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A new extraction method of creep exponents and coefficients from an indentation creep test by multiaxial stress analysis
Journal Pre-proofs A New Extraction Method of Creep Exponents and Coefficients from an Indentation Creep Test by Multiaxial Stress Analysis Masao Sakane, Akihiko Hirano, Naomi Hamada, Yukari Hoya, Takahiro Oka, Masataka Furukawa, Takamoto Itoh PII: DOI: Reference:
S0167-8442(19)30559-2 https://doi.org/10.1016/j.tafmec.2020.102522 TAFMEC 102522
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
3 October 2019 3 February 2020 8 February 2020
Please cite this article as: M. Sakane, A. Hirano, N. Hamada, Y. Hoya, T. Oka, M. Furukawa, T. Itoh, A New Extraction Method of Creep Exponents and Coefficients from an Indentation Creep Test by Multiaxial Stress Analysis, Theoretical and Applied Fracture Mechanics (2020), doi: https://doi.org/10.1016/j.tafmec.2020.102522
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A New Extraction Method of Creep Exponents and Coefficients from an Indentation Creep Test by Multiaxial Stress Analysis Masao Sakanea, Akihiko Hiranob, Naomi Hamadac, Yukari Hoyaa, Takahiro Okaa, Masataka Furukawaa, and Takamoto Itoha
aRitsumeikan University,
1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan Sangyo University, 3-1-1 Nakagaichi, Daito, Osaka 574-8530, Japan cHiroshima Kokusai Gakuin University, 6-20-1 Nakano, Akiku, Hiroshima 739-0321, Japan bOsaka
Abstract. In this paper, we propose a method for analytically extracting creep exponents and coefficients in Norton’s law from the results of a ball indentation creep test using the finite element method. Elastic-creep finite element analyses were performed with the creep exponent and coefficient varied widely, and a simple method of obtaining three principal stresses at a point of the tested material in the vicinity of an indenter ball tip was developed. Using the three principal stresses in combination with a creep constitutive equation, we propose an analytical method to derive creep exponents and coefficients. This method is verified to determine the two material constants in Norton’s law with a satisfactory accuracy via finite element analyses. The indentation creep tests were performed using Sn-37Pb solder to demonstrate the validity of the proposed method in experiments. The proposed method successfully extracted the creep exponent and coefficient of the solder in experiments.
Keywords: Indentation; Creep; Creep coefficient; Creep exponent; Constitutive equation; Solder
1. Introduction In this paper, we discuss a method for extracting creep exponents and coefficients in Norton’s law from the results of an indentation creep test. Indentation tests have long been widely used to assess the mechanical properties of materials [1]. A recent topic in indentation studies is to extract material constants in uniaxial tension tests, such as Young’s modulus, strain hardening exponent, and coefficient, from the results of hardness tests. Among these studies, hardness studies to extract creep exponents and coefficients have attracted the interest of researchers. For example, Hyde et al. [2, 3] described a method for obtaining creep exponents and coefficients in a uniaxial creep test from the results of an indentation creep test by introducing conversion parameters for stress and strain. They showed that the conversion parameters obtained from the results of finite element method (FEM) analyses were independent of the material. Takagi et al. [4, 5] also discussed a method for extracting uniaxial creep parameters using conversion factors for stress and strain. These conversion factors are used to convert the indentation force to equivalent stress in a material under an indenter, and the indentation displacement rate is used to determine the equivalent strain rate there. Ginder et al. [6] extracted creep exponents and coefficients in the steady-state creep equation using a void expansion model proposed by Hill under the assumption of a heavy triaxial stressed region beneath an indenter.
In addition to the aforementioned studies, numerous researchers have attempted to develop models to extract creep exponents and coefficients in uniaxial creep testing from the results of indentation creep testing [7–13]. The overall fruits of these investigations are summarized as follows. The creep exponent in uniaxial creep tests can be successfully extracted from the results of indentation creep tests using simple conversion methods. Extraction of the creep coefficient in uniaxial creep tests requires sophisticated procedures, and the creep coefficients extracted by simple procedures using Eqs.(3) and (4) that will be introduced in Chapter 3 tend to be substantially smaller than those experimentally determined in uniaxial creep tests. Despite the great progress by the aforementioned researchers, open questions remain in the research on extracting creep exponents and coefficients obtained in uniaxial tests from indentation tests. The objective of this paper is to develop a new method for extracting creep exponents and coefficients from the results of an indentation creep test using a different approach than those used in previous works. This study focuses on multiaxial stresses at a small part of the tested material beneath the tip of a ball indenter in the indentation creep process. Three principal stresses at this part are analyzed using an elastic-creep FEM with the creep exponent and coefficient widely varied, and an empirical formula is derived to express the principal stresses under the indenter. Using an inelastic-creep constitutive equation in combination with the derived principal stresses, we derive a method of extracting creep exponents and coefficients. The creep exponents and coefficients derived by this method are compared with those input into the FEM analyses. A new indentation creep testing machine is developed to examine the validity of the new method experimentally. Creep exponents and coefficients obtained by the proposed method are then compared with those in uniaxial creep tests for Sn-37Pb eutectic solder.
2. Finite Element Analysis In the present study, finite element analyses were used to simulate indentation creep tests using the finite element mesh shown in Fig. 1. The right panel in Fig. 1 is a high-magnification view of the circled region A in the left panel. The figure also shows the coordinate system used in this study. The origin of the coordinate system is the tip of the indenter ball. The model consists of a specimen, an indentation ball with a 0.79 mm radius, and a ball holder. The element type used in the FEM analysis is an axisymmetric isoparametric four-node element. The numbers of elements and nodes of the model are 1,603 and 1,765, respectively. As the constraint conditions, the axisymmetric line (the 𝑥-axis) of the model is fixed in the 𝑟-direction and the bottom line of the specimen is fixed in the 𝑥-direction. The Norton-type creep constitutive equation [Eq. (1)] was used to relate equivalent creep strain rates to equivalent stresses: 𝜀 = 𝛼𝜎𝛽,
(1)
where 𝜀 is the von Mises equivalent creep strain rate (mm/h), 𝛼 is the creep coefficient, σ is the von Mises equivalent stress (MPa), and 𝛽 is the creep exponent. Elastic-creep FEM analyses were performed using a Young’s modulus of 150 GPa, and stresses and strains of an element of the tested specimen and the 𝑥-directional displacement of the holder top were output with time. The location of the element was two elements away in the 𝑥-direction near the 𝑥-axis (reference
2
element). A finite element program, MARC, was used to carry out the FEM analyses using the finite strain, large deformation, and contact option. The contact option was only applied to the surface of the indenter ball and to the top surface of the specimen. The preliminary FEM analysis revealed that the friction between the indenter ball and the specimen surface weakly affected the FEM results; this friction was, therefore, not introduced in subsequent analyses. The combinations of 𝛼 and 𝛽 used in the FEM analyses are tabulated in Table 1. The values of 𝛼 ranged from 10−12 to 10−20 and those of 𝛽 ranged from 3 to 11. These ranges of the creep exponent and coefficient mostly encompass the 𝛼- and 𝛽-values of high-temperature materials used in practical applications.
3. Analytical Approach to Extract Creep Exponents and Coefficients 3.1. Stress–strain rate relationship in indenter creep testing From the geometric relationship, the projected area of the contact surface of the ball indenter (𝐴) and the specimen is expressed by with the ball radius (𝑅) and the current indentation depth (ℎ) as follows: 𝐴 = 𝜋ℎ(2𝑅 ― ℎ). (2) Using the result from Eq. (2), we can evaluate the indentation stress (𝜎𝐼𝑛𝑑) under the indenter ball by 𝑃
𝑃
𝜎𝐼𝑛𝑑 = 𝐴 = 𝜋ℎ(2𝑅 ― ℎ), (3) where 𝑃 is the load applied to the indenter. The indentation stress evaluated by Eq. (3) directs in the 𝑥-direction; thus, the indentation stress is identical to the 𝑥-directional stress. The strain rates of the test material just under the ball indenter in the indentation creep tests (indentation strain rate) (𝜀𝐼𝑛𝑑) can be calculated from the following equation [14, 15]: 1𝑑ℎ
𝜀𝐼𝑛𝑑 = ℎ 𝑑𝑡 . (4) In order to confirm the validity of Eq. (3), the stress distribution under the indenter for the creep exponent and coefficient of SUS 304 stainless steel at 923 K (𝛽 = 6.29, 𝛼 = 3.05 × 10 ―19) at ℎ = 0.0075 mm is illustrated in Fig. 2. The solid line in the figure indicates the stress amplitude estimated by Eq. (3), and the dots are the results of the FEM analyses. The figure shows that the stress amplitudes estimated by Eq. (3) closely agree with those from the FEM analyses from the symmetrical axis to 𝑟 = 0.2 mm in this case. However, Eq. (3) overestimates the absolute values of the stress amplitudes at 𝑟 > 0.2 mm. Notably, Eq. (3) well estimates the 𝑥-directional stresses of the tested material near the symmetrical axis, but the estimation loses accuracy near the peripheral part of the contacted part. In Fig. 3, the 𝑥-directional strain rates estimated by Eq. (4) are compared with the FEM data of the reference element for 𝛽 = 5.00 and α = 1 × 10 ―16. The strain rates estimated by Eq. (4) are larger than those obtained from the FEM analyses (finite strain rate) in the time range less than 103 h; however, the former approaches the latter in the time range beyond 103 h. The same trend observed in Figs. 2 and 3 was also found in the other combinations of creep exponents and coefficients shown in Table 1. Therefore, the stress amplitudes estimated by Eq.
