
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13687-0
10.1016/j.heliyon.2024.e37656
e37656
Research Article
Experimental investigations on characterizing the uniaxial mechanical property variations along the thickness of hydrogenation reactor welded joints by spherical indentation tests
Ma Xin a
Ge Zhiqiang ba
Zhang Tairui tairui_zhang@seu.edu.cn
c⁎
Zheng Weiwei c
a Special equipment safety supervision inspection institute of Jiangsu province, Nanjing, China
b School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing, China
c School of Mechanical Engineering, Southeast University, Nanjing, China
⁎ Corresponding author. tairui_zhang@seu.edu.cn
10 9 2024
30 9 2024
10 9 2024
10 18 e376569 5 2024
28 7 2024
7 9 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
To meet the demands for non-destructive testing of uniaxial mechanical properties of welded joints of thick-wall hydrogenation reactors, this study provides an experimental investigation on whether the spherical indentation tests (SITs) can accurately characterize the uniaxial mechanical property variations along the thickness of welded joints. The phenomenologically summarized empirical method (i.e., the Kwon method) and the semi-analytical method (i.e., the simplified-IIEM) were selected as representatives, and their reliability was judged from the viewpoints of stress-strain prediction, the inversion accuracy and repeatability of strength, and the ability to characterize the variation of uniaxial mechanical properties along the thickness of welded joints. Characteristics of the inverse predictions were analyzed, and the source of errors in each method were extensively investigated. This study provides a theoretical and technical guidance for the engineering application and promotion of SITs.

Keywords

Uniaxial mechanical properties
Welded joints
Hydrogenation reactor
Spherical indentation tests
Semi-analytical method
==== Body
pmc1 Introduction

Hydrogenation reactors are the key parts of equipment used in petroleum product processing and are widely used in pillar industries of the national economy such as petroleum and chemical. Once a safety accident occurs in the hydrogenation reactor, it will cause serious consequences, e.g., casualties, property damage, and pollution. Welded joints are the weakest part of hydrogenation reactors. They are used in harsh environments, such as high temperature, high pressure, hydrogen, and corrosive media, and are subjected to unstable vibration during start-up and shut-down, which can lead to failures caused by creep, hydrogen embrittlement, fatigue, and so on. The occurrence of these above-mentioned failures is mostly related to the degradation of their mechanical performance, which cannot be detected by conventional non-destructive tests depending on acoustic and optical signals.

Uniaxial mechanical property is one of the most basic mechanical performance indicators of metals and the basis for obtaining other mechanical properties, e.g., fracture toughness. However, conventional uniaxial mechanical property tests, such as uniaxial tension, uniaxial compression, and pure shear, not only involve sophisticated sampling and testing procedures, but also require large-scale destructive sampling. Therefore, these conventional tests are incapable of providing a rapid evaluation of the newly manufactured welded joints or a non-destructive evaluation in the deterioration of in-service ones. In contrast, the spherical indentation test (SIT), due to its non-destructive characteristic, has been considered a promising substitute of conventional tests [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10]].

In the past decades, a variety of theories has been proposed to predict the uniaxial mechanical property from SITs, including the stress-strain characterization based on phenomenological summarization of experimental results (also referred as the empirical method) [11,12], the phenomenological summarization of finite element (FE) results (also referred as the numerical method) [4,[13], [14], [15]], and the analytical (or semi-analytical) derivations (also referred as the analytical or semi-analytical method) [[16], [17], [18], [19], [20], [21]]. Especially, the empirical method summarized from SITs on brass, aluminum alloys, carbon steels, and low alloy steels, has been promoted to a certain extent in engineering applications due to its simple form and high repeatability [[22], [23], [24]]. However, over simplification in its phenomenological summarization can be a source of errors, which is also hard to be improved through analytical (or numerical) investigations. The analytical (or semi-analytical) method can achieve the stress-strain prediction from a single indentation cycle, but its prediction accuracy highly depends on the accuracy of displacement acquisition [25], which may result in poor reliability and repeatability in field tests.

