# Strain Range Dependent Cyclic Hardening of 08Ch18N10T Stainless Steel—Experiments and Simulations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiments

#### 2.1. Experimental Setup

#### 2.2. Experimental Program

## 3. Constitutive Model with Strain Range Dependency

#### 3.1. Cyclic Plasticity and Memory Surface

#### 3.2. Isotropic Hardening

#### 3.3. Kinematic Hardening

#### 3.4. Modification for Torsional Loading

## 4. Identification of Material Parameters

## 5. FE Simulations

## 6. Experimental and Simulation Results

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DIC | digital image correlation |

IDF | identification specimen series |

FE | finite element |

FEM | finite element method |

UG | uniform-gage |

## Appendix A. Boundary Conditions of Simulations

Specimen Name | Geometry Type | $\Delta {\mathit{L}}_{\mathit{ext}}$$\left[\mathbf{mm}\right]$ | ${\mathit{N}}_{\mathit{d}}$ |
---|---|---|---|

IDF-1 | UG | 0.030 | 37509 |

IDF-2 | UG | 0.050 | 4285 |

IDF-3 | UG | 0.075 | 916 |

IDF-4 | UG | 0.100 | 580 |

IDF-5 | UG | 0.125 | 254 |

IDF-6 | E9 | 0.132 | 159 |

IDF-7 | E9 | 0.154 | 381 |

IDF-8 | E9 | 0.176 | 370 |

IDF-9 | E9 | 0.198 | 161 |

IDF-10 | E9 | 0.245 | 156 |

IDF-11 | E9 | 0.264 | 124 |

IDF-12 | E9 | 0.353 | 93 |

Specimen Name | Geometry Type | $\Delta {\mathit{L}}_{\mathit{ext}}$$\left[\mathbf{mm}\right]$ | ${\mathit{N}}_{\mathit{d}}$ |
---|---|---|---|

E9-1 | E9 | 0.0447 | 13382 |

E9-2 | E9 | 0.0446 | 15104 |

E9-3 | E9 | 0.0662 | 4053 |

E9-4 | E9 | 0.0662 | 3887 |

E9-5 | E9 | 0.0881 | 1529 |

E9-6 | E9 | 0.0880 | 1853 |

E9-7 | E9 | 0.1100 | 1158 |

E9-8 | E9 | 0.1100 | 631 |

E9-9 | E9 | 0.1320 | 748 |

E9-10 | E9 | 0.1540 | 546 |

E9-11 | E9 | 0.1770 | 406 |

E9-12 | E9 | 0.1980 | 332 |

E9-13 | E9 | 0.2200 | 253 |

E9-14 | E9 | 0.2420 | 181 |

E9-15 | E9 | 0.2420 | 195 |

E9-16 | E9 | 0.2640 | 220 |

E9-17 | E9 | 0.3520 | 128 |

Specimen Name | Geometry Type | $\Delta {\mathit{\varphi}}_{\mathit{ext}}$${[}^{\circ}]$ | ${\mathit{N}}_{\mathit{d}}$ |
---|---|---|---|

NT-1 | NT | 0.8703 | 5006 |

NT-2 | NT | 0.8694 | 6894 |

NT-3 | NT | 1.1423 | 2222 |

NT-4 | NT | 1.1414 | 2289 |

NT-5 | NT | 1.4031 | 2045 |

NT-6 | NT | 1.3772 | 1532 |

NT-7 | NT | 1.6554 | 1170 |

NT-8 | NT | 2.1492 | 925 |

Specimen Name | Geometry Type | $\Delta {\mathit{L}}_{\mathit{ext}}$$\left[\mathbf{mm}\right]$ | ${\mathit{N}}_{\mathit{d}}$ |
---|---|---|---|

R1.2-1 | R1.2 | 0.0245 | 1429 |

R1.2-2 | R1.2 | 0.0246 | 946 |

R1.2-3 | R1.2 | 0.0326 | 715 |

R1.2-4 | R1.2 | 0.0406 | 523 |

R1.2-5 | R1.2 | 0.0407 | 490 |

R1.2-6 | R1.2 | 0.0489 | 290 |

R1.2-7 | R1.2 | 0.0485 | 356 |

R1.2-8 | R1.2 | 0.0560 | 241 |

R1.2-9 | R1.2 | 0.0563 | 256 |

R1.2-10 | R1.2 | 0.0639 | 134 |

R1.2-11 | R1.2 | 0.0642 | 202 |

R1.2-12 | R1.2 | 0.0721 | 171 |

R1.2-13 | R1.2 | 0.0718 | 164 |

R1.2-14 | R1.2 | 0.0794 | 112 |

R1.2-15 | R1.2 | 0.0868 | 145 |

R1.2-16 | R1.2 | 0.0869 | 114 |

R1.2-17 | R1.2 | 0.0945 | 96 |

R1.2-18 | R1.2 | 0.0944 | 105 |

Specimen Name | Geometry Type | $\Delta {\mathit{L}}_{\mathit{ext}}$$\left[\mathbf{mm}\right]$ | ${\mathit{N}}_{\mathit{d}}$ |
---|---|---|---|

