Experimental Study on the Constitutive Relation of Austenitic Stainless Steel S31608 Under Monotonic and Cyclic Loading

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1 Experimental Study on the Constitutive Relation of Austenitic Stainless Steel S3168 Under Monotonic and Cyclic Loading Y.Q. Wang a,*. T. Chang a, Y.J. Shi a, H.X. Yuan a, D.F. Liao b a Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 184, PR China b Measure Stick-Curtain Wall Structural Fittings Co., Ltd., Shenzhen 518, PR China Abstract In order to study the constitutive relation of domestic austenitic stainless steel S3168 (AISI 316, EN 1.441) under monotonic and cyclic loading, different types of specimens were tested. Based on the Ramberg-Osgood model, modified by Gardner and Nethercot, the parameters that described stress-strain relationship under monotonic loading were obtained. Comparison between data obtained using different types of specimens was made and the influence of the welding and rolling direction was discussed. The cyclic skeleton curves were obtained by fitting the Ramberg-Osgood model to the curves. Parameters of the hardening model of cyclic plasticity were calibrated from test data and the test was simulated with ABAQUS. The results show that, stainless steel exhibit remarkable nonlinearity, the Ramberg- Osgood model modified by Gardner and Nethercot provide excellent agreement with experimental data and welding and rolling direction have a direct influence on the stress-strain relationship. Under cyclic loading, with the increase of cyclic loops, stainless steel exhibit cyclic hardening behaviour, and the simulated curves and the test curves agree fairly well. Therefore, the influence of welding and rolling direction on the stress-strain relationship should be taken into consideration in projects and the constitutive relation under cyclic loading should be used if the structure is subjected to cyclic loading. Keywords stainless steel; constitutive relation; monotonic loading; cyclic loading; finite element analysis 1 Introduction Under a seismic event, the structural members, especially the dissipative elements, usually experience a small number of heavy cyclic loads accompanied with large plastic deformation within a short time-frame. The response of the element is mainly depended on the geometric dimension and the hysteretic behaviour of the material, which can be studied by large strain extremely low cyclic fatigue testing [1-4]. In the seismic design of steel structures, numerical simulation is widely used because of the high cost of the tests. Besides, the constitutive relation under cyclic loading is different from that under monotonic loading and is more complex in numerical simulation. Therefore, an accurate constitutive relation and calculation model of steel under cyclic loading is quite important to the numerical simulation [5]. Several simplified calculation models have been proposed by foreign researchers for simulating the behaviour of steel under cyclic loading. Comparisons have been made between tests and numerical simulations to verify the precision and reliability of these calculation models [6,7]. In recent years, more stainless steel structures have been built because stainless steel has many advantages, such as the attractive appearance, good corrosion resistance, ease of maintenance and low life cycle cost [8,9]. Several researchers have carried out a series of tests and numerical simulation to study the low cycle fatigue behaviour of stainless steel [1-13], but the tests on the large strain extremely low cycle fatigue are quite limited. However, because of the limitations in the development of the industry and economy in China, there are some differences in the quality, material properties and processing technology of stainless steel at home and aboard. The calculation models proposed by foreign researchers are relative with the processing technology and material properties of stainless steel, which may not apply to domestic ones. Therefore, to give reference to the design of stainless steel structures and seismic response, a total of 29 domestic stainless steel specimens were tested in the monotonic and cyclic loading to study the behaviour of the material and the tests were also simulated by finite element software ABAQUS. 2 Stress-strain relationships of stainless steel 2.1 Stress-strain relationships under monotonic loading The stress-strain relationship of stainless steel is different from that of carbon steel which exhibits a rounded stressstrain curve with no definitely yielding point, considerable stain hardening, and high ductility [14]. Since the absence of a definitely yielding point, the stress at.2% plastic strain is adopted as the equivalent yield strength (the.2% proof Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 1

