Non-linear model of stress relaxation curve at high temperatures of Gr. 91 steel

*Woo-Gon Kim1), Jae-Young Park2) and Hyeong-Yean Lee3)

1), 3)

Korea Atomic Energy Research Institute (KAERI), Daejeon 305-353, Korea 2) Dept. of Mechanical Engineering, Pukyong National Univ., Busan 608-739, Korea

1) wgkim@kaeri.re.kr

ABSTRACT

Two non-linear models were proposed for modeling stress relaxation curves of high temperatures of Gr. 91 steel, which is used as structural materials of Gen-IV reactor systems. Stress relaxation data were obtained from a series of the SRTs conducted under a constant strain of 0.6% at 500, 550, 600, and 650oC. To model the stress relaxation curves well, a spring-dashpot model and Peleg’s model were applied. The parameter values used in two models were determined through a nonlinear least square fit to experimental stress relaxation curves. The results showed that the spring-dashpot model was not in accordance with the experimental stress relaxation curves at 500 and 550oC, although it was agreement at 600 and 650oC. On the other hand, Peleg’s model was in good accordance regardless of the temperatures, and the parameters followed temperature dependence well because its model was simple as four parameters. It is thus suggested that Peleg’s can be used well to model the stress relaxation curves of Gr. 91 steel. 1. INTRODUCTION

Modified 9Cr-1Mo steel (ASME Grade 91, hereafter Gr. 91 steel) is regarded as a promising candidate for structural materials such as steam generators (SG), intermediate heat exchangers (IHX), and hot pipes in sodium-cooled fast reactors (SFR). The modified version is obtained by the addition of strong carbide/nitride forming elements such Nb and V and nitrogen in plain 9Cr–1Mo steel and is also known as 9Cr–1Mo–V–Nb steel designated as P91 or T91 by ASTM standards. The selection of Gr. 91 steel is mainly based on its high creep and low cycle fatigue resistance than those exhibited by its counterparts such as plain 9Cr-1Mo and 2.25-1Mo steels (Choudhary 2011, Mannon 2003, Raj 2010, Kim 2013, Srinivasan 2012). Presently, the KAERI (Korea Atomic Energy Research Institute) has established an R&D program to develop a Prototype Gen-IV SFR (PGSFR) by 2028, and key technologies for constructing its reactor are being developed step by step through a long-term R&D program (Kim 2011). The two main materials of the PGSFR consist of Gr. 91 and 316

1,3)

Principal Researcher 2)

Graduate Student

stainless steels. Its structural components will be designed for service of up to 60 years at elevated temperatures.

In order to design the components, the data for stress relaxation referring to a a material response whereby stress under constraint is relaxed by creep strain are required. In a common experiment test, the total strain of a uniaxial test specimen is fixed and load (stress) is monitored as it is gradually reduced by conversion of elastic to inelsatic strain at a constant temperature. The engineering applications of relaxation data cover a wide range of displacement or strain controlled loading such as bolted joint, thermal stresses giving rise to secondary loads or to a creep-fatigue cycle and welding residual stresses. In addition, stress relaxation offers a means of evaluationg the deformation dynamics of materials. However, unlike creep rupture data which is used for design under primary stresses, most stress relaxation models are limited in applicability by either strain range, temperature or both. Therefore, it is necessary to investigate a suitable model for modelling relaxation data.

In the present investigation, the stress relaxation tests are conducted under a constant strain of 0.6% at 500, 550, 600, and 650oC for Gr. 91 steel. In order to well describe the stress relaxation curves experimentally obtained, two non-linear models of a spring-dashpot and Peleg’s models are introduced among various models and applied to Gr. 91 steel. The parameter values used in two models are determined to give the best fit to experimental relaxation curves. A temperature dependence of the parameter values is investigated to find a suitable model describing well the stress relaxation behavior.

2. EXPERIMENTAL PROCEDURES

A commercial grade hot-rolled Gr. 91 steel plate was used for the testing material. The heat treatment was normalized and tempered at 1050°C/1mim/mm and 770°C/3mim/mm. A stress relaxation test (SRT) is to perform by slowly applying a finite

amount of initial strain, o (total or global strain) to a specimen and then maintaining it

constant. While the specimen is continued to be loaded to the initial constraint, the

stress drops with time. In this study, the SRTs were carried out under a controlled constant tensile strain of 0.6% at the four temperatures of 500, 550, 600, and 650oC for Gr. 91 steel. The initial loading rate reaching up to 0.6% was applied with the strain rate of 3.33x10-4 (1/s). After reaching to the target strain level of 0.6%, the strain was a constantly maintained with a stop, and then the relaxation data with elapsed time were automatically obtained using PC. A three-zone furnace was used to provide the constant temperature environment. The jigs and pull rods attached to the specimen were used with Ni-based superalloy to prevent from the deformation during test period at the high temperature. The SRTs were performed according to the ASTM E328 procedures (ASTM 1986). The test specimens were prepared with a sufficient size recommended in the ASTM standard as a round type with 6.0 mm in diameter and 105 mm in gage length.

