Appendix

Here are the equations and functions used in the AMM.

Data Preparation:
  • Young's Module
  • Smooth
  • Add Mean

Curve Fitting

Table 1. Curve Fitting Equations
Voce
σ = K 0 + Q ( 1 e B p l ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaJecqaHdp WCcqGH9aqpcaWGlbGaaGimaiabgUcaRiaadgfacaGGOaGaaGymaiab gkHiTiaadwgadaahaaWcbeqaaiabgkHiTiaadkeacqGHiiIZaaGcda ahaaWcbeqaamaaBaaameaacaWGWbGaamiBaaqabaaaaOGaaiykaaaa @45EB@
Swift
σ=Α ε pl + ε 0 n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaeuyKde0aaeWaaeaacqaH1oqzdaWgaaWcbaGaamiCaiaadYga aeqaaOGaey4kaSIaeqyTdu2aaSbaaSqaaiaaicdaaeqaaaGccaGLOa GaayzkaaWaaWbaaSqabeaacaWGUbaaaaaa@4405@
Sherby
σ=Qs(QsQ0) e m( ε n ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaamyuaiaadohacqGHsislcaGGOaGaamyuaiaadohacqGHsisl caWGrbGaaGimaiaacMcacaWGLbWaaWbaaSqabeaacqGHsislcaWGTb Gaaiikaiabew7aLnaaCaaameqabaGaamOBaaaaliaacMcaaaaaaa@4841@
Ghosh
σ = K ( ε o ε p l ) n p MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jaam4saiaacIcacqaH1oqzdaWgaaWcbaGaam4BaaqabaGccqGH sislcqaH1oqzdaWgaaWcbaGaamiCaiaadYgaaeqaaOGaaiykamaaCa aaleqabaGaamOBaiabgkHiTiaadchaaaaaaa@456C@
EIMagd
σ = Q 0 + A ε p l + B ( 1 e ) c ε p l MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaamyuaiaaicdacqGHRaWkcaWGbbGaey4fIOIaeqyTdu2aaSba aSqaaiaadchacaWGSbaabeaakiabgUcaRiaadkeacaGGOaGaaGymai abgkHiTiaadwgacaGGPaWaaWbaaSqabeaacqGHsislcaWGJbGaey4f IOIaeqyTdu2aaSbaaWqaaiaadchacaWGSbaabeaaaaaaaa@4CE9@
Johnson
σ=A+B ε pl n MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaamyqaiabgUcaRiaadkeacqGHxiIkcqaH1oqzdaqhaaWcbaGa amiCaiaadYgaaeaacaWGUbaaaaaa@40CB@
Gsell
σ=A+B(1 e (c ε pl ) )(1+d ε pl +f ε pl 2 ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaamyqaiabgUcaRiaadkeacaGGOaGaaGymaiabgkHiTiaadwga daahaaWcbeqaaiaacIcacqGHsislcaWGJbGaey4fIOIaeqyTdu2aaS baaWqaaiaadchacaWGSbaabeaaliaacMcaaaGccaGGPaGaey4fIOIa aiikaiaaigdacqGHRaWkcaWGKbGaey4fIOIaeqyTdu2aaSbaaSqaai aadchacaWGSbaabeaakiabgUcaRiaadAgacqGHxiIkcqaH1oqzdaWg aaWcbaGaamiCaiaadYgaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaai ykaaaa@5825@
Polymer
σ = A + B * ε p l c + D * ε p l E * e - F * ε p l  
Spline  

Strain Rate

Table 2. Strain Rate Equations
Johnson & Cook
σ = ( A + B ε p l n ) ( 1 + c 1 n ε ˙ ε ˙ 0 ) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaaiikaiaadgeacqGHRaWkcaWGcbGaey4fIOIaeqyTdu2aa0ba aSqaaiaadchacaWGSbaabaGaamOBaaaakiaacMcacqGHxiIkcaGGOa GaaGymaiabgUcaRiaadogacaaIXaGaamOBamaalaaabaGafqyTduMb aiaaaeaacuaH1oqzgaGaamaaBaaaleaacaaIWaaabeaaaaGccaGGPa aaaa@4D09@
Cowper Symonds
σ=(A+B ε pl n ) 1+ ε ˙ C 1 p MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaaiikaiaadgeacqGHRaWkcaWGcbGaey4fIOIaeqyTdu2aa0ba aSqaaiaadchacaWGSbaabaGaamOBaaaakiaacMcacqGHxiIkdaqada qaaiaaigdacqGHRaWkdaWcaaqaaiqbew7aLzaacaaabaGaam4qaaaa aiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaadc haaaaaaaaa@4AB8@
FOSTA
σ = A + C ε p l m n m MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaamyqaiabgUcaRiaadoeacqGHxiIkcqaH1oqzdaWgaaWcbaGa amiCaiaadYgaaeqaaOWaaWbaaSqabeaacaWGTbGaeyOeI0IaamOBaa aakmaaCaaaleqabaWaaWbaaWqabeaacaWGTbaaaaaaaaa@4438@

Failure

Table 3. Failure Equations
Johnson&Cook
εf=D1+D2exp(D3σ*) MathType@MTEF@5@5@+= feaahGart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaam Ozaiabg2da9iaadseacaaIXaGaey4kaSIaamiraiaaikdacqGHxiIk ciGGLbGaaiiEaiaacchacaGGOaGaamiraiaaiodacqGHxiIkcqaHdp WCcaGGQaGaaiykaaaa@4783@
BiQuad