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Fractal analysis of cyclic creep fractured surfaces of two high temperature alloys
Materials Science and Engineering A266 (1999) 250 – 254
Fractal analysis of cyclic creep fractured surfaces of two high temperature alloys X. Wang *, H. Zhou, Z.H. Wang, M.S. Tian, Y.S. Liu, Q.P. Kong Institute of Solid State Physics, Academia Sinica, Hefei230031, PR China Received 25 September 1998; received in revised form 10 November 1998
Abstract The fractal dimensions of cyclic creep fractured surfaces of two high-temperature alloys (Nimonic 75 and A-286 types) have been measured by using secondary electron line scanning method. All the fractures are intergranular. It is found that the fractal dimensions nearly linearly increase with the increase in fracture strain. The upper limits of measuring units for the linear range of fractal curve are below respective grain sizes. The results indicate that a larger amount of deformation results in a larger extent of irregularity of fractured surfaces caused by the propagation and linkage of intergranular cracks and the distortion near grain boundaries, and hence a higher fractal dimension. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Cyclic creep; Fractal dimension; Fractured surfaces
1. Introduction Fractal analysis is an effective tool to get useful information on the process of fracture in materials [1 –8]. It is known that there are several available methods for measuring the fractal dimension [4,5], such as: (a) the slit island method; (b) the vertical section method, and; (c) the secondary electron line scanning method. For the first two methods, the fractured samples should be ground and polished along the horizontal or vertical direction of fractured surfaces. For the third method, the secondary electron scanning line from a fractured surface in scanning electron microscopy (SEM) is used to express the profile roughness. This method does not need to grind and polish the fractured sample and is more convenient. Pande, Richards and Smith [4] first employed the secondary electron line scanning method to measure the fractal dimension of fractured surfaces. They adopted the above mentioned three methods to a titanium alloy after tensile tests, and found that the fractal dimensions measured by the three methods were reasonably consistent with each other. Using the three methods to the * Corresponding author. Tel.: +86-551-5591427; fax: + 86-5515591434.
impact fractured surfaces of a CK45 steel, Huang, Tian and Wang [6–8] observed that the fractal dimensions measured by different methods increased with impact toughness in a similar manner. They found for the same fractured surfaces that, the fractal dimension obtained by secondary electron line scanning method was a little higher than that obtained by vertical section method, but a little lower than that obtained by the slit island method; the relative deviation among the absolute values obtained by different methods was about 10%. There has been numerous fractal analyses on fractured surfaces in the literature. Nevertheless, the relation between fractal dimension and material property is not clear enough. Different scientists connect the fractal dimension to different mechanical properties, e.g. impact energy [1], dynamic tear energy [3], critical crack extension force [9–11], fracture strain [12–15] and so forth. The present paper is aimed to make a fractal analysis of cyclic creep fractured surfaces of two high temperature alloys by using the secondary electron line scanning method. Based on four series of cyclic creep tests with different cyclic periods, a correlation between the fractal dimension and the fracture strain has been observed. The experimental procedure and results are reported below.
