Comparative study between crack closure model
and Willenborg model for fatigue prediction under
overload effects
Jiang Shan, Zhang Wei
*, He Jingjing, Wang Zili
Science and Technology on Reliability and Environmental Engineering Laboratory, School of Reliability and Systems Engineering, Beihang University, Beijing 100083, China
Received 13 October 2015; revised 21 April 2016; accepted 24 August 2016
KEYWORDS Crack closure; Fatigue crack growth; Interaction effects; Overloads; Plastic zone
Abstract A comparative study is performed between a crack closure model and the Willenborg model, which can calculate the fatigue crack growth rate under the overload effects. The modified virtual crack annealing (VCA) model is briefly reviewed, which is based on the equivalent plastic zone concept. In this method, the retardation phenomenon is explained by the crack closure level variation, which is derived from the interactions between forward and reverse plastic zones ahead of the crack tip. As a comparison, the Forman equation in conjunction with the Willenborg model is also reviewed. The retardation phenomenon is described by directly modifying the stress intensity factor. It is known that the large plastic zone created by the overload can decelerate the fatigue crack growth rate until the crack grows beyond this region. A relationship between the plastic zone and the modified stress intensity factor is developed, which is a mathematical fitting equation instead of physical-based formulation. The experimental data in aluminum alloys are used to val-idate these two models. Overall, good agreement is observed between the model predictions and the testing data. It is noted that the approach based on modified VCA model can give more accurate prediction curves than the Willenborg model.
Ó 2016 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
For many aircraft structures, understanding the fatigue crack growth under service loading conditions is of great signifi-cance. Many structure components are frequently subjected to constant amplitude loading with occasional high peak loads, which are called overloads. Due to the constant air current and occasional turbulence during the flight, the fatigue crack * Corresponding author. Tel.: +86 10 82314649.
E-mail addresses: [email protected] (S. Jiang), zhangwei. [email protected](W. Zhang).
Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
Chinese Journal of Aeronautics, (2016), xxx(xx): xxx–xxx
Chinese Society of Aeronautics and Astronautics
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growth of the aircraft structural component is always under the influence of overloads. It is necessary to understand deeply and predict precisely the fatigue crack growth behavior under the constant amplitude loading with overloads. And many experimental, theoretical and numerical studies are conducted to investigate it.1–3Additionally, the retardation phenomenon caused by the overload is a typical load sequence interaction effect.4,5The intensive study on overload effect is a prerequi-site for theoretically solving the nonlinear crack propagation problem under variable amplitude loading.
Plentiful models were proposed to quantify the nonlinear fatigue crack growth with the overload effect.6–8 These mod-els could be divided into two main categories: the crack-closure-based models and the plasticity-based models. For the crack-closure-based model, the crack closure level increases as a result of overload, which retards the subse-quent fatigue crack growth. Elber9introduced the crack clo-sure concept firstly and developed the relationship between the crack growth rate and the effective intensity factor range. In his study, this approach is only validated under the con-stant amplitude loading. And then, many modified models are proposed to analyze the crack closure variation under overload effects.10–14However, in most of the existing mod-els, the crack closure is evaluated either by the finite element analysis or by the indirect experimental measurements. The former is relatively time consuming due to involving the highly nonlinear analysis of cyclic plasticity and contact, while the latter is not suitable for variable amplitude load-ing.15,16 Zhang and Liu17,18 preformed an in-situ scanning electron microscope (SEM) experiment and directly observed the crack closure phenomenon. The virtual crack annealing (VCA) model is developed, which is inspired from this exper-iment observation. In this paper, this model is modified by introducing the equivalent plastic zone concept which could quantify the previous load sequence effects. For the second category, the methods are based on the assumption that the large monotonic plastic zone ahead of the crack tip leads to the crack growth retardation phenomenon. In these mod-els, the plastic zone size is considered as a modification fac-tor of the fatigue crack growth rate. For example, Wheeler19 proposed a series of empirical models to evaluate the over-load effects. And Willenborg et al.20modified these models, which do not incorporate any meaningless parameter. Wil-lenborg et al. described the retardation phenomenon by proposing a relationship between the effective stress intensity factor and the plastic zone size, which is a mathematical fit-ting equation instead of physical-based formulation. Addi-tionally, this model presents the limitation of estimating the crack arrest for overloads with magnitude ROLP 2 and the
load sequence effects owing to underloads.21
This investigation aims to compare the Willenborg approach with the modified VCA model for fatigue crack growth prediction under the constant amplitude loading with overload effects. The paper is organized as follows. First, the modified VCA model and the Willenborg model are reviewed. Then, the fatigue testing data of D16 aluminum alloy and Al7075-T6 under constant amplitude loading with/without overloads are employed to validate these two models. Finally, the comparative analysis and some conclusions are given based on the current investigation.
