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Creep life analysis by an energy model of small punch creep test
Materials and Design 91 (2016) 98–103
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
Creep life analysis by an energy model of small punch creep test Sisheng Yang a, Xiang Ling a,⁎, Yangyan Zheng a,b, Rongbiao Ma a a b
Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China Jiangsu Special Equipment Safety Supervision Inspection Institute, Nanjing 211178, China
a r t i c l e
i n f o
Article history: Received 27 September 2015 Received in revised form 21 November 2015 Accepted 23 November 2015 Available online xxxx Keywords: Creep Failure analysis Small punch creep test Life prediction
a b s t r a c t Currently most creep life prediction methods by small punch test are focused on the steady creep deformation phase which is similar to conventional uniaxial creep tests. This paper presents a novel creep life extrapolation approach by small punch creep test. An energy model believed to have advantages in modeling creep deformation process of small punch tests is introduced. Linear expression between general strain and creep deflection was recommended as the deformation criterion in the analytical prediction. Meanwhile, the combination of ball radius and hole size of the lower die was analyzed to accommodate the difference of experimental devices. Based on the small punch creep data with different materials and creep environments, the creep life in the multiaxial form of large deformation was analyzed. Finally, by selecting different creep displacements from initial creep phase, the results were in agreement with the expected trend. The creep life extrapolation method was verified. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Engineering components often operate at high temperature and neutron irradiation environment which may easily result in creep damage and structure degradation [1]. Consequently, the experimental and analytical investigation on the failure mechanics and remaining life of structural materials is paramount to their safe operation. For the test of damage level and residual life analysis of in-service component, the miniature sample tests have received growing concern compared to the conventional testing methods for over twenty years. The reason is that a relatively smaller volume of sample is easier to obtain and economical for practical engineering, especially for the critical region with small volumes [2,3]. Among various micro-sample experiment methods, the small punch test (SPT) technology has been widely applied to acquire the creep properties and estimate the effects of long-term service. However, previous theoretical analysis is usually based on empirical relationships with a number of restrictions, which will limit further application of SPT in actual engineering [4,5]. In order to obtain enough information about deformation behavior of SPT, Yang and Wang [6] proposed a relation between small punch creep strain and center displacement based on the Chakrabarty model [7] by simplifying the deformation process and selecting appropriate geometry parameters. As a large deformation process, the complex non-linear relation is responsible for the difficult comprehensive analysis of stress and strain states, which restricts the development of SPT. Considering the creep ⁎ Corresponding author. E-mail address: [email protected] (X. Ling).
http://dx.doi.org/10.1016/j.matdes.2015.11.079 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
process which is similar to those obtained in the uniaxial creep test, numerous experimental results have been used to show how the load, deflection and minimum deflection rate correspond with the uniaxial creep tests [8–10]. On this basis, some extrapolation methods of small punch creep tests (SPCT) have been applied to estimate the remaining life of components [11]. Besides, the analyses of parameters such as activation energy provide a wide range of possibilities to analyze the interfaces between the uniaxial and multi-axial creep behaviors [12,13]. Moreover, the finite element method (FEM) was also applied to interpret the stress distribution and damage evolution with different constitutive equations [14,15]. It was clearly shown that different geometric factors of SPT would affect the results of creep properties [16]. However, the lack of consideration on apparatus geometric parameters and complication of the pre-existing expression may restrict the application of these methods. Consequently, this paper devotes to obtain the corresponding multi-axial stress parameters of SPCT by a new energy model [17]. As for the small punch creep life prediction, most of the literatures have been focused on the deflection rate and three stages of deformation, especially the steady creep phase. However in this paper, the law of energy conservation in creep deformation was introduced to establish a simple creep life prediction expression for the constant load deformation. Based on the results of a thorough investigation into the deformation process of small punch circular specimens pressed by constant force, the relationship between creep life and mechanical work was proposed. Meanwhile, the geometrical parameters of devices and specimen were taken into account. To approach a reasonable creep life analysis, the expression discussed in this work may provide some insights for SPCT.
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2. Experimental Process As shown in Fig. 1, all the tests were performed on a self-made small punch creep system which consisted of ceramic ball, punch, temperature controller system, upper die and lower die. Before testing, the circular thin disks, 0.5 mm in thickness and 10 mm in diameter, were ground and polished for lowering surface roughness. The upper and lower dies would be clamped after the sample was put in the specimen holder. It was important to note that the dimension of center hole in the lower die which was mostly related to failure of specimen. Considering the relation of geometric parameters suggested, the lower die contained a receiving hole with diameter 4 mm was used in this paper. 12Cr1MoV with a composition (in wt%): 0.095C, 0.23Si, 0.558Mn, 0.963Cr, 0.25Mo, 0.17 V, 0.001S, 0.0127P was employed as the test material in this study, which was widely used in the high temperature components. The inservice materials of Cr5Mo were extracted from a furnace tube. The service temperature was 550 °C and the service time was ten years [18]. Carbon content of in-service material (0.12 C) increased apparently compared to that of new materials (0.095 C). During the test, the load was transferred onto disk by a 2.4 mm ceramic ball extruded by punch. Near the fixture of SPT, three thermocouples were installed for controlling the temperature in the heating furnace. The deflection of specimen was monitored and recorded by displacement transducer during the creep deformation. 3. Model of Small Punch Creep Test Considering a thin circular disk subjected to constant force by a rigid ball, as depicted in Fig. 2. The creep process obeys the law of energy conservation and momentum conservation, the relationship between mechanical work, internal energy and kinetic energy can be represented by Eq. (1) [17]: dεij de þ hij −ργ ¼ 0 ρ −σ ij dt dt
ð1Þ
where ρ is the density, e is the internal energy unit mass, εij is the strain, σij is the stress, γ is the heat supply unit mass, hij is the heat flux and t is time. Due to the constant temperature, there is no effect of heat fluxes on local deformation behavior of materials in a steady-state environment
Fig. 2. Schematic diagram of deformation form of small punch creep specimen.
