Compressive Behavior and Constitutive Model of Austenitic ... - MDPI

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Compressive Behavior and Constitutive Model of Austenitic Stainless Steel S30403 in High Strain Range Yang Peng, Jiang Chu and Jun Dong * College of Civil Engineering, Nanjing Tech University, Nanjing 211816, China; [email protected] (Y.P.); [email protected] (J.C.) * Correspondence: [email protected]; Tel.: +86-136-0140-7837 Received: 31 March 2018; Accepted: 12 June 2018; Published: 15 June 2018







Abstract: Material anisotropy for tension and compression is a significant characteristic of austenitic stainless steel compared to carbon steel. Due to limitations during the testing of the restrained jig, the maximum strain value of compressive experiments of austenitic stainless steel is around 2%. This value cannot satisfy the requirements of accurate finite simulation on austenitic stainless steel columns and beams in the high compressive strain range. In this study, a new type of compressive specimen that satisfies the high compressive strain test was designed. The stress-strain response of austenitic stainless steel S30403 (JISCO, Gansu, China) was investigated in the high compressive strain range up to 10%, and constitutive models were compared with the experimental data. It was found that the new type specimen with length-to-diameter ratio of 1:1 can reliably obtain the stress-strain response of austenitic stainless steel S30403 in the high compressive strain range. It was found that the material anisotropy of austenitic stainless steel S30403 is remarkable in the high compressive strain range up to 10%. The strain-hardening curve of the austenitic stainless steel S30403 can be represented by a straight line in the high compressive strain range. Our study also found that the Quach constitutive model accurately describes the two-stage strain-hardening phenomenon in the high compressive strain range up to 10%. Keywords: compressive behavior in high strain range; stress-strain response; compressive constitutive model; material anisotropy; stainless steel

1. Introduction Owing to its easy maintenance and ability to resist corrosion as well as its attractive appearance and excellent structural performance, stainless steel has been increasingly utilized in structural applications. In addition to its low proportional limits, stainless steel is characterized by material anisotropy, round stress-strain behavior, and different constitutive behavior in tensile and compressive loading [1]. Compressive constitutive models for stainless steel have a significant impact on the prediction of the structural behavior of stainless steel columns and beams by numerical simulation [2–6]. The predicted difference in the ultimate axial load of concrete-filled stainless-steel tubular columns is 16.9% when the strain-hardening behavior in tension and compression is ignored [5,6]. As a commonly used material of stainless steel, austenitic stainless steel is initially entirely composed of austenite. The austenite can be transformed into martensite under loading, resulting in a change to the mechanical behavior of austenitic stainless steel. Different constitutive models for austenitic stainless steel have been suggested based on different approaches. Physically based constitutive models have been developed [7–9], which can reveal the strain dependence of the kinetics of transformation of the austenitic stainless steel under loading. These models have many material

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parameters and the identification of these material parameters is time-consuming. However, they are popular in materials engineering research. Two constitutive models proposed by Gardner and Ashraf [10] and Quach et al. [11] are suitable for accurately describing the compressive behavior of austenitic stainless steel [2–6]. These constitutive models were verified by comparing their results with the experimental data from compressive tests published in the literature [10–12]. These models are phenomenological; some of the parameters of these models are empirical and obtained by regression. They are popular in engineering applications. However, the reliability of the constitutive models verified by these compressive experimental data in the high strain range is unknown. Constitutive models are therefore needed to verify the experimental data in the high strain range. Numerous experimental studies of compressive behavior of austenite stainless steel have been performed. Johnson and Kelsen [13] studied the compressive behavior of the 302 and 304 austenitic stainless steel and the maximum strain was less than 1%. They found that austenitic stainless steel was anisotropic. Korvink et al. [14] tested the compressive behavior of ferritic stainless steel 430 and the maximum strain was 0.6%. Gardner et al. [12] tested the compressive behavior of austenitic stainless steel 304, and their results indicated that the compressive stress-strain curves were in agreement with the tensile stress-strain curve with a strain less than 2%. Buchanan et al. [15] tested the additive manufactured austenitic stainless steel 316L under compressive stress, and the maximum strain was 1%. They found that the strength and Young’s modulus of additive manufactured austenitic stainless steel was lower than those of conventional austenitic stainless steel. Overall, the maximum compressive strain was less than 2% of all the compressive experimental data. The specimens buckled under high strain range. To prevent buckling, a lateral support jig was used in the compressive experiments. However, the gap between the lateral support jig and the specimen was limited. Owing to the Poisson effect and volume expansion, the specimen contracted the lateral support jig under high strain, which restricted the deformation of the specimens. Therefore, high compressive strain could not be attained by this test approach. Due to the limitation of these experiments, the constitutive model could not be verified by comparison with the high compressive strain data. In this study, a new configuration of compressive specimen made with austenitic stainless steel S30403 was designed to be suitable for high compressive strain experiments, and the feasibility of the new specimen was studied. The compressive mechanical properties of austenitic stainless steel S30403 were determined and the material anisotropy under tension and compression was studied. Furthermore, Gardner and Ashraf constitutive model and the Quach model were compared to the compressive test results, and the reliability of these models is discussed. 2. Experimental Details The test material was a low-carbon austenitic stainless steel S30403 (022Cr19Ni10, equivalent to AISI304L and EN 1.4306) that is commonly used in welded structures. The Chinese standard GB/T 4237-2015 [16] provided the chemical composition shown in Table 1 and mechanical properties shown in Table 2. Table 1. Chemical composition specified by GB/T 4237-2015 (weight percentage). w(C)

w(Si)

w(Mn)

w(P)

w(S)

w(Cr)

w(Ni)

0.03

0.75

2.00

0.045

0.030

17.50

8.00

Table 2. Mechanical properties specified by GB/T 4237-2015.

