flights was under estimated. 1. 2. The CORPUS model gives much importance to a rarely occurring negative load if that load is more compressive (gust load) than the frequently occurring ground stress level. The prediction is inaccurate but conservative in that case. 3. Some improvements have to be done on the load sequence, as in some cases with simple load sequences, a sequence effect was predicted but it was not observed in the test series and sometimes it occurred in tests but CORPUS model did not predict it. 4. The CORPUS model predicts a higher crack growth rate for a lower yield stress if the other material constants are not changed. The latter condition is not realistic, but it indicates that relevant CA crack growth rates are essential for good predictions.
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5.
The CORPUS model does not consider the multiple overload effects on the 7075 alloy.
Padmadinata [38] proposes a modification which is intended primarily to remove the influence of the most severe gust load, which is more negative than the ground load. As in the CORPUS model, the severe underload flattened the dominant hump lowering the Sop-level of the dominant hump. The Sop-level is associated with the highest overload and the lowest underload. Since there is no mechanism in the CORPUS model to remove the influence of lowest underload even it can last for the entire flight simulation block depending on the plastic zone size of the dominant hump. In the modified CORPUS model Padmadinata introduced an underload affected zone to uncouple the effect of the underload from the maximum overload. The second modification he proposed is the procedure to select the Sop-level. In the original CORPUS model, the Sop is selected from the maximum historic operating stress SHop of the PPZ and the SPZ. The modified CORPUS model does not consider the SPZ, instead the local Sop related to the current stress level Smax,i,Smin,i is introduced. The local Sop is applied to the next cycle only if it exceeds SHop. With this procedure the selection of Sop is simplified, because no historical SPZ has to be stored. In the modified CORPUS model, Padmadinata applied the multiple overload interaction also to 7075-T6. The relaxation parameter δ for 7075-T6 was chosen to be 0.15. This value was obtained by comparing prediction results of different values of δ. The prediction results of the modified CORPUS model are summarized as: 1. Similar to the CORPUS model, the predictions are fairly accurate, but with a different ratio of predicted and tested results. (Standard deviation 0.182 for modified CORPUS, 0.275 for CORPUS). 2. Load sequence effect was correctly predicted. 3. Crack extension in the most severe flight is inadequately predicted. 4. The application of multiple overload interaction to 7075-T6 apply appears to be justified.
5.2.3. The PREFFAS Model The PREFFAS (PREvision de la Fissuration en Fatigue, AeroSpatiale) model was proposed by Aliaga, Davy and Schaff [42-44]. The main objective of this model was to make a simple model with a few crack growth calibration tests to characterize the material response and its sensitivity to overload effects. The load history will be considered as cycles of the stress intensity factor K. Each cycle is supposed to start with a maximum Kmax,i followed by a minimum Kmin,i. Kop is calculated for each cycle using the corresponding Kmax and Kmin. Maximum reversed plasticity should be considered which occurs at Kmin,k (figure 9). Kmax,j and Kmin,k must be considered to calculate Kop,i. According to the PREFFAS model, the Figure 9 Sequence of K cycles maximum Kop should be used for cycle i. For the calculation of Kop,i,j (K-opening value in cycle i as it is affected by cycle j) the empirical Elber approach is adopted:
- 12 -
K max, j − K op ,i , j = U ( K max, j − K min,k )
(15)
where U is the linear function of the stress ratio R, material parameters A and B. K min,k U = A + BR = A + B K max, j
(16)
For the Kop calculation a limited number of historical values of Kmax and Kmin have to be remembered. The largest Kmax and the lowest Kmin value will erase all historical effects of previous cycles. The Kop,i value will be determined by a KHmax value and the following KHmin which form a pair. Each pair determines a “historical” Kop level (KHop). The authors of the PREFFAS model have introduced the rain-flow concept in order to avoid un-conservative predictions due to small intermediate load variations. However, instead of considering Kmax and Kmin, the relevant levels are Kmax and Kop. This is a logical consequence of the crack closure concept, i.e., only the K-variations during which the crack is open are relevant. The predictions from PREFFAS model without considering the rain-flow effect might be un-conservative. In the PREFFAS model, the crack length for the calculation of K is assumed to be constant during a period of the flight load spectrum. It implies that the stress intensity factor K is only a function of stress history without an influence of the crack length variation. This leads to a rather simple solution for crack growth calculations. If ∆a is the crack growth increment in one period of the load spectrum, it is: n
[
n
∆a = ∑ ∆a i =C ∑ ∆K effm ,i = C f (a ) πa i =1
i =1
[
∆a da dN = n = C f ( a ) πa av
]
m
] ∑ ∆S m n
i =1
m eff ,i
EF n
In the PREFFAS model there are four parameters i.e. C and m are Paris relation constants, A and B determine the effective stress range U. According to Aliaga et al. [42-44], their experience has shown that the relationship A+B=1 can be used for aluminium and steels used in aircraft structures. Another relation between A and B is obtained experimentally from CA tests (R=0.1) with periodic overload every 1000 cycles while KOL/Kmax=1.7 (figure 10). The value 1.7 is selected as it is supposed to be realistic especially for wing spectra and secondly by experience it gives good prediction results. But there is no information available about the prediction for fuselage spectra. The last unknown constant EF, can be calculated
[
= C f (a ) πa
]
m
EF
(17)
(18)
Figure 10 CA loading tests with and without periodic OLs as a calibration for U=A+BR.
