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Analysis of crack geometry and location in notched bars by means of a three-probe potential drop technique
International Journal of Fatigue 124 (2019) 167–187
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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Analysis of crack geometry and location in notched bars by means of a threeprobe potential drop technique
T
Alberto Campagnoloa, Jürgen Bärb, Giovanni Meneghettia,
⁎
a b
Department of Industrial Engineering, University of Padova, Via Venezia, 1 – 35131 Padova, Italy Institute for Materials Science, University of the Bundeswehr Munich, D-85577 Neubiberg, Germany
ARTICLE INFO
ABSTRACT
Keywords: Potential drop method Calibration curves Crack shape Steel Titanium alloy
It is often necessary to define the crack initiation life of a fatigue tested component, generally at a given (short) crack length. The size of an initiated crack can be estimated by employing different experimental methods, one of which is the direct current potential drop (DCPD) technique. In the case of notched bars subjected to fatigue loadings, the crack configuration (i.e. circumferential or semi-elliptical) and location cannot be singled out by means of the potential drop method (PDM) operating with a single potential probe. In the present contribution, three potential probes are adopted to overcome this issue. The calibration curves reporting the three potential drops as a function of the crack size are derived by means of 3-dimensional electrical FE analyses. Two different crack configurations are analyzed: (i) circumferential and (ii) semi-elliptical surface cracks. The calibration curves have been validated by systematic comparison with experimental results, generated by fatigue testing of sharp as well as blunt notched specimens made of steel and a titanium alloy under pure axial loading. Finally, a procedure to assess the area, the configuration and the location of the initiated fatigue crack starting from the experimentally measured potential drops is discussed.
1. Introduction Crack monitoring is often adopted to investigate damage initiation and spreading in specimens or structural components undergoing fatigue loading [1]. The majority of local approaches for fatigue life assessment require a quantitative estimation of the crack size, at which the reference fatigue life is defined. As an example, the experimental results obtained by fatigue testing of sharp as well as blunt notched bars made of TiAl6V4 under multiaxial loading have been recently analysed [2] by adopting the local approach based on the strain energy density (SED) averaged over a material structural volume, surrounding the crack initiation point and having size R0 [3]. Strictly speaking, a physical definition of “crack initiation” is necessary to apply the SED approach, which should be defined when an initiated crack has a depth a = R0, i.e. when the propagating (short) crack leads to the structural volume failure. The size of a propagating crack can be estimated by employing different experimental techniques, one of which is the direct current potential drop (DCPD) method. Calibration of the potential drop method (PDM), i.e. determining the potential drop change as a function of the crack depth a, has been performed in the literature by employing experimental, analytical and numerical techniques. Generally, the ⁎
experimental calibration [4–11] requires comparison with a different (previously calibrated) method. Experimental calibration has been performed primarily by optically measuring the surface crack advance [8], by machining pre-cracks of increasing depth in a tested sample [9] or by marking the fracture surface by applying overloads or by changing the nominal load ratio [4]. Electrolytic tank simulation [10], cutting of graphitized papers [11] or thin aluminium foils [4] to simulate a propagating crack are other available experimental techniques. Calibration can also be performed by solving the Laplace equation, i.e. 2 V = 0 , in which V is the electrical potential, for given specimen geometry and PDM operating conditions. However, closed-form solutions of the Laplace equation are available only for few simple cases [4,12]; an example is the Johnson’s formula, which has been widely adopted for single edge notch (SEN), single edge bend (SEB), compact tension (CT) and disc compact tension (DCT) specimens [13–15]. The conformal mapping approach [12,16] has been employed to derive analytical solutions of the Laplace equation in the case of notched specimens. The finite element method (FEM) can also be used to calibrate the potential drop method, as was suggested by Ritchie and Bathe [9] and by Aronson and Ritchie [17]. Calibration by FEM has the following
Corresponding author. E-mail address: [email protected] (G. Meneghetti).
https://doi.org/10.1016/j.ijfatigue.2019.02.045 Received 14 January 2019; Received in revised form 20 February 2019; Accepted 27 February 2019 Available online 01 March 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature
Δ
a Acrack Anet c c/a F I Nf Pi
δV θ
Pavg R R0 ROL r ΔVi,0 ΔVi
crack depth crack area specimen net-section area half surface crack length crack aspect ratio axial load electric current number of loading cycles to failure potential drop ratio, i.e. ΔVi/ΔVi,0, at i-th probe (i = 1, 2, 3) average potential drop ratio, i.e. (P1 + P2 + P3)/3 nominal load ratio size of the structural volume for averaged SED calculation load ratio of overload specimen net-section radius potential drop of the un-cracked (a = 0) specimen at i-th probe (i = 1, 2, 3) potential drop of a cracked specimen at i-th probe (i = 1, 2, 3)
θ′ ρ
Abbreviations BN CT CTS DCPD DCT FE FEM OL PDM SEB SED SEN SN
Symbols 2α
range of the considered cyclic quantity (maximum value minus minimum value) distance between the potential probes crack angular position, defined from potential probe 1 towards potential probe 2 crack angular position, defined from potential probe A towards potential probe B notch tip radius
notch opening angle
advantages: (i) it allows treating specimen geometries and PDM operating conditions having any degree of complexity and (ii) it is less complicated and less time-consuming than analytical and experimental calibrations, respectively. Moreover, FE calibration allows to evaluate the effects of different geometrical, operational and environmental parameters, and possibly to optimize them. In this context, the effect of the location of both current and potential probes on calibration of SEN and compact tension shear (CTS) samples and on V-notched specimens was analysed by Ritchie et al. [11,17] and by Clark and Knott [12], respectively. Afterwards, Wilson [18] applied the PDM to CT samples and investigated the effects of: notch geometry; concentrated or distributed current input; and density of the mesh adopted in the FE analyses. All previous contributions employed 2-dimensional FE
Blunt Notch Compact Tension Compact Tension Shear Direct Current Potential Drop Disc Compact Tension Finite Element Finite Element Method Overload Potential Drop Method Single Edge Bend Strain Energy Density Single Edge Notch Sharp Notch
models, while 3-dimensional FE models have more recently been adopted [19–22], which enable to analyse the effect of the crack front shape. Gandossi et al. [23] compared in detail the numerical calibration performed by 2-dimensional and 3-dimensional electrical FE models. In a recent contribution focused on the PDM calibration applied to cracked bars [22], the configuration (i.e. circumferential or semi-elliptical) and the location of the initiated crack could not be singled out by means of the traditional PDM, i.e. operating with a single potential probe attached to the tested specimen. To overcome this issue, in the present paper a particular PDM experimental set-up is adopted, consisting of three potential probes attached to the same tested specimen, by following the idea of using multiple potential probes as was proposed in [24,25]. Calibration is performed by means of 3-dimensional
Table 1 Summary of the specimen geometry (see Fig. 1). Specimen code^
ρ (mm)
dnet (mm)
dgross (mm)
δV# (mm)
SN_ρ0.1 BN_ρ4
0.1 4
13 13
25 25
13 13
^ #
SN = sharp-notch, BN = blunt-notch. Distance between the potential probes (see Fig. 3).
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electrical FE analyses, which are validated against experimental results generated by fatigue testing on sharp as well as blunt notched specimens made of steel and TiAl6V4 and by analysing the initiated crack through the three-probe PDM. The effects of the notch geometry and the crack shape are investigated. Finally, a procedure to assess the area, the configuration (i.e. circumferential or semi-elliptical) and the location of the crack starting from the experimentally measured potential drops is discussed. 2. Specimen geometry and experimental set-up Specimen geometry is summarised in Table 1 and reported in Fig. 1, along with details of the circumferential notches characterized by different values of the notch tip radius ρ. The chemical compositions of the AISI 304L austenitic steel and of TiAl6V4 titanium grade 5 alloy are reported in Tables 2 and 3, respectively; while the main mechanical and physical properties [26,27] are summarised in Table 4. Fully reversed (R = −1) axial fatigue tests have been performed on a Schenck PSB servo-hydraulic testing machine (see Fig. 2) with a load capacity of 100 kN, equipped with a DOLI EDC580V digital controller. Concerning the PDM operating conditions, a constant current (I in Fig. 3) of approximately 5 A in case of steel and 2.2 A for titanium has been conducted through the specimen ends leading to an electric potential of approximately 0.25–0.3 mV for the un-cracked configuration (ΔVi,0). Three individual potential probes have been used on each specimen according to the experimental set-up shown in Fig. 2 and sketched in Fig. 3. Therefore, copper wires with a diameter of 0.2 mm were laser welded 0.5 mm above and below the notch flanks, evenly distributed over the circumference, i.e. with an angle of 120° in between. The potential drops ΔVi (i = 1, 2 and 3) have been measured at a distance δV from each other (see Fig. 3 and Table 1) by using amplifiers (type 4FAD) of the control electronics operating with a sampling rate of
Fig. 1. Geometry of circumferentially notched specimens (dimensions in mm).