0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 9 8 ) 0 1 1 2 7 - 7
X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
2. Materials and experimental procedures The samples were made of two types of high temperature alloys. One is a Nimonic 75 type alloy with main composition (wt.%): 20.13 Cr, 0.064 C and the balance nickel. Another is an A-286 type superalloy, of which two heats I and II were used. The main composition of A-286 (I) is (wt.%): 27.04 Ni, 14.08 Cr, 2.05 Ti, 0.25 Al, 1.34 Mo, 0.33 V, 0.059 C and the balance iron. The main composition of A-286 (II) is (wt.%): 26.29 Ni, 14.19 Cr, 2.18 Ti, 0.17 Al, 1.28 Mo, 0.28 V, 0.034 C and the balance iron. After solid solution and aging treatments, the average grain sizes of Nimonic 75, A-286 (I) and A-286 (II) are 100 ,75 and 36 mm, respectively. The load-controlled cyclic creep tests were performed on an Instron 1362 machine especially designed for combined creep-fatigue testing. The waveform of stress as a function of time is of trapezoidal shape, the details were described in [16 – 18]. Each series of cyclic creep was tested at a constant temperature and a constant peak stress, but with different hold times at peak stress. The duration of loading and unloading in each cycle was 1 min in total. The six different hold times at peak stress were 0 (pure fatigue), 0.33, 1, 3, 15 min and (pure creep). That is, the six different cyclic periods for each cycle were respectively 1 (pure fatigue), 1.33, 2, 4, 16 min and (pure creep). The fractured surfaces after ultrasonic cleaning were line-scanned in a scanning electron microscope of Amary-1000B type. For each fractured surface, five scanning lines with equal separation were taken at a fixed magnification along each of two perpendicular directions (i.e. totally 5×2 = 10 scanning lines). During the whole scanning process, the incident electron beam current and diameter, the contrast and the tilt angle of the specimen to the electron beam were kept constant to ensure a comparable basis among the scanning lines from fractured surfaces. The micrographs of the scanning lines were digitized with a fractal image processing system (FIPS), and the length L(h) of each scanning line was measured as a function of measuring unit h. The fractal dimension D was evaluated by means of the following equation [2]: L(h)= L0h 1 − D
251
ported in next section is the average of the data evaluated from all the scanning lines of the same fractured surface. The mean square deviation of the reported D data is within 91%.
3. Experimental results Four series of cyclic creep tests at constant temperature and constant peak stress, but with different cyclic periods were conducted with the two type alloys. All the cyclic creep fractures are of intergranular type. Fig. 1 shows SEM photograghs of cyclic creep fractured surfaces of (a) the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa with cyclic period 1.33 min, and (b) the A-286 (I) alloy tested at 650°C and peak stress 510 MPa with cyclic period 2 min, respectively. Fig. 2 is an example of a secondary electron scanning line taken from a cyclic creep fractured surface. Fig. 3 shows the double logarithms of the length L(h) of a scanning line and the reciprocal of measuring unit (1/h). One can see that there is a linear range in the middle portion of the curve. The fractal dimension D is then evaluated from the slope of the straight line according to Eq. (2). The measuring units of the linear range for the Nimonic 75, A-286 (I) and A-286 (II) alloys are 4–22,
(1)
where L0 is the projected length of the scanning line. Take logarithm of Eq. (1), we have:
ln L(h)=C+ (D −1)ln
1 h
(2)
where C is a constant. Through linear regression for the nearly straight central portion of the curve ln L(h) ln (1/h), the fractal dimension D was obtained from the slope. The correlation coefficients for all the linear regression are higher than 0.99. The value of D re-
Fig. 1. SEM photographs of cyclic creep fractured surfaces of (a) the Nimonic 75 alloy, and (b) the A-286 (I) alloy.
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X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
Fig. 2. An example of a secondary electron scanning line taken from a cyclic creep fractured surface.
Fig. 3. Double logarithmic plot of length L(h) and the reciprocal of the measuring unit (1/h) for a scanning line.
8 –75 and 5–32 mm, respectively. The upper limits are below respective average grain sizes. The implication of the limited linear range of measuring unit will be discussed in Section 4. Figs. 4–7 show the fracture time tf, fracture strain of and fractal dimension D versus the cyclic period for: the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa; the A-286 (I) alloy tested at 500°C and peak stress 862 MPa; the A-286 (I) alloy tested at 650°C and peak stress 510 MPa, and; the A-286 (II) alloy tested at 650°C and peak stress 510 MPa, respectively.
Fig. 5. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the A-286 (I) alloy tested at 500°C and peak stress 862 MPa.
Fig. 6. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the A-286 (I) alloy tested at 650°C and peak stress 510 MPa.