Grade

Nominal Yield Strength σ 0.2 /MPa

Tensile Strength σ u /MPa

Elongation after Fracture εu /%

Longitudinal/Transverse Strain Reinforcement Coefficient n

S30403

180

485

40

6/8

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In stress compression tests, distributed, specimens may exhibit under high strain, including elastic and gauge is unevenly making the buckling test result ineffective [17]. To avoid buckling in inelastic buckling. After buckling, the specimens undergo bending or torsional deformation and compressive tests, the specimens are generally surrounded by a bracing jig and smeared with the gauge stress is unevenly distributed, test result However, ineffectiveowing [17]. To buckling lubricating paste—a method that has beenmaking widelythe used [12,15]. to avoid the limited gap in compressive tests, the specimens are generally surrounded by a bracing jig and smeared with between the bracing jig and the specimen, the stress-strain curve cannot be measured for high strain lubricating paste—a methodthis thatapproach has beencannot widelyobtain used [12,15]. However,behavior owing tointhe gap up to 3% [12,15]. Therefore, the compressive thelimited high strain between range. the bracing jig and the specimen, the stress-strain curve cannot be measured for high strain up to 3%Another [12,15]. Therefore, this obtain the compressive behavior in thereduce high strain range. approach is toapproach keep the cannot specimen’s cross-sectional area unchanged, the gauge Another approach is to keep the specimen’s cross-sectional unchanged, thethe gauge length of the specimen, and thus reduce the slenderness ratio of area the specimen andreduce increase test length of the specimen, and thus reduce the slenderness ratio of the specimen and increase the test specimen’s buckling strength [18]. This approach can obtain compressive behavior in the high strain specimen’s buckling strength [18].that This can obtain ratio compressive behaviorsteel in the strain range. Marsh et al. [19] proposed theapproach length-to-diameter of the structural testhigh specimen range. Marsh et al. [19] of proposed thatsection the length-to-diameter ratio the structural steel test (the ratio of the length the gauge to the diameter of theof gauge section) should bespecimen less than (the ratio of the length of the gauge section to the diameter of the gauge section) should be less or equal to 1.3 to prevent the specimen from buckling. Zhou et al. [20] conducted the structural than steel or equal totest 1.3 to prevent the specimen from buckling. Zhou et al. the structural hysteresis and concluded that the length-to-diameter ratio of [20] the conducted test specimens should besteel less hysteresis testtoand concluded that the ratio of therecommends test specimens be less than or equal 1.13. Compression test length-to-diameter standard GB/T 7314-2005 [17] thatshould the specimen than or equal to 1.13. Compression standard [17] recommends that the specimen length-to-diameter ratio be betweentest 1 and 2. The GB/T tension7314-2005 and compressive properties of stainless steel length-to-diameter ratio beand between 1 and 2. The tension and compressive properties ofisstainless are significantly different the strain-hardening phenomenon under compression higher steel than are different andstrain the strain-hardening phenomenon compression is higher than thatsignificantly when pulled [21]. The hardening of the austenitic under stainless steel is higher than thatthat of when pulled [21]. The strain hardening of the austenitic stainless steel is higher than that of structural structural steel [1]. This implies that the force imposed on compressive specimens is higher than that steel [1]. This implies that the force imposed on compressive specimens is higher than that of structural of structural steel specimens when the compressive strain is identical. Austenitic stainless steel steel specimens when the compressive strain is identical. Austenitic stainless steel specimens require specimens require higher buckling strength compared to structural steel specimen when a higher higher structural specimen when a higher load is imposed on load isbuckling imposedstrength on the compared austenitic to stainless steelsteel specimen under the same compressive strain. the austeniticthe stainless steeldue specimen undercompressive the same compressive Considering variations Considering variations to different propertiesstrain. between austenitic the stainless steel due to different compressive properties between austenitic stainless steel and and structural steel, this test adopted a smaller value than that recommended bystructural Zhou et alsteel, [20]. this The test adopted a smaller value thanspecimen that recommended by Zhou ettoal.the [20]. The limit length-to-diameter length-to-diameter ratio of the was 1:1, according lower value of the ratio of the specimen was 1:1, according to the lower of the testthe standard compression test standard GB/T 7314-2005 [17]; the limit gaugevalue length wascompression 12.5 mm and gauge GB/T 7314-2005 [17]; the gauge length was 12.5 mm and the gauge diameter was 12.5 mm (Figure 1). diameter was 12.5 mm (Figure 1).

Figure 1. Small-gauge-length specimen.