Table 2 Chemical composition of the austenitic stainless steel, AISI 304L. C (%)
Mn (%)
Si (%)
Cr (%)
Ni (%)
P (%)
S (%)
N (%)
Fe (%)
0.02
1.82
0.30
18.27
8.20
0.028
0.022
0.087
Balance
Table 3 Chemical composition of the titanium grade 5 alloy, TiAl6V4. Fe (%)
O (%)
C (%)
N (%)
Al (%)
V (%)
H (%)
Residuals (%)
Ti (%)
0.13
0.15
0.02
0.03
6.05
3.98
0.001
< 0.40
Balance
Table 4 Mechanical and physical properties of the considered materials. Property
AISI 304L
TiAl6V4
Young’s modulus [GPa] Poisson’s ratio Yield stress [MPa] Ultimate tensile stress [MPa] Electrical resistivity* at 20 °C [10−7 Ω·m]
206 0.3 277 579 7.13
110 0.3 915 980 17.02
* Values taken from [26,27].
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Fig. 2. (a) Schenck PSB servo-hydraulic testing machine adopted for pure axial fatigue tests. (b) Operating conditions of the direct current potential drop method: location of the current connections and the three 120°-spaced potential probes.
a
a
a
r
r
Fig. 3. Experimental setup of the direct current potential drop method with three 120°-spaced potential probes. δV is the distance between the two ends of each potential probe. Examples of a circumferential and a semi-elliptical surface crack initiated at the notch tip. 170
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a r
a r
Fig. 4. 3D electrical FE analyses to calibrate the potential drop method for a circumferential crack. Considered case: sharp V-notched specimen (SN_ρ0.1), a = 3 mm.
1 kHz. The minimum and maximum values of the potential drop during each loading cycle (i.e. min(ΔVi) and max(ΔVi) in Fig. 3) have been recorded. Accordingly, the potential drop ratios Pi = max(ΔVi)/ΔVi,0, according to ASTM E647 [1], have been calculated (i = 1, 2 and 3) as a function of the number of loading cycles.
analyses simulating the three-probe PDM (see Figs. 2 and 3) applied to the specimen geometries shown in Fig. 1. According to previous experimental observations following fatigue testing of notched bars under pure axial loading [2,28], in the present paper two different crack shapes are considered: (i) a circumferential crack (see Fig. 4) and (ii) a semi-elliptical surface crack (see Fig. 5). It should be noted that the crack shapes sketched in Figs. 4 and 5 are idealizations of the actual more complicated material cracking behaviour, reported elsewhere [2,28]. Concerning semi-elliptical surface cracks, the crack aspect ratio
3. FE calibration of the three-probe potential drop method Calibration is performed by means of 3-dimensional electrical FE
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a r
a
r
Fig. 5. 3D electrical FE analyses to calibrate the potential drop method in the case of a single semi-elliptical surface crack located at a generic angle θ (θ = 30° in the figure). Considered case: sharp V-notched specimen (SN_ρ0.1), a = 3 mm. Table 5 Summary of the electrical FE analyses carried out to derive the calibration curves of the potential drop method. Specimen code^
a (mm)
r (mm)
Crack configuration
c/a (–)
θ (°)
SN_ρ0.1
0, 0.1–5
6.5
BN_ρ4
0, 0.1–5
6.5
Circumferential Semi-elliptical Circumferential Semi-elliptical
– 1, 3, 5, 10 – 1, 3, 5, 10
– 0, 30, 60 – 0, 30, 60
^
SN = sharp-notch, BN = blunt-notch.
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a Fig. 6. Calibration curves for a circumferential crack in the case of sharp and blunt notched specimens, a pre-cracked specimen with the same net-section area and a smooth specimen having outer radius r. ΔVi,0 is the electrical potential of the reference un-cracked specimen (a = 0).
c/a and the crack angular position θ have been investigated, the latter being measured from potential probe 1 counter clockwise (see Figs. 3 and 5). In particular, the aspect ratio c/a was varied between 1 and 10, such a range being appropriate under pure axial fatigue loading, as is illustrated in [2] and in Fig. 10c-e. The angle θ has been fixed at 0°, 30° or 60°, all the same crack locations with respect to the other potential probes being included for symmetry reasons. FE analyses have been performed by adopting 3-dimensional, 10node, tetrahedral, electric solid elements (SOLID 232 of Ansys® element library). A rather coarse FE mesh consisting of a global element size d = 1 mm and a local one of about 0.5 mm has been employed in all numerical models (see Figs. 4 and 5). In fact the electrical potential evaluated far from the notch tip is almost insensitive to the mesh density, according to the pioneering contribution by Wilson [18]. Taking advantage of the XZ anti-symmetry plane (Figs. 4 and 5), half geometry is modelled and a 0-V-electrical-potential is applied only to the un-cracked portion of the net-section, to simulate the absence of electric contact between crack faces. A constant electrical current is conducted through the specimen’s end in all numerical models (I in Figs. 4 and 5). First, the un-cracked configuration (a = 0) is analyzed and the electrical potential values Vi,0 are evaluated at the 120°-spaced-potential-probes (i = 1, 2 and 3), as is illustrated in Figs. 4 and 5. Obviously,
for the un-cracked configuration it results V1,0 = V2,0 = V3,0. Then, the potential drop values ΔVi,0 are calculated as 2·Vi,0, due to the antisymmetry boundary conditions. Afterwards, several FE models are analyzed with surface cracks having depth a in the range between 0.1 and 5 mm. The electrical potential values Vi of Figs. 4 and 5 are calculated again at the 120°-spacedpotential-probes (i = 1, 2 and 3). Finally, the potential drop values ΔVi = 2·Vi are calculated. Table 5 summarises all electrical FE analyses carried out. Since calibration curves report the normalised potential drop Pi = ΔVi/ΔVi,0 versus the crack depth a, they depend only on the cracked specimen geometry and neither on the applied electrical current nor on the material electrical resistivity. Results of calibration for a circumferential crack are reported in Fig. 6, where all potential probes are equally affected by the circumferential crack so that ΔV1 = ΔV2 = ΔV3, i.e. all calibration curves are overlapped. Fig. 6 shows that the smaller the notch tip radius ρ, the more sensitive the potential drop method. The same behaviour was observed elsewhere [22,29]. For comparison, Fig. 6 includes also the calibration curves relevant to both a smooth specimen having outer radius r and a pre-cracked specimen with net-section radius equal to r; it is seen that they include those of notched specimens because they represent the upper and lower bounds of calibration, as was previously
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a
a Fig. 7. Calibration curves for a semi-elliptical surface crack in the case SN_ρ0.1: (a) θ = 30°, (b) c/a = 1, (c) c/a = 3, (d) c/a = 5 and (e) c/a = 10. ΔVi,0 is the electrical potential of the reference un-cracked specimen (a = 0).
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a
a Fig. 7. (continued)
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a Fig. 7. (continued)
observed [29]. The results for semi-elliptical surface cracks initiated at the tip of sharp and blunt notches are reported in Figs. 7 and 8, respectively. Comparison of Figs. 7 and 8 with previous Fig. 6 highlights that, obviously, for a given crack depth a, the maximum sensitivity of the PDM holds for the circumferential crack, the reduction of the specimen transverse section being maximum as compared to the elliptical crack case for the same a. For semi-elliptical surface cracks and a fixed angular position θ, Figs. 7a and 8a show that the higher the crack aspect ratio c/a, the more sensitive the potential drop method, because the greater is the cross section reduction. For the sake of brevity, only the case of θ = 30° is reported in Figs. 7a and 8a, the results for the other cracked configurations being similar. Moreover, Figs. 7b–e and 8b–e show that:
It should be noted that the distance δV between the potential probes (see Fig. 3 and Table 1) greatly affects the sensitivity of the potential drop to the crack angular position θ: the greater the distance δV, the lower the sensitivity to the θ-angle. 4. Experimental validation of the PDM calibration curves Some of the calibration curves derived in the previous Section have been validated by testing AISI 304L and TiAl6V4 specimens (see Fig. 1) under axial fatigue loading and by monitoring the tests with the threeprobe PDM (Fig. 2). In particular, three sharp V-notched specimens (SN_ρ0.1), two made of steel and one of titanium alloy, and two blunt notched steel specimens (BN_ρ4) have been fatigue tested in standard laboratory environment. In all cases, a sinusoidal load cycle with a load frequency f = 20 Hz has been imposed under closed-loop load control with a nominal load ratio R equal to −1. The fatigue tests have been interrupted at specimen complete failure. To mark the crack extension on the fracture surface, overloads have been introduced at defined intervals of 5,000 or 10,000 cycles (depending on the expected cyclic lifetime) by doubling the maximum load and by using a frequency of 1 Hz. It is well-known that overloads have a significant influence on the cyclic lifetime [30–33], however they generate markers on the crack surface indicating the actual crack extension.
