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Fatigue crack initiation life of fine grain brass H62
Theoretical and Applied Fracture Mechanics 54 (2010) 105–109
Contents lists available at ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Fatigue crack initiation life of fine grain brass H62 M. Zheng a,⇑, M.X. Tong b, H.P. Cai b, C.Z. Xu b, M.Q. Huang b a b
Inst. for Cond. Matt. Phys. & Mater., Northwest Univ., 710069 Xi’an, China School of Mater. Sci. & Eng., Xi’an Jiaotong Univ., 710049 Xi’an, China
a r t i c l e
i n f o
Article history: Available online 13 October 2010 Keywords: Fine grain brass Size effect Fatigue Inter-grain cracking Energetic approach Crack initiation
a b s t r a c t Fine grain alloys possess excellent properties entailing high strength and toughness. Fine brass H62 is made by re-crystallization with grain size ranging from 5 to 10 lm. Fatigue initiation life is investigated from specimens tested on Instron 1341 machine a frequency of 30 Hz. Furthermore fatigue crack initiation life of this fine brass H62 is predicted by the energetic approach. It is found that: (1) the finer the grain size, the longer the fatigue life, which is due to its higher toughness; (2) intergranular cracking is the main mechanism of fatigue failure. Concave pits were found in the zone of fatigue crack propagation; and (3) the energetic approach gave acceptable fatigue crack initiation life estimation. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Fine-grained materials (FGM) have received much attention with reference to their high static strength and details of fine microstructure. Relatively little is known with regard to their fatigue properties and microstructure change during cyclic loading. Reduction of grain size can increase the strength and ductility at the same time, and hence enhancing the formability. The corrosion resistance is also increased as a results of the homogenization of the distribution for impurities in the grain boundary. The super plasticity has been found in fine grain Al-1420 (Al– 5.5Mg–2.2Li–0.12Zr), the ductility reaches even to 950% at rate of 1 s1 and high temperature. The S–N curves shows that fatigue life is higher for fine grain Cu (0.2 lm) than those with coarse grains (35 lm) [1]. However, the fatigue strength of fine grain Al–Mg alloys (0.2 lm) remains almost the same as those of the coarse grains Al–Mg alloys (35 lm), while its fatigue life in high stress region increases [1]. The fatigue limit of fine grain Ti (0.3 lm) increases to 380 MPa, while that of the coarse grain (35 lm) is 238 MPa [1]. All these results indicate a complex behavior of fine grain alloy. The procedure for producing fine grain materials includes re-crystallization, equal-channel angular pressing (ECAP), gas condensation, ball milling, torsion straining under high pressure, etc. The weakest link of an engineering structure determines the life of the structure. The fatigue life of a component can be divided into crack initiation (FCI) life and crack propagation (FCP) [2–11]. There are more than 50 models of cumulative fatigue damage and life ⇑ Corresponding author. E-mail address: [email protected] (M. Zheng). 0167-8442/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2010.10.005
prediction, all of which can be further improved [12]. Extensive empirical studies [13–16] have been made [13–16] in the low cycle fatigue (LCF) regime on the basis of the Manson–Coffin relation given by
Dep Naf ¼ C;
ð1Þ
in which Dep and Nf are the plastic strain range and the number of cycles to failure, respectively; a and C are the material constants. Eq. (1) represents a linear relation on the log–log coordinates of the Dep and Nf. It is worthwhile to say that there is no fatigue limit in above relation. It was reported in [17,18] that for high strength and low-ductility materials failure occurred at strain cycles less than those expected from the Manson–Coffin relation for the small level of Dep. In order to elucidate the reduction in fatigue life, the following modified Manson–Coffin relation applies to local plastic deformation in the low Dep regime [17,18]:
Dep Na ¼ C: 1 eaep f
ð2Þ
Eq. (2), however, makes no precise reference to the size scale of material damage except for the reference of plastic strain range Dep> The recent works in [19,20] proposed a dual scale micro/macro line crack model that consists of three essential parameters d (micro/macro length), l (micro/macro material constants) and r (external/internal stress ratios). The model can accommodate two incidental variables (u1, u2) derived for the double singularity solution for micro-cracks to account for specific micro-structural behavior. More complex models involving more than two incidental variables can also be developed if the need arises. Otherwise, the function with (u1, u2) can be absorbed into one of the empirical
106
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109
parameters used for correlating the da/dN relation. That is the microscopic effects are reflected indirectly although not sensitively by the macroscopic parameter. This approach established for dual scaling applies equally well to micro/nano cracks such that when connected with the macro/micro-cracks it can be further extended to the development of triple scale models involving macro/micro/ nano cracks [19,20]. A local strain fracture model [2,21] has also been proposed to predict fatigue crack initiation life empirically, the fatigue limit was taken into consideration, and reasonable results were obtained. Moreover, energetic based modification to the local strain fracture model has been used [22] to account for coarse grain 16Mn steels and LY 12 CZ aluminum alloys at room and low temperatures. However, the applicability of such energetic approach to other materials is lacking. In what follows, the re-crystallization technique is employed to make fine grain Cu alloys H62 for improving the fatigue crack initiation life. Fig. 1. Optical replica for 1st group sample.
