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Experimental evaluation of the effect of overloads on fatigue crack growth by analysing crack tip displacement fields
Engineering Fracture Mechanics 166 (2016) 82–96
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Experimental evaluation of the effect of overloads on fatigue crack growth by analysing crack tip displacement fields J.M. Vasco-Olmo ⇑, F.A. Díaz Universidad de Jaén, Escuela Politécnica Superior de Jaén, Departamento de Ingeniería Mecánica y Minera, Campus Las Lagunillas, Jaén 23071, Spain
a r t i c l e
i n f o
Article history: Received 1 April 2015 Received in revised form 25 July 2016 Accepted 29 August 2016 Available online 30 August 2016 Keywords: Fatigue crack Crack shielding Overload Stress intensity factor Digital image correlation
a b s t r a c t In this work, the shielding effect on growing fatigue cracks under different overload levels is evaluated by estimating stress intensity factors from the analysis of crack tip displacement fields measured by digital image correlation (DIC). A novel model called CJP is implemented to characterise crack tip displacement fields. The retardation effect induced by overloads is quantified from the calculation of the crack opening load. Moreover, a compliance based method is employed to compare and validate those results obtained by DIC. Results show a good level of agreement, highlighting that CJP model is a powerful tool to evaluate fracture mechanics problems. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Providing quantitative answers to problems related to contained cracks in mechanical elements has been an interesting topic of research since long time. In particular, fatigue cracks have been one of the main sources of structural failure in real service structures, where fracture mechanics has contributed to a better understanding of fatigue crack growth mechanisms. However there are still some issues that remain unresolved or misunderstood, one of these aspects is fatigue crack shielding phenomenon. This misunderstanding comes mainly from problems in its measurement and interpretation [1]. Fatigue crack growth at constant amplitude loading is reasonably well-understood through the shielding mechanisms at the vicinity of the crack tip. However, there is controversy about the possible mechanisms that can explain the retardation effect induced on fatigue crack growth due to the application of overloads. Three possible mechanisms have been proposed to explain retardation following an overload [2]. The first mechanism establishes plasticity-induced crack closure as retardation effect in the study of overloads [3,4]. The second retardation mechanism relapses on crack tip blunting [5]. Finally, residual compressive stresses are established as the third retardation mechanism in the study of overloads [6]. Plasticityinduced crack closure decreases fatigue crack growth rate by reducing the effective stress intensity factor range due to the contact between crack surfaces as the crack starts closing in a premature way [2]. Perhaps the most compelling evidence in favour of the closure argument is the phenomenon of delay retardation. That is, retardation generally does not occur immediately following the application of an overload. In some cases, the crack growth rate actually accelerates for a brief period after the overload. Closure mechanism provides a plausible explanation for the momentary acceleration of crack growth rate following an overload. If closure is occurring during constant amplitude loading, the effective stress intensity factor range is lower than the nominal stress intensity factor range and the crack growth rate is less than it would be in ⇑ Corresponding author. E-mail address: [email protected] (J.M. Vasco-Olmo). http://dx.doi.org/10.1016/j.engfracmech.2016.08.026 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.
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Nomenclature A0 , . . . , F 0 a a0, b0 c0 E FAx, FAy FC FPx, FPy FS FT G i j Kcl KF KI Kop KR KS R r, h s2 T t u, v W z DKeff DKnom DP
l j m
coefficients on CJP model for describing crack tip displacement fields crack length terms accounting for the horizontal and vertical rigid body translation term accounting for the rigid body rotation Young’s modulus reactions forces to the applied remote load contact force between crack flanks forces induced by the compatibility requirements on the elastic-plastic boundary shear force on the elastic-plastic boundary force due to the T-stress shear modulus square root of 1 jth collected data point closing stress intensity factor opening mode stress intensity factor in CJP model mode I stress intensity factor opening stress intensity factor retardation stress intensity factor shear stress intensity factor ratio between the minimum and the maximum applied load polar coordinates of the collected data points variance of the mathematical fitting T-stress specimen thickness horizontal and vertical displacements specimen width complex coordinate of the collected points around the crack tip effective stress intensity factor range nominal stress intensity factor range applied loading range mean of the mathematical fitting function of Poisson’s ratio Poisson’s ratio
the absence of closure. When an overload is applied, closure does not occur immediately following the overload, so the crack growth rate is momentarily higher than it was prior to the overload. In addition, the overload produces a larger plastic stretch in the wake of the crack tip as the fatigue crack propagates through the overload plastic zone. The contact of the fracture surfaces causes an enhancement in the level of plasticity-induced crack closure in the post-overload regime which promotes a retardation of crack growth rate. In the study of fracture mechanics problems, the analysis of the stress or displacement fields at the vicinity of the crack tip plays a significant role. Recently, full field experimental optical techniques such as photoelasticity [7,8] and thermoelasticity [9,10] have contributed to a better understanding of the different fatigue mechanisms and the retardation effect that can be induced during fatigue crack growth. In addition, digital image correlation (DIC) seems to be a very suitable technique for quantifying crack shielding phenomenon since displacement fields around the crack tip can be quantified with high level of accuracy [11]. Thus, experimental analysis of stress or displacement fields around a fatigue crack tip together with the experimental determination of stress intensity factors (SIFs) have constituted a major area of research for many years. Traditionally, it has been known in Linear Elastic Fracture Mechanics (LEFM) [12] that displacement fields at the vicinity of the crack tip can be characterised by SIFs. New methodologies for the calculation of SIFs from the analysis of the crack tip displacement fields have been developed [13–15]. Some works were based on Williams’s expansion series [13,14] and others on Muskhelishvili’s complex potentials [15]. All these methodologies were based on the Multi-Point Over-Deterministic Method (MPODM) developed by Sanford and Dally [16] for SIF calculation. Recently, special interest has been focused on the development of a novel mathematical model named CJP to describe crack tip stress/displacement fields. This model [17–19] considers the crack shielding effect on the elastic stress field induced by plasticity generated around growing fatigue cracks. Although DIC is globally accepted as a powerful technique for the analysis of fatigue and fracture problems, relatively little work has been done on the evaluation of crack shielding in growing fatigue cracks. The use of DIC for studying fracture mechanics problems started in 1980s, therefore most of the research work developed on the evaluation of fatigue crack shielding is quite recent. The first application of DIC to the measurement of fatigue crack closure was published by Sutton
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et al. [20] in 1999. The authors developed a methodology for the estimation of crack closing loads measuring the crack opening displacements behind the crack tip. More recently, de Mattos and Nowell [21] evaluated the effect of the specimen thickness during closure using 6082-T6 aluminium alloy CT specimens. Crack closure was assessed using a back face strain gauge, a crack mouth gauge and DIC. The authors used DIC to measure the relative displacements between pairs of points located behind the crack tip at each flank of the fracture. The procedure described was efficient and seemed to be a good alternative to conventional surface gauges. Another work on crack closure analysis using DIC was performed by López-Crespo et al. [22], who developed an approach to evaluate fatigue crack closure and incorporate the effect of crack tip plasticity on the effective stress intensity factors. The authors conducted closure tests on centre-cracked plate (CCP) specimens manufactured in Al7010-T7651 subjected to different mixed-mode loads. For the determination of SIFs the authors employed a method based on Muskhelishvili’s complex potentials [23]. The evaluation of the effective mixed-mode SIF allowed them to quantify the contact effects as a consequence of the interlocking of fracture surface asperities and plastic deformation of the asperities. Further work was also performed by Vasco-Olmo et al. [24], who proposed a methodology for the experimental evaluation of fatigue crack shielding using 2D-DIC. This methodology was based on the analysis of the SIFs calculated from the displacement fields measured around the crack tip. The authors performed a comparison between four different mathematical models describing crack tip displacement fields (namely Wetergaard’s, Williams’s, Muskhelishvili’s and CJP models). They established that CJP model was the most suitable for the evaluation of fatigue crack shielding from the analysis of SIFs KF and KR. In addition, results obtained from DIC were compared with those obtained using compliance-based methods from the analysis of the data recorded with an extensometer located on the mouth of Al2024-T3 CT specimens. The authors calculated crack opening and closing loads, obtaining a very good level of agreement between results from DIC technique and from compliance methods. Another topic related to the evaluation of fatigue crack shielding that has focused the attention of many researchers has been the evaluation of the effects produced by the application of overloads during fatigue crack growth. However, little work has been conducted on the study of overloads using DIC. Nowell and de Matos [25] presented results of crack growth rate and crack opening loads before and after a single overload that was applied during constant amplitude fatigue experiments. The authors employed DIC together with conventional strain gauges, comparing both techniques through the calculation of crack opening load. Recently, López-Crespo et al. [26] performed a study to evaluate the effect of an overload on fatigue crack tip fields by surface and bulk analysis. DIC was employed to obtain crack opening displacement (COD) data at several locations behind the crack tip by subtracting the vertical displacements of the upper and the lower flanks of the crack. In the current work, the effect induced on fatigue crack growth by the application of overloads is evaluated. In this work the effect of overloads is quantified from the analysis of stress intensity factors. In this way, the experimental methodology developed in [24] for estimating opening and closing loads from the opening SIF KF is employed for the quantification of crack shielding effect induced by the application of overloads. For the purpose of this work, several fatigue experiments at constant amplitude loading on 2024-T3 aluminium CT specimens subjected to different overload levels were conducted. In addition, the results obtained from DIC are compared with those obtained by employing a compliance based technique. The strain offset calculation method is implemented using COD data collected with an extensometer located at the mouth of the CT specimens. Results highlight that CJP model is a powerful tool to evaluate fracture mechanics problems such as plasticity-induced crack shielding and the effect of overloads.