It can be seen from Figs. 4–6 that the fractal dimension D increases with the increase of fracture strain of. In Fig. 7, the of is little changed with cyclic period, and the D keeps almost constant. Nevertheless, the D is not correlated with the fracture time tf. One can see from Fig. 4 that although the tf keeps nearly constant, the D decrease monotonously with cyclic period; while from Fig. 7 that when the tf decrease monotonously, the D keeps nearly constant. Based on the data in Figs. 4–6, the fractal dimension D versus fracture strain of are plotted in Figs. 8–10, respectively. One can see that the D nearly linearly increases with the increase of of in all the three cases. Since both of and D are almost constant in Fig. 7, no such concurrent increase can be plotted.
4. Discussion
Fig. 4. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa.
It has been shown in last section that, the factal dimension D of fractured surfaces is closely related to the fracture strain of for the four series of cyclic creep tests of the Nimonic 75, A-286 (I) and A-286 (II) alloys, and the D increases nearly linearly with the increase of of.
X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
The phenomenon that the fractal dimension increases with increasing fracture strain was also observed by other scientists for a variety of fractures, such as after tensile deformation of a low alloy steel [13], after hot-working of a Cu – Zn – Al alloy [14], and after superplastic deformation of an Al alloy [15]. Especially, Yi and his co-workers reported a linear increase of fractal dimension of cyclic creep fractured surfaces with increasing fracture strain in a low alloy steel [12], similar to the observations in this study. It seems that the concurrent increase of D and of has its somewhat generality. The concurrent increase of D and symbol of can be qualitatively explained as follows: a larger amount of deformation results in a larger extent of irregularity of fractured surfaces, and hence a higher fractal dimension. The more exact explanation for this correlation is to be further investigated. It has been described in last section that a linear fractal curve and constant D exists only in a limited range of measuring unit, and the upper limits of the linear ranges of measuring unit are below respective average grain sizes. This limitation and the occurrence of intergranular fracture indicate that the fractal dimen-
253
Fig. 9. The fractal dimension D vs. the fracture strain of in the A-286 (I) alloy tested at 500°C and peak stress 862 MPa.
Fig. 10. The fractal dimension D vs. the fracture strain of in the A-286 (I) alloy tested at 650°C and peak stress 510 MPa.
sion estimated in this study is an indication of the irregularity of fractured surfaces caused by the propagation and linkage of intergranular cracks and the distortion near grain boundaries. Fig. 7. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the A-286 (II) alloy tested at 650°C and peak stress 510 MPa.
Fig. 8. The fractal dimension D vs. the fracture strain of in the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa.
5. Summary and conclusions Four series of cyclic creep tests at constant temperature and constant peak stress, but with different cyclic periods have been conducted with two high temperature alloys (i.e. Nimonic 75 type and A-286 type). All the fractures are of intergranular type. The fractal dimension D of the fractured surfaces has been measured by the secondary electron line scanning method. It is found that when the fracture strain varies with the variation of cyclic period, the fractal dimension D increases with the increase of fracture strain (Figs. 4–6), and nearly linearly with the increase of fracture strain (Figs. 8–10); while the fracture strain does not vary markedly with the variation of cyclic period, the fractal dimension almost keeps constant (Fig. 7). It is observed that the linear fractal curve and constant fractal dimension exist only in a limited range of
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X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
measuring unit, i.e. the character of ‘self-similarity’ of the profile of fractured surfaces is only valid in the linear range of measuring unit. The upper limits of the linear range of measuring unit are below respective grain sizes. Hence we suggest that the fractal dimension estimated in this study is an indication of the irregularity of fractured surfaces caused by the propagation and linkage of intergranular cracks and the distortion near grain boundaries.
Acknowledgements This project has been supported by the National Natural Science Foundation of China under Grant 59671036. The authors are grateful to Professor Z.Q. Wu, University of Science and Technology of China, for his help in fractal analysis.
References [1] B.B. Mandelbrot, D.E. Passoja, A.J. Paullay, Nature 308 (1984) 721.
.
