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A modified reference area method to estimate creep behaviour of service-exposed Cr5Mo based on spherical indentation creep test
Vacuum 169 (2019) 108923
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A modified reference area method to estimate creep behaviour of serviceexposed Cr5Mo based on spherical indentation creep test
T
Wenbin Lu, Xiang Ling∗, Sisheng Yang∗∗ Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, No. 30 Pu Zhu South Road, Nanjing, 211816, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Spherical indentation creep test Modified reference area method Creep deformation Service-exposed Cr5Mo
In this study, indentation creep tests under different conditions were conducted using a self-designed vacuum high-temperature indentation creep device to investigate the creep properties of service-exposed Cr5Mo. A microstructure test was also conducted and a prominent pile-up was found on the surface of the specimen. We performed a comprehensive finite element analysis to evaluate the non-uniform deformation characteristics of the spherical indentation creep test. The stress distribution and stress evolution for indenters with different diameters were also estimated during the test. Subsequently, considering the difficulty of calculating the equivalent stress owing to the area variation during spherical indentation, a modified reference area method was proposed. Based on a load relaxation simulation and geometric factor analysis, a general formula for the reference areas of indenters with different diameters was given. Thus, the equivalent stress during spherical indentation could be calculated, and the creep behaviour of service-exposed Cr5Mo was estimated. The results when using the modified method in this study demonstrated good consistency with the results obtained by the standard method. To sum up, the spherical indentation creep test has prominent advantages over the uniaxial tensile creep test in the analysis of in-service components.
1. Introduction It has been a critical issue how to evaluate creep properties of materials properly, as the structures facing severe operation conditions in aerospace, nuclear power and petrochemical industries [1]. Generally speaking, the uniaxial tensile creep test is the standard technique for estimating the creep behaviour of materials. However, in some situations, the large size of a specimen restricts the use of the conventional technique. For instance, the creep performance of in-service furnace pipes cannot be estimated using this measurement method. The indentation creep test evolved from the conventional indentation test [2] and was proposed as a technique to evaluate the creep properties of miniature specimens [3,4]. In addition to its non-destructive nature [5], the indentation creep test also has the advantage of being able to determine the creep properties of specific materials or materials in localised regions, such as bulk metallic glasses [6], the heat-affected zone of welds [7], and so on [8]. To popularise the engineering application of this technique, it is
critical to establish rational correlations between the properties obtained from the indentation creep test and conventional test. Chu et al. [9] compared the activation energy values obtained from cylindrical flat indentation and conventional creep tests, and found that the measurements agreed. Based on a conversion factor, Liu et al. [10] presented a method that set up the relationship between the creep parameters of the two tests. However, the contact area continuously rises when a non-flat indenter is being pressed in by degrees, which makes it difficult to estimate the creep behaviour. In this regard, a functional relationship between the contact area and indentation depth for a Berkovich indenter was established by Brotzen [11], which allowed the stress at an arbitrary time to be calculated. Besides, a trend of pile-up normally exists during non-flat indentation, resulting in a complicated variation of the contact area. Therefore, some researchers have employed the incremental plasticity theory to revise the contact area during spherical indentation [12]. Furthermore, Arai et al. [13] proposed a procedure that can estimate the creep exponent and coefficient from the impression size rather than the penetration depth. Their
∗ Corresponding author. Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, No. 30 Pu Zhu South Road, Nanjing, 211816, China. ∗∗ Corresponding author. Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, No. 30 Pu Zhu South Road, Nanjing, 211816, China. E-mail addresses: [email protected] (X. Ling), [email protected] (S. Yang).
https://doi.org/10.1016/j.vacuum.2019.108923 Received 9 July 2019; Received in revised form 6 August 2019; Accepted 2 September 2019 Available online 04 September 2019 0042-207X/ © 2019 Elsevier Ltd. All rights reserved.
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method avoided the problem of the complicated calculation of the equivalent stress due to area variation. On the other hand, Takita et al. [14] proposed a method based on a stress relaxation numerical test, which made it possible to evaluate the equivalent stress in spite of the area changing problem. Not only that, various research methods were used for the instrumentation indentation test, which made the technique more practical. For example, elastic-plastic cavity expansion model and slip-line limit method have been developed to analyse the indentation deformation of plastic materials [15,16]. Ai et al. [17] proposed a new modified expanding cavity model for characterising the indentation behaviour of the bulk metallic glass. Besides, inverse method has also been commonly used for indentation tests [18]. To systematically estimate the creep behaviours of materials based on spherical indentation creep tests, it is vital to rationally evaluate the equivalent stress. However, the pile-up phenomenon and geometric factors during the test make the stress calculation complex. In addition, the oxidation of the material under high temperature test conditions may also affect the accuracy of the measurement method. Thus, corresponding studies ought to be conducted to verify the universality and accuracy of the spherical indentation creep test. In our previous study, another batch of service-exposed materials were investigated by the small punch creep test [19]. It was proved that the non-uniform and large deformation did not affect the accuracy of the measurement results. Although different service conditions may lead to changes in material characteristics, it can still be concluded that the creep properties of materials can be reasonably evaluated by microtest technique. Therefore, the indentation creep test is considered in present study due to its non-destructive property on specimens. Considering the discussion above, this study aims to estimate the creep behaviour of service-exposed Cr5Mo on the basis of the spherical indentation creep test. A vacuum atmosphere was set outside the fixture to avoid interference of oxidation. A microstructure test and finite element analysis were also performed to obtain a rational evaluation of the deformation characteristics during the test. Based on the simulation and geometric factor analysis, a modified reference area method is proposed. A general formula for the reference area during spherical indentation is given, which makes it possible to calculate the equivalent stress at a certain load and time. Finally, the creep behaviour of Cr5Mo is precisely estimated. We believe that the results of this study would be meaningful in the analysis of in-service components and further investigation of the spherical indention creep test.
Fig. 1. Schematic diagram of indentation creep test.
up with two heating elements. When the thermal expansion was stable, a constant load was applied to the indenter and transmitted to the specimen. By monitoring the creep deformation, indentation depth versus time curves under different loads could be obtained. 2.2. Experimental curves and microstructure Typical creep curves for the tensile creep test are shown in Fig. 2. Three distinct stages are observed, with the nominal stresses ranging from 160 MPa to 220 MPa. Fig. 3(a) shows spherical indentation depth versus time curves, with the applied load ranging from 250 N to 550 N. Similarities and differences exist between the creep curves obtained by the two measurement methods. In the primary creep stage, the work hardening rate is greater than the recovery rate for creep deformation, resulting in a decrease in the creep rate with time. Subsequently, the work hardening and recovery rates gradually reach a relatively balanced state during the secondary creep stage, which results in a fairly constant creep rate. For the conventional creep curves, the tertiary stage is observed because of the specimens necking and forming creep voids [20]. However, no substantial fracture will occur to the specimen in the indentation creep test, and it explains the absence of tertiary stage in the creep curves. Compare to other micro-test measurement methods, a fundamental summary is made. Owing to the destructive behaviour of specimens in the tertiary creep stage, certain micro-test measurement results such as small punch creep curves are more similar to the uniaxial creep curves than the indentation creep curves in the form of expressions. But the creep characteristics still differ, as the contribution of the secondary creep time to the overall creep life is larger in the micro-test measurement results and the tertiary stage occupies the majority of the rupture strain in the uniaxial creep results.
