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JMEPEG (2016) 25:1284–1290 DOI: 10.1007/s11665-016-1968-9
Fatigue Life Prediction Based on Local Strain Energy for Healed Copper Film by Laser Irradiation Feng-Zhu Liu, De-Guang Shang, Chong-Gang Ren, and Yu-Juan Sun (Submitted September 30, 2015; in revised form January 19, 2016; published online February 23, 2016) Changes of total cyclic strain energy at the notch for copper film specimen were analyzed before and after laser irradiation treatment. The results showed that laser irradiation can increase total cyclic strain energy and the effect of increase is more evident for the damaged copper specimen. Based on the damage-healing mechanism, an enhancement parameter and a healing parameter were defined by the local cyclic strain energy. A new model based on local strain energy was proposed to predict residual fatigue life for the damaged copper film specimen after laser irradiation. The predicted results by the proposed model agree well with the experimental lives. Keywords
enhancement parameter, healing parameter, laser irradiation, residual fatigue life prediction, total cyclic strain energy
1. Introduction Fatigue cracks usually initiate at the position of the stress concentration due to the complex geometry, which result in fatigue failure because of the cyclic loading and tiny defects. Some methods to heal fatigue damages had been studied by many researchers, such as D.C. electro-pulsing, annealing, plasma nitriding, laser surface treatment, and so on (Ref 1-5). Because of the many advantages of laser irradiation, it has been widely used for surface treatment of metal materials. With regard to the fatigue healing, Liu et al. (Ref 6, 7) identified two basic mechanisms. One is the dissipated energy enhancement mechanism, which improves the fatigue life caused by laser shock stress, and the other is the healing mechanism, which results in a further improvement. Based on the two mechanisms, Liu et al. (Ref 6) and Zhang et al. (Ref 8) proposed fatigue life prediction methods for copper film. However, these methods only took into account the plastic strain energy consumption before and after laser shock peening. As well known, the plastic strain energy cannot be used as a damage parameter in the high-cycle fatigue regime, where the material behaves elastically and the fatigue life is significantly influenced by the mean stresses (Ref 9). Ellyin and coauthors (Ref 10-12) thought the cyclic elastic strain energy with tensile stress part could promote crack propagation, because tensile stress can exacerbate the concentration degree at the microscopic defects or micro-cracks, which can be weakened by compressive stress. Therefore, in order to predict fatigue life in both the low-cycle and high-cycle regimes, total cyclic strain
Nomenclature
Kf KD0 D DWp DWeþ DWt E Dr rm rmax rmin K¢ n¢ a b De Dee Dep N0 Ns Nx DWt0 DWts DWtx Wt0 Wts Wtx Y Q
The fatigue notch factor Cycle strength coefficient for damaged specimen Damage degree Plastic strain energy density Elastic strain energy density Total cyclic strain energy density Elastic modulus Stress range Mean stress Maximum stress Minimum stress Cycle strength coefficient Cyclic strain hardening exponent Material constant for the fitted equation of cycle strength coefficient Another material constant for the fitted equation of cycle strength coefficient Total strain range Elastic strain range Plastic strain range Fatigue life for original specimen Fatigue life for undamaged specimens after laser irradiation Fatigue life for damaged specimens after laser irradiation Total cyclic strain energy density for original specimen Total cyclic strain energy density for undamaged specimens after laser irradiation Total cyclic strain energy density for damaged specimens after laser irradiation Total cyclic strain energy for original specimen Total cyclic strain energy for undamaged specimens after laser irradiation Total cyclic strain energy for damaged specimens after laser irradiation Healing parameter Enhancement parameter
Feng-Zhu Liu, De-Guang Shang, Chong-Gang Ren, and Yu-Juan Sun, College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China. Contact e-mails:
[email protected] and
[email protected].
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2. Experimental Procedure
energy model had been developed, in which elastic strain energy was added to the plastic strain energy (Ref 9, 13). The main aims of this study are to analyze the changes of total cyclic strain energies at the notch for the original specimen, the undamaged specimen after laser irradiation, and the damaged specimen after laser irradiation and to propose a new model based on total cyclic strain energy to predict residual fatigue life for the healed copper film by laser irradiation.