[2] E.E. Underwood, K. Banerji, Mater. Sci. Eng. 80 (1986) 1. [3] C.S. Pande, L.E. Richards, N. Louat, B.D. Dempsey, A.J. Schwoeble, Acta Metall. 35 (1987) 1633. [4] C.S. Pande, L.R. Richards, S. Smith, J. Mater. Sci. Lett. 6 (1987) 295. [5] R.H. Dauskardt, F. Haubensak, R.O. Ritchie, Acta Metall. Mater. 38 (1990) 143. [6] Z.H. Huang, J.F. Tian, Z.G. Wang, Mater. Sci. Eng. A118 (1989) 19. [7] Z.H. Huang, J.F. Tian, Z.G. Wang, Mater. Sci. Eng. A129 (1990) 1. [8] Z.H. Huang, J.F. Tian, Z.G. Wang, Scripta Metall. Mater. 24 (1990) 967. [9] C.W. Lung: in L. Pietronero, E. Tosatti (Eds.), Fractals in Physics, North-Holland, Amsterdam, 1986, p.187. [10] L.K. Dong, X.W. Wang, Acta Metall. Sin. 26 (1990) 145. [11] W.S. Lei, B.S. Chen, Acta Metall. Sin. 30 (1994) 269. [12] J.Z. Yi, Z.A. Yang, S.J. Zhu, Z.G. Wang, Acta Metall. Sin. 28 (1992) 530. [13] H. Su, Y.G. Zhang, Z.Q. Yan, Mater. Sci. Prog. 4 (1990) 339. [14] E. Hombogen, Z. Metallkde 78 (1987) 622. [15] J. Xinggang, C. Jianzhong, M. Longxiang, J. Mater. Sci. Lett. 12 (1993) 1616. [16] X. Wang, H. Zhou, Q.H. Ni, Q.P. Kong, N.B. Zhou, Mater. Sci. Eng. A122 (1989) 9. [17] X. Wang, Q.H. Ni, H. Zhou, Q.P. Kong, Mater. Sci. Eng. A123 (1990) 207. [18] D.N. Wang, X. Wang, Q.P. Kong, Mater. Sci. Eng. A124 (1991) 157.
Fractal analysis of cyclic creep fractured surfaces of two high temperature alloys X. Wang *, H. Zhou, Z.H. Wang, M.S. Tian, Y.S. Liu, Q.P. Kong Institute of Solid State Physics, Academia Sinica, Hefei230031, PR China Received 25 September 1998; received in revised form 10 November 1998
Abstract The fractal dimensions of cyclic creep fractured surfaces of two high-temperature alloys (Nimonic 75 and A-286 types) have been measured by using secondary electron line scanning method. All the fractures are intergranular. It is found that the fractal dimensions nearly linearly increase with the increase in fracture strain. The upper limits of measuring units for the linear range of fractal curve are below respective grain sizes. The results indicate that a larger amount of deformation results in a larger extent of irregularity of fractured surfaces caused by the propagation and linkage of intergranular cracks and the distortion near grain boundaries, and hence a higher fractal dimension. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Cyclic creep; Fractal dimension; Fractured surfaces
1. Introduction Fractal analysis is an effective tool to get useful information on the process of fracture in materials [1 –8]. It is known that there are several available methods for measuring the fractal dimension [4,5], such as: (a) the slit island method; (b) the vertical section method, and; (c) the secondary electron line scanning method. For the first two methods, the fractured samples should be ground and polished along the horizontal or vertical direction of fractured surfaces. For the third method, the secondary electron scanning line from a fractured surface in scanning electron microscopy (SEM) is used to express the profile roughness. This method does not need to grind and polish the fractured sample and is more convenient. Pande, Richards and Smith [4] first employed the secondary electron line scanning method to measure the fractal dimension of fractured surfaces. They adopted the above mentioned three methods to a titanium alloy after tensile tests, and found that the fractal dimensions measured by the three methods were reasonably consistent with each other. Using the three methods to the * Corresponding author. Tel.: +86-551-5591427; fax: + 86-5515591434.