2. Fundamentals of the model characterising crack tip displacement fields The current work has as main objective the experimental evaluation of the effect of the application of overloads during fatigue crack growth from the analysis of the stress intensity factors. Therefore, the implementation of a mathematical model characterising crack tip displacement fields constitutes a fundamental issue. In the literature, a large amount of analytical models can be found [13–15,19]. However, in a previous work [24] it was established that CJP model was the most suitable to evaluate fatigue crack shielding from the analysis of SIFs. According to this, CJP model is adopted to describe crack tip displacement fields and evaluate the effect of overloads. A brief description of CJP model is presented in this epigraph. CJP model is a novel mathematical model developed by Christopher, James and Patterson [17–19] for the characterisation of stress/displacement fields at the vicinity of the crack tip based on Muskhelishvili’s complex potentials [23] in combination with the Williams’s expansion series [27] using two terms. Additionally, the resulting expression is modified to include the effect of plasticity on crack tip elastic stress fields. According to this model, the plastic enclave existing at a fatigue crack tip and along the crack flanks shields the crack from the applied elastic stress field. Thus, the crack tip shielding mechanism includes the effect of the contact forces at the crack flanks and a compatibility-induced interfacial shear stress at the elastic-plastic boundary. In Fig. 1 a schematic illustration of the forces acting at the interface between the plastic zone and the surrounding elastic field is shown. FAx and FAy are the reaction forces to the applied remote load. FT is the force induced by the T-stress, which is also generated by the remote applied load and whose sense is dependent on the geometry of the specimen. The above forces generate a plastic zone ahead of the crack tip such that the material will become permanently deformed inducing during unloading forces FPx and FPy. These forces arise from the interaction between the plastic wake generated during fatigue crack growth and the surrounding elastic field, resulting therefore from the permanent deformation in the plastic zone around the crack tip, which is extensive in the direction perpendicular to the crack and contractive along the crack due to the effect of the Poisson’s ratio. FS is the induced force by the interfacial shear at the elastic-plastic
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Fig. 1. Scheme illustrating the forces acting at the elastic-plastic boundary [17].
boundary of the crack wake. FC is the induced force by the elastic field that originates the contact between crack flanks and the consequent crack closure. Therefore, the joint action between FP and FC induces a shield effect on the crack tip. According to this model, the horizontal and vertical displacements around the crack tip are described as follows:
C0 F0 C0 F0 2Gðu þ iv Þ ¼ j 2ðB0 þ 2E0 Þz1=2 þ 4z1=2 2E0 z1=2 ln z z z ðB0 þ 2E0 Þz1=2 E0 z1=2 ln z 4 4 0 0 C þ F z A0 z1=2 þ D0 z1=2 ln z 2D0 z1=2 þ 4
ð1Þ
E where G ¼ 2ð1þ mÞ is the shear modulus, E and m are the Young’s modulus and the Poisson’s ratio of the material respectively,
m for plane stress or j ¼ 3 4m for plane strain, z is the complex coordinate and A0 , B0 , C0 , D0 , E0 and F0 are unknown j ¼ 3 1þm
coefficients. This model defines four parameters to characterise crack tip stress/displacement fields through the forces represented in Fig. 1. An opening mode stress intensity factor (KF), a retardation stress intensity factor (KR), a shear stress intensity factor (KS) and the T-stress (T). KF is characterised by the driving crack growth force FA generated by the remote load, which originates crack tip stress fields traditionally characterised by classical KI. This quantity is defined from the asymptotic limit of ry as x ? +0, along y = 0, i.e. towards the crack tip from the front along the crack line:
rffiffiffiffi hpffiffiffiffiffiffiffiffiffi i p 0 K F ¼ lim 2pr ry þ 2E0 r 1=2 ln r ¼ ðA 3B0 8E0 Þ r!0 2
ð2Þ
KR is characterised by FC and FP, which induce a shielding effect on the crack tip and therefore a retardation effect on fatigue crack growth. This quantity characterises the direct stresses acting parallel to the crack growth direction and is obtained by evaluating rx in the limit as x ? 0, along y = 0, i.e. towards the crack tip from behind along the crack flank:
hpffiffiffiffiffiffiffiffiffi i p3=2 K R ¼ lim 2prrx ¼ pffiffiffi ðD0 3E0 Þ r!0 2
ð3Þ
KS is characterised by FS, which acts in the interfacial boundary between the plastic zone and the surrounding elastic field and is derived from the asymptotic limit of rxy as x ? 0, along y = 0, i.e. towards the crack tip from behind along the crack flank:
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ffi hpffiffiffiffiffiffiffiffiffi i rffiffiffi p 0 0 K S ¼ lim 2pr rxy ¼ ðA þ B Þ r!0 2
ð4Þ
Finally, T-stress is characterised by FT, which is generated by the remote load:
T x ¼ C 0 ;
T y ¼ F 0
ð5Þ
3. Experimental procedure For the different fatigue experiments conducted in this work, 2 mm thick CT specimens manufactured on 2024-T3 aluminium alloy were employed (Fig. 2a). All fatigue tests were conducted on a servohidraulic machine (MTS, model 370.10) with a loading capacity of 100 kN (Fig. 2b), employing a cycling frequency of 10 Hz. The purpose of the current work is the experimental evaluation of the effect derived of the application of different overloads during fatigue crack growth. Three specimens were tested. Experimental conditions and specimen references are presented in Table 1. The same loading amplitude was employed for all the tests. Thus, Al_CT1 specimen was not subjected to any overload, while Al_CT2 and Al_CT3 specimens were subjected to different single overloads. In the case of overload tests, it has been detailed the overload percentage respect to the maximum applied load, the overload value and the crack length and the number of cycles at which the overload was applied. In addition, it can be observed that a low R-ratio (R = 0) was defined to perform fatigue experiments. In this way, plasticity-induced crack shielding and the consequent retardation effect on fatigue crack growth could be evaluated in two different ways: that induced by the natural propagation of the crack and that induced by the application of overloads. During fatigue experiments, the test was momentarily stopped after a certain number of cycles and a sequence of images from a reference state to a deformed state at different load levels was captured using a monochromatic CCD camera (Allied Vision Technologies, model Stingray F-504B/C), placed perpendicularly to the specimen surface with a 75 mm lens controlled by an E56510 Intel Core i5 Dell-Latitude laptop through a FWB-EC3402 video card. The field of view obtained to conduct the fatigue experiments was 80.5 67.5 mm (2452 2056 pixels). The commercial software package Vic-2D [28] was employed to process the captured images to obtain in-plane displacements. Thus, 21 pixels were employed as size for the subset, with an overlap of 5 pixels to process the images. In addition, a shape function that allows translation, rotation, shear, normal strains and combinations of the subset was employed to assume that the shape of the reference subset can change in the deformed image. This setup allows measurement of the displacements of less than 0.01 mm. In addition, for the correct implementation of DIC, the surfaces of the specimens were prepared to obtain a grey intensity distribution (i.e. a random speckle pattern). Thus, the speckle was generated by spraying black paint over the specimen surface with a previously
Fig. 2. Geometry and dimensions of the CT specimens tested (a) and experimental setup employed during fatigue testing (b).
Table 1 Experimental conditions defined for fatigue tests. Specimen reference
Al_CT1 Al_CT2 Al_CT3
Loading conditions Pmin (N)
Pmax (N)
5
600
Ratio R
0
Overload conditions %
POL (N)
aOL (mm)
NOL (cycles)
– 100 125
– 1200 1350
– 26.12 26.72
– 280,000 200,000
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Fig. 3. Detail of the extensometer employed for crack COD measurement.
painted white background (Fig. 3). Also, an extra camera (Allied Vision Technologies, model Pike F-032B/C) with a 25 mm lens was employed to track the crack tip and monitor the crack growth during the fatigue experiments. As indicated above, results obtained from DIC were compared to those obtained from the interpretation of the compliance traces. The compliance evaluation was performed from the COD measurements using an extensometer (model 632.03F-30) located at the mouth of the notch manufactured in the CT specimens with a data sampling of 5 Hz.
4. Description of the experimental methodology for stress intensity factor calculation In this work stress intensity factors are analysed to evaluate the induced effect on fatigue crack growth by the application of overloads. According to this, it is essential the development of an experimental methodology to calculate SIFs from the analysis of the displacement fields measured at the vicinity of the crack tip. The different steps followed to implement the proposed methodology are shown in Fig. 4. The first step in the methodology is to capture a sequence of images during fatigue experiments between a reference state (undeformed) and a deformed state at different loading steps. Once the images have been captured, the next step is to select the region of interest for the subsequent image processing. According to this, the areas that do not have relevant information must be avoided. Thus, the areas corresponding to the notch of the CT specimens and the grips were removed for the correct processing of the captured images. The next step in the methodology consists on the image processing. The sequence of images was processed using a crosscorrelation algorithm [29] implemented in the commercial software package Vic-2D [28]. Fig. 5 shows the horizontal and vertical displacement fields for a 34.96 mm crack at a load level of 600 N. Once the displacement fields have been obtained, in the next step a set of data points are collected around the crack tip (Fig. 5). According to this, it is important to establish that the adopted model of crack tip displacement fields is based on LEFM and SIFs are derived from the MPODM developed by Sanford and Dally [16]. Therefore, the way to define the region around the crack tip constitutes a significant aspect. Thus, an annular region was adopted for the data point collection, selecting two main parameters to define it, an inner radius and another outer. In this way, the inner radius must be defined avoiding the collection of data points too close to the crack tip where the effects of the plastic zone exist. According to this, the adopted inner radius was defined as the Dugdale plastic zone size to guarantee the removal of any plastic deformation around the crack. Moreover, the outer radius is less easy to define and usually it is adopted a distance from the crack tip where the stress fields no longer describe the state of stresses. According to this, the outer radius was defined from the influence zone of the crack tip. This information was extracted from the vertical displacement field where it is observed a change in the orientation of the displacements due to the interaction between the influence zone of the crack tip and that of the free
Fig. 4. Experimental methodology for the calculation of stress intensity factors.