2. Fine grain Brass H62 alloy and mechanical property The copper alloy employed in this research is Brass, H62 (in Chinese standard), of which the chemical composition is in Table 1. Re-crystallization technique is employed to prepare fine grain Cu alloys. Two rolling processes, three heat treatment procedures are applied to the commercial copper alloy to obtain fine-grained samples. The initial thickness of the plank of copper alloy is 25 mm. After 1st annealing the plank is rolled to the thickness of about 11 mm, then a 2nd annealing is applied. Thereafter 3rd rolling is employed to get a sheet with the thickness of 2–3 mm followed by a 3rd heat treatment. The time for each heat treatment is about 50 min, which is followed by air-cooling. Figs. 1–3 show the optical microscopic photographs of samples. The average grain sizes of the 1st, 2nd, and 3rd group sheets are about 7 lm, 10 lm, and 5 lm, respectively. The heat treatment and processing for each group of samples are shown in Table 2. The typical stress–strain curve is shown in Fig. 4, and the corresponding mechanical properties are listed in Table 3, respectively. 3. Energetic approach for fatigue crack initiation life prediction
Fig. 2. Optical replica for 2nd group sample.
3.1. Local strain model In most engineering applications, the nominal stress is kept below the yield stress of the metal, while the structural member, as a whole, is elastic. Local plastic deformation is limited to the notch root in Fig. 5 [9]. This is sometimes known as small scale yielding where the applied stress amplitude is limited to 50% of the yield strength according to the ASTM requirement. A fatigue endurance limit strain can also be used where a threshold for the hypothetical element is used such that no fracture occurs if the cyclic strain/ deformation is smaller than the limit strain. This equivalent to defining unlimited fatigue life greater than 107–108. Moreover, the applied strain range DeP could be divided into two parts, i.e., the critical part DeC and the fatigue damage part DeD such that DeP = DeC + DeD. Local cyclic plastic deformation is responsible for fatigue crack formation in despite of micro-crack occurring at gain boundary or 2nd phase grain breaking. The fatigue life of a smooth testing
Table 1 Chemical composition of copper alloy H62 (wt.%).
Fig. 3. Optical replica for 3rd group sample.
Cu
Fe
Pb
Sb
Bi
P
Zn
60.5–63.5
0.15
0.08
0.005
0.002
0.01
Balance
specimen can be taken as the FCI life of a notched element if the fatigue element at the notch root undergoes the same stress–strain
107
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109 Table 2 Heat treatment and processing for each group. Group No.
1
2
3
Thickness (mm) Annealing 1 Thickness after rolling 1 (mm) Annealing 2 Thickness after rolling 2 (mm) Annealing 3
25 400 °C 1 h 11 450 °C 1 h 2 420 °C 50 min + 430 °C 60 min
25 400 °C 1 h 11 450 °C 1 h 2 380 °C 50 min + 460 °C 45 min
25 400 °C 1 h 11 450 °C 1 h 2 440 °C 50 min
Fig. 5. Schematic illustration of the plastic zone and the hypothetical fatigue element at the notch root [9].
Fig. 4. Stress–strain curve for tensile specimen 1–4.