R2.5-1 | R2.5 | 0.0228 | 5875 |

R2.5-2 | R2.5 | 0.0341 | 1245 |

R2.5-3 | R2.5 | 0.0340 | 1041 |

R2.5-4 | R2.5 | 0.0454 | 607 |

R2.5-5 | R2.5 | 0.0454 | 761 |

R2.5-6 | R2.5 | 0.0568 | 378 |

R2.5-7 | R2.5 | 0.0567 | 429 |

R2.5-8 | R2.5 | 0.0718 | 242 |

R2.5-9 | R2.5 | 0.0679 | 346 |

R2.5-10 | R2.5 | 0.0794 | 265 |

R2.5-11 | R2.5 | 0.0791 | 212 |

R2.5-12 | R2.5 | 0.0904 | 210 |

R2.5-13 | R2.5 | 0.0903 | 221 |

R2.5-14 | R2.5 | 0.1015 | 205 |

R2.5-15 | R2.5 | 0.1015 | 163 |

R2.5-16 | R2.5 | 0.1126 | 189 |

R2.5-17 | R2.5 | 0.1126 | 156 |

R2.5-18 | R2.5 | 0.1237 | 132 |

R2.5-19 | R2.5 | 0.1237 | 129 |

R2.5-20 | R2.5 | 0.1419 | 106 |

R2.5-21 | R2.5 | 0.1346 | 114 |

Specimen Name | Geometry Type | $\Delta {\mathit{L}}_{\mathit{ext}}$$\left[\mathbf{mm}\right]$ | ${\mathit{N}}_{\mathit{d}}$ |
---|---|---|---|