2. Experimental In order to verify the universality of the indentation creep test, a service-exposed Cr5Mo steel sample obtained from a furnace pipe was used in this study. The pipe was in service at 343 °C and 0.26 MPa for approximately ten years. Considering the maximum service temperature of the pipes in the same process, all of the following tests were conducted at 550 °C to obtain a reasonable assessment of the material. 2.1. Experimental procedure The indentation creep tests were performed using a self-designed vacuum high-temperature indentation creep device. With a miniature specimen, the oxidation cannot be neglected because of the tiny creep deformation. Thus, a vacuum atmosphere was set up to avoid interference from oxidation to the maximum extent. The fixture structure and assembly are shown in Fig. 1. A cylindrical flat indenter with the diameter of 1 mm and spherical indenters with the diameters of 1 mm and 2 mm were employed in the experiments. Before the test, specimens were shaped by wire cutting with a size of 10 mm × 10 mm × 3 mm and then ground with a succession of abrasive paper until using 1200 grit silicon carbide abrasive paper. Thereafter, the furnaces in which the fixture and specimen were placed on were vacuum pumped and then heated. To ensure the heating efficiency, the heating apparatus was set
Fig. 2. Uniaxial tensile creep curves. 2
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Fig. 4. Optical micrograph of 550 N spherical indentation creep deformation: (a) overall morphology and (b) local region around indenter.
postponed with an increase in the applied load, and the ratio of the transient indentation depth to the total indentation depth progressively increases during the same dwell time. It is noteworthy that the ideal contact stress should be stable when the spherical indenter is completely pressed into the specimen, yet the displacement rate still shows a tiny tendency to change by degrees. We considered that a trend of pile-up [22] existed on the surface of a specimen during spherical indentation, resulting in the growth of the true contact area. Thereby, it brought about the decrease of the equivalent stress and explained the decrease in the displacement rate. To verify the theoretical analysis and characterise the creep deformation, we examined the microstructure under an optical microscope, as shown in Fig. 4. A prominent pile-up can be seen on the surface of the specimen, which corresponds to the analysis above. This pile-up phenomenon was also observed in the traditional indentation test. Researchers studied the hardness of materials using the conventional indention test and proposed that the pile-up effect that results in the deviation of the contact area cannot be ignored [17]. The situation in this study was very similar because the equivalent stress could not easily be calculated as a result of the area variation. The following demonstration could explain the phenomenon to some extent: during the test, the plastic flow existed and was constrained in a large elastic/ rigid volume in the material. The flow was directed upward and then appeared as a pile-up on the surface of the specimen [23]. Referring to the dislocation distribution observed in the vicinity of the indenter and the creep coefficient mentioned below, the creep deformation mechanism of the service-exposed Cr5Mo may be described as dislocation movement-controlled creep [24]. Besides, two distinguishable regions could be observed as shown in Fig. 4(b). In the region beneath the indenter, the material was found to experience a massive shear deformation. The other region, which was not beneath the spherical indenter, exhibited no change in the shape of the grains. This revealed the absence of plastic deformation away from the indentation, showing the
Fig. 3. Experimental curves of indentation creep tests: (a) spherical indenter, and (b) comparison between cylindrical flat and spherical indenters.
The difference in the form does not affect the accuracy of the analysis. In contrast, the absence of the tertiary stage in the indentation creep test makes the deformation more stable and convenient to evaluate. A comparison between the spherical and cylindrical flat indentation results is shown in Fig. 3(b). For the flat cylindrical indentation, the indentation stress is constant during the creep test, and a steady state can be achieved after a transient primary stage [21]. It is not hard to understand that the overall stress level of the flat indentation creep is lower than the spherical one at the same load. Because of the growth of the contact area, a decrease in the contact stress is inevitable in the spherical indentation creep test. What we discussed above basically explains the phenomenon of the spherical indentation creep velocity being greater than the other one at the same load, but its curve demonstrates a more remarkable trend to drop. On the other hand, although the steady-state velocity could not be obtained from the spherical indentation creep test, the experimental curves still show that the creep velocity of the spherical indentation tends to be stable by degrees, similar to the flat one. This indicates that an approximate steady stage can be achieved in the spherical indentation creep test, because the growth of the contact area becomes increasingly insignificant after a sufficiently long creep time. In addition, for both sets of curves, we can see that the time required to enter the steady stage is gradually
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Fig. 6. Comparison of experimental and simulation curves. Fig. 5. Finite element model of spherical indentation creep test.
localised nature of the indentation creep deformation [25]. 3. Finite element simulation The aim of the finite element simulation was to further characterise the creep deformation mechanism of the material. To make a systematic comparison of the indenter shapes, a spherical indentation creep test with an indenter diameter of 2 mm was also carried out. 3.1. Finite element model Based on the symmetrical geometry of the test, a two-dimensional finite element model was established as shown in Fig. 5. In particular, the size of the model was consistent with the experiment. The indenter was set as a rigid body, and the specimen was deformable. The contact type between the indenter and the specimen was surface-to-surface contact, and the friction coefficient was set as 0 between them. Based on fitting using the previously mentioned uniaxial creep curves, B and n were defined as 3.72 × 10−20 and 7.4, respectively. The plastic parameters were described by defining the true plastic strain. The other theoretical values included a Young's modulus E = 153000 MPa and Poisson's ratio μ = 0.29. Meanwhile, a similar finite element model with an indenter diameter of 2 mm was also established for a later analysis. 3.2. Creep deformation analysis Fig. 6 shows the indentation depth versus time curves at different loads for both the experiment and simulation. In general, the simulation curves display trends similar to those of the experimental curves. It can be found that the two sets of curves at 300 N almost entirely overlap. The other simulation curves are slightly lower than the experimental curves, but they still demonstrate good consistency in the creep stage. The initial deviation during loading may account for the difference. In addition, the internal defects of service-exposed materials can also lead to dispersions of creep results. Therefore, the finite element model established in this study was rational. Fig. 7 shows the evolution of the pile-up on the surface of the Cr5Mo specimen. We can see that the maximum pile-up on the surface of the specimen continues to grow with increases in the load and dwell time. For the case where the diameter of the indenter was 1 mm, as shown in Fig. 7(a), it was found that when the indenter was gradually pressed in, the maximum pile-up occurred at a distance of one radius to twice the
Fig. 7. Evolution of pile-up on surface of Cr5Mo specimen: (a) time-dependent and load-dependent evolution with indenter diameter of 1 mm, and (b) timedependent evolution at 900 N with indenter diameter of 2 mm.