2.1 Fatigue Tests In this investigation, the smooth and notched copper film specimens with thickness of 25 lm were fabricated by a highprecision stamping process. The chemical compositions and mechanical properties of the material are listed in Tables 1 and 2. The shape and dimension are shown in Fig. 1.
Table 1 Chemical compositions of copper film specimen (wt.%) Cu 99.95
Sn
Pb
P
Zn
0.006
0.003
0.007
0.005
Table 2 Mechanical properties of copper film specimen Yield stress, MPa 252
Fig. 1
Ultimate tensile stress, MPa
Elongation to fracture, %
Elastic modulus, MPa
PoissonÕs ratio
418
6.0
50000
0.33
Shape and dimension of copper film specimen (millimeter): (a) notched specimen and (b) smooth specimen
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was clammed by the grips of the fatigue testing machine to obtain approximately the strain values. The data of elastic strain and plastic strain per cycle can be recorded by the fatigue testing system. The obtained cyclic stress–strain hysteresis curves are shown in Fig. 2 for the original specimen, the undamaged specimen after laser irradiation, and the damaged specimen after laser irradiation, and the evolution of plastic deformation with the number of cycles for the notched specimens is shown in Fig. 3.
400 350
Stress(MPa)
300 250 200 150
2.2 Laser Irradiation Tests
100
Original specimen Undamaged specimen after laser irradiation Damaged specimen(D=0.5) after laser irradiation
50 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Plastic Strain% Fig. 2 Cyclic stress–strain hysteresis curves of the original specimen, the undamaged specimen after laser irradiation, and the damaged specimen after laser irradiation
In this investigation, all of the laser irradiation treatments were carried out using an ultraviolet excimer gas laser with appropriate laser processing parameters, including energy density of 7 9 103 J/m2, the pulse number of 1, wavelength of 0.248 lm, and pulse width of 30 ns. The shadow regions in Fig. 1 are the treated areas by laser irradiation and its dimension is 45 mm 9 1.5 mm. References 14, 15 showed the details of laser irradiation treatment.
3. Experimental Results and Discussion 0.08
0.06
Plastic Strain (%)
3.1 Fatigue Lives for the Notched Specimens Before and After Laser Irradiation
Original specimen Undamaged specimen after laser irradiation Damaged specimen after laser irradiation
0.07
According to the tests for the notched specimens, the fatigue lives for the original specimens, the undamaged specimens after laser irradiation, and the damaged specimens after laser irradiation are listed in Table 3. The local stress and strain ranges at the notch root are obtained by NeuberÕs rule. In the investigation, the fatigue notch factor Kf obtained by the ratio between the fatigue limits of smooth and notched specimen is 2.7. Figure 4 shows the local stress ranges at the root of the notched specimens.
0.05 0.04 0.03 0.02 0.01 0.00 0
5000
10000
15000
20000
Number of cycles
Fig. 3 Evolution of plastic deformation with the number of cycles for notched specimens with different states
The fatigue tests were conducted on MMT-11N micromechanical fatigue testing machine with load-controlled mode. The limit load and displacement for the machine are 10 N and 2 mm, respectively. Because the copper film specimens cannot be subjected to compressive stress, all of the fatigue tests were conducted under pulsating cyclic tensile load (stress ratio R = 0) with a frequency of 20 Hz at room temperature in air. Fatigue tests were carried out for smooth and notched copper film specimens, respectively. For the notched specimens, some original specimens were first tested to obtain the damaged specimens. The damage degree is defined as the ratio between the current number of