To improve the reliability and repeatability of analytical (or semi-analytical) methods, it is suggested to determine the proportional limit of specimen material by introducing digital image correlation (DIC) measurements [[26], [27], [28]], and then derive the stress-strain relationship after yielding through the energy increment of each indentation cycle, leading to the incremental indentation energy method (IIEM). Experimental investigations [29] proved that for different kinds of metals, the IIEM well reproduces all their hardening behaviors and can be regarded as a highly universal inversion prediction method. However, the above-mentioned IIEM taking the proportional limit as the starting point and thus accuracy of the following stress-strain calculation largely depends on the DIC measurements, which demands a series of sophisticated operations [30], e.g., preparing the speckle pattern, leading to restrictions on its engineering application. In additional, the inhomogeneous characteristic of the welded joint may induce the indentation plastic region deviated from the ideal circular shape [31], and thus cause significant errors in determining the plastic zone radius.

The purpose of this study lies in experimentally investigating whether the SITs can be used to characterize the uniaxial mechanical property variations along the thickness of welded joints of hydrogenation reactor. On this basis, the phenomenological summarized empirical method (i.e., Kwon method) and the semi-analytical method (i.e., simplified-IIEM) were selected as representatives, and their reliability was judged from the viewpoints of stress-strain prediction, the inversion accuracy and repeatability of strength, and the ability to characterize the variation of uniaxial mechanical properties along the thickness of welded joints. Characteristics of the inverse predictions were analyzed, and the source of errors in each method were extensively investigated. This study provides a theoretical and technical guidance for the engineering application and promotion of SITs.

2 Theoretical basis for the uniaxial mechanical property predictions from SITs

2.1 Kwon method from phenomenological summarization

In the 1990s, to satisfy the demands for the life extension of nuclear power plants, researches [32,33] from the Metals and Ceramics Division of Oak Ridge National Laboratory extensively summarized the experimental results on brass, aluminum alloys, carbon steels, and low alloy steels. It was concluded that for a SIT contains N loading-unloading cycles, as shown in Fig. 1, the true stress σt(i)- true strain εt(i) of the ith indentation cycle, corresponding to that from uniaxial tensile tests, can be estimated using the empirical functions shown in Eq. (1) and Eq. (2), respectively [34].(1) εt(i)=0.2a(i)R

(2) σt(i)=Pmax(i)3π(a(i))2

where R is the radius of the spherical indenter, Pmax(i) is the maximum load of the ith indentation cycle, and a(i) is the contact radius of the ith indentation cycle that corresponds to Pmax(i).Fig. 1 Illustration of the load-depth curve from a SIT contains N loading-unloading cycles.

Fig. 1

With further investigations on SITs and the developments of finite element (FE) simulations, it was realized that the phenomenon of pile-up (or sink-in) around the indentation can cause the actual indentation depth deviated from that measured by the displacement sensor [35]. To provide a more accurate prediction of the contact radius a, through systematic FE calculations on metals following the Holloman constitutive law shown in Eq. (3), Kwon [34] suggested using Eq. (4) to take the phenomenon of pile-up (or sink-in) into consideration.(3) {σt=Eεtεt≤ε0σt=Eε01−nεtnεt>ε0

(4) (a(i))2=52(2−n)(4+n)[2R(hmax(i)−0.75Pmax(i)S(i))−(hmax(i)−0.75Pmax(i)S(i))2]

Where ε0 and n are the proportional limit (in strain) and the hardening exponent of the specimen material, respectively, and hmax(i) and S(i) are the maximum indentation depth and the unloading slope of the ith indentation cycle.