The stress state of the small-gauge-length specimen may be in a triaxial stress state under high The stress state of the small-gauge-length specimen may be in a triaxial stress state under high strain when the grip section and the fillet are larger than the gauge length. The strain distribution is strain when the grip section and the fillet are larger than the gauge length. The strain distribution nonuniform in the triaxial stress state, and the nonuniform strain distribution cannot satisfy the is nonuniform in the triaxial stress state, and the nonuniform strain distribution cannot satisfy the requirement of the extensometer. In this study, the finite element analysis software ABAQUS was requirement of the extensometer. In this study, the finite element analysis software ABAQUS was used used to analyze the strain distribution within the gauge length section of the specimen under to analyze the strain distribution within the gauge length section of the specimen under compression. compression. The finite element analysis uses a solid element. The constitutive model was the The finite element analysis uses a solid element. The constitutive model was the kinematic-hardening kinematic-hardening model and the specified value was selected for the material parameter from the model and the specified value was selected for the material parameter from the standard CECS 410: standard CECS 410: 2015 [22]. The simulation results in Figure 2 show that when a −10% compressive 2015 [22]. The simulation results in Figure 2 show that when a −10% compressive strain was imposed strain was imposed on the small-gauge-length specimen, the strain was evenly distributed along the on the small-gauge-length specimen, the strain was evenly distributed along the gauge length section, gauge length section, and the influence of the grip section and the fillet could be ignored. The strain and the influence of the grip section and the fillet could be ignored. The strain distribution therefore distribution therefore satisfied the requirements of the extensometer. satisfied the requirements of the extensometer. Zhou et al. [20] suggested that the feasibility of small-gauge-length specimen can be verified by comparing the tensile test results of conventional-gauge-length specimens with those of smallgauge-length specimens. The tensile specimen was designed according to national design standard

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GB/T 228.1-2010 [23], as shown in Figure 3, in which the gauge length was 70 mm and gauge diameter 2018, 11, x 4 of 11 wasMaterials 12.5 mm.

Figure 2. Strain distribution along the gauge length under high compressive strain (ε = −10%).

Zhou et al. [20] suggested that the feasibility of small-gauge-length specimen can be verified by comparingFigure the tensile test results ofalong conventional-gauge-length with those of small-gauge2. Strain distribution the gauge length under highspecimens compressive strain (ε = −10%). length specimens. The tensile specimen was designed according to national design standard GB/T Figure 2.al. Strain distribution along the gauge under high compressive strain (εgauge = −10%). Zhou [20] suggested that feasibility of small-gauge-length can verified 228.1-2010 [23], as shown in Figure 3,the in which the gauge length was 70 specimen mm and was Figure 2.etStrain distribution along the gaugelength length under high compressive strain (εbe=diameter −10%). by comparing the tensile test results of conventional-gauge-length specimens with those of small-gauge12.5 mm. Zhou et al. [20] suggested the feasibility of small-gauge-length specimen can be verifiedGB/T by length specimens. The tensilethat specimen was designed according to national design standard comparing the tensile test results of conventional-gauge-length specimens with those of small-gauge228.1-2010 [23], as shown in Figure 3, in which the gauge length was 70 mm and gauge diameter was length specimens. The tensile specimen was designed according to national design standard GB/T 12.5 mm. 228.1-2010 [23], as shown in Figure 3, in which the gauge length was 70 mm and gauge diameter was 12.5 mm.

Figure3.3.Conventional-gauge-length Conventional-gauge-length specimen. Figure specimen.

