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Nondestructive evaluation of the fatigue damage accumulation process around a notch using a digital image measurement system
ARTICLE IN PRESS
Optics and Lasers in Engineering 41 (2004) 477–487
Nondestructive evaluation of the fatigue damage accumulation process around a notch using a digital image measurement system F.V. D!ıaza, A.F. Armasa,b, G.H. Kaufmanna,b,*, G.E. Galizzia a
Instituto de F!ısica Rosario (CONICET-UNR), Bvd. 27 de Febrero 210 bis, S2000EZP Rosario, Argentina b Departamento de F!ısica, Facultad de Ciencias Exactas, Ingenier!ıa y Agrimensura, Universidad Nacional de Rosario, S2000PTB Rosario, Argentina Received 22 October 2002; received in revised form 3 February 2003; accepted 18 February 2003
Abstract The fatigue damage accumulation process around a notch is studied using a noncontact digital image measurement system. This system incorporates a contrast correlation method to evaluate the level of plastic damage at each point of the studied area of the specimen from two images acquired before and after the introduction of fatigue deformation. A compact tension specimen of 304 stainless steel with a notch radius of 1 mm is analyzed during the first 1000 cycles of the crack growth stage. During this period, the externally given work not only impels the crack growth and its local plastic zone but also generates plastic damage around the notch. The obtained results are used to explain the behavior of cracks emanating from notches. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Digital image measurement; Cyclic deformation
Light
scattering;
Low-cycle
fatigue;
Plastic
damage;
1. Introduction When a metal component is subjected to loads that generate plastic deformation, perceptible changes at the surface roughness are introduced. This effect is due to microstructural mechanisms such as slip system rotation and grain boundary sliding, *Corresponding author. Instituto de F!ısica Rosario (CONICET-UNR), Bvd. 27 de Febrero 210 bis, S2000EZP Rosario, Argentina. Tel.: +54-341-485-3200-20; fax: +54-341-482-1772. E-mail address: guille@ifir.edu.ar (G.H. Kaufmann). 0143-8166/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-8166(03)00018-6
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which act during plastic deformation [1,2]. In the last years, several researchers have experimentally investigated the roughness changes generated in metal components submitted to different loading modes. Kienzle and Mietzner [3] reported on the relation between the increase in the surface roughness and the growth in plastic deformation in metal sheets tested in tension and compression. Yamaguchi and Mellor [4] verified that the free surface roughness of a metal component increases in proportion to both the magnitude of the effective strain and the grain size of the material. More recently, Dautzemberg and Kals [5] introduced the bulge test and reported a linear relationship between surface roughness and effective strain. Furthermore, Thomson and Shafer [6] observed that the rate of roughening of specimens with a low initial roughness is independent of the mode of loading. On the other hand, several optical techniques have been developed to obtain information from such surface plastic changes in notched metal specimens. Lee et al. [7] developed an image-processing method to quantitatively extract the level of plastic deformation in flat specimens under tension using white light. Later, Dai et al. [8] reported on a nondestructive method for the determination of the plastic zone in specimens under tension using a laser speckle pattern correlation technique. More recently, D!ıaz et al. [9,10] showed the feasibility of a digital image measurement system for monitoring the plastic changes generated in specimens submitted to cyclic loads using white-light scattering. The purpose of the present paper is to show the usefulness of a digital image measurement system to evaluate nondestructively the plastic damage accumulation process on the surface of a notched steel specimen submitted to low-cycle fatigue. This system uses white-light scattering caused by the plastically induced surface roughness on the metal specimen polished surface. The determination of the plastically strained region around the notch is based upon the loss of correlation between two images acquired before and after the introduction of the fatigue damage. It must be noted that all experimental studies carried out to evaluate the fatigue damage accumulation in notched specimens were realized using destructive methods like the recrystallization technique [11,12]. In this work, the tests were performed using a compact tension specimen of 304 stainless steel with a notch radius of 1 mm. The damage accumulation process was evaluated during the first 1000 cycles of the crack growth stage. During this period, a crack emanating from the notch was propagated through a plastically damaged zone that had been previously introduced. The obtained results show the influence of the notch in the plastic damage generation and its relation with the growth rate changes of the evaluated crack.
2. Experimental A scheme of the optical system used in this work is shown in Fig. 1. The whitelight beam emitted by the lamp (L) was first collimated by a lens (CL) and then deviated by a beam-splitter (BS) to illuminate perpendicularly the test specimen (S). To minimize light intensity changes, the lamp was fed by means of a regulated dc
ARTICLE IN PRESS F.V. D!ıaz et al. / Optics and Lasers in Engineering 41 (2004) 477–487
479
z SH CL
x
S
L
BS PC CCD Fig. 1. Optical system: L, lamp; BS, beam-splitter; CL, collimating lens; S, specimen; SH, specimen holder; CCD, video camera; PC, digital image processing system.