3
(3) and the strain rates calculated by Eq. (4) estimate those at the reference element with a reasonable accuracy; to conserve print space, the other cases are not graphically presented here. Figure 4 illustrates the variation of the indentation depth at three loads as a function of time, as obtained from the FEM analyses, where the creep exponent and coefficient used in the FEM analyses are noted in the figure. A higher indentation rate is found at larger loads. The shape of the three curves resembles that of uniaxial creep curves, consisting of primary and secondary stages, where primary stage is the stage with a descending strain rate with time and the secondary stage is the stage with a constant strain rate. However, no clear secondary stage is observed in the indentation curves and the rate of the indentation depth decreases continuously with time over the full indentation duration. The indentation strain rates and indentation stresses calculated from the data in Fig. 4 using Eqs. (3) and (4) are illustrated in Fig. 5 as hollow circles. In order to minimize the scatter of the indentation strain rates, a quadratic curve was fitted by the least-squares method using seven successive data points in Fig. 4 and an indentation strain rate was obtained as a rate at a middle time of the seven data points after differentiating the curve by time. In Fig. 5, a linear relationship is observed between the indentation stress and the indentation strain rate; thus, the following equation holds between them: 𝜀IND = 𝛾𝜎𝑘IND. (5) In this equation, the slope of the indentation data is denoted as 𝑘 and the coefficient is denoted as γ. The slope obtained from the indentation data in Fig. 5 is 𝑘 = 5.00, which, when input into the analysis, yields 𝛽 = 5.00. Both values are in perfect agreement, consistent with previously reported results [7, 9]. However, the indentation creep strain rate is smaller by approximately two orders of magnitude than the uniaxial creep strain rate input into the FEM analysis at the same stress; thus, the creep coefficient γ obtained from the indentation creep test is smaller than the uniaxial one (i.e., 𝛾 < 𝛼). This trend of the relationship between the indentation strain rate and the uniaxial strain rate has been reported by other researchers [6, 10]. 3.2. Theoretical derivation of creep exponents and coefficients from an indentation creep test The comparison revealing the disagreement of the creep coefficient between the indentation test and the uniaxial creep test results from Eqs. (3) and (4) does not consider the stress multiaxiality in the tested material under the indenter ball. Creep strain rates are strongly dependent on stress multiaxiality; thus, stress multiaxiality should be taken into account for extracting the coefficient from an indentation test. Figure 6(a) depicts the variation of the stress ratios with indention depth for six 𝛽-values and 𝛼 = 1.8 × 10 ―18 at the reference element obtained from the FEM analyses. In the figure, the stress ratios of 𝑤𝑟 and 𝑤𝜃 are the ratios of the radial stress and the tangential stress to the 𝑥directional stress, respectively: 𝜎𝑟𝑟
𝜎𝜃𝜃
𝑤𝜃 = 𝜎𝑥𝑥, (6)
𝑤𝑟 = 𝜎𝑥𝑥,
where 𝜎𝑟𝑟 is the radial principal stress and 𝜎𝜃𝜃 is the tangential principal stress at the reference element. The FEM model used in the present study is an axisymmetric model, and the location of the reference element is very close to the symmetrical axis; thus, the radial principal stress has
4
the same amplitude as that of the tangential principal stress at the reference element. Figure 6(a) indicates that the stress ratios have values of approximately 0.7 immediately after the indentation starts; however, they decrease as the indenter depth increases, depending on 𝛽. Figure 6(b) illustrates the variation of the two stress ratios with indentation depth for 𝛽 = 5 and six 𝛼-values. The stress ratios take a stabilized value of approximately 0.6 in the indentation depth range beyond 0.025 mm, independent of the value of 𝛼 in this case. Similar stabilized values were also found for the other cases of 𝛽-values with different stabilized stress ratios. The stabilized stress ratios shown in Figs. 6(a) and (b) may result from stress distributions not substantially changing with time (i.e., the indentation depth). Figure 7 illustrates the stress ratios at the two creeping times against the distance from the tip of the indenter. The stress distribution at 20,000 h does not differ largely from that at 3,000 h. In particular, the stress ratios near the indenter ball at 20,000 h perfectly agree with those at 3,000 h. Therefore, over time, a stabilized stress ratio results from the similar stress ratio distribution under the indenter ball. Notably, the absolute value of the 𝑥-directional stress as well as the other two principal stresses under the indenter decreases with time because the contact area increases with time. However, the stress ratios maintain similar values over time. The relationship between the stabilized stress ratio and 𝛽 is shown in Fig. 8. Variations of the stabilized stress ratios are expressed by the following cubic function: 𝛽 = 3448𝑤3𝑟 ―5580𝑤2𝑟 +3024𝑤𝑟 ―546.
(7)
From the relationship shown in Fig. 8, the stabilized stress ratios can be determined from the creep exponent 𝛽 obtained from indentation creep tests. Because the 𝑥-directional stress can be calculated from the indentation creep tests by Eq. (3), we can obtain all three principal stresses at the reference element using 𝛽 in the indentation test and Eq. (7). The 𝑥-directional creep strain rate (𝜀𝑥𝑥) under multiaxial stress states is related to the von Mises stress (𝜎) and deviatoric stress (𝑆𝑥𝑥) by the following equation from the theory of plasticity [16]: 3
𝜀𝑥𝑥 = 2𝛼𝜎(𝛽 ― 1)𝑆𝑥𝑥, (8) where 𝛽 is the creep exponent and 𝛼 is the creep coefficient, as shown in Eq. (1). The von Mises equivalent stress and the 𝑥-directional deviatoric stress under axisymmetric conditions are expressed by Eqs. (9) and (10), with the principal stresses of 𝜎𝑥𝑥, 𝜎𝑟𝑟, and 𝜎𝜃𝜃: 𝜎=
1 2
(𝜎𝑥𝑥 ― 𝜎𝑟𝑟)2 + (𝜎𝑟𝑟 ― 𝜎𝜃𝜃)2 + (𝜎𝜃𝜃 ― 𝜎𝑥𝑥)2,
(9)
1
𝑆𝑥𝑥 = 𝜎𝑥𝑥 ― 3(𝜎𝑥𝑥 + 𝜎𝑟𝑟 + 𝜎𝜃𝜃). (10) Substituting 𝜎𝑟𝑟 and 𝜎𝜃𝜃 in Eq. (6) into Eq. (9), we can express the equivalent stress (𝜎) with only 𝜎𝑥𝑥 setting 𝑤𝑟 = 𝑤𝜃 = 𝑤 as follows: 𝜎 = (1 ― 𝑤)𝜎𝑥𝑥.
5
(11)
Similarly, by substituting 𝜎𝑟𝑟 and 𝜎𝜃𝜃 obtained from Eq. (6) into Eq. (10), we can express the deviatoric stress in the 𝑥-direction with 𝜎𝑥𝑥 as 2
(12)
𝑆𝑥𝑥 = 3(1 ― 𝑤)𝜎𝑥𝑥. Combining Eq. (8), (11), and (12) yields 𝜀𝑥𝑥 = 𝛼[(1 ― 𝑤)𝜎𝑥𝑥]𝛽.