3.2. Formulation of fatigue crack initiation life
Table 3 Mechanical properties of copper alloy sheet H62. Specimen no. 1–1 1–2 1–3 1–4 2–1 2–2 2–3 2–4 3–1 3–2 3–3 3–4 Specimen no. 1–1 1–2 1–3 1–4 2–1 2–2 2–3 2–4 3–1 3–2 3–3 3–4
A0 (mm2)
A (mm2)
27.8 28.2 28.4 27.8 29.5 29.2 28.9 29.5 28.0 28.1 28.6 27.5
r0.2 (MPa)
230 240 230 241 230 270 250 240 240
13.6 13.6 13.1 13.6 14.2 13.0 13.8 13.5 13.2 13.0 12.0 13.1
ef
w 0.512 0.518 0.539 0.512 0.519 0.556 0.524 0.543 0.528 0.538 0.581 0.524
0.718 0.730 0.775 0.717 0.732 0.812 0.742 0.784 0.751 0.772 0.870 0.741
rb (MPa)
rf (MPa)
HB
416.6 417.6 417.9 420.5 413.1 413.7 422.4 407.8 461.0 436.6 425.8 430.5
763.3 768.5 790.1 775.0 789.2 816.6 776.7 798.8 867.6 826.9 884.7 806.7
39.6 39.4 39.6 38.9 38.9 41.5 34.3 38.2 37.7 38.6 38.5 39.6
h i2 Ni ¼ C ðDr Þ2=ð1þnÞ ðDrc Þ2=ð1þnÞ ; 2=ð1þnÞ 2 f ðEKÞ
C ¼ 0:25e
h
½2=ð1 þ nÞ
2=ð1þnÞ
; i0:5 ; 2ð1þnÞ ð1 RÞð1nÞ
Dr ¼ K t Dr0 h i0:5 ; Drc ¼ K t Drc 2ð1þnÞ ð1 RÞð1nÞ
ð4Þ ð5Þ ð6Þ ð7Þ
Drc ¼ 2r1 E ef =103:5 ; corresponding to fatigue limit N ¼ 107 ; n
r ¼ Ke :
history as the metal at the notch tip. Many investigators used this hypothesis for the FCI life of notched elements. In this vein, the local strain range should be correlated to the nominal cyclic stress applied to the notched elements. It follows that the Manson–Coffin formula can be modified for predicting fatigue crack initiation life [2,21] as
DeD ¼ DeP DeC ¼ aN bf ; b 0:5;
In the energetic approach [22], the stress range Dr0, the endurance limit Drc, the equivalent stress range Dr, the equivalent endurance limit Drc , the facture ductility ef, the elastic modulus E, the stress ratio R, the stress concentration factor Kt, the workhardening coefficient K and the work-hardening exponent n are all employed for FCI life Ni prediction
ð3Þ
This was employed in [2,21] by application of a critical local strain or fatigue endurance limit.
ð8Þ ð9Þ
The notations used in the above equation are Dr0 is the stress range; Drc, the endurance limit; Dr, the equivalent stress range; Drc , the equivalent endurance limit; ef, the facture ductility; E, the elastic modulus; R, the stress ratio; Kt, the stress concentration factor; K, the work-hardening coefficient; and n is the work-hardening exponent. It may be worthwhile to note that C, the FCI resistance coefficient, may be considered to be the equivalent stress amplitude when Ni = 1/4 cycle, equivalent to the crack initiation at the notch root during monotonic tensile test. The FCI threshold, Drc, is the upper limit of the equivalent stress amplitude, below or equal to which no fatigue damage to occur, and FCI life tends to be infinite correspondingly. Both C and Drc are material constants, and can be predicted from the tensile properties. So long as the values of C and Drc are given, FCI life can be obtained from above expression. This energetic approach has been employed to prognosticate the fatigue
108
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109 Table 4 Comparison of the fitted C and Drcpred with the predictions (R = 0.1)*. Group 1
Group 2 11
Group 3 11
Cfit Cpred Drcfit (MPa) Dr0cpred: (MPa)
3.11 10 2.70 1011 226.44 264.83
2.59 10 3.05 1011 229.30 260.85
4.31 1011 5.66 1011 220.55 277.19
Dr00cpred: (MPa)
225.34
221.85
235.82
*
Fig. 6. Schematics of fatigue specimen.
The data in Table 3 and r1 = 0.35 rb are employed for Dr0cpred: estimation; The data in Table 3 and r1 = 0.30 rb are employed for Dr00cpred: estimation.
Fig. 7. S–N curve for 1st group materials.
Fig. 10. Comparison of fatigue fracture surfaces for specimens 1–1 (upper) and 1–2 (lower).
Fig. 8. S–N curve for 2nd group materials.
Fig. 11. Comparison of fatigue fracture surfaces for specimens 2–1 (upper) and 2–8 (lower).
Fig. 9. S–N curve for 3rd group materials.
crack initiation lives of 16 Mn steels and LY12CZ aluminum alloys with coarse grain [22], satisfied results are obtained. 4. Fatigue behavior of Brass H62 The specimen for fatigue test is machined from the three groups of materials with the geometry shown in Fig. 6. The material is prepared by the above mentioned re-crystallization procedure in the previous paragraph. Instron 1341 fatigue machine was employed to conduct the fatigue tests. Crack initiation is defined by the
Fig. 12. Comparison of fatigue fracture surfaces for specimens 3–1 (upper) and 3–7 (lower).
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109
The energetic approach for fatigue crack initiation life assessment could be employed to predict the fatigue crack initiation life of fine grain Cu alloys; by analyzing the fitted results of the fatigue data, a modified empirical correlation between r1 and rb is proposed for fine grain copper alloy, i.e., r1 = 0.30 rb.