2. Methodology
2.1. Modified virtual crack annealing model
Elber9introduced the crack closure concept firstly and devel-oped the relationship between the fatigue crack growth rate and the effective stress intensity factor, which can be expressed as da dN¼ CðDKeffÞ m DKeff¼ Kmax Kop ð1Þ where da/dN is the crack growth rate; DKeff is the effective
stress intensity factor; Kmax is the stress intensity factor of
the peak load; Kopis the stress intensity factor of crack closure
level; C and m are calibration parameters.
Since the fatigue crack growth is significantly influenced by plasticity ahead of the crack tip, the load sequence effect can be correlated with the plastic deformation in the vicinity of the crack tip. In order to investigate this effect, the plastic state caused by the previous loads should be traced. Therefore, the equivalent plastic zone concept is proposed herein, and the general expression can be written as
a0þ Xi j¼1 dajþDeq;i¼max a0þ Xi j¼1 dajþdi;a0þ Xi1 j¼1 dajþDeq;i1 ! ð2Þ where Deq,iis the equivalent plastic zone size in the ith cycle; a0
the initial crack length; da the crack increment; dithe current
plastic zone size in the ith cycle; a0þ
Pi
j¼1dajthe crack length
in the ith cycle; i the current cycle number. A schematic sketch is given to illustrate the equivalent plastic zone concept. The loading sequential process and the corresponding plastic state variation are shown inFig. 1. The dashed zigzag lines repre-sent the loading history. The large plastic zones have been formed at t1, and the crack tip is O1 at that moment. The
monotonic and reverse plastic zones can be expressed as10
dm¼p8 Krmax y 2 dr¼p8 Kmax2rKop y 2 8 > < > : ð3Þ
where dmis the monotonic plastic zone size; drthe reverse
plas-tic zone size; ry is tensile yield strength. The current load is
applied at t2 and the new crack tip is O2. The large forward
and reverse plastic zones, which are the dotted ellipses, form during the largest load cycle in the previous loading history. Before t2, the following plastic zones do not reach their
bound-aries respectively even though the crack grows. The solid ellipses represent the equivalent plastic zones ahead of the crack tip O2. In addition, the actual contour of plastic zone
is butterfly-shape instead of circle, but theoretically their diameters along the crack direction are identical (Fig. 1). In current study, the plastic zone effects are calculated by the equivalent plastic zone which is considered to be in direct pro-portion to the circular diametric distance. This propro-portional relation is indicated by the geometry modification factor which is equal to or slightly greater than 1. Eqs. (2) and (3)can be rewritten as
a0þ Xi j¼1 dajþ Dm;eq;i¼ max a0þ Xi j¼1 dajþ dm;i;a0 þXi1 j¼1 dajþ Dm;eq;i1 ! dm;i¼ wp8 Kmax;i ry 2 a0þ Xi j¼1 dajþ Dr;eq;i¼ max a0þ Xi j¼1 dajþ dr;i;a0 þXi1 j¼1 dajþ Dr;eq;i1 ! dr;i¼ wp8 Kmax;iKop;i 2ry 2 8 > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : ð4Þ
where Dm,eq,i and Dr,eq,i are the equivalent monotonic and
reverse plastic zone in ith cycle respectively;w is the geometry modification factor of plastic zone.