such as SPCT. Any change of entropy must be linked to changes of the mechanical work. Thus, the entropy change can be expressed as: ρT
dεij ds dv dv de dW −hij þ ργ ¼ ρT þ ρ −σ ij ¼f ¼ ρT dt dt dt dt dt dt
ð2Þ
where s is the entropy per mass, v⁎ is the entropy generation per mass, T is the temperature, W is the mechanical work. It can be found that only the irreversible part of the entropy will lead to the damage of the material, thus: ρT
dsi dW dsr dW ¼f ¼η −ρT dt dt dt dt
ð3Þ
where si is the irreversible entropy increase per mass, sr is the reversible entropy increase per mass. According to the assumption of conventional creep tests, there is a ratio of conversion form mechanical work to the entropy increase. Here, η was used to express the ratio which can be defined as the function of time [17]. Meanwhile, mechanical work per unit is the function of stress σ and strain ε, thus: ρT
dsi dW σdε ¼η ¼ Ct q dt dt dt
ð4Þ
where C and q are constants. It is worthy to note that the internal energy is a state parameter and the irreversible entropy is a constant value in an identical external condition. Consequently, Eq. (5) is capable to describe the deformation process from initial creep to final failure: Z 0
tr
η
dW dt ¼ dt
Z 0
tr
Ct q
σdε dt ¼ dt
Z 0
tr
Ct q σdε ¼ const
ð5Þ
In such instances, the estimation of equivalent stress and strain value must be performed through a robust analytical approach. Previous investigations have proved that there is a linear relationship between equivalent stress and load applied to small punch specimens [19]. A general expression provided by the CEN (The European Committee for Standardization) [5] is given by Eq. (6): F ¼ b1 rb2 Rb3 h0 σ
Fig. 1. Schematic representation of small punch creep test.
ð6Þ
where R is the radius of the lower hole, r is the radius of ball and h0 is the initial thickness of specimen. After introducing a correlation factor (Ksp) to account for the influence of material, temperature and necking in creep deformation, a
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more comprehensive and widely accepted linear form between the force and equivalent stress is F ¼ 3:33K sp r 1:2 h0 R−0:2 σ
ð7Þ
The strain distribution presents non-uniform variations in the small punch specimen, especially near the edge of contact region, at which position the strain is highly greater than the average level of whole specimen. In view of this, the general strain proposed by Hyde et al. provided an important insight into the mean strain for interpretation of the small punch creep deformation [4]. In the range of creep deflection, the change of specimen volume can be neglected and the thickness is constant based on the visco-plastic mechanism [19]. With the assumption of cone and geometrical relation, the general strain can be defined as:
1 r r π εm ¼ ln − þ −α sin α R tan α R 2
ð8Þ
1 r − tan α R
1 −1 sin α
ε_ m ¼ BL_ m
ð10Þ
where B is the constant, L_ m is the minimum rate. As discussed above, some linearity exists when the deflection of SPCT is above 0.5 mm. Strain variation is inversely proportional to the thickness and proportional to the deflection. According to Norton-Bailey model, creep strain is a function of time and stress. As discussed above, equivalent stress is a constant related to force. Thus, the term tm is introduced in Eq. (11) below, where A, A′ and m are constants [22,23]: A0 L At m ¼ : h0 h0
ε¼
ð11Þ
According to Eq. (7) and Eq. (11), Eq. (5) can be rearranged and is valid for the geometric sizes range:
Meanwhile, the deflection of center is given by: L¼R
specimen thickness on strain and found a similar relationship between strain and deflection. Compared to the conventional creep results, the deflection rate and creep rate can be described by the Monkman-Grant relation [21] and the minimum strain rate (ε_ m ) can be derived to:
ð9Þ
The general strain–deflection curve calculated in the above way with different geometric sizes is depicted in Fig. 3. Of the curves, a typical linear feature is able to be found when the center creep deflection is larger than 0.5 mm. For most engineering materials, creep tends to occur at this stage. Thus, strain can be defined as a linear function of deflection based on the above observations. It is worthy to be noted that a third order polynomial correlation between strain and deflection is also obtained based on the membrane stretching theory [19]. This is because that the assumptions about cone and uniform thickness are not applicable in the acceleration and fracture region. However, steady state comprises most of the life of small punch specimen. Therefore, we believe that the linear correlation discussed in this paper is reasonable. In the small punch deformation problem the stress triaxiality is changing, thus finite element simulation is another suitable method to describe the complex strain variation and yield some accurate analysis [20]. Based on the modified Kachanov-Rabotnov (K-R) creep damage constitutive equations, Ling et al. [16] analyzed the influence of
η
dW FR0:2 t m−1 R0:2 F mþq−1 mA ¼n 2 t ¼ Ct q 1:2 h0 dt 3:33K sp r h0 h r 1:2
ð12Þ
0
Creep fracture occurs at the maximum deflection. Hence, Eq. (13) can be obtained when we integrate the right side of (12): Z
tr
n 0
R0:2 F 2 h0 r 1:2
t mþq−1 dt ¼
n R0:2 F mþq R0:2 FL q tr ¼u 2 t r ¼ const 2 m þ q h r 1:2 h r 1:2 0
ð13Þ
0
The parameter u, n, q are constants depending on materials and test environment. Correlation also can be obtained by the rupture time and the geometrical parameters of experimental fixture in Eq. (13) which can be used for different devices. Generally, in the same set of device, the sizes of lower die and small ball remain unchanged. Thus, a parameter P is used here: R0:2 r 1:2
P¼
ð14Þ
Expression (14) can be rewritten as: uP
W sp
W sp 2
h0
2
h0
t qr ¼ const
t qr ¼ const
ð15Þ
ð16Þ
The parameter n, u, q and P are all constants, Wsp is the mechanical work in SPCT. Thus, Eq. (16) can effectively be utilized to predict the creep life under mechanical work introduced by external load in SPCT. 4. Results and Discussion
Fig. 3. General strain variation with deflection for the widely used sizes of small punch creep tests.