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Fig. 5. Horizontal (a) and vertical (b) displacement fields measured for a crack length of 34.96 mm and a load level of 600 N, and data point collection around the crack tip.
edge of the specimen. In addition, a sensitivity analysis was performed varying the value for the outer radius and it was observed that the SIF values were more sensitive when the data collection reached the zone dominated by the free edge of the specimen, while they were more consistent when the data were collected in the region dominated by the crack tip. For the displacement fields shown in Fig. 5, the inner radius used was 2.5 mm and the outer radius 15 mm. The SIF values were more sensitive for inner radius values, especially when data were collected close to the crack tip because they were in the zone dominated by plasticity. Moreover, another important aspect is the crack tip location. It was inferred by the direct observation over the measured displacement fields. The data collection was composed by 230 data-points, using 5° as separation angle between radial lines and taking 10 points per radial line. However, no significant differences in the SIF values were observed in the sensitivity for the data density. After data point collection, the next step in the methodology was to use the data points collected to fit the equations defining the mathematical model and estimate the SIFs. According to CJP model, the displacement fields were directly related to the unknown coefficients from which the SIFs can be determined; hence the resultant system of equations was linear. An error function (Eq. (6)) was defined to relate the experimental data with the mathematical expression describing the adopted model. At this point, it is important to indicate that in Eq. (6), rigid body motion terms were considered. Three additional terms were included in the formulation: horizontal translation (a0), vertical translation (b0) and rotation (c0).
(
g u;j
(
) ¼
g v ;j
f u;j ðA0 ; B0 ; C 0 ; D0 ; E0 ; F 0 ; z; G; m; a0 ; c0 r cos hÞ f v ;j ðA0 ; B0 ; C 0 ; D0 ; E0 ; F 0 ; z; G; m; b0 ; c0 r sin hÞ
)
uj
vj
¼0
ð6Þ
where the sub-index j indicates the values of the error function evaluated at the jth data point with polar coordinates (r, h). Moreover, to aid in the calculation process of the stress intensity factors, a graphical computer interface has been built using MatlabÒ. In addition, the quality of the mathematical fitting was assessed by two statistical parameters, such as the mean and the variance [30]:
l¼
Pp
hi ¼
i¼1 hi
p
g u;i g v ;i
1X 2 ðhi lÞ p i¼1 p
;
s2 ¼
!,
ui
ð7Þ
vi
ð8Þ
where p is the total number of collected data points. Finally, the experimental SIFs were compared with those nominal values predicted by ASTM E647-99 [31] according to the following expression:
a 2 a 3 a 4 2 þ Wa DP a 13:32 DK nom ¼ pffiffiffiffiffiffi 0:886 þ 4:64 þ 14:72 5:6 W W W W t W 1 Wa 3=2
ð9Þ
where DP is the loading range, t and W are the thickness and the width of the specimen, respectively, and a is the crack length.
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5. Methodology for calculating opening/closing loads from compliance traces In Section 4, the experimental methodology for the calculation of the stress intensity factors from displacements measured at the vicinity of the crack tip using DIC was described. However, DIC is a modern technique that has not been established yet as standard method for the evaluation of crack shielding. To support the idea that DIC can provide information about real prediction of fatigue crack growth rates, in this section the experimental methodology for the calculation of opening and closing loads from the analysis of compliance traces is described. Compliance based techniques have been one of the most extensively reported techniques for the measurement of crack opening/closing loads when crack closure or shielding is present [32]. The compliance concept is derived from crack closure phenomenon. Thus, the contact between crack flanks results in a change of the specimen stiffness and consequently in the specimen compliance. Therefore, compliance based techniques estimate the crack opening and closing loads by analysing the change in the specimen stiffness as the crack starts opening or closing. Different strain methods [32] to quantify crack closure can be found in the literature. These techniques employ strain gauges or extensometers placed at different locations to determine the near crack tip strains or the COD. The method adopted in this work is the strain offset calculation [33], which estimates the opening and closing loads by recording COD data as a function of the applied load using an extensometer. According to this method, the signal collected from the extensometer was divided into two data sets corresponding to the loading and unloading branches of the applied loading cycle. In addition, to minimise the noise influence, the collected signal was filtered employing an incremental polynomial method similar to that described by ASTM E647-99 [31]. The two branches were represented on a COD versus load plot, and a least squares straight line was fitted to experimental data at the part of the loading cycle at which the crack was fully open. For this purpose, a segment spanning a range of the 25% of the loading range starting just below the maximum load was selected to represent the fully open crack configuration. The fitted straight line was employed to estimate the theoretical COD values for a particular load value. Therefore, the strain offset was obtained as the difference between theoretical and experimental COD values. Finally, the strain offset was presented as a function of the applied load, estimating the opening (Pop) and closing (Pcl) loads from the loading and unloading branches respectively, as the load value at which the strain offset starts deviating from zero value.
6. Results and discussion In this section, results obtained by employing DIC and those obtained from the analysis of the compliance traces are presented and discussed. The first way to study the influence originated by the application of overloads has been from the analysis of the variation of the crack length with the number of cycles. The plots corresponding to the crack length versus the number of cycles for the specimens reported in Table 1 have been represented in Fig. 6. It can be clearly observed the retardation effect induced on crack growth for the two specimens subjected to overloads compared with that at which any overload was not applied. The retardation effect induced by 100% overload (1.2 kN) remained during 415,000 cycles (Da 2.11 mm); while the retardation remained during 420,000 cycles (Da 3.42 mm) for the case of the specimen subjected to 125% overload (1.35 kN).
Fig. 6. Evolution of the crack length with the number of cycles for the specimens collected in Table 1.
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After the study of the overload influence along the number of cycles, an important aspect investigated was the ability to evaluate the retardation effect on fatigue crack growth from the analysis of KF and KR SIFs according to the reported methodology in [24] for the estimation of opening and closing loads. Thus, the evolution of these SIFs along the loading cycle for the previous and subsequent cycles to the application of 125% overload has been represented in Figs. 7 and 8, respectively. From the analysis of KF, it is observed a change in the trend of the values for the lower part of the loading cycle from a particular load value for both the loading and unloading branches. According to this, Kop and Kcl values can be inferred from KF trend as that value corresponding to the minimum value of the loading range. In this way, it is observed that Kop and Kcl values are higher after the application of the overload as illustrated in Fig. 8 from the observation of a more evident change in KF values. Moreover, from the analysis of KR, a change in its sign is observed for the same portion of the cycle above analysed from KF.
Fig. 7. Experimental stress intensity factors obtained for the previous cycle to the application of 125% overload.
Fig. 8. Experimental stress intensity factors obtained in the subsequent cycle to the application of the overload for the specimen subjected to 125% overload level.
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This change in the trends of KF and KR highlights the existence of shielding effect at low load levels, therefore this detected portion divides the loading cycle into two intervals, that at which the crack is closed and that at which is open. These results highlight that the application of an overload increases the shielding effect on crack tip, and hence an increase in the fatigue life of the analysed component. Once KF and KR have been evaluated prior and after the application of an overload, Kop and DKeff are calculated. DKeff is obtained as the difference between KF at the maximum load and Kop. Fig. 9 shows the variation of Kop and DKeff for the two analysed tests with overload. In addition, Kmin corresponding to the minimum applied load and DK obtained from the nominal loading range have been also presented. It can be observed that Kop shows a gradual increase with the crack length, with values always above Kmin. This behaviour highlights that the crack opens at a load value higher than the minimum applied load, illustrating evidence of crack shielding effect. In addition, this evidence of plasticity-induced crack shielding is also observed from the analysis of DKeff, with values always below DKnom that highlights a decrease in the fatigue crack growth rate. Moreover, it is observed an increase in Kop in the subsequent cycle to the application of the overloads, decreasing gradually until it approaches again the trend prior to the application of the overload. It is believed that this increase in Kop is due to the plasticity increase after the overload which has as consequence that the required applied load to open the crack was higher. In addition, an opposite behaviour is observed for DKeff, decreasing immediately after the application of the overload to subsequently increase until it approaches the trend prior to the overload. Thus, a higher retardation effect was observed for the highest applied overload. Therefore, these behaviours highlight an increase in the retardation effect in the zones affected by overloads as a direct consequence of the plasticity increase at the crack tip due to the applied overload. The analysis of Kop and DKeff can explain how different overloads could affect in the shielding level. This issue can be more clearly quantified from the calculation of the crack opening load and its subsequent analysis. Hence, Pop was calculated from Kop using Eq. (9). Fig. 10 shows the variation of Pop along the crack length for the two conducted tests with the application of overloads. In addition, the zones affected by the different overloads have been also marked. It is observed that Pop values are always above the minimum applied load, taking a value around 100 N in the regions before and after the zone affected by the applied overload. This behaviour was previously observed for Kop, which highlights that a shielding effect due to plasticity was induced during crack growth. Therefore, as it is illustrated in the results, the opening load is not influenced by the crack length in that range established by ASTM E647-99 [31] unlike the opening stress intensity factor, which increases with the crack length since Eq. (9) marked by the standard [31] to obtain the nominal stress intensity factor is directly proportional to the crack length. Moreover, this behaviour was observed by Sehitoglu [34], McClung and Sehitoglu [35], and Wei and James [36]. All they reported that opening and closing loads remained unchanged with increasing crack length. In addition, Pop values experienced an increase immediately after the application of the overloads, decreasing later progressively until it reaches the trend prior to the overload. Thus, the increase experienced by Pop was different depending on the overload level. In the case of a 100% overload an increase until 194.41 N was experienced, which corresponds to a 31.83% of the applied loading range. However, in the case of a 125% overload Pop increased until 256.42 N, which supposes a 42.25% with respect to the applied loading range. All results indicate that the crack opens at load levels above the minimum applied load throughout the crack length, while for the overload affected zones the crack opens at higher loads than those corresponding to the zones not affected by the overloads. Moreover, the changes observed in Pop values at crack lengths of 24.5 and 25 mm for the specimen Al_CT3 are attributed to the normal variability in experimental results. There are different
Fig. 9. Variation of Kop and DKeff with the crack length for the two conducted tests with overload application.