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Fig. 8. Stress distributions on surface of specimen with dwell time of 30 h: (a) D = 1 mm, F = 450 N; (b) D = 2 mm, F = 900 N.
strain was at the centre of the contact region and decreased with the radial distance from the axis of symmetry to the contact edge [27]. This basically corresponded to the simulation results obtained in this study. For the latter case, there was a stress concentration at the bulged region under the contact area of the indenter, which hindered the further growth of the pile-up. This explained the phenomenon that the evolution of the pile-up with the indenter diameter of 2 mm had a slower tendency, as previously mentioned. It was concluded that the largest strain of the specimen appeared in the necking region by means of certain micro-test technique in our previous research. However, the ductile fracture did not occur to the specimen during the indentation creep test, and the corresponding appearance was a pile-up hillock. Moreover, the consistency between the finite element analysis results and microstructure again verified the rationality of the model. To obtain a more intuitive view of the stress evolution during spherical indentation, the Mises stress along the axial direction was extracted separately for the two cases (Fig. 9). In general, with an increase in the creep time, there was a state where the Mises stress gradually decreased as a result of the increase in the contact area. Stress redistribution with creep time can be observed and the difference in creep rates during the test is found to be the cause for the phenomenon [23]. In the former case, we find that the curves in Fig. 9(a) can be
radius from the symmetric centre. Then, the pile-up decreased with an increase in the distance and became negligible by degrees after triple the radius distance. However, the situation was not exactly the same when the diameter of the indenter was 2 mm, as shown in Fig. 7(b). Because of the smaller deformation, the indenter was not fully pressed in. The maximum pile-up occurred at approximately the distance of one radius from the symmetric centre, and its growth tendency was not as significant as the former. The pile-up hillocks obtained in the finite element simulation were somewhat greater than those obtained experimentally, and similar phenomenon was also observed and illustrated by Kucharski [26]. Thus, the accuracy of the finite element predictions of the pile-up in the vicinity of the indenter could be regarded as satisfactory. The size of the pile-up hillock is not quantitatively described in this paper, but it is generally believed that the specific value is related to the strain hardening exponent and the ratio of yield stress to elastic modulus of the material, and the maximum indentation depth to the radius of the spherical indenter during the test. The stress distributions on the specimen surfaces for both cases are shown in Fig. 8. It is observed that a great stress concentration occurs in the centre of the contact region and the stress spreads out in the radial direction from the centre of the specimen. Karthik also studied the spherical indentation test and concluded that the maximum plastic 5
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4. A modified reference area method It is well known that metallic materials display time-dependent creep behaviours at high homologous temperatures, and uniaxial creep tests are generally conducted to determine the relationship between steady-state strain rate ε˙ c and applied stress σ using the Norton equation as follows:
ε˙ c = Bσ n
(1)
where the constants B and n uniquely characterise this power-law creep response. Using the uniaxial measurement method, the creep parameters can be determined by plotting the measured strain rates against the applied stresses on logarithmic scales. For the spherical indentation creep test, the most troublesome problem is that the contact area continuously changes with the growth of the indentation depth. In addition, the influence of pile-up, which results in a complicated variation of the true contact area, cannot be neglected. It has been found that the reference stress at a certain time cannot be easily calculated. To precisely estimate the creep behaviour despite the complicated stress diversification, a modified reference area method is proposed. We studied this method using the following procedures on the basis of a load relaxation simulation. In the case of uniaxial loading, the total strain increment is the sum of the elastic strain increment, plastic strain increment, and creep strain increment. In the strain maintenance process, the total strain increment is zero, and the plastic strain remains undeveloped [14]. Taking Hooke's law into consideration, the creep strain rate ε˙tc can be expressed as follows：
1 ε˙tc = − σ˙ t E
(2)
where the subscript t denotes a uniaxial condition, E is Young's modulus and σ˙ t is the stress relaxation rate. Using Eq. (2), the creep strain rate ε˙tc can be calculated from the stress relaxation rate which is obtained as the slope of the stress relaxation curve. Because the true contact area increases as result of the pile-up trend and geometric factor during the spherical indentation, we should give a rational definition of the reference area, to calculate the corresponding equivalent stress. The equivalent stress is obtained with the following procedure using the new reference area Sr . The stress due to the indentation is defined as follows: Fig. 9. Evolution of Mises stress on surface of specimen: (a) D = 1 mm, F = 450 N; (b) D = 2 mm, F = 900 N.
σt =
F Sr
(3)
where F is the applied force, and the stress is the value after translating the multiaxial stress into the tensile stress. In other words, the stress in Eq. (3) can be regarded as the equivalent stress for the indentation creep test. Using time differentiation, Eq. (3) is rewritten as follows:
divided into three regions indicated with Ⅰ, Ⅱ, and Ⅲ respectively. The first stage shows the trend of slowly decreasing as the contact area increases by degrees. The second region coincides well with the position where the maximum pile-up exists, and the rapid decrease in stress in this region is due to the plastic flow [28]. For the case with the indenter diameter of 2 mm, as shown in Fig. 9(b), it can be found that the maximum stress point occurs slightly before the position of maximum pile-up in the radius distance, and there is a significant stress wave around the bulged region [29]. The following demonstration may account for the difference of the stress distributions: the indentation depth has reached a remarkable value in the elastic-plastic stage for the former case, which is almost half of the radius of the indenter. However, for the later case, the initial indentation depth is a smaller value compared with the radius of the indenter. It means that the indenter is far from fully pressed in, which results in the stress concentration at the edge of the contact area between the indenter and the specimen and the obvious stress diversification. Besides, the tertiary stage shows a slowly decreasing trend once again, and the stress becomes insignificant away from the indenter, which agrees with the localised nature of the indentation creep deformation.
1 σ˙ = F˙ Sr
(4)
The load relaxation rate F˙ in Eq. (4) can be obtained as the slope of the load relaxation curves which can be derived from the following finite element analysis. The indentation depth maintenance simulation used the finite element model established in section 3.1. Ten kinds of load conditions were employed throughout the process of the spherical indentation. In the simulation procedure, the indenter was pushed into the specimen with an equal indentation rate until the predetermined depth under each load condition. Subsequently, the indentation depths were kept constant for 2 h, and the load relaxation curves were obtained (Fig. 10). By substituting Eq. (4) for Eq. (2), the creep strain rate could be obtained from the load relaxation curve as:
ε˙tc = −
1 ˙ F ESr
(5)
Then, using Eq. (3) to calculate the stress for the indentation creep 6
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Fig. 10. Load relaxation curves during indentation depth maintenance.