impact fractured surfaces of a CK45 steel, Huang, Tian and Wang [6–8] observed that the fractal dimensions measured by different methods increased with impact toughness in a similar manner. They found for the same fractured surfaces that, the fractal dimension obtained by secondary electron line scanning method was a little higher than that obtained by vertical section method, but a little lower than that obtained by the slit island method; the relative deviation among the absolute values obtained by different methods was about 10%. There has been numerous fractal analyses on fractured surfaces in the literature. Nevertheless, the relation between fractal dimension and material property is not clear enough. Different scientists connect the fractal dimension to different mechanical properties, e.g. impact energy [1], dynamic tear energy [3], critical crack extension force [9–11], fracture strain [12–15] and so forth. The present paper is aimed to make a fractal analysis of cyclic creep fractured surfaces of two high temperature alloys by using the secondary electron line scanning method. Based on four series of cyclic creep tests with different cyclic periods, a correlation between the fractal dimension and the fracture strain has been observed. The experimental procedure and results are reported below.
0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 9 8 ) 0 1 1 2 7 - 7
X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
2. Materials and experimental procedures The samples were made of two types of high temperature alloys. One is a Nimonic 75 type alloy with main composition (wt.%): 20.13 Cr, 0.064 C and the balance nickel. Another is an A-286 type superalloy, of which two heats I and II were used. The main composition of A-286 (I) is (wt.%): 27.04 Ni, 14.08 Cr, 2.05 Ti, 0.25 Al, 1.34 Mo, 0.33 V, 0.059 C and the balance iron. The main composition of A-286 (II) is (wt.%): 26.29 Ni, 14.19 Cr, 2.18 Ti, 0.17 Al, 1.28 Mo, 0.28 V, 0.034 C and the balance iron. After solid solution and aging treatments, the average grain sizes of Nimonic 75, A-286 (I) and A-286 (II) are 100 ,75 and 36 mm, respectively. The load-controlled cyclic creep tests were performed on an Instron 1362 machine especially designed for combined creep-fatigue testing. The waveform of stress as a function of time is of trapezoidal shape, the details were described in [16 – 18]. Each series of cyclic creep was tested at a constant temperature and a constant peak stress, but with different hold times at peak stress. The duration of loading and unloading in each cycle was 1 min in total. The six different hold times at peak stress were 0 (pure fatigue), 0.33, 1, 3, 15 min and (pure creep). That is, the six different cyclic periods for each cycle were respectively 1 (pure fatigue), 1.33, 2, 4, 16 min and (pure creep). The fractured surfaces after ultrasonic cleaning were line-scanned in a scanning electron microscope of Amary-1000B type. For each fractured surface, five scanning lines with equal separation were taken at a fixed magnification along each of two perpendicular directions (i.e. totally 5×2 = 10 scanning lines). During the whole scanning process, the incident electron beam current and diameter, the contrast and the tilt angle of the specimen to the electron beam were kept constant to ensure a comparable basis among the scanning lines from fractured surfaces. The micrographs of the scanning lines were digitized with a fractal image processing system (FIPS), and the length L(h) of each scanning line was measured as a function of measuring unit h. The fractal dimension D was evaluated by means of the following equation [2]: L(h)= L0h 1 − D
251
ported in next section is the average of the data evaluated from all the scanning lines of the same fractured surface. The mean square deviation of the reported D data is within 91%.
3. Experimental results Four series of cyclic creep tests at constant temperature and constant peak stress, but with different cyclic periods were conducted with the two type alloys. All the cyclic creep fractures are of intergranular type. Fig. 1 shows SEM photograghs of cyclic creep fractured surfaces of (a) the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa with cyclic period 1.33 min, and (b) the A-286 (I) alloy tested at 650°C and peak stress 510 MPa with cyclic period 2 min, respectively. Fig. 2 is an example of a secondary electron scanning line taken from a cyclic creep fractured surface. Fig. 3 shows the double logarithms of the length L(h) of a scanning line and the reciprocal of measuring unit (1/h). One can see that there is a linear range in the middle portion of the curve. The fractal dimension D is then evaluated from the slope of the straight line according to Eq. (2). The measuring units of the linear range for the Nimonic 75, A-286 (I) and A-286 (II) alloys are 4–22,
(1)
where L0 is the projected length of the scanning line. Take logarithm of Eq. (1), we have:
ln L(h)=C+ (D −1)ln
1 h
(2)
where C is a constant. Through linear regression for the nearly straight central portion of the curve ln L(h) ln (1/h), the fractal dimension D was obtained from the slope. The correlation coefficients for all the linear regression are higher than 0.99. The value of D re-
Fig. 1. SEM photographs of cyclic creep fractured surfaces of (a) the Nimonic 75 alloy, and (b) the A-286 (I) alloy.