R5-1 | R5 | 0.0308 | 4427 |

R5-2 | R5 | 0.0461 | 1700 |

R5-3 | R5 | 0.0457 | 1072 |

R5-4 | R5 | 0.0603 | 733 |

R5-5 | R5 | 0.0589 | 953 |

R5-6 | R5 | 0.0727 | 623 |

R5-7 | R5 | 0.0747 | 527 |

R5-8 | R5 | 0.0893 | 342 |

R5-9 | R5 | 0.0869 | 543 |

R5-10 | R5 | 0.1050 | 297 |

R5-12 | R5 | 0.1010 | 374 |

R5-13 | R5 | 0.1154 | 264 |

R5-14 | R5 | 0.1156 | 290 |

R5-15 | R5 | 0.1146 | 228 |

R5-16 | R5 | 0.1287 | 152 |

R5-17 | R5 | 0.1276 | 272 |

R5-18 | R5 | 0.1418 | 179 |

R5-19 | R5 | 0.1467 | 155 |

R5-20 | R5 | 0.1403 | 177 |

R5-21 | R5 | 0.1540 | 163 |

R5-22 | R5 | 0.1531 | 174 |

R5-23 | R5 | 0.1663 | 144 |

R5-24 | R5 | 0.1685 | 189 |

R5-25 | R5 | 0.1652 | 163 |

## Appendix B. Abaqus USDFLD Subroutine

#### Appendix B.1. Full Fortran Code of Abaqus USDFLD Subroutine

C Material model by Miro Fumfera C C version 2019-11-10 C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C USDFLD Subroutine for 08Ch18N10T Austenitic Stainless Steel C Original modely by Radim Halama C modified by Miro Fumfera for 08Ch18N10T SUBROUTINE USDFLD(FIELD,STATEV,PNEWDT,DIRECT,T,CELENT, 1 TIME,DTIME,CMNAME,ORNAME,NFIELD,NSTATV,NOEL,NPT,LAYER, 2 KSPT,KSTEP,KINC,NDI,NSHR,COORD,JMAC,JMATYP,MATLAYO,LACCFLA) INCLUDE 'ABA_PARAM.INC' CHARACTER*80 CMNAME,ORNAME CHARACTER*3 FLGRAY(15) DIMENSION FIELD(NFIELD),STATEV(NSTATV),DIRECT(3,3),T(3,3),TIME(2) DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),COORD(*) parameter ZERO=0D0,ONE=1D0,TWO=2D0,THREE=3D0,TOLER=1D-12, + NTENS=6 !NTENS=4 for Axisymetric, NTENS=6 for 3D real*8 RMused,RM,RMmax,RMmin,oRM,dRM, RMRused,RMR,oRMR,dRMR, + heavisideG,DDP,G,DirVec(NTENS),DirVecR(NTENS) real*8 ALPHAv(NTENS),dALPHA1v(NTENS),ALPHA1v(NTENS), + dALPHA2v(NTENS),ALPHA2v(NTENS),dALPHA3v(NTENS),ALPHA3v(NTENS), + dALPHAv(NTENS),oALPHAv(NTENS),magALPHAv real*8 ALPHAr(NTENS),dALPHA1r(NTENS),ALPHA1r(NTENS), + dALPHA2r(NTENS),ALPHA2r(NTENS),dALPHA3r(NTENS),ALPHA3r(NTENS), + dALPHAr(NTENS),oALPHAr(NTENS),magALPHAr real*8 EPLAS(NTENS),oEPLAS(NTENS),dEPLAS(NTENS),EQPLAS,oEQPLAS, + dEQPLAS,FLOW(NTENS) real*8 R,oR,dR,AR,BR,CR,ER real*8 PhiInfty,dPHIcyc,PHIcyc,oPHIcyc,PHI0,PHI real*8 AInfty,BInfty,CInfty,DInfty,EInfty real*8 AOmega,BOmega,COmega real*8 KShear real*8 C1,GAMMA1,C2,GAMMA2,C3,GAMMA3 integer K1,iEPLAS,iALPHA1v,iALPHA2v,iALPHA3v,iEQPLAS,iRM,iPHI, + iPHIcyc,iALPHAv,iR,iFIELD1,iFIELD2,iALPHA1r,iALPHA2r,iALPHA3r, + iRMR,iALPHAr parameter(iEPLAS=7,iALPHA1v=31,iALPHA2v=37,iALPHA3v=43,iEQPLAS=49, + iR=50,iRM=51,iPHI=52,iPHIcyc=53,iPhiInfty=54,iRMR=61,iALPHA1r=71, + iALPHA2r=77,iALPHA3r=83,iRMRused=95,iRMused=96,iALPHAv=97, + iALPHAr=94,iFIELD1=98,iFIELD2=99) C Material parameters C1 = 6.339971e+04 GAMMA1 = 1.485569e+02 C2 = 9.999778e+03 GAMMA2 = 9.113512e+02 C3 = 2000 GAMMA3 = 0 SYIELD = 150 PHI0 = 2.317802e+00 AInfty = -1.312737e-09 BInfty = 1.798138e-06 CInfty = -8.670490e-04 DInfty = 1.667770e-01 EInfty = -1.060028e+01 RMmin = 1.305410e+02 RMmax = 5.065918e+02 BR = 3.011316e-01 CR = 1.486489e-01 ER = 1.181843e-02 AOmega = 0 BOmega = 2.002387e-13 COmega = -4.859126e+00 KShear = 1.50 C get PE components