Fig. 11. Relationship between reference area and ideal contact area.
test, the following equation can be obtained with Eqs. (1), (3) and (5) as:
In addition, Eq. (8) can be simplified into Eq. (10):
Sc = πh (D − h)
ni
−
1 ˙ F F = Bi ⎛ ⎞ ESr ⎝ Sr ⎠ ⎜
⎟
Eq. (9) can be rewritten as Eq. (11):
(6)
Sr = k2 h (D − h)
where Bi and ni are the creep coefficient and creep exponential obtained on the basis of the indentation creep test, respectively. The reference area is defined as the area where the stress is kept constant during the load relaxation. If the stress is obtained in the reference area, Sr , Norton's law should give the same values for both the indentation creep test and uniaxial creep test. Consequently, using the material constants B and n in Norton's law, the new reference area Sr is expressed as the following equation:
(7)
In other words, the reference area corresponding to an indentation depth can be calculated using Eq. (7). Meanwhile, another series of numerical analyses with an indenter diameter of 2 mm was also conducted following the same steps. Taking the geometric factor into consideration, there should be a certain relationship between the reference area and the ideal contact area. Each reference area calculated by Eq. (7) and obtained from the specific relationship with the ideal contact area should have the same value at a given indentation depth. This implies that we can plot a scatter diagram of the reference area versus the ideal contact area to verify the relationship between them. For the spherical indenter, the ideal contact area is calculated by Eq. (8):
Sc = π (r 2 − (r − h)2)
(11)
The diameter D of a particular spherical indenter is a fixed value. Numerous researchers have applied indentation creep tests to evaluate the superficial properties of specimens. If the indentation depth h is a small value compared to the diameter D, the reference area can be linearly fitted as a function of indentation depth h. Choosing the values until the indenter is half pressed, it is found that the reference area and indentation depth still have a proportional relationship with a small deviation, and the fitting results share a nearly common coefficient as before (Fig. 12). In some cases such as thick wall in-service pipes, the creep properties of a material may not be accurately evaluated with a small indentation depth. Thus, it is also necessary to obtain a rational evaluation of the creep performance of materials with a large indentation depth, and the corresponding reference area needs to be obtained. Based on the fitting results shown in Fig. 13, the modified reference area for spherical indentation is defined as follows:
1
−F˙ ⎫1 − n Sr = ⎧ n⎬ ⎨ BEF ⎩ ⎭
(10)
(8)
where r is the radius of the indenter and h is the indentation depth. The scatter diagrams of the reference area versus ideal contact area of cases with different diameters are shown in Fig. 11. We can see that the reference area is proportional to the ideal contact area, with a small deviation. Thus, it can be claimed that during the indentation creep test, the reference area normally depends on the ideal contact area Sc , as follows:
Sr = k1 Sc
(9)
where k1 is a constant related to the material properties and test conditions. By the substitution of Eqs. (7)–(9), the constant k1 can be obtained, and both the fitting results have similar values. This verifies the reliability of the linear relationship and proves the versatility of the method for distinct indenters.
Fig. 12. Relationship between reference area and indentation depth till indenters half pressed. 7
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Fig. 14. Comparison of relationship between stress and creep strain rate.
particularity of the service-exposed material investigated in this study, it should be noted that more experimental tests for other materials and more in depth physical characterisation and more accurate physically based numerical modelling are needed in order to further evaluate the feasibility of the modified reference area method before putting it to practical application. Conclusively, by applying the previously discussed procedures, it was possible to obtain the creep parameters of the service-exposed Cr5Mo, which corresponded to the fitted value from the conventional method. Because the indentation creep test is a miniature test, it can effectively and economically evaluate creep properties of in-service components. However, the problem of the complicated stress calculation restricts the engineering application of the spherical indention creep test in related fields. Therefore, we believe that the modified reference area method reported in this paper can provide new insights for the study of the spherical indentation creep test. 5. Conclusions This paper aims to estimate the creep behaviour of service-exposed Cr5Mo and evaluate its creep properties using a modified reference area method. Above all, indentation and uniaxial creep tests were conducted under different conditions. Subsequently, finite element simulations and a series of theoretical analyses were carried out to obtain reasonable evaluations of the material. The main conclusions are as follows:
Fig. 13. Determination of modified reference area: (A) D = 1 mm and (B) D = 2 mm.
Sr = krh
(12)
where k is a constant related to the material properties and test conditions, r is the indenter radius and h is the indentation depth. Considering the indentation depth versus time curves under different loads are obtained using indentation creep tests, the equivalent stress at a certain load and time can be calculated. The creep strain rate for spherical indentation can be evaluated by the following equation [30]:
ε˙ c =
h˙ h
(1). Experiments with the service-exposed Cr5Mo under different conditions were conducted. The results showed that the creep performance of the material could be reasonably evaluated using the spherical indentation creep test. (2). A microstructure test and finite element analysis were performed to evaluate the non-uniform deformation characteristics of the spherical indentation creep test. (3). An estimation of creep stress during the test was performed. The stress distributions and stress evolutions of indenters with different diameters were distinct, accounting for the difference in the evolution of the pile-up. (4). A modified reference area method was proposed based on the load relaxation simulation and geometric factor analysis. A simple linear relationship was established between the reference area and indentation depth, and the equivalent stresses for distinct indenters during spherical indentation at arbitrary times could be calculated. (5). The modified reference area method reported in this paper made it possible to precisely evaluate the creep exponential of service-exposed Cr5Mo based on the spherical indentation creep test.
(13)
where h˙ is the displacement rate obtained from the experimental curve. From now on, we can establish the relationship between the creep strain rate and equivalent stress at a certain load using Eqs. (1), (3), (12) and (13). Using the modified reference area presented in Fig. 13(a), we can determine the creep parameters by plotting the creep strain rates against the equivalent stresses on logarithmic scales, and make a comparison with the conventional measurement (Fig. 14). The creep constant B and creep exponential n for the spherical indentation creep test are 9.08 × 10−19 and 7.04, while the values obtained from the uniaxial creep test are 3.72 × 10−20 and 7.4, respectively. It is noteworthy that the creep exponentials obtained by both measurement methods demonstrate good consistency. The creep constant remains some deviations, but within a reasonable range. Considering the 8
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Acknowledgements
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This work is supported by National Natural Science Foundation Project of China (Grant No. 51805245) and China Postdoctoral Science Foundation (2017M621738). References [1] Y.B. Hu, L. Zhang, C.Q. Cheng, P.T. Zhao, T.S. Cao, G.P. Guo, J. Zhao, Influence of specimen thickness on the creep behavior of a directional solidification nickel-based superalloy, Vacuum 150 (2018) 105–115 https://doi.org/10.1016/j.vacuum.2018. 01.032. [2] A.C. Fischer-Cripps, A review of analysis methods for sub-micron indentation testing, Vacuum 58 (2000) 569–585 https://doi.org/10.1016/S0042-207X(00) 00377-8. [3] D.H. Sastry, Impression creep technique—an overview, Math. Sci. Eng. 409 (1) (2005) 67–75 https://doi.org/10.1016/j.msea.2005.05.110. [4] S.N.G. Chu, J.C.M. LI, Impression creep; a new creep test, J. Mater. Sci. 12 (1977) 2200–2208 https://doi.org/10.1007/BF00552241. [5] B.X. Xu, X.M. Wang, B. Zhao, Z.F. Yue, Study of crystallographic creep parameters of nickel-based single crystal superalloys by indentation method, Math. Sci. Eng. 478 (2008) 187–194 https://doi.org/10.1016/j.msea.2007.07.038. [6] Y.D. Sun, Z.Q. Li, J.S. Liu, M.Q. Cong, J.Y. Qin, Indentation creep behaviors of Mg61Cu28Gd11 and (Mg61Cu28Gd11)99.5Sb0.5 bulk metallic glasses at room temperature, J. Rare Earths 29 (2011) 253–258 https://doi.org/10.1016/S10020721(10)60441-3. [7] W. Sun, T.H. Hyde, A potential method for determining the haz properties in welds using two-material impression creep test, Acta. Metall. Sin-Engl. 17 (4) (2004) 591–596 http://www.amse.org.cn/EN/Y2004/V17/I4/591. [8] R. Seltzer, Y.-W. Mai, Depth sensing indentation of linear viscoelastic–plastic solids: a simple method to determine creep compliance, Eng. Fract. Mech. 75 (17) (2008) 4852–4862 https://doi.org/10.1016/j.engfracmech.2008.06.012. [9] S.N.G. Chu, J.C.M. LI, Impression creep of β-tin single crystals, Math. Sci. Eng. 39 (1979) 1–10 https://doi.org/10.1063/1.1722705. [10] Y.J. Liu, B. Zhao, B.X. Xu, Z.F. Yue, Experimental and numerical study of the method to determine the creep parameters from the indentation creep testing, Math. Sci. Eng. 456 (1–2) (2007) 103–108 https://doi.org/10.1016/j.msea.2006. 11.098. [11] F.R. Brotzen, Mechanical testing of thin films, Int. Mater. Rev. 39 (1) (1994) 24–45 https://doi.org/10.1109/SENSOR.1991.149045. [12] H. Lee, J.H. Lee, G.M. Pharr, A numerical approach to spherical indentation techniques for material property evaluation, J. Mech. Phys. Solids 53 (9) (2005) 2037–2069 https://doi.org/10.1016/j.jmps.2005.04.007. [13] M. Arai, High-temperature creep property of high-Cr ferritic heat-resisting steel identified by indentation test, J. Press. Vess-T ASME. 139 (2) (2016) 021403https://doi.org/10.1115/1.4033943. [14] A. Takita, K. Sasaki, K.-i. Ohguchi, A method to identify steady creep strain from indentation creep using a new reference area of indentation, J. Electron. Mater. 43 (7) (2014) 2530–2539 https://doi.org/10.1007/s11664-014-3196-0.