(13)
Equation (13) relates the 𝑥-directional principal stress and 𝑥-directional strain rate at the reference element. As shown in the results in Figs. 2 and 3, because the 𝑥-directional stress is calculable by Eq. (3) and the 𝑥-directional strain rate is calculable by Eq. (4), Eq. (13) is rewritten as Eq. (14) without losing accuracy. That is, substituting the 𝑥-directional indentation stress (𝜎𝐼𝑛𝑑) in Eq. (3) and the indentation strain rate (𝜀𝐼𝑛𝑑) in Eq. (4) into Eq. (13), we obtain 𝜀𝐼𝑛𝑑 = 𝛼(1 ― 𝑤)𝛽𝜎𝛽𝐼𝑛𝑑.
(14)
Comparing Eq. (14) with Eq. (5) reveals that the creep exponent (𝑘) obtained in the indentation test is identical to the exponent (𝛽) in the uniaxial creep test. Because of this agreement, we can use the creep exponent obtained in the indentation test as that in the uniaxial test. The creep coefficient can also be calculated by comparing Eq. (14) to Eq. (5) if the indentation strain rate (𝜀𝐼𝑛𝑑), indentation stress (𝜎𝐼𝑛𝑑), stress ratio (𝑤), and 𝛾 (=𝛽), as obtained from the indentation test, are known: 𝛾
α = (1 ― 𝑤)𝛽.
(15)
In Fig. 9, the creep exponents and coefficients obtained by Eqs. (14) and (15) from the FEM indentation simulations using the material constants shown in Table 1 are compared with those in uniaxial creep data input into the FEM analysis. The creep exponents derived from the indentation simulation agree with those in the uniaxial creep data input into the FEM analyses within a factor of 1.1 [Fig. 9(a)]. In addition, the estimated creep coefficients agree with those in the uniaxial creep data within a factor of two [Fig. 9(b)]. Therefore, we concluded that the extraction method proposed in this paper, in conjunction with an FEM simulation, estimates creep exponents and coefficients obtained in uniaxial creep tests with a satisfactory accuracy.
4. Experimental Approach to Extract Creep Exponents and Coefficients In order to confirm the validity of the proposed method of extracting creep exponents and coefficients in experiments, we developed a new indentation creep test machine. A schematic of this machine is shown in Fig. 10. A deadweight system was used as a loading device, and the indentation depth was measured using an LVTD connected to a loading rod. The temperature of the specimen was raised by small ceramic heaters attached to the inner wall of the furnace. The radius of the indenter was 0.79 mm, which is the same radius used in the FEM simulation. The material tested was Sn-37Pb eutectic solder with an approximate composition of 37 wt.% Pb, 63 wt.% Sn, and other elements in trace concentrations. This Sn-37Pb solder has been used in indentation creep studies by other researchers for extracting creep exponents and coefficients [17, 18]. The specimen used in the present study was prepared according to the procedure in the “Creep
6
Standard Testing for Solders” standard issued by the Society of Materials Science, Japan [19]. The solder was first cast to a solid cylinder at 550 K (𝑇m + 100 K) and turned into a cylindrical specimen shape with a 10 mm diameter and a 10 mm height. The cylinder was then annealed at 0.87𝑇m for 1 h to stabilize the microstructure formed during the casting process (𝑇m is the absolute melting temperature of the solder, i.e., 456 K). Figure 11 illustrates the variation of the indentation depth with time for the Sn-37Pb eutectic solder at 2.5, 5.0, and 10.0 N at 313 K. The shape of the three curves resembles that of the tensile creep curve of conventional heat-resistant steels, consisting of transient and secondary creep stages; the shape of the curves in this figure is very similar to that of the curves obtained from the FEM analyses (Fig. 4). The creep strain rates are higher in the initial stage after loading starts but decrease with time. Using the three data points at 2.5, 5.0, and 10.0 N in Fig. 11, we plotted the indentation stresses–indentation creep strain rates in Fig. 12 using Eqs. (3) and (4), with the previously mentioned least-squares fitting results indicated by solid dots. The uniaxial tensile creep data are plotted as circular and triangular open dots [20, 21]. Two kinds of uniaxial tensile creep data were obtained for the same material but different heat batches. Because the tensile solder specimens were made by casting, the microstructures of the two batches presumably differed somewhat, resulting in different creep strain rates, as shown in Fig. 12, even though the two batches were prepared following the aforementioned standard testing method. The difference in the creep strain rate between the two batches, however, was within the range of acceptable scatter. Solid lines refer to data converted from the indentation experiments using Eq. (15). At indentation loads of 2.5 and 5.0 N shown in Figs. 12(a) and 12(b), the slopes obtained in the indentation tests are 3.65 and 4.46, respectively, which are similar to those from the creep database as shown in Table 2 (4.48 and 4.00, respectively). However, the creep strain rates obtained from Eqs. (3) and (4) are far smaller than those from the database because Eqs. (3) and (4) do not account for the stress multiaxiality of the material under the indenter. The creep strain rates extracted using Eq. (15), which are represented as a solid line, fall between the strain rates of the two databases denoted by the dashed lines. The values of the exponent and coefficient extracted from the indentation tests at the two loading levels and those in uniaxial tensile creep tests are tabulated in Table 2, along with the results of the other cases investigated here. At the two loading levels, the extracted creep parameters are sufficiently similar to the uniaxial data in practical use. At 10 N [Fig. 12(c)], however, the indentation data denoted by solid dots have a steeper slope compared to the two uniaxial creep data points. Because of this steeper slope, the creep strain rates obtained by Eq. (15) slightly overestimate the uniaxial creep data [Fig. 12(c)]. The reason for this overestimation is not immediately clear; however, the testing load of 10 N may be too large for the Sn-37Pb solder tested here. Similar indentation creep tests were performed at 353 and 398 K. The results are not graphically presented here to avoid the repetition of similar graphs; however, the results are summarized in Table 2. The results obtained at 353 K are similar to those obtained at 313 K. The creep strain rates extracted from the indentation tests at 2.5 and 5.0 N are intermediate between those in the uniaxial data; however, the rates extracted from the tests at 10 N are greater than the values listed in the databases. At 398 K, all of the data extracted from the indentation tests are outside the range of the uniaxial data, as shown in Table 2. These results indicate that the indentation creep test using a ball indenter may lose accuracy when the indentation load is too large or the test temperature is too high. These limitations should be further addressed in the future.
4. Conclusions
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(1) A new method of extracting creep exponents and coefficients from indentation creep tests was proposed with the assistance of extensive finite element creep indentation simulations. The creep exponents and coefficients extracted using the new method agree well with those of uniaxial creep tests input into the finite element analysis. (2) A new indentation creep test machine was developed to confirm the validity of the new extraction method of creep exponents and coefficients. The creep exponents and coefficients of Sn-37Pb solder extracted from the indentation creep tests agree well with those obtained in uniaxial tests in most of the investigated load and temperature cases. Acknowledgment This work was supported by JSPS KAKENHI Grant Number JP17K06068. References [1] S. Arunkumar, A Review of Indentation Theory, Mater. Today Proc. 5 (2018) 23664– 23673. doi:10.1016/j.matpr.2018.10.156. [2] T.H. Hyde, K.A. Yehia, A.A. Becker, Interpretation of impression creep data using a reference stress approach, Int. J. Mech. Sci. 35 (1993) 451–462. doi:10.1016/00207403(93)90035-S. [3] T.H. Hyde, W. Sun, Evaluation of conversion relationships for impression creep test at elevated temperatures, Int. J. Press. Vessels Pip. 86 (2009) 757–763. doi:10.1016/j.ijpvp.2009.07.001. [4] H. Takagi, M. Fujiwara, Set of conversion coefficients for extracting uniaxial creep data from pseudo-steady indentation creep test results, Mater. Sci. Eng. A. 602 (2014) 98–104. doi:10.1016/j.msea.2014.02.060. [5] H. Takagi, M. Dao, M. Fujiwara, M. Otsuka, Experimental and computational creep characterization of Al–Mg solid-solution alloy through instrumented indentation, Philos. Mag. 83 (2003) 3959–3976. doi:10.1080/14786430310001616045. [6] R.S. Ginder, W.D. Nix, G.M. Pharr, A simple model for indentation creep, J. Mech. Phys. Solids. 112 (2018) 552–562. doi:10.1016/j.jmps.2018.01.001. [7] S.N.G. Chu, J.C.M. Li, Impression creep; a new creep test, J. Mater. Sci. 12 (1977) 2200– 2208. doi:10.1007/BF00552241. [8] P.M. Sargent, M.F. Ashby, Indentation creep, Mater. Sci. Technol. 8 (1992) 594–601. doi:10.1179/mst.1992.8.7.594. [9] V. Raman, R. Berriche, An investigation of the creep processes in tin and aluminum using a depth-sensing indentation technique, J. Mater. Res. 7 (1992) 627–638. doi:10.1557/JMR.1992.0627. [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, Mater. Sci. Eng. A. 456 (2007) 103–108. doi:10.1016/j.msea.2006.11.098. [11]N.Q. Chinh, P. Szommer, Mathematical description of indentation creep and its application for the determination of strain rate sensitivity, Mater. Sci. Eng. A. 611 (2014) 333–336. doi:10.1016/j.msea.2014.06.011. [12]M. Kim, K.P. Marimuthu, S. Jung, H. Lee, Contact size-independent method for estimation of creep properties with spherical indentation, Comput. Mater. Sci. 113 (2016) 211–220. doi:10.1016/j.commatsci.2015.11.044. [13]M. Arai, On Estimation of Creep Constitutive Equation for Welded Joint by HighTemperature Indentaion Creep Testing Method, J. Mater. Scinece Jpn. 68 (2019) 607–613. doi:10.2472/jsms.68.607.