The stainless steel plates were cut in the rolled direction. The test specimens were manufactured Figure 3. Conventional-gauge-length The stainless steel plates were cut in the rolled direction. specimen. The test specimens were manufactured from stainless steel plates into round test coupons. The reduced section and round transition zone from stainless steel plates into round test coupons. The reduced section and round transition zone Figure 3. cut Conventional-gauge-length were machined using numerically controlled ofwere the specimens were The stainless steelthe plates were in the rolled equipment. direction.specimen. TheThe testsurfaces specimens manufactured were machined usingusing the numericallygrit controlled equipment. The surface surfacesdefects. of the specimens were carefully polished eliminate The test was from stainless steel platesdifferent into round testsandpapers coupons. Thetoreduced section and round transition zone carefully polished using different grit sandpapers to eliminate surface defects. The test was performed The stainless steel plates were cut in the rolled direction. The test specimens were manufactured performed on an MTS 370 hydraulic material machine with a maximum were machined using Landmark the numerically controlledservo equipment. Thetesting surfaces of the specimens were from stainless into roundservo test The reduced section and round transition zone on loading an MTS Landmark 370 hydraulic material testing machine with adefects. maximum loading force force ofsteel 500 plates kN. Computer-controlled loading procedures were used during the The carefully polished using different grit coupons. sandpapers to eliminate surface Thetest. test wastest were machined using the numerically controlled equipment. The surfaces of the specimens were of 500 kN. Computer-controlled loading procedures were used during the test. Thea test procedure performed on an MTS 370 hydraulic servo material testing machine with maximum procedure complied withLandmark GB/T 7314-2005 [17]. During the test, the strains of the conventional-gaugecarefully polished using different sandpapers toprocedures eliminate surface defects. Thetest. testThe was loading force of 500 kN. Computer-controlled loading were used during the test a complied with GB/T 7314-2005 [17].grit During the test, the strains of the conventional-gauge-length length specimens and small-gauge-length specimens were determined by an extensometer with performed on an MTS Landmark 370 hydraulic servo material testing machine with a maximum procedure GB/T 7314-2005 [17]. During the theand strains ofextensometer thelength conventional-gaugespecimens andcomplied small-gauge-length specimens were determined byaan a gauge gauge length of 50 mmwith (strain measurement range −10% totest, +50%) gauge of 10with mm (strain loading force of 500 kN. Computer-controlled loading procedures were used the test. Thewith test a length and small-gauge-length were determined byduring an extensometer measurement −10% to +20%), respectively. Thetoextensometer used were 634.25Flength of 50specimens mmrange (strain measurement rangespecimens −10% +50%) and models a gauge length ofMTS 10 mm (strain procedure complied with GB/T 7314-2005 [17]. During the test, the strains of the conventional-gaugelength of 50 mm (strain measurement −10% to +50%) andmodels aisgauge length of 10MTS mm (strain 24 gauge and MTS 634.31F-21. The arrangement ofrange theThe measuring device shown in were Figure 4. The force measurement range − 10% to +20%), respectively. extensometer used 634.25F-24 length specimens and −10% small-gauge-length specimens determined by anused extensometer with a measurement range to +20%), respectively. Thewere extensometer models were MTS 634.25Fwas measured by a The force sensor of the machine and all data werelength recorded by a (strain computer. and MTS 634.31F-21. arrangement oftesting the measuring device isashown in Figure 4. The force was gauge length of 50 mm (strain measurement range −10% to +50%) and gauge of 10 mm 24 and MTS 634.31F-21. The arrangement of the measuring device is shown in Figure 4. The force When the was less of than the nominal yield strength, thewere loading rate was mm/min. Whenthe measured by stress a force sensor the testing machine and all data recorded by a0.1 computer. When measurement range +20%), The extensometer models were MTS 634.25Fwas measured by a−10% forcetosensor ofrespectively. the testing machine and all data wereused recorded by a computer. the stress was greater than the nominal strength, the was loading rate 0.54. mm/min. The 24When and MTS 634.31F-21. Thethan arrangement ofyield the measuring device is 0.1 shown in was Figure The stress was less the yield strength, the loading rate mm/min. theforce stress was the than stress wasnominal less the nominal yield strength, the loading rate was 0.1When mm/min. When recording frequency of the extensometer was 10 Hz. was measured by a force sensor of the testing machine and all data were recorded by a computer. greater than the nominal yield strength, the loading rate was 0.5 mm/min. The recording frequency of the stress was greater than the nominal yield strength, the loading rate was 0.5 mm/min. The

the stress was10 less than the nominalwas yield the loading rate was 0.1 mm/min. When the When extensometer was recording frequency ofHz. the extensometer 10strength, Hz.

the stress was greater than the nominal yield strength, the loading rate was 0.5 mm/min. The recording frequency of the extensometer was 10 Hz.

Grips

Grips Grips

Specimen

Specimen Specimen

Grips

Grips Grips

Extensometer

Extensometer Extensometer

Figure 4. Arrangement of the extensometer.

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Figure 4. Arrangement of the extensometer.

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When specimens are subjected to high non-elastic strains, the gauge length section has a radial When specimens are subjected to high non-elastic strains, the gauge length section has a radial contraction or expansion deformation, and the nominal stress and strain cannot accurately describe contraction or expansion deformation, and the nominal stress and strain cannot accurately describe the “true” stress and strain states on the specimen [24]. In this study, the nominal stress and strain the “true” stress and strain states on the specimen [24]. In this study, the nominal stress and strain were converted into true stress and strain using Equations (1) and (2) [20]: were converted into true stress and strain using Equations (1) and (2) [20]:

 true  ln(1   nom )

(1) (1)

 true   nom (1   nom )

(2) (2)

ε true = ln(1 + ε nom )

σtrue = σnom (1 + ε nom )

andε  trueareare true stressand andstrain, strain,respectively, respectively,and andσ  nom and  are In the true and In the formula, formula, σtrue true stress true nom and ε nom nom are nominal stress strain, respectively. nominal stress andand strain, respectively. 3. Results and Discussion 3.1. Comparison of of Tensile 3.1. Comparison Tensile Test TestResults Results Figure ofof the specimens is Figure 55 shows shows photographs photographsof ofthe thetwo twospecimens specimensafter afterfracture. fracture.The Thenecking necking the specimens obvious before fracture. It shows that the had a large before fracture, is obvious before fracture. It shows thattwo thespecimens two specimens had aplastic large deformation plastic deformation before which represents the true performance of the stainless steel. Figure 6 shows the true stress-true strain fracture, which represents the true performance of the stainless steel. Figure 6 shows the true stresscurves of the conventional-gauge-length specimen and the small-gauge-length specimen specimen obtained true strain curves of the conventional-gauge-length specimen and the small-gauge-length from the tensile test. The 10% strain was consistent for both samples and the test curve of the obtained from the tensile test. The 10% strain was consistent for both samples and the test curve of small-gauge-length specimen was slightly higher than the conventional-gauge-length specimen’s the small-gauge-length specimen was slightly higher than the conventional-gauge-length specimen’s test curve. This Thisshows showsthat that the small-gauge-length specimen uniformly deformed within the test curve. the small-gauge-length specimen waswas uniformly deformed within the 10% 10% strain of the gauge length section. Table 3 shows the nominal yield strength, tensile strength, strain of the gauge length section. Table 3 shows the nominal yield strength, tensile strength, and and elongation fracture thespecimens. two specimens. The measured strength the two test elongation after after fracture of theoftwo The measured strength valuesvalues of the of two test pieces pieces were almost theand same the elongation after breaking was different. The elongation after were almost the same theand elongation after breaking was different. The elongation after fracture fracture was inversely proportional to the length of the gauge length section; thus, the elongation after was inversely proportional to the length of the gauge length section; thus, the elongation after fracture fracture of of the the small-gauge-length small-gauge-length specimen specimen was was greater greater than than that that of of the the conventional-gauge-length conventional-gauge-length specimen [25]. specimen [25].