cycles and the number of cycles to failure under corresponding stress level. Fatigue tests were carried out for the original specimens, the undamaged specimens after laser irradiation, and the damaged specimens after laser irradiation to obtain the corresponding fatigue lives. For the smooth specimens, the processes of the tests were the same as that of the notched specimens. MMT fatigue testing system can directly measure the strain of tiny film specimens by detecting the displacement between the grips of the machine. Since the copper film specimen cannot install an extensometer, the gage section for the specimens
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3.2 Fatigue Response for the Smooth Copper Film Specimens Before and After Laser Irradiation The uniaxial cyclic stress and strain data for the smooth specimens were drawn from fatigue testing recorder. The cyclic strength coefficient K¢ and the cyclic strain hardening exponent n¢ can be determined by fitting the cyclic stress–strain curves for different specimen conditions. For the original specimen, the undamaged specimen after laser irradiation, and the damaged specimen with the damage of 0.3, 0.5, and 0.7, respectively, K 0 are 4541, 5807, 6471, 6939, and 7407 MPa, respectively, and n¢ are 0.40874, 0.4231, 0.4231, 0.4231, and 0.4231, respectively. It can be found that the values of n¢ are almost identical. According to the data, the changes of the plastic strain of the smooth specimens in different stress levels are shown in Fig. 5, in which Fig. 5(a) shows that laser irradiation can decrease the plastic strain of the copper film specimens and the degree of decrease is more obvious for the damaged specimens. Figure 5(b) shows that all the lines are nearly parallel, which means that the cyclic strain hardening exponents of the copper specimens are basically identical. In order to simplify the calculation, the linear relationship between cycle strength coefficient and damage degree shown in Fig. 6 is given as follows: KD0 ¼ a D þ b;
ðEq 1Þ
where a and b are the constants of the material, a = 2341 MPa, b = 5769 MPa.
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Table 3 Fatigue lives for the notched specimens Fatigue life for the original specimen N0, cycle
Fatigue life for the undamaged specimen after laser irradiation Ns, cycle
90
690919
757167
100
253497
379646
Nominal stress, MPa
Fatigue life for the damaged specimen after laser irradiation, Nx, cycle
Damage degree, D 0.5 0.7 0.3
354931 172759 397922 355433 389343 212662 330582 417703 157674 397146 174369 202456 130294 186944 162799 68313 74113 100158 48977 89885 87321
0.5 0.7 110
102448
203312
0.5 0.7
120
44760
114965
140
10322
41871
0.3 0.5 0.7 0.3 0.5 0.7
380
Local stress (MPa)
360 340
(
Original specimen Undamaged specimen after laser irradiation Damaged specimen (D=0.3) after laser irradiation Damaged specimen (D=0.5) after laser irradiation Damaged specimen (D=0.7) after laser irradiation
DWeþ
¼
300 280 260 240
DWp ¼
220 100
110
120
130
140
Nominal stress (MPa) Fig. 4 Local stresses of the notched specimens calculated by NeuberÕs rule
4. Residual Fatigue Life Prediction 4.1 Total Cyclic Strain Energy Density Total cyclic strain energy model has been developed in which elastic strain energy is added to the plastic strain energy to predict fatigue life. Golos and Ellyin (Ref 9) proposed a key expression in the total energy-based approach as follows: DWt ¼ DWp þ DWeþ with
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for
rmin 0
for
rmin > 0
;
ðEq 3Þ
where DWeþ and DWp are the positive elastic strain energy density and the plastic strain energy density, respectively, E is the elastic modulus, Dr is the stress range, rm is the mean stress, and rmax, rmin are the maximum stress and the minimum stress, respectively. The Halford-Morrow equation is commonly used to calculate the plastic strain energy density, that is, the area of hysteresis loop (Ref 16). It is expressed as