2 strength). Besides, stainless steel exhibits anisotropy and a difference in stress-strain curves between tension and compression. The most widely used constitutive model was proposed by Ramberg and Osgood [7] and modified by Hill [15], as Eq. (1) σ σ ε = +.2 E σ n (1) where E is the material Young s modulus, σ.2 is the.2% proof stress, and n is the strain-hardening exponent which can be calculated as n=(ln2)/[ln(σ.2 /σ.1 )]. At higher strains, a large deviation has been observed between experimental data and Ramberg-Osgood model, therefore Mirambell and Real [16], Rasmussen [17] modified Ramberg-Osgood model by a two-stage model, as Eq. (2) n σ σ +.2 σ σ E σ.2 ε = m σ σ σ σ ε + ε σ > σ u E.2 σ σ u where E.2 =E /(1+.2nE /σ.2 ), m=1+3.5σ.2 /σ u. Gardner and Nethercot [14,18,19] pointed out that the use of ultimate stress σ u had two drawbacks. At first, the strain at the ultimate stress is much higher than that in the general structural response, which results in a great deviation between the measured stress-strain curve and the model when the stain is low. Then, the model cannot apply to the compressive stress-strain curve because there is no ultimate stress in compression. Therefore, σ 1. was proposed to replace σ u, as Eq. (3). (2) n σ σ +.2 σ σ E σ ε =.2 ' n.2,1. σ σ σ σ σ σ E E σ σ ε σ > σ t.2,.2.2 (3) where n.2,1. is the strain-hardening coefficient. Eq. (3) can be applied for design and numerical simulation because the experimental data is in excellent agreement with it, both in tension and compression and up to strain about 1% [14, 18, 19]. At higher strains, the original propose is more accurate [16]. At higher stain, the results obtained via the two-stage model still deviated from experimental data, so Quach et al. [2] proposed a three-stage model, as Eq. (4). n σ σ.2 E σ.2 σ σ 1 1 σ σ ( ) E E E σ σ σ a b σ + σ σ ' n.2,1. ε = + + σ σ + ε σ < σ σ σ > σ.2 2. (4) where a=σ 2. (1±ε 2. )-bε 2., b=[σ u (1±ε u )-σ 2. (1±ε 2. )]/(ε u- ε 2. ), positive denotes tension and negative compression. The three-stage model above is based on a large number of experiments, including austenitic stainless steel, ferritic stainless steel and duplex stainless steel, both in tension and compression. Therefore, it is applicable to the most of stainless steel used in structures and exhibits high precision in the entire range of strain [2, 21]. 2.2 Cyclic skeleton curves Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 2

3 The response of stainless steel under cyclic loading is different from that under monotonic loading, which can be expressed by cyclic skeleton curves. One of the widely used models is Ramberg-Osgood model [7] as Eq. (5). The symmetric cyclic loading about ε= is described with Eq. (6). (5) (6) where Δε is the total strain amplitude, Δε e is the elastic strain amplitude, Δε p is the plastic strain amplitude, Δσ is the stress amplitude, K is the cyclic strength coefficient and n is the cyclic strain hardening exponent. 3 Experimental study 3.1 Materials The domestic austenitic stainless steel S3168 (AISI 316, EN 1.441) was tested in the experiments. The chemical compositions are shown in Table 1. Table 1 Chemical compositions of austenitic stainless steel S3168 Chemical compositions C Si Mn P S Cr Mo Ni Cu wt % Monotonic loading experiments The size of the specimens used for monotonic loading is shown in Fig. 1. There are two categories, differ only in thickness. One is 6 mm in thick, the other is 8 mm. To study the influence of the rolling direction and welding, the specimens were taken from both base material and welded section, longitudinally and transversely. The tensile tests were carried out according to GB/T Metallic materials-tensile testing at ambient temperature [22]. The loading device was the hydraulic universal testing machine as shown in Fig.2. The strain was measured by strain gauge and extensometer. Fig. 1. Size of specimen used for monotonic loading Fig. 2. Experimental setups of monotonic loading 3.3 Cyclic loading experiments The size of specimen used for cyclic loading is shown in Fig. 3, taking from the welded section, longitudinally. The testing device was Instron 881, a hydraulic fatigue testing machine as shown in Fig. 4. The tests were strain-controlled with the instrument extensometer (gauge length of 12.5 mm) and loading frequency of.5 Hz. Because of the limitation of the gauge length and the harm to the instrument when the specimen buckles suddenly, the loading process was stopped when obvious buckling took place. As different instruments were used in the monotonic and cyclic loading experiments, another 3 specimens (H1 and H2) were monotonically loaded in this part. There were 8 different cyclic loading systems, as shown in Fig. 5. The load amplitude of each step was determined according to the actual situation. Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 3