(a)

(b) Figure 5. 5. Tensile test failure failure morphology; morphology; (a) Figure Tensile test (a) Conventional-gauge-length Conventional-gauge-length specimen; specimen; (b) (b) Small-gaugeSmall-gaugelength specimen. length specimen.

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True stress true(MPa)

600 500

si pres m o C

400

e Tensil

300 200

CT-1 CT-2 CT-3

100 0 0

on

0.02

0.04

ST-1 ST-2 ST-3

0.06

0.08

SC-1 SC-2 SC-3

0.10

0.12

True strain true(-) Figure 6. True stress-true strain curves for tensile and compression tests. (CT denotes tensile test of Figure 6. True stress-true strain curves for tensile and compression tests. (CT denotes tensile test of the the conventional-gauge-length specimen, and ST and SC denote tensile test and compressive test of conventional-gauge-length specimen, and ST and SC denote tensile test and compressive test of the the small-gauge-length specimen). small-gauge-length specimen). Table 3. Test results. Table 3. Test results. Nominal Initial Proof Stress of Proof Stress of Ultimate Elongation Secant ModulusNominal Yield Proof Stress Initial Secant Proof Stress Test No. Modulus 1.0 10.0% Strain Ultimate u aElongation 1.0% Strain σ Strength σ after Fracture Yield of 10.0% E0.2/MPa Strength σ0.2 Test No. Modulus Modulus of 1.0%aStrain Strength after Fracture a /MPa /MPa 10.0σ u/% E0/MPa /MPa σ ε Strength Strain a/MPa 10.0 E0 /MPa E0.2 /MPa σ 1.0 a /MPa σ u a /MPa εu /% a /MPa σ 0.2 a /MPa CT-1 203,626 16,382 233 275 395 721 57 203,626 16,382 233 236 275 278 395396 721 715 57 56 CT-2CT-1 194,100 14,981 194,100 14,981 236 240 278 277 396396 715 725 56 59 CT-3CT-2 195,331 15,214 CT-3 195,331 15,214 240 277 396 725 59 mean value 197,686 15,526 236 277 396 720 57 mean value 197,686 15,526 236 277 396 720 57 Standard deviation 4230 613 Standard deviation 4230 613 2.872.87 1.251.25 0.470.47 4.114.11 1.25 1.25 ST-1 ST-1 194,689 15,596 194,689 15,596 242 242 290 290 406406 720 720 94 94 196,896 16,330 244 244 295 295 409409 727 727 93 93 ST-2 ST-2 196,896 16,330 195,460 15,784 245 245 292 292 407407 723 723 92 92 ST-3 ST-3 195,460 15,784 mean value 195,682 15,903 244 292 407 723 93 93 mean value 195,682 15,903 244 292 407 723 Standard deviation 915 311 1.25 2.05 1.25 2.87 0.82 Standard deviation 915 311 1.25 2.05 1.25 2.87 0.82 SC-1 206,405 15,196 267 317 600 b b SC-1 206,405 15,196 267 317 600 b SC-2 195,471 16,014 267 317 604 b SC-2SC-3 195,471 16,014 604 b 203,590 17,046 269 267 314 317 600 b b SC-3 203,590 17,046 mean value 201,822 16,085 268 269 316 314 - 601600 mean value 16,085 601 b Standard deviation 201,822 4636 757 0.94268 1.41316 1.89 - Standardadeviation 4636 757 0.94 1.41 1.89 b σ0.2 , σ1.0 , σ10.0 and σu are the nominal strength values; For the description of compressive stress-strain behavior, a σ0.2, is there no, σultimate compression due to thevalues; absenceb of thethe necking phenomenon, so we usedstress-strain σ10.0 instead σ1.0 10.0 and stress σu areinthe nominal strength For description of compressive of σu . behavior, there is no ultimate stress in compression due to the absence of the necking phenomenon,

so we used σ10.0 instead of σu.

3.2. Compression Test

3.2. Compression Test Since the maximum measured value of the extensometer compressive strain is 10%, the test was terminated when the compressive reached 10% and nocompressive buckling occurred the entire Since the maximum measured strain value of the extensometer strain isduring 10%, the test was loading process. at the endreached of loading is and shown Figure 7. The gauge section the terminated whenThe the specimen compressive strain 10% no in buckling occurred during the of entire specimen bulged under compression, and the diameter increased from 12.50 mm to 13.16 mm, or by loading process. The specimen at the end of loading is shown in Figure 7. The gauge section of the 5.3%. Thebulged new type of the specimen with a small gauge length did not12.50 require bracing jigorand specimen under compression, and the diameter increased from mmthe to 13.16 mm, by lubricating paste, as shown in Figure 4. Thisa simplifies thelength compressive preparation and reduces 5.3%. The new type of the specimen with small gauge did nottest require the bracing jig and the testing cost. The strain of thesimplifies test can reach 10%. lubricating paste, as compressive shown in Figure 4. This the compressive test preparation and reduces

the testing cost. The compressive strain of the test can reach 10%.