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A modified reference area method to estimate creep behaviour of serviceexposed Cr5Mo based on spherical indentation creep test
T
Wenbin Lu, Xiang Ling∗, Sisheng Yang∗∗ Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, No. 30 Pu Zhu South Road, Nanjing, 211816, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Spherical indentation creep test Modified reference area method Creep deformation Service-exposed Cr5Mo
In this study, indentation creep tests under different conditions were conducted using a self-designed vacuum high-temperature indentation creep device to investigate the creep properties of service-exposed Cr5Mo. A microstructure test was also conducted and a prominent pile-up was found on the surface of the specimen. We performed a comprehensive finite element analysis to evaluate the non-uniform deformation characteristics of the spherical indentation creep test. The stress distribution and stress evolution for indenters with different diameters were also estimated during the test. Subsequently, considering the difficulty of calculating the equivalent stress owing to the area variation during spherical indentation, a modified reference area method was proposed. Based on a load relaxation simulation and geometric factor analysis, a general formula for the reference areas of indenters with different diameters was given. Thus, the equivalent stress during spherical indentation could be calculated, and the creep behaviour of service-exposed Cr5Mo was estimated. The results when using the modified method in this study demonstrated good consistency with the results obtained by the standard method. To sum up, the spherical indentation creep test has prominent advantages over the uniaxial tensile creep test in the analysis of in-service components.
1. Introduction It has been a critical issue how to evaluate creep properties of materials properly, as the structures facing severe operation conditions in aerospace, nuclear power and petrochemical industries [1]. Generally speaking, the uniaxial tensile creep test is the standard technique for estimating the creep behaviour of materials. However, in some situations, the large size of a specimen restricts the use of the conventional technique. For instance, the creep performance of in-service furnace pipes cannot be estimated using this measurement method. The indentation creep test evolved from the conventional indentation test [2] and was proposed as a technique to evaluate the creep properties of miniature specimens [3,4]. In addition to its non-destructive nature [5], the indentation creep test also has the advantage of being able to determine the creep properties of specific materials or materials in localised regions, such as bulk metallic glasses [6], the heat-affected zone of welds [7], and so on [8]. To popularise the engineering application of this technique, it is
critical to establish rational correlations between the properties obtained from the indentation creep test and conventional test. Chu et al. [9] compared the activation energy values obtained from cylindrical flat indentation and conventional creep tests, and found that the measurements agreed. Based on a conversion factor, Liu et al. [10] presented a method that set up the relationship between the creep parameters of the two tests. However, the contact area continuously rises when a non-flat indenter is being pressed in by degrees, which makes it difficult to estimate the creep behaviour. In this regard, a functional relationship between the contact area and indentation depth for a Berkovich indenter was established by Brotzen [11], which allowed the stress at an arbitrary time to be calculated. Besides, a trend of pile-up normally exists during non-flat indentation, resulting in a complicated variation of the contact area. Therefore, some researchers have employed the incremental plasticity theory to revise the contact area during spherical indentation [12]. Furthermore, Arai et al. [13] proposed a procedure that can estimate the creep exponent and coefficient from the impression size rather than the penetration depth. Their
∗ Corresponding author. Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, No. 30 Pu Zhu South Road, Nanjing, 211816, China. ∗∗ Corresponding author. Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing Tech University, No. 30 Pu Zhu South Road, Nanjing, 211816, China. E-mail addresses: [email protected] (X. Ling), [email protected] (S. Yang).
https://doi.org/10.1016/j.vacuum.2019.108923 Received 9 July 2019; Received in revised form 6 August 2019; Accepted 2 September 2019 Available online 04 September 2019 0042-207X/ © 2019 Elsevier Ltd. All rights reserved.
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method avoided the problem of the complicated calculation of the equivalent stress due to area variation. On the other hand, Takita et al. [14] proposed a method based on a stress relaxation numerical test, which made it possible to evaluate the equivalent stress in spite of the area changing problem. Not only that, various research methods were used for the instrumentation indentation test, which made the technique more practical. For example, elastic-plastic cavity expansion model and slip-line limit method have been developed to analyse the indentation deformation of plastic materials [15,16]. Ai et al. [17] proposed a new modified expanding cavity model for characterising the indentation behaviour of the bulk metallic glass. Besides, inverse method has also been commonly used for indentation tests [18]. To systematically estimate the creep behaviours of materials based on spherical indentation creep tests, it is vital to rationally evaluate the equivalent stress. However, the pile-up phenomenon and geometric factors during the test make the stress calculation complex. In addition, the oxidation of the material under high temperature test conditions may also affect the accuracy of the measurement method. Thus, corresponding studies ought to be conducted to verify the universality and accuracy of the spherical indentation creep test. In our previous study, another batch of service-exposed materials were investigated by the small punch creep test [19]. It was proved that the non-uniform and large deformation did not affect the accuracy of the measurement results. Although different service conditions may lead to changes in material characteristics, it can still be concluded that the creep properties of materials can be reasonably evaluated by microtest technique. Therefore, the indentation creep test is considered in present study due to its non-destructive property on specimens. Considering the discussion above, this study aims to estimate the creep behaviour of service-exposed Cr5Mo on the basis of the spherical indentation creep test. A vacuum atmosphere was set outside the fixture to avoid interference of oxidation. A microstructure test and finite element analysis were also performed to obtain a rational evaluation of the deformation characteristics during the test. Based on the simulation and geometric factor analysis, a modified reference area method is proposed. A general formula for the reference area during spherical indentation is given, which makes it possible to calculate the equivalent stress at a certain load and time. Finally, the creep behaviour of Cr5Mo is precisely estimated. We believe that the results of this study would be meaningful in the analysis of in-service components and further investigation of the spherical indention creep test.