11
y 2 48
x
33 R1
10 50 Fig. 2. Test specimen. The units are in mm and the thickness is 5.8 mm.
power supply of 12 V. Rigid body motions were minimized using a specimen holder (SH) which is extremely stable and has no moving parts [13]. The specimen rests by gravity by leaning it slightly (5 ) backwards. The holder allows an effective specimen reposition with an accuracy of 0.5 mm. To acquire the images on the specimen surface, a CCD camera (Pulnix TM-620) was used. The camera output was fed to a frame grabber (Matrox Pulsar) located inside a personal computer (PC), which digitized the images with a resolution of 512 512 pixels 8 bits (256 gray levels). The optical system magnification yielded a physical size of 149 pixels/mm on the specimen surface in both horizontal and vertical directions. Fig. 2 shows the test specimen, which was made from a commercially available type 304 stainless steel plate of longitudinal elastic modulus E ¼200 GPa, Poisson’s ratio n ¼0.29 and yield stress s0 ¼311 MPa. The specimen was machined so that the loading direction was parallel to the rolling direction (y axis). The notch, with a radius R1 ¼1 mm, was made by electric-discharge wire cutting to avoid surface
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(a)
(b)
Fig. 3. Images of the acquired specimen surface: (a) before the first cycle and (b) after 1550 cycles.
deformation during machining. The notch root was carefully mechanically polished using emery papers numbers 320, 800 and 1500. The specimen surfaces were first polished using emery papers of grade numbers up to 1500 under water cooling, followed by wet grinding with high purity alumina powder until a well-polished surface was produced. After polishing, an average roughness value Ra of 0.05 mm was measured on the specimen surfaces. Before the mechanical test, the specimen was positioned in the specimen holder and a reference image of the region of interest was acquired. Afterwards, the specimen was transferred to the fatigue testing machine (Instron 1362). The lowcycle fatigue test was performed at room temperature (20 C) under fully reversed load control of the maximum apparent stress intensity factor Krmax ¼34.4 MN m3/2 and the minimum value of Krmin ¼ 34.4 MN m3/2 [14]. The specimen was loaded using a triangular load profile at a nominal frequency of 0.1 Hz. The duration of the fatigue test was of 1550 cycles, but the analysis was performed during the first 1000 cycles of the crack growth stage, from Ni ¼550 to Nf ¼1550 cycles. The fatiguecrack initiation limit Ni was defined as the number of cycles after which a 1 mm-long crack on the notch root surface is visible at 20-fold magnification. The test was interrupted at various number of cycles. After each interruption, which was performed after unloading from the negative part of the last cycle, the specimen was put back into the plate holder and a new image was acquired. The region of interest that covered the images was of 3.4 mm 3.4 mm. Fig. 3 shows the images acquired before the first cycle and after 1550 cycles.
3. Correlation coefficient determination A subimage-correlation technique was used to characterize the evaluated surface, which contains the notch and the elastic and plastic regions. As previously
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mentioned, through the use of a specimen holder it is possible to minimize rigid body translations and rotations, which could cause decorrelation between the images acquired before and after the introduction of the fatigue damage. Therefore, as surface morphology only changes under plastic or extreme elastic conditions, the decorrelation region will be a good indicator of the size and shape of the plastically strained region. In this work, a contrast correlation coefficient was used to determine the decorrelation region. This correlation coefficient was calculated from two contrast distributions. Using the light intensity values of all the pixels within a window centered at each pixel, the contrast distribution F ðx; yÞ of an image recorded by the CCD camera can be determined as [10] ½sv ðx; yÞ1=2
F ðx; yÞ ¼
½aðx; yÞ1=4
ð1Þ
;
where sv ðx; yÞ ¼
yþM=2 X
1 M2
xþM=2 X
% yÞ2 ; ½Iðm; nÞ Iðx;
ð2Þ
n¼yM=2 m¼xM=2
1 2 M ½sv ðx; yÞ2
aðx; yÞ ¼
% yÞ ¼ Iðx;
1 M2
yþM=2 X
yþM=2 X
xþM=2 X
% yÞ4 ; ½Iðm; nÞ Iðx;
ð3Þ
n¼yM=2 m¼xM=2
xþM=2 X
Iðm; nÞ:
ð4Þ
n¼yM=2 m¼xM=2
In the former equations, sv is the variance, a is the kurtosis and Iðm; nÞ is the light intensity value corresponding to a pixel at the location ðm; nÞ in a window of dimensions M M; where M is an odd integer. The contrast correlation coefficient Cðx; yÞ can be calculated as [10] Cðx; yÞ ¼
Fb ðx; yÞ ; Fa ðx; yÞ
ð5Þ
where Fb ðx; yÞ and Fa ðx; yÞ are the contrast distributions of two images acquired before and after the fatigue deformation, respectively. In those regions away from the notch where the elastic deformation predominates, both images will show similar contrast values. Therefore, the correlation coefficient will be very near to the unity. On the contrary, in regions near to the notch, the surface texture generated by fatigue will decrease the intensity values and will increase the contrast values of the second image. Then, the correlation coefficient will be smaller than one.