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Figure 7. Specimen under −10% compressive strain. Figure 7. Specimen under under −10% −10%compressive compressivestrain. strain. Figure 7. Specimen

The strain-hardening behavior of the austenitic stainless steel in compression tests is affected by The behavior of austenitic stainless steel tests is by The strain-hardening strain-hardening of the thethe austenitic stainless process steel in in compression compression testsstainless is affected affected by the manufacturing process.behavior Considering manufacturing of the austenitic steel, the manufacturing process. Considering the manufacturing process of the austenitic stainless steel, the manufacturing process. results Considering manufacturing the austenitic stainless steel, the published experimental can bethe classified into threeprocess grades:of annealed condition, tempered the published experimental results can be classified into three grades: annealed condition, tempered the published experimental results can be classified into three annealed condition, tempered condition, and cold-formed condition. The material anisotropy is higher in the tempered condition condition, and condition. The anisotropy is in tempered condition condition, and cold-formed cold-formed condition. The material material anisotropy is higher higher in the the tempered condition and cold-formed condition [1,13]. We compared the mean stress-strain curve of the compressive test and cold-formed condition [1,13]. We compared the mean stress-strain curve of the compressive test and cold-formed conditionexperimental [1,13]. We compared thethe mean stress-strain curve of processes the compressive results with the published results for different manufacturing (Figuretest 8). results with published experimental results for the manufacturing processes (Figure 8). results with the thestainless published experimental results forwas thedifferent different manufacturing 8). The austenitic steel adopted in this study manufactured under the processes hot-rolled(Figure annealed The austenitic stainless steel adopted in this study was manufactured under the hot-rolled annealed The austenitic steel adopted this study was manufactured under the hot-rolled annealed condition. The stainless results obtained underin tempered and cold-formed conditions [12,14,26] are above the condition. The results obtained under tempered and cold-formed conditions [12,14,26] are above the condition. The results obtained under tempered and cold-formed conditions [12,14,26] are above compressive stress-strain curve obtained in this study (Figure 8). In Figure 8, the compressive stresscompressive stress-strain curvecurve obtained in thisinstudy (Figure 8). In Figure 8, the compressive stressthe compressive obtained this study (Figure In Figure 8, the compressive strain curve is in stress-strain agreement with the stress-strain curve under the 8). annealed condition reported by strain curve is in agreement with the stress-strain curve under the annealed condition reported by stress-strain is in[13] agreement with the stress-strain under the annealed reported Johnson andcurve Kelsen and Rasmussen et al. [26]. curve The test results representcondition the compressive Johnson andand Kelsen [13][13] and Rasmussen et al.al.[26]. by Johnson Kelsen and Rasmussen [26].The Thetest testresults resultsrepresent represent the the compressive compressive behavior of austenitic stainless steel under theetannealed condition. behavior behavior of of austenitic austenitic stainless steel under the annealed condition.

True (MPa) TrueStress Stresstrue true (MPa)

1400 1400 1200 1200 1000 1000 800 800 600 600 400 400 200 200 0 00 0

Tempered Tempered

Cold rolled Cold rolled Annealed Annealed

0.02 0.02

Test result Test result Gardner & Nethercot [12] Gardner Nethercot [12] Korvink & [14] Korvink [14]1 [26] Rasmussen Rasmussen Rasmussen 12 [26] [26] Rasmussen 2 [26] STP454 1 [13] STP454 STP454 12 [13] [13] STP454 STP454 23 [13] [13] STP454 3 [13]

0.04 0.06 0.08 0.04 0.06 0.08 True Strain (-) true True Straintrue (-)

0.10 0.10

0.12 0.12

Figure 8. Comparison of compressive test results with the compressive test data in literature. Figure Figure8. 8. Comparison Comparisonof ofcompressive compressive test test results results with with the the compressive compressive test test data data in in literature. literature.