In this section, the results of new experiment and our previous research works [18,24,25] were combined to examine the validity of energy model in practical engineering. In addition to the creep environment, different materials and specimen thickness factors were also discussed. At 550 °C, SPCT of 12Cr1MoV without protective gas are shown in Fig. 4. The deflection-time plots of SPCT can be divided to three regions: a decaying primary stage, a long secondary region and a well chiseled tertiary stage. A good agreement between the creep deformation trends of SPCT and conventional creep tests provides an important validation of the creep life prediction by SPCT [19]. It can be inferred from Fig. 5
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101
Fig. 4. Small punch creep curves for 12Cr1MoV at 550 °C.
that local necking happens at the fracture region which comprises little life of creep specimen. Thus, the assumption of cone with constant thickness does not introduce a substantial error in the analytical solution of energy model. The ultimate goal of this research is to establish the relationship between mechanical work and creep life. Considering the loading process, localized stresses around small contact area will lead to an initial large deformation as shown in Fig. 4. Thus, the displacement of initial elastic–plastic deformation should be subtracted when the mechanical work is calculated in SPCT. In this paper, the primary creep stage of specimen is defined as a starting point of creep deflection with the absence of initial loading displacement. From starting point of primary creep stage to different creep deformation regions, different creep displacements can be chosen. Multiply the displacement by load of SPCT, mechanical work is able to be calculated. For 12Cr1MoV, three different displacements (fracture displacement, 0.3 mm and 0.4 mm) were chosen. Meanwhile, the relationships between the mechanical work and fracture time based on Eq. (16) were established. In the identical experimental device, the variation of volumes and initial thickness were not considered since the diameter and thickness of small punch specimens remained unchanged. From Fig. 6, the linear relationship can be achieved by taking logarithm of mechanical work and time. In most instances, creep environment may affect creep deformation including fracture time, displacement and minimum creep rate. Therefore, the SPCT are often accomplished in protective gas [26]. As shown in Fig. 7, four load levels (513 N, 482 N, 463 N, 443 N) were chosen for SUS304 stainless steel and the temperature was 650 °C in the
Fig. 5. Deformation specimen of small punch test.
Fig. 6. The relationship between work and time for 12Cr1MoV without protective gas.
experiment. It is demonstrated that the proposed model is in well agreement with the calculated results of different creep displacements. Compared the creep results of new Cr5Mo and in-service Cr5Mo at 550 °C in Fig. 8 (424 N, 404 N, 375 N), the feasibility of energy model was testified. The same conclusion could also be found in the analysis of small punch creep results of single crystal alloy (CMSX-4) and 316 L (N) as shown in the Fig. 9 [27,28]. Meanwhile, the data of base metal (BM), welded metal (WM), fine grain heat affected zone (FGHAZ) and coarse grain heat affected zone (CGHAZ) of P92 steel weld joint in Fig. 10 (650 °C) were also found to obey energy model [29]. In this paper, the precision of fitting results calculated by fracture displacement is lower than that of certain creep displacement. A possible explanation for this phenomenon is that the condition for fracture is assumed to be ideal in energy model. However, as a complicated deformation process with multi-axial stress state, the failure mechanism is caused by an interaction between propagation of microcracks, creep damage evolution in tertiary acceleration stage [19]. The complicated failure mechanism and cutty creep acceleration phase will led to error and uncertainty in the measurement of final fracture deflection. Thus, how to define the actual rupture displacement is another problem. Considering the transient tertiary phase of SPCT, it is recommended that deformation before accelerated creep stage can be chosen as the creep displacement in the energy model.
Fig. 7. The relationship between work and time for SUS304.
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Fig. 10. The relationship between work and time for different zones in P92 steel weld joint. Fig. 8. The relationship between work and time for Cr5Mo (new material and in-service material).
Although the thickness of 0.5 mm have been widely accepted for obtaining the uniform parameters, the minor deviation or variation in thickness is inevitable, especially for the critical area which can not be easily extracted. Traditionally, it is generally accepted that the actual specimen thickness has an apparent effect on the creep properties. Several methods have been reported to establish the relation between fracture time and thickness. Comparing to the experiment, the finite element simulation may eliminate the influence of oxidation on the disk. Therefore, deflection - time curves obtained from simulations was used to analyze the influence of thickness. The creep temperature was 650 °C and the specimen thicknesses were 0.4 mm, 0.45 mm, 0.5 mm. Based on the results described in Reference [16], a linear relationship between log(Wsp/h20) and log(tr) can be found with the identical creep load level (463 N) as shown in Fig. 11. It proves that the proposed method is capable of producing extreme extrapolation results in creep analysis with different specimen thicknesses. It is important to note that the slope of straight lines in Figs. 6–10 is – q. It follows from the conversion ratio from mechanical work to the entropy increase. The apparent difference between new materials and in-service materials has lead to the belief that material will affect the value of q. An explanation for this phenomenon is that a higher conversation ratio can be obtained with the increase of q according to Eq. (4). A larger slope leads to a higher conversation ratio. The increase of irreversible entropy causes a rapid increase in deformation ratio and the
Fig. 9. The relationship between work and time for 316 L (N) and CMSX-4.