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4. Results From the images shown in Fig. 3, the contrast values of F ðx; yÞ were calculated along the symmetrical axis of the notch using a window size M ¼31 pixels. Fig. 4(a) shows both profiles. The contrast values of these profiles are similar at regions far from the notch-tip. The curve corresponding to 1550 cycles has a higher gradient near the notch-tip, where the plastic damage is expected to occur. Fig. 4(b) shows a plot of the contrast correlation coefficient Cðx; yÞ along the axis y ¼0 obtained from the contrast distributions shown in Fig. 4(a) by using Eq. (5). This profile contains information about the plastic damage generated along the performed fatigue test. As expected, a high gradient is observed near the notch edge and far away it decreases to a nearly constant value. The contrast profiles shown in Fig. 5(a) were obtained from the images recorded for 550 and 1550 cycles. In this case, a vertical line located at 0.2 mm from the notch tip was evaluated using also a window size M ¼31 pixels. The high contrast values of the central part of this plot correspond to the notch axis zone, where the higher plastic damage is expected to occur. Fig. 5(b) show the correlation profile obtained from the contrast distributions shown in Fig. 5(a). This profile contains information
10
II
8
F
6 4
I
2 0 0.0
0.5
1.0
1.5
x (mm)
(a)
1.00
C
0.75 0.50 0.25 0.00 0.0
(b)
0.5
1.0
1.5
x (mm)
Fig. 4. (a) Plot of the contrast distribution F ðx; yÞ along the axis y ¼0 corresponding to the images acquired: (I) before the first cycle and (II) after 1550 cycles. (b) Profile of the coefficient Cðx; yÞ:
ARTICLE IN PRESS F.V. D!ıaz et al. / Optics and Lasers in Engineering 41 (2004) 477–487
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10
II
1.00
8 0.75
C
F
6
4
0.50
I
2 -2
(a)
-1
0
y (mm)
0.25 1
2
-2
(b)
-1
0
1
2
y (mm)
Fig. 5. (a) Plot of the distribution F ðx; yÞ along the vertical line y ¼0.2 mm corresponding to the images acquired after (I) 550 and (II) 1550 cycles. (b) Profile of the coefficient Cðx; yÞ:
Fig. 6. Calibration curve obtained from a tensile test.
about the plastic damage introduced during the first 1000 cycles of the crack growth stage. The decrease of the correlation coefficient beyond a certain level can therefore provide an indication of the onset of plastic deformation and hence of the location of the plastic zone. For obtaining the threshold value of the correlation coefficient to determine the boundary of the plastic zone, a calibration curve was traced using a tensile specimen. The threshold value of accumulated plastic deformation eac was chosen as the value for which the correlation coefficient in the fatigue-damaged specimen was the same that in the tensile specimen monotonically strained to the same amount of true tensile strain e: The strain eac is an equivalent strain because the strain state around a notch is triaxial [15]. The calibration curve obtained from a tensile test is shown in Fig. 6. The same material and experimental arrangements that were used in the fatigue procedure were utilized to obtain the calibration curve. Besides, both fatigue and tensile specimens were machined from the same plate, and the plate rolling direction was
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484
chosen as the loading direction for all specimens. Each data point of the curve shown in Fig. 6 was obtained using two images of the tensile specimen surface, which corresponded to different deformation states. The contrast value of each image was obtained from the average of the contrast values from all pixels within the image. The contrast value at a particular pixel was calculated using the same procedure utilized in the fatigue test, that is from a window of size M ¼31 pixels centered at each pixel. From two deformation states, the contrast correlation coefficient C was calculated as [10] C¼
Fl ; Fm
ð6Þ
where Fl is the image contrast value corresponding to the less deformed state and Fm is the image contrast value associated to the more deformed state. In this work, the boundary of the plastic region growth is defined at points where the variation of the true plastic strain is 0.1%. From the correlation coefficient-strain variation relation, we found that the 0.1% strain variation corresponds to a 0.986 contrast coefficient value. Then, this value was used as the threshold to determine the extent of the plastic zone. Using the correlation coefficient Cðx; yÞ obtained from the contrast distributions of the images acquired before the test and after 550 cycles, the extent of the plastic zone Z1 developed around the notch root was determined. Fig. 7(a) shows this plastic zone, which represents the fatigue damage in the crack initiation stage. In this stage, the damage grows not only ahead of the notch tip but also in the y direction and along the notch root. Fig. 7(b) shows the size and the shape of the plastic zone Z551 developed around the notch root between 551 and 1550 cycles. This zone, obtained using the same procedure, was generated in the crack growth stage within a pre-existing damaged-accumulated zone. The extent of Z551 indicates that part of the externally given work is used to increase the plastic damage far from the local plastic zone around the crack. This influence of the notch would generate a decrease of the fatigue crack-growth rate. y (mm)
y (mm)
1.6 1.4 1.2 1.0 0.8
0.2
0.6
1.0
1.4
0.4
x (mm)
x (mm)
Z551
Z1
(a)
0.8 1.2
(b)
Fig. 7. Plot of the plastic zone at: (a) the crack initiation stage (Z1 ); (b) the crack growth stage (Z551 ).