The mechanical properties of compressive specimens are shown in Table 3. The tensile and The mechanical properties of compressive specimens are shown in Table 3. The tensile and compressive deformation modulus values (including initial modulus and wereand the The mechanical properties of compressive specimens are shown in secant Table modulus) 3. The tensile compressive deformation modulus values (including initial modulus and secant modulus) were the same, as shown in Table 3. The true stress-true strain curves obtained by the compressive tests up to compressive deformation modulus values (including initial modulus and secant modulus) were the same, as shown in Table 3. The true stress-true strain curves obtained by the compressive tests up to 10% strain are shown 6.stress-true There were significant differences in the strength tests values, same, as shown in Tablein3. Figure The true strain curves obtained by the compressive up as to 10% strain are shown in Figure 6. There were significant differences in the strength values, as observed from the compression and tension test results. The proof stress values of the small-gauge10% strain are shown in Figure 6. There were significant differences in the strength values, as observed observed from the compression and tension test results. The proof stress values of the small-gaugelengththe specimens at 1%and strain were test 292 MPa andThe 316 proof MPa (i.e., difference 8%)small-gauge-length and at 10% strain, from compression tension results. stress values ofofthe length specimens at 1% strain were 292 MPa and 316 MPa (i.e., difference of 8%) and at 10% strain, the values at were 407 MPa and MPa of 48%) underoftensile and loading, specimens 1% strain were 292601 MPa and(difference 316 MPa (i.e., difference 8%) and at compressive 10% strain, the values the values were 407 MPa and 601 MPa (difference of 48%) under tensile and compressive loading, respectively. The material anisotropy was found to be more remarkable at a higher level of imposed were 407 MPa and 601 MPa (difference of 48%) under tensile and compressive loading, respectively. respectively. The material anisotropy was found to be more remarkable at a higher level of imposed strain. This characteristic canfound be attributed to remarkable the strain-induced transformation phenomenon of The material anisotropy was to be more at a higher level of imposed strain. This strain. This characteristic can be attributed to the strain-induced transformation phenomenon of austenite to martensite in stainless steelstrain-induced [8]. Since the formation of martensite is suppressed under characteristic can be attributed to the transformation phenomenon of austenite to austenite to martensite in stainless steel [8]. Since the formation of martensite is suppressed under tensile loading but is steel promoted under loading [27,28], the proportion of martensite martensite in stainless [8]. Since the compressive formation of martensite is suppressed under tensile loading tensile loading but is promoted under compressive loading [27,28], the proportion of martensite increases at high compressive strains, which enhances the strain hardenability of the austenitic increases at high compressive strains, which enhances the strain hardenability of the austenitic stainless steel and postpones the onset of necking [8]. Therefore, the higher proportion of martensite stainless steel and postpones the onset of necking [8]. Therefore, the higher proportion of martensite

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but is promoted under compressive loading [27,28], the proportion of martensite increases at high compressive strains, which enhances the strain hardenability of the austenitic stainless steel and postpones the onset of necking [8]. Therefore, the higher proportion of martensite caused by the high compressive loading is the main reason for the material anisotropy of stainless steel. In our study, when the true strain was 1%, the true stress-true strain curve of the tensile test and compression test of the small-gauge-length specimen began to separate, and with the increase in true strain, the difference in true stress between the two increased. When the true strain reached 10%, the true stress difference between the two exceeded 80 MPa (Figure 6). The comparison of the strength values measured in the compression test and tensile test of the small-gauge-length specimens, shown in Table 3, reveals that the nominal yield strength measured by the compression test of the small-gauge-length specimens was higher than the nominal yield strength measured by the tensile test of the conventional-gauge-length specimens. The elongation after fracture measured by the tensile test of the small-gauge-length specimens was much greater than that of the conventional-gauge-length specimens, and the difference in elongation after fracture between the conventional-gauge-length specimens and small-gauge-length specimens was due to the large difference in the ratio of the contraction segment to the standardization segment of the two specimens. The difference between the tensile and compressive stress-strain curves increased with increase of the strain. This is related to the different strain-hardening behavior being attributed to different microstructure evolution under the tensile and compressive loading. 3.3. Compressive Constitutive Model of Austenite Stainless Steel 3.3.1. Constitutive Models In the field of civil engineering, there are two constitutive models that describe the compressive properties of stainless steel: (1) Gardner and Nethercot revised model based on the Ramberg–Osgood model [12] and (2) Quach et al.’s three-stage model [11]. The modified model of Gardner and Nethercot is a combination of two Ramberg–Osgood models and is in good agreement with the stress-strain curve of stainless steel:   n   Eσ + 0.002 σσ σ ≤ σ0.2 0 0.2     n0 0.2,1.0 (3) ε= σ −σ0.2 σ0.2  1 1  σ− + ε σ > σ + 0.008 + σ − σ − · ( ) 0.2 0.2 0.2 1.0 E0.2 E0 E0.2 σ −σ0.2 1.0

where σ1.0 is the proof stress of 1.0% strain, n is the first curve’s hardening index, ε 0.2 is the total strain at stress σ0.2 , and n0 0.2,1.0 is the second curve’s hardening index expressed by σ0.2 and σ1.0 . Compared with the experimental data, the Gardner and Nethercot model has high accuracy when the compressive strain is less than 2% [11]. Olsson [29] conducted many experiments on uniaxial tensile and biaxial proportional loading and studied the plastic yield criterion and the hardening model of stainless steel. The stress-strain curves under uniaxial tension had two-stage strain hardening and the second stage strain hardening could be represented by a straight line [29]. Quach et al. [11] adopted Olsson’s conclusion and modified the Gardner and Nethercot model by adding a straight-line stage after the 2% proof stress σ0.2 . The Quach model is in good agreement with the stress-strain curve of the compressive behavior (Equation (4)):  n + 0.002 σσ0.2   σ −σ0.2 ε= 1 + 0.008 + σ − σ ( ) 0.2 1.0  E E0 − 0.2    σ− a     

σ E0

b∓σ

σ ≤ σ0.2   n0 0.2,1.0 1 · σσ−−σσ0.20.2 + ε 0.2 E0.2 1.0

σ0.2 < σ ≤ σ2.0

(4)

σ > σ2.0

where σ2.0 is the proof stress of 2.0% strain, n is the first curve’s hardening index, ε 0.2 is the total strain at stress σ0.2 , n0 0.2,1.0 is the second curve’s hardening index expressed by σ0.2 and σ1.0 , and a and b are