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Fig. 10. Plots showing Pop variation with the crack length for the two different tests conducted with the application of overloads.
factors and parameters that may affect the results, such as image capturing parameters, the image processing issues or the mathematical approach for SIF calculation, among others. Once the retardation effect induced on fatigue crack growth has been quantified from the analysis of Pop, this effect has been also studied through the Paris law. In Fig. 11 it is plotted the curve corresponding to Paris law from the nominal stress intensity factor range for the specimen subjected to the highest overload level. It is observed a normal crack growth until the application of the overload (line 1–2). Then, a brief acceleration in the crack growth rate is experienced immediately after the overload was applied (line 2–3), followed by a retardation effect (line 4–5). Finally, as the crack grows, the crack growth rate approaches that observed for constant amplitude loading prior to the overload application. The above plot has been employed to explain the behaviour of fatigue crack growth due to the application of an overload. However, it is of particular interest to focus the overload problem. For this purpose, Paris law from the overload application has been plotted. According to this, point 3 corresponding with the acceleration generated on fatigue crack growth due to the application of the overload has not been plotted. Both the nominal and effective Paris law curves obtained after the overload have been presented in Fig. 12. It can be observed that immediately after the overloads, the effective curves are located to the left of the nominal one, which highlights the induced retardation effect on crack growth rate. Subsequently a progressive increase in DKeff is observed as the crack approaches crack growth rate prior to the overload. In addition, it is observed that DKeff is smaller when the applied overload level is higher, which highlights that a longer and bigger retardation effect is induced at higher overload levels.
Fig. 11. Paris law obtained from the nominal stress intensity factor range for the specimen subjected to the highest overload level.
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Fig. 12. Nominal and effective Paris law curves focusing after the applied overloads for the two conducted tests.
Once the influence due to the application of overloads on fatigue crack growth has been evaluated from the analysis of the SIFs were compared with those obtained from the analysis of the compliance traces plotted by using COD data recorded with an extensometer. The strain offset calculation method has been implemented to obtain crack opening load as a function of the crack length. As it was previously done for the analysis of the SIFs along the loading cycle in the case of the specimen under the highest overload, in Figs. 13 and 14 the compliance traces prior and after the application of the overload have been plotted. Plots located on the left correspond to results for the loading branch, while those located on the right correspond to the unloading branch. Top plots show the applied load versus the measured COD for loading (left) and unloading (right) branches. In both plots, it can be observed raw data, filtered data and a fitted line corresponding to no closure. Bottom plots show the applied load versus the strain offset for loading (left) and unloading (right) branches. Prior to the overload (Fig. 13), opening and closing loads obtained were 100 and 110 N, respectively from the analysis of loading and unloading braches. In addition, after the overload (Fig. 14), opening and closing loads were 255 and 260, respectively. Therefore, the influence of the overload is clearly established with an increase of opening and closing loads.
Fig. 13. Results obtained from the strain offset calculation method for the previous cycle to the application of the overload for the specimen subjected to the highest overload. Plots corresponding to applied load versus COD for loading (a) and unloading branches (b) and plots corresponding to applied load versus strain offset for loading (c) and unloading branches (d).
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Fig. 14. Results obtained from the strain offset calculation method for the subsequent cycle to the application of the overload for the specimen subjected to the highest overload. Plots corresponding to applied load versus COD for loading (a) and unloading branches (b) and plots corresponding to applied load versus strain offset for loading (c) and unloading branches (d).
Results obtained in the above figures can be extrapolated to analyse the evolution of Pop along the crack length. Fig. 15 shows the variation of Pop with the crack length for the two overload levels studied using both DIC and the strain offset calculation method. It can be observed that a good level of agreement between results is achieved. Therefore, this comparative study highlights that the proposed methodology to estimate Kop from the analysis of KF constitutes a real alternative to evaluate the effect induced on fatigue crack growth by the application of overloads. The authors believe that the proposed methodology using the CJP model combined with DIC measurements has not been previously employed in the literature considering the effect of overloads in the opening and closing loads. In other similar works reported in the literature [25,26], they measured the COD employing the relative displacements between the crack flanks; while in this work, the authors have evaluated the effect of the overloads from the analysis of the SIFs defined in the CJP model. Nevertheless, the obtained results show a great agreement with those obtained from the compliance method
Fig. 15. Variation of Pop with the crack length for the two overload levels studied using DIC and the strain offset calculation method.
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indicated above. Moreover, another important aspect is the applied load. In other previous works reported in the literature, the maximum applied load was much higher than the maximum load adopted for the current work, which can also explain differences with other authors. It is convenient indicate that in the reported works here [25,26], the authors observed little or no closure in the cycle or two cycles after the overload. However, this has not been that observed in the current work. In the view of the authors, the difference regarding to those results referenced in the literature can be due to the applied overload level, being much higher that those applied in the current work. Therefore, the blunting phenomenon can exist in the results reported in the references, while in the current work the applied overload was not enough to generate this effect. The authors, in the current work, have focused their investigations on the evaluation of the effect derived of the application of overloads in thin metal specimens. Therefore, the study of this phenomenon on thicker specimens will be part of future work focusing the attention on the effect of overloads for plane strain conditions. 7. Conclusions In this work the effect induced on fatigue crack growth by the applications of different overload levels has been evaluated by analysing the stress intensity factors calculated from the displacement fields measured at the vicinity of the crack tip using DIC. This involves a novel contribution regarding to the research work previously published on the overload effect evaluation using DIC, where it was employed as a technique for COD measurement. Plasticity-induced crack shielding and the consequent retardation effect on fatigue crack growth have been successfully quantified in two different ways: that induced by the natural propagation of the crack and that induced by the application of overloads. CJP model has been implemented for the characterisation of crack tip displacement fields, showing an enormous potential for the evaluation of plasticity-induced crack shielding on growing fatigue cracks due to the application of overloads. The crack opening load has been estimated from the analysis of KF. Several fatigue tests at constant amplitude loading on 2024-T3 aluminium CT specimens subjected to different overload levels were conducted. Results obtained from DIC have been compared with those obtained from the analysis of the compliance traces. Results show a high level of agreement, highlighting that CJP model constitutes a real alternative for the evaluation of fracture mechanics problems such as plasticity-induced crack shielding and the retardation effect induced by overloads. With this work, the authors have tried to contribute to a better understanding on fatigue crack growth mechanisms and all the existent controversy about crack closure phenomenon. In the view of the authors, shielding phenomenon is a more general concept than closure. Thus, closure phenomenon is a consequence of shielding since the plastic enclave generated during fatigue crack growth will shield the crack from the influence of the surrounding elastic stress field. Therefore, the crack flanks will contact due to the action of the elastic stress fields on the plastic zone which will produce that the crack closes. For this reason, in this work the CJP model has been employed to evaluate the plasticity-induced crack shielding phenomenon and, one way can be through the quantification of the closure effect. Acknowledgements The current work has been conducted with financial support from Junta de Andalucía through the project ‘Proyecto de Investigación de Excelencia de la Junta de Andalucía TEP 2009-51770 , funded with FEDER funds. References [1] James MN. Some unresolved issues with fatigue crack closure – measurement, mechanism and interpretation problems. Ninth international conference on fracture, vol. 5. Sydney (Australia): Pergamon Press; 1997. p. 2403–14. [2] Suresh S. Fatigue of materials. Cambridge: Cambridge University Press; 2001. [3] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2(1):37–45. [4] Elber W. The significance of fatigue crack closure. Philadelphia: Damage Tolerance in Aircraft Structures, ASTM STP 486, American Society for Testing and Materials; 1971. p. 230–42. [5] Rice JR. Mechanics of crack tip deformation and extension by fatigue. Philadelphia: Fatigue Crack Propagation, ASTM STP 415, American Society for Testing and Materials; 1967. p. 247–309. [6] Wheeler OE. Spectrum loading and crack growth. J Basic Eng 1972;94:181–6. [7] Vasco-Olmo JM, Díaz FA. Experimental evaluation of plasticity-induced crack shielding from isochromatic data. Opt Eng 2015;54(8). http://dx.doi.org/ 10.1117/1.OE.54.8.08120. [8] Colombo C, Du Y, James MN, Patterson EA, Vergani L. On crack tip shielding due to plasticity-induced closure during an overload. Fatigue Fract Eng Mater Struct 2010;33:766–77. [9] Díaz FA, Patterson EA, Yates JR. Assessment of effective stress intensity factors using thermoelastic stress analysis. J Strain Anal Eng Des 2009;44:621–31. [10] Patki AS, Patterson EA. Thermoelastic stress analysis of fatigue cracks subject to overloads. Fatigue Fract Eng Mater Struct 2010;33(12):809–21. [11] Chu TC, Ranson WF, Sutton MA, Peters WH. Applications of digital-image correlation technique to experimental mechanics. Exp Mech 1985;25:232–44. [12] Ewalds HL, Wanhill RJH. Fracture mechanics. London: Edward Arnold; 1991. [13] Yoneyama S, Morimoto Y, Takahasi M. Automatic evaluation of mixed-mode stress intensity factors utilizing digital image correlation. Strain 2006;42:21–9. [14] Yates JR, Zanganeh M, Tai YH. Quantifying crack tip displacement fields with DIC. Eng Fract Mech 2010;77:2063–76. [15] López-Crespo P, Shterenlikht A, Patterson EA, Withers PJ, Yates JR. The stress intensity factors of mixed mode cracks determined by digital image correlation. J Strain Anal Eng Des 2008;43:769–80.