reduction of failure time. Therefore, the value of q can also be used to analyze the creep properties of different materials. Compared to the conventional test, SPCT can be used to analyze the material behavior of in-service components and localized structures. Thus, how to complement the deficiency of common methods and improve the accuracy of prediction has important significance in the development of SPCT. The proposed methodology holds a key advantage over the other methods that it can estimate remaining creep life convenient without the limitation of geometrical parameters of SPCT devices. Meanwhile, this methodology based on the energy model need not calculate the complicated multi-axial stress parameters in the deformation. Hence, we believe that this approach can provide some insight for the engineering in related fields. 5. Conclusion In this paper, an energy model of SPCT is used to extrapolate the creep life of materials at constant temperature. An approximate stress function depending on the load and device shape was introduced. Strain variation has been proven to be inversely proportional to the thickness and proportional to the deflection. The geometric factors including specimen thickness, hole radius of the lower die and radius of ball were considered in the model. Thus, this method is adaptable to different technical devices and specimen sizes. The results of different materials including 12Cr1MoV, new Cr5Mo, inservice Cr5Mo, SUS304, CMSX-4, 316 L (N) and different thicknesses were used in order to verify the accuracy of energy model. Meanwhile,
Fig. 11. The relationship between work and time with different specimen thicknesses.
S. Yang et al. / Materials and Design 91 (2016) 98–103
it can be found that this method is able to be applied to predict the creep life in different test environments. As a simple and convenient model of SPCT, the conclusion is capable to be widely used to extrapolate the creep life of materials. Acknowledgements The authors hope to acknowledge the support provided by the Research Innovation Project for College Graduates of Jiangsu Province (KYZZ15_0229) and National high technology research and development plan (863 plan) of China (Grant No. 2012AA040105). References [1] F. Hou, H. Xu, Y.C. Wang, L. Zhang, Determination of creep property of 1.25Cr0.5Mo pearlitic steels by small punch test, Eng. Fail. Anal. 28 (2013) 215–221. [2] S. Haroush, E. Priel, D. Moreno, A. Busiba, I. Silverman, A. Turgeman, et al., Evaluation of the mechanical properties of SS-316 L thin foils by small punch testing and finite element analysis, Mater. Des. 83 (2015) 75–84. [3] T. Bai, K.S. Guan, Evaluation of stress corrosion cracking susceptibility of nanocrystallized stainless steel 304 L welded joint by small punch test, Mater. Des. 52 (2013) 849–860. [4] T.H. Hyde, M. Stoyanov, W. Sun, C.J. Hyde, On the interpretation of results from small punch creep tests, J. Strain Anal. Eng. Des. 45 (2010) 141–164. [5] CEN Workshop Agreement, CWA 15627:2006 E, small punch test method for metallic materials: Part A, CEN, Brussels, 2006. [6] Z. Yang, Z.W. Wang, Relationship between strain and central deflection in small punch creep specimens, Int. J. Press. Vessel. Pip. 80 (2003) 397–404. [7] J. Chakrabarty, A theory of stretch forming over hemispherical punch heads, Int. J. Mech. Sci. 12 (1970) 315–325. [8] K. Milička, F. Dobeš, Small punch testing of P91 steel, Int. J. Press. Vessel. Pip. 83 (2006) 625–634. [9] B. Ule, T. Šuštar, F. Dobeš, K. Milička, V. Bicego, S. Tettamanti, et al., Small punch test method assessment for the determination of the residual creep life of service exposed components: outcomes from an interlaboratory exercise, Nucl. Eng. Des. 192 (1999) 1–11. [10] S.I. Komazaki, T. Kato, Y. Kohno, H. Tanigawa, Creep property measurements of welded joint of reduced-activation ferritic steel by the small-punch creep test, Mater. Sci. Eng. A 510-511 (2009) 229–233. [11] F. Dobeš, K. Milička, Application of creep small punch testing in assessment of creep lifetime, Mater. Sci. Eng. A 510-511 (2009) 440–443.
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[12] A. Nagode, L. Kosec, B. Ule, Uni-axial and multi-axial creep behaviour of P91-type steel under constant load, Eng. Fail. Anal. 18 (2011) 61–67. [13] F. Dobeš, K. Milička, Comparison of conventional and small punch creep tests of mechanically alloyed Al-C-O alloys, Mater. Charact. 59 (2008) 961–964. [14] I.I. Cuesta, C. Rodriquez, F.J. Belzunce, J.M. Alegre, Analysis of different techniques for obtaining pre-cracked/notched small punch test specimens, Eng. Fail. Anal. 18 (2011) 2282–2287. [15] S. Foletti, M. Madia, A. Cammi, G. Torsello, Characterization of the behavior of a turbine rotor steel by inverse analysis on the small punch test, Procedia Eng. 10 (2011) 3628–3635. [16] Z.X. Zhou, Y.Y. Zheng, X. Ling, R.M. Hu, J.Q. Zhou, A study on influence factors of small punch creep test by experimental investigation and finite element analysis, Mater. Sci. Eng. A 527 (2010) 2784–2789. [17] H. Liu, F.Z. Xuan, A new model for life estimation of creep, Nucl. Eng. Des. 240 (2010) 2528–2531. [18] J.H. Shan, X. Ling, Z.M. Qian, Residual life assessment of in-service high temperature components by small punch creep test, ASME 2007 Pressure Vessels and Piping Conference Volume 9: Eighth International Conference on Creep and Fatigue at Elevated Temperatures 2007, pp. 427–432. [19] J.P. Rouse, F. Cortellino, W. Sun, T.H. Hyde, J. Shingledecker, Small punch creep testing: review on modelling and data interpretation, Mater. Sci. Technol. 29 (2013) 1328–1345. [20] S. Goyal, K. Laha, C.R. Das, S. Panneerselvi, M.D. Mathew, Finite element analysis of effect of triaxial state of stress on creep cavitation and rupture behaviour of 2.25Cr–1Mo steel, Int. J. Mech. Sci. 75 (2013) 233–243. [21] F. Dobeš, K. Milička, On the monkman-grant relation for small punch test data, Mater. Sci. Eng. A 336 (2002) 245–248. [22] F. Dobeš, M. Besterci, B. Ballóková, K. Sűlleiová, P. Dymáček, Analysis of creep fracture in Al-Al4C3 composite after ECAP, Mater. Sci. Eng. A 532 (2012) 567–572. [23] B.J. Kim, Y.B. Sim, J.H. Lee, M.K. Kim, B.S. Lim, Application of small punch creep test for Inconel 617 alloy weldment, Procedia Eng. 10 (2011) 2579–2584. [24] Y.X. Chen, X. Ling, S.T. Tu, Study on small punch creep test for SUS304, J. Exp. Mech. 20 (2005) 219–225 (in Chinese). [25] X. Ling, Y.Y. Zheng, Y.J. You, Y.X. Chen, Creep damage in small punch creep specimens of type 304 stainless steel, Int. J. Press. Vessel. Pip. 84 (2007) 304–309. [26] T. Auger, I. Serre, G. Lorang, Z. Hamouche, D. Gorse, J.B. Vogt, Role of oxidation on LME of T91 steel studied by small punch test, J. Nucl. Mater. 