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[14]M.J. Mayo, W.D. Nix, A micro-indentation study of superplasticity in Pb, Sn, and Sn-38 wt% Pb, Acta Metall. 36 (1988) 2183–2192. doi:10.1016/0001-6160(88)90319-7. [15]W.H. Poisl, W.C. Oliver, B.D. Fabes, The relationship between indentation and uniaxial creep in amorphous selenium, J. Mater. Res. 10 (1995) 2024–2032. doi:10.1557/JMR.1995.2024. [16]S. Murakami, Introduction of Solid Mechanics, J. Soc. Mater. Sci. Jpn. 28 (1979) 251–257. doi:10.2472/jsms.28.251. [17]M. Fujiwara, M. Otsuka, Indentation creep of β-Sn and Sn–Pb eutectic alloy, Mater. Sci. Eng. A. 319–321 (2001) 929–933. doi:10.1016/S0921-5093(01)01079-6. [18]R. Mahmudi, A.R. Geranmayeh, H. Khanbareh, N. Jahangiri, Indentation creep of lead-free Sn–9Zn and Sn–8Zn–3Bi solder alloys, Mater. Des. 30 (2009) 574–580. doi:10.1016/j.matdes.2008.05.058. [19]Japan Society of Materials Science, Creep Standard Testing for Solders, (2004). [20]Japan Society of Materials Science, Factual Database on Creep and Creep-Fatigue Properties of Sn-37Pb and Sn-3.5Ag Solders, Japan Society of Materials Science, 2004. http://www.jsms.jp/book/handacreepdb.htm (accessed March 23, 2019). [21]Japan Society of Materials Science, Factual Database on Tensile, Creep, Low Cycle Fatigue and Creep-fatigue of Lead and Lead-free Solders, Japan Society of Materials Science, 2013. http://www.jsms.jp/book/handahdb.htm (accessed March 23, 2019).
9
Table 1: Creep coefficients and exponents used in the finite element analysis. 1.00×10−12 1.00×10−14 1.00×10−16
3 5 7 9 11
○ ○ ○ ○ ○
○ ○ ○ ○ ○
○ ○ ○ ○ ○
1.00×10−18 ○ ○ ○ ○ ○
1.00×10−20 ○ ○ ○ ○ ○
Table 2: Comparison of creep exponents and coefficients between the indentation and uniaxial tensile creep tests. Temperature , 𝑇 (K) 313
353
398
Load, 𝑃 (N) 2.5 5 10 2.5 5 10 2.5 5 10
Creep database Ⅰ
𝛼 Creep database Ⅱ
5.92 × 10−8
1.20 × 10−6
7.56 × 10−6
6.32 × 10−5
2.95 × 10−4
2.93 × 10−4
10
𝛽 Creep Creep Indentation database database Indentation Ⅰ Ⅱ −7 7.99 × 10 3.65 4.48 4.00 2.31 × 10−7 4.46 −8 6.18 × 10 5.78 −5 3.28 × 10 3.08 6.42 × 10−6 3.80 3.54 2.70 −5 3.51 × 10 3.59 −3 2.64 × 10 3.27 −4 2.68 2.82 2.37 × 10 5.28 7.61 × 10−7 11.81
. Ａ
r
x
Fig. 1. Finite element mesh of the indenter and the material.
11
x diretional stress MPa xx, xx x-directional stress , MPa
SUS304 923K ( =3.05x10-19, =6.29)
1000
Indenter displacement(mm) 7.54E-02
500 0 -500 -1000 -1500 0.0
0.2
0.4
0.6
0.8
1.0
Distancefrom from symmetrical indentation center Distance axisR,r,mm mm
Indentation strain rate ind , 1/h
Fig. 2. Comparison of the 𝑥-directional stress between the indentation test and finite element analysis.
・
10-2 10
= 1x10 -16, = 5 Eq.(4)
-3
FEM
10-4 10-5 10-6 2 10
103 Time t , h
104
Fig. 3. Comparison of the indentation creep strain rate with the FEM-calculated strain rate under the indenter.
12
Fig. 4. Variation of the indentation depth with time at three loads, as calculated by FEM.
Fig. 5. Comparison of the indentation strain rate with the uniaxial creep strain rate input into the FEM analysis.
13
=1.8x10-18
Stress ratio wrw ,w Stress ratio r,w
0.8
3 5 6 7 9 11
0.7
0.6
0.5 0.05
0.10
0.15
0.20
0.25
Indentation depth h, , mm Indenter displacement mm (a) Variations of the stress ratios with 𝛽 for 𝛼 = 1.8 × 10−18. =5
Stress w Stressratio ratio w wrr,,w
0.8
1x10-12 1x10-14 1x10-16 1x10-17 1x10-18 1x10-20
0.7
0.6
0.5 0.05
0.10
0.15
0.20
0.25
Indenter displacement , mm Indentation depth h, mm (b) Variations of the stress ratios with 𝛼 for 𝛽 = 5.0. Fig. 6. Reduction in the stress ratios with indenter displacement.
14
= 1x10 -16, = 5
Stress ratio wr , w
2
1
0
-1
wr w wr Time=20000h w 0.1 0.2 0.3 0.4
Time=3000h
0
0.5
0.6
Distance from symmetrical axis Rr , mm Fig. 7. Variations of stress ratios with distance at 3,000 h and 20,000 h.
= 3448wr3 - 5580wr2+3024wr3 - 546
Fig. 8. Dependence of the principal stress ratios on the creep exponent.
15
Factor of 1.1
(a) Comparison of creep exponents
Factor of 2
(b) Comparison of creep coefficients Fig. 9. Comparison of creep exponents derived from indentation tests with those in uniaxial creep tests input into the FEM analysis.
16
Fig. 10. A schematic of an indentation creep tester for solders.
17
Indenter displacement h, mm
Sn37Pb, 313K
0.2
2.5N 5N 10N 0.1
0
0
20
40
60
80
Time t, h Fig. 11. Indentation depth–time relationship for Sn-37Pb solder at 313 K.
18
100
Indentation strain rate , 1/hr
10
-1
Sn37Pb, 313K, 2.5N
・ 10-2 10-3 10-4 10-5 0 10
101
102
Indentation stress xx, MPa (a) 2.5 N
Indentation strain rate , 1/hr
10-1
Sn37Pb, 313K, 5N
10-2 10-3 10-4 10-5 0 10
101 Indentation stress xx, MPa (b) 5.0 N
19
102
Indentation strain rate ， 1/hr
10
-1
Sn37Pb, 313K, 10N
10-2 10-3 10-4 10-5 0 10
101 Indentation stress xx， MPa (c) 10.0 N
Fig. 12. Comparison of the creep strain ratees between indentation tests and uniaxial creep tests for Sn-37Pb solder at 313 K.