In this study, the empirical conclusions that modifying the contact radius by considering the phenomenon of pile-up (or sink-in) around the indentation was referred as the Kwon method, and the flow chart of its calculation was shown in Fig. 2. By assuming the hardening exponent n as 0.1, the contact radius considering the phenomenon of pile-up (or sink-in) around the indentation can be calculated using Eq. (4). Then, the true stress σt(i)- true strain εt(i) was estimated using the empirical formulas shown in Eq. (1) and Eq. (2), respectively. For a SIT contains N loading-unloading cycles, a total number of N sets of σt(i)- εt(i) data points can be obtained and used in the Holloman constitutive law shown in Eq. (3) to fit the ε0 and n. The updated n was compared with the previously assumed value to verify whether the convergence criterion is satisfied. If the convergence criterion is met, the updated ε0 and n can be regarded as the values that reflect the actual hardening behavior of specimen materials. Otherwise, the above-mentioned calculation shall be repeated till the convergence criterion is satisfied.Fig. 2 Flow chart for the Kwon method.

Fig. 2

2.2 Simplified-IIEM from semi-analytical derivations

In analytical analyses [31], deformation of the specimen material is described with the simplified expanding cavity model (ECM) shown in Fig. 3. According to the stages of elastoplastic deformation, the specimen is divided into the elastic and plastic zones, characterized with a hemisphere elastoplastic boundary with a radius of rp. Solidarity of the ECM has been extensively investigated in the past decades, through both experiments and FE calculations [27,29,36,37]. It was proved that the plastic zone may not fully develop at a small indentation depth (hmax smaller than 0.16R) [29], leading to errors in hardening behavior predictions. When hmax reaches 0.24R, its further increase will not continue to bring beneficial effects on tensile property estimations. In our previous study [27], an extensive analytical analysis on the energy consumption during a SIT has been conducted, and the equivalent stress-strain were calculated from the elastic and plastic portion of the external work, respectively. For the ith indentation cycle, as shown in Fig. 4, the external indentation work can be divided into the elastic portion We(i) and the plastic portion Wp(i). It deserves noted that the ith indentation cycle refers to the complete P-h curve with previous unloading curves removed, rather that the P-h increment from the (i-1)th indentation cycle. As the specimen is assumed following von Mises isotropic hardening criterion, there is no difference between the σeq - εeq relationship from SITs and the σt - εt relationship from uniaxial mechanical tests.Fig. 3 Illustration of the expanding cavity model [31].

Fig. 3

Fig. 4 Illstration of stress-strain calculation from energy analysis.

Fig. 4

For the ith indentation cycle, a correlation between the equivalent stress σeq(i) and the elastic strain energy of the plastic zone of the specimen Upe(i) can be established, and the Upe(i) can be determined through deducting the elastic strain energy of the spherical indenter Uind(i) and elastic strain energy of the elastic zone of the specimen Ue(i) from the elastic work We(i), as shown in Eq. (5) [27].(5) σeq(i)=2Eeff(i)(We(i)−Uind(i)−Ue(i))[2π3(3Pmax(i)2πσ0)3/2]

where Eeff(i) is the effective Young's modulus of the ith indentation cycle and its value may continuously decrease with the increasing indentation depth, corresponding to the gradual accumulation of damage with the increasing indentation depth. Since the indentation-induced damage at the 1st indentation cycle is negligible, the Eeff(1) can be regarded as the Young's modulus of virgin material E0. It also deserves noted that it is the Eeff(i) rather than the E0 should be used in calculating the equivalent stress to take the effect of indentation damage on the elastic work. If the Young's modulus of virgin material E0 is used, an additional term [38], that is the damage dissipation energy of the ith indentation cycle UD(i), should be incorporated into We(i). The equivalent strain increment from the (i-1)th to the ith indentation cycle can be derived from the plastic dissipation energy increment in the plastic zone, as shown in Eq. (6) [27].(6) Δεeq−M(i)=1σeq−M(i){[WP(i)2π3(3Pmax(i)2πσ0)−WP(i−1)2π3(3Pmax(i−1)2πσ0)]+[(σeq(i))22Eeff(i)−(σeq(i−1))22Eeff(i−1)]}

The equivalent strain corresponds to the ith indentation cycle εeq(i) can then be determined as:(7) εeq(i)=εeq(i−1)+Δεeq(i)