Table 5 Fatigue loads for some specimens (R = 0.1). Specimen no.
Maximum fatigue load (MPa)
1–1 1–2 2–1 2–8 3–1 3–7
250.90 292.72 331.54 290.10 301.95 325.68
Acknowledgement The author would like to show their thanks to Prof. G.C. Sih for his encouragement and great help in preparing this paper, ChineseCzech bilateral project entitled ‘‘fatigue behavior of ultra-fine grain Cu and Mg Alloys” is acknowledged as well.
appearance of crack with the length of millimeter. S–N curves are drawn in Figs. 7–9 for the three group materials, respectively. The S–N curve can be fitted by the energetic approach for fatigue crack initiation prediction proposed in [22]. The comparison of the fitted results of C for above three group materials and those of the theoretical prediction are listed in Table 4. It can be seen that the fitted fatigue resistant factor Cfit agrees with those of the predictions excellently. The fatigue endurance limit (stress) Dr0cpred: is predicted with Eq. (8) and the empirical correlation r1 = 0.35 rb [2] for copper alloy as well as the data in Table 3. It can be seen from Table 4 that Dr0cpred: is greater than Drcfit, which is due to the employing of the empirical formula r1 = 0.35 rb for r1 estimation for this fine grain copper alloy in such prediction. In order to reduce the difference between Dr0cpred: and Drcfit, the empirical relation between r1 with rb is modified as Eq. (10) for fine grain Cu alloy, and a new prediction for fatigue the endurance limit (stress), says Dr00cpred: is performed with the relevant processes, of which the result is shown in Table 4 as well. It can be seen that Dr00cpred: is close to Drcfit.
r1 ¼ 0:30 rb :
109
ð10Þ
5. Characteristic of fatigue fracture surface Figs. 10–12 show the characteristics of fatigue fracture surface. From these figures it can be seen that the characteristic of fatigue fracture surface depends on the material properties and fatigue load. Table 5 shows the fatigue load for some specimens. The size of fatigue process region decreases with fatigue load in each comparative group. The size of fatigue process region varies with the toughness of the materials from group to group for similar load level. Intergranular cracking is the main mechanism in the fatigue source zones. Concave pits exist in the latter fatigue propagation zones. 6. Concluding remarks Finer-grained H62 has longer fatigue life, which is due to its higher toughness. The characteristic of fatigue fracture surface depends on the material properties and fatigue load. The size of fatigue process region decreases with fatigue load in each comparative group. The size of fatigue process region varies with the toughness of the materials from group to group for similar load level. Intergrain cracking is the main mechanism in the fatigue source zones. Concave pits exist in the latter fatigue propagation zones.
References [1] A. Vinogradov, S. Hashimoto, Multi-scale phenomena in fatigue of ultra-fine grain materials, an overview, Materials Transactions (2001) 42. [2] X.L. Zheng, Quantitative Theory of Metal Fatigue, Northwestern Polytechnic University (NPU) Press, Xi’an, China, 1994 (in Chinese). [3] M.P. Zhao, Handbook of Strain Fatigue Analysis, Science Press, Beijing, China, 1988 (in Chinese). [4] Y.H. Kim, S.M. Speaker, D.E. Gordon, S.D. Manning, R.P. Wei, Development of fatigue and crack propagation design and analysis methodology in a corrosive environment for typical mechanically-fastened joints, in: Report No. NADC83126-60, vol. I, 1983. (AD-A136414). [5] Y.H. Kim, S.M. Speaker, D.E. Gordon, S.D. Manning, R.P. Wei, Development of fatigue and crack propagation design and analysis methodology in a corrosive environment for typical mechanically-fastened joints (assessment of art state), in: Report No. NADC-83126-60, vol. 2, 1983. (AD - A136415). [6] R.J.H. Wanhill, J.J. De Luccia, M.T. Russo, The fatigue in aircraft corrosion testing (FACT) program, in: AGARD Report No. 713, 1989. (AD-A208-359). [7] A. Buch, Prediction of fatigue life under aircraft loading with and without use of material memory rules, International Journal of Fatigue 11 (2) (1989) 97– 106. [8] D. Dupart, A. Davy, R. Boetsch, R. Boudet, Fatigue damage calculation in stress concentration fields under uniaxial stress, International Journal of Fatigue 18 (4) (1996) 245–253. [9] W. Schutz, The prediction of fatigue life in the crack initiation and propagation stages – a state of the art survey, Engineering Fracture Mechanics 11 (3) (1979) 405–421. [10] P.J. Laz, B.A. Craig, S.M. Rohrbaugh, B.M. Hillberry, The development of a total fatigue life approach accounting for nucleation and propagation, in: Proceedings of the 7th International Fatigue Congress, FATIGUE ’99, Beijing, June 1999, High Educational Press, EMAS, 1999, pp. 833–838. [11] H. Nowack, Development in variable amplitude prediction methods for light weight structures, in: Proceedings of