• for θ = 0° (the crack is located exactly at probe 1) probes 2 and 3 are • •
equally affected by the crack, i.e. calibration curves 2 and 3 are overlapped (ΔV2 = ΔV3); for θ = 30°, the crack is located between probes 1 and 2, closer to probe 1 and farther from probe 2, so that the potential drops are in the relationship ΔV1 > ΔV2 > ΔV3; for θ = 60°, the crack is located exactly at the middle between probe 1 and probe 2, therefore calibration curves 1 and 2 are overlapped (ΔV1 = ΔV2).
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a
a Fig. 8. Calibration curves for a semi-elliptical surface crack in the case BN_ρ4: (a) θ = 30°, (b) c/a = 1, (c) c/a = 3, (d) c/a = 5 and (e) c/a = 10. ΔVi,0 is the electrical potential of the reference un-cracked specimen (a = 0).
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a
a Fig. 8. (continued)
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a Fig. 8. (continued)
As an example, Fig. 9 reports the potential drop ratios measured during the tests of two steel specimens: a sharp one and a blunt one. It should be noted that the condition Pi = 1 is met at the beginning of each fatigue test. To reduce the scatter, the mean potential of the first 10 cycles is used as ΔVi,0. Table 6 summarizes the loading parameters and the results of the experimental fatigue tests. The crack surfaces are analyzed using a Keyence VHX-2000 digital microscope with a magnification of 50-times. The results are shown in Fig. 10, where it can be seen that a circumferential crack is initiated in sharp V-notched steel specimens (Fig. 10a, b), a single semi-elliptical crack is initiated in a blunt notched steel specimen (Fig. 10c), while three semi-elliptical cracks are initiated both in a blunt notched steel specimen (Fig. 10d) and in the sharp V-notched specimen made of titanium alloy (Fig. 10e). Fig. 10 reports also the crack sizes and positions, measured taking advantage of the markers (on the crack surfaces) produced by the overloads. In the case of circumferential cracks, the remaining un-cracked area is approximated by a circle. Beside the radius, also the position of the center of the circle is determined. For isolated, semi-elliptical cracks, the angular position θ, the area, the half-surface length c and the crack depth a are measured. After measuring the crack depths a from the overload-markers visible on the fracture surfaces as is shown in Fig. 10, the relevant potential drop ratios can be determined from the experimental curves reporting Pi versus loading cycles (see examples in Fig. 9a, b).
Dedicated FE analyses have been carried out by modelling circumferential or semi-elliptical cracks having the actual crack sizes and positions, observed experimentally in Fig. 10. Fig. 11 compares the potential drop curves obtained numerically with the experimental potential drops Pi (i = 1, 2 and 3), as a function of the normalized crack depth a/r. It should be noted that in the case of multiple initiated cracks (see Fig. 10d, e) the average crack depth aaverage = (a1 + a2 + a3)/3 has been adopted in Fig. 11d, e. A quite good agreement between numerical and experimental results can be observed from Fig. 11a, c–e: given the experimental potential drops Pi = ΔVi/ΔVi,0, the maximum deviation between the measured and the estimated crack depths is approximately ± 20%, similar deviations having been obtained elsewhere [21]. The deviation increases to approximately ± 30% for the case in Fig. 11b. 5. Predictions based on FE results 5.1. Assessment of the crack area All FE calibration curves reported in previous Figs. 6–8 and 11 are summarised in Fig. 12 in terms of the average potential drop ratio Pavg = (P1 + P2 + P3)/3 versus the normalised crack area Acrack/Anet. Fig. 12 shows that the effects of the notch geometry, i.e. sharp or blunt notch, and of the crack configuration, i.e. circumferential or semi-
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Fig. 9. Potential drop ratios measured during fatigue tests of (a) sharp (SN_ρ0.1_2) and (b) blunt (BN_ρ4_1) notched specimens under pure axial loading. Nf is the total fatigue life. Table 6 Summary of the loading parameters and the results of experimental fatigue tests under pure axial loading. Specimen code^
Material
SN_ρ0.1_1 SN_ρ0.1_2 BN_ρ4_1 BN_ρ4_2 SN_ρ0.1_3
AISI 304L
^
TiAl6V4
ΔF [kN]
40 40 60 65 50
R
Overload
−1
SN = sharp-notch, BN = blunt-notch.
180
Nf [cycles]
ΔF [kN]
ROL
Interval [cycles]
80 40 60 65 100
−1 0 0 0 −1
10,000 5,000 10,000 10,000 10,000
610,000 182,000 490,345 300,000 120,000
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Fig. 10. Experimental measurement of the crack depth a, the aspect ratio c/a and the angle θ of cracks generated by fatigue testing (a) and (b) sharp V-notched (SN_ρ0.1) and (c) and (d) blunt notched (BN_ρ4) steel specimens, and (e) a sharp V-notched (SN_ρ0.1) titanium specimen under pure axial loading. The loading parameters are reported in Table 6.
elliptical, are quite limited, since all curves are included in a scatter band with a maximum deviation of 16% in terms of predicted crack area Acrack for a fixed average potential drop ratio Pavg. Concerning the case of semi-elliptical surface cracks, Fig. 12 shows also that calibration curves are almost overlapped regardless the crack angular position θ and aspect ratio c/a. Accordingly, Fig. 12 suggests to adopt the scatter band consisting of a lower bound and an upper bound to predict the cracked area with a maximum deviation of 16%, once known the average potential drop ratio Pavg experimentally measured in a fatigue tested specimen, regardless of the notch geometry, the crack shape and location. The scatter band of Fig. 12 is validated against previous experimental results (see Figs. 10 and 11) generated by fatigue testing of sharp as well as blunt notched specimens made of AISI 304L and TiAl6V4. A quite
good agreement can be observed in Fig. 13, the great majority of experimental results falling inside the scatter band, while only those relevant to the case SN_2 being slightly on the unsafe side. 5.2. Assessment of the crack configuration and the crack angular position The numerical results reported in Figs. 6–8 suggest to define a procedure to assess the configuration of the crack, i.e. circumferential or semi-elliptical, and the angular position θ in the case of semi-elliptical cracks. Let us consider a notched bar undergoing a fatigue test monitored by the potential drop method sketched in Fig. 3. The potential drops Pi (i = 1, 2 and 3) are measured as a function of the number of loading cycles. The condition Pi = 1 is met at the beginning of the fatigue test;
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V1
(d)
θ
V2
V3
Specimen code: BN_ρ4_2 N [cycles] 0 280000 290000
crack-1 c1 [mm] 0 1.41 2.34
Material: AISI 304L
θ = 28° a1 [mm] 0 1.00 1.42
c1/a1 1.41 1.65
crack-2 c2 [mm] 0 1.01 1.47
θ = 74° a2 [mm] 0 1.11 1.54
c2/a2 0.91 0.96
crack-3 c3 [mm] 0 1.30 1.60
θ = 122° a3 [mm] 0 1.09 1.48
c3/a3 1.20 1.08
Fig. 10. (continued)
V1
therefore fatigue crack initiation at the notch root is detected when at least one of the three potential drops Pi increases above 1. Let us indicate PA ≥ PB ≥ PC and assume that only a single crack is initiated at the notch root. The configuration of the initiated fatigue crack and the crack angular position can be assessed as follows:
(e)
• if P ≈ P ≈ P , then a circumferential crack is initiated (see Fig. 14a); • if P ≈ P ≠ P , then a semi-elliptical crack is initiated with an angular location θ′ = 0° with respect to probe A; • if P ≈ P ≠ P , then a semi-elliptical crack is initiated with an angular location θ′ = 60° defined from probe A towards probe B; if • P > P > P , then a semi-elliptical crack is initiated with an A
θ
B
C
B
C
A
A
B
C
A
B
C
angular location θ′, defined from probe A towards probe B, in the range between 0° and 60°; it can be approximately estimated as θ′ ≈ 30° with an error of ± 30° (see Fig. 14b).