- 13 -
using the equation 18. The material and thickness is incorporated in PREFFAS using the B variable. A large B value implies more crack closure and more retardation. The data given in Aliaga et al. [42-44] showed lower B values for thicker material, which correlates with less crack closure and less retardation. Schijve [45] has tabulated some thickness variation trends in aluminium 2024 and 7075. If the thickness ranges from 1 mm to 15 mm the value of B varies from 0.45 to 0.35. Similarly for 7075 B varies from 0.35 to 0.3 for the same thickness range. In short, PREFFAS is a simple model based on the Elber crack closure concept. The crack closure is calculated with a cycle-by-cycle approach. The model was developed for stationary VA load histories with a short recurrence period. The model includes the “Rain-Flow” approach. A few limitations are: 1. In the PREFFAS model compressive stresses are truncated to zero (S=0). In reality, the ground-air-cycle in the compressive range has a systematic effect on crack growth [13]. It is obvious that the truncation could not be avoided in the model, because of the linear (Elber type) U(R) relation since U=A+BR cannot be reconciled with compressive stresses [26]. However if another U(R) relation (Equation. 7) is adopted it should not be difficult to include the negative loads in the PREFFAS model. 2. The separation of variables (ie. crack length and load history) is only possible using a Paris relation (da/dN=C∆Keffm). For other crack growth relations separation of the variables will become impossible (unless the Paris relation is replaced by a higher order polynomial equation) and will require more computational power. 3. As mentioned by the authors, the model is not applicable when crack growth rates are high due to ductile tearing at high Kmax values. It is due to the fact that ∆a in one load spectrum period must be significantly smaller than the plastic zone size associated with the maximum load of the spectrum. Otherwise, the KHop values can no longer be reliable. 4. In most of the prediction models plastic zone size calculations are an important issue. This aspect is not a. Yield in a Narrow Strip explicitly described in PREFFAS model.
5.3. Strip yield models Strip yield models are based on the Dugdale model [52]. The Dugdale model was used to estimate the size of the plastic zone rp at the tip of the crack. Dugdale assumes that yielding occurs in a narrow strip ahead of the crack tip (figure 11.a). The material response to plastic deformation is rigid-perfectly plastic, which leads to a constant stress (yield stress) in the plastic zone. Dugdale assumes that the situation in figure 11.a is equivalent to the stress system in figure 11.b where a larger fictitious crack length a*=a+rp is present with crack edge loading between a and a* with stress σ=σy. The equivalence also requires that the stress at the fictitious crack tip is nonsingular (i.e., equal to σy).