call GETVRM('PE',ARRAY,JARRAY,FLGRAY,JRCD,JMAC,JMATYP, + MATLAYO,LACCFLA) C EQPLAS EQPLAS = ARRAY(7) oEQPLAS = STATEV(iEQPLAS) dEQPLAS = EQPLAS - oEQPLAS C get PE do K1=1,NTENS oEPLAS(K1) = STATEV(iEPLAS-1+K1) EPLAS(K1) = ARRAY(K1) dEPLAS(K1) = EPLAS(K1) - oEPLAS(K1) enddo C get ALPHAv do K1=1,NTENS ALPHA1v(K1) = STATEV(iALPHA1v-1+K1) ALPHA2v(K1) = STATEV(iALPHA2v-1+K1) ALPHA3v(K1) = STATEV(iALPHA3v-1+K1) oALPHAv(K1) = STATEV(iALPHAv-1+K1) ALPHA1r(K1) = STATEV(iALPHA1r-1+K1) ALPHA2r(K1) = STATEV(iALPHA2r-1+K1) ALPHA3r(K1) = STATEV(iALPHA3r-1+K1) oALPHAr(K1) = STATEV(iALPHAr-1+K1) enddo C get FLOW vector if(dEQPLAS.gt.ZERO) then do K1=1,NDI FLOW(K1) = dEPLAS(K1)/dEQPLAS enddo do K1=NDI+1,NTENS FLOW(K1) = dEPLAS(K1)/TWO/dEQPLAS enddo else do K1=1,NTENS FLOW(K1) = ZERO enddo endif C RM RM = STATEV(iRM) oRM = RM C dALPHAv do K1=1, NDI dALPHA1v(K1) = (TWO/THREE*C1*dEQPLAS*FLOW(K1) - + GAMMA1*ALPHA1v(K1)*dEQPLAS)/(ONE+GAMMA1*dEQPLAS) dALPHA2v(K1) = (TWO/THREE*C2*dEQPLAS*FLOW(K1) - + GAMMA2*ALPHA2v(K1)*dEQPLAS)/(ONE+GAMMA2*dEQPLAS) dALPHA3v(K1) = (TWO/THREE*C3*dEQPLAS*FLOW(K1) - + GAMMA3*ALPHA3v(K1)*dEQPLAS)/(ONE+GAMMA3*dEQPLAS) ALPHAv(K1) = (ALPHA1v(K1)+dALPHA1v(K1)) + + (ALPHA2v(K1)+dALPHA2v(K1)) + (ALPHA3v(K1)+dALPHA3v(K1)) !dALPHAv(K1) = ALPHAv(K1)-oALPHAv(K1) dALPHAv(K1) = dALPHA1v(K1) + dALPHA2v(K1) + dALPHA3v(K1) enddo do K1=NDI+1, NTENS dALPHA1v(K1) = (TWO/THREE*C1*dEQPLAS*FLOW(K1) - + GAMMA1*KShear*ALPHA1v(K1)*dEQPLAS)/(ONE+GAMMA1*dEQPLAS) dALPHA2v(K1) = (TWO/THREE*C2*dEQPLAS*FLOW(K1) - + GAMMA2*KShear*ALPHA2v(K1)*dEQPLAS)/(ONE+GAMMA2*dEQPLAS) dALPHA3v(K1) = (TWO/THREE*C3*dEQPLAS*FLOW(K1) - + GAMMA3*KShear*ALPHA3v(K1)*dEQPLAS)/(ONE+GAMMA3*dEQPLAS) ALPHAv(K1) = (ALPHA1v(K1)+dALPHA1v(K1)) + + (ALPHA2v(K1)+dALPHA2v(K1)) + (ALPHA3v(K1)+dALPHA3v(K1)) !dALPHAv(K1) = ALPHAv(K1)-oALPHAv(K1) dALPHAv(K1) = dALPHA1v(K1) + dALPHA2v(K1) + dALPHA3v(K1) enddo do K1=1, NTENS ALPHA1v(K1) = ALPHA1v(K1) + dALPHA1v(K1) ALPHA2v(K1) = ALPHA2v(K1) + dALPHA2v(K1) ALPHA3v(K1) = ALPHA3v(K1) + dALPHA3v(K1) ALPHAv(K1) = ALPHA1v(K1) + ALPHA2v(K1) + ALPHA3v(K1) enddo C magALPHAv magALPHAv = ZERO do K1=1, NDI magALPHAv = magALPHAv + ALPHAv(K1)**2 enddo do K1=NDI+1, NTENS magALPHAv = magALPHAv + TWO*ALPHAv(K1)**2 enddo magALPHAv = sqrt(THREE/TWO*magALPHAv) C G function G = magALPHAv - RM if(magALPHAv.gt.ZERO) then do K1 = 1, NTENS DirVec(K1)=ALPHAv(K1)/magALPHAv enddo else do K1 = 1, NTENS DirVec(K1) = ZERO enddo endif C double dot product DDP DDP = ZERO do K1 = 1, NDI DDP = DDP+DirVec(K1)*dALPHAv(K1) enddo do K1 = NDI+1, NTENS DDP = DDP+TWO*DirVec(K1)*dALPHAv(K1) enddo C heaviside function of G if (G.gt.ZERO) then heavisideG = ONE elseif (abs(G).lt.TOLER) then heavisideG = ONE/TWO else heavisideG = ZERO endif C memory surface RM dRM = heavisideG*abs(DDP) RM = oRM + dRM if (RM.lt.RMmin) then RMused = RMmin elseif (RM.gt.RMmax) then RMused = RMmax else RMused = RM endif C RMR RMR = STATEV(iRMR) oRMR = RMR do K1=1, NDI