ARTICLE IN PRESS F.V. D!ıaz et al. / Optics and Lasers in Engineering 41 (2004) 477–487 A
485
a B
Rx 1100-1350 cycles Rx=0.55 mm a=18 µm CA =0.9, CB =0.87
1450-1500 cycles Rx=0.49 mm a=130 µm CA =0.92, CB =0.91
1350-1400 cycles Rx=0.44 mm a=57 µm CA =0.97, CB =0.96
1400-1450 cycles Rx=0.47 mm a=100 µm CA =0.97, CB =0.95
1500-1550 cycles Rx=0.51 mm a=155 µm CA =0.9, CB =0.88
Fig. 8. Plot of the evolution of the plastic zone size and the length of the crack between 1100 and 1550 cycles.
A crack emanating from the notch was propagated through the evaluated surface after 1100 cycles. Using the coefficient Cðx; yÞ obtained from different image pairs acquired between 1100 and 1550 cycles, the damage around the notch accompanying the growth of this crack was evaluated. At Nf ; the length of the crack was of a ¼155 mm. Fig. 8 shows the growth of the plastic zone around the notch at each considered substage. It must be noted that each zone is formed within a previously damaged zone and that in all substages the size and the shape of the zones are similar. On the other hand, the values of the correlation coefficient evaluated at the points A(0.1,0.4) and B(0.1,0.4) are CA and CB ; respectively. Fig. 8 also shows these values, which contain information about the plastic damage level generated away from the crack influence.
5. Discussion The 304 stainless steel used in this investigation has high strain-hardening coefficient (n ¼0.5) at room temperature. Therefore, the accumulated plastic zone size around the notch increases significantly during the crack initiation stage (see Fig. 7(a)). Within the plastic zone, the larger damage is introduced at the notch-tip region. When the local accumulated plastic work at the initiation site reaches a critical value, the crack is initiated. At the first part of the crack growth stage, the evaluated crack grows through the damaged zone forming its own plastic zone on the former one. The externally given work impels the crack growth and the generation of the local plastic zone around the
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crack. It has been shown that the local damage generation plays an important role in the crack growth rate [11]. However, it has not been still studied the effect of the damage accumulated around the notch, away from the crack influence. Fig. 7(b) shows that the effect of the notch exists, probably because of the shortness of the fatigue crack. Accordingly, part of the plastic work is used to increase the damage around the notch root. Fig. 8 shows that the damage generation is cyclic and also reveals that a low cycle number is enough to generate a new plastic zone around the notch root. Furthermore, the damage generation impelled by the notch produces modifications at the crack growth rate. Fig. 8 shows the values of a at the end of each substage. Until 1450 cycles, the values of the crack growth rate da=dN are similar to the average of the crack initiation stage, which was estimated to be approximately 0.8 mm/cycle [11]. Afterwards, da=dN decreases until 0.6 mm/cycle (from 1450 cycles) and then it reaches a value of 0.5 mm/cycle (from 1500 cycles). These values show that in the damaged zone near the notch tip da=dN is greater than in the zone with less damage. Furthermore, at those substages where da=dN decreases, CA and CB also decrease, which indicates that the plastic damage level away from the crack increases (see Fig. 8). This increment supposes a softening effect of the material. Accordingly, part of the externally given energy that first serves to accelerate the crack, afterwards is used to accumulate damage around the notch. Therefore, this softening effect acts to decrease the crack-growth rate.
6. Conclusions 1. A noncontact digital image measurement system was found to be very useful to study the fatigue damage accumulation around a notch in 304 stainless steel. 2. During the first part of the crack growth stage, the externally given work not only impels the growth of the evaluated crack and its local plastic zone but also generates plastic damage around the notch, away from the crack influence. 3. The growth rate of the evaluated crack is abnormal. First, the rate is fast and then it decelerates. 4. The damage generation around the notch is cyclic. A low cycle number is enough to generate a new plastic zone, which grows not only ahead of the notch tip but also in the y direction and along the notch root. 5. A softening effect of the material around the notch is produced by cycled. This effect help to explain the changes operating at the growth rate of the evaluated crack.