20
102
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
21
Authors Contribution Sheet Masao Sakane : Basic idea presentation and whole management of the research Akihiko Hirano : Conducting FEM analyses and data analysis Naomi Hamada : Conducting FEM analyses and data analysis Yukari Hoya : FEM data analysis Takahiro Oka : FEM data analysis Masataka Furukawa : Data analysis and experiment Takamoto Itoh : Experiment and writing the paper
22
S0167-8442(19)30559-2 https://doi.org/10.1016/j.tafmec.2020.102522 TAFMEC 102522
To appear in:
Theoretical and Applied Fracture Mechanics
Received Date: Revised Date: Accepted Date:
3 October 2019 3 February 2020 8 February 2020
Please cite this article as: M. Sakane, A. Hirano, N. Hamada, Y. Hoya, T. Oka, M. Furukawa, T. Itoh, A New Extraction Method of Creep Exponents and Coefficients from an Indentation Creep Test by Multiaxial Stress Analysis, Theoretical and Applied Fracture Mechanics (2020), doi: https://doi.org/10.1016/j.tafmec.2020.102522
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© 2020 Published by Elsevier Ltd.
A New Extraction Method of Creep Exponents and Coefficients from an Indentation Creep Test by Multiaxial Stress Analysis Masao Sakanea, Akihiko Hiranob, Naomi Hamadac, Yukari Hoyaa, Takahiro Okaa, Masataka Furukawaa, and Takamoto Itoha
aRitsumeikan University,
1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan Sangyo University, 3-1-1 Nakagaichi, Daito, Osaka 574-8530, Japan cHiroshima Kokusai Gakuin University, 6-20-1 Nakano, Akiku, Hiroshima 739-0321, Japan bOsaka
Abstract. In this paper, we propose a method for analytically extracting creep exponents and coefficients in Norton’s law from the results of a ball indentation creep test using the finite element method. Elastic-creep finite element analyses were performed with the creep exponent and coefficient varied widely, and a simple method of obtaining three principal stresses at a point of the tested material in the vicinity of an indenter ball tip was developed. Using the three principal stresses in combination with a creep constitutive equation, we propose an analytical method to derive creep exponents and coefficients. This method is verified to determine the two material constants in Norton’s law with a satisfactory accuracy via finite element analyses. The indentation creep tests were performed using Sn-37Pb solder to demonstrate the validity of the proposed method in experiments. The proposed method successfully extracted the creep exponent and coefficient of the solder in experiments.
Keywords: Indentation; Creep; Creep coefficient; Creep exponent; Constitutive equation; Solder
1. Introduction In this paper, we discuss a method for extracting creep exponents and coefficients in Norton’s law from the results of an indentation creep test. Indentation tests have long been widely used to assess the mechanical properties of materials [1]. A recent topic in indentation studies is to extract material constants in uniaxial tension tests, such as Young’s modulus, strain hardening exponent, and coefficient, from the results of hardness tests. Among these studies, hardness studies to extract creep exponents and coefficients have attracted the interest of researchers. For example, Hyde et al. [2, 3] described a method for obtaining creep exponents and coefficients in a uniaxial creep test from the results of an indentation creep test by introducing conversion parameters for stress and strain. They showed that the conversion parameters obtained from the results of finite element method (FEM) analyses were independent of the material. Takagi et al. [4, 5] also discussed a method for extracting uniaxial creep parameters using conversion factors for stress and strain. These conversion factors are used to convert the indentation force to equivalent stress in a material under an indenter, and the indentation displacement rate is used to determine the equivalent strain rate there. Ginder et al. [6] extracted creep exponents and coefficients in the steady-state creep equation using a void expansion model proposed by Hill under the assumption of a heavy triaxial stressed region beneath an indenter.
In addition to the aforementioned studies, numerous researchers have attempted to develop models to extract creep exponents and coefficients in uniaxial creep testing from the results of indentation creep testing [7–13]. The overall fruits of these investigations are summarized as follows. The creep exponent in uniaxial creep tests can be successfully extracted from the results of indentation creep tests using simple conversion methods. Extraction of the creep coefficient in uniaxial creep tests requires sophisticated procedures, and the creep coefficients extracted by simple procedures using Eqs.(3) and (4) that will be introduced in Chapter 3 tend to be substantially smaller than those experimentally determined in uniaxial creep tests. Despite the great progress by the aforementioned researchers, open questions remain in the research on extracting creep exponents and coefficients obtained in uniaxial tests from indentation tests. The objective of this paper is to develop a new method for extracting creep exponents and coefficients from the results of an indentation creep test using a different approach than those used in previous works. This study focuses on multiaxial stresses at a small part of the tested material beneath the tip of a ball indenter in the indentation creep process. Three principal stresses at this part are analyzed using an elastic-creep FEM with the creep exponent and coefficient widely varied, and an empirical formula is derived to express the principal stresses under the indenter. Using an inelastic-creep constitutive equation in combination with the derived principal stresses, we derive a method of extracting creep exponents and coefficients. The creep exponents and coefficients derived by this method are compared with those input into the FEM analyses. A new indentation creep testing machine is developed to examine the validity of the new method experimentally. Creep exponents and coefficients obtained by the proposed method are then compared with those in uniaxial creep tests for Sn-37Pb eutectic solder.
2. Finite Element Analysis In the present study, finite element analyses were used to simulate indentation creep tests using the finite element mesh shown in Fig. 1. The right panel in Fig. 1 is a high-magnification view of the circled region A in the left panel. The figure also shows the coordinate system used in this study. The origin of the coordinate system is the tip of the indenter ball. The model consists of a specimen, an indentation ball with a 0.79 mm radius, and a ball holder. The element type used in the FEM analysis is an axisymmetric isoparametric four-node element. The numbers of elements and nodes of the model are 1,603 and 1,765, respectively. As the constraint conditions, the axisymmetric line (the 𝑥-axis) of the model is fixed in the 𝑟-direction and the bottom line of the specimen is fixed in the 𝑥-direction. The Norton-type creep constitutive equation [Eq. (1)] was used to relate equivalent creep strain rates to equivalent stresses: 𝜀 = 𝛼𝜎𝛽,
(1)
where 𝜀 is the von Mises equivalent creep strain rate (mm/h), 𝛼 is the creep coefficient, σ is the von Mises equivalent stress (MPa), and 𝛽 is the creep exponent. Elastic-creep FEM analyses were performed using a Young’s modulus of 150 GPa, and stresses and strains of an element of the tested specimen and the 𝑥-directional displacement of the holder top were output with time. The location of the element was two elements away in the 𝑥-direction near the 𝑥-axis (reference
2
element). A finite element program, MARC, was used to carry out the FEM analyses using the finite strain, large deformation, and contact option. The contact option was only applied to the surface of the indenter ball and to the top surface of the specimen. The preliminary FEM analysis revealed that the friction between the indenter ball and the specimen surface weakly affected the FEM results; this friction was, therefore, not introduced in subsequent analyses. The combinations of 𝛼 and 𝛽 used in the FEM analyses are tabulated in Table 1. The values of 𝛼 ranged from 10−12 to 10−20 and those of 𝛽 ranged from 3 to 11. These ranges of the creep exponent and coefficient mostly encompass the 𝛼- and 𝛽-values of high-temperature materials used in practical applications.
3. Analytical Approach to Extract Creep Exponents and Coefficients 3.1. Stress–strain rate relationship in indenter creep testing From the geometric relationship, the projected area of the contact surface of the ball indenter (𝐴) and the specimen is expressed by with the ball radius (𝑅) and the current indentation depth (ℎ) as follows: 𝐴 = 𝜋ℎ(2𝑅 ― ℎ). (2) Using the result from Eq. (2), we can evaluate the indentation stress (𝜎𝐼𝑛𝑑) under the indenter ball by 𝑃
𝑃
𝜎𝐼𝑛𝑑 = 𝐴 = 𝜋ℎ(2𝑅 ― ℎ), (3) where 𝑃 is the load applied to the indenter. The indentation stress evaluated by Eq. (3) directs in the 𝑥-direction; thus, the indentation stress is identical to the 𝑥-directional stress. The strain rates of the test material just under the ball indenter in the indentation creep tests (indentation strain rate) (𝜀𝐼𝑛𝑑) can be calculated from the following equation [14, 15]: 1𝑑ℎ
𝜀𝐼𝑛𝑑 = ℎ 𝑑𝑡 . (4) In order to confirm the validity of Eq. (3), the stress distribution under the indenter for the creep exponent and coefficient of SUS 304 stainless steel at 923 K (𝛽 = 6.29, 𝛼 = 3.05 × 10 ―19) at ℎ = 0.0075 mm is illustrated in Fig. 2. The solid line in the figure indicates the stress amplitude estimated by Eq. (3), and the dots are the results of the FEM analyses. The figure shows that the stress amplitudes estimated by Eq. (3) closely agree with those from the FEM analyses from the symmetrical axis to 𝑟 = 0.2 mm in this case. However, Eq. (3) overestimates the absolute values of the stress amplitudes at 𝑟 > 0.2 mm. Notably, Eq. (3) well estimates the 𝑥-directional stresses of the tested material near the symmetrical axis, but the estimation loses accuracy near the peripheral part of the contacted part. In Fig. 3, the 𝑥-directional strain rates estimated by Eq. (4) are compared with the FEM data of the reference element for 𝛽 = 5.00 and α = 1 × 10 ―16. The strain rates estimated by Eq. (4) are larger than those obtained from the FEM analyses (finite strain rate) in the time range less than 103 h; however, the former approaches the latter in the time range beyond 103 h. The same trend observed in Figs. 2 and 3 was also found in the other combinations of creep exponents and coefficients shown in Table 1. Therefore, the stress amplitudes estimated by Eq.