Ref.18derived a crack closure model named as VCA model in the continuous unloading-loading process, which is vali-dated under constant amplitude loading only. In this study, the equivalent plastic zones are introduced to replace the plas-tic zone formed in the unloading process. The modified VCA model is derived below and a schematic illustration is shown inFig. 2. The crack tip is O after the unloading process. The current crack length is a and the crack closure length is b. The reversed plastic zone appears ahead of the crack tip and has a diameter of dr. According to the assumption of crack
annealing, the virtual crack length can be considered as ab and the diameter of reversed plastic zone ahead of the crack tip O0will be Dr. The equation Dr= b + drcan be established.
In the following loading process, the crack closure length b gradually reduces. When the value of b becomes zero, the crack is fully open and the forward plastic zone with diameter df
appears. The above equation can be written as df= Dr dr.
The equivalent plastic zones substitute for the original plastic zone effects. Therefore, the equation can be expressed as
Dr;eq dr¼ df;eq ð5Þ wp 8 rmax;eq rmin 2ry 2 pða bÞ dr ¼ wp 8 rop rmin;eq 2ry 2 pða bÞ ð6Þ
wherermax,eq the inverse computation maximum stress level
from the above equivalent monotonic plastic zone df;rmin,eq
the inverse computation minimum stress level from the above equivalent reverse plastic zone; rop the stress level of crack
closure.
Eq.(6)can be rewritten using the reverse plastic zone size and monotonic zone size as
wp 8 dm;eqp28aY 0:5r y rmin 2ry !2 pða bÞ dr ¼ wp 8 rop rmin;eq ry 2 pða bÞ ð7Þ
where dm,eqis the equivalent monotonic zone size. Based on
that, the theoretical solution ofropcan be expressed as Fig. 2 Schematic illustration of real crack and virtual crack model.
rop¼ rmin;eqþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dm;eqp28aY 0:5r y rmin 2 !2 8 p2aYr 2 ydr v u u t ð8Þ
The fatigue crack growth model can be developed based on the modified VCA model. The crack length can be derived from Eq.(1): an¼ a0þ Xn i¼1 dai ¼ a0þ Xn i¼1
C K i maxðKop;i; Kmin;iÞðKmax;i Kop;iÞm1 ð9Þ
where a0is the initial crack length and anis the crack length in
the nth cycle.
2.2. Willenborg model
Noteworthy research on overload effect prediction was done by Wheeler.19 He recognized that the plastic zone has an important effect on retardation phenomenon. Current plastic zones are created inside the large plastic zone due to an over-load. The model proposed by Willenborg is also based on the plasticity concept and does not contain any meaningless parameter.20The deceleration of crack growth after the
over-load is accounted for by a reduction in stress intensity factor. The effective stress intensity factor is defined by the following expression:
Kmax;eff;i¼ Kmax;i Kred ð10Þ
Kmin;eff;i¼
Kmin;i Kred Kmin;i> Kred
0 Kmin;i6 Kred
ð11Þ
DKeff;i¼ Kmax;eff;i Kmin;eff;i ð12Þ
where Kmax,eff,i and Kmax,i are the maximum effective and
apparent (under constant amplitude) stress intensity factors in each load cycle i, respectively; Kmin,eff,i and Kmin,i are the
corresponding minimum values;DKeff,iis the effective stress
intensity factor range. When Kmin,iis larger than Kred, DKeff
is equal toDK. Otherwise, DKeff is equal to Kmax,eff,i. Kredis
the modified stress intensity factor, which characterizes the retardation phenomenon. Kredcan be written as
Kred¼ ðKmaxÞOL 1
Da ZOL
0:5
Kmax ð13Þ
where (Kmax)OL is the stress intensity factor of the overload
cycle,Da the amount crack growth length since the overload cycle, ZOLthe plastic zone size created by overload, and Kmax
the maximum stress intensity factor for cycle i. In order to esti-mate the plastic zone size, the formulation proposed by John-son is employed, which can be expressed as
ZOL¼ a
ðKmaxÞOL
ry
2
ð14Þ wherea is the plastic zone size factor. A parametric function fora proposed by Voorwald et al.22is employed. Huang et al.23 performed an elastic plastic analysis to verify the precision of this model by using finite element method. It is noted that the method is effective, which can be given by the following equation: a ¼ 1 6p tP 2:5 Krmaxy 2 1 p t6p1 Krmaxy 2 1 6pþ6p5 2:5tðKmax=ryÞ2 2:5p1 1 p Krmaxy 2 6 t 6 2:5 Kmax ry 2 8 > > > > > < > > > > > : ð15Þ where t is the specimen thickness. The Forman equation in conjunction with the Willenborg model can be expressed as
da dN¼ CF ðDKeffÞmF ð1 ReffÞKC DKeff ð16Þ Reff¼ Kmin;eff Kmax;eff ð17Þ
where CF and mF are the calibration parameters obtained
under constant amplitude loading, KC is the critical stress
intensity factor, and Reffis the effective stress ratio.