the 7th International Fatigue Congress, FATIGUE ’99, Beijing, June 1999, High Educational Press, EMAS, 1999, pp. 991– 1000. [12] A. Fatemi, L. Yang, International Journal of Fatigue 20 (1998) 9–34. [13] S.S. Manson, Behavior of materials under thermal stress, NACATN 2933 (1953). [14] S.S. Manson, Discussion, Transactions of the ASME 84 (1962) 537–541. [15] L.F. Coffin Jr., A study of the effects of cyclic thermal stresses on a ductile metal, Transactions of the ASME 76 (1954) 931–950. [16] J.F. Tavernelli, L.F. Coffin Jr., Experimental support for generalized equation predicting low cycle fatigue, Transactions of the ASME 84 (1962) 533–537. [17] K. Hatanaka, T. Fujimitsu, Transactions of the Japan Society for Mechanical Engineering A50 (1984) 291–300 (in Japanese). [18] K. Hatanaka, Transactions of the Japan Society for Mechanical Engineering A50 (1984) 831–838 (in Japanese). [19] G.C. Sih, X.S. Tang, Micro/macro-crack growth due to creep-fatigue dependency on time-temperature material behavior, Theoretical and Applied Fracture Mechanics 50 (2008) 9–22. [20] G.C. Sih, X.S. Tang, Form-invariant representation of fatigue crack growth rate enabling linearization of multiscale data, Theoretical and Applied Fracture Mechanics 47 (2007) 1–14. [21] X. Zheng, On some basic problems of fatigue research in engineering, International Journal of Fatigue 23 (2001) 751–766. [22] M. Zheng, E. Niemi, X. Zheng, An energetic approach for predict fatigue crack initiation life of LY 12 CZ aluminum and 16 Mn steel, Theoretical and Applied Fracture Mechanics 26 (1997) 23–28.
Contents lists available at ScienceDirect
Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec
Fatigue crack initiation life of fine grain brass H62 M. Zheng a,⇑, M.X. Tong b, H.P. Cai b, C.Z. Xu b, M.Q. Huang b a b
Inst. for Cond. Matt. Phys. & Mater., Northwest Univ., 710069 Xi’an, China School of Mater. Sci. & Eng., Xi’an Jiaotong Univ., 710049 Xi’an, China
a r t i c l e
i n f o
Article history: Available online 13 October 2010 Keywords: Fine grain brass Size effect Fatigue Inter-grain cracking Energetic approach Crack initiation
a b s t r a c t Fine grain alloys possess excellent properties entailing high strength and toughness. Fine brass H62 is made by re-crystallization with grain size ranging from 5 to 10 lm. Fatigue initiation life is investigated from specimens tested on Instron 1341 machine a frequency of 30 Hz. Furthermore fatigue crack initiation life of this fine brass H62 is predicted by the energetic approach. It is found that: (1) the finer the grain size, the longer the fatigue life, which is due to its higher toughness; (2) intergranular cracking is the main mechanism of fatigue failure. Concave pits were found in the zone of fatigue crack propagation; and (3) the energetic approach gave acceptable fatigue crack initiation life estimation. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Fine-grained materials (FGM) have received much attention with reference to their high static strength and details of fine microstructure. Relatively little is known with regard to their fatigue properties and microstructure change during cyclic loading. Reduction of grain size can increase the strength and ductility at the same time, and hence enhancing the formability. The corrosion resistance is also increased as a results of the homogenization of the distribution for impurities in the grain boundary. The super plasticity has been found in fine grain Al-1420 (Al– 5.5Mg–2.2Li–0.12Zr), the ductility reaches even to 950% at rate of 1 s1 and high temperature. The S–N curves shows that fatigue life is higher for fine grain Cu (0.2 lm) than those with coarse grains (35 lm) [1]. However, the fatigue strength of fine grain Al–Mg alloys (0.2 lm) remains almost the same as those of the coarse grains Al–Mg alloys (35 lm), while its fatigue life in high stress region increases [1]. The fatigue limit of fine grain Ti (0.3 lm) increases to 380 MPa, while that of the coarse grain (35 lm) is 238 MPa [1]. All these results indicate a complex behavior of fine grain alloy. The procedure for producing fine grain materials includes re-crystallization, equal-channel angular pressing (ECAP), gas condensation, ball milling, torsion straining under high pressure, etc. The weakest link of an engineering structure determines the life of the structure. The fatigue life of a component can be divided into crack initiation (FCI) life and crack propagation (FCP) [2–11]. There are more than 50 models of cumulative fatigue damage and life ⇑ Corresponding author. E-mail address: [email protected] (M. Zheng). 0167-8442/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2010.10.005
prediction, all of which can be further improved [12]. Extensive empirical studies [13–16] have been made [13–16] in the low cycle fatigue (LCF) regime on the basis of the Manson–Coffin relation given by
Dep Naf ¼ C;
ð1Þ