320
90
Dr 2 r2 1 max 2E 2 þ rm ¼ 2E 2 ðrmax rmin Þ 2E
ðEq 2Þ
1 n0 Dr Dep : 1 þ n0
ðEq 4Þ
The equation of uniaxial cyclic stress–strain curve can be described as 1=n0 De Dee Dep Dr Dr ¼ þ þ ¼ ; ðEq 5Þ 2 2E 2K 0 2 2 where K¢ and n¢ are the cyclic strength coefficient and the cyclic strain hardening exponent, respectively, and De, Dee, and Dep are the total strain range, the elastic strain range, and the plastic strain range, respectively. Owing to the copper material being approximately a Masing behavior, according to Eq 4 and 5, the plastic strain energy can be expressed as 0 0 1 n0 1=n0 ðDr=2Þð1þn Þ=n : ðEq 6Þ ðK 0 Þ DWp ¼ 4 1 þ n0 Because the fatigue tests are carried out with stress ratio of 0, that is, rmin ¼ 0, the total cyclic energy can be obtained by the following equation:
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8
350
300
250
Original specimen Undamaged specimen after laser irradiation Damaged specimen (D=0.7) after laser irradiation 200 -4 2.0x10
-4
-4
4.0x10
(a)
-4
6.0x10
-3
8.0x10
Ratio of total cyclic strain energy
Maximum stress (MPa)
400
7
Undamaged specimen Damaged specimen (D=0.3)
6 5 4 3 2 1
1.0x10
0
Plastic strain range
100
110
120
130
140
Nominal stress (MPa)
400
Fig. 7 Ratio of total cyclic strain energies for undamaged/damaged specimen after laser irradiation and original specimen Maximum Stress (MPa)
350
DWt ¼ DWp þ DWeþ ¼ 4 300
þ
1 n0 1 þ n0
ðK 0 Þ
1=n0
r2max : 2E ðEq 7Þ
Damaged specimen (D=0.2) after laser irradiation Damaged specimen (D=0.3) after laser irradiation Damaged specimen (D=0.5) after laser irradiation Damaged specimen (D=0.6) after laser irradiation Damaged specimen (D=0.7) after laser irradiation
250
0
4.2 Comparison and Analysis of Total Cyclic Strain Energy -4
-4
2x10
(b)
3x10
-4
4x10
-4
5x10
-4
6x10
-4
7x10
Plastic strain range
Fig. 5 Plastic strain ranges of the smooth specimens in different stress levels: (a) Original specimens, undamaged specimens after laser irradiation, and specimens with damage degree of 0.7 after laser irradiation and (b) Specimens with different damage degrees after laser irradiation
8000
Cyclic strength coefficient
7500
7000
6500
6000
5500 0.0
0.2
0.4
0.6
0.8
1.0
Damage degree D
Fig. 6 Relation between cyclic strength coefficients with different damage degrees after laser irradiation
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To simplify calculation, it was assumed that the consumed strain energy during each cycle is constant, that is, the strain energy density was obtained under the condition of a stabilized cyclic stress–strain behavior. The total cyclic strain energy for copper film specimen can be calculated by Wta ¼ DWta Na
ða ¼ 0; s; xÞ;
0
ðDr=2Þð1þn Þ=n
ðEq 8Þ
where Ns, Nx, and Nx are the fatigue lives for original specimen, undamaged, and damaged specimens after laser irradiation, respectively; DWt0, DWts, and DWtx are the total cyclic strain energy densities for original specimen, undamaged, and damaged specimens after laser irradiation, respectively; and Wt0, Wts, Wtx are the total cyclic strain energies for original specimen, undamaged, and damaged specimens after laser irradiation, respectively. Figure 7 shows that the total cyclic strain energy of undamaged specimen after laser irradiation is larger than that of the original specimen. In addition, the total cyclic strain energy of damaged specimen after laser irradiation is larger than that of the undamaged specimen after laser irradiation. It means that laser irradiation can increase the total cyclic strain energy of copper film and the increase is more evident for the damaged specimen.