3
(3) and the strain rates calculated by Eq. (4) estimate those at the reference element with a reasonable accuracy; to conserve print space, the other cases are not graphically presented here. Figure 4 illustrates the variation of the indentation depth at three loads as a function of time, as obtained from the FEM analyses, where the creep exponent and coefficient used in the FEM analyses are noted in the figure. A higher indentation rate is found at larger loads. The shape of the three curves resembles that of uniaxial creep curves, consisting of primary and secondary stages, where primary stage is the stage with a descending strain rate with time and the secondary stage is the stage with a constant strain rate. However, no clear secondary stage is observed in the indentation curves and the rate of the indentation depth decreases continuously with time over the full indentation duration. The indentation strain rates and indentation stresses calculated from the data in Fig. 4 using Eqs. (3) and (4) are illustrated in Fig. 5 as hollow circles. In order to minimize the scatter of the indentation strain rates, a quadratic curve was fitted by the least-squares method using seven successive data points in Fig. 4 and an indentation strain rate was obtained as a rate at a middle time of the seven data points after differentiating the curve by time. In Fig. 5, a linear relationship is observed between the indentation stress and the indentation strain rate; thus, the following equation holds between them: 𝜀IND = 𝛾𝜎𝑘IND. (5) In this equation, the slope of the indentation data is denoted as 𝑘 and the coefficient is denoted as γ. The slope obtained from the indentation data in Fig. 5 is 𝑘 = 5.00, which, when input into the analysis, yields 𝛽 = 5.00. Both values are in perfect agreement, consistent with previously reported results [7, 9]. However, the indentation creep strain rate is smaller by approximately two orders of magnitude than the uniaxial creep strain rate input into the FEM analysis at the same stress; thus, the creep coefficient γ obtained from the indentation creep test is smaller than the uniaxial one (i.e., 𝛾 < 𝛼). This trend of the relationship between the indentation strain rate and the uniaxial strain rate has been reported by other researchers [6, 10]. 3.2. Theoretical derivation of creep exponents and coefficients from an indentation creep test The comparison revealing the disagreement of the creep coefficient between the indentation test and the uniaxial creep test results from Eqs. (3) and (4) does not consider the stress multiaxiality in the tested material under the indenter ball. Creep strain rates are strongly dependent on stress multiaxiality; thus, stress multiaxiality should be taken into account for extracting the coefficient from an indentation test. Figure 6(a) depicts the variation of the stress ratios with indention depth for six 𝛽-values and 𝛼 = 1.8 × 10 ―18 at the reference element obtained from the FEM analyses. In the figure, the stress ratios of 𝑤𝑟 and 𝑤𝜃 are the ratios of the radial stress and the tangential stress to the 𝑥directional stress, respectively: 𝜎𝑟𝑟
𝜎𝜃𝜃
𝑤𝜃 = 𝜎𝑥𝑥, (6)
𝑤𝑟 = 𝜎𝑥𝑥,
where 𝜎𝑟𝑟 is the radial principal stress and 𝜎𝜃𝜃 is the tangential principal stress at the reference element. The FEM model used in the present study is an axisymmetric model, and the location of the reference element is very close to the symmetrical axis; thus, the radial principal stress has
4
the same amplitude as that of the tangential principal stress at the reference element. Figure 6(a) indicates that the stress ratios have values of approximately 0.7 immediately after the indentation starts; however, they decrease as the indenter depth increases, depending on 𝛽. Figure 6(b) illustrates the variation of the two stress ratios with indentation depth for 𝛽 = 5 and six 𝛼-values. The stress ratios take a stabilized value of approximately 0.6 in the indentation depth range beyond 0.025 mm, independent of the value of 𝛼 in this case. Similar stabilized values were also found for the other cases of 𝛽-values with different stabilized stress ratios. The stabilized stress ratios shown in Figs. 6(a) and (b) may result from stress distributions not substantially changing with time (i.e., the indentation depth). Figure 7 illustrates the stress ratios at the two creeping times against the distance from the tip of the indenter. The stress distribution at 20,000 h does not differ largely from that at 3,000 h. In particular, the stress ratios near the indenter ball at 20,000 h perfectly agree with those at 3,000 h. Therefore, over time, a stabilized stress ratio results from the similar stress ratio distribution under the indenter ball. Notably, the absolute value of the 𝑥-directional stress as well as the other two principal stresses under the indenter decreases with time because the contact area increases with time. However, the stress ratios maintain similar values over time. The relationship between the stabilized stress ratio and 𝛽 is shown in Fig. 8. Variations of the stabilized stress ratios are expressed by the following cubic function: 𝛽 = 3448𝑤3𝑟 ―5580𝑤2𝑟 +3024𝑤𝑟 ―546.
(7)
From the relationship shown in Fig. 8, the stabilized stress ratios can be determined from the creep exponent 𝛽 obtained from indentation creep tests. Because the 𝑥-directional stress can be calculated from the indentation creep tests by Eq. (3), we can obtain all three principal stresses at the reference element using 𝛽 in the indentation test and Eq. (7). The 𝑥-directional creep strain rate (𝜀𝑥𝑥) under multiaxial stress states is related to the von Mises stress (𝜎) and deviatoric stress (𝑆𝑥𝑥) by the following equation from the theory of plasticity [16]: 3
𝜀𝑥𝑥 = 2𝛼𝜎(𝛽 ― 1)𝑆𝑥𝑥, (8) where 𝛽 is the creep exponent and 𝛼 is the creep coefficient, as shown in Eq. (1). The von Mises equivalent stress and the 𝑥-directional deviatoric stress under axisymmetric conditions are expressed by Eqs. (9) and (10), with the principal stresses of 𝜎𝑥𝑥, 𝜎𝑟𝑟, and 𝜎𝜃𝜃: 𝜎=
1 2
(𝜎𝑥𝑥 ― 𝜎𝑟𝑟)2 + (𝜎𝑟𝑟 ― 𝜎𝜃𝜃)2 + (𝜎𝜃𝜃 ― 𝜎𝑥𝑥)2,
(9)
1
𝑆𝑥𝑥 = 𝜎𝑥𝑥 ― 3(𝜎𝑥𝑥 + 𝜎𝑟𝑟 + 𝜎𝜃𝜃). (10) Substituting 𝜎𝑟𝑟 and 𝜎𝜃𝜃 in Eq. (6) into Eq. (9), we can express the equivalent stress (𝜎) with only 𝜎𝑥𝑥 setting 𝑤𝑟 = 𝑤𝜃 = 𝑤 as follows: 𝜎 = (1 ― 𝑤)𝜎𝑥𝑥.
5
(11)
Similarly, by substituting 𝜎𝑟𝑟 and 𝜎𝜃𝜃 obtained from Eq. (6) into Eq. (10), we can express the deviatoric stress in the 𝑥-direction with 𝜎𝑥𝑥 as 2
(12)
𝑆𝑥𝑥 = 3(1 ― 𝑤)𝜎𝑥𝑥. Combining Eq. (8), (11), and (12) yields 𝜀𝑥𝑥 = 𝛼[(1 ― 𝑤)𝜎𝑥𝑥]𝛽.