4 Fig. 3. Size of specimen used for cyclic loading Fig. 4. Experimental setups of cyclic loading Fig. 5 Cyclic loading systems: (a) H3; (b) H4; (c) H5; (d) H6; (e) H7; (f) H8; (g) H9; (h) H1 4 Analysis of the experimental results 4.1 Experimental results of monotonic loading Failure mode No apparent necking phenomenon was observed in the experiments. The angle between the glossy fracture surface and the tensile axis was 45, which is a little different from carbon steel (Fig. 6) Experimental results Eq. (3) was used to fit the stress-strain curves. Table 2 and Table 3 give the parameters of the stress-strain curves of specimens of different thickness and Fig.7 is the corresponding stress-strain curves. Fig. 7 (a) and (b) are typical curves showing the comparison between the experimental curves and modified Ramberg-Osgood model and (c) ~ (e) illustrate the influence of rolling direction and welding to the stress-strain curves. Each specimen is labelled to identify the thickness, rolling direction and welding condition. The first part is the thickness of the specimen in the tensile experiments (6mm or 8mm in thick); the second part denotes the orientation in which the specimen is taken: L means the longitudinal direction from base material, i.e. the rolling direction, T means the transverse direction from base material, i.e. perpendicular to the rolling direction, W means the longitudinal direction from welded section. Table 2 Parameters of the stress-strain curves of specimens (6 mm in thick) Types of specimens E / MPa σ.1 / MPa σ.2 / MPa σ 1. / MPa σ 2. / MPa σ u / MPa n E.2 n, T6-L T6-T T6-W T6-T / T6-L T6-W / T6-L Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 4

5 Table 3 Parameters of the stress-strain curves of specimens (8 mm in thick) Types of specimens E / MPa σ.1 / MPa σ.2 / MPa σ 1. / MPa σ 2. / MPa σ u / MPa n E.2 n, T8-L T8-W T8-W / T8-L Fig. 6. Fracture surface Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 5

6 Modified R-O model Modified R-O model T6-L-1 T6-L-2 T6-L-3 T6-T-1 T6-T-2 T6-T T6-L-1 T6-L-2 T6-L-3 T6-W-1 T6-W-2 T6-W ε / % T8-L-1 T8-L-2 T8-L-3 T8-W-1 T8-W-2 T8-W ε / % Fig. 7. Stress-strain curves: (a) stress-strain curve of specimen T6- L-1; (b) stress-strain curve of specimen T8- L-1; (c) the influence of rolling direction (6mm); (d) the influence of welding (6mm); (e) The influence of welding (8mm). From the above figures and tables: (1) As shown in Table 2 and Fig. 7, the elastic modulus of T6-T is larger than that of T6-L by 4.4%. The strength level of T6-T is higher than T6-L, and for instance, the.2% proof strength σ.2 larger by 15.5% and the ultimate tensile strength σ u larger by 8.6%. These mean that rolling direction has an effect on the mechanical behaviour of stainless steel and the material of transverse direction is better than that of longitudinal direction. (2) Compared with T6-L, the elastic modulus of T6-W is higher than that of T6-L by 13.3%. The strength level of T6- W, except for σ.1, is lower than that of T6-L up to 4% which can be neglected. Compared with T8-L, the elastic modulus of T8-W is higher than that of T8-L by 6.4%. The strength level of T8-W is lower than that of T8-L, the.2% proof strength σ.2 of T8-W lower by 11.9% which cannot be neglected. Since the influence of welding is different for specimens of different thickness and no unified conclusion can be made, the influence of welding should be taken into consideration for safety. (3) The lower the percentage non-proportional extension, the more scattered the corresponding stress values are. σ.1 of specimen T8-L-5 cannot be calibrated from the experimental data which may be resulted from the unsteady data when Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 6