4.3 Healing and Enhancement Parameters According to the above analysis, the schematic diagram of total cyclic strain energy distribution by laser irradiation is shown in Fig. 8. The enhancement effect for the undamaged specimen is assumed to be the same as that of the damaged
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specimen. According to Eq 8 and the schematic diagram, the enhanced strain energy and the healing strain energy can be calculated by WQH ¼ Wts Wt0 ¼ DWts Ns DWt0 N0 ;
WYH ¼ WtD Wts ¼ D DWt0 N0 þ Wtx Wts ;
ðEq 10Þ
where D is the damage degree which is defined as the ratio of the current cycles and the cycles to failure. Then healing parameter Y and enhancement parameter Q are defined as follows:
ðEq 9Þ
Y ¼
WYH WQH þ WYH
ðEq 11Þ
Q¼
WQH : WQH þ WYH
ðEq 12Þ
Obviously, Y þ Q ¼ 1:
According to Eq 11, 12, and 13, the relationship between the healing strain energy and the enhanced strain energy can be expressed as
Fig. 8 Schematic diagram of total cyclic strain energy distribution by laser irradiation treatment
WYH ¼ ½Y =ð1 Y Þ WQH :
ðEq 14Þ
According to Eq 9, 10, and 11, the healing parameter can be expressed as
1.0 0.9
Y ¼
0.8
Healing parameter
ðEq 13Þ
0.7
D DWt0 N0 þ Wtx Wts : ðD 1Þ DWt0 N0 þ Wtx
ðEq 15Þ
4.4 Proposed Residual Fatigue Life Prediction Model
0.6
According to the schematic diagram shown in Fig. 6, the total cyclic strain energy of damaged specimen after laser irradiation can be expressed as
0.5 0.4 0.3
Wtx ¼ ð1 DÞ Wt0 þ WQH þ WYH :
D=0.3 D=0.5 D=0.7
0.2 0.1
Equations 9 and 14 are brought into Eq 16, so the total cyclic strain energy of damaged specimen after laser irradiation can also be expressed as
0.0 100
120
ðEq 16Þ
140
Wtx ¼ ð1 DÞ DWt0 N0 þ ½1=ð1 Y Þ ðDWts Ns DWt0 N0 Þ: ðEq 17Þ
Nominal stress (MPa) Fig. 9 Healing parameters of damaged specimens with different damage degrees in different stress levels
According to Eq 8 and 17, the proposed residual fatigue life of damaged specimen after laser irradiation can be expressed as
Table 4 The calculated total cyclic strain energy density of copper film specimen Total cyclic strain energy density for the original specimen, DWt0
Total cyclic strain energy density for the undamaged specimen, DWts
90
0.585
0.586
100
0.722
0.723
110
0.872
0.873
120
1.037
1.038
140
1.407
1.410
Nominal stress, MPa
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Damage degree, D
Total cyclic strain energy density for the damaged specimen, DWtx
0.5 0.7 0.3 0.5 0.7 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7
0.587 0.588 0.724 0.725 0.725 0.876 0.877 1.040 1.042 1.030 1.414 1.416 1.417
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(3)
Predicted fatigue life (cycles)
106
According to the relationship of total cyclic strain energy, healing parameter, and damage degree, a new method was proposed to predict the residual fatigue life for the damaged copper film specimens after laser irradiation. The predicted lives by the proposed prediction method agreed well with the experimental data.
105
Acknowledgments D=0.3 D=0.5 D=0.7 104 104
105
106
Experimental fatigue life (cycles)
For this work, the authors gratefully acknowledge the financial support from the Major International Joint Research Project of the National Natural Science Foundation of China (51010006), the National Natural Science Foundation of China (51535001, 11572008, 11272019), and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20131103120024).
Fig. 10 Comparison of predicted and experimental residual lives for damaged copper specimens after laser irradiation
Nx ¼
References
½ð1 Y Þ ð1 DÞ 1 DWt0 N0 þ DWts Ns : ð1 Y Þ DWtx ðEq 18Þ
4.5 Verification of Residual Fatigue Life Prediction Model The values of healing parameter Y for the damaged specimens with different damage degrees in different stress levels are shown in Fig. 9 by substituting the corresponding parameters in Tables 3 and 4 into Eq 15. It can be found that the healing parameters are independent of stress levels and damage degrees, because the range of healing parameters is always 0.43 to 0.63. Therefore, the mean value of the healing parameter is taken as 0.53. In other words, the healing effect for the damaged specimen after laser irradiation accounts for 53% of the sum of healing and enhancement effects, that is, Y = 0.53. The predicted residual fatigue lives for the damaged specimens after laser irradiation can be calculated by substituting the basic parameters shown in Tables 3 and 4 into Eq 18. Figure 10 shows the comparison of the predicted and experimental residual lives for the damaged copper specimens after laser irradiation. It can be found that the predicted residual fatigue lives agree well with the experimental data, in which all the predicted residual fatigue life errors are within a factor of 2.