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[16] Sanford RJ, Dally JW. A general method for determining the mixed-mode stress intensity factors from isochromatic fringe patterns. Eng Fract Mech 1979;11:621–33. [17] Christopher CJ, James MN, Patterson EA, Tee KF. Towards a new model of crack tip stress fields. Int J Fract 2007;148:361–71. [18] Christopher CJ, James MN, Patterson EA, Tee KF. A quantitative evaluation of fatigue crack shielding forces using photoelasticity. Eng Fract Mech 2008;75:4190–9. [19] James MN, Christopher CJ, Lu Y, Patterson EA. Local crack plasticity and its influences on the global elastic stress field. Int J Fatigue 2013;46:4–15. [20] Sutton MA, Zhao W, McNeill SR, Helm JD, Piascik RS, Ridell WT. Local crack closure measurements: development of a measurement system using computer vision and a far-field microscope. Philadelphia: Advances in Fatigue Crack Closure Measurement and Analysis, ASTM STP 1343, American Society for Testing and Materials; 1999. p. 145–56. [21] de Mattos PFP, Nowell D. Experimental and numerical investigations of thickness effects in plasticity-induced fatigue crack closure. Int J Fatigue 2008;31:1795–804. [22] López-Crespo P, Shterenlikht A, Yates JR, Patterson EA, Withers PJ. Some experimental observations on crack closure and crack-tip plasticity. Fatigue Fract Eng Mater Struct 2009;32:418–29. [23] Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity. Groningen: Noordhoff International Publishing; 1977. [24] Vasco-Olmo JM, Díaz FA, García-Collado A, Dorado R. Experimental evaluation of crack shielding during fatigue crack growth using digital image correlation. Fatigue Fract Eng Mater Struct 2015;38(2):223–37. [25] Nowell D, de Matos PFP. Application of digital image correlation to the investigation of crack closure following overloads. Proc Eng 2010;2:2722–7. [26] López-Crespo P, Whiters PJ, Yusof F, Dai H, Steuwer A, Kelleher JF, Buslaps T. Overload effects on fatigue crack-tip fields under plane stress conditions: surface and bulk analysis. Fatigue Fract Eng Mater Struct 2013;36(1):75–84. [27] Williams ML. On the stress distribution at the base of a stationary crack. J Appl Mech 1957;24:109–14. [28] http://www.correlatedsolutions.com/vic-2d/. [29] Sutton MA, Orteau JJ, Schreier HW. Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications. New York: Springer Science + Business Media; 2009. [30] Nurse AD, Patterson EA. Determination of predominantly mode II stress intensity factors from isochromatic data. Fatigue Fract Eng Mater Struct 1993;16(12):1339–54. [31] E647-99. Standard test method for measurement of fatigue crack growth rates. Philadelphia: American Society for Testing and Materials; 1999. [32] Stoychev S, Kujawski D. Methods for crack opening load and crack tip shielding determination: a review. Fatigue Fract Eng Mater Struct 2003;26:1053–67. [33] Skorupa M, Beretta S, Carboni M, Machniewicz T. An algorithm for evaluating fatigue crack closure from local compliance measurements. Fatigue Fract Eng Mater Struct 2002;25:261–73. [34] Sehitoglu H. Crack opening and closure in fatigue. Eng Fract Mech 1985;21(2):329–39. [35] McClung RC, Sehitoglu H. On the finite element analysis of fatigue crack closure – 2. Numerical results. Eng Fract Mech 1989;33(2):253–72. [36] Wei L-W, James MN. A study of fatigue crack closure in polycarbonate CT specimens. Eng Fract Mech 2000;66:223–42.
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Experimental evaluation of the effect of overloads on fatigue crack growth by analysing crack tip displacement fields J.M. Vasco-Olmo ⇑, F.A. Díaz Universidad de Jaén, Escuela Politécnica Superior de Jaén, Departamento de Ingeniería Mecánica y Minera, Campus Las Lagunillas, Jaén 23071, Spain
a r t i c l e
i n f o
Article history: Received 1 April 2015 Received in revised form 25 July 2016 Accepted 29 August 2016 Available online 30 August 2016 Keywords: Fatigue crack Crack shielding Overload Stress intensity factor Digital image correlation
a b s t r a c t In this work, the shielding effect on growing fatigue cracks under different overload levels is evaluated by estimating stress intensity factors from the analysis of crack tip displacement fields measured by digital image correlation (DIC). A novel model called CJP is implemented to characterise crack tip displacement fields. The retardation effect induced by overloads is quantified from the calculation of the crack opening load. Moreover, a compliance based method is employed to compare and validate those results obtained by DIC. Results show a good level of agreement, highlighting that CJP model is a powerful tool to evaluate fracture mechanics problems. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Providing quantitative answers to problems related to contained cracks in mechanical elements has been an interesting topic of research since long time. In particular, fatigue cracks have been one of the main sources of structural failure in real service structures, where fracture mechanics has contributed to a better understanding of fatigue crack growth mechanisms. However there are still some issues that remain unresolved or misunderstood, one of these aspects is fatigue crack shielding phenomenon. This misunderstanding comes mainly from problems in its measurement and interpretation [1]. Fatigue crack growth at constant amplitude loading is reasonably well-understood through the shielding mechanisms at the vicinity of the crack tip. However, there is controversy about the possible mechanisms that can explain the retardation effect induced on fatigue crack growth due to the application of overloads. Three possible mechanisms have been proposed to explain retardation following an overload [2]. The first mechanism establishes plasticity-induced crack closure as retardation effect in the study of overloads [3,4]. The second retardation mechanism relapses on crack tip blunting [5]. Finally, residual compressive stresses are established as the third retardation mechanism in the study of overloads [6]. Plasticityinduced crack closure decreases fatigue crack growth rate by reducing the effective stress intensity factor range due to the contact between crack surfaces as the crack starts closing in a premature way [2]. Perhaps the most compelling evidence in favour of the closure argument is the phenomenon of delay retardation. That is, retardation generally does not occur immediately following the application of an overload. In some cases, the crack growth rate actually accelerates for a brief period after the overload. Closure mechanism provides a plausible explanation for the momentary acceleration of crack growth rate following an overload. If closure is occurring during constant amplitude loading, the effective stress intensity factor range is lower than the nominal stress intensity factor range and the crack growth rate is less than it would be in ⇑ Corresponding author. E-mail address: [email protected] (J.M. Vasco-Olmo). http://dx.doi.org/10.1016/j.engfracmech.2016.08.026 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.
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Nomenclature A0 , . . . , F 0 a a0, b0 c0 E FAx, FAy FC FPx, FPy FS FT G i j Kcl KF KI Kop KR KS R r, h s2 T t u, v W z DKeff DKnom DP
l j m
coefficients on CJP model for describing crack tip displacement fields crack length terms accounting for the horizontal and vertical rigid body translation term accounting for the rigid body rotation Young’s modulus reactions forces to the applied remote load contact force between crack flanks forces induced by the compatibility requirements on the elastic-plastic boundary shear force on the elastic-plastic boundary force due to the T-stress shear modulus square root of 1 jth collected data point closing stress intensity factor opening mode stress intensity factor in CJP model mode I stress intensity factor opening stress intensity factor retardation stress intensity factor shear stress intensity factor ratio between the minimum and the maximum applied load polar coordinates of the collected data points variance of the mathematical fitting T-stress specimen thickness horizontal and vertical displacements specimen width complex coordinate of the collected points around the crack tip effective stress intensity factor range nominal stress intensity factor range applied loading range mean of the mathematical fitting function of Poisson’s ratio Poisson’s ratio
the absence of closure. When an overload is applied, closure does not occur immediately following the overload, so the crack growth rate is momentarily higher than it was prior to the overload. In addition, the overload produces a larger plastic stretch in the wake of the crack tip as the fatigue crack propagates through the overload plastic zone. The contact of the fracture surfaces causes an enhancement in the level of plasticity-induced crack closure in the post-overload regime which promotes a retardation of crack growth rate. In the study of fracture mechanics problems, the analysis of the stress or displacement fields at the vicinity of the crack tip plays a significant role. Recently, full field experimental optical techniques such as photoelasticity [7,8] and thermoelasticity [9,10] have contributed to a better understanding of the different fatigue mechanisms and the retardation effect that can be induced during fatigue crack growth. In addition, digital image correlation (DIC) seems to be a very suitable technique for quantifying crack shielding phenomenon since displacement fields around the crack tip can be quantified with high level of accuracy [11]. Thus, experimental analysis of stress or displacement fields around a fatigue crack tip together with the experimental determination of stress intensity factors (SIFs) have constituted a major area of research for many years. Traditionally, it has been known in Linear Elastic Fracture Mechanics (LEFM) [12] that displacement fields at the vicinity of the crack tip can be characterised by SIFs. New methodologies for the calculation of SIFs from the analysis of the crack tip displacement fields have been developed [13–15]. Some works were based on Williams’s expansion series [13,14] and others on Muskhelishvili’s complex potentials [15]. All these methodologies were based on the Multi-Point Over-Deterministic Method (MPODM) developed by Sanford and Dally [16] for SIF calculation. Recently, special interest has been focused on the development of a novel mathematical model named CJP to describe crack tip stress/displacement fields. This model [17–19] considers the crack shielding effect on the elastic stress field induced by plasticity generated around growing fatigue cracks. Although DIC is globally accepted as a powerful technique for the analysis of fatigue and fracture problems, relatively little work has been done on the evaluation of crack shielding in growing fatigue cracks. The use of DIC for studying fracture mechanics problems started in 1980s, therefore most of the research work developed on the evaluation of fatigue crack shielding is quite recent. The first application of DIC to the measurement of fatigue crack closure was published by Sutton
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et al. [20] in 1999. The authors developed a methodology for the estimation of crack closing loads measuring the crack opening displacements behind the crack tip. More recently, de Mattos and Nowell [21] evaluated the effect of the specimen thickness during closure using 6082-T6 aluminium alloy CT specimens. Crack closure was assessed using a back face strain gauge, a crack mouth gauge and DIC. The authors used DIC to measure the relative displacements between pairs of points located behind the crack tip at each flank of the fracture. The procedure described was efficient and seemed to be a good alternative to conventional surface gauges. Another work on crack closure analysis using DIC was performed by López-Crespo et al. [22], who developed an approach to evaluate fatigue crack closure and incorporate the effect of crack tip plasticity on the effective stress intensity factors. The authors conducted closure tests on centre-cracked plate (CCP) specimens manufactured in Al7010-T7651 subjected to different mixed-mode loads. For the determination of SIFs the authors employed a method based on Muskhelishvili’s complex potentials [23]. The evaluation of the effective mixed-mode SIF allowed them to quantify the contact effects as a consequence of the interlocking of fracture surface asperities and plastic deformation of the asperities. Further work was also performed by Vasco-Olmo et al. [24], who proposed a methodology for the experimental evaluation of fatigue crack shielding using 2D-DIC. This methodology was based on the analysis of the SIFs calculated from the displacement fields measured around the crack tip. The authors performed a comparison between four different mathematical models describing crack tip displacement fields (namely Wetergaard’s, Williams’s, Muskhelishvili’s and CJP models). They established that CJP model was the most suitable for the evaluation of fatigue crack shielding from the analysis of SIFs KF and KR. In addition, results obtained from DIC were compared with those obtained using compliance-based methods from the analysis of the data recorded with an extensometer located on the mouth of Al2024-T3 CT specimens. The authors calculated crack opening and closing loads, obtaining a very good level of agreement between results from DIC technique and from compliance methods. Another topic related to the evaluation of fatigue crack shielding that has focused the attention of many researchers has been the evaluation of the effects produced by the application of overloads during fatigue crack growth. However, little work has been conducted on the study of overloads using DIC. Nowell and de Matos [25] presented results of crack growth rate and crack opening loads before and after a single overload that was applied during constant amplitude fatigue experiments. The authors employed DIC together with conventional strain gauges, comparing both techniques through the calculation of crack opening load. Recently, López-Crespo et al. [26] performed a study to evaluate the effect of an overload on fatigue crack tip fields by surface and bulk analysis. DIC was employed to obtain crack opening displacement (COD) data at several locations behind the crack tip by subtracting the vertical displacements of the upper and the lower flanks of the crack. In the current work, the effect induced on fatigue crack growth by the application of overloads is evaluated. In this work the effect of overloads is quantified from the analysis of stress intensity factors. In this way, the experimental methodology developed in [24] for estimating opening and closing loads from the opening SIF KF is employed for the quantification of crack shielding effect induced by the application of overloads. For the purpose of this work, several fatigue experiments at constant amplitude loading on 2024-T3 aluminium CT specimens subjected to different overload levels were conducted. In addition, the results obtained from DIC are compared with those obtained by employing a compliance based technique. The strain offset calculation method is implemented using COD data collected with an extensometer located at the mouth of the CT specimens. Results highlight that CJP model is a powerful tool to evaluate fracture mechanics problems such as plasticity-induced crack shielding and the effect of overloads.