252
X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
Fig. 2. An example of a secondary electron scanning line taken from a cyclic creep fractured surface.
Fig. 3. Double logarithmic plot of length L(h) and the reciprocal of the measuring unit (1/h) for a scanning line.
8 –75 and 5–32 mm, respectively. The upper limits are below respective average grain sizes. The implication of the limited linear range of measuring unit will be discussed in Section 4. Figs. 4–7 show the fracture time tf, fracture strain of and fractal dimension D versus the cyclic period for: the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa; the A-286 (I) alloy tested at 500°C and peak stress 862 MPa; the A-286 (I) alloy tested at 650°C and peak stress 510 MPa, and; the A-286 (II) alloy tested at 650°C and peak stress 510 MPa, respectively.
Fig. 5. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the A-286 (I) alloy tested at 500°C and peak stress 862 MPa.
Fig. 6. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the A-286 (I) alloy tested at 650°C and peak stress 510 MPa.
It can be seen from Figs. 4–6 that the fractal dimension D increases with the increase of fracture strain of. In Fig. 7, the of is little changed with cyclic period, and the D keeps almost constant. Nevertheless, the D is not correlated with the fracture time tf. One can see from Fig. 4 that although the tf keeps nearly constant, the D decrease monotonously with cyclic period; while from Fig. 7 that when the tf decrease monotonously, the D keeps nearly constant. Based on the data in Figs. 4–6, the fractal dimension D versus fracture strain of are plotted in Figs. 8–10, respectively. One can see that the D nearly linearly increases with the increase of of in all the three cases. Since both of and D are almost constant in Fig. 7, no such concurrent increase can be plotted.
4. Discussion
Fig. 4. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa.
It has been shown in last section that, the factal dimension D of fractured surfaces is closely related to the fracture strain of for the four series of cyclic creep tests of the Nimonic 75, A-286 (I) and A-286 (II) alloys, and the D increases nearly linearly with the increase of of.
X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
The phenomenon that the fractal dimension increases with increasing fracture strain was also observed by other scientists for a variety of fractures, such as after tensile deformation of a low alloy steel [13], after hot-working of a Cu – Zn – Al alloy [14], and after superplastic deformation of an Al alloy [15]. Especially, Yi and his co-workers reported a linear increase of fractal dimension of cyclic creep fractured surfaces with increasing fracture strain in a low alloy steel [12], similar to the observations in this study. It seems that the concurrent increase of D and of has its somewhat generality. The concurrent increase of D and symbol of can be qualitatively explained as follows: a larger amount of deformation results in a larger extent of irregularity of fractured surfaces, and hence a higher fractal dimension. The more exact explanation for this correlation is to be further investigated. It has been described in last section that a linear fractal curve and constant D exists only in a limited range of measuring unit, and the upper limits of the linear ranges of measuring unit are below respective average grain sizes. This limitation and the occurrence of intergranular fracture indicate that the fractal dimen-
253
Fig. 9. The fractal dimension D vs. the fracture strain of in the A-286 (I) alloy tested at 500°C and peak stress 862 MPa.
Fig. 10. The fractal dimension D vs. the fracture strain of in the A-286 (I) alloy tested at 650°C and peak stress 510 MPa.
sion estimated in this study is an indication of the irregularity of fractured surfaces caused by the propagation and linkage of intergranular cracks and the distortion near grain boundaries. Fig. 7. Fracture time tf, fracture strain of and fractal dimension D vs. the cyclic period for the A-286 (II) alloy tested at 650°C and peak stress 510 MPa.