V2
V3
Specimen code: SN_ρ0.1_3 N [cycles] 0 60000 70000 80000 90000 100000 110000
crack-1 c1 [mm] 0 0.61 0.75 1.03 1.48 2.24 3.37
θ = 30° a1 [mm] 0 0.55 0.73 0.97 1.26 1.64 2.20
To reduce the estimation error of the crack angular position, it would be necessary to increase the number of potential probes connected to the tested specimen or to investigate the relationship between the potential drop ratios Pi (i = 1, 2 and 3) and the crack angular position θ with smaller steps of θ. It should be noted that the case of multiple crack initiation is more complex and cannot be treated with the above procedure, but new developments would be needed. The proposed procedure is validated against previous experimental results generated by fatigue testing AISI 304L and TiAl6V4 specimens under axial fatigue loading. Fig. 11a and b show that the experimental potential drops are P1 ≈ P2 ≈ P3 with a reduced deviation in the order of 2%, the actual crack centre being not coincident with the specimen axis. A fatigue crack having a circumferential configuration would have been anticipated, which agrees with the experimental observations
Material: TiAl6V4 c1/a1 1.12 1.03 1.06 1.18 1.36 1.53
crack-2 c2 [mm] 0 0.00 0.00 0.00 0.00 0.00 2.88
θ = 105° a2 [mm] 0 0.00 0.00 0.00 0.00 0.00 1.12
c2/a2 0.00 0.00 0.00 0.00 0.00 2.57
crack-3 c3 [mm] 0 0.87 1.20 1.58 1.93 2.32 3.29
θ = 247° a3 [mm] 0 0.52 0.75 1.08 1.44 1.74 2.22
c3/a3 1.68 1.60 1.46 1.34 1.33 1.48
Fig. 10. (continued) 182
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a
a Fig. 11. Comparison between numerical calibration curves and experimental measurements performed in Fig. 10: (a) and (b) sharp V-notched (SN_ρ0.1) and (c) and (d) blunt notched (BN_ρ4) steel specimens, and (e) a sharp V-notched (SN_ρ0.1) titanium specimen tested under pure axial fatigue loading.
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a
a Fig. 11. (continued)
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a Fig. 11. (continued)
Acrack A Fig. 12. Calibration curves for circumferential and semi-elliptical surface cracks in the cases SN_ρ0.1 and BN_ρ4 taken from previous Figs. 6–8 and 11 and expressed in terms of average potential drop ratio Pavg = (P1 + P2 + P3)/3 versus the normalised cracked area Acrack/Anet.
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Acrack A Fig. 13. Comparison between scatter band calibrated in Fig. 12 and experimental measurements performed in Fig. 10.
Fig. 14. Procedure to assess the crack shape and the crack angular position based on the potential drop ratios evaluated at three 120°-spaced potential probes: (a) circumferential and (b) semi-elliptical crack at an angular location θ′. Pi = ΔVi/ΔVi,0.
reported in Fig. 10a and b. On the other hand, Fig. 11c shows that the experimental potential drop ratios are in the relationship P1 > P3 > P2, with differences in the order of 4–5%. According to the procedure reported above, a semi-elliptical surface crack at an angular location θ′ 30° from probe 1 towards probe 3 is predicted, which is in good agreement with the experimental observations shown in Fig. 10c. The cases reported in Fig. 11d, e are more complex to be analysed, and they cannot be treated with the proposed procedure, multiple semi-elliptical cracks having been initiated as is shown in Fig. 10d, e.
angular positions θ between 0° and 60° with respect to one potential probe have been analysed. The following conclusions can be drawn:
• Concerning the effect of the notch geometry, for a given crack depth,
•
6. Conclusions A three-probe potential drop method (PDM) has been used to estimate crack area, location, and configuration (i.e. circumferential or semi-elliptical) in fatigue tested notched bars. Calibration of the adopted PDM configuration has been performed by employing 3-dimensional electrical FE models of sharply and bluntly notched bars adopted in the experimental tests. A range of circumferential and semielliptical surface cracks has been analyzed; concerning semi-elliptical surface cracks, aspect ratios c/a in the range between 1 and 10 and
•
186
the smaller the notch tip radius ρ, the more sensitive the potential drop method. Dealing with the effect of the crack shape for a given crack depth, the maximum sensitivity of the PDM is for a circumferential crack. For semi-elliptical surface cracks, the higher the crack aspect ratio c/a, the more sensitive the PDM. The numerical analyses adopted for calibration have been validated by comparison with experimental results obtained by fatigue testing sharply as well as bluntly notched specimens made of steel and titanium alloy under axial loading. A quite good agreement has been obtained, the deviation between the experimentally measured crack depth and that estimated from the calibration curves derived by dedicated FE analyses is approximately ± 20% in most cases. The numerical results suggest an approximated procedure to assess the area, the configuration as well as the angular position of the crack, starting from the potential drop ratios measured at the three probes. The average potential drop ratio Pavg = (PA + PB + PC)/3 is
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correlated to the normalised cracked area Acrack/Anet with a maximum deviation of 16%, regardless of the notch geometry, the crack shape and location. Concerning the crack configuration and location, the procedure is valid for a single crack, the multiple crack case not being included in the present analysis. When PA ≈PB ≈ PC, a circumferential crack is initiated. As to the crack location, assuming PA ≥ PB ≥ PC, when PB ≈ PC ≠ PA, then the crack is at probe A; when PA ≈ PB ≠ PC, then the crack is in the mid-position between probe A and probe B; finally, when PA > PB > PC then the crack is between probe A and probe B, in the angular range from 0° to 60° from probe A. The proposed procedure has been validated by comparison with experimental results, allowing to estimate with good approximation the area, the configuration (either circumferential or elliptical), and the location in case of a single semi-elliptical surface crack.
[10] You CP, Knott JF. Electrolytic tank simulation of the potential drop technique for crack length determinations. Int J Fract 1983;23:R139–41. [11] Ritchie RO, Garrett GG, Knott JP. Crack-growth monitoring: Optimisation of the electrical potential technique using an analogue method. Int J Fract Mech 1971;7:462. [12] Clark G, Knott JF. Measurement of fatigue cracks in notched specimens by means of theoretical electrical potential calibrations. J Mech Phys Solids 1975;23:265–76. [13] Johnson HH. Calibrating the electrical potential method for studying slow crack growth. Mater Res Stand 1965:442–5. [14] Schwalbe K, Hellmann D. Application of the electrical potential method to crack length measurements using Johnson’s formula. J Test Eval 1981;9:218–20. [15] Černý I. The use of DCPD method for measurement of growth of cracks in large components at normal and elevated temperatures. Eng Fract Mech 2004;71:837–48. [16] Green DA, Kendall JM, Knott JF. Analytic and analogue techniques for determining potential distributions around angled cracks. Int J Fract 1988;37:R3–12. [17] Aronson G, Ritchie R. Optimization of the electrical potential technique for crack growth monitoring in compact test pieces using finite element analysis. J Test Eval 1979;7:208–15. [18] Wilson WK. On the electrical potential analysis of a cracked fracture mechanics test specimen using the finite element method. Eng Fract Mech 1983;18:349–58. [19] Nath S, Rudolphi T, Lord W. Three dimensional modeling of the DC potential drop method using finite element and boundary element analysis. Rev Prog Quant Nondestruct Eval 1992;11:553–60. [20] Harrington DS, Bell R, Tan CL. Calibration of the localized DCPD method for crack shape measurement using the boundary element method. Fatigue Fract Eng Mater Struct 1995;18:875–84. [21] Doremus L, Nadot Y, Henaff G, Mary C, Pierret S. Calibration of the potential drop method for monitoring small crack growth from surface anomalies – crack front marking technique and finite element simulations. Int J Fatigue 2015;70:178–85. [22] Campagnolo A, Meneghetti G, Berto F, Tanaka K. Calibration of the potential drop method by means of electric FE analyses and experimental validation for a range of crack shapes. Fatigue Fract Eng Mater Struct 2018:1–16. [23] Gandossi L, Summers S, Taylor N, Hurst R, Hulm B, Parker J. The potential drop method for monitoring crack growth in real components subjected to combined fatigue and creep conditions: application of FE techniques for deriving calibration curves. Int J Press Vessel Pip 2001;78:881–91. [24] Černý I. Measurement of subcritical growth of defects in large components of nuclear power plants at elevated temperatures. Int J Press Vessel Pip 2001;78:893–902. [25] Černý I. Estimation of crack depth and profile using DCPD method in full scale railway axle loaded by rotating bending. Appl Mech Mater 2015;732:171–4. [26] Ho CY, Chu TK. Electrical resistivity and thermal conductivity of nine selected AISI stainless steels. CINDAS Rep 1977:45. [27] Milck JT. Electrical resistivity data and bibliography on titanium and titanium alloys. Rep Hughes Aircr Co 1970. [28] Berto F, Campagnolo A, Lazzarin P. Fatigue strength of severely notched specimens made of Ti-6Al-4V under multiaxial loading. Fatigue Fract Eng Mater Struct 2015;38:503–17. [29] Campagnolo A, Meneghetti G, Berto F, Tanaka K. Crack initiation life in notched steel bars under torsional fatigue: synthesis based on the averaged strain energy density approach. Int J Fatigue 2017;100:563–74. [30] Schijve J, Broek D. Crack propagation: the results of a test programme based on a gust spectrum with variable amplitude loading. Aircr Eng Aerosp Technol 1962;34:314–6. [31] DuQuesnay DL, MacDougall C, Dabayeh A, Topper TH. Notch fatigue behaviour as influenced by periodic overloads. Int J Fatigue 1995;17:91–9. [32] Bär J, Wilhelm G. Influence of overloads on the crack formation and propagation in EN AW 7475-T761, in: 19th Eur Conf Fract Fract Mech Durability, Reliab Safety, ECF 2012, European Conference on Fracture, ECF; 2012. [33] Tiedemann D, Bär J, Gudladt H-J. The crack propagation rate according to notches and overload levels. Proc Mater Sci 2014;3:1359–64.