376 (2008) 336–340. [27] S.P. Jeffs, R.J. Lancaster, Elevated temperature creep deformation of a single crystal superalloy through the small punch creep method, Mater. Sci. Eng. A 626 (2015) 330–337. [28] J. Ganesh Kumar, K. Laha, Small punch creep deformation and rupture behavior of 316 L (N) stainless steel, Mater. Sci. Eng. A 641 (2015) 315–322. [29] L. Zhao, H.Y. Jing, L.Y. Xu, Y.D. Han, J.J. Xiu, Y.X. Qiao, Evaluating of creep property of distinct zones in P92 steel welded joint by small punch creep test, Mater. Des. 47 (2013) 677–686.
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
Creep life analysis by an energy model of small punch creep test Sisheng Yang a, Xiang Ling a,⁎, Yangyan Zheng a,b, Rongbiao Ma a a b
Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China Jiangsu Special Equipment Safety Supervision Inspection Institute, Nanjing 211178, China
a r t i c l e
i n f o
Article history: Received 27 September 2015 Received in revised form 21 November 2015 Accepted 23 November 2015 Available online xxxx Keywords: Creep Failure analysis Small punch creep test Life prediction
a b s t r a c t Currently most creep life prediction methods by small punch test are focused on the steady creep deformation phase which is similar to conventional uniaxial creep tests. This paper presents a novel creep life extrapolation approach by small punch creep test. An energy model believed to have advantages in modeling creep deformation process of small punch tests is introduced. Linear expression between general strain and creep deflection was recommended as the deformation criterion in the analytical prediction. Meanwhile, the combination of ball radius and hole size of the lower die was analyzed to accommodate the difference of experimental devices. Based on the small punch creep data with different materials and creep environments, the creep life in the multiaxial form of large deformation was analyzed. Finally, by selecting different creep displacements from initial creep phase, the results were in agreement with the expected trend. The creep life extrapolation method was verified. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Engineering components often operate at high temperature and neutron irradiation environment which may easily result in creep damage and structure degradation [1]. Consequently, the experimental and analytical investigation on the failure mechanics and remaining life of structural materials is paramount to their safe operation. For the test of damage level and residual life analysis of in-service component, the miniature sample tests have received growing concern compared to the conventional testing methods for over twenty years. The reason is that a relatively smaller volume of sample is easier to obtain and economical for practical engineering, especially for the critical region with small volumes [2,3]. Among various micro-sample experiment methods, the small punch test (SPT) technology has been widely applied to acquire the creep properties and estimate the effects of long-term service. However, previous theoretical analysis is usually based on empirical relationships with a number of restrictions, which will limit further application of SPT in actual engineering [4,5]. In order to obtain enough information about deformation behavior of SPT, Yang and Wang [6] proposed a relation between small punch creep strain and center displacement based on the Chakrabarty model [7] by simplifying the deformation process and selecting appropriate geometry parameters. As a large deformation process, the complex non-linear relation is responsible for the difficult comprehensive analysis of stress and strain states, which restricts the development of SPT. Considering the creep ⁎ Corresponding author. E-mail address: [email protected] (X. Ling).
http://dx.doi.org/10.1016/j.matdes.2015.11.079 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
process which is similar to those obtained in the uniaxial creep test, numerous experimental results have been used to show how the load, deflection and minimum deflection rate correspond with the uniaxial creep tests [8–10]. On this basis, some extrapolation methods of small punch creep tests (SPCT) have been applied to estimate the remaining life of components [11]. Besides, the analyses of parameters such as activation energy provide a wide range of possibilities to analyze the interfaces between the uniaxial and multi-axial creep behaviors [12,13]. Moreover, the finite element method (FEM) was also applied to interpret the stress distribution and damage evolution with different constitutive equations [14,15]. It was clearly shown that different geometric factors of SPT would affect the results of creep properties [16]. However, the lack of consideration on apparatus geometric parameters and complication of the pre-existing expression may restrict the application of these methods. Consequently, this paper devotes to obtain the corresponding multi-axial stress parameters of SPCT by a new energy model [17]. As for the small punch creep life prediction, most of the literatures have been focused on the deflection rate and three stages of deformation, especially the steady creep phase. However in this paper, the law of energy conservation in creep deformation was introduced to establish a simple creep life prediction expression for the constant load deformation. Based on the results of a thorough investigation into the deformation process of small punch circular specimens pressed by constant force, the relationship between creep life and mechanical work was proposed. Meanwhile, the geometrical parameters of devices and specimen were taken into account. To approach a reasonable creep life analysis, the expression discussed in this work may provide some insights for SPCT.