Fig. 1. Schematic diagram of indentation creep test.
up with two heating elements. When the thermal expansion was stable, a constant load was applied to the indenter and transmitted to the specimen. By monitoring the creep deformation, indentation depth versus time curves under different loads could be obtained. 2.2. Experimental curves and microstructure Typical creep curves for the tensile creep test are shown in Fig. 2. Three distinct stages are observed, with the nominal stresses ranging from 160 MPa to 220 MPa. Fig. 3(a) shows spherical indentation depth versus time curves, with the applied load ranging from 250 N to 550 N. Similarities and differences exist between the creep curves obtained by the two measurement methods. In the primary creep stage, the work hardening rate is greater than the recovery rate for creep deformation, resulting in a decrease in the creep rate with time. Subsequently, the work hardening and recovery rates gradually reach a relatively balanced state during the secondary creep stage, which results in a fairly constant creep rate. For the conventional creep curves, the tertiary stage is observed because of the specimens necking and forming creep voids [20]. However, no substantial fracture will occur to the specimen in the indentation creep test, and it explains the absence of tertiary stage in the creep curves. Compare to other micro-test measurement methods, a fundamental summary is made. Owing to the destructive behaviour of specimens in the tertiary creep stage, certain micro-test measurement results such as small punch creep curves are more similar to the uniaxial creep curves than the indentation creep curves in the form of expressions. But the creep characteristics still differ, as the contribution of the secondary creep time to the overall creep life is larger in the micro-test measurement results and the tertiary stage occupies the majority of the rupture strain in the uniaxial creep results.
2. Experimental In order to verify the universality of the indentation creep test, a service-exposed Cr5Mo steel sample obtained from a furnace pipe was used in this study. The pipe was in service at 343 °C and 0.26 MPa for approximately ten years. Considering the maximum service temperature of the pipes in the same process, all of the following tests were conducted at 550 °C to obtain a reasonable assessment of the material. 2.1. Experimental procedure The indentation creep tests were performed using a self-designed vacuum high-temperature indentation creep device. With a miniature specimen, the oxidation cannot be neglected because of the tiny creep deformation. Thus, a vacuum atmosphere was set up to avoid interference from oxidation to the maximum extent. The fixture structure and assembly are shown in Fig. 1. A cylindrical flat indenter with the diameter of 1 mm and spherical indenters with the diameters of 1 mm and 2 mm were employed in the experiments. Before the test, specimens were shaped by wire cutting with a size of 10 mm × 10 mm × 3 mm and then ground with a succession of abrasive paper until using 1200 grit silicon carbide abrasive paper. Thereafter, the furnaces in which the fixture and specimen were placed on were vacuum pumped and then heated. To ensure the heating efficiency, the heating apparatus was set
Fig. 2. Uniaxial tensile creep curves. 2
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Fig. 4. Optical micrograph of 550 N spherical indentation creep deformation: (a) overall morphology and (b) local region around indenter.
postponed with an increase in the applied load, and the ratio of the transient indentation depth to the total indentation depth progressively increases during the same dwell time. It is noteworthy that the ideal contact stress should be stable when the spherical indenter is completely pressed into the specimen, yet the displacement rate still shows a tiny tendency to change by degrees. We considered that a trend of pile-up [22] existed on the surface of a specimen during spherical indentation, resulting in the growth of the true contact area. Thereby, it brought about the decrease of the equivalent stress and explained the decrease in the displacement rate. To verify the theoretical analysis and characterise the creep deformation, we examined the microstructure under an optical microscope, as shown in Fig. 4. A prominent pile-up can be seen on the surface of the specimen, which corresponds to the analysis above. This pile-up phenomenon was also observed in the traditional indentation test. Researchers studied the hardness of materials using the conventional indention test and proposed that the pile-up effect that results in the deviation of the contact area cannot be ignored [17]. The situation in this study was very similar because the equivalent stress could not easily be calculated as a result of the area variation. The following demonstration could explain the phenomenon to some extent: during the test, the plastic flow existed and was constrained in a large elastic/ rigid volume in the material. The flow was directed upward and then appeared as a pile-up on the surface of the specimen [23]. Referring to the dislocation distribution observed in the vicinity of the indenter and the creep coefficient mentioned below, the creep deformation mechanism of the service-exposed Cr5Mo may be described as dislocation movement-controlled creep [24]. Besides, two distinguishable regions could be observed as shown in Fig. 4(b). In the region beneath the indenter, the material was found to experience a massive shear deformation. The other region, which was not beneath the spherical indenter, exhibited no change in the shape of the grains. This revealed the absence of plastic deformation away from the indentation, showing the
Fig. 3. Experimental curves of indentation creep tests: (a) spherical indenter, and (b) comparison between cylindrical flat and spherical indenters.
The difference in the form does not affect the accuracy of the analysis. In contrast, the absence of the tertiary stage in the indentation creep test makes the deformation more stable and convenient to evaluate. A comparison between the spherical and cylindrical flat indentation results is shown in Fig. 3(b). For the flat cylindrical indentation, the indentation stress is constant during the creep test, and a steady state can be achieved after a transient primary stage [21]. It is not hard to understand that the overall stress level of the flat indentation creep is lower than the spherical one at the same load. Because of the growth of the contact area, a decrease in the contact stress is inevitable in the spherical indentation creep test. What we discussed above basically explains the phenomenon of the spherical indentation creep velocity being greater than the other one at the same load, but its curve demonstrates a more remarkable trend to drop. On the other hand, although the steady-state velocity could not be obtained from the spherical indentation creep test, the experimental curves still show that the creep velocity of the spherical indentation tends to be stable by degrees, similar to the flat one. This indicates that an approximate steady stage can be achieved in the spherical indentation creep test, because the growth of the contact area becomes increasingly insignificant after a sufficiently long creep time. In addition, for both sets of curves, we can see that the time required to enter the steady stage is gradually
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Fig. 6. Comparison of experimental and simulation curves. Fig. 5. Finite element model of spherical indentation creep test.