in which Dep and Nf are the plastic strain range and the number of cycles to failure, respectively; a and C are the material constants. Eq. (1) represents a linear relation on the log–log coordinates of the Dep and Nf. It is worthwhile to say that there is no fatigue limit in above relation. It was reported in [17,18] that for high strength and low-ductility materials failure occurred at strain cycles less than those expected from the Manson–Coffin relation for the small level of Dep. In order to elucidate the reduction in fatigue life, the following modified Manson–Coffin relation applies to local plastic deformation in the low Dep regime [17,18]:
Dep Na ¼ C: 1 eaep f
ð2Þ
Eq. (2), however, makes no precise reference to the size scale of material damage except for the reference of plastic strain range Dep> The recent works in [19,20] proposed a dual scale micro/macro line crack model that consists of three essential parameters d (micro/macro length), l (micro/macro material constants) and r (external/internal stress ratios). The model can accommodate two incidental variables (u1, u2) derived for the double singularity solution for micro-cracks to account for specific micro-structural behavior. More complex models involving more than two incidental variables can also be developed if the need arises. Otherwise, the function with (u1, u2) can be absorbed into one of the empirical
106
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109
parameters used for correlating the da/dN relation. That is the microscopic effects are reflected indirectly although not sensitively by the macroscopic parameter. This approach established for dual scaling applies equally well to micro/nano cracks such that when connected with the macro/micro-cracks it can be further extended to the development of triple scale models involving macro/micro/ nano cracks [19,20]. A local strain fracture model [2,21] has also been proposed to predict fatigue crack initiation life empirically, the fatigue limit was taken into consideration, and reasonable results were obtained. Moreover, energetic based modification to the local strain fracture model has been used [22] to account for coarse grain 16Mn steels and LY 12 CZ aluminum alloys at room and low temperatures. However, the applicability of such energetic approach to other materials is lacking. In what follows, the re-crystallization technique is employed to make fine grain Cu alloys H62 for improving the fatigue crack initiation life. Fig. 1. Optical replica for 1st group sample.
2. Fine grain Brass H62 alloy and mechanical property The copper alloy employed in this research is Brass, H62 (in Chinese standard), of which the chemical composition is in Table 1. Re-crystallization technique is employed to prepare fine grain Cu alloys. Two rolling processes, three heat treatment procedures are applied to the commercial copper alloy to obtain fine-grained samples. The initial thickness of the plank of copper alloy is 25 mm. After 1st annealing the plank is rolled to the thickness of about 11 mm, then a 2nd annealing is applied. Thereafter 3rd rolling is employed to get a sheet with the thickness of 2–3 mm followed by a 3rd heat treatment. The time for each heat treatment is about 50 min, which is followed by air-cooling. Figs. 1–3 show the optical microscopic photographs of samples. The average grain sizes of the 1st, 2nd, and 3rd group sheets are about 7 lm, 10 lm, and 5 lm, respectively. The heat treatment and processing for each group of samples are shown in Table 2. The typical stress–strain curve is shown in Fig. 4, and the corresponding mechanical properties are listed in Table 3, respectively. 3. Energetic approach for fatigue crack initiation life prediction
Fig. 2. Optical replica for 2nd group sample.
3.1. Local strain model In most engineering applications, the nominal stress is kept below the yield stress of the metal, while the structural member, as a whole, is elastic. Local plastic deformation is limited to the notch root in Fig. 5 [9]. This is sometimes known as small scale yielding where the applied stress amplitude is limited to 50% of the yield strength according to the ASTM requirement. A fatigue endurance limit strain can also be used where a threshold for the hypothetical element is used such that no fracture occurs if the cyclic strain/ deformation is smaller than the limit strain. This equivalent to defining unlimited fatigue life greater than 107–108. Moreover, the applied strain range DeP could be divided into two parts, i.e., the critical part DeC and the fatigue damage part DeD such that DeP = DeC + DeD. Local cyclic plastic deformation is responsible for fatigue crack formation in despite of micro-crack occurring at gain boundary or 2nd phase grain breaking. The fatigue life of a smooth testing
Table 1 Chemical composition of copper alloy H62 (wt.%).
Fig. 3. Optical replica for 3rd group sample.