As the idea of calculus has been introduced into the above-mentioned calculation of equivalent strain, it is referred as the incremental indentation energy method (IIEM). It should be noted that the above-mentioned equivalent stress-strain calculation must take the proportional limit (i.e., σ0- ε0) of specimen material as the starting point. Therefore, it is suggested to use optical measurements, such as DIC [27], to measure the plastic zone radius, and then the σ0 (a material constant that is independent of the indentation depth) can be obtained by Eq. (8). It has been proved that errors in the proportional limit estimation will not affect the overall development of stress-strain calculations in IIEM [27], indicating a basically constant hardening exponent n can be expected. In this case, when optical measurements are not available, a regression function, phenomenologically summarized from FE calculations, is suggested to update the proportional limit, leading to a simplified-IIEM independent of rp measurements. A detailed illustration on the flow chart of stress-strain calculations in the simplified-IIEM is shown in Fig. 5.Fig. 5 Flow chart for the simplified-IIEM.

Fig. 5

Firstly, the ε0 is assumed to be 0.2 %, and the σeq - εeq can be calculated with Eq. (5) and Eq. (7), respectively. For a SIT contains N loading-unloading cycles, N sets of σeq - εeq data points can be calculated and then used in fitting the ε0 and n in Eq. (3). It should be noted that the ε0 from current fitting largely depends on the pre-assumed value, but it has been demonstrated in our previous study that n is almost independent of the pre-assumed ε0. Therefore, the n from current fitting can be substitute into Eq. (9), together with loading portion of the P-h curve to update the ε0 [31]. If the updated and pre-assumed ε0 satisfy the convergence criterion, the updated ε0 and n can be regarded as the values that reflect the actual hardening behavior of specimen materials. Otherwise, the above-mentioned calculation shall be repeated till the convergence criterion is satisfied.

The advantage of this simplified-IIEM relies in evaluating the overall stress-strain developments (corresponding to the n) from the elastic and plastic portion of the external indentation work of each indentation cycle, while updating the ε0 from the overall development trend of the loading curve, achieving a full utilization of the P-h data.(8) σ0=3Pmax(N)2πrp2

(9) P=ξhψ

(10) ξ=(a00+a01n+a02n2)+(a10+a11n+a12n2)ε0+(a10+a11n+a12n2)ε02

(11) ψ=0.5878n+1.0215

where aij (i, j = 0, 1, 2) are the fitting coefficients, with detailed values shown in Table 1.Table 1 Values of the fitting coefficients.

Table 1coefficients	a00	a01	a02	a10	a11	
values	7.63E+01	−4.93E+03	4.61E+04	2.02E+06	4.05E+06	
coefficients	a12	a20	a21	a22		
values	3.19E+07	−1.99E+07	3.62E+08	−4.29E+09		

3 Experiments

3.1 Welded joint of thick-wall hydrogenation reactor

A schematic illustration of the welded joint is shown in Fig. 6 (a). Cr-Mo steel (2.25Cr1Mo) manufactured by forging processing and then annealing treated was used as the base, with its detailed chemical compositions provided in Table 2. The welding processes in this study were designed in accordance with that used in manufacturing hick-wall hydrogenation reactors, that is using submerged arc welding (SAW) for backing welding and filling, while using shielded metal arc welding (SMAW) for local repair welding and root cleaning, with detailed welding parameters provided in Table 3. Magnetic particle testing (MT) was performed on the groove before welding to guarantee its integrity, and then the base was preheated at a temperature of 150 °C. After completing the front welding, carbon arc air gouging was performed to clean the back at preheating temperature (i.e., 150 °C), and then the cleaned area was polished to remove the carburized layer until a metallic luster is visible. SMAW was conducted after root cleaning position passing the flaw detection (100 % MT). Hydrogen removal treatment (300 °C–350 °C for 2 h) was carried out immediately after welding, followed by the post weld heat treatment (690 ± 14 °C for 8 h). The welded joint was delivered after passing MT, ultrasonic testing, as well as time of flight diffraction testing.Fig. 6 Welded joint of thick-wall hydrogenation reactor: (a) geometric details, (b) cutting strategy, (c) sampling position.