dALPHA1r(K1) = (TWO/THREE*C1*dEQPLAS*FLOW(K1) - + GAMMA1*ALPHA1r(K1)*dEQPLAS)/(ONE+GAMMA1*dEQPLAS) dALPHA2r(K1) = (TWO/THREE*C2*dEQPLAS*FLOW(K1) - + GAMMA2*ALPHA2r(K1)*dEQPLAS)/(ONE+GAMMA2*dEQPLAS) dALPHA3r(K1) = (TWO/THREE*C3*dEQPLAS*FLOW(K1) - + GAMMA3*ALPHA3r(K1)*dEQPLAS)/(ONE+GAMMA3*dEQPLAS) ALPHAr(K1) = (ALPHA1r(K1)+dALPHA1r(K1)) + + (ALPHA2r(K1)+dALPHA2r(K1)) + (ALPHA3r(K1)+dALPHA3r(K1)) !dALPHAr(K1) = ALPHAr(K1)-oALPHAr(K1) dALPHAr(K1) = dALPHA1r(K1) + dALPHA2r(K1) + dALPHA3r(K1) enddo do K1=NDI+1, NTENS dALPHA1r(K1) = (TWO/THREE*C1*dEQPLAS*FLOW(K1) - + GAMMA1*KShear*ALPHA1r(K1)*dEQPLAS)/(ONE+GAMMA1*dEQPLAS) dALPHA2r(K1) = (TWO/THREE*C2*dEQPLAS*FLOW(K1) - + GAMMA2*KShear*ALPHA2r(K1)*dEQPLAS)/(ONE+GAMMA2*dEQPLAS) dALPHA3r(K1) = (TWO/THREE*C3*dEQPLAS*FLOW(K1) - + GAMMA3*KShear*ALPHA3r(K1)*dEQPLAS)/(ONE+GAMMA3*dEQPLAS) ALPHAr(K1) = (ALPHA1r(K1)+dALPHA1r(K1)) + + (ALPHA2r(K1)+dALPHA2r(K1)) + (ALPHA3r(K1)+dALPHA3r(K1)) !dALPHAr(K1) = ALPHAr(K1)-oALPHAr(K1) dALPHAr(K1) = dALPHA1r(K1) + dALPHA2r(K1) + dALPHA3r(K1) enddo do K1=1, NTENS ALPHA1r(K1) = ALPHA1r(K1) + dALPHA1r(K1) ALPHA2r(K1) = ALPHA2r(K1) + dALPHA2r(K1) ALPHA3r(K1) = ALPHA3r(K1) + dALPHA3r(K1) ALPHAr(K1) = ALPHA1r(K1) + ALPHA2r(K1) + ALPHA3r(K1) enddo C magALPHAr magALPHAr = ZERO do K1=1, NDI magALPHAr = magALPHAr + ALPHAr(K1)**2 enddo do K1=NDI+1, NTENS magALPHAr = magALPHAr + TWO*ALPHAr(K1)**2 enddo magALPHAr = sqrt(THREE/TWO*magALPHAr) C G function G = magALPHAr - RMR if(magALPHAr.gt.ZERO) then do K1 = 1, NTENS DirVecR(K1)=ALPHAr(K1)/magALPHAr enddo else do K1 = 1, NTENS DirVecR(K1) = ZERO enddo endif C double dot product DDP DDP = ZERO do K1 = 1, NDI DDP = DDP+DirVecR(K1)*dALPHAr(K1) enddo do K1 = NDI+1, NTENS DDP = DDP+TWO*DirVecR(K1)*dALPHAr(K1) enddo C heaviside function of G if (G.gt.ZERO) then heavisideG = ONE elseif (abs(G).lt.TOLER) then heavisideG = ONE/TWO else heavisideG = ZERO endif C memory surface RMR dRMR = heavisideG*abs(DDP) RMR = oRMR + dRMR if (RMR.lt.RMmin) then RMRused = RMmin elseif (RMR.gt.RMmax) then RMRused = RMmax else RMRused = RMR endif C R oR = STATEV(iR) AR = CR*exp(ER*RMRused) dR = AR*((EQPLAS+dEQPLAS)**BR-EQPLAS**BR) R = oR + dR; C PHIinfty PhiInfty = AInfty*RMused**4 + BInfty*RMused**3 + + CInfty*RMused**2 + DInfty*RMused + EInfty C Omega OMEGA∼= AOmega+BOmega*(RMused)**-COmega C PHIcyc oPHIcyc = STATEV(iPHIcyc) dPHIcyc = OMEGA*(PhiInfty-oPHIcyc)*DEQPLAS PHIcyc = oPHIcyc + dPHIcyc C PHI PHI = PHI0 + PHIcyc C save STATEV STATEV(iEQPLAS) = EQPLAS do K1=1,NTENS STATEV(iEPLAS-1+K1) = EPLAS(K1) STATEV(iALPHA1v-1+K1) = ALPHA1v(K1) STATEV(iALPHA2v-1+K1) = ALPHA2v(K1) STATEV(iALPHA3v-1+K1) = ALPHA3v(K1) STATEV(iALPHAv-1+K1) = ALPHAv(K1) STATEV(iALPHA1r-1+K1) = ALPHA1r(K1) STATEV(iALPHA2r-1+K1) = ALPHA2r(K1) STATEV(iALPHA3r-1+K1) = ALPHA3r(K1) STATEV(iALPHAr-1+K1) = ALPHAr(K1) STATEV(120+K1) = dALPHAv(K1) enddo STATEV(iR) = R STATEV(iRM) = RM STATEV(iRMR) = RMR STATEV(iRMused) = RMused STATEV(iRMRused) = RMRused STATEV(iPHIcyc) = PHIcyc STATEV(iPHI) = PHI STATEV(iPhiInfty) = PhiInfty STATEV(127) = SYIELD+R STATEV(128) = DDP C FIELD(1) FIELD(1) = SYIELD+R STATEV(iFIELD1) = FIELD(1) C FIELD(2) FIELD(2) = PHI STATEV(iFIELD2) = FIELD(2) RETURN END