2. Fundamentals of the model characterising crack tip displacement fields The current work has as main objective the experimental evaluation of the effect of the application of overloads during fatigue crack growth from the analysis of the stress intensity factors. Therefore, the implementation of a mathematical model characterising crack tip displacement fields constitutes a fundamental issue. In the literature, a large amount of analytical models can be found [13–15,19]. However, in a previous work [24] it was established that CJP model was the most suitable to evaluate fatigue crack shielding from the analysis of SIFs. According to this, CJP model is adopted to describe crack tip displacement fields and evaluate the effect of overloads. A brief description of CJP model is presented in this epigraph. CJP model is a novel mathematical model developed by Christopher, James and Patterson [17–19] for the characterisation of stress/displacement fields at the vicinity of the crack tip based on Muskhelishvili’s complex potentials [23] in combination with the Williams’s expansion series [27] using two terms. Additionally, the resulting expression is modified to include the effect of plasticity on crack tip elastic stress fields. According to this model, the plastic enclave existing at a fatigue crack tip and along the crack flanks shields the crack from the applied elastic stress field. Thus, the crack tip shielding mechanism includes the effect of the contact forces at the crack flanks and a compatibility-induced interfacial shear stress at the elastic-plastic boundary. In Fig. 1 a schematic illustration of the forces acting at the interface between the plastic zone and the surrounding elastic field is shown. FAx and FAy are the reaction forces to the applied remote load. FT is the force induced by the T-stress, which is also generated by the remote applied load and whose sense is dependent on the geometry of the specimen. The above forces generate a plastic zone ahead of the crack tip such that the material will become permanently deformed inducing during unloading forces FPx and FPy. These forces arise from the interaction between the plastic wake generated during fatigue crack growth and the surrounding elastic field, resulting therefore from the permanent deformation in the plastic zone around the crack tip, which is extensive in the direction perpendicular to the crack and contractive along the crack due to the effect of the Poisson’s ratio. FS is the induced force by the interfacial shear at the elastic-plastic
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Fig. 1. Scheme illustrating the forces acting at the elastic-plastic boundary [17].
boundary of the crack wake. FC is the induced force by the elastic field that originates the contact between crack flanks and the consequent crack closure. Therefore, the joint action between FP and FC induces a shield effect on the crack tip. According to this model, the horizontal and vertical displacements around the crack tip are described as follows:
C0 F0 C0 F0 2Gðu þ iv Þ ¼ j 2ðB0 þ 2E0 Þz1=2 þ 4z1=2 2E0 z1=2 ln z z z ðB0 þ 2E0 Þz1=2 E0 z1=2 ln z 4 4 0 0 C þ F z A0 z1=2 þ D0 z1=2 ln z 2D0 z1=2 þ 4
ð1Þ
E where G ¼ 2ð1þ mÞ is the shear modulus, E and m are the Young’s modulus and the Poisson’s ratio of the material respectively,
m for plane stress or j ¼ 3 4m for plane strain, z is the complex coordinate and A0 , B0 , C0 , D0 , E0 and F0 are unknown j ¼ 3 1þm
coefficients. This model defines four parameters to characterise crack tip stress/displacement fields through the forces represented in Fig. 1. An opening mode stress intensity factor (KF), a retardation stress intensity factor (KR), a shear stress intensity factor (KS) and the T-stress (T). KF is characterised by the driving crack growth force FA generated by the remote load, which originates crack tip stress fields traditionally characterised by classical KI. This quantity is defined from the asymptotic limit of ry as x ? +0, along y = 0, i.e. towards the crack tip from the front along the crack line:
rffiffiffiffi hpffiffiffiffiffiffiffiffiffi i p 0 K F ¼ lim 2pr ry þ 2E0 r 1=2 ln r ¼ ðA 3B0 8E0 Þ r!0 2
ð2Þ
KR is characterised by FC and FP, which induce a shielding effect on the crack tip and therefore a retardation effect on fatigue crack growth. This quantity characterises the direct stresses acting parallel to the crack growth direction and is obtained by evaluating rx in the limit as x ? 0, along y = 0, i.e. towards the crack tip from behind along the crack flank:
hpffiffiffiffiffiffiffiffiffi i p3=2 K R ¼ lim 2prrx ¼ pffiffiffi ðD0 3E0 Þ r!0 2
ð3Þ
KS is characterised by FS, which acts in the interfacial boundary between the plastic zone and the surrounding elastic field and is derived from the asymptotic limit of rxy as x ? 0, along y = 0, i.e. towards the crack tip from behind along the crack flank:
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ffi hpffiffiffiffiffiffiffiffiffi i rffiffiffi p 0 0 K S ¼ lim 2pr rxy ¼ ðA þ B Þ r!0 2
ð4Þ
Finally, T-stress is characterised by FT, which is generated by the remote load:
T x ¼ C 0 ;
T y ¼ F 0
ð5Þ
3. Experimental procedure For the different fatigue experiments conducted in this work, 2 mm thick CT specimens manufactured on 2024-T3 aluminium alloy were employed (Fig. 2a). All fatigue tests were conducted on a servohidraulic machine (MTS, model 370.10) with a loading capacity of 100 kN (Fig. 2b), employing a cycling frequency of 10 Hz. The purpose of the current work is the experimental evaluation of the effect derived of the application of different overloads during fatigue crack growth. Three specimens were tested. Experimental conditions and specimen references are presented in Table 1. The same loading amplitude was employed for all the tests. Thus, Al_CT1 specimen was not subjected to any overload, while Al_CT2 and Al_CT3 specimens were subjected to different single overloads. In the case of overload tests, it has been detailed the overload percentage respect to the maximum applied load, the overload value and the crack length and the number of cycles at which the overload was applied. In addition, it can be observed that a low R-ratio (R = 0) was defined to perform fatigue experiments. In this way, plasticity-induced crack shielding and the consequent retardation effect on fatigue crack growth could be evaluated in two different ways: that induced by the natural propagation of the crack and that induced by the application of overloads. During fatigue experiments, the test was momentarily stopped after a certain number of cycles and a sequence of images from a reference state to a deformed state at different load levels was captured using a monochromatic CCD camera (Allied Vision Technologies, model Stingray F-504B/C), placed perpendicularly to the specimen surface with a 75 mm lens controlled by an E56510 Intel Core i5 Dell-Latitude laptop through a FWB-EC3402 video card. The field of view obtained to conduct the fatigue experiments was 80.5 67.5 mm (2452 2056 pixels). The commercial software package Vic-2D [28] was employed to process the captured images to obtain in-plane displacements. Thus, 21 pixels were employed as size for the subset, with an overlap of 5 pixels to process the images. In addition, a shape function that allows translation, rotation, shear, normal strains and combinations of the subset was employed to assume that the shape of the reference subset can change in the deformed image. This setup allows measurement of the displacements of less than 0.01 mm. In addition, for the correct implementation of DIC, the surfaces of the specimens were prepared to obtain a grey intensity distribution (i.e. a random speckle pattern). Thus, the speckle was generated by spraying black paint over the specimen surface with a previously
Fig. 2. Geometry and dimensions of the CT specimens tested (a) and experimental setup employed during fatigue testing (b).
Table 1 Experimental conditions defined for fatigue tests. Specimen reference
Al_CT1 Al_CT2 Al_CT3
Loading conditions Pmin (N)
Pmax (N)
5
600
Ratio R
0
Overload conditions %
POL (N)
aOL (mm)
NOL (cycles)
– 100 125
– 1200 1350
– 26.12 26.72
– 280,000 200,000
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Fig. 3. Detail of the extensometer employed for crack COD measurement.
painted white background (Fig. 3). Also, an extra camera (Allied Vision Technologies, model Pike F-032B/C) with a 25 mm lens was employed to track the crack tip and monitor the crack growth during the fatigue experiments. As indicated above, results obtained from DIC were compared to those obtained from the interpretation of the compliance traces. The compliance evaluation was performed from the COD measurements using an extensometer (model 632.03F-30) located at the mouth of the notch manufactured in the CT specimens with a data sampling of 5 Hz.