Fig. 6

Table 2 Chemical compositions of the base and welding metals (%).

Table 2Metals	C	Si	Mn	Ni	Cr	others	
Base	0.150	0.173	0.470	0.188	2.41	0.880	
Weld	0.142	0.188	0.740	0.174	2.30	0.930	

Table 3 Welding parameters.

Table 3Layer	Welding metals	Welding method	Current (A)	Voltage (V)	Speed (cm/min)	Maximum heat input (kJ/cm)	Inter-pass temp (°C)	
Type	Size (mm)	
Filling	US521S	Ф4.0	SAW	500–520	29–31	42–48	22.5	150–250	
Repair	CMA-106N	Ф4.0	SMAW	160–180	24–27	15–20	19.4	150–250	
Gouging	CMA-106N	Ф5.0	SMAW	175–220	24–28	19–24	19.5	150–250	

Due to the excessive thickness of the welded joint, it was cut into four blocks, as shown in Fig. 6 (b), after finishing the above-mentioned welding, heat treatment, and integrity inspection. Among the four blocks, block-1# corresponds to the upper surface of the weld, while block-4# corresponds to the bottom surface of the weld. Sampling positions of the metallographic, indentation (and hardness), and mini-sized uniaxial tensile specimens from the three blocks are shown in Fig. 6 (c). It deserves noted that block-2# was not used in experiments as its spacing between block-1# and block-3# is small.

3.2 Metallographic observation

From the metallographic observations shown in Fig. 7, it was found that the welded joint can be roughly divided into four zones, namely the base, the heat affected zone (HAZ), the interface, and the weld. As the cooling speed of weld is too fast, the formation of ferrite is suppressed, leading to the formation of lath martensite with high strength and toughness, as shown in Fig. 7 (e). In addition, a small amount of interlayered pearlite structure and feathery bainite structure is also observed. At the junction between the melted weld metal and the unmelted base metal (i.e., the interface), large differences in structure between the weld and the base metals can be found, with coarse grains and uneven micro-structure distributions, as shown in Fig. 7 (d). From the judgements of grain size, the HAZ can be further divided into two zones, namely the coarse-grained and fine-grained zones, as shown in Fig. 7(b) and (c). Among which, the coarse-grained zone corresponds to the overheated zone of the HAZ, while the fine-grained zone corresponds to the phase transformation recrystallization zone of the HAZ. The grain size of the base is relatively uniform, consisting of ferrite and a small amount of pearlite, as shown in Fig. 7 (a).Fig. 7 Metallographic observations on: (a) base, (b) HAZ (fine grain),

(c) HAZ (coarse grain), (d) interface, and (e) weld.

Fig. 7

Comparing the metallography of three blocks, it was found that grain size gradually decreases from the upper surface to the bottom of the weld, with a corresponding decrease in the martensite content. This can be understood from the differences in temperature during the cooling of the molten pool. The upper surface has the best heat dissipation condition, leading to a preferable condition to the formation of martensitic structure. The interface of block-1# (i.e., upper surface) is not obvious, while that from block-4# (i.e., bottom surface) is clearer, with a narrower weld width compared to its upper surface counterpart. The HAZ of block-4# is also characterized with more significant differences between coarse-grained and fine-grained zones. There is no significant change in the microstructure and grain size of the bases along the thickness of the welded joint.

3.3 Hardness tests

To provide a quick estimation on whether the uniaxial mechanical property varies along the thickness direction of the welded joint (i.e., judging the necessity of sampling from different blocks of the welded joint), and to estimate the approximate range of the HAZ (providing reference for sampling). The indented surface of the specimen was polished and then corroded with nitric acid alcohol to distinguish the different zones of joints. The Vickers hardness distributions perpendicular to the welding direction were tested, with the indentation positions and numbering strategy shown in Fig. 8 (a), and the test results were shown in Fig. 8 (b).Fig. 8 Hardness distribution (HV0.3).