Acknowledgement The authors thank Prof. Filippo Berto from the Department of Engineering Design and Materials, NTNU in Trondheim, (Norway) for providing the specimen material. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijfatigue.2019.02.045. References [1] ASTM. ASTM Standard E647 – 15e1, Standard test method for measurement of fatigue crack growth rates; 2015. [2] Meneghetti G, Campagnolo A, Berto F, Tanaka K. Notched Ti-6Al-4V titanium bars under multiaxial fatigue: synthesis of crack initiation life based on the averaged strain energy density. Theor Appl Fract Mech 2018;96:509–33. [3] Lazzarin P, Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int J Fract 2001;112:275–98. [4] Hicks MA, Pickard AC. A comparison of theoretical and experimental methods of calibrating the electrical potential drop technique for crack length determination. Int J Fract 1982;20:91–101. [5] Smith RA. Calibrations for the electrical potential method of crack growth measurement by a direct electrical analogy. Strain 1974;10:183–7. [6] Soboyejo WO, Reed RC, Knott JF. On the calibration of the direct current potential difference method for the determination of semi-elliptical crack lengths. Int J Fract 1990;44:27–41. [7] Belloni G, Gariboldi E, Lo Conte A, Tono M, Speranzoso P. On the experimental calibration of a potential drop system for crack length measurements in a compact tension specimen. J Test Eval 2002;30:11149. [8] Hill MR, Stuart DH. Direct current potential difference correlation for open-hole, single-crack coupons. Eng Fract Mech 2013;99:141–6. [9] Ritchie RO, Bathe KJ. On the calibration of the electrical potential technique for monitoring crack growth using finite element methods. Int J Fract 1979;15:47–55.
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Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Analysis of crack geometry and location in notched bars by means of a threeprobe potential drop technique
T
Alberto Campagnoloa, Jürgen Bärb, Giovanni Meneghettia,
⁎
a b
Department of Industrial Engineering, University of Padova, Via Venezia, 1 – 35131 Padova, Italy Institute for Materials Science, University of the Bundeswehr Munich, D-85577 Neubiberg, Germany
ARTICLE INFO
ABSTRACT
Keywords: Potential drop method Calibration curves Crack shape Steel Titanium alloy
It is often necessary to define the crack initiation life of a fatigue tested component, generally at a given (short) crack length. The size of an initiated crack can be estimated by employing different experimental methods, one of which is the direct current potential drop (DCPD) technique. In the case of notched bars subjected to fatigue loadings, the crack configuration (i.e. circumferential or semi-elliptical) and location cannot be singled out by means of the potential drop method (PDM) operating with a single potential probe. In the present contribution, three potential probes are adopted to overcome this issue. The calibration curves reporting the three potential drops as a function of the crack size are derived by means of 3-dimensional electrical FE analyses. Two different crack configurations are analyzed: (i) circumferential and (ii) semi-elliptical surface cracks. The calibration curves have been validated by systematic comparison with experimental results, generated by fatigue testing of sharp as well as blunt notched specimens made of steel and a titanium alloy under pure axial loading. Finally, a procedure to assess the area, the configuration and the location of the initiated fatigue crack starting from the experimentally measured potential drops is discussed.
1. Introduction Crack monitoring is often adopted to investigate damage initiation and spreading in specimens or structural components undergoing fatigue loading [1]. The majority of local approaches for fatigue life assessment require a quantitative estimation of the crack size, at which the reference fatigue life is defined. As an example, the experimental results obtained by fatigue testing of sharp as well as blunt notched bars made of TiAl6V4 under multiaxial loading have been recently analysed [2] by adopting the local approach based on the strain energy density (SED) averaged over a material structural volume, surrounding the crack initiation point and having size R0 [3]. Strictly speaking, a physical definition of “crack initiation” is necessary to apply the SED approach, which should be defined when an initiated crack has a depth a = R0, i.e. when the propagating (short) crack leads to the structural volume failure. The size of a propagating crack can be estimated by employing different experimental techniques, one of which is the direct current potential drop (DCPD) method. Calibration of the potential drop method (PDM), i.e. determining the potential drop change as a function of the crack depth a, has been performed in the literature by employing experimental, analytical and numerical techniques. Generally, the ⁎
experimental calibration [4–11] requires comparison with a different (previously calibrated) method. Experimental calibration has been performed primarily by optically measuring the surface crack advance [8], by machining pre-cracks of increasing depth in a tested sample [9] or by marking the fracture surface by applying overloads or by changing the nominal load ratio [4]. Electrolytic tank simulation [10], cutting of graphitized papers [11] or thin aluminium foils [4] to simulate a propagating crack are other available experimental techniques. Calibration can also be performed by solving the Laplace equation, i.e. 2 V = 0 , in which V is the electrical potential, for given specimen geometry and PDM operating conditions. However, closed-form solutions of the Laplace equation are available only for few simple cases [4,12]; an example is the Johnson’s formula, which has been widely adopted for single edge notch (SEN), single edge bend (SEB), compact tension (CT) and disc compact tension (DCT) specimens [13–15]. The conformal mapping approach [12,16] has been employed to derive analytical solutions of the Laplace equation in the case of notched specimens. The finite element method (FEM) can also be used to calibrate the potential drop method, as was suggested by Ritchie and Bathe [9] and by Aronson and Ritchie [17]. Calibration by FEM has the following
Corresponding author. E-mail address: [email protected] (G. Meneghetti).
https://doi.org/10.1016/j.ijfatigue.2019.02.045 Received 14 January 2019; Received in revised form 20 February 2019; Accepted 27 February 2019 Available online 01 March 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature
Δ
a Acrack Anet c c/a F I Nf Pi
δV θ
Pavg R R0 ROL r ΔVi,0 ΔVi
crack depth crack area specimen net-section area half surface crack length crack aspect ratio axial load electric current number of loading cycles to failure potential drop ratio, i.e. ΔVi/ΔVi,0, at i-th probe (i = 1, 2, 3) average potential drop ratio, i.e. (P1 + P2 + P3)/3 nominal load ratio size of the structural volume for averaged SED calculation load ratio of overload specimen net-section radius potential drop of the un-cracked (a = 0) specimen at i-th probe (i = 1, 2, 3) potential drop of a cracked specimen at i-th probe (i = 1, 2, 3)
θ′ ρ
Abbreviations BN CT CTS DCPD DCT FE FEM OL PDM SEB SED SEN SN
Symbols 2α
range of the considered cyclic quantity (maximum value minus minimum value) distance between the potential probes crack angular position, defined from potential probe 1 towards potential probe 2 crack angular position, defined from potential probe A towards potential probe B notch tip radius
notch opening angle
advantages: (i) it allows treating specimen geometries and PDM operating conditions having any degree of complexity and (ii) it is less complicated and less time-consuming than analytical and experimental calibrations, respectively. Moreover, FE calibration allows to evaluate the effects of different geometrical, operational and environmental parameters, and possibly to optimize them. In this context, the effect of the location of both current and potential probes on calibration of SEN and compact tension shear (CTS) samples and on V-notched specimens was analysed by Ritchie et al. [11,17] and by Clark and Knott [12], respectively. Afterwards, Wilson [18] applied the PDM to CT samples and investigated the effects of: notch geometry; concentrated or distributed current input; and density of the mesh adopted in the FE analyses. All previous contributions employed 2-dimensional FE
Blunt Notch Compact Tension Compact Tension Shear Direct Current Potential Drop Disc Compact Tension Finite Element Finite Element Method Overload Potential Drop Method Single Edge Bend Strain Energy Density Single Edge Notch Sharp Notch
models, while 3-dimensional FE models have more recently been adopted [19–22], which enable to analyse the effect of the crack front shape. Gandossi et al. [23] compared in detail the numerical calibration performed by 2-dimensional and 3-dimensional electrical FE models. In a recent contribution focused on the PDM calibration applied to cracked bars [22], the configuration (i.e. circumferential or semi-elliptical) and the location of the initiated crack could not be singled out by means of the traditional PDM, i.e. operating with a single potential probe attached to the tested specimen. To overcome this issue, in the present paper a particular PDM experimental set-up is adopted, consisting of three potential probes attached to the same tested specimen, by following the idea of using multiple potential probes as was proposed in [24,25]. Calibration is performed by means of 3-dimensional
Table 1 Summary of the specimen geometry (see Fig. 1). Specimen code^
ρ (mm)
dnet (mm)
dgross (mm)
δV# (mm)
SN_ρ0.1 BN_ρ4
0.1 4
13 13
25 25
13 13
^ #
SN = sharp-notch, BN = blunt-notch. Distance between the potential probes (see Fig. 3).