(13)
Equation (13) relates the 𝑥-directional principal stress and 𝑥-directional strain rate at the reference element. As shown in the results in Figs. 2 and 3, because the 𝑥-directional stress is calculable by Eq. (3) and the 𝑥-directional strain rate is calculable by Eq. (4), Eq. (13) is rewritten as Eq. (14) without losing accuracy. That is, substituting the 𝑥-directional indentation stress (𝜎𝐼𝑛𝑑) in Eq. (3) and the indentation strain rate (𝜀𝐼𝑛𝑑) in Eq. (4) into Eq. (13), we obtain 𝜀𝐼𝑛𝑑 = 𝛼(1 ― 𝑤)𝛽𝜎𝛽𝐼𝑛𝑑.
(14)
Comparing Eq. (14) with Eq. (5) reveals that the creep exponent (𝑘) obtained in the indentation test is identical to the exponent (𝛽) in the uniaxial creep test. Because of this agreement, we can use the creep exponent obtained in the indentation test as that in the uniaxial test. The creep coefficient can also be calculated by comparing Eq. (14) to Eq. (5) if the indentation strain rate (𝜀𝐼𝑛𝑑), indentation stress (𝜎𝐼𝑛𝑑), stress ratio (𝑤), and 𝛾 (=𝛽), as obtained from the indentation test, are known: 𝛾
α = (1 ― 𝑤)𝛽.
(15)
In Fig. 9, the creep exponents and coefficients obtained by Eqs. (14) and (15) from the FEM indentation simulations using the material constants shown in Table 1 are compared with those in uniaxial creep data input into the FEM analysis. The creep exponents derived from the indentation simulation agree with those in the uniaxial creep data input into the FEM analyses within a factor of 1.1 [Fig. 9(a)]. In addition, the estimated creep coefficients agree with those in the uniaxial creep data within a factor of two [Fig. 9(b)]. Therefore, we concluded that the extraction method proposed in this paper, in conjunction with an FEM simulation, estimates creep exponents and coefficients obtained in uniaxial creep tests with a satisfactory accuracy.
4. Experimental Approach to Extract Creep Exponents and Coefficients In order to confirm the validity of the proposed method of extracting creep exponents and coefficients in experiments, we developed a new indentation creep test machine. A schematic of this machine is shown in Fig. 10. A deadweight system was used as a loading device, and the indentation depth was measured using an LVTD connected to a loading rod. The temperature of the specimen was raised by small ceramic heaters attached to the inner wall of the furnace. The radius of the indenter was 0.79 mm, which is the same radius used in the FEM simulation. The material tested was Sn-37Pb eutectic solder with an approximate composition of 37 wt.% Pb, 63 wt.% Sn, and other elements in trace concentrations. This Sn-37Pb solder has been used in indentation creep studies by other researchers for extracting creep exponents and coefficients [17, 18]. The specimen used in the present study was prepared according to the procedure in the “Creep
6
Standard Testing for Solders” standard issued by the Society of Materials Science, Japan [19]. The solder was first cast to a solid cylinder at 550 K (𝑇m + 100 K) and turned into a cylindrical specimen shape with a 10 mm diameter and a 10 mm height. The cylinder was then annealed at 0.87𝑇m for 1 h to stabilize the microstructure formed during the casting process (𝑇m is the absolute melting temperature of the solder, i.e., 456 K). Figure 11 illustrates the variation of the indentation depth with time for the Sn-37Pb eutectic solder at 2.5, 5.0, and 10.0 N at 313 K. The shape of the three curves resembles that of the tensile creep curve of conventional heat-resistant steels, consisting of transient and secondary creep stages; the shape of the curves in this figure is very similar to that of the curves obtained from the FEM analyses (Fig. 4). The creep strain rates are higher in the initial stage after loading starts but decrease with time. Using the three data points at 2.5, 5.0, and 10.0 N in Fig. 11, we plotted the indentation stresses–indentation creep strain rates in Fig. 12 using Eqs. (3) and (4), with the previously mentioned least-squares fitting results indicated by solid dots. The uniaxial tensile creep data are plotted as circular and triangular open dots [20, 21]. Two kinds of uniaxial tensile creep data were obtained for the same material but different heat batches. Because the tensile solder specimens were made by casting, the microstructures of the two batches presumably differed somewhat, resulting in different creep strain rates, as shown in Fig. 12, even though the two batches were prepared following the aforementioned standard testing method. The difference in the creep strain rate between the two batches, however, was within the range of acceptable scatter. Solid lines refer to data converted from the indentation experiments using Eq. (15). At indentation loads of 2.5 and 5.0 N shown in Figs. 12(a) and 12(b), the slopes obtained in the indentation tests are 3.65 and 4.46, respectively, which are similar to those from the creep database as shown in Table 2 (4.48 and 4.00, respectively). However, the creep strain rates obtained from Eqs. (3) and (4) are far smaller than those from the database because Eqs. (3) and (4) do not account for the stress multiaxiality of the material under the indenter. The creep strain rates extracted using Eq. (15), which are represented as a solid line, fall between the strain rates of the two databases denoted by the dashed lines. The values of the exponent and coefficient extracted from the indentation tests at the two loading levels and those in uniaxial tensile creep tests are tabulated in Table 2, along with the results of the other cases investigated here. At the two loading levels, the extracted creep parameters are sufficiently similar to the uniaxial data in practical use. At 10 N [Fig. 12(c)], however, the indentation data denoted by solid dots have a steeper slope compared to the two uniaxial creep data points. Because of this steeper slope, the creep strain rates obtained by Eq. (15) slightly overestimate the uniaxial creep data [Fig. 12(c)]. The reason for this overestimation is not immediately clear; however, the testing load of 10 N may be too large for the Sn-37Pb solder tested here. Similar indentation creep tests were performed at 353 and 398 K. The results are not graphically presented here to avoid the repetition of similar graphs; however, the results are summarized in Table 2. The results obtained at 353 K are similar to those obtained at 313 K. The creep strain rates extracted from the indentation tests at 2.5 and 5.0 N are intermediate between those in the uniaxial data; however, the rates extracted from the tests at 10 N are greater than the values listed in the databases. At 398 K, all of the data extracted from the indentation tests are outside the range of the uniaxial data, as shown in Table 2. These results indicate that the indentation creep test using a ball indenter may lose accuracy when the indentation load is too large or the test temperature is too high. These limitations should be further addressed in the future.
4. Conclusions
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(1) A new method of extracting creep exponents and coefficients from indentation creep tests was proposed with the assistance of extensive finite element creep indentation simulations. The creep exponents and coefficients extracted using the new method agree well with those of uniaxial creep tests input into the finite element analysis. (2) A new indentation creep test machine was developed to confirm the validity of the new extraction method of creep exponents and coefficients. The creep exponents and coefficients of Sn-37Pb solder extracted from the indentation creep tests agree well with those obtained in uniaxial tests in most of the investigated load and temperature cases. Acknowledgment This work was supported by JSPS KAKENHI Grant Number JP17K06068. References [1] S. Arunkumar, A Review of Indentation Theory, Mater. Today Proc. 5 (2018) 23664– 23673. doi:10.1016/j.matpr.2018.10.156. [2] T.H. Hyde, K.A. Yehia, A.A. Becker, Interpretation of impression creep data using a reference stress approach, Int. J. Mech. Sci. 35 (1993) 451–462. doi:10.1016/00207403(93)90035-S. [3] T.H. Hyde, W. Sun, Evaluation of conversion relationships for impression creep test at elevated temperatures, Int. J. Press. Vessels Pip. 86 (2009) 757–763. doi:10.1016/j.ijpvp.2009.07.001. [4] H. Takagi, M. Fujiwara, Set of conversion coefficients for extracting uniaxial creep data from pseudo-steady indentation creep test results, Mater. Sci. Eng. A. 602 (2014) 98–104. doi:10.1016/j.msea.2014.02.060. [5] H. Takagi, M. Dao, M. Fujiwara, M. Otsuka, Experimental and computational creep characterization of Al–Mg solid-solution alloy through instrumented indentation, Philos. Mag. 83 (2003) 3959–3976. doi:10.1080/14786430310001616045. [6] R.S. Ginder, W.D. Nix, G.M. Pharr, A simple model for indentation creep, J. Mech. Phys. Solids. 112 (2018) 552–562. doi:10.1016/j.jmps.2018.01.001. [7] S.N.G. Chu, J.C.M. Li, Impression creep; a new creep test, J. Mater. Sci. 12 (1977) 2200– 2208. doi:10.1007/BF00552241. [8] P.M. Sargent, M.F. Ashby, Indentation creep, Mater. Sci. Technol. 8 (1992) 594–601. doi:10.1179/mst.1992.8.7.594. [9] V. Raman, R. Berriche, An investigation of the creep processes in tin and aluminum using a depth-sensing indentation technique, J. Mater. Res. 7 (1992) 627–638. doi:10.1557/JMR.1992.0627. [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, Mater. Sci. Eng. A. 456 (2007) 103–108. doi:10.1016/j.msea.2006.11.098. [11]N.Q. Chinh, P. Szommer, Mathematical description of indentation creep and its application for the determination of strain rate sensitivity, Mater. Sci. Eng. A. 611 (2014) 333–336. doi:10.1016/j.msea.2014.06.011. [12]M. Kim, K.P. Marimuthu, S. Jung, H. Lee, Contact size-independent method for estimation of creep properties with spherical indentation, Comput. Mater. Sci. 113 (2016) 211–220. doi:10.1016/j.commatsci.2015.11.044. [13]M. Arai, On Estimation of Creep Constitutive Equation for Welded Joint by HighTemperature Indentaion Creep Testing Method, J. Mater. Scinece Jpn. 68 (2019) 607–613. doi:10.2472/jsms.68.607.