7 the stress is relatively low. The mean value of this group was used to fit the stress-strain curves and the modified Ramberg-Osgood model agree with the experimental data fairly well. (4) Up to 1% of strain, there is an excellent agreement between the experimental data and modified Ramberg-Osgood model SEM of the fractured surface The fractured surfaces were observed by scanning electron microscope with the magnification of 1 times and 2 times. Some of the results are shown in Fig. 8 which (a) and (c) enlarged 1 times and (b) and (d) 2 times. Obvious dimples can be observed which indicates ductile failure. 6μm 3μm 6μm 3μm a b c d Fig. 8. SEM of tensile fractured surfaces: (a) T6-T-3 (x1); (b) T6-T-3 (x2); (c) T8-W-1 (x1); (d) T8- W-1 (x2) 4.2 Experimental results of cyclic loading Monotonic loading The parameters of monotonic loading are shown in Table 4 and the stress-strain curves are shown in Fig. 9 (a) and (b). A little difference exists between part 4.1 and 4.2, which may be due to the use of different experiment instruments. It is important to note that stainless steel exhibit high ductility and the strain can reach 4%-6% [8]. But as the specimen is thin in thick and high in slenderness ratio, the buckling load is quite low and the strength and ductility of stainless steel are not fully developed in this part. Table 4 Main mechanical parameters of monotonic loading No. of specimen E / MPa σ.2 / MPa ε.2 / % note H The tensile strain is design to 1% H The tensile strain is design to 2% H The compressive strain is design to 1% Cyclic loading Table 5 gives the main mechanical behaviour of specimen under cyclic loading and Fig. 9 (c) ~ (j) gives the stress-strain curves. Table 5 Main mechanical parameters of cyclic loading No. of specimen E / MPa σ.2 / MPa ε.2 / % number of cyclic loops H H H H H H H H Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 7

8 Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 8

9 Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 9

10 Fig. 9 Stress-strain curves: (a) H1; (b) H2; (c) H3; (d) H4; (e) H5; (f) H6; (g) H7; (h) H8; (i) H9; (j) H1 From Fig. 9 we can conclude that, (1) After the material yields, with the increase of cyclic loops, cyclic hardening takes place. (2) In the first several loops, the maximum tensile stress is smaller than the maximum compressive stress because the cross sectional area increases under compression which makes the compressive load increases; then opposite phenomenon occurs because the decrease of compressive bearing capacity due to buckling of specimens. (3) After the material yields, the stress-strain curves of different loops under constant cyclic loading overlap with each other, as specimen H7 and H8. (4) The behaviour of one cycle is dependent on the stress and strain amplitude of the previous cycle which means the loading history has a direct influence on the stress-strain curves of stainless steel under cyclic loading Cyclic skeleton curves Table 6 gives the main parameters of cyclic hardening and Fig. 1 gives the cyclic skeleton curves. Table 6 Main parameters of cyclic hardening No. of Specimen E / MPa K / MPa N H H H H H H Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 1