localised nature of the indentation creep deformation [25]. 3. Finite element simulation The aim of the finite element simulation was to further characterise the creep deformation mechanism of the material. To make a systematic comparison of the indenter shapes, a spherical indentation creep test with an indenter diameter of 2 mm was also carried out. 3.1. Finite element model Based on the symmetrical geometry of the test, a two-dimensional finite element model was established as shown in Fig. 5. In particular, the size of the model was consistent with the experiment. The indenter was set as a rigid body, and the specimen was deformable. The contact type between the indenter and the specimen was surface-to-surface contact, and the friction coefficient was set as 0 between them. Based on fitting using the previously mentioned uniaxial creep curves, B and n were defined as 3.72 × 10−20 and 7.4, respectively. The plastic parameters were described by defining the true plastic strain. The other theoretical values included a Young's modulus E = 153000 MPa and Poisson's ratio μ = 0.29. Meanwhile, a similar finite element model with an indenter diameter of 2 mm was also established for a later analysis. 3.2. Creep deformation analysis Fig. 6 shows the indentation depth versus time curves at different loads for both the experiment and simulation. In general, the simulation curves display trends similar to those of the experimental curves. It can be found that the two sets of curves at 300 N almost entirely overlap. The other simulation curves are slightly lower than the experimental curves, but they still demonstrate good consistency in the creep stage. The initial deviation during loading may account for the difference. In addition, the internal defects of service-exposed materials can also lead to dispersions of creep results. Therefore, the finite element model established in this study was rational. Fig. 7 shows the evolution of the pile-up on the surface of the Cr5Mo specimen. We can see that the maximum pile-up on the surface of the specimen continues to grow with increases in the load and dwell time. For the case where the diameter of the indenter was 1 mm, as shown in Fig. 7(a), it was found that when the indenter was gradually pressed in, the maximum pile-up occurred at a distance of one radius to twice the
Fig. 7. Evolution of pile-up on surface of Cr5Mo specimen: (a) time-dependent and load-dependent evolution with indenter diameter of 1 mm, and (b) timedependent evolution at 900 N with indenter diameter of 2 mm.
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Fig. 8. Stress distributions on surface of specimen with dwell time of 30 h: (a) D = 1 mm, F = 450 N; (b) D = 2 mm, F = 900 N.
strain was at the centre of the contact region and decreased with the radial distance from the axis of symmetry to the contact edge [27]. This basically corresponded to the simulation results obtained in this study. For the latter case, there was a stress concentration at the bulged region under the contact area of the indenter, which hindered the further growth of the pile-up. This explained the phenomenon that the evolution of the pile-up with the indenter diameter of 2 mm had a slower tendency, as previously mentioned. It was concluded that the largest strain of the specimen appeared in the necking region by means of certain micro-test technique in our previous research. However, the ductile fracture did not occur to the specimen during the indentation creep test, and the corresponding appearance was a pile-up hillock. Moreover, the consistency between the finite element analysis results and microstructure again verified the rationality of the model. To obtain a more intuitive view of the stress evolution during spherical indentation, the Mises stress along the axial direction was extracted separately for the two cases (Fig. 9). In general, with an increase in the creep time, there was a state where the Mises stress gradually decreased as a result of the increase in the contact area. Stress redistribution with creep time can be observed and the difference in creep rates during the test is found to be the cause for the phenomenon [23]. In the former case, we find that the curves in Fig. 9(a) can be
radius from the symmetric centre. Then, the pile-up decreased with an increase in the distance and became negligible by degrees after triple the radius distance. However, the situation was not exactly the same when the diameter of the indenter was 2 mm, as shown in Fig. 7(b). Because of the smaller deformation, the indenter was not fully pressed in. The maximum pile-up occurred at approximately the distance of one radius from the symmetric centre, and its growth tendency was not as significant as the former. The pile-up hillocks obtained in the finite element simulation were somewhat greater than those obtained experimentally, and similar phenomenon was also observed and illustrated by Kucharski [26]. Thus, the accuracy of the finite element predictions of the pile-up in the vicinity of the indenter could be regarded as satisfactory. The size of the pile-up hillock is not quantitatively described in this paper, but it is generally believed that the specific value is related to the strain hardening exponent and the ratio of yield stress to elastic modulus of the material, and the maximum indentation depth to the radius of the spherical indenter during the test. The stress distributions on the specimen surfaces for both cases are shown in Fig. 8. It is observed that a great stress concentration occurs in the centre of the contact region and the stress spreads out in the radial direction from the centre of the specimen. Karthik also studied the spherical indentation test and concluded that the maximum plastic 5
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4. A modified reference area method It is well known that metallic materials display time-dependent creep behaviours at high homologous temperatures, and uniaxial creep tests are generally conducted to determine the relationship between steady-state strain rate ε˙ c and applied stress σ using the Norton equation as follows:
ε˙ c = Bσ n
(1)
where the constants B and n uniquely characterise this power-law creep response. Using the uniaxial measurement method, the creep parameters can be determined by plotting the measured strain rates against the applied stresses on logarithmic scales. For the spherical indentation creep test, the most troublesome problem is that the contact area continuously changes with the growth of the indentation depth. In addition, the influence of pile-up, which results in a complicated variation of the true contact area, cannot be neglected. It has been found that the reference stress at a certain time cannot be easily calculated. To precisely estimate the creep behaviour despite the complicated stress diversification, a modified reference area method is proposed. We studied this method using the following procedures on the basis of a load relaxation simulation. In the case of uniaxial loading, the total strain increment is the sum of the elastic strain increment, plastic strain increment, and creep strain increment. In the strain maintenance process, the total strain increment is zero, and the plastic strain remains undeveloped [14]. Taking Hooke's law into consideration, the creep strain rate ε˙tc can be expressed as follows：
1 ε˙tc = − σ˙ t E
(2)
where the subscript t denotes a uniaxial condition, E is Young's modulus and σ˙ t is the stress relaxation rate. Using Eq. (2), the creep strain rate ε˙tc can be calculated from the stress relaxation rate which is obtained as the slope of the stress relaxation curve. Because the true contact area increases as result of the pile-up trend and geometric factor during the spherical indentation, we should give a rational definition of the reference area, to calculate the corresponding equivalent stress. The equivalent stress is obtained with the following procedure using the new reference area Sr . The stress due to the indentation is defined as follows: Fig. 9. Evolution of Mises stress on surface of specimen: (a) D = 1 mm, F = 450 N; (b) D = 2 mm, F = 900 N.
σt =
F Sr
(3)
where F is the applied force, and the stress is the value after translating the multiaxial stress into the tensile stress. In other words, the stress in Eq. (3) can be regarded as the equivalent stress for the indentation creep test. Using time differentiation, Eq. (3) is rewritten as follows:
divided into three regions indicated with Ⅰ, Ⅱ, and Ⅲ respectively. The first stage shows the trend of slowly decreasing as the contact area increases by degrees. The second region coincides well with the position where the maximum pile-up exists, and the rapid decrease in stress in this region is due to the plastic flow [28]. For the case with the indenter diameter of 2 mm, as shown in Fig. 9(b), it can be found that the maximum stress point occurs slightly before the position of maximum pile-up in the radius distance, and there is a significant stress wave around the bulged region [29]. The following demonstration may account for the difference of the stress distributions: the indentation depth has reached a remarkable value in the elastic-plastic stage for the former case, which is almost half of the radius of the indenter. However, for the later case, the initial indentation depth is a smaller value compared with the radius of the indenter. It means that the indenter is far from fully pressed in, which results in the stress concentration at the edge of the contact area between the indenter and the specimen and the obvious stress diversification. Besides, the tertiary stage shows a slowly decreasing trend once again, and the stress becomes insignificant away from the indenter, which agrees with the localised nature of the indentation creep deformation.