3. RESULTS AND DISCUSSION

3.1 Proposition of non-linear stress relaxation models

Here, two non-linear stress relaxation models of a spring-dashpot and Peleg’s model are considered to model the stress relaxation behavior for Gr. 91 steel. The non-linear equations for the two models are described as follows.

First, the spring-dashpot model (Lubahn 1961)) was derived from a single paralleled spring-dashpot combination, as shown in Fig. 1 (a). Under a constant strain condition, the equation of the spring-dashpot model can be given by

))(1( /texpa

eEeSS a

a (1)

where, S is the elastic part of the stress and Sa is the anelastic part. This equation was derived by converting into stress from strain basically. In addition, Eq. (1) can be modified into multi terms to determine the correct parameter values in a non-linear least square fit (NLSF). The modified equation can be given by

))(1())(1( /texpa

e

/texpa

eEeSS aa

a (2)

))(1())(1( 6

5

3

21

/texpp

p

/texpp

ppy (3)

Eq. (2) means that metal behaves in the same way as a spring in series with two parallel spring-dashpot combination, as shown in Fig. 1 (b). The stress relaxation values of y-axis can be obtained with the time variations of x-axis by determining the values for seven parameters: P1, P2, P3, P4, P5, P6, and P7, as given in Eq. (3).

(a) single spring-dashpot system (b) multi spring-dashpot system

Fig. 1 Schematic drawing of simplified spring-dashpot analogue to anelastic strain

Second, the Peleg’s model was proposed by Peleg (1976). This model was known as a rheological model of non-linear stress relaxation. The equation is given by

,tkk

tS)t(S

n

n

max

21

1 (4)

where, Smax is the maximum stress, t is the time, and k1, k2, n are the rheological parameters. This model has been known to well describe viscoelastic properties of materials (Myhan 2015)). In the present investigation, the parameter values for the two models were determined using the NLSF method on the results of the SRTs obtained under a constant strain of 0.6% at 500, 550, 600, and 650oC for Gr. 91 steel Each model was compared and evaluated to investigate a better model for Gr. 91 steel.

3.2 Application of two models Fig. 2 shows the stress relaxation curves obtained under a constant strain of 0.6%

at 500, 550, 600, and 650oC for Gr. 91 steel. The four relaxation curves present the temperature dependence well. The stress drops abruptly at initial stage, and then the stress decrease slightly. The recovery amount of elastic stress is very large for of Gr. 91 steel. The recovery of anelastic stress occurs largely from an initial stage, and then the stress relaxation by creep recovery is smooth with time variations. As the temperature increases, the recovery rate is fast: a low temperature of 550oC is slower in the recovery rate than a high temperature of 650oC. Based on the results of the stress relaxation curves, Optimum values of each parameter for two models considered in the present study are determined using the NLSF method to give the best fit to experimental data.

Fig. 2 Stress relaxation curves obtained at each temperature

0 10 20 30 40 50 600

100

200

300

400

500

600

650oC

600oC

550oC

Str

ess(M

Pa)

Time(h)

500oC

Fig.3 NLSF results of the spring-dashpot model to experimental data at 500, 550, 600 and 650oC of Gr. 91 steel

Figs. 3 and 4 show the NLSF results of the spring-dashpot model and Peleg’s model to experimental data at 500, 550, 600 and 650oC of Gr. 91 steel. For the spring-dashpot model, although the NLSF results shows good agreement with the experimental data at 600 and 650oC, but the results do not match with the experimental at 600 and 650oC. It was also investigated that seven parameters did not obey the temperature dependence well. On the other hand, for the Peleg’s model, the NLSF results reveal good match with the experimental data for all the temperature regardless of four temperatures, as shown in Fig. 4. Thus, it is found that the Peleg’s model was superior in modeling of the stress relaxation curves of Gr. 91 steel to the spring-dashpot model. From the results, the optimum values for each parameter in the Peleg’s model were drawn through many iteration processes to give the best regression to the experimental data. The summary of the parameter values of Smax, k1, k2, and n is given in Table 1. As listed in Table 1, the R2 value (coefficient of determination, COD) is closer to unity. It means that the Peleg’s model can describe well the stress relaxation curves of Gr. 91 steel. Using the results of Table 1, the temperature dependence for

0 10 20 30 40 50100

200

300

400

500

600

700

Data: Data1_B

Model: SR2

Equation: y=P1+P2/(P3*(1-exp(-x/P4)))+P5/(P6*(1-exp(-x/P7)))