Acknowledgements The authors acknowledge the financial support of Consejo Nacional de ! Cient!ıfica Investigaciones Cient!ıficas y Te! cnicas and Agencia Nacional de Promocion ! y Tecnologica of Argentina.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15]
Friedel J. Dislocations. Oxford: Pergamon Press, 1964. Mendelson AS. Plasticity: theory and application. Mahbar: Krieger Publishing, 1983. Kienzle O, Mietzner K. Atlas umformter metallischer oberflachen. Berlin: Springer, 1967. Yamaguchi K, Mellor PB. Thickness and grain size dependence of limit strains in sheet metal stretching. Int J Mech Sci 1976;18:85–90. Dautzenberg JH, Kals JA. Stress state and surface roughness. Adv Technol Plasticity, Japan 1984;1:18–26. Thomson PF, Shafer BV. The roughening of free surface during plastic working. Int J Mach Tool Des Res 1982;22:261–4. Lee C, Chao YJ, Sutton MA, Peters WH, Ranson WF. Determination of plastic strains at notches by image-processing methods. Exp Mech 1989;29:214–20. Dai YZ, Tay CJ, Chiang FP. Determination of the plastic zone by laser-speckle correlation. Exp Mech 1991;31:348–52. D!ıaz FV, Kaufmann GH, Armas AF, Galizzi GE. Optical measurement of the plastic zone size in a notched metal specimen subjected to low-cycle fatigue. Opt Lasers Eng 2001;35:325–33. D!ıaz FV, Armas AF, Kaufmann GH, Galizzi GE. Fatigue damage accumulation around a notch using a digital image measurement system. Exp Mech (in review). Iino Y. Local fatigue damage accumulation around notch attending crack initiation. Metall Mater Trans A 1995;26A:1419–30. Iino Y. Effect of stress ratio on local fatigue damage accumulation around notch. In: Rie K, Portella P, editors. Low cycle fatigue and elasto plastic behaviour of materials. Oxford: Elsevier Science Ltd, 1998. p. 585–90. Kaufmann GH, Feugeas JN, Marino B, Galizzi GE. Holographic investigation of residual deformations induced by a pulsed ion implanter. Appl Opt 1991;30:85–9. Srawley JE. Wide range stress intensity factor expressions for ASTM E399 standard fracture toughness specimens. Int J Fract 1976;12:475–9. Rosakis AJ, Ravi-Chandar K. On crack tip stress state: an experimental evaluation of threedimensional effects. Int J Solids Struct 1986;22:121–34.
Optics and Lasers in Engineering 41 (2004) 477–487
Nondestructive evaluation of the fatigue damage accumulation process around a notch using a digital image measurement system F.V. D!ıaza, A.F. Armasa,b, G.H. Kaufmanna,b,*, G.E. Galizzia a
Instituto de F!ısica Rosario (CONICET-UNR), Bvd. 27 de Febrero 210 bis, S2000EZP Rosario, Argentina b Departamento de F!ısica, Facultad de Ciencias Exactas, Ingenier!ıa y Agrimensura, Universidad Nacional de Rosario, S2000PTB Rosario, Argentina Received 22 October 2002; received in revised form 3 February 2003; accepted 18 February 2003
Abstract The fatigue damage accumulation process around a notch is studied using a noncontact digital image measurement system. This system incorporates a contrast correlation method to evaluate the level of plastic damage at each point of the studied area of the specimen from two images acquired before and after the introduction of fatigue deformation. A compact tension specimen of 304 stainless steel with a notch radius of 1 mm is analyzed during the first 1000 cycles of the crack growth stage. During this period, the externally given work not only impels the crack growth and its local plastic zone but also generates plastic damage around the notch. The obtained results are used to explain the behavior of cracks emanating from notches. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Digital image measurement; Cyclic deformation
Light
scattering;
Low-cycle
fatigue;
Plastic
damage;
1. Introduction When a metal component is subjected to loads that generate plastic deformation, perceptible changes at the surface roughness are introduced. This effect is due to microstructural mechanisms such as slip system rotation and grain boundary sliding, *Corresponding author. Instituto de F!ısica Rosario (CONICET-UNR), Bvd. 27 de Febrero 210 bis, S2000EZP Rosario, Argentina. Tel.: +54-341-485-3200-20; fax: +54-341-482-1772. E-mail address: guille@ifir.edu.ar (G.H. Kaufmann). 0143-8166/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-8166(03)00018-6
ARTICLE IN PRESS 478
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which act during plastic deformation [1,2]. In the last years, several researchers have experimentally investigated the roughness changes generated in metal components submitted to different loading modes. Kienzle and Mietzner [3] reported on the relation between the increase in the surface roughness and the growth in plastic deformation in metal sheets tested in tension and compression. Yamaguchi and Mellor [4] verified that the free surface roughness of a metal component increases in proportion to both the magnitude of the effective strain and the grain size of the material. More recently, Dautzemberg and Kals [5] introduced the bulge test and reported a linear relationship between surface roughness and effective strain. Furthermore, Thomson and Shafer [6] observed that the rate of roughening of specimens with a low initial roughness is independent of the mode of loading. On the other hand, several optical techniques have been developed to obtain information from such surface plastic changes in notched metal specimens. Lee et al. [7] developed an image-processing method to quantitatively extract the level of plastic deformation in flat specimens under tension using white light. Later, Dai et al. [8] reported on a nondestructive method for the determination of the plastic zone in specimens under tension using a laser speckle pattern correlation technique. More recently, D!ıaz et al. [9,10] showed the feasibility of a digital image measurement system for monitoring the plastic changes generated in specimens submitted to cyclic loads using white-light scattering. The purpose of the present paper is to show the usefulness of a digital image measurement system to evaluate nondestructively the plastic damage accumulation process on the surface of a notched steel specimen submitted to low-cycle fatigue. This system uses white-light scattering caused by the plastically induced surface roughness on the metal specimen polished surface. The determination of the plastically strained region around the notch is based upon the loss of correlation between two images acquired before and after the introduction of the fatigue damage. It must be noted that all experimental studies carried out to evaluate the fatigue damage accumulation in notched specimens were realized using destructive methods like the recrystallization technique [11,12]. In this work, the tests were performed using a compact tension specimen of 304 stainless steel with a notch radius of 1 mm. The damage accumulation process was evaluated during the first 1000 cycles of the crack growth stage. During this period, a crack emanating from the notch was propagated through a plastically damaged zone that had been previously introduced. The obtained results show the influence of the notch in the plastic damage generation and its relation with the growth rate changes of the evaluated crack.