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2. Experimental Process As shown in Fig. 1, all the tests were performed on a self-made small punch creep system which consisted of ceramic ball, punch, temperature controller system, upper die and lower die. Before testing, the circular thin disks, 0.5 mm in thickness and 10 mm in diameter, were ground and polished for lowering surface roughness. The upper and lower dies would be clamped after the sample was put in the specimen holder. It was important to note that the dimension of center hole in the lower die which was mostly related to failure of specimen. Considering the relation of geometric parameters suggested, the lower die contained a receiving hole with diameter 4 mm was used in this paper. 12Cr1MoV with a composition (in wt%): 0.095C, 0.23Si, 0.558Mn, 0.963Cr, 0.25Mo, 0.17 V, 0.001S, 0.0127P was employed as the test material in this study, which was widely used in the high temperature components. The inservice materials of Cr5Mo were extracted from a furnace tube. The service temperature was 550 °C and the service time was ten years [18]. Carbon content of in-service material (0.12 C) increased apparently compared to that of new materials (0.095 C). During the test, the load was transferred onto disk by a 2.4 mm ceramic ball extruded by punch. Near the fixture of SPT, three thermocouples were installed for controlling the temperature in the heating furnace. The deflection of specimen was monitored and recorded by displacement transducer during the creep deformation. 3. Model of Small Punch Creep Test Considering a thin circular disk subjected to constant force by a rigid ball, as depicted in Fig. 2. The creep process obeys the law of energy conservation and momentum conservation, the relationship between mechanical work, internal energy and kinetic energy can be represented by Eq. (1) [17]: dεij de þ hij −ργ ¼ 0 ρ −σ ij dt dt
ð1Þ
where ρ is the density, e is the internal energy unit mass, εij is the strain, σij is the stress, γ is the heat supply unit mass, hij is the heat flux and t is time. Due to the constant temperature, there is no effect of heat fluxes on local deformation behavior of materials in a steady-state environment
Fig. 2. Schematic diagram of deformation form of small punch creep specimen.
such as SPCT. Any change of entropy must be linked to changes of the mechanical work. Thus, the entropy change can be expressed as: ρT
dεij ds dv dv de dW −hij þ ργ ¼ ρT þ ρ −σ ij ¼f ¼ ρT dt dt dt dt dt dt
ð2Þ
where s is the entropy per mass, v⁎ is the entropy generation per mass, T is the temperature, W is the mechanical work. It can be found that only the irreversible part of the entropy will lead to the damage of the material, thus: ρT
dsi dW dsr dW ¼f ¼η −ρT dt dt dt dt
ð3Þ
where si is the irreversible entropy increase per mass, sr is the reversible entropy increase per mass. According to the assumption of conventional creep tests, there is a ratio of conversion form mechanical work to the entropy increase. Here, η was used to express the ratio which can be defined as the function of time [17]. Meanwhile, mechanical work per unit is the function of stress σ and strain ε, thus: ρT
dsi dW σdε ¼η ¼ Ct q dt dt dt
ð4Þ
where C and q are constants. It is worthy to note that the internal energy is a state parameter and the irreversible entropy is a constant value in an identical external condition. Consequently, Eq. (5) is capable to describe the deformation process from initial creep to final failure: Z 0
tr
η
dW dt ¼ dt
Z 0
tr
Ct q
σdε dt ¼ dt
Z 0
tr
Ct q σdε ¼ const
ð5Þ
In such instances, the estimation of equivalent stress and strain value must be performed through a robust analytical approach. Previous investigations have proved that there is a linear relationship between equivalent stress and load applied to small punch specimens [19]. A general expression provided by the CEN (The European Committee for Standardization) [5] is given by Eq. (6): F ¼ b1 rb2 Rb3 h0 σ
Fig. 1. Schematic representation of small punch creep test.
ð6Þ
where R is the radius of the lower hole, r is the radius of ball and h0 is the initial thickness of specimen. After introducing a correlation factor (Ksp) to account for the influence of material, temperature and necking in creep deformation, a
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more comprehensive and widely accepted linear form between the force and equivalent stress is F ¼ 3:33K sp r 1:2 h0 R−0:2 σ
ð7Þ
The strain distribution presents non-uniform variations in the small punch specimen, especially near the edge of contact region, at which position the strain is highly greater than the average level of whole specimen. In view of this, the general strain proposed by Hyde et al. provided an important insight into the mean strain for interpretation of the small punch creep deformation [4]. In the range of creep deflection, the change of specimen volume can be neglected and the thickness is constant based on the visco-plastic mechanism [19]. With the assumption of cone and geometrical relation, the general strain can be defined as:
1 r r π εm ¼ ln − þ −α sin α R tan α R 2
ð8Þ
1 r − tan α R
1 −1 sin α
ε_ m ¼ BL_ m
ð10Þ
where B is the constant, L_ m is the minimum rate. As discussed above, some linearity exists when the deflection of SPCT is above 0.5 mm. Strain variation is inversely proportional to the thickness and proportional to the deflection. According to Norton-Bailey model, creep strain is a function of time and stress. As discussed above, equivalent stress is a constant related to force. Thus, the term tm is introduced in Eq. (11) below, where A, A′ and m are constants [22,23]: A0 L At m ¼ : h0 h0
ε¼
ð11Þ
According to Eq. (7) and Eq. (11), Eq. (5) can be rearranged and is valid for the geometric sizes range:
Meanwhile, the deflection of center is given by: L¼R
specimen thickness on strain and found a similar relationship between strain and deflection. Compared to the conventional creep results, the deflection rate and creep rate can be described by the Monkman-Grant relation [21] and the minimum strain rate (ε_ m ) can be derived to:
ð9Þ