Fig. 8

It was found that block-4# (i.e., the weld bottom) exhibits the lowest hardness, which is consistent with the metallographic observations shown in Fig. 7, ie., the martensite content decreases from the upper surface to the bottom of the weld. However, there is basically no difference in the hardness of the bases from the three blocks. In addition, from the hardness distributions, it can be seen that the HAZ only exists in a small region (roughly around 15 mm away from the interface), and the hardness of the remaining parts exhibit no significant difference from the base.

3.4 Mini-sized uniaxial tensile tests

Due to the volume limitation of the welded joint of the hydrogenation reactor, uniaxial mechanical property distributions of the welded joint were obtained using mini-sized uniaxial tensile specimens. According to the hardness distributions shown in Fig. 8, sampling position of these specimens was determined (the sampling and numbering strategy for the three blocks are completely consistent), as shown in Fig. 9. The sampling positions cover the base metal, HAZ, interface, and weld. Geometric details of the mini-sized uniaxial tensile specimens were shown in Fig. 10. Experiments were conducted on the INSTRON 8803 universal testing machine with the fixtures provided in our previous study [31]. Extension of the specimen was measured through an extensometer with a gauge length of 10 mm, and the tests were displacement control, with a rate 0.2 mm/min until it breaks.Fig. 9 Sampling position of the mini-sized tensile specimens.

Fig. 9

Fig. 10 Geometric details of the mini-sized uniaxial tensile specimen.

Fig. 10

3.5 Spherical indentation tests

A tungsten carbide spherical indenter with a radius of R = 0.38 mm (Eind = 710 GPa, vind = 0.21) was used in spherical indentation tests. The indentation positions and numbering strategy of SITs are consistent with the hardness tests shown in Fig. 8 (a). The SITs were conducted on the Stress-strain micro-probe system SSM-B4000, equipped with a load cell (a range of 1.1 kN and a resolution of 0.11N) and a displacement sensor (a range of 1 mm and a resolution of 0.025 μm) to obtain the continuous indentation load P-displacement h curve.

As both the Kwon method and simplified-IIEM obtain a set of stress-strain data point from each loading-unloading cycle, SITs with 12 equally spaced intervals were conducted to ensure a sufficient number of data points (setting hmax(12) to 0.24R). The loading process of the indentation test was displacement-control with a rate of 0.02 mm/min, while the unloading process was load-control with a rate of 1 kN/min. Indentation P-h curves of the specimens from block 1#, 3#, and 4# were shown in Fig. 11, Fig. 12, Fig. 13, respectively. As there exists a preload of around 5N, the depth of the 1st indentation cycle is significantly greater than that from the subsequent cycles. Even for the 1st indentation cycle, the contact diameter is no smaller than 120 μm, effectively cover more than three grains, and thus the indentation size effect was not discussed [[39], [40], [41], [42]].Fig. 11 Indentation P-h curves of the Block-1#:

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 11

Fig. 12 Indentation load-depth curves of the Block-3#:

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 12

Fig. 13 Indentation P-h curves of the Block-4#:

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 13

4 Results and discussions

4.1 Results from mini-sized tensions

The stress-strain curves from mini-sized tensions were shown in Fig. 14, with the corresponding proof strength Rp0.2 (i.e., the engineering strength at a plastic strain of 0.2 %) and the tensile strength Rm (i.e., the ultimate engineering strength during the uniaxial tension) shown in Fig. 15(a) and (b), respectively. From Fig. 14, it was found that bases of block-1#, 3#, and 4# exhibit almost identically the same hardening behavior, specifically for SP-1 and SP-2 that far from the weld, the necking and fracture moments of specimens from the three blocks are basically the same, indicating there is no difference in their strength and plasticity, which is consistent with the hardness test results shown in Fig. 8 (b).Fig. 14 Stress-strain curves from mini-sized uniaxial tensile tests:

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 14

Fig. 15 Strength distributions from mini-sized tension:

(a) proof strenght Rp0.2 and (b) tensile strenght Rm.