2. Experimental A scheme of the optical system used in this work is shown in Fig. 1. The whitelight beam emitted by the lamp (L) was first collimated by a lens (CL) and then deviated by a beam-splitter (BS) to illuminate perpendicularly the test specimen (S). To minimize light intensity changes, the lamp was fed by means of a regulated dc
ARTICLE IN PRESS F.V. D!ıaz et al. / Optics and Lasers in Engineering 41 (2004) 477–487
479
z SH CL
x
S
L
BS PC CCD Fig. 1. Optical system: L, lamp; BS, beam-splitter; CL, collimating lens; S, specimen; SH, specimen holder; CCD, video camera; PC, digital image processing system.
11
y 2 48
x
33 R1
10 50 Fig. 2. Test specimen. The units are in mm and the thickness is 5.8 mm.
power supply of 12 V. Rigid body motions were minimized using a specimen holder (SH) which is extremely stable and has no moving parts [13]. The specimen rests by gravity by leaning it slightly (5 ) backwards. The holder allows an effective specimen reposition with an accuracy of 0.5 mm. To acquire the images on the specimen surface, a CCD camera (Pulnix TM-620) was used. The camera output was fed to a frame grabber (Matrox Pulsar) located inside a personal computer (PC), which digitized the images with a resolution of 512 512 pixels 8 bits (256 gray levels). The optical system magnification yielded a physical size of 149 pixels/mm on the specimen surface in both horizontal and vertical directions. Fig. 2 shows the test specimen, which was made from a commercially available type 304 stainless steel plate of longitudinal elastic modulus E ¼200 GPa, Poisson’s ratio n ¼0.29 and yield stress s0 ¼311 MPa. The specimen was machined so that the loading direction was parallel to the rolling direction (y axis). The notch, with a radius R1 ¼1 mm, was made by electric-discharge wire cutting to avoid surface
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(a)
(b)
Fig. 3. Images of the acquired specimen surface: (a) before the first cycle and (b) after 1550 cycles.
deformation during machining. The notch root was carefully mechanically polished using emery papers numbers 320, 800 and 1500. The specimen surfaces were first polished using emery papers of grade numbers up to 1500 under water cooling, followed by wet grinding with high purity alumina powder until a well-polished surface was produced. After polishing, an average roughness value Ra of 0.05 mm was measured on the specimen surfaces. Before the mechanical test, the specimen was positioned in the specimen holder and a reference image of the region of interest was acquired. Afterwards, the specimen was transferred to the fatigue testing machine (Instron 1362). The lowcycle fatigue test was performed at room temperature (20 C) under fully reversed load control of the maximum apparent stress intensity factor Krmax ¼34.4 MN m3/2 and the minimum value of Krmin ¼ 34.4 MN m3/2 [14]. The specimen was loaded using a triangular load profile at a nominal frequency of 0.1 Hz. The duration of the fatigue test was of 1550 cycles, but the analysis was performed during the first 1000 cycles of the crack growth stage, from Ni ¼550 to Nf ¼1550 cycles. The fatiguecrack initiation limit Ni was defined as the number of cycles after which a 1 mm-long crack on the notch root surface is visible at 20-fold magnification. The test was interrupted at various number of cycles. After each interruption, which was performed after unloading from the negative part of the last cycle, the specimen was put back into the plate holder and a new image was acquired. The region of interest that covered the images was of 3.4 mm 3.4 mm. Fig. 3 shows the images acquired before the first cycle and after 1550 cycles.
3. Correlation coefficient determination A subimage-correlation technique was used to characterize the evaluated surface, which contains the notch and the elastic and plastic regions. As previously
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mentioned, through the use of a specimen holder it is possible to minimize rigid body translations and rotations, which could cause decorrelation between the images acquired before and after the introduction of the fatigue damage. Therefore, as surface morphology only changes under plastic or extreme elastic conditions, the decorrelation region will be a good indicator of the size and shape of the plastically strained region. In this work, a contrast correlation coefficient was used to determine the decorrelation region. This correlation coefficient was calculated from two contrast distributions. Using the light intensity values of all the pixels within a window centered at each pixel, the contrast distribution F ðx; yÞ of an image recorded by the CCD camera can be determined as [10] ½sv ðx; yÞ1=2
F ðx; yÞ ¼
½aðx; yÞ1=4
ð1Þ
;
where sv ðx; yÞ ¼
yþM=2 X
1 M2
xþM=2 X
% yÞ2 ; ½Iðm; nÞ Iðx;
ð2Þ
n¼yM=2 m¼xM=2
1 2 M ½sv ðx; yÞ2
aðx; yÞ ¼
% yÞ ¼ Iðx;
1 M2
yþM=2 X
yþM=2 X
xþM=2 X
% yÞ4 ; ½Iðm; nÞ Iðx;
ð3Þ
n¼yM=2 m¼xM=2
xþM=2 X
Iðm; nÞ:
ð4Þ
n¼yM=2 m¼xM=2
In the former equations, sv is the variance, a is the kurtosis and Iðm; nÞ is the light intensity value corresponding to a pixel at the location ðm; nÞ in a window of dimensions M M; where M is an odd integer. The contrast correlation coefficient Cðx; yÞ can be calculated as [10] Cðx; yÞ ¼
Fb ðx; yÞ ; Fa ðx; yÞ
ð5Þ
where Fb ðx; yÞ and Fa ðx; yÞ are the contrast distributions of two images acquired before and after the fatigue deformation, respectively. In those regions away from the notch where the elastic deformation predominates, both images will show similar contrast values. Therefore, the correlation coefficient will be very near to the unity. On the contrary, in regions near to the notch, the surface texture generated by fatigue will decrease the intensity values and will increase the contrast values of the second image. Then, the correlation coefficient will be smaller than one.