#### Appendix B.2. Material Parameters Definition in the Abaqus Input File

*Material, name=Material-1 *Depvar 128 *Elastic 210000.0,0.3 *Plastic, dependencies=2, hardening=COMBINED, datatype=PARAMETERS, number backstresses=3 ** Material data as∼a∼function of FIELD1 and∼FIELD2 follows: SYIELD,C1,GAMMA1,C2,GAMMA2,C3,GAMMA3,FIDEL1,FIELD2 %%

** Material data as∼a∼function of FIELD1 and∼FIELD2 follows: ** ... 250.0,63399.70889,222.83539,9999.77788,1367.02686,2000.0,0.0,250.0,1.5 150.0,63399.70889,237.69108,9999.77788,1458.16199,2000.0,0.0,150.0,1.6 151.0,63399.70889,237.69108,9999.77788,1458.16199,2000.0,0.0,151.0,1.6 ** ...

## References

- Halama, R.; Sedlák, J.; Šofer, M. Phenomenological Modelling of Cyclic Plasticity. In Numerical Modelling; Miidla, P., Ed.; IntechOpen: Rijeka, Croatia, 2012; pp. 329–354. [Google Scholar]
- Halama, R.; Fumfera, J.; Gál, P.; Kumar, T.; Makropulos, A. Modeling the Strain-Range Dependent Cyclic Hardening of SS304 and 08Ch18N10T Stainless Steel with a Memory Surface. Metals
**2019**, 9, 832. [Google Scholar] [CrossRef] [Green Version] - Jiang, Y.; Zhang, J. Benchmark experiments and characteristic cyclic plastic deformation behavior. Int. J. Plast.
**2008**, 24, 1481–1515. [Google Scholar] [CrossRef] - Facheris, G.; Janssens, K.G.F.; Foletti, S. Multiaxial fatigue behavior of AISI 316L subjected to strain-controlled and ratcheting paths. Int. J. Fatigue
**2014**, 68, 195–208. [Google Scholar] [CrossRef] - Kim, C. Nondestructive Evaluation of Strain-Induced Phase Transformation and Damage Accumulation in Austenitic Stainless Steel Subjected to Cyclic Loading. Metals
**2018**, 8, 14. [Google Scholar] [CrossRef] [Green Version] - Borodii, M.; Shukayev, S. Additional cyclic strain hardening and its relation to material structure, mechanical characteristics, and lifetime. Int. J. Fatigue
**2007**, 29, 1184–1191. [Google Scholar] [CrossRef] - Jin, D.; Tian, D.J.; Li, J.H.; Sakane, M. Low-cycle fatigue of 316L stainless steel under proportional and nonproportional loadings. Fatigue Fract. Eng. Mater. Struct.
**2016**, 39, 850–858. [Google Scholar] [CrossRef] - Xing, R.; Dunji, Y.; Shouwen, S.; Xu, C. Cyclic deformation of 316L stainless steel and constitutive modeling under non-proportional variable loading path. Int. J. Plast.
**2019**, 120, 127–146. [Google Scholar] [CrossRef] - Srnec Novak, J.; Benasciutti, D.; De Bona, F.; Stanojevic, A.S.; De Luca, A.; Raffaglio, Y. Estimation of Material Parameters in Nonlinear Hardening Plasticity Models and Strain Life Curves for CuAg Alloy. IOP Conf. Ser. Mater. Sci. Eng.
**2016**, 119, 012020. [Google Scholar] [CrossRef] - Benallal, A.; Marquis, D. Constitutive Equations for Nonproportional Cyclic Elasto-Viscoplasticity. J. Eng. Mater. Technol.
**1987**, 109, 326–336. [Google Scholar] [CrossRef] - Tanaka, E. A nonproportionality parameter and a cyclic viscoplastic constitutive model taking into account amplitude dependences and memory effects of isotropic hardening. Eur. J. Mech. A/Solids
**1994**, 13, 155–173. [Google Scholar] - Jiang, Y.; Sehitoglu, H. Modeling of cyclic ratchetting plasticity, part I: Development of constitutive relations. J. Appl. Mech.
**1996**, 63, 720–725. [Google Scholar] [CrossRef] - ASTM Standard E606-92. Standard Practise for Strain-Controlled Fatigue Testing; ASTM International: West Conshohocken, PA, USA, 1998. [Google Scholar]
- Fumfera, J.; Halama, R.; Kuželka, J.; Španiel, M. Strain-Range Dependent Cyclic Plasticity Material Model Calibration for the 08Ch18N10T Steel. In Proceedings of the 33rd Conference with International Participation on Computational Mechanics, Špičák, Czech Republic, 6–8 November 2017. [Google Scholar]
- Abdel-Karim, M.; Ohno, N. Kinematic hardening model suitable for ratchetting with steady-state. Int. J. Plast.
**2000**, 16, 225–240. [Google Scholar] [CrossRef] - Peč, M.; Šebek, F.; Zapletal, J.; Petruška, J.; Hassan, T. Automated calibration of advanced cyclic plasticity model parameters with sensitivity analysis for aluminium alloy 2024-T351. Adv. Mech. Eng.
**2019**, 11, 1–14. [Google Scholar] [CrossRef] [Green Version] - Halama, R.; Pagáč, M.; Paška, Z.; Pavlíček, P.; Chen, X. Ratcheting Behaviour of 3D Printed and Conventionally Produced SS316L Material. In Proceedings of the ASME 2019 Pressure Vessels & Piping Conference PVP2019, San Antonio, TX, USA, 14–19 July 2019. Paper Number PVP2019-93384. [Google Scholar]