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electrical FE analyses, which are validated against experimental results generated by fatigue testing on sharp as well as blunt notched specimens made of steel and TiAl6V4 and by analysing the initiated crack through the three-probe PDM. The effects of the notch geometry and the crack shape are investigated. Finally, a procedure to assess the area, the configuration (i.e. circumferential or semi-elliptical) and the location of the crack starting from the experimentally measured potential drops is discussed. 2. Specimen geometry and experimental set-up Specimen geometry is summarised in Table 1 and reported in Fig. 1, along with details of the circumferential notches characterized by different values of the notch tip radius ρ. The chemical compositions of the AISI 304L austenitic steel and of TiAl6V4 titanium grade 5 alloy are reported in Tables 2 and 3, respectively; while the main mechanical and physical properties [26,27] are summarised in Table 4. Fully reversed (R = −1) axial fatigue tests have been performed on a Schenck PSB servo-hydraulic testing machine (see Fig. 2) with a load capacity of 100 kN, equipped with a DOLI EDC580V digital controller. Concerning the PDM operating conditions, a constant current (I in Fig. 3) of approximately 5 A in case of steel and 2.2 A for titanium has been conducted through the specimen ends leading to an electric potential of approximately 0.25–0.3 mV for the un-cracked configuration (ΔVi,0). Three individual potential probes have been used on each specimen according to the experimental set-up shown in Fig. 2 and sketched in Fig. 3. Therefore, copper wires with a diameter of 0.2 mm were laser welded 0.5 mm above and below the notch flanks, evenly distributed over the circumference, i.e. with an angle of 120° in between. The potential drops ΔVi (i = 1, 2 and 3) have been measured at a distance δV from each other (see Fig. 3 and Table 1) by using amplifiers (type 4FAD) of the control electronics operating with a sampling rate of
Fig. 1. Geometry of circumferentially notched specimens (dimensions in mm).
Table 2 Chemical composition of the austenitic stainless steel, AISI 304L. C (%)
Mn (%)
Si (%)
Cr (%)
Ni (%)
P (%)
S (%)
N (%)
Fe (%)
0.02
1.82
0.30
18.27
8.20
0.028
0.022
0.087
Balance
Table 3 Chemical composition of the titanium grade 5 alloy, TiAl6V4. Fe (%)
O (%)
C (%)
N (%)
Al (%)
V (%)
H (%)
Residuals (%)
Ti (%)
0.13
0.15
0.02
0.03
6.05
3.98
0.001
< 0.40
Balance
Table 4 Mechanical and physical properties of the considered materials. Property
AISI 304L
TiAl6V4
Young’s modulus [GPa] Poisson’s ratio Yield stress [MPa] Ultimate tensile stress [MPa] Electrical resistivity* at 20 °C [10−7 Ω·m]
206 0.3 277 579 7.13
110 0.3 915 980 17.02
* Values taken from [26,27].
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Fig. 2. (a) Schenck PSB servo-hydraulic testing machine adopted for pure axial fatigue tests. (b) Operating conditions of the direct current potential drop method: location of the current connections and the three 120°-spaced potential probes.
a
a
a
r
r
Fig. 3. Experimental setup of the direct current potential drop method with three 120°-spaced potential probes. δV is the distance between the two ends of each potential probe. Examples of a circumferential and a semi-elliptical surface crack initiated at the notch tip. 170
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a r
a r
Fig. 4. 3D electrical FE analyses to calibrate the potential drop method for a circumferential crack. Considered case: sharp V-notched specimen (SN_ρ0.1), a = 3 mm.
1 kHz. The minimum and maximum values of the potential drop during each loading cycle (i.e. min(ΔVi) and max(ΔVi) in Fig. 3) have been recorded. Accordingly, the potential drop ratios Pi = max(ΔVi)/ΔVi,0, according to ASTM E647 [1], have been calculated (i = 1, 2 and 3) as a function of the number of loading cycles.
analyses simulating the three-probe PDM (see Figs. 2 and 3) applied to the specimen geometries shown in Fig. 1. According to previous experimental observations following fatigue testing of notched bars under pure axial loading [2,28], in the present paper two different crack shapes are considered: (i) a circumferential crack (see Fig. 4) and (ii) a semi-elliptical surface crack (see Fig. 5). It should be noted that the crack shapes sketched in Figs. 4 and 5 are idealizations of the actual more complicated material cracking behaviour, reported elsewhere [2,28]. Concerning semi-elliptical surface cracks, the crack aspect ratio
3. FE calibration of the three-probe potential drop method Calibration is performed by means of 3-dimensional electrical FE
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a r
a
r
Fig. 5. 3D electrical FE analyses to calibrate the potential drop method in the case of a single semi-elliptical surface crack located at a generic angle θ (θ = 30° in the figure). Considered case: sharp V-notched specimen (SN_ρ0.1), a = 3 mm. Table 5 Summary of the electrical FE analyses carried out to derive the calibration curves of the potential drop method. Specimen code^
a (mm)
r (mm)
Crack configuration
c/a (–)
θ (°)
SN_ρ0.1
0, 0.1–5
6.5
BN_ρ4
0, 0.1–5
6.5
Circumferential Semi-elliptical Circumferential Semi-elliptical
– 1, 3, 5, 10 – 1, 3, 5, 10
– 0, 30, 60 – 0, 30, 60
^
SN = sharp-notch, BN = blunt-notch.
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a Fig. 6. Calibration curves for a circumferential crack in the case of sharp and blunt notched specimens, a pre-cracked specimen with the same net-section area and a smooth specimen having outer radius r. ΔVi,0 is the electrical potential of the reference un-cracked specimen (a = 0).
c/a and the crack angular position θ have been investigated, the latter being measured from potential probe 1 counter clockwise (see Figs. 3 and 5). In particular, the aspect ratio c/a was varied between 1 and 10, such a range being appropriate under pure axial fatigue loading, as is illustrated in [2] and in Fig. 10c-e. The angle θ has been fixed at 0°, 30° or 60°, all the same crack locations with respect to the other potential probes being included for symmetry reasons. FE analyses have been performed by adopting 3-dimensional, 10node, tetrahedral, electric solid elements (SOLID 232 of Ansys® element library). A rather coarse FE mesh consisting of a global element size d = 1 mm and a local one of about 0.5 mm has been employed in all numerical models (see Figs. 4 and 5). In fact the electrical potential evaluated far from the notch tip is almost insensitive to the mesh density, according to the pioneering contribution by Wilson [18]. Taking advantage of the XZ anti-symmetry plane (Figs. 4 and 5), half geometry is modelled and a 0-V-electrical-potential is applied only to the un-cracked portion of the net-section, to simulate the absence of electric contact between crack faces. A constant electrical current is conducted through the specimen’s end in all numerical models (I in Figs. 4 and 5). First, the un-cracked configuration (a = 0) is analyzed and the electrical potential values Vi,0 are evaluated at the 120°-spaced-potential-probes (i = 1, 2 and 3), as is illustrated in Figs. 4 and 5. Obviously,
for the un-cracked configuration it results V1,0 = V2,0 = V3,0. Then, the potential drop values ΔVi,0 are calculated as 2·Vi,0, due to the antisymmetry boundary conditions. Afterwards, several FE models are analyzed with surface cracks having depth a in the range between 0.1 and 5 mm. The electrical potential values Vi of Figs. 4 and 5 are calculated again at the 120°-spacedpotential-probes (i = 1, 2 and 3). Finally, the potential drop values ΔVi = 2·Vi are calculated. Table 5 summarises all electrical FE analyses carried out. Since calibration curves report the normalised potential drop Pi = ΔVi/ΔVi,0 versus the crack depth a, they depend only on the cracked specimen geometry and neither on the applied electrical current nor on the material electrical resistivity. Results of calibration for a circumferential crack are reported in Fig. 6, where all potential probes are equally affected by the circumferential crack so that ΔV1 = ΔV2 = ΔV3, i.e. all calibration curves are overlapped. Fig. 6 shows that the smaller the notch tip radius ρ, the more sensitive the potential drop method. The same behaviour was observed elsewhere [22,29]. For comparison, Fig. 6 includes also the calibration curves relevant to both a smooth specimen having outer radius r and a pre-cracked specimen with net-section radius equal to r; it is seen that they include those of notched specimens because they represent the upper and lower bounds of calibration, as was previously
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a
a Fig. 7. Calibration curves for a semi-elliptical surface crack in the case SN_ρ0.1: (a) θ = 30°, (b) c/a = 1, (c) c/a = 3, (d) c/a = 5 and (e) c/a = 10. ΔVi,0 is the electrical potential of the reference un-cracked specimen (a = 0).