3. Comparative study under overload effect
In this section, the crack growth predictions by the Willenborg model and the modified VCA model are examined with exper-imental data in aluminum alloys under overload effects from the open literatures.
3.1. D16 aluminum alloy under single overload
Experimental data on D16 aluminum alloy under constant amplitude loading with/without multiple overloads spectrums are used for model validation.24,25The standard chemical com-position is shown inTable 1. And the mechanical properties of this material in longitudinal-transverse (L-T) orientation were as follows: ry= 347 MPa, ultimate tensile strength
rUTS= 460 MPa and percentage elongation is 12%. KC, the
critical stress intensity factor in Eq.(12), is 72.5 MPam0.5. In both of the two models, there are several unknown parameters which need to be calibrated. One set of da/dN-DK testing data under the constant amplitude loading (R = 0) are used to determine these parameters (Fig. 3).24
Table 1 Standard chemical composition of D16 aluminum alloy.17
Element Cu Mg Mn Si Fe Zn
Weight (%) 3.8–4.9 1.2–1.8 0.3–0.9 0.5 0.5 0.3
The fitting coefficients of the modified VCA model and Willen-borg model are identified as C = 1.2 109, m = 2.4534 and CF= 4.46 108, mF= 2.1394, respectively.
Manjunatha and Parida25 collected fatigue data for D16 aluminum alloy. The details are shown below. The specimens are with nominal dimensions of length = 180 mm, width = 45 mm and thickness = 1.5 mm. The initial edge crack length is a = 4.0 mm. These specimens are subjected to two different loading spectra: (1) constant amplitude loading withrmax= 60 MPa, R = 0.1; (2) constant amplitude loading
with single overload applied at the certain crack length (over-load ratio is 2).
Model predictions are compared with testing data inFig. 4. The single overloads are applied at the crack length of 5.5, 9.0,
12.0 and 16.0 mm. The circlets and squares are the experimen-tal data under constant amplitude (CA) loading with/without single overloads, respectively. The solid and dashed lines are the predictions of the modified VCA model and the Willenborg model under the above two conditions, respectively. It is noted that both of the two models can predict the crack growth well under constant amplitude loading. However, in the overload case, the predicted crack growth curve of the Willenborg model stops growing after the single overload is applied, which cannot match the testing data. According to many researches, the Willenborg model presents the limitation of predicting crack arrest for overloads with magnitude KOLP 2Kmax,i.
Actually, when the overload ratio is larger than 2, the fatigue cracks can still grow for many kinds of materials. Obviously, the modified VCA model can give a reasonable and better pre-diction under the constant amplitude loading with overloads.
3.2. Al7075-T6 under block loading
Porter26 collected the fatigue test data on center-notched Al7075-T6 specimens under different types of variable ampli-tude loading, some of which are also employed to validate the two models. The mechanical properties of this material in L-T orientation were as follows: ry= 520 MPa,
rUTS= 575 MPa, elastic modulus E= 72 GPa, and
KC= 71.5 MPa m0.5.