Fig. 15

By comparison, significant differences can be found for tensile properties in HAZ and interface from block-1#, 3#, and 4#. This is not only manifested by a decrease in the strength of HAZ from block-4# compared to its block-1# counterpart, but also by a significant decrease in its plasticity. This phenomenon also inspires future investigations on the fracture toughness variations along the thickness of hydrogenation reactor welded joints. The hardening behavior of the weld of all three blocks are basically consistent, but the mini-sized uniaxial tensile specimens taken from block-4# exhibit greater dispersion on the moment of necking and fracture. From the strength distribution shown in Fig. 15, it can be found that the overall strength of block-1# is higher than that of block-4#, both from the viewpoints of proof strength and tensile strength. This phenomenon is particularly evident in the weak part of the welded joints, that is the HAZ and interface, leading to a decrease of more than 10 % in strength from block-1# to block-4#, which is also consistent with the hardness test results shown in Fig. 8 (b). The above-mentioned results suggest that a fast and accurate evaluation on the tensile property variations through the thickness of welded joints by SITs is necessary.

4.2 Predictions from SITs

The stress-strain predictions from the Kwon method for block-1#, 3#, and 4# are shown in Fig. 16, Fig. 17, Fig. 18, respectively, with the proof and tensile strength predictions provided in Fig. 19(a) and (b), respectively. It was found that due to the over simplification in developing the phenomenological summarized empirical functions shown in Eq. (1) and Eq. (2), the stress-strain predictions from different zones of the welded joint always approximate to a roughly linear relationship, deviated from the actual hardening behavior of the specimen materials. In additional, the empirical functions were developed from a rough fitting of the entire indentation process, which is not capable of accurately predicting the stress-strain relationship at the beginning of indentation (i.e., the first three indentation cycles). It is mainly because the deformation at the beginning of indentation is jointly affected by elastic deformation and plastic flow, which is significantly different from the plastic flow dominated indentation deformation at a large indentation depth [29]. The above-mentioned errors in stress-strain calculation are mainly manifested in the proof strength prediction. Compared to the tensile strength predictions, more inconsistence and incoherence can be found from the proof strength predictions, e.g., the difference between the two proof strength predictions in the weld center of block-3# and block-4# exceeds 20 %, as shown in Fig. 19(a) and (b). It should be pointed out that the prediction results from Kwon method have high stability, except for the beginning of indentation, which can be confirmed by the stress-strain predictions for the base metals and the weld (except for point-13 of block-1# and the point-14 of block-4#) and the tensile strength predictions.Fig. 16 Stress-strain relationship predictions from Kwon method (for block-1#):

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 16

Fig. 17 Stress-strain relationship predictions from Kwon method (for block-3#):

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 17

Fig. 18 Stress-strain relationship predictions from Kwon method (for block-4#):

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 18

Fig. 19 Strength distribution predictions from SITs (Kwon method):

(a) proof strenght Rp0.2 and (b) tensile strenght Rm.

Fig. 19

The stress-strain predictions from the simplified-IIEM for block 1#, 3#, and 4# are shown in Fig. 20, Fig. 21, Fig. 22, respectively, with the proof and tensile strength predictions provided in Fig. 23(a) and (b), respectively. Compared with the phenomenological summarized Kwon method, the simplified-IIEM were developed by semi-analytical derivation, characterized with a more robust physical background, and thus can better predict the actual hardening behavior of specimen materials. This advantage of the simplified-IIEM is particularly obvious for characterizing the stress-strain relationship at the beginning of indentation deformation, leading to more stable and coherent proof strength predictions.Fig. 20 Stress-strain relationship predictions from simplified-IIEM (for block-1#):

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 20

Fig. 21 Stress-strain relationship predictions from simplified-IIEM (for block-3#):

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 21

Fig. 22 Stress-strain relationship predictions from simplified-IIEM (for block-4#):

(a) base, (b) interface and HAZ, and (c) weld.

Fig. 22

Fig. 23 Strength distribution predictions from SITs (simplified-IIEM):

(a) proof strenght Rp0.2 and (b) tensile strenght Rm.