4. Description of the experimental methodology for stress intensity factor calculation In this work stress intensity factors are analysed to evaluate the induced effect on fatigue crack growth by the application of overloads. According to this, it is essential the development of an experimental methodology to calculate SIFs from the analysis of the displacement fields measured at the vicinity of the crack tip. The different steps followed to implement the proposed methodology are shown in Fig. 4. The first step in the methodology is to capture a sequence of images during fatigue experiments between a reference state (undeformed) and a deformed state at different loading steps. Once the images have been captured, the next step is to select the region of interest for the subsequent image processing. According to this, the areas that do not have relevant information must be avoided. Thus, the areas corresponding to the notch of the CT specimens and the grips were removed for the correct processing of the captured images. The next step in the methodology consists on the image processing. The sequence of images was processed using a crosscorrelation algorithm [29] implemented in the commercial software package Vic-2D [28]. Fig. 5 shows the horizontal and vertical displacement fields for a 34.96 mm crack at a load level of 600 N. Once the displacement fields have been obtained, in the next step a set of data points are collected around the crack tip (Fig. 5). According to this, it is important to establish that the adopted model of crack tip displacement fields is based on LEFM and SIFs are derived from the MPODM developed by Sanford and Dally [16]. Therefore, the way to define the region around the crack tip constitutes a significant aspect. Thus, an annular region was adopted for the data point collection, selecting two main parameters to define it, an inner radius and another outer. In this way, the inner radius must be defined avoiding the collection of data points too close to the crack tip where the effects of the plastic zone exist. According to this, the adopted inner radius was defined as the Dugdale plastic zone size to guarantee the removal of any plastic deformation around the crack. Moreover, the outer radius is less easy to define and usually it is adopted a distance from the crack tip where the stress fields no longer describe the state of stresses. According to this, the outer radius was defined from the influence zone of the crack tip. This information was extracted from the vertical displacement field where it is observed a change in the orientation of the displacements due to the interaction between the influence zone of the crack tip and that of the free
Fig. 4. Experimental methodology for the calculation of stress intensity factors.
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Fig. 5. Horizontal (a) and vertical (b) displacement fields measured for a crack length of 34.96 mm and a load level of 600 N, and data point collection around the crack tip.
edge of the specimen. In addition, a sensitivity analysis was performed varying the value for the outer radius and it was observed that the SIF values were more sensitive when the data collection reached the zone dominated by the free edge of the specimen, while they were more consistent when the data were collected in the region dominated by the crack tip. For the displacement fields shown in Fig. 5, the inner radius used was 2.5 mm and the outer radius 15 mm. The SIF values were more sensitive for inner radius values, especially when data were collected close to the crack tip because they were in the zone dominated by plasticity. Moreover, another important aspect is the crack tip location. It was inferred by the direct observation over the measured displacement fields. The data collection was composed by 230 data-points, using 5° as separation angle between radial lines and taking 10 points per radial line. However, no significant differences in the SIF values were observed in the sensitivity for the data density. After data point collection, the next step in the methodology was to use the data points collected to fit the equations defining the mathematical model and estimate the SIFs. According to CJP model, the displacement fields were directly related to the unknown coefficients from which the SIFs can be determined; hence the resultant system of equations was linear. An error function (Eq. (6)) was defined to relate the experimental data with the mathematical expression describing the adopted model. At this point, it is important to indicate that in Eq. (6), rigid body motion terms were considered. Three additional terms were included in the formulation: horizontal translation (a0), vertical translation (b0) and rotation (c0).
(
g u;j
(
) ¼
g v ;j
f u;j ðA0 ; B0 ; C 0 ; D0 ; E0 ; F 0 ; z; G; m; a0 ; c0 r cos hÞ f v ;j ðA0 ; B0 ; C 0 ; D0 ; E0 ; F 0 ; z; G; m; b0 ; c0 r sin hÞ
)
uj
vj
¼0
ð6Þ
where the sub-index j indicates the values of the error function evaluated at the jth data point with polar coordinates (r, h). Moreover, to aid in the calculation process of the stress intensity factors, a graphical computer interface has been built using MatlabÒ. In addition, the quality of the mathematical fitting was assessed by two statistical parameters, such as the mean and the variance [30]:
l¼
Pp
hi ¼
i¼1 hi
p
g u;i g v ;i
1X 2 ðhi lÞ p i¼1 p
;
s2 ¼
!,
ui
ð7Þ
vi
ð8Þ
where p is the total number of collected data points. Finally, the experimental SIFs were compared with those nominal values predicted by ASTM E647-99 [31] according to the following expression:
a 2 a 3 a 4 2 þ Wa DP a 13:32 DK nom ¼ pffiffiffiffiffiffi 0:886 þ 4:64 þ 14:72 5:6 W W W W t W 1 Wa 3=2
ð9Þ
where DP is the loading range, t and W are the thickness and the width of the specimen, respectively, and a is the crack length.
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5. Methodology for calculating opening/closing loads from compliance traces In Section 4, the experimental methodology for the calculation of the stress intensity factors from displacements measured at the vicinity of the crack tip using DIC was described. However, DIC is a modern technique that has not been established yet as standard method for the evaluation of crack shielding. To support the idea that DIC can provide information about real prediction of fatigue crack growth rates, in this section the experimental methodology for the calculation of opening and closing loads from the analysis of compliance traces is described. Compliance based techniques have been one of the most extensively reported techniques for the measurement of crack opening/closing loads when crack closure or shielding is present [32]. The compliance concept is derived from crack closure phenomenon. Thus, the contact between crack flanks results in a change of the specimen stiffness and consequently in the specimen compliance. Therefore, compliance based techniques estimate the crack opening and closing loads by analysing the change in the specimen stiffness as the crack starts opening or closing. Different strain methods [32] to quantify crack closure can be found in the literature. These techniques employ strain gauges or extensometers placed at different locations to determine the near crack tip strains or the COD. The method adopted in this work is the strain offset calculation [33], which estimates the opening and closing loads by recording COD data as a function of the applied load using an extensometer. According to this method, the signal collected from the extensometer was divided into two data sets corresponding to the loading and unloading branches of the applied loading cycle. In addition, to minimise the noise influence, the collected signal was filtered employing an incremental polynomial method similar to that described by ASTM E647-99 [31]. The two branches were represented on a COD versus load plot, and a least squares straight line was fitted to experimental data at the part of the loading cycle at which the crack was fully open. For this purpose, a segment spanning a range of the 25% of the loading range starting just below the maximum load was selected to represent the fully open crack configuration. The fitted straight line was employed to estimate the theoretical COD values for a particular load value. Therefore, the strain offset was obtained as the difference between theoretical and experimental COD values. Finally, the strain offset was presented as a function of the applied load, estimating the opening (Pop) and closing (Pcl) loads from the loading and unloading branches respectively, as the load value at which the strain offset starts deviating from zero value.
6. Results and discussion In this section, results obtained by employing DIC and those obtained from the analysis of the compliance traces are presented and discussed. The first way to study the influence originated by the application of overloads has been from the analysis of the variation of the crack length with the number of cycles. The plots corresponding to the crack length versus the number of cycles for the specimens reported in Table 1 have been represented in Fig. 6. It can be clearly observed the retardation effect induced on crack growth for the two specimens subjected to overloads compared with that at which any overload was not applied. The retardation effect induced by 100% overload (1.2 kN) remained during 415,000 cycles (Da 2.11 mm); while the retardation remained during 420,000 cycles (Da 3.42 mm) for the case of the specimen subjected to 125% overload (1.35 kN).
Fig. 6. Evolution of the crack length with the number of cycles for the specimens collected in Table 1.
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After the study of the overload influence along the number of cycles, an important aspect investigated was the ability to evaluate the retardation effect on fatigue crack growth from the analysis of KF and KR SIFs according to the reported methodology in [24] for the estimation of opening and closing loads. Thus, the evolution of these SIFs along the loading cycle for the previous and subsequent cycles to the application of 125% overload has been represented in Figs. 7 and 8, respectively. From the analysis of KF, it is observed a change in the trend of the values for the lower part of the loading cycle from a particular load value for both the loading and unloading branches. According to this, Kop and Kcl values can be inferred from KF trend as that value corresponding to the minimum value of the loading range. In this way, it is observed that Kop and Kcl values are higher after the application of the overload as illustrated in Fig. 8 from the observation of a more evident change in KF values. Moreover, from the analysis of KR, a change in its sign is observed for the same portion of the cycle above analysed from KF.
Fig. 7. Experimental stress intensity factors obtained for the previous cycle to the application of 125% overload.
Fig. 8. Experimental stress intensity factors obtained in the subsequent cycle to the application of the overload for the specimen subjected to 125% overload level.