**Figure 1.**Experiment: (

**a**) Experimental Setup, (

**b**) digital image correlation (DIC) Snapshot of Specimen IDF-6.

**Figure 5.**Original model under torsional loading: (

**a**) specimen NT-1 (low loading level), (

**b**) specimen NT-6 (high loading level).

$\mathit{E}\phantom{\rule{0.17em}{0ex}}\left[\mathbf{MPa}\right]$ | $\mathit{\nu}$ | ${\mathit{\sigma}}_{\mathit{y}}\phantom{\rule{0.17em}{0ex}}\left[\mathbf{MPa}\right]$ | ${\mathit{C}}_{1}$ | ${\mathit{\gamma}}_{1}$ | ${\mathit{C}}_{2}\phantom{\rule{0.17em}{0ex}}\left[\mathbf{MPa}\right]$ |

210,000 | 0.3 | 150 | 63,400 | 148.6 | 10,000 |

${\mathit{\gamma}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}\phantom{\rule{0.17em}{0ex}}\mathbf{\left[}\mathbf{MPa}\mathbf{\right]}$ | ${\mathit{\gamma}}_{\mathbf{3}}$ | ${\mathit{A}}_{\mathbf{\infty}}$ | ${\mathit{B}}_{\mathbf{\infty}}$ | ${\mathit{C}}_{\mathbf{\infty}}$ |

911.4 | 2000 | 0 | $-1.3127\times {10}^{-9}$ | $1.7981\times {10}^{-6}$ | $-8.6705\times {10}^{-4}$ |

${\mathit{D}}_{\mathbf{\infty}}$ | ${\mathit{F}}_{\mathbf{\infty}}$ | ${\mathit{A}}_{\mathit{R}}\phantom{\rule{0.17em}{0ex}}\mathbf{\left[}{\mathbf{MPa}}^{\mathbf{-}\mathbf{1}}\mathbf{\right]}$ | ${\mathit{B}}_{\mathit{R}}$ | ${\mathit{C}}_{\mathit{R}}\phantom{\rule{0.17em}{0ex}}\mathbf{\left[}\mathbf{MPa}\mathbf{\right]}$ | ${\mathit{R}}_{\mathbf{M}}^{\mathit{min}}\phantom{\rule{0.17em}{0ex}}\mathbf{\left[}\mathbf{MPa}\mathbf{\right]}$ |

$1.6678\times {10}^{-1}$ | $-10.600$ | $3.0113\times {10}^{-1}$ | $1.4865\times {10}^{-1}$ | $1.1818\times {10}^{-2}$ | 130.54 |

${\mathit{A}}_{\mathit{\omega}}$ | ${\mathit{B}}_{\mathit{\omega}}$ | ${\mathit{C}}_{\mathit{\omega}}$ | ${\mathit{R}}_{\mathit{M}}^{\mathit{max}}\phantom{\rule{0.17em}{0ex}}\mathbf{\left[}\mathbf{MPa}\mathbf{\right]}$ | ${\mathit{\varphi}}_{\mathbf{0}}$ | ${\mathit{K}}_{\mathit{shear}}$ |

0 | $2.0024\times {10}^{-13}$ | −4.8591 | 506.59 | 2.3178 | 1.5 |

**Table 2.**Mean error of all E9 specimens tested—experiment vs. simulations [2].

Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] | Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] |
---|---|---|---|---|---|

E9-1 | 2.9226 | 1.8207 | E9-10 | 7.8144 | 8.8757 |

E9-2 | 2.3311 | 1.2756 | E9-11 | 2.5028 | 3.9003 |

E9-3 | 2.4027 | 1.1938 | E9-12 | 4.3523 | 6.5915 |

E9-4 | 1.6977 | 0.7773 | E9-13 | 4.0929 | 3.4343 |

E9-5 | 8.0687 | 7.0447 | E9-14 | 2.1610 | 3.8515 |

E9-6 | 8.8658 | 7.4521 | E9-15 | 2.9195 | 2.9485 |

E9-7 | 11.7310 | 10.4229 | E9-16 | 1.8601 | 2.7524 |

E9-8 | 3.8241 | 3.9171 | E9-17 | 4.9766 | 2.7579 |

E9-9 | 9.8245 | 9.5508 |

Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] | Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] |
---|---|---|---|---|---|

NT-1 | 1.9100 | 4.0682 | NT-5 | 14.2137 | 1.3947 |

NT-2 | 0.8367 | 5.9823 | NT-6 | 15.5549 | 2.2815 |

NT-3 | 11.2048 | 1.3797 | NT-7 | 13.1168 | 1.5014 |

NT-4 | 11.1021 | 1.0934 | NT-8 | 8.8054 | 4.7887 |

Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] | Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] |
---|---|---|---|---|---|