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4. Results From the images shown in Fig. 3, the contrast values of F ðx; yÞ were calculated along the symmetrical axis of the notch using a window size M ¼31 pixels. Fig. 4(a) shows both profiles. The contrast values of these profiles are similar at regions far from the notch-tip. The curve corresponding to 1550 cycles has a higher gradient near the notch-tip, where the plastic damage is expected to occur. Fig. 4(b) shows a plot of the contrast correlation coefficient Cðx; yÞ along the axis y ¼0 obtained from the contrast distributions shown in Fig. 4(a) by using Eq. (5). This profile contains information about the plastic damage generated along the performed fatigue test. As expected, a high gradient is observed near the notch edge and far away it decreases to a nearly constant value. The contrast profiles shown in Fig. 5(a) were obtained from the images recorded for 550 and 1550 cycles. In this case, a vertical line located at 0.2 mm from the notch tip was evaluated using also a window size M ¼31 pixels. The high contrast values of the central part of this plot correspond to the notch axis zone, where the higher plastic damage is expected to occur. Fig. 5(b) show the correlation profile obtained from the contrast distributions shown in Fig. 5(a). This profile contains information
10
II
8
F
6 4
I
2 0 0.0
0.5
1.0
1.5
x (mm)
(a)
1.00
C
0.75 0.50 0.25 0.00 0.0
(b)
0.5
1.0
1.5
x (mm)
Fig. 4. (a) Plot of the contrast distribution F ðx; yÞ along the axis y ¼0 corresponding to the images acquired: (I) before the first cycle and (II) after 1550 cycles. (b) Profile of the coefficient Cðx; yÞ:
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10
II
1.00
8 0.75
C
F
6
4
0.50
I
2 -2
(a)
-1
0
y (mm)
0.25 1
2
-2
(b)
-1
0
1
2
y (mm)
Fig. 5. (a) Plot of the distribution F ðx; yÞ along the vertical line y ¼0.2 mm corresponding to the images acquired after (I) 550 and (II) 1550 cycles. (b) Profile of the coefficient Cðx; yÞ:
Fig. 6. Calibration curve obtained from a tensile test.
about the plastic damage introduced during the first 1000 cycles of the crack growth stage. The decrease of the correlation coefficient beyond a certain level can therefore provide an indication of the onset of plastic deformation and hence of the location of the plastic zone. For obtaining the threshold value of the correlation coefficient to determine the boundary of the plastic zone, a calibration curve was traced using a tensile specimen. The threshold value of accumulated plastic deformation eac was chosen as the value for which the correlation coefficient in the fatigue-damaged specimen was the same that in the tensile specimen monotonically strained to the same amount of true tensile strain e: The strain eac is an equivalent strain because the strain state around a notch is triaxial [15]. The calibration curve obtained from a tensile test is shown in Fig. 6. The same material and experimental arrangements that were used in the fatigue procedure were utilized to obtain the calibration curve. Besides, both fatigue and tensile specimens were machined from the same plate, and the plate rolling direction was
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chosen as the loading direction for all specimens. Each data point of the curve shown in Fig. 6 was obtained using two images of the tensile specimen surface, which corresponded to different deformation states. The contrast value of each image was obtained from the average of the contrast values from all pixels within the image. The contrast value at a particular pixel was calculated using the same procedure utilized in the fatigue test, that is from a window of size M ¼31 pixels centered at each pixel. From two deformation states, the contrast correlation coefficient C was calculated as [10] C¼
Fl ; Fm
ð6Þ
where Fl is the image contrast value corresponding to the less deformed state and Fm is the image contrast value associated to the more deformed state. In this work, the boundary of the plastic region growth is defined at points where the variation of the true plastic strain is 0.1%. From the correlation coefficient-strain variation relation, we found that the 0.1% strain variation corresponds to a 0.986 contrast coefficient value. Then, this value was used as the threshold to determine the extent of the plastic zone. Using the correlation coefficient Cðx; yÞ obtained from the contrast distributions of the images acquired before the test and after 550 cycles, the extent of the plastic zone Z1 developed around the notch root was determined. Fig. 7(a) shows this plastic zone, which represents the fatigue damage in the crack initiation stage. In this stage, the damage grows not only ahead of the notch tip but also in the y direction and along the notch root. Fig. 7(b) shows the size and the shape of the plastic zone Z551 developed around the notch root between 551 and 1550 cycles. This zone, obtained using the same procedure, was generated in the crack growth stage within a pre-existing damaged-accumulated zone. The extent of Z551 indicates that part of the externally given work is used to increase the plastic damage far from the local plastic zone around the crack. This influence of the notch would generate a decrease of the fatigue crack-growth rate. y (mm)
y (mm)
1.6 1.4 1.2 1.0 0.8
0.2
0.6
1.0
1.4
0.4
x (mm)
x (mm)
Z551
Z1
(a)
0.8 1.2
(b)
Fig. 7. Plot of the plastic zone at: (a) the crack initiation stage (Z1 ); (b) the crack growth stage (Z551 ).