The general strain–deflection curve calculated in the above way with different geometric sizes is depicted in Fig. 3. Of the curves, a typical linear feature is able to be found when the center creep deflection is larger than 0.5 mm. For most engineering materials, creep tends to occur at this stage. Thus, strain can be defined as a linear function of deflection based on the above observations. It is worthy to be noted that a third order polynomial correlation between strain and deflection is also obtained based on the membrane stretching theory [19]. This is because that the assumptions about cone and uniform thickness are not applicable in the acceleration and fracture region. However, steady state comprises most of the life of small punch specimen. Therefore, we believe that the linear correlation discussed in this paper is reasonable. In the small punch deformation problem the stress triaxiality is changing, thus finite element simulation is another suitable method to describe the complex strain variation and yield some accurate analysis [20]. Based on the modified Kachanov-Rabotnov (K-R) creep damage constitutive equations, Ling et al. [16] analyzed the influence of
η
dW FR0:2 t m−1 R0:2 F mþq−1 mA ¼n 2 t ¼ Ct q 1:2 h0 dt 3:33K sp r h0 h r 1:2
ð12Þ
0
Creep fracture occurs at the maximum deflection. Hence, Eq. (13) can be obtained when we integrate the right side of (12): Z
tr
n 0
R0:2 F 2 h0 r 1:2
t mþq−1 dt ¼
n R0:2 F mþq R0:2 FL q tr ¼u 2 t r ¼ const 2 m þ q h r 1:2 h r 1:2 0
ð13Þ
0
The parameter u, n, q are constants depending on materials and test environment. Correlation also can be obtained by the rupture time and the geometrical parameters of experimental fixture in Eq. (13) which can be used for different devices. Generally, in the same set of device, the sizes of lower die and small ball remain unchanged. Thus, a parameter P is used here: R0:2 r 1:2
P¼
ð14Þ
Expression (14) can be rewritten as: uP
W sp
W sp 2
h0
2
h0
t qr ¼ const
t qr ¼ const
ð15Þ
ð16Þ
The parameter n, u, q and P are all constants, Wsp is the mechanical work in SPCT. Thus, Eq. (16) can effectively be utilized to predict the creep life under mechanical work introduced by external load in SPCT. 4. Results and Discussion
Fig. 3. General strain variation with deflection for the widely used sizes of small punch creep tests.
In this section, the results of new experiment and our previous research works [18,24,25] were combined to examine the validity of energy model in practical engineering. In addition to the creep environment, different materials and specimen thickness factors were also discussed. At 550 °C, SPCT of 12Cr1MoV without protective gas are shown in Fig. 4. The deflection-time plots of SPCT can be divided to three regions: a decaying primary stage, a long secondary region and a well chiseled tertiary stage. A good agreement between the creep deformation trends of SPCT and conventional creep tests provides an important validation of the creep life prediction by SPCT [19]. It can be inferred from Fig. 5
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101
Fig. 4. Small punch creep curves for 12Cr1MoV at 550 °C.
that local necking happens at the fracture region which comprises little life of creep specimen. Thus, the assumption of cone with constant thickness does not introduce a substantial error in the analytical solution of energy model. The ultimate goal of this research is to establish the relationship between mechanical work and creep life. Considering the loading process, localized stresses around small contact area will lead to an initial large deformation as shown in Fig. 4. Thus, the displacement of initial elastic–plastic deformation should be subtracted when the mechanical work is calculated in SPCT. In this paper, the primary creep stage of specimen is defined as a starting point of creep deflection with the absence of initial loading displacement. From starting point of primary creep stage to different creep deformation regions, different creep displacements can be chosen. Multiply the displacement by load of SPCT, mechanical work is able to be calculated. For 12Cr1MoV, three different displacements (fracture displacement, 0.3 mm and 0.4 mm) were chosen. Meanwhile, the relationships between the mechanical work and fracture time based on Eq. (16) were established. In the identical experimental device, the variation of volumes and initial thickness were not considered since the diameter and thickness of small punch specimens remained unchanged. From Fig. 6, the linear relationship can be achieved by taking logarithm of mechanical work and time. In most instances, creep environment may affect creep deformation including fracture time, displacement and minimum creep rate. Therefore, the SPCT are often accomplished in protective gas [26]. As shown in Fig. 7, four load levels (513 N, 482 N, 463 N, 443 N) were chosen for SUS304 stainless steel and the temperature was 650 °C in the
Fig. 5. Deformation specimen of small punch test.
Fig. 6. The relationship between work and time for 12Cr1MoV without protective gas.
experiment. It is demonstrated that the proposed model is in well agreement with the calculated results of different creep displacements. Compared the creep results of new Cr5Mo and in-service Cr5Mo at 550 °C in Fig. 8 (424 N, 404 N, 375 N), the feasibility of energy model was testified. The same conclusion could also be found in the analysis of small punch creep results of single crystal alloy (CMSX-4) and 316 L (N) as shown in the Fig. 9 [27,28]. Meanwhile, the data of base metal (BM), welded metal (WM), fine grain heat affected zone (FGHAZ) and coarse grain heat affected zone (CGHAZ) of P92 steel weld joint in Fig. 10 (650 °C) were also found to obey energy model [29]. In this paper, the precision of fitting results calculated by fracture displacement is lower than that of certain creep displacement. A possible explanation for this phenomenon is that the condition for fracture is assumed to be ideal in energy model. However, as a complicated deformation process with multi-axial stress state, the failure mechanism is caused by an interaction between propagation of microcracks, creep damage evolution in tertiary acceleration stage [19]. The complicated failure mechanism and cutty creep acceleration phase will led to error and uncertainty in the measurement of final fracture deflection. Thus, how to define the actual rupture displacement is another problem. Considering the transient tertiary phase of SPCT, it is recommended that deformation before accelerated creep stage can be chosen as the creep displacement in the energy model.