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This change in the trends of KF and KR highlights the existence of shielding effect at low load levels, therefore this detected portion divides the loading cycle into two intervals, that at which the crack is closed and that at which is open. These results highlight that the application of an overload increases the shielding effect on crack tip, and hence an increase in the fatigue life of the analysed component. Once KF and KR have been evaluated prior and after the application of an overload, Kop and DKeff are calculated. DKeff is obtained as the difference between KF at the maximum load and Kop. Fig. 9 shows the variation of Kop and DKeff for the two analysed tests with overload. In addition, Kmin corresponding to the minimum applied load and DK obtained from the nominal loading range have been also presented. It can be observed that Kop shows a gradual increase with the crack length, with values always above Kmin. This behaviour highlights that the crack opens at a load value higher than the minimum applied load, illustrating evidence of crack shielding effect. In addition, this evidence of plasticity-induced crack shielding is also observed from the analysis of DKeff, with values always below DKnom that highlights a decrease in the fatigue crack growth rate. Moreover, it is observed an increase in Kop in the subsequent cycle to the application of the overloads, decreasing gradually until it approaches again the trend prior to the application of the overload. It is believed that this increase in Kop is due to the plasticity increase after the overload which has as consequence that the required applied load to open the crack was higher. In addition, an opposite behaviour is observed for DKeff, decreasing immediately after the application of the overload to subsequently increase until it approaches the trend prior to the overload. Thus, a higher retardation effect was observed for the highest applied overload. Therefore, these behaviours highlight an increase in the retardation effect in the zones affected by overloads as a direct consequence of the plasticity increase at the crack tip due to the applied overload. The analysis of Kop and DKeff can explain how different overloads could affect in the shielding level. This issue can be more clearly quantified from the calculation of the crack opening load and its subsequent analysis. Hence, Pop was calculated from Kop using Eq. (9). Fig. 10 shows the variation of Pop along the crack length for the two conducted tests with the application of overloads. In addition, the zones affected by the different overloads have been also marked. It is observed that Pop values are always above the minimum applied load, taking a value around 100 N in the regions before and after the zone affected by the applied overload. This behaviour was previously observed for Kop, which highlights that a shielding effect due to plasticity was induced during crack growth. Therefore, as it is illustrated in the results, the opening load is not influenced by the crack length in that range established by ASTM E647-99 [31] unlike the opening stress intensity factor, which increases with the crack length since Eq. (9) marked by the standard [31] to obtain the nominal stress intensity factor is directly proportional to the crack length. Moreover, this behaviour was observed by Sehitoglu [34], McClung and Sehitoglu [35], and Wei and James [36]. All they reported that opening and closing loads remained unchanged with increasing crack length. In addition, Pop values experienced an increase immediately after the application of the overloads, decreasing later progressively until it reaches the trend prior to the overload. Thus, the increase experienced by Pop was different depending on the overload level. In the case of a 100% overload an increase until 194.41 N was experienced, which corresponds to a 31.83% of the applied loading range. However, in the case of a 125% overload Pop increased until 256.42 N, which supposes a 42.25% with respect to the applied loading range. All results indicate that the crack opens at load levels above the minimum applied load throughout the crack length, while for the overload affected zones the crack opens at higher loads than those corresponding to the zones not affected by the overloads. Moreover, the changes observed in Pop values at crack lengths of 24.5 and 25 mm for the specimen Al_CT3 are attributed to the normal variability in experimental results. There are different
Fig. 9. Variation of Kop and DKeff with the crack length for the two conducted tests with overload application.
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Fig. 10. Plots showing Pop variation with the crack length for the two different tests conducted with the application of overloads.
factors and parameters that may affect the results, such as image capturing parameters, the image processing issues or the mathematical approach for SIF calculation, among others. Once the retardation effect induced on fatigue crack growth has been quantified from the analysis of Pop, this effect has been also studied through the Paris law. In Fig. 11 it is plotted the curve corresponding to Paris law from the nominal stress intensity factor range for the specimen subjected to the highest overload level. It is observed a normal crack growth until the application of the overload (line 1–2). Then, a brief acceleration in the crack growth rate is experienced immediately after the overload was applied (line 2–3), followed by a retardation effect (line 4–5). Finally, as the crack grows, the crack growth rate approaches that observed for constant amplitude loading prior to the overload application. The above plot has been employed to explain the behaviour of fatigue crack growth due to the application of an overload. However, it is of particular interest to focus the overload problem. For this purpose, Paris law from the overload application has been plotted. According to this, point 3 corresponding with the acceleration generated on fatigue crack growth due to the application of the overload has not been plotted. Both the nominal and effective Paris law curves obtained after the overload have been presented in Fig. 12. It can be observed that immediately after the overloads, the effective curves are located to the left of the nominal one, which highlights the induced retardation effect on crack growth rate. Subsequently a progressive increase in DKeff is observed as the crack approaches crack growth rate prior to the overload. In addition, it is observed that DKeff is smaller when the applied overload level is higher, which highlights that a longer and bigger retardation effect is induced at higher overload levels.
Fig. 11. Paris law obtained from the nominal stress intensity factor range for the specimen subjected to the highest overload level.
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Fig. 12. Nominal and effective Paris law curves focusing after the applied overloads for the two conducted tests.
Once the influence due to the application of overloads on fatigue crack growth has been evaluated from the analysis of the SIFs were compared with those obtained from the analysis of the compliance traces plotted by using COD data recorded with an extensometer. The strain offset calculation method has been implemented to obtain crack opening load as a function of the crack length. As it was previously done for the analysis of the SIFs along the loading cycle in the case of the specimen under the highest overload, in Figs. 13 and 14 the compliance traces prior and after the application of the overload have been plotted. Plots located on the left correspond to results for the loading branch, while those located on the right correspond to the unloading branch. Top plots show the applied load versus the measured COD for loading (left) and unloading (right) branches. In both plots, it can be observed raw data, filtered data and a fitted line corresponding to no closure. Bottom plots show the applied load versus the strain offset for loading (left) and unloading (right) branches. Prior to the overload (Fig. 13), opening and closing loads obtained were 100 and 110 N, respectively from the analysis of loading and unloading braches. In addition, after the overload (Fig. 14), opening and closing loads were 255 and 260, respectively. Therefore, the influence of the overload is clearly established with an increase of opening and closing loads.
Fig. 13. Results obtained from the strain offset calculation method for the previous cycle to the application of the overload for the specimen subjected to the highest overload. Plots corresponding to applied load versus COD for loading (a) and unloading branches (b) and plots corresponding to applied load versus strain offset for loading (c) and unloading branches (d).
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Fig. 14. Results obtained from the strain offset calculation method for the subsequent cycle to the application of the overload for the specimen subjected to the highest overload. Plots corresponding to applied load versus COD for loading (a) and unloading branches (b) and plots corresponding to applied load versus strain offset for loading (c) and unloading branches (d).
Results obtained in the above figures can be extrapolated to analyse the evolution of Pop along the crack length. Fig. 15 shows the variation of Pop with the crack length for the two overload levels studied using both DIC and the strain offset calculation method. It can be observed that a good level of agreement between results is achieved. Therefore, this comparative study highlights that the proposed methodology to estimate Kop from the analysis of KF constitutes a real alternative to evaluate the effect induced on fatigue crack growth by the application of overloads. The authors believe that the proposed methodology using the CJP model combined with DIC measurements has not been previously employed in the literature considering the effect of overloads in the opening and closing loads. In other similar works reported in the literature [25,26], they measured the COD employing the relative displacements between the crack flanks; while in this work, the authors have evaluated the effect of the overloads from the analysis of the SIFs defined in the CJP model. Nevertheless, the obtained results show a great agreement with those obtained from the compliance method
Fig. 15. Variation of Pop with the crack length for the two overload levels studied using DIC and the strain offset calculation method.
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indicated above. Moreover, another important aspect is the applied load. In other previous works reported in the literature, the maximum applied load was much higher than the maximum load adopted for the current work, which can also explain differences with other authors. It is convenient indicate that in the reported works here [25,26], the authors observed little or no closure in the cycle or two cycles after the overload. However, this has not been that observed in the current work. In the view of the authors, the difference regarding to those results referenced in the literature can be due to the applied overload level, being much higher that those applied in the current work. Therefore, the blunting phenomenon can exist in the results reported in the references, while in the current work the applied overload was not enough to generate this effect. The authors, in the current work, have focused their investigations on the evaluation of the effect derived of the application of overloads in thin metal specimens. Therefore, the study of this phenomenon on thicker specimens will be part of future work focusing the attention on the effect of overloads for plane strain conditions. 7. Conclusions In this work the effect induced on fatigue crack growth by the applications of different overload levels has been evaluated by analysing the stress intensity factors calculated from the displacement fields measured at the vicinity of the crack tip using DIC. This involves a novel contribution regarding to the research work previously published on the overload effect evaluation using DIC, where it was employed as a technique for COD measurement. Plasticity-induced crack shielding and the consequent retardation effect on fatigue crack growth have been successfully quantified in two different ways: that induced by the natural propagation of the crack and that induced by the application of overloads. CJP model has been implemented for the characterisation of crack tip displacement fields, showing an enormous potential for the evaluation of plasticity-induced crack shielding on growing fatigue cracks due to the application of overloads. The crack opening load has been estimated from the analysis of KF. Several fatigue tests at constant amplitude loading on 2024-T3 aluminium CT specimens subjected to different overload levels were conducted. Results obtained from DIC have been compared with those obtained from the analysis of the compliance traces. Results show a high level of agreement, highlighting that CJP model constitutes a real alternative for the evaluation of fracture mechanics problems such as plasticity-induced crack shielding and the retardation effect induced by overloads. With this work, the authors have tried to contribute to a better understanding on fatigue crack growth mechanisms and all the existent controversy about crack closure phenomenon. In the view of the authors, shielding phenomenon is a more general concept than closure. Thus, closure phenomenon is a consequence of shielding since the plastic enclave generated during fatigue crack growth will shield the crack from the influence of the surrounding elastic stress field. Therefore, the crack flanks will contact due to the action of the elastic stress fields on the plastic zone which will produce that the crack closes. For this reason, in this work the CJP model has been employed to evaluate the plasticity-induced crack shielding phenomenon and, one way can be through the quantification of the closure effect. Acknowledgements The current work has been conducted with financial support from Junta de Andalucía through the project ‘Proyecto de Investigación de Excelencia de la Junta de Andalucía TEP 2009-51770 , funded with FEDER funds. References [1] James MN. Some unresolved issues with fatigue crack closure – measurement, mechanism and interpretation problems. Ninth international conference on fracture, vol. 5. Sydney (Australia): Pergamon Press; 1997. p. 2403–14. [2] Suresh S. Fatigue of materials. Cambridge: Cambridge University Press; 2001. [3] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2(1):37–45. [4] Elber W. The significance of fatigue crack closure. 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