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a
a Fig. 7. (continued)
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a Fig. 7. (continued)
observed [29]. The results for semi-elliptical surface cracks initiated at the tip of sharp and blunt notches are reported in Figs. 7 and 8, respectively. Comparison of Figs. 7 and 8 with previous Fig. 6 highlights that, obviously, for a given crack depth a, the maximum sensitivity of the PDM holds for the circumferential crack, the reduction of the specimen transverse section being maximum as compared to the elliptical crack case for the same a. For semi-elliptical surface cracks and a fixed angular position θ, Figs. 7a and 8a show that the higher the crack aspect ratio c/a, the more sensitive the potential drop method, because the greater is the cross section reduction. For the sake of brevity, only the case of θ = 30° is reported in Figs. 7a and 8a, the results for the other cracked configurations being similar. Moreover, Figs. 7b–e and 8b–e show that:
It should be noted that the distance δV between the potential probes (see Fig. 3 and Table 1) greatly affects the sensitivity of the potential drop to the crack angular position θ: the greater the distance δV, the lower the sensitivity to the θ-angle. 4. Experimental validation of the PDM calibration curves Some of the calibration curves derived in the previous Section have been validated by testing AISI 304L and TiAl6V4 specimens (see Fig. 1) under axial fatigue loading and by monitoring the tests with the threeprobe PDM (Fig. 2). In particular, three sharp V-notched specimens (SN_ρ0.1), two made of steel and one of titanium alloy, and two blunt notched steel specimens (BN_ρ4) have been fatigue tested in standard laboratory environment. In all cases, a sinusoidal load cycle with a load frequency f = 20 Hz has been imposed under closed-loop load control with a nominal load ratio R equal to −1. The fatigue tests have been interrupted at specimen complete failure. To mark the crack extension on the fracture surface, overloads have been introduced at defined intervals of 5,000 or 10,000 cycles (depending on the expected cyclic lifetime) by doubling the maximum load and by using a frequency of 1 Hz. It is well-known that overloads have a significant influence on the cyclic lifetime [30–33], however they generate markers on the crack surface indicating the actual crack extension.
• for θ = 0° (the crack is located exactly at probe 1) probes 2 and 3 are • •
equally affected by the crack, i.e. calibration curves 2 and 3 are overlapped (ΔV2 = ΔV3); for θ = 30°, the crack is located between probes 1 and 2, closer to probe 1 and farther from probe 2, so that the potential drops are in the relationship ΔV1 > ΔV2 > ΔV3; for θ = 60°, the crack is located exactly at the middle between probe 1 and probe 2, therefore calibration curves 1 and 2 are overlapped (ΔV1 = ΔV2).
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a
a Fig. 8. Calibration curves for a semi-elliptical surface crack in the case BN_ρ4: (a) θ = 30°, (b) c/a = 1, (c) c/a = 3, (d) c/a = 5 and (e) c/a = 10. ΔVi,0 is the electrical potential of the reference un-cracked specimen (a = 0).
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a
a Fig. 8. (continued)
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a Fig. 8. (continued)
As an example, Fig. 9 reports the potential drop ratios measured during the tests of two steel specimens: a sharp one and a blunt one. It should be noted that the condition Pi = 1 is met at the beginning of each fatigue test. To reduce the scatter, the mean potential of the first 10 cycles is used as ΔVi,0. Table 6 summarizes the loading parameters and the results of the experimental fatigue tests. The crack surfaces are analyzed using a Keyence VHX-2000 digital microscope with a magnification of 50-times. The results are shown in Fig. 10, where it can be seen that a circumferential crack is initiated in sharp V-notched steel specimens (Fig. 10a, b), a single semi-elliptical crack is initiated in a blunt notched steel specimen (Fig. 10c), while three semi-elliptical cracks are initiated both in a blunt notched steel specimen (Fig. 10d) and in the sharp V-notched specimen made of titanium alloy (Fig. 10e). Fig. 10 reports also the crack sizes and positions, measured taking advantage of the markers (on the crack surfaces) produced by the overloads. In the case of circumferential cracks, the remaining un-cracked area is approximated by a circle. Beside the radius, also the position of the center of the circle is determined. For isolated, semi-elliptical cracks, the angular position θ, the area, the half-surface length c and the crack depth a are measured. After measuring the crack depths a from the overload-markers visible on the fracture surfaces as is shown in Fig. 10, the relevant potential drop ratios can be determined from the experimental curves reporting Pi versus loading cycles (see examples in Fig. 9a, b).
Dedicated FE analyses have been carried out by modelling circumferential or semi-elliptical cracks having the actual crack sizes and positions, observed experimentally in Fig. 10. Fig. 11 compares the potential drop curves obtained numerically with the experimental potential drops Pi (i = 1, 2 and 3), as a function of the normalized crack depth a/r. It should be noted that in the case of multiple initiated cracks (see Fig. 10d, e) the average crack depth aaverage = (a1 + a2 + a3)/3 has been adopted in Fig. 11d, e. A quite good agreement between numerical and experimental results can be observed from Fig. 11a, c–e: given the experimental potential drops Pi = ΔVi/ΔVi,0, the maximum deviation between the measured and the estimated crack depths is approximately ± 20%, similar deviations having been obtained elsewhere [21]. The deviation increases to approximately ± 30% for the case in Fig. 11b. 5. Predictions based on FE results 5.1. Assessment of the crack area All FE calibration curves reported in previous Figs. 6–8 and 11 are summarised in Fig. 12 in terms of the average potential drop ratio Pavg = (P1 + P2 + P3)/3 versus the normalised crack area Acrack/Anet. Fig. 12 shows that the effects of the notch geometry, i.e. sharp or blunt notch, and of the crack configuration, i.e. circumferential or semi-
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Fig. 9. Potential drop ratios measured during fatigue tests of (a) sharp (SN_ρ0.1_2) and (b) blunt (BN_ρ4_1) notched specimens under pure axial loading. Nf is the total fatigue life. Table 6 Summary of the loading parameters and the results of experimental fatigue tests under pure axial loading. Specimen code^
Material
SN_ρ0.1_1 SN_ρ0.1_2 BN_ρ4_1 BN_ρ4_2 SN_ρ0.1_3
AISI 304L
^
TiAl6V4
ΔF [kN]
40 40 60 65 50
R
Overload
−1
SN = sharp-notch, BN = blunt-notch.
180
Nf [cycles]
ΔF [kN]
ROL
Interval [cycles]
80 40 60 65 100
−1 0 0 0 −1
10,000 5,000 10,000 10,000 10,000
610,000 182,000 490,345 300,000 120,000
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Fig. 10. Experimental measurement of the crack depth a, the aspect ratio c/a and the angle θ of cracks generated by fatigue testing (a) and (b) sharp V-notched (SN_ρ0.1) and (c) and (d) blunt notched (BN_ρ4) steel specimens, and (e) a sharp V-notched (SN_ρ0.1) titanium specimen under pure axial loading. The loading parameters are reported in Table 6.
elliptical, are quite limited, since all curves are included in a scatter band with a maximum deviation of 16% in terms of predicted crack area Acrack for a fixed average potential drop ratio Pavg. Concerning the case of semi-elliptical surface cracks, Fig. 12 shows also that calibration curves are almost overlapped regardless the crack angular position θ and aspect ratio c/a. Accordingly, Fig. 12 suggests to adopt the scatter band consisting of a lower bound and an upper bound to predict the cracked area with a maximum deviation of 16%, once known the average potential drop ratio Pavg experimentally measured in a fatigue tested specimen, regardless of the notch geometry, the crack shape and location. The scatter band of Fig. 12 is validated against previous experimental results (see Figs. 10 and 11) generated by fatigue testing of sharp as well as blunt notched specimens made of AISI 304L and TiAl6V4. A quite
good agreement can be observed in Fig. 13, the great majority of experimental results falling inside the scatter band, while only those relevant to the case SN_2 being slightly on the unsafe side. 5.2. Assessment of the crack configuration and the crack angular position The numerical results reported in Figs. 6–8 suggest to define a procedure to assess the configuration of the crack, i.e. circumferential or semi-elliptical, and the angular position θ in the case of semi-elliptical cracks. Let us consider a notched bar undergoing a fatigue test monitored by the potential drop method sketched in Fig. 3. The potential drops Pi (i = 1, 2 and 3) are measured as a function of the number of loading cycles. The condition Pi = 1 is met at the beginning of the fatigue test;
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V1
(d)
θ
V2
V3
Specimen code: BN_ρ4_2 N [cycles] 0 280000 290000
crack-1 c1 [mm] 0 1.41 2.34
Material: AISI 304L
θ = 28° a1 [mm] 0 1.00 1.42
c1/a1 1.41 1.65
crack-2 c2 [mm] 0 1.01 1.47
θ = 74° a2 [mm] 0 1.11 1.54
c2/a2 0.91 0.96
crack-3 c3 [mm] 0 1.30 1.60
θ = 122° a3 [mm] 0 1.09 1.48
c3/a3 1.20 1.08
Fig. 10. (continued)
V1
therefore fatigue crack initiation at the notch root is detected when at least one of the three potential drops Pi increases above 1. Let us indicate PA ≥ PB ≥ PC and assume that only a single crack is initiated at the notch root. The configuration of the initiated fatigue crack and the crack angular position can be assessed as follows:
(e)
• if P ≈ P ≈ P , then a circumferential crack is initiated (see Fig. 14a); • if P ≈ P ≠ P , then a semi-elliptical crack is initiated with an angular location θ′ = 0° with respect to probe A; • if P ≈ P ≠ P , then a semi-elliptical crack is initiated with an angular location θ′ = 60° defined from probe A towards probe B; if • P > P > P , then a semi-elliptical crack is initiated with an A
θ
B
C
B
C
A
A
B
C
A
B
C
angular location θ′, defined from probe A towards probe B, in the range between 0° and 60°; it can be approximately estimated as θ′ ≈ 30° with an error of ± 30° (see Fig. 14b).