- 14 -
b. Longer Fictitious Crack
c. Plastic Deformation in Fictitious Crack Tip Figure 11. Aspects of Dug dale model and Strip Yield Concept
The strip yield model is initially proposed [30-32, 47-49] to solve the elastic-plastic crack problems approximately for the plane stress state. Satisfactory results were often obtained as most of the cracks in real structures are under a stress state between the plane stress and plane strain, but further efforts were made to extend the concept to the plane strain or at least near to plane strain conditions. In Dill and Saff [30], Rice’s assumptions [50] that the plastic zone size and crack opening displacement are roughly half as large for the plane strain conditions as those for the plane stress conditions were used to modify the Dugdale solutions for applications under plane strain conditions. Their results show a fair correlation with experimental data. Similarly, Newman [49] introduced a plastic constraint factor α [23] to the tension yield stress ahead of the crack tip to account for the three-dimensional effects at the crack tip under plane strain conditions or near plane strain conditions. The results were confirmed by FE calculations [51]. This verification shows that the value of constraint factor α=1 (plane stress) and α=3 (plane strain) show reasonable agreement with experimental data. The latest work was presented by Ibrahim et al. [48]. Their model was more or less based on the perfect-plastic material behaviour assumption. In the work of Dill and Saff [30] and Budiansky and Hutchinson [31], a constant “height” plastically deformed wake was assumed. In Fuehring and Seeger [32, 47] and Ibrahim et al. [48], a trapezoid plastically deformed wake was assumed. These assumptions bring theoretical approximations on the plastic deformation in the crack wake. Numerical solutions for the analytical Dugdale model will be better, especially for variable amplitude loading, because the theoretical assumptions are minimized for the plastic deformations around the crack but on the cost of time and effort. The most famous strip yield models were developed by Newman [28] and De Koning [53]. The main difference between these two models is the definition of the constraint factor. Newman assumes that the state of stress depends on crack growth rate, with low crack velocities being under plane strain conditions and high rates being under plane stress. The value for the constraint factor is assumed to be constant along the elements of plastic zone. In contrast to Newman, De Koning defines the constraint factor under tension as a parabolic function along these elements, whereas a plane stress value is assumed at the end of plastic zone. Schijve [18] mentioned that in the strip yield model, the plane stress/plane strain transition is still covered by assumptions. This objection is valid as the strip yield model uses a constraint factor α to account for the 3D effects at the crack tip; this is an approximate method to deal with the 3D constraint at the crack tip. α is initially used to determine the plastic zone size for the crack under plane strain conditions for small scale plastic yielding at the crack tip. By stress analyses and von Mises yield criterion, α is equal to 3 under plane strain conditions [52], but Irwin’s constraint factor α=1.73 seems to be nearer to many plane strain experimental observations. The stress state at the crack tip is determined both by the 3D geometry constraint and the plastic zone size, and is usually between the plane stress and plane strain states. The plane stress state prevails when there are small 3D geometry constraints and large crack tip plastic deformation, and plane strain prevails when there are large 3D geometry constraints and small crack tip plastic deformation. α will be between 1 and 3 (or 1.73). The value of α has a very strong effect on the prediction results of the strip yield model. A correct estimation of α must be made before performing predictions. The initial and simplified assumption for α is a constant value throughout the crack growth process from initiation to failure. The drawback of using a constant α is that the gradual change of the stress condition at the crack tip with the propagation of the crack is averaged. - 15 -
Wang and Blom [54] proposed a variable α to account for the gradual changes of the stress state at the crack tip, especially for plates. α is determined by the maximum plastic zone size at the crack tip and the thickness of the relevant plate. The selection of useful α values continues till now. The latest software NASGRO has defined two options and allows the user to opt for either a constant or variable α based on the requirement. In the first option, the tensile constraint factor α is assumed constant along the elements of the plastic zone, but its value depends on the state of stress, ranging from plane strain to plane stress. This constraint loss is based on the observation that cracks which start initially with a flat face eventually grow in a slant face mode. This assumption is widely used by NASA, FAA and their contractors. The second option is that α varies along the elements of the plastic zone according to a parabolic expression derived from finiteelement analyses. The constraint decays spatially from its value at the crack tip αtip to a plane stress Figure 12. Crack propagation behaviour of 2024-T3 following a single OL. Comparison between experiment and prediction results value of 1.15 at the forward end of [54] the plastic zone. Furthermore the compressive constraint factors in the plastic zone and in the crack wake are spatially constant, and their values are given by αtip/αnew and 1/αnew, respectively, where the material parameter αnew characterizes the ratio of tensile tip constraint to compressive constraint. This assumption is widely used by ESA and its contractors [48]. According to Skorupa [55] the strip yield type models are capable to qualitatively describe crack growth transients and many trends corresponding to changes in loading parameters observed for simple VA loads sequences (Newman [28], Koning and Liefting [56], Dexter et al. [57], Ward-Close et al. [58], Wang and Blom [54]). Skorupa [55] mentioned that the strip yield concept adequately predicts delayed retardation in fatigue crack growth and the overload effects as well as the detrimental influence of an underload following an overload. Figure 12 highlights the strip yield models as most accurate to predict fatigue crack growth, being the only model predicting delayed retardation. The quantitative agreement between predictions and experiments, however, is not always so encouraging. Newman and Dawicke [59] have applied FASTRAN (a constant α option) to predict Sop levels in fatigue tests by Zhang et. al.[60] on a high strength aluminium alloy under VA load sequences. Only in two out of four cases, a close agreement between the estimated and measured is found. In the light of demands on the prediction quality, the strip yield model performance can be considered good or acceptable. A lower prediction accuracy typically shown in applications to the thicker specimens is usually ascribed to an inappropriate handling of the 3D crack closure mechanism. Rationalizing the constraint factor, which currently is often used for a tuning in - 16 -
the predictions to test results, is indispensable. However, improving the prediction quality seems to be also conditioned by incorporating other mechanisms into the model, especially roughness-induced crack closure.