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[14]M.J. Mayo, W.D. Nix, A micro-indentation study of superplasticity in Pb, Sn, and Sn-38 wt% Pb, Acta Metall. 36 (1988) 2183–2192. doi:10.1016/0001-6160(88)90319-7. [15]W.H. Poisl, W.C. Oliver, B.D. Fabes, The relationship between indentation and uniaxial creep in amorphous selenium, J. Mater. Res. 10 (1995) 2024–2032. doi:10.1557/JMR.1995.2024. [16]S. Murakami, Introduction of Solid Mechanics, J. Soc. Mater. Sci. Jpn. 28 (1979) 251–257. doi:10.2472/jsms.28.251. [17]M. Fujiwara, M. Otsuka, Indentation creep of β-Sn and Sn–Pb eutectic alloy, Mater. Sci. Eng. A. 319–321 (2001) 929–933. doi:10.1016/S0921-5093(01)01079-6. [18]R. Mahmudi, A.R. Geranmayeh, H. Khanbareh, N. Jahangiri, Indentation creep of lead-free Sn–9Zn and Sn–8Zn–3Bi solder alloys, Mater. Des. 30 (2009) 574–580. doi:10.1016/j.matdes.2008.05.058. [19]Japan Society of Materials Science, Creep Standard Testing for Solders, (2004). [20]Japan Society of Materials Science, Factual Database on Creep and Creep-Fatigue Properties of Sn-37Pb and Sn-3.5Ag Solders, Japan Society of Materials Science, 2004. http://www.jsms.jp/book/handacreepdb.htm (accessed March 23, 2019). [21]Japan Society of Materials Science, Factual Database on Tensile, Creep, Low Cycle Fatigue and Creep-fatigue of Lead and Lead-free Solders, Japan Society of Materials Science, 2013. http://www.jsms.jp/book/handahdb.htm (accessed March 23, 2019).
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Table 1: Creep coefficients and exponents used in the finite element analysis. 1.00×10−12 1.00×10−14 1.00×10−16
3 5 7 9 11
○ ○ ○ ○ ○
○ ○ ○ ○ ○
○ ○ ○ ○ ○
1.00×10−18 ○ ○ ○ ○ ○
1.00×10−20 ○ ○ ○ ○ ○
Table 2: Comparison of creep exponents and coefficients between the indentation and uniaxial tensile creep tests. Temperature , 𝑇 (K) 313
353
398
Load, 𝑃 (N) 2.5 5 10 2.5 5 10 2.5 5 10
Creep database Ⅰ
𝛼 Creep database Ⅱ
5.92 × 10−8
1.20 × 10−6
7.56 × 10−6
6.32 × 10−5
2.95 × 10−4
2.93 × 10−4
10
𝛽 Creep Creep Indentation database database Indentation Ⅰ Ⅱ −7 7.99 × 10 3.65 4.48 4.00 2.31 × 10−7 4.46 −8 6.18 × 10 5.78 −5 3.28 × 10 3.08 6.42 × 10−6 3.80 3.54 2.70 −5 3.51 × 10 3.59 −3 2.64 × 10 3.27 −4 2.68 2.82 2.37 × 10 5.28 7.61 × 10−7 11.81
. Ａ
r
x
Fig. 1. Finite element mesh of the indenter and the material.
11
x diretional stress MPa xx, xx x-directional stress , MPa
SUS304 923K ( =3.05x10-19, =6.29)
1000
Indenter displacement(mm) 7.54E-02
500 0 -500 -1000 -1500 0.0
0.2
0.4
0.6
0.8
1.0
Distancefrom from symmetrical indentation center Distance axisR,r,mm mm
Indentation strain rate ind , 1/h
Fig. 2. Comparison of the 𝑥-directional stress between the indentation test and finite element analysis.
・
10-2 10
= 1x10 -16, = 5 Eq.(4)
-3
FEM
10-4 10-5 10-6 2 10
103 Time t , h
104
Fig. 3. Comparison of the indentation creep strain rate with the FEM-calculated strain rate under the indenter.
12
Fig. 4. Variation of the indentation depth with time at three loads, as calculated by FEM.
Fig. 5. Comparison of the indentation strain rate with the uniaxial creep strain rate input into the FEM analysis.
13
=1.8x10-18
Stress ratio wrw ,w Stress ratio r,w
0.8
3 5 6 7 9 11
0.7
0.6
0.5 0.05
0.10
0.15
0.20
0.25
Indentation depth h, , mm Indenter displacement mm (a) Variations of the stress ratios with 𝛽 for 𝛼 = 1.8 × 10−18. =5
Stress w Stressratio ratio w wrr,,w
0.8
1x10-12 1x10-14 1x10-16 1x10-17 1x10-18 1x10-20
0.7
0.6
0.5 0.05
0.10
0.15
0.20
0.25
Indenter displacement , mm Indentation depth h, mm (b) Variations of the stress ratios with 𝛼 for 𝛽 = 5.0. Fig. 6. Reduction in the stress ratios with indenter displacement.
14
= 1x10 -16, = 5
Stress ratio wr , w
2
1
0
-1
wr w wr Time=20000h w 0.1 0.2 0.3 0.4
Time=3000h
0
0.5
0.6
Distance from symmetrical axis Rr , mm Fig. 7. Variations of stress ratios with distance at 3,000 h and 20,000 h.
= 3448wr3 - 5580wr2+3024wr3 - 546
Fig. 8. Dependence of the principal stress ratios on the creep exponent.
15
Factor of 1.1
(a) Comparison of creep exponents
Factor of 2
(b) Comparison of creep coefficients Fig. 9. Comparison of creep exponents derived from indentation tests with those in uniaxial creep tests input into the FEM analysis.
16
Fig. 10. A schematic of an indentation creep tester for solders.
17
Indenter displacement h, mm
Sn37Pb, 313K
0.2
2.5N 5N 10N 0.1
0
0
20
40
60
80
Time t, h Fig. 11. Indentation depth–time relationship for Sn-37Pb solder at 313 K.
18
100
Indentation strain rate , 1/hr
10
-1
Sn37Pb, 313K, 2.5N
・ 10-2 10-3 10-4 10-5 0 10
101
102
Indentation stress xx, MPa (a) 2.5 N
Indentation strain rate , 1/hr
10-1
Sn37Pb, 313K, 5N
10-2 10-3 10-4 10-5 0 10
101 Indentation stress xx, MPa (b) 5.0 N
19
102
Indentation strain rate ， 1/hr
10
-1
Sn37Pb, 313K, 10N
10-2 10-3 10-4 10-5 0 10
101 Indentation stress xx， MPa (c) 10.0 N
Fig. 12. Comparison of the creep strain ratees between indentation tests and uniaxial creep tests for Sn-37Pb solder at 313 K.
20
102
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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Authors Contribution Sheet Masao Sakane : Basic idea presentation and whole management of the research Akihiko Hirano : Conducting FEM analyses and data analysis Naomi Hamada : Conducting FEM analyses and data analysis Yukari Hoya : FEM data analysis Takahiro Oka : FEM data analysis Masataka Furukawa : Data analysis and experiment Takamoto Itoh : Experiment and writing the paper
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