V2
V3
Specimen code: SN_ρ0.1_3 N [cycles] 0 60000 70000 80000 90000 100000 110000
crack-1 c1 [mm] 0 0.61 0.75 1.03 1.48 2.24 3.37
θ = 30° a1 [mm] 0 0.55 0.73 0.97 1.26 1.64 2.20
To reduce the estimation error of the crack angular position, it would be necessary to increase the number of potential probes connected to the tested specimen or to investigate the relationship between the potential drop ratios Pi (i = 1, 2 and 3) and the crack angular position θ with smaller steps of θ. It should be noted that the case of multiple crack initiation is more complex and cannot be treated with the above procedure, but new developments would be needed. The proposed procedure is validated against previous experimental results generated by fatigue testing AISI 304L and TiAl6V4 specimens under axial fatigue loading. Fig. 11a and b show that the experimental potential drops are P1 ≈ P2 ≈ P3 with a reduced deviation in the order of 2%, the actual crack centre being not coincident with the specimen axis. A fatigue crack having a circumferential configuration would have been anticipated, which agrees with the experimental observations
Material: TiAl6V4 c1/a1 1.12 1.03 1.06 1.18 1.36 1.53
crack-2 c2 [mm] 0 0.00 0.00 0.00 0.00 0.00 2.88
θ = 105° a2 [mm] 0 0.00 0.00 0.00 0.00 0.00 1.12
c2/a2 0.00 0.00 0.00 0.00 0.00 2.57
crack-3 c3 [mm] 0 0.87 1.20 1.58 1.93 2.32 3.29
θ = 247° a3 [mm] 0 0.52 0.75 1.08 1.44 1.74 2.22
c3/a3 1.68 1.60 1.46 1.34 1.33 1.48
Fig. 10. (continued) 182
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a
a Fig. 11. Comparison between numerical calibration curves and experimental measurements performed in Fig. 10: (a) and (b) sharp V-notched (SN_ρ0.1) and (c) and (d) blunt notched (BN_ρ4) steel specimens, and (e) a sharp V-notched (SN_ρ0.1) titanium specimen tested under pure axial fatigue loading.
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a
a Fig. 11. (continued)
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a Fig. 11. (continued)
Acrack A Fig. 12. Calibration curves for circumferential and semi-elliptical surface cracks in the cases SN_ρ0.1 and BN_ρ4 taken from previous Figs. 6–8 and 11 and expressed in terms of average potential drop ratio Pavg = (P1 + P2 + P3)/3 versus the normalised cracked area Acrack/Anet.
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Acrack A Fig. 13. Comparison between scatter band calibrated in Fig. 12 and experimental measurements performed in Fig. 10.
Fig. 14. Procedure to assess the crack shape and the crack angular position based on the potential drop ratios evaluated at three 120°-spaced potential probes: (a) circumferential and (b) semi-elliptical crack at an angular location θ′. Pi = ΔVi/ΔVi,0.
reported in Fig. 10a and b. On the other hand, Fig. 11c shows that the experimental potential drop ratios are in the relationship P1 > P3 > P2, with differences in the order of 4–5%. According to the procedure reported above, a semi-elliptical surface crack at an angular location θ′ 30° from probe 1 towards probe 3 is predicted, which is in good agreement with the experimental observations shown in Fig. 10c. The cases reported in Fig. 11d, e are more complex to be analysed, and they cannot be treated with the proposed procedure, multiple semi-elliptical cracks having been initiated as is shown in Fig. 10d, e.
angular positions θ between 0° and 60° with respect to one potential probe have been analysed. The following conclusions can be drawn:
• Concerning the effect of the notch geometry, for a given crack depth,
•
6. Conclusions A three-probe potential drop method (PDM) has been used to estimate crack area, location, and configuration (i.e. circumferential or semi-elliptical) in fatigue tested notched bars. Calibration of the adopted PDM configuration has been performed by employing 3-dimensional electrical FE models of sharply and bluntly notched bars adopted in the experimental tests. A range of circumferential and semielliptical surface cracks has been analyzed; concerning semi-elliptical surface cracks, aspect ratios c/a in the range between 1 and 10 and
•
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the smaller the notch tip radius ρ, the more sensitive the potential drop method. Dealing with the effect of the crack shape for a given crack depth, the maximum sensitivity of the PDM is for a circumferential crack. For semi-elliptical surface cracks, the higher the crack aspect ratio c/a, the more sensitive the PDM. The numerical analyses adopted for calibration have been validated by comparison with experimental results obtained by fatigue testing sharply as well as bluntly notched specimens made of steel and titanium alloy under axial loading. A quite good agreement has been obtained, the deviation between the experimentally measured crack depth and that estimated from the calibration curves derived by dedicated FE analyses is approximately ± 20% in most cases. The numerical results suggest an approximated procedure to assess the area, the configuration as well as the angular position of the crack, starting from the potential drop ratios measured at the three probes. The average potential drop ratio Pavg = (PA + PB + PC)/3 is
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correlated to the normalised cracked area Acrack/Anet with a maximum deviation of 16%, regardless of the notch geometry, the crack shape and location. Concerning the crack configuration and location, the procedure is valid for a single crack, the multiple crack case not being included in the present analysis. When PA ≈PB ≈ PC, a circumferential crack is initiated. As to the crack location, assuming PA ≥ PB ≥ PC, when PB ≈ PC ≠ PA, then the crack is at probe A; when PA ≈ PB ≠ PC, then the crack is in the mid-position between probe A and probe B; finally, when PA > PB > PC then the crack is between probe A and probe B, in the angular range from 0° to 60° from probe A. The proposed procedure has been validated by comparison with experimental results, allowing to estimate with good approximation the area, the configuration (either circumferential or elliptical), and the location in case of a single semi-elliptical surface crack.
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Acknowledgement The authors thank Prof. Filippo Berto from the Department of Engineering Design and Materials, NTNU in Trondheim, (Norway) for providing the specimen material. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijfatigue.2019.02.045. References [1] ASTM. ASTM Standard E647 – 15e1, Standard test method for measurement of fatigue crack growth rates; 2015. [2] Meneghetti G, Campagnolo A, Berto F, Tanaka K. Notched Ti-6Al-4V titanium bars under multiaxial fatigue: synthesis of crack initiation life based on the averaged strain energy density. Theor Appl Fract Mech 2018;96:509–33. [3] Lazzarin P, Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int J Fract 2001;112:275–98. [4] Hicks MA, Pickard AC. A comparison of theoretical and experimental methods of calibrating the electrical potential drop technique for crack length determination. Int J Fract 1982;20:91–101. [5] Smith RA. Calibrations for the electrical potential method of crack growth measurement by a direct electrical analogy. Strain 1974;10:183–7. [6] Soboyejo WO, Reed RC, Knott JF. On the calibration of the direct current potential difference method for the determination of semi-elliptical crack lengths. Int J Fract 1990;44:27–41. [7] Belloni G, Gariboldi E, Lo Conte A, Tono M, Speranzoso P. On the experimental calibration of a potential drop system for crack length measurements in a compact tension specimen. J Test Eval 2002;30:11149. [8] Hill MR, Stuart DH. Direct current potential difference correlation for open-hole, single-crack coupons. Eng Fract Mech 2013;99:141–6. [9] Ritchie RO, Bathe KJ. On the calibration of the electrical potential technique for monitoring crack growth using finite element methods. Int J Fract 1979;15:47–55.
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