Weighting:

y No weighting

Chi^2/DoF = 169.98683

R^2 = 0.91983

P1 286.99536 ±0.58412

P2 -24.6709 ±2220.9035

P3 -1905.2853 ±174303.48741

P4 2644.41319 ±105111.73633

P5 17.41978 ±--

P6 0.0662 ±--

P7 -0.1301±0.00154

Str

ess(M

Pa)

Time(h)

Temp.: 500oC

Spring-dashpot model

0 10 20 30 40 500

100

200

300

400

500

600

Temp.: 550oC

Data: Data1_B

Model: SR2

Equation: y=P1+P2/(P3*(1-exp(-x/P4)))+P5/(P6*(1-exp(-x/P7)))

Weighting:

y No weighting

Chi^2/DoF = 293.54395

R^2 = 0.90084

P1 196.91463 ±0.46014

P2 -21.16875 ±3704.47772

P3 -2157.39967 ±374661.13609

P4 2323.9991 ±43935.8189

P5 18.26603 ±6431.62954

P6 0.06277 ±22.21325

P7 -0.07834 ±0.00002

Str

ess(M

Pa

)

Time(h)

Spring-dashpot model

0 10 20 30 40 500

100

200

300

400

500

600

Temp.: 600oC

Data: Data1_B

Model: SR2

Equation: y=P1+P2/(P3*(1-exp(-x/P4)))+P5/(P6*(1-exp(-x/P7)))

Weighting:

y No weighting

Chi^2/DoF = 321.38791

R^2 = 0.84876

P1 105.07094 ±0.67098

P2 -21.23309 ±3226.1236

P3 -2125.73036 ±306743.04745

P4 2348.78102 ±110301.55795

P5 16.45899 ±--

P6 0.06547 ±--

P7 -0.09326 ±0.00149

Str

ess(M

Pa)

Time(h)

Spring-dashpot model

0 10 20 30 40 500

100

200

300

400

500

600

Temp.: 650oC Data: Data1_B

Model: SR2

Equation: y=P1+P2/(P3*(1-exp(-x/P4)))+P5/(P6*(1-exp(-x/P7)))

Weighting:

y No weighting

Chi^2/DoF = 236.62995

R^2 = 0.8205

P1 70.79009 ±0.69003

P2 -16.73725 ±5651.12531

P3 -2644.55142 ±1333231.51414

P4 1888.83261 ±320425.58826

P5 13.03817 ±79537.54853

P6 0.07397 ±453.51006

P7 -0.06777 ±0.00005

Str

ess(M

Pa

)

Time(h)

Spring-dashpot model

four parameters was investigated as shown in Fig. 5. The parameter Smax follows the 2nd polynomial, and the parameters of k1, k2, and n follows a linear relation well when given by semi-log coordinate. Therefore, it is suggested that this result was reasonable because the parameters of the Peleg’s model follows the temperature dependence well. Each equation on temperature dependence of four parameters was obtained as follows,

Smax = – 603.4622+4.93966 T – 0.0055 T2 (5)

Log (k1) = 3.60155 – 0.00686 T (6)

Log (k2) = 0.54185 – 6.79297E-4 T (7)

Log (n) = –1.03594 0.00104 T (8)

Using the four equations obtained on the temperature dependence, it can be estimated for the stress relaxation curves at different temperature ranges for Gr. 91 steel.

Fig. 4 NLSF results of the Peleg’s model to experimental data at 500, 550, 600 and 650oC of Gr. 91 steel

0 10 20 30 40 50100

200

300

400

500

600

700

Data: Data1_B

Model: user31

Equation: y=489.31666*(1-(x^n/(k1+k2*x^n)))

Weighting:

y No weighting

Chi^2/DoF = 0.98404

R^2 = 0.99954

k1 1.60296 ±0.00201

k2 1.65558 ±0.00166

n 0.30364 ±0.0003

Str

ess(M

Pa)

Time(h)

Temp.: 500oC

Peleg's model

0 10 20 30 40 500

100

200

300

400

500

600

Data: Data1_B

Model: user31

Equation: y=453.22263*(1-(x^n/(k1+k2*x^n)))

Weighting:

y No weighting

Chi^2/DoF = 5.90654

R^2 = 0.998

k1 0.65013 ±0.00164

k2 1.41801 ±0.00147

n 0.34578 ±0.00059

Str

ess(M

Pa

)

Time(h)

Temp.: 550oC

Peleg's model

0 10 20 30 40 500

100

200

300

400

500

600

Data: Data1_B

Model: user31

Equation: y=375.11832*(1-(x^n/(k1+k2*x^n)))

Weighting:

y No weighting

Chi^2/DoF = 8.19791

R^2 = 0.99614

k1 0.26541 ±0.00104

k2 1.31173 ±0.00105

n 0.39847 ±0.00085

Str

ess(M

Pa)

Time(h)