Cu
Fe
Pb
Sb
Bi
P
Zn
60.5–63.5
0.15
0.08
0.005
0.002
0.01
Balance
specimen can be taken as the FCI life of a notched element if the fatigue element at the notch root undergoes the same stress–strain
107
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109 Table 2 Heat treatment and processing for each group. Group No.
1
2
3
Thickness (mm) Annealing 1 Thickness after rolling 1 (mm) Annealing 2 Thickness after rolling 2 (mm) Annealing 3
25 400 °C 1 h 11 450 °C 1 h 2 420 °C 50 min + 430 °C 60 min
25 400 °C 1 h 11 450 °C 1 h 2 380 °C 50 min + 460 °C 45 min
25 400 °C 1 h 11 450 °C 1 h 2 440 °C 50 min
Fig. 5. Schematic illustration of the plastic zone and the hypothetical fatigue element at the notch root [9].
Fig. 4. Stress–strain curve for tensile specimen 1–4.
3.2. Formulation of fatigue crack initiation life
Table 3 Mechanical properties of copper alloy sheet H62. Specimen no. 1–1 1–2 1–3 1–4 2–1 2–2 2–3 2–4 3–1 3–2 3–3 3–4 Specimen no. 1–1 1–2 1–3 1–4 2–1 2–2 2–3 2–4 3–1 3–2 3–3 3–4
A0 (mm2)
A (mm2)
27.8 28.2 28.4 27.8 29.5 29.2 28.9 29.5 28.0 28.1 28.6 27.5
r0.2 (MPa)
230 240 230 241 230 270 250 240 240
13.6 13.6 13.1 13.6 14.2 13.0 13.8 13.5 13.2 13.0 12.0 13.1
ef
w 0.512 0.518 0.539 0.512 0.519 0.556 0.524 0.543 0.528 0.538 0.581 0.524
0.718 0.730 0.775 0.717 0.732 0.812 0.742 0.784 0.751 0.772 0.870 0.741
rb (MPa)
rf (MPa)
HB
416.6 417.6 417.9 420.5 413.1 413.7 422.4 407.8 461.0 436.6 425.8 430.5
763.3 768.5 790.1 775.0 789.2 816.6 776.7 798.8 867.6 826.9 884.7 806.7
39.6 39.4 39.6 38.9 38.9 41.5 34.3 38.2 37.7 38.6 38.5 39.6
h i2 Ni ¼ C ðDr Þ2=ð1þnÞ ðDrc Þ2=ð1þnÞ ; 2=ð1þnÞ 2 f ðEKÞ
C ¼ 0:25e
h
½2=ð1 þ nÞ
2=ð1þnÞ
; i0:5 ; 2ð1þnÞ ð1 RÞð1nÞ
Dr ¼ K t Dr0 h i0:5 ; Drc ¼ K t Drc 2ð1þnÞ ð1 RÞð1nÞ
ð4Þ ð5Þ ð6Þ ð7Þ
Drc ¼ 2r1 E ef =103:5 ; corresponding to fatigue limit N ¼ 107 ; n
r ¼ Ke :
history as the metal at the notch tip. Many investigators used this hypothesis for the FCI life of notched elements. In this vein, the local strain range should be correlated to the nominal cyclic stress applied to the notched elements. It follows that the Manson–Coffin formula can be modified for predicting fatigue crack initiation life [2,21] as
DeD ¼ DeP DeC ¼ aN bf ; b 0:5;
In the energetic approach [22], the stress range Dr0, the endurance limit Drc, the equivalent stress range Dr, the equivalent endurance limit Drc , the facture ductility ef, the elastic modulus E, the stress ratio R, the stress concentration factor Kt, the workhardening coefficient K and the work-hardening exponent n are all employed for FCI life Ni prediction
ð3Þ
This was employed in [2,21] by application of a critical local strain or fatigue endurance limit.
ð8Þ ð9Þ
The notations used in the above equation are Dr0 is the stress range; Drc, the endurance limit; Dr, the equivalent stress range; Drc , the equivalent endurance limit; ef, the facture ductility; E, the elastic modulus; R, the stress ratio; Kt, the stress concentration factor; K, the work-hardening coefficient; and n is the work-hardening exponent. It may be worthwhile to note that C, the FCI resistance coefficient, may be considered to be the equivalent stress amplitude when Ni = 1/4 cycle, equivalent to the crack initiation at the notch root during monotonic tensile test. The FCI threshold, Drc, is the upper limit of the equivalent stress amplitude, below or equal to which no fatigue damage to occur, and FCI life tends to be infinite correspondingly. Both C and Drc are material constants, and can be predicted from the tensile properties. So long as the values of C and Drc are given, FCI life can be obtained from above expression. This energetic approach has been employed to prognosticate the fatigue
108
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109 Table 4 Comparison of the fitted C and Drcpred with the predictions (R = 0.1)*. Group 1
Group 2 11
Group 3 11
Cfit Cpred Drcfit (MPa) Dr0cpred: (MPa)
3.11 10 2.70 1011 226.44 264.83
2.59 10 3.05 1011 229.30 260.85
4.31 1011 5.66 1011 220.55 277.19
Dr00cpred: (MPa)
225.34
221.85
235.82
*
Fig. 6. Schematics of fatigue specimen.