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where

 2.0

is the proof stress of 2.0% strain,

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n is the first curve’s hardening index,  0.2 is the

total strain at stress  0.2 , n ,1.0 is the second curve's hardening index expressed by  0.2 and material constants. The model0.2 was verified by comparing the results with the compressive test results  1.0 , and a and b are material constants. The model was verified by comparing the results with the under ≤2.4% strain [11]. compressive test results under ≤2.4% strain [11]. 3.3.2. Reliability of Constitutive Models 3.3.2.We Reliability of Constitutive fitted the Gardner andModels Nethercot model and the Quach model with the results of the 2 of the regression compressive tests. The fitting parameters aremodel shownand in Table 4. The coefficients We fitted the Gardner and Nethercot the Quach model withR the results of the equations were 0.988 and 0.997, respectively, suggesting that the two models perfectly correlate with 2 compressive tests. The fitting parameters are shown in Table 4. The coefficients R of the regression the compressive test results. equations were 0.988 and 0.997, respectively, suggesting that the two models perfectly correlate with the compressive test results. Table 4. Compressive test fitting parameters using two models. The comparisons between model predictions and compressive experimental data are shown in Figure 9. The compressive stress-strain curve is the two-stage strain-hardening curve, wherein the Model E0 /GPa σ 0.2 /MPa n n0 0.2,1.0 a b second strain-hardening stage is represented by a straight line. This is the same as Olsson’s conclusion G&N 8.5 1.5 obtained from the experimental 202 tensile data. 268 The stress-strain curves predicted by these models Quach 8.5 1.7 296 2390 coincide with the compressive experimental data. There is a slight difference between the predictions of the two models. The Quach model can accurately describe the second strain-hardening behavior, between predictions and compressive experimental data are shown in whileThe thecomparisons prediction curve of themodel Gardner and Nethercot model deviates from the compressive stressFigure 9. The Therefore, compressive curve is the two-stage strain-hardening curve, wherein strain curve. thestress-strain Quach model is in a better agreement with the test results than the the second strain-hardening stage is represented by a straight line. This is the same as Olsson’s conclusion Gardner and Nethercot model under high compressive strain. obtained from the experimental tensile data. The stress-strain curves predicted by these models coincide with the compressive experimental There is a slight difference between the predictions Table 4. Compressive testdata. fitting parameters using two models. of the two models. The Quach model can accurately describe the second strain-hardening behavior,  model adeviates n n0.2,1.0 Modelof the E0/GPa σ0.2/MPa b from the compressive while the prediction curve Gardner and Nethercot stress-strain curve. Therefore, a better1.5 agreement - with - the test results than the G & N the Quach model is in 8.5 202high compressive 268 Gardner and NethercotQuach model under strain. 8.5 1.7 296 2390

True stress true(MPa)

600 500 400 300 200

Test result Gardner & Nethercot model Quach model

100 0

0

0.02

0.04

0.06

0.08

0.10

0.12

True strain true(-) Figure 9. Comparison between the test and two fitting stress-strain curves. Figure 9. Comparison between the test and two fitting stress-strain curves.

4. Conclusions 4. Conclusions Compressive mechanical properties and constitutive model of the austenitic stainless steel Compressive mechanical properties and constitutive model of the austenitic stainless steel S30403 S30403 were studied. A new type of compressive specimen not aided by a bracing jig or a lubricating were studied. A new type of compressive specimen not aided by a bracing jig or a lubricating paste paste was then designed. A series of tensile and compressive tests were performed on austenitic was then designed. A series of tensile and compressive tests were performed on austenitic stainless stainless steel S30403 to obtain the stress-strain curves. The material anisotropy and strain-hardening steel S30403 to obtain the stress-strain curves. The material anisotropy and strain-hardening behavior behavior of austenitic stainless steel S30403 were also analyzed. In addition, the existing constitutive of austenitic stainless steel S30403 were also analyzed. In addition, the existing constitutive models models of austenitic stainless steel were discussed by comparing them with the test results. The of austenitic stainless steel were discussed by comparing them with the test results. The following following conclusions can be drawn from the results: conclusions can be drawn from the results:

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(1)

(2)

(3)

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When the length-to-diameter ratio of the specimen is 1:1, the specimens do not buckle under high compressive strain of 10%. The small-gauge-length specimen with length-to-diameter ratio 1:1 is suitable for compression experiment in the high strain range. There is an obvious material anisotropy of the stress-strain response in the tension and compression of austenitic stainless steel under high strain conditions up to 10%. The compressive nominal yield strength is higher than the nominal tensile yield strength. The compressive initial Young’s modulus is close to the tensile initial Young’s modulus. The compressive stress-strain curve is a two-stage strain-hardening curve and the second strain-hardening stage is represented by a straight line. The Quach model agrees with the test results and can accurately describe the strain-hardening behavior of austenitic stainless steel under high compressive strains up to 10%.

Author Contributions: Y.P. contributed to the overall process of the theoretical analysis, experiments, data analysis, and preparation of this manuscript. J.C. took part in the theoretical analysis, experiments, data analysis, and preparation of this manuscript. J.D. supervised the project and the student with contributions to guide the research program. Acknowledgments: The support of the National Natural Science Foundation of China is greatly acknowledged (Grand No. 51408307). Conflicts of Interest: The authors declare no conflict of interest.

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