5. Conclusions (1)
(2)
Laser irradiation can increase total cyclic strain energy of the copper film. Compared to the undamaged specimens, the effect of increase for the damaged specimens after laser irradiation is more evident. It means that laser irradiation treatment also generates a healing effect except the enhancement effect for the damaged specimen. Healing parameter is independent of stress levels and damage degrees. The healing effect for the damaged specimen after laser irradiation is about 53% of the sum of healing and enhancement effects.
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1. Q. Shengru, L. YanLi, L. Yun, and Z. Chengyu, Damage Healing of Aluminum Alloys by D.C. Electropulsing and Evaluation by Resistance, Rare Met. Mater. Eng., 2009, 38(4), p 570–573 2. B. Du, N. Li, B. Ma, and W. Zhu, Mechanism of Prolonging Life and Effect for Fatigue Damaged Steel Specimens, J. Mech. Strength, 2006, 28(1), p 40–45 3. B. Du, W. Zhu, B. Ma, and N. Li, Prolonging Fatigue Life of a Damaged Steel by Annealing, Int. J. Fatigue, 2004, 26(9), p 1017– 1023 4. A. Alsaran, I. Kaymaz, A. Celik, F. Yetim, and M. Karakan, A Repair Process for Fatigue Damage Using Plasma Nitriding, Surf. Coat. Technol., 2004, 186(3), p 333–338 5. E. Altus and E. Konstantiono, Optimum Laser Surface Treatment of Fatigue Damaged Ti-6Al-4V Alloy, Mater. Sci. Eng. A, 2001, 302(1), p 100–105 6. X.D. Liu, D.G. Shang, L.H. Zhang, Y.J. Sun, and T. Chen, Residual Life Prediction for Healing Fatigue Damaged Copper Film by Laser Shock Peening, Fatigue Fract. Eng. Mater. Struct., 2014, 37(4), p 427– 435 7. X.D. Liu, D.G. Shang, M. Li, J. Jin, T. Chen, Y.B. Guo et al., Healing Fatigue Damage by Laser Shock Peening for Copper Film, Int. J. Fatigue, 2013, 54, p 127–132 8. L.H. Zhang, D.G. Shang, X.D. Liu, Y.J. Sun, Y.B. Guo, F.Z. Liu et al., A Nonlinear Fatigue Damage-Healing Model for Copper Film by LSP, Fatigue Fract. Eng. Mater. Struct., 2014, 37(11), p 1232–1241 9. K. Golos and F. Ellyin, A Total Strain Energy Density Theory for Cumulative Fatigue Damage, J. Press. Vessel Technol., 1988, 110(1), p 36–41 10. F. Ellyin and D. Kujawski, An Energy-Based Fatigue Failure Criterion, Microstruct. Mech. Behav. Mater., 1986, 11, p 541–600 11. F. Ellyin, Fatigue Life Prediction Under Multi-axial Stress Conditions, Dev. Eng. Mech., 1987, 1987, p 133–158 12. K. Golos and F. Ellyin, Generalization of Cumulative Damage Criterion to Multilevel Cyclic Loading, Theor. Appl. Fract. Mech., 1987, 7, p 169–176 13. P. Jinso and D. Nelson, Evaluation of an Energy-Based Approach and a Critical Plane Approach for Predicting Constant Amplitude Multiaxial Fatigue Life, Int. J. Fatigue, 2000, 22(1), p 23–39 14. D.G. Shang, C.G. Ren, H. Liu, and L. Wang, Approach to Recovery of Fatigue Damage for Copper Film by Pulsed Laser Irradiation, Surf. Eng., 2013, 29(7), p 536–542 15. C.G. Ren, D.G. Shang, L. Wang, and Y.B. Guo, Effect Of Single-Pulse Laser Irradiation Energy on Healing Fatigue Damage for Copper Film, J. Mech. Sci. Technol., 2014, 28(4), p 1257–1264 16. G.R. Halford, The Energy Required for Fatigue, J. Mater., 1966, 1(1), p 3–18
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