The data in Table 3 and r1 = 0.35 rb are employed for Dr0cpred: estimation; The data in Table 3 and r1 = 0.30 rb are employed for Dr00cpred: estimation.
Fig. 7. S–N curve for 1st group materials.
Fig. 10. Comparison of fatigue fracture surfaces for specimens 1–1 (upper) and 1–2 (lower).
Fig. 8. S–N curve for 2nd group materials.
Fig. 11. Comparison of fatigue fracture surfaces for specimens 2–1 (upper) and 2–8 (lower).
Fig. 9. S–N curve for 3rd group materials.
crack initiation lives of 16 Mn steels and LY12CZ aluminum alloys with coarse grain [22], satisfied results are obtained. 4. Fatigue behavior of Brass H62 The specimen for fatigue test is machined from the three groups of materials with the geometry shown in Fig. 6. The material is prepared by the above mentioned re-crystallization procedure in the previous paragraph. Instron 1341 fatigue machine was employed to conduct the fatigue tests. Crack initiation is defined by the
Fig. 12. Comparison of fatigue fracture surfaces for specimens 3–1 (upper) and 3–7 (lower).
M. Zheng et al. / Theoretical and Applied Fracture Mechanics 54 (2010) 105–109
The energetic approach for fatigue crack initiation life assessment could be employed to predict the fatigue crack initiation life of fine grain Cu alloys; by analyzing the fitted results of the fatigue data, a modified empirical correlation between r1 and rb is proposed for fine grain copper alloy, i.e., r1 = 0.30 rb.
Table 5 Fatigue loads for some specimens (R = 0.1). Specimen no.
Maximum fatigue load (MPa)
1–1 1–2 2–1 2–8 3–1 3–7
250.90 292.72 331.54 290.10 301.95 325.68
Acknowledgement The author would like to show their thanks to Prof. G.C. Sih for his encouragement and great help in preparing this paper, ChineseCzech bilateral project entitled ‘‘fatigue behavior of ultra-fine grain Cu and Mg Alloys” is acknowledged as well.
appearance of crack with the length of millimeter. S–N curves are drawn in Figs. 7–9 for the three group materials, respectively. The S–N curve can be fitted by the energetic approach for fatigue crack initiation prediction proposed in [22]. The comparison of the fitted results of C for above three group materials and those of the theoretical prediction are listed in Table 4. It can be seen that the fitted fatigue resistant factor Cfit agrees with those of the predictions excellently. The fatigue endurance limit (stress) Dr0cpred: is predicted with Eq. (8) and the empirical correlation r1 = 0.35 rb [2] for copper alloy as well as the data in Table 3. It can be seen from Table 4 that Dr0cpred: is greater than Drcfit, which is due to the employing of the empirical formula r1 = 0.35 rb for r1 estimation for this fine grain copper alloy in such prediction. In order to reduce the difference between Dr0cpred: and Drcfit, the empirical relation between r1 with rb is modified as Eq. (10) for fine grain Cu alloy, and a new prediction for fatigue the endurance limit (stress), says Dr00cpred: is performed with the relevant processes, of which the result is shown in Table 4 as well. It can be seen that Dr00cpred: is close to Drcfit.
r1 ¼ 0:30 rb :
109
ð10Þ
5. Characteristic of fatigue fracture surface Figs. 10–12 show the characteristics of fatigue fracture surface. From these figures it can be seen that the characteristic of fatigue fracture surface depends on the material properties and fatigue load. Table 5 shows the fatigue load for some specimens. The size of fatigue process region decreases with fatigue load in each comparative group. The size of fatigue process region varies with the toughness of the materials from group to group for similar load level. Intergranular cracking is the main mechanism in the fatigue source zones. Concave pits exist in the latter fatigue propagation zones. 6. Concluding remarks Finer-grained H62 has longer fatigue life, which is due to its higher toughness. The characteristic of fatigue fracture surface depends on the material properties and fatigue load. The size of fatigue process region decreases with fatigue load in each comparative group. The size of fatigue process region varies with the toughness of the materials from group to group for similar load level. Intergrain cracking is the main mechanism in the fatigue source zones. Concave pits exist in the latter fatigue propagation zones.
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