R1.2-1 | 2.8075 | 2.4172 | R1.2-10 | 1.6518 | 1.7538 |

R1.2-2 | 3.7011 | 3.1679 | R1.2-11 | 2.0827 | 2.2332 |

R1.2-3 | 2.2438 | 2.2027 | R1.2-12 | 3.9411 | 3.2028 |

R1.2-4 | 2.8530 | 2.7056 | R1.2-13 | 2.5308 | 3.1540 |

R1.2-5 | 2.8984 | 2.7105 | R1.2-14 | 1.4521 | 1.8444 |

R1.2-6 | 4.7877 | 4.4405 | R1.2-15 | 3.6781 | 2.6435 |

R1.2-7 | 1.4888 | 1.4897 | R1.2-16 | 1.5820 | 1.9106 |

R1.2-8 | 7.1382 | 6.7943 | R1.2-17 | 1.6089 | 2.5930 |

R1.2-9 | 2.4171 | 2.2355 | R1.2-18 | 1.2789 | 2.2219 |

Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] | Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] |
---|---|---|---|---|---|

R2.5-1 | 7.3714 | 7.1025 | R2.5-12 | 2.1944 | 1.6489 |

R2.5-2 | 8.1586 | 7.6327 | R2.5-13 | 1.2466 | 1.0057 |

R2.5-3 | 9.1468 | 8.6587 | R2.5-14 | 8.7778 | 9.1473 |

R2.5-4 | 6.8139 | 6.8130 | R2.5-15 | 2.6624 | 3.0678 |

R2.5-5 | 6.6714 | 6.6118 | R2.5-16 | 1.4643 | 1.3563 |

R2.5-6 | 9.9838 | 9.1708 | R2.5-17 | 0.9873 | 1.5697 |

R2.5-7 | 4.3249 | 3.4860 | R2.5-18 | 1.4020 | 1.4515 |

R2.5-8 | 3.8551 | 3.8250 | R2.5-19 | 1.6099 | 2.6423 |

R2.5-9 | 1.0034 | 0.9027 | R2.5-20 | 0.9634 | 2.4069 |

R2.5-10 | 4.7921 | 4.9816 | R2.5-21 | 4.1944 | 3.2605 |

R2.5-11 | 1.9673 | 2.1464 |

Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] | Specimen Name | Orig. Model Mean Err. [%] | Mod. Model Mean Err. [%] |
---|---|---|---|---|---|

R5-1 | 2.1303 | 1.4186 | R5-13 | 6.7479 | 6.8700 |

R5-2 | 2.0673 | 1.8112 | R5-14 | 5.1055 | 5.4414 |

R5-3 | 0.7021 | 0.8284 | R5-15 | 1.3043 | 1.4251 |

R5-4 | 0.9757 | 0.9284 | R5-16 | 1.1829 | 1.3661 |

R5-5 | 1.4847 | 1.4209 | R5-17 | 3.6903 | 3.6048 |

R5-6 | 1.7435 | 1.6993 | R5-18 | 3.1399 | 2.9518 |

R5-7 | 2.9066 | 2.7548 | R5-19 | 6.1649 | 6.1226 |

R5-8 | 5.3372 | 5.4106 | R5-20 | 2.8263 | 2.6683 |

R5-9 | 4.9004 | 4.5530 | R5-21 | 1.0485 | 1.2882 |

R5-10 | 2.3623 | 2.6227 | R5-22 | 8.2167 | 7.6119 |

R5-11 | 7.0110 | 6.8065 | R5-23 | 2.2011 | 1.6441 |

R5-12 | 2.3912 | 3.1025 | R5-24 | 3.5803 | 3.1425 |

Geometry | The Original Nodel [2] Total Error [%] | The Modified Model Total Error [%] |
---|---|---|

E9 | 4.84 | 4.61 |

NT | 9.60 | 2.85 |

R1.2 | 2.79 | 2.76 |

R2.5 | 4.27 | 4.23 |

R5 | 3.30 | 3.23 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fumfera, J.; Halama, R.; Procházka, R.; Gál, P.; Španiel, M.
Strain Range Dependent Cyclic Hardening of 08Ch18N10T Stainless Steel—Experiments and Simulations. *Materials* **2019**, *12*, 4243.
https://doi.org/10.3390/ma12244243

**AMA Style**

Fumfera J, Halama R, Procházka R, Gál P, Španiel M.
Strain Range Dependent Cyclic Hardening of 08Ch18N10T Stainless Steel—Experiments and Simulations. *Materials*. 2019; 12(24):4243.
https://doi.org/10.3390/ma12244243

**Chicago/Turabian Style**

Fumfera, Jaromír, Radim Halama, Radek Procházka, Petr Gál, and Miroslav Španiel.
2019. "Strain Range Dependent Cyclic Hardening of 08Ch18N10T Stainless Steel—Experiments and Simulations" *Materials* 12, no. 24: 4243.
https://doi.org/10.3390/ma12244243