Fig. 8. The fractal dimension D vs. the fracture strain of in the Nimonic 75 alloy tested at 600°C and peak stress 392 MPa.
5. Summary and conclusions Four series of cyclic creep tests at constant temperature and constant peak stress, but with different cyclic periods have been conducted with two high temperature alloys (i.e. Nimonic 75 type and A-286 type). All the fractures are of intergranular type. The fractal dimension D of the fractured surfaces has been measured by the secondary electron line scanning method. It is found that when the fracture strain varies with the variation of cyclic period, the fractal dimension D increases with the increase of fracture strain (Figs. 4–6), and nearly linearly with the increase of fracture strain (Figs. 8–10); while the fracture strain does not vary markedly with the variation of cyclic period, the fractal dimension almost keeps constant (Fig. 7). It is observed that the linear fractal curve and constant fractal dimension exist only in a limited range of
254
X. Wang et al. / Materials Science and Engineering A266 (1999) 250–254
measuring unit, i.e. the character of ‘self-similarity’ of the profile of fractured surfaces is only valid in the linear range of measuring unit. The upper limits of the linear range of measuring unit are below respective grain sizes. Hence we suggest that the fractal dimension estimated in this study is an indication of the irregularity of fractured surfaces caused by the propagation and linkage of intergranular cracks and the distortion near grain boundaries.
Acknowledgements This project has been supported by the National Natural Science Foundation of China under Grant 59671036. The authors are grateful to Professor Z.Q. Wu, University of Science and Technology of China, for his help in fractal analysis.
References [1] B.B. Mandelbrot, D.E. Passoja, A.J. Paullay, Nature 308 (1984) 721.
.
[2] E.E. Underwood, K. Banerji, Mater. Sci. Eng. 80 (1986) 1. [3] C.S. Pande, L.E. Richards, N. Louat, B.D. Dempsey, A.J. Schwoeble, Acta Metall. 35 (1987) 1633. [4] C.S. Pande, L.R. Richards, S. Smith, J. Mater. Sci. Lett. 6 (1987) 295. [5] R.H. Dauskardt, F. Haubensak, R.O. Ritchie, Acta Metall. Mater. 38 (1990) 143. [6] Z.H. Huang, J.F. Tian, Z.G. Wang, Mater. Sci. Eng. A118 (1989) 19. [7] Z.H. Huang, J.F. Tian, Z.G. Wang, Mater. Sci. Eng. A129 (1990) 1. [8] Z.H. Huang, J.F. Tian, Z.G. Wang, Scripta Metall. Mater. 24 (1990) 967. [9] C.W. Lung: in L. Pietronero, E. Tosatti (Eds.), Fractals in Physics, North-Holland, Amsterdam, 1986, p.187. [10] L.K. Dong, X.W. Wang, Acta Metall. Sin. 26 (1990) 145. [11] W.S. Lei, B.S. Chen, Acta Metall. Sin. 30 (1994) 269. [12] J.Z. Yi, Z.A. Yang, S.J. Zhu, Z.G. Wang, Acta Metall. Sin. 28 (1992) 530. [13] H. Su, Y.G. Zhang, Z.Q. Yan, Mater. Sci. Prog. 4 (1990) 339. [14] E. Hombogen, Z. Metallkde 78 (1987) 622. [15] J. Xinggang, C. Jianzhong, M. Longxiang, J. Mater. Sci. Lett. 12 (1993) 1616. [16] X. Wang, H. Zhou, Q.H. Ni, Q.P. Kong, N.B. Zhou, Mater. Sci. Eng. A122 (1989) 9. [17] X. Wang, Q.H. Ni, H. Zhou, Q.P. Kong, Mater. Sci. Eng. A123 (1990) 207. [18] D.N. Wang, X. Wang, Q.P. Kong, Mater. Sci. Eng. A124 (1991) 157.