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a B
Rx 1100-1350 cycles Rx=0.55 mm a=18 µm CA =0.9, CB =0.87
1450-1500 cycles Rx=0.49 mm a=130 µm CA =0.92, CB =0.91
1350-1400 cycles Rx=0.44 mm a=57 µm CA =0.97, CB =0.96
1400-1450 cycles Rx=0.47 mm a=100 µm CA =0.97, CB =0.95
1500-1550 cycles Rx=0.51 mm a=155 µm CA =0.9, CB =0.88
Fig. 8. Plot of the evolution of the plastic zone size and the length of the crack between 1100 and 1550 cycles.
A crack emanating from the notch was propagated through the evaluated surface after 1100 cycles. Using the coefficient Cðx; yÞ obtained from different image pairs acquired between 1100 and 1550 cycles, the damage around the notch accompanying the growth of this crack was evaluated. At Nf ; the length of the crack was of a ¼155 mm. Fig. 8 shows the growth of the plastic zone around the notch at each considered substage. It must be noted that each zone is formed within a previously damaged zone and that in all substages the size and the shape of the zones are similar. On the other hand, the values of the correlation coefficient evaluated at the points A(0.1,0.4) and B(0.1,0.4) are CA and CB ; respectively. Fig. 8 also shows these values, which contain information about the plastic damage level generated away from the crack influence.
5. Discussion The 304 stainless steel used in this investigation has high strain-hardening coefficient (n ¼0.5) at room temperature. Therefore, the accumulated plastic zone size around the notch increases significantly during the crack initiation stage (see Fig. 7(a)). Within the plastic zone, the larger damage is introduced at the notch-tip region. When the local accumulated plastic work at the initiation site reaches a critical value, the crack is initiated. At the first part of the crack growth stage, the evaluated crack grows through the damaged zone forming its own plastic zone on the former one. The externally given work impels the crack growth and the generation of the local plastic zone around the
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crack. It has been shown that the local damage generation plays an important role in the crack growth rate [11]. However, it has not been still studied the effect of the damage accumulated around the notch, away from the crack influence. Fig. 7(b) shows that the effect of the notch exists, probably because of the shortness of the fatigue crack. Accordingly, part of the plastic work is used to increase the damage around the notch root. Fig. 8 shows that the damage generation is cyclic and also reveals that a low cycle number is enough to generate a new plastic zone around the notch root. Furthermore, the damage generation impelled by the notch produces modifications at the crack growth rate. Fig. 8 shows the values of a at the end of each substage. Until 1450 cycles, the values of the crack growth rate da=dN are similar to the average of the crack initiation stage, which was estimated to be approximately 0.8 mm/cycle [11]. Afterwards, da=dN decreases until 0.6 mm/cycle (from 1450 cycles) and then it reaches a value of 0.5 mm/cycle (from 1500 cycles). These values show that in the damaged zone near the notch tip da=dN is greater than in the zone with less damage. Furthermore, at those substages where da=dN decreases, CA and CB also decrease, which indicates that the plastic damage level away from the crack increases (see Fig. 8). This increment supposes a softening effect of the material. Accordingly, part of the externally given energy that first serves to accelerate the crack, afterwards is used to accumulate damage around the notch. Therefore, this softening effect acts to decrease the crack-growth rate.
6. Conclusions 1. A noncontact digital image measurement system was found to be very useful to study the fatigue damage accumulation around a notch in 304 stainless steel. 2. During the first part of the crack growth stage, the externally given work not only impels the growth of the evaluated crack and its local plastic zone but also generates plastic damage around the notch, away from the crack influence. 3. The growth rate of the evaluated crack is abnormal. First, the rate is fast and then it decelerates. 4. The damage generation around the notch is cyclic. A low cycle number is enough to generate a new plastic zone, which grows not only ahead of the notch tip but also in the y direction and along the notch root. 5. A softening effect of the material around the notch is produced by cycled. This effect help to explain the changes operating at the growth rate of the evaluated crack.
Acknowledgements The authors acknowledge the financial support of Consejo Nacional de ! Cient!ıfica Investigaciones Cient!ıficas y Te! cnicas and Agencia Nacional de Promocion ! y Tecnologica of Argentina.
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