Fig. 7. The relationship between work and time for SUS304.
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Fig. 10. The relationship between work and time for different zones in P92 steel weld joint. Fig. 8. The relationship between work and time for Cr5Mo (new material and in-service material).
Although the thickness of 0.5 mm have been widely accepted for obtaining the uniform parameters, the minor deviation or variation in thickness is inevitable, especially for the critical area which can not be easily extracted. Traditionally, it is generally accepted that the actual specimen thickness has an apparent effect on the creep properties. Several methods have been reported to establish the relation between fracture time and thickness. Comparing to the experiment, the finite element simulation may eliminate the influence of oxidation on the disk. Therefore, deflection - time curves obtained from simulations was used to analyze the influence of thickness. The creep temperature was 650 °C and the specimen thicknesses were 0.4 mm, 0.45 mm, 0.5 mm. Based on the results described in Reference [16], a linear relationship between log(Wsp/h20) and log(tr) can be found with the identical creep load level (463 N) as shown in Fig. 11. It proves that the proposed method is capable of producing extreme extrapolation results in creep analysis with different specimen thicknesses. It is important to note that the slope of straight lines in Figs. 6–10 is – q. It follows from the conversion ratio from mechanical work to the entropy increase. The apparent difference between new materials and in-service materials has lead to the belief that material will affect the value of q. An explanation for this phenomenon is that a higher conversation ratio can be obtained with the increase of q according to Eq. (4). A larger slope leads to a higher conversation ratio. The increase of irreversible entropy causes a rapid increase in deformation ratio and the
Fig. 9. The relationship between work and time for 316 L (N) and CMSX-4.
reduction of failure time. Therefore, the value of q can also be used to analyze the creep properties of different materials. Compared to the conventional test, SPCT can be used to analyze the material behavior of in-service components and localized structures. Thus, how to complement the deficiency of common methods and improve the accuracy of prediction has important significance in the development of SPCT. The proposed methodology holds a key advantage over the other methods that it can estimate remaining creep life convenient without the limitation of geometrical parameters of SPCT devices. Meanwhile, this methodology based on the energy model need not calculate the complicated multi-axial stress parameters in the deformation. Hence, we believe that this approach can provide some insight for the engineering in related fields. 5. Conclusion In this paper, an energy model of SPCT is used to extrapolate the creep life of materials at constant temperature. An approximate stress function depending on the load and device shape was introduced. Strain variation has been proven to be inversely proportional to the thickness and proportional to the deflection. The geometric factors including specimen thickness, hole radius of the lower die and radius of ball were considered in the model. Thus, this method is adaptable to different technical devices and specimen sizes. The results of different materials including 12Cr1MoV, new Cr5Mo, inservice Cr5Mo, SUS304, CMSX-4, 316 L (N) and different thicknesses were used in order to verify the accuracy of energy model. Meanwhile,
Fig. 11. The relationship between work and time with different specimen thicknesses.
S. Yang et al. / Materials and Design 91 (2016) 98–103
it can be found that this method is able to be applied to predict the creep life in different test environments. As a simple and convenient model of SPCT, the conclusion is capable to be widely used to extrapolate the creep life of materials. Acknowledgements The authors hope to acknowledge the support provided by the Research Innovation Project for College Graduates of Jiangsu Province (KYZZ15_0229) and National high technology research and development plan (863 plan) of China (Grant No. 2012AA040105). References [1] F. Hou, H. Xu, Y.C. Wang, L. Zhang, Determination of creep property of 1.25Cr0.5Mo pearlitic steels by small punch test, Eng. Fail. Anal. 28 (2013) 215–221. [2] S. Haroush, E. Priel, D. Moreno, A. Busiba, I. Silverman, A. Turgeman, et al., Evaluation of the mechanical properties of SS-316 L thin foils by small punch testing and finite element analysis, Mater. Des. 83 (2015) 75–84. [3] T. Bai, K.S. Guan, Evaluation of stress corrosion cracking susceptibility of nanocrystallized stainless steel 304 L welded joint by small punch test, Mater. Des. 52 (2013) 849–860. [4] T.H. Hyde, M. Stoyanov, W. Sun, C.J. Hyde, On the interpretation of results from small punch creep tests, J. Strain Anal. Eng. Des. 45 (2010) 141–164. [5] CEN Workshop Agreement, CWA 15627:2006 E, small punch test method for metallic materials: Part A, CEN, Brussels, 2006. [6] Z. Yang, Z.W. Wang, Relationship between strain and central deflection in small punch creep specimens, Int. J. Press. Vessel. Pip. 80 (2003) 397–404. [7] J. Chakrabarty, A theory of stretch forming over hemispherical punch heads, Int. J. Mech. Sci. 12 (1970) 315–325. [8] K. Milička, F. Dobeš, Small punch testing of P91 steel, Int. J. Press. Vessel. Pip. 83 (2006) 625–634. [9] B. Ule, T. Šuštar, F. Dobeš, K. Milička, V. Bicego, S. Tettamanti, et al., Small punch test method assessment for the determination of the residual creep life of service exposed components: outcomes from an interlaboratory exercise, Nucl. Eng. Des. 192 (1999) 1–11. [10] S.I. Komazaki, T. Kato, Y. Kohno, H. Tanigawa, Creep property measurements of welded joint of reduced-activation ferritic steel by the small-punch creep test, Mater. Sci. Eng. A 510-511 (2009) 229–233. [11] F. Dobeš, K. Milička, Application of creep small punch testing in assessment of creep lifetime, Mater. Sci. Eng. A 510-511 (2009) 440–443.
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