Fig. 23

However, unlike the Kwon method, which is capable of providing the stress-strain relationship till a strain of 0.14 at the currently maximum indentation depth (i.e., 0.24R), strain predictions from the simplified-IIEM are significantly smaller. This is because the true strain in Kwon method is phenomenologically summarized from the strain at the edge of indentation, which characterizes the strain of metals squeezed out of the specimen surface and its value is continuously increased with the increasing indentation depth. By contrast, the equivalent strain in the simplified-IIEM indicates the average strain of the plastic zone, as shown in Fig. 3. The external indentation work continues to increase with the increasing indentation depth, but the volume of the plastic zone correspondingly increases, leading to a smaller growth rate of the average strain in the plastic zone than the true strain at the edge of indentation. This characteristic of simplified-IIEM, that is lacking the stress-strain data points at large strains, resulting in less stability and coherence in tensile strength prediction than that from the Kwon method.

5 Concluding remarks

To meet the demands for non-destructive testing of uniaxial mechanical properties of welded joints of thick-wall hydrogenation reactors, this study provides an experimental investigation on whether the SITs can accurately characterize the uniaxial mechanical property variations along the thickness of welded joints. The phenomenologically summarized the Kwon method and the semi-analytically derived simplified-IIEM were selected as representatives. Their reliability was judged from the viewpoints of stress-strain prediction, the inversion accuracy and repeatability of strength, and the ability to characterize the variation of uniaxial mechanical properties along the thickness of welded joints. Characteristics of the inverse predictions were analyzed, and the source of errors in each method were extensively investigated. The following remarks can be drawn from the mentioned-above investigations.1) Bases of block-1#, 3#, and 4# exhibit almost identically the same hardening behavior, with less than 3 % differences in their strength. By contrast, a 10 % decrease in strength from the upper surface to bottom of the weld can be observed for the HAZ and interface.

2) The Kwon method, developed from a rough fitting of the entire indentation process, is incapable of accurately predicting the stress-strain relationship at the beginning of indentation (i.e., the first three indentation cycles), leading to more than 20 % errors in proof strength predictions.

3) The simplified-IIEM is characterized with a more robust physical background, evaluating the n from elastic and plastic portion of the external indentation work of each cycle, while updating the ε0 from the overall development trend of the loading curve, achieving a full utilization of the P-h data. Therefore, it can better predict the actual hardening behavior of the welded joints, especially for the stress-strain relationship at the beginning of indentation deformation, leading to more reliable proof strength predictions (errors smaller than 10 %).

4) Equivalent strain in simplified-IIEM indicates the average strain of the plastic zone, yielding a value around 6 % at the maximum indentation depth 0.24R. This strain value is only half of that from Kwon method, resulting in the absence of stress-strain data points at large strains and thus leading to a maximum of 5 % more errors in tensile strength predictions than the Kwon method.

CRediT authorship contribution statement

Xin Ma: Writing – original draft, Methodology, Investigation, Funding acquisition. Zhiqiang Ge: Writing – original draft, Methodology, Investigation, Formal analysis. Tairui Zhang: Writing – original draft, Supervision, Methodology, Conceptualization. Weiwei Zheng: Writing – original draft, Validation.

Declaration of competing interest

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.

Acknowledgement

This research was funded by 10.13039/501100001809 National Natural Science Foundation of China (Grant/Award Number: 52305149 ), 10.13039/501100012166 National Key Research and Development Program of China (Grant/Award Number: 2021YFC3001804 ), State Administration for Market Regulation (Grant/Award Number: 2023MK042 ), Jiangsu Administration for Market Regulation (Grant/Award Number: KJ2023003 ), Jiangsu Province Special Equipment Safety Supervision Inspection Institute (Grant/Award Number: KJ(Y)202429 ), Jiangsu Province Special Equipment Safety Supervision Inspection Institute (Grant/Award Number: KJ(YJ)2023001 ), and the 10.13039/501100012226 Fundamental Research Funds for the Central Universities .
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