11 Cyclic test data Hysteresis skeleton curve Monotonic curve Cyclic test data Hysteresis skeleton curve Monotonic curve Cyclic test data Hysteresis skeleton curve Cyclic test data Hysteresis skeleton curve Cyclic test data Hysteresis skeleton curve Monotonic curve Cyclic test data Hysteresis skeleton curve Monotonic curve Fig. 1 Cyclic skeleton curves: (a) H3; (b) H4; (c) H5; (d) H6; (e) H9; (f) H1 From Fig. 1 we can conclude that the response of stainless steel under cyclic loading is different from that under monotonic loading; Ramberg-Osgood model [7] fits the cyclic skeleton curves well; with the increase of the cyclic loops, the strength increases, especially in the later stage of cyclic loading. 5 Finite element analysis of the cyclic loading experiments Finite element software ABAQUS was used to simulate the cyclic loading experiments. The von Mises flow rule and the mixed model combining both isotropic hardening and non-linear kinematic hardening [23] were adopted in the simulation. The parameters of cyclic hardening was calibrated from the experimental data according to the help files of ABAQUS, as shown in Table 7, where σ is the yielding stress at zero plastic strain (taken as σ.1 ), Q is the maximum change in the size of yielding surface, b iso is the rate at which the size of yielding surface changes as plastic straining develops, C kin is the initial kinematic hardening moduli and γ is the rate at which the kinematic hardening moduli decrease with the increasing plastic deformation. Fig. 9 gives the comparison between the simulated curves and experimental curves. The simulated curves agree well with the experimental data which means that the parameters of cyclic loading can be used in the simulation of stainless steel under cyclic loading. Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 11

12 Table 7 Parameters of cyclic hardening used in ABAQUS σ / MPa Q / MPa b iso C kin / MPa γ Conclusion Experimental investigation and numerical simulation of stainless steel S3168 under monotonic and cyclic loading have been presented in this paper. It is shown that stainless steel exhibits remarkable nonlinearity. The modified Ramberg- Osgood model (Eq. (3)) and Ramberg-Osgood model (Eq. (5) and (6)) have an excellent agreement with the stressstrain curves under monotonic and cyclic loading, respectively. Rolling direction and welding process have an influence on the constitutive relation of stainless steel which should not be neglected in the engineering practice. It is also shown that the response of stainless steel under cyclic loading is different from that under monotonic loading. Under cyclic loading, stainless steel exhibits remarkable cyclic hardening and excellent cyclic behaviour. The simulated curves by ABAQUS agree well with the experimental data and the cyclic hardening parameters obtained in this paper can be used in the engineering practice. Acknowledgement This work was financially supported by Beijing Natural Science Foundation (No ), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No ) and the National Natural Science Foundation of China (No ). Nomenclature b iso rate at which the size of yielding surface changes as plastic straining develops σ 1. σ 2. σ u stress at 1.% plastic strain stress at 2.% plastic strain ultimate stress C kin initial kinematic hardening moduli σ yielding stress at zero plastic strain E material Young s modulus ε u ultimate strain K cyclic strength coefficient ε.2 strain at equivalent yield strength N strain-hardening exponent Δε total strain amplitude n.2,1. strain-hardening coefficient Δε e elastic strain amplitude n cyclic strain hardening exponent Δε p plastic strain amplitude Q σ.1 σ.2 maximum change in the size of yielding surface stress at.1% plastic strain equivalent yield strength Δσ γ stress amplitude rate at which the kinematic hardening moduli decrease with the increasing plastic deformation References Nip KH, Gardner L, Davies CM, Elghazouli AY. Extremely low cycle fatigue tests on structural carbon steel and stainless steel. Journal of Constructional Steel Research 21; 66(1): Shi YJ, Wang M, Wang YQ. Experimental study of structural steel constitutive under cyclic loading. Journal of Building Materials 212; 15(3): Shi YJ, Wang M, Wang YQ. Experimental and constitutive study of structural steel under cyclic loading. Journal of Constructional Steel Research 211; 67(8): Shi G, Wang F, Dai GX, Shi YJ, Wang YQ. Experimental study on high strength structural steel Q46D under cyclic loading. China Civil Engineering Journal 212; 45(7): Gu Q. The Hysteretic Behaviour and Seismic Design of Steel Structures. China Architecture & Building Press, Beijing, 29. Dong YT, Zhang YC. Cyclic plasticity constitutive model of structural steel. Journal of Harbin University of Civil Engineering and Architecture 1993; 26(5): Paper Presented by Huanxin Yuan - yuanhx9@gmail.com Y.Q. Wang, T. Chang, Y.J. Shi, H.X. Yuan & D.F. Liao, Tsinghua University 12

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