1 σ˙ = F˙ Sr
(4)
The load relaxation rate F˙ in Eq. (4) can be obtained as the slope of the load relaxation curves which can be derived from the following finite element analysis. The indentation depth maintenance simulation used the finite element model established in section 3.1. Ten kinds of load conditions were employed throughout the process of the spherical indentation. In the simulation procedure, the indenter was pushed into the specimen with an equal indentation rate until the predetermined depth under each load condition. Subsequently, the indentation depths were kept constant for 2 h, and the load relaxation curves were obtained (Fig. 10). By substituting Eq. (4) for Eq. (2), the creep strain rate could be obtained from the load relaxation curve as:
ε˙tc = −
1 ˙ F ESr
(5)
Then, using Eq. (3) to calculate the stress for the indentation creep 6
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Fig. 10. Load relaxation curves during indentation depth maintenance.
Fig. 11. Relationship between reference area and ideal contact area.
test, the following equation can be obtained with Eqs. (1), (3) and (5) as:
In addition, Eq. (8) can be simplified into Eq. (10):
Sc = πh (D − h)
ni
−
1 ˙ F F = Bi ⎛ ⎞ ESr ⎝ Sr ⎠ ⎜
⎟
Eq. (9) can be rewritten as Eq. (11):
(6)
Sr = k2 h (D − h)
where Bi and ni are the creep coefficient and creep exponential obtained on the basis of the indentation creep test, respectively. The reference area is defined as the area where the stress is kept constant during the load relaxation. If the stress is obtained in the reference area, Sr , Norton's law should give the same values for both the indentation creep test and uniaxial creep test. Consequently, using the material constants B and n in Norton's law, the new reference area Sr is expressed as the following equation:
(7)
In other words, the reference area corresponding to an indentation depth can be calculated using Eq. (7). Meanwhile, another series of numerical analyses with an indenter diameter of 2 mm was also conducted following the same steps. Taking the geometric factor into consideration, there should be a certain relationship between the reference area and the ideal contact area. Each reference area calculated by Eq. (7) and obtained from the specific relationship with the ideal contact area should have the same value at a given indentation depth. This implies that we can plot a scatter diagram of the reference area versus the ideal contact area to verify the relationship between them. For the spherical indenter, the ideal contact area is calculated by Eq. (8):
Sc = π (r 2 − (r − h)2)
(11)
The diameter D of a particular spherical indenter is a fixed value. Numerous researchers have applied indentation creep tests to evaluate the superficial properties of specimens. If the indentation depth h is a small value compared to the diameter D, the reference area can be linearly fitted as a function of indentation depth h. Choosing the values until the indenter is half pressed, it is found that the reference area and indentation depth still have a proportional relationship with a small deviation, and the fitting results share a nearly common coefficient as before (Fig. 12). In some cases such as thick wall in-service pipes, the creep properties of a material may not be accurately evaluated with a small indentation depth. Thus, it is also necessary to obtain a rational evaluation of the creep performance of materials with a large indentation depth, and the corresponding reference area needs to be obtained. Based on the fitting results shown in Fig. 13, the modified reference area for spherical indentation is defined as follows:
1
−F˙ ⎫1 − n Sr = ⎧ n⎬ ⎨ BEF ⎩ ⎭
(10)
(8)
where r is the radius of the indenter and h is the indentation depth. The scatter diagrams of the reference area versus ideal contact area of cases with different diameters are shown in Fig. 11. We can see that the reference area is proportional to the ideal contact area, with a small deviation. Thus, it can be claimed that during the indentation creep test, the reference area normally depends on the ideal contact area Sc , as follows:
Sr = k1 Sc
(9)
where k1 is a constant related to the material properties and test conditions. By the substitution of Eqs. (7)–(9), the constant k1 can be obtained, and both the fitting results have similar values. This verifies the reliability of the linear relationship and proves the versatility of the method for distinct indenters.
Fig. 12. Relationship between reference area and indentation depth till indenters half pressed. 7
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Fig. 14. Comparison of relationship between stress and creep strain rate.
particularity of the service-exposed material investigated in this study, it should be noted that more experimental tests for other materials and more in depth physical characterisation and more accurate physically based numerical modelling are needed in order to further evaluate the feasibility of the modified reference area method before putting it to practical application. Conclusively, by applying the previously discussed procedures, it was possible to obtain the creep parameters of the service-exposed Cr5Mo, which corresponded to the fitted value from the conventional method. Because the indentation creep test is a miniature test, it can effectively and economically evaluate creep properties of in-service components. However, the problem of the complicated stress calculation restricts the engineering application of the spherical indention creep test in related fields. Therefore, we believe that the modified reference area method reported in this paper can provide new insights for the study of the spherical indentation creep test. 5. Conclusions This paper aims to estimate the creep behaviour of service-exposed Cr5Mo and evaluate its creep properties using a modified reference area method. Above all, indentation and uniaxial creep tests were conducted under different conditions. Subsequently, finite element simulations and a series of theoretical analyses were carried out to obtain reasonable evaluations of the material. The main conclusions are as follows:
Fig. 13. Determination of modified reference area: (A) D = 1 mm and (B) D = 2 mm.
Sr = krh
(12)
where k is a constant related to the material properties and test conditions, r is the indenter radius and h is the indentation depth. Considering the indentation depth versus time curves under different loads are obtained using indentation creep tests, the equivalent stress at a certain load and time can be calculated. The creep strain rate for spherical indentation can be evaluated by the following equation [30]:
ε˙ c =
h˙ h
(1). Experiments with the service-exposed Cr5Mo under different conditions were conducted. The results showed that the creep performance of the material could be reasonably evaluated using the spherical indentation creep test. (2). A microstructure test and finite element analysis were performed to evaluate the non-uniform deformation characteristics of the spherical indentation creep test. (3). An estimation of creep stress during the test was performed. The stress distributions and stress evolutions of indenters with different diameters were distinct, accounting for the difference in the evolution of the pile-up. (4). A modified reference area method was proposed based on the load relaxation simulation and geometric factor analysis. A simple linear relationship was established between the reference area and indentation depth, and the equivalent stresses for distinct indenters during spherical indentation at arbitrary times could be calculated. (5). The modified reference area method reported in this paper made it possible to precisely evaluate the creep exponential of service-exposed Cr5Mo based on the spherical indentation creep test.
(13)
where h˙ is the displacement rate obtained from the experimental curve. From now on, we can establish the relationship between the creep strain rate and equivalent stress at a certain load using Eqs. (1), (3), (12) and (13). Using the modified reference area presented in Fig. 13(a), we can determine the creep parameters by plotting the creep strain rates against the equivalent stresses on logarithmic scales, and make a comparison with the conventional measurement (Fig. 14). The creep constant B and creep exponential n for the spherical indentation creep test are 9.08 × 10−19 and 7.04, while the values obtained from the uniaxial creep test are 3.72 × 10−20 and 7.4, respectively. It is noteworthy that the creep exponentials obtained by both measurement methods demonstrate good consistency. The creep constant remains some deviations, but within a reasonable range. Considering the 8
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Acknowledgements
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