Temp.: 600oC Peleg's model

0 10 20 30 40 500

100

200

300

400

500

600

Temp.: 650oC

Data: Data7_B

Model: user31

Equation: y=284*(1-(x^n/(k1+k2*x^n)))

Weighting:

y No weighting

Chi^2/DoF = 8.98301

R^2 = 0.99318

k1 0.15537 ±0.00112

k2 1.30947 ±0.00122

n 0.43072 ±0.00146

Str

ess(M

Pa)

Time(h)

Peleg's model

Table 1 Summary of each parameter of Peleg's model

Temp. (oC) Smax K1 K2 n R2

500 489.316 1.602 1.655 0.303 0.999

550 453.222 0.650 1.418 0.345 0.998

600 375.118 0.265 1.311 0.398 0.996

650 284.000 0.155 1.309 0.431 0.993

(a) Smax (b) k1, k2, and n

Fig. 5 Temperature dependence of Smax, k1, k2, and n in the Peleg's model

4. CONCLUSIONS

Stress relaxation data were obtained from a series of the SRTs conducted under a constant strain of 0.6% at 500, 550, 600, and 650oC. Using the experimental data, to model the stress relaxation curves well, a spring-dashpot model and Peleg’s model were applied. The parameter values in two models were reasonably determined through a nonlinear least square fit to experimental stress relaxation curves. The results showed that the spring-dashpot model was not in accordance with the experimental data at 500 and 550oC, although it was agreement at 600 and 650oC. On the other hand, the Peleg’s model was in good accordance with the experimental data regardless of the temperatures, and the parameters followed temperature dependence well because its model was simple as four parameters. It is thus suggested that Peleg’s can be used well to model the stress relaxation curves of Gr. 91 steel. ACKNOWLEDGEMENTS

This study was supported by Nuclear Research & Development Program of the National Research Foundation of Korea (NRF) grant (2012M2A8A2010652) and the International Research & Development Program Foundation NRF grant (2013K1A3A7A

450 500 550 600 650 700200

250

300

350

400

450

500

550

600

Exp. data

Fitted curve

Y =-603.4622+4.93966 X-0.0055 X2

Ma

xim

um

str

ess (

MP

a)

Temperature (oC)

450 500 550 600 650 7000.01

0.1

1

10

Y =-1.03594+0.00104 X

Y =0.54185-6.79297E-4 X

Y =3.60155-0.00686 X

Pa

ram

ete

rs

Temperature(oC)

K1

K2

n

Fitted for K1

Fitted for K2

Fitted for n

03078195) funded by the Ministry of Science, ICT and Future Planning. The authors would like to acknowledge with gratitude to Mr. Seok-Woo Hong and professor Chang-Sung Seok of Sungunkwan university for experimental contributions in this work. REFERENCES ASTM E328-86, (1986), “Standard methods for stress relaxation tests for materials and

structures”, 353-364. Choudhary, B.K. and Issac Samual, E. (2011), “Creep behavior of modified 9Cr-1Mo

ferritic steel”, Journal of Nuclear Materials, 412, 82-89. Kim, W.G., Park, J.Y., Kim, S.J. and Jang J. (2013), “Reliability assessment of creep

rupture life for Gr. 91 steel”, Materials and Design, 51 1045-1051. Kim, W.G., Park, J.Y., Yin, S.N., Kim, D.W., Park, J.Y. and Kim, S.J. (2011),

“Reliability prediction of long-term creep strength of Gr. 91 steel for next generation reactor structure materials”, Kor J Met Mater, 29, 275-280.

Lubahn, J.D. and Felgar, R.P. (1961), “Plasticity and creep of metals”, John Willy & Sons, Inc, New York, 363-404.

Mannan, S.L., Chetal, S.C., Raj, B. and Bhoje, S.B. (2003), “Selection of materials for prototype fast breeder reactor”, Trans Indian Inst Met, 56, 155-78.

Myhan, R., Markowski, M., Daszkiewicz, T., Zapotoczny, P. and Sadowski, P. (2015), “Non-linear stress relaxation model as a tool for evaluating the viscoelastic properties of meat products”, Journal of Food Engineering, 146, 107-115.

Peleg, M. (1976), “Considerations of a general rheological model for the mechanical behavior of viscoelastic solid food materials”, J. Text Stud., 7 (2), 243–255.

Raj, B. and Choudhary, B. K. (2010), “A perspective on creep and fatigue issues in sodium cooled fast reactors”, Trans Indian Inst Met, 63, 75–84.

Srinivasan, V.S., Choudhary, B.K., Mathew, M.D. and Jayakumar,T. (2012), “Long-term creep-rupture strength prediction for modified 9Cr–1Mo ferritic steel and type 316L(N) austenitic stainless steel”, Materials at High Temperatures, 29, 41-48.