6. Discussion
SINGLE UNDERLOAD
MULTIPLE UNDERLOAD
DELAYED RETARDATION
Truncation of High Loads
-
-
-
-
-
-
RETARDATION BEYOND YIELD ZONE
OVERLOAD INTERACTION
-
EFFECT OF THICKNESS
MULTIPLE OVERLOAD
[MODEL TYPE]
CRACK ARREST
SINGLE OVER LOAD
--
DESCRIPTION
PLANE STRAIN PLANE STRESS TRANSITION CRACK ACCELERATION
PHYSICALLY SOUND
Table 1 is summarizing the discussion about the different models presented in this review, together with the advantages and disadvantages reported in the literature. It can be seen that as soon as the physical phenomena such as crack closure, plastic zones and retardation are being involved in the modelling, an agreement is present in prediction and experiments.
-
-
-
-
-
-
-
-
Global Analysis Model RMS Model Linear Damage Accumulation Models
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
-
-
+
-
-
Yield Zone Models Wheeler Model Willenborg Model
-
-
-
-
-
-
-
-
-
Modified Willenborg Models
-
+
-
-
-
-
-
-
+
ONERA Model
+
+
+
+
+
-
-
+
+
+
+
+
CORPUS Model
+
+
+
+
+
-
-
+
+
+
+
-
+
-
-
+
-
+
Crack Closure Models
PREFFAS Model Strip-Yield Models
+
+
-
-
+
-
-
+
+
++
+
+
+
+
-
+
+
+
+ = SATISFACTORY/GOOD
- = NOT CONSIDERED / POOR
Table 1 Qualitative Comparison of the fatigue crack growth prediction model
It is evident from the work reported in the literature, that the global analysis model and linear damage accumulation models (Non-interaction) can not be used for the prediction of fatigue crack growth under VA loading. The lack of physical phenomenon like crack closure, plastic zone results in very inaccurate prediction as compared to experimental results. The yield zone models are not strong contestants in prediction due to weak physical basis and spectrum dependent parameter requirements. Nevertheless these models are still famous in case of quick estimation or just to get a rough estimate about the retardation and plastic zone sizes. But some specialists [64] declared these models unsuitable, even for trend predictions while using aircraft spectra. In order to have an accurate crack growth prediction, it is necessary to rely on the models including the physical phenomena like crack closure.
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The use of advance crack closure models generally results in good prediction of fatigue crack growth under flight spectrum loading. Even inclusion of the crack closure phenomenon in these prediction models, the delayed retardation remains unpredicted. The models show accurate crack growth rather than interpolation of fitted test results. However, it is still necessary to improve the model by checking and refining the assumptions and extending the predictive capability (i.e. 3D crack geometries). Strip yield models are complex due to non-linear material behaviour and the changing geometry of crack tip region. Applicability is confined only to through cracks under plane stress. Even the plastic constraint factor is a virtually fitting parameter. The accuracy regarding plane strain and complicated geometries is still vague and need improvement.
7. Conclusions Aircraft structural components are generally exposed to the variable amplitude loading. The method of predicting life under VA loading becomes very complex and complicated if one aims for an accurate assessment. Introduction of features related with the VA loadings in the models such as like interaction (retardation and acceleration), plastic zone formation and crack closure make the prediction very accurate, but on the expense of complexity and complicated algorithms. In addition, the magnitude of these effects depends on loading variables, specimen geometry, material properties, microstructure and environment. In this paper several crack growth prediction models and concepts have been discussed and evaluated. Investigations show that the predictions are strongly influenced by the parameters (empirical, material, assumed, etc.) which have to be fitted to experimental data. It is clear from the prediction model analyses that curve fitting used to be an important procedure to correlate the predictions with the experimental data. Sometimes accuracy is mentioned as Npred/Nexp=1 with out taking into account the shape of curve. As a result, a lot of precious work done on analytical modelling and mathematical estimation of crack growth under VA loading is physically doubtful, because they lack validated and generalized formulations and concepts.
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