Similarly, the da/dN-DK test data of R = 0 are employed to calibrate these unknown parameters (Fig. 5).26The calibrated parameters of the modified VCA model and the Willenborg model are C= 1.9125 1010, m= 3.5990 and CF=
2.3918 109, mF= 3.4846.
The specimen configuration for these tests is width = 305 mm, length = 915 mm and thickness = 4.1 mm. The initial fatigue crack length is 2a = 12.7 mm. A schematic illustration of the loadings is shown inFig. 6. The specimen is subjected to (a) two kinds of constant loadings with rmax= 103.43 MPa, R= 0.1 and rmax= 68.95 MPa,
R= 0.1, respectively; (b) constant loading with overloads, in which rmax= 68.95 MPa, R = 0.1 and roverload= 103.43
-MPa; and (c) block of loading, in which q controls the number of cycles at the high amplitude. In this case, the value of q can be 3, 6, 10, 25 and 50.
The model predictions are compared to Al7075-T6 test data, as shown inFig. 7. The circlets are the experimental data. The solid and dashed lines are the predictions of the modified VCA model and the Willenborg model, respectively. Overall, both of the two models can match the testing data well, and the prediction curves of the former model are all closer to the testing data than those of the latter one. For the constant
Fig. 4 a-N curves for periodic spike loading.
Fig. 5 Figure of da/dN-DK calibration (Al7075-T6).
amplitude loading, the two models give satisfactory results. For the overload condition, good agreement is observed in general: when q is 1 and 3, the fatigue crack growth curves are slightly over-predicted; when q is 6 and 10, the two meth-ods agree with the testing data perfectly; when q is 25 and 50, the predictions are conservative. One probable explanation for these deviations is that the transient acceleration effect due to overload is neglected in both models. Physically, right after the single overload, the crack growth rate will have a transient acceleration before the relatively long-term retardation takes place. As the crack continues growing, the retardation will reduce gradually until vanish completely. Compared with the retardation, the effect of transient acceleration process can be ignored. In other words, the detailed instantaneous fatigue crack growth rate of the real crack after single overload is dif-ferent from the model simulations, but the total overload effects are approximately equivalent. For the block loading cases, however, these subtle differences are repeated and accu-mulated, which may lead to the deviations mentioned above.
4. Conclusions
This paper presents a comprehensive comparative study on the crack-closure-based model and the Willenborg model under the constant amplitude loading with/without overloads. In the modified VCA model, the fatigue crack growth rate is con-trolled by crack closure level, which is determined by forward and reverse plastic zone. In the Willenborg model, a monotonic plastic zone size is correlated with the effective stress intensity factor by a mathematical fitting equation. Detailed comparison is performed to investigate the applicabil-ity of these two models for the fatigue crack growth prediction under the constant amplitude loading with/without overloads in two aluminum alloys. The following conclusions can be given:
Generally, both of the two models can predict the fatigue crack growth under overload effects appropriately. In most of the cases, the modified VCA model shows better perfor-mance than the other. But when the overload ratio exceeds 2, the Willenborg model becomes nonfunctional.
The modified VCA model is based on the physical mecha-nism, which can be directly observed and verified by the in-situ SEM experiment.17,18 Moreover, it can be further improved based on micro/meso-scale experimental observa-tions in the future. However, the Willenborg model is more empirical rather than physical, which makes it difficult to ben-efit from the advanced experimental observations.
Additionally, the proposed modified VCA model can also mathematically deal with the tiny acceleration caused by the underload, whereas the Willenborg model cannot estimate the load sequence interaction effects which involve the underload.
Acknowledgements
The research was financially supported by the National Natu-ral Science Foundation of China (No. 51405009) and the Fun-damental Research Funds for the Central Universities of China.
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Zhang Wei received his Ph.D. degree in engineering from Clarkson University in 2012. He worked in Arizona State University as an assistant research scientist in 2013. Now he is an associate professor in reliability and systems engineering at Beihang University. His research areas include fatigue of material and structure.
