/MAT/LAW110 (VEGTER)

Block Format Keyword Elasto-plastic constitutive law using the interpolated yield criterion of Corus-Vegter and the hardening law of Vegter, accounting for strain rate dependency and thermal effect.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW110/mat_ID/unit_ID or /MAT/VEGTER/mat_ID/unit_ID
mat_title
ρ i                
E v Ires      
I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ TAB_YLD MAT_Xscale MAT_Yscale ƒ b i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby 6aaSbaaSqaaiaadkgacaWGPbaabeaaaaa@39BC@ ρ b i 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaDaaaleaapeGaamOyaiaadMgaa8aabaWdbiaaicda aaaaaa@3AC3@
σ y l d 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaDaaaleaapeGaamyEaiaadYgacaWGKbaapaqaa8qa caaIWaaaaaaa@3BC9@ Δ σ m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiabeo8aZ9aadaWgaaWcbaWdbiaad2gaa8aabeaaaaa@3A32@ β Ω MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdaaa@3738@ η
ε 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38C4@ σ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaDaaaleaapeGaaGimaaWdaeaatuuDJXwAK1uy0Hwm aeHbfv3ySLgzG0uy0Hgip5wzaGqba8qacqWFgls5aaaaaa@44CD@ Δ G 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaadEeapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3903@ ε ˙ 0 m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyBaaaa@36FB@
Tini Chard Fcut VP Ismooth TAB_TEMP  
I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 1: Read N a n g l e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadggacaWGUbGaam4zaiaadYgacaWG Lbaapaqabaaaaa@3BD6@ (Number of experimental angles, at least 1) cards
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
f u n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaaaaa@393B@ R f p s 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIXaaa aaaa@3A07@ f p s 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIYaaa aaaa@3A08@ f s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3933@
I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 2: Read N a n g l e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadggacaWGUbGaam4zaiaadYgacaWG Lbaapaqabaaaaa@3BD6@ (Number of experimental angles, at least 1) cards
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
f u n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaaaaa@393B@ R f p s 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIXaaa aaaa@3A07@ α p s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySde2damaaBaaaleaapeGaamiCaiaadohaa8aabeaaaaa@39EF@ f s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3933@
I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 3: Read the parameters ( N a n g l e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadggacaWGUbGaam4zaiaadYgacaWG Lbaapaqabaaaaa@3BD6@ is locked to 3 in this case, 0, 45, and 90)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
R m 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaqhaaWcbaWdbiaad2gaa8aabaWdbiaaicdaaaaaaa@38F7@ R m 45 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaqhaaWcbaWdbiaad2gaa8aabaWdbiaaisdacaaI1aaa aaaa@39BA@ R m 90 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaqhaaWcbaWdbiaad2gaa8aabaWdbiaaiMdacaaIWaaa aaaa@39BA@ A g 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaqhaaWcbaWdbiaadEgaa8aabaWdbiaaicdaaaaaaa@38E0@ A g 45 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaqhaaWcbaWdbiaadEgaa8aabaWdbiaaisdacaaI1aaa aaaa@39A3@
A g 90 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaqhaaWcbaWdbiaadEgaa8aabaWdbiaaiMdacaaIWaaa aaaa@39A3@ R 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaahaaWcbeqaa8qacaaIWaaaaaaa@37E6@ R 45 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaahaaWcbeqaa8qacaaI0aGaaGynaaaaaaa@38A9@ R 90 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaahaaWcbeqaa8qacaaI5aGaaGimaaaaaaa@38A9@
I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 4: Read N a n g l e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadggacaWGUbGaam4zaiaadYgacaWG Lbaapaqabaaaaa@3BD6@ cards
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
f u n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaaaaa@393B@ R w p s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadchacaWGZbaapaqabaaaaa@394C@ w s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3944@

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID Optional unit identifier.

(Integer, maximum 10 digits)

mat_title Material title.

(Character, maximum 100 characters)

ρ i Initial density.

(Real)

[ kg m 3 ]
E Young‘s modulus.

(Real)

[ Pa ]
v Poisson's ratio.

(Real)

Ires Return mapping algorithm flag.
= 1
Nice explicit method.
= 2 (Default)
Newton iterative semi-implicit method – Cutting Plane.

(Integer)

I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ Vegter formulation selection:
= 1
Classic Vegter formulation.
= 2
Standard Vegter formulation.
= 3
Vegter 2017 formulation.
= 4
Simplified Vegter-Lite formulation.

(Integer)

TAB_YLD Tabulated yield stress – plastic strain - strain rate function identifier.

(Integer)

MAT_Xscale X scale factor of the tabulated yield – plastic strain - strain rate function.

Default = 10 (Real)

MAT_Yscale Y scale factor of the tabulated yield – plastic strain - strain rate function.

Default = 1.0 (Real)

ƒ b i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby 6aaSbaaSqaaiaadkgacaWGPbaabeaaaaa@39BC@ Biaxial scale factor.

(Real > 0.0)

ρ b i 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaDaaaleaapeGaamOyaiaadMgaa8aabaWdbiaaicda aaaaaa@3AC3@ Biaxial strain rate ratio in the 0 degree direction with respect to RD.

(Real > 0.0)

σ y l d 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaDaaaleaapeGaamyEaiaadYgacaWGKbaapaqaa8qa caaIWaaaaaaa@3BC9@ Initial yield stress.

(Real)

[ Pa ]
Δ σ m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiabeo8aZ9aadaWgaaWcbaWdbiaad2gaa8aabeaaaaa@3A32@ Hardening stress increment.

(Real)

[ Pa ]
β Large strain hardening parameter.

(Real)

Ω MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdaaa@3738@ Small strain hardening parameter.

(Real)

η Hardening exponent.

(Real)

ε 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38C4@ Initial plastic strain.

(Real)

σ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaDaaaleaapeGaaGimaaWdaeaatuuDJXwAK1uy0Hwm aeHbfv3ySLgzG0uy0Hgip5wzaGqba8qacqWFgls5aaaaaa@44CD@ Limit dynamic flow stress.

(Real)

[ Pa ]
Δ G 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaadEeapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3903@ Maximum activation enthalpy.

(Real)

[ eV ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaamWaa8aabaWdbiaabwgacaqGwbaacaGLBbGaayzxaaaaaa@39DE@
ε ˙ 0 Limit strain rate for thermal activated movement.

(Real)

[Hz]
m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyBaaaa@36FB@ Strain rate behavior exponent.

(Real)

Tini Initial temperature.

(Real)

[ K ]
Chard Hardening coefficient.
= 0
Hardening is a full isotropic model.
= 1
Hardening uses the kinematic Prager-Ziegler model.
= value between 0 and 1
Hardening is interpolated between the two models.

(Real)

Fcut Cutoff frequency for strain rate filtering.

Default = 1.0 x 1020 (Real)

[Hz]
VP Strain rate choice flag.
= 1
Strain rate effect on yield stress depends on the plastic strain rate.
= 2 (Default)
Strain rate effect on yield depends on the total strain rate.
= 3
Strain rate effect on yield depends on the deviatoric strain rate.

(Integer)

Ismooth Interpolation type (in case of tabulated yield function).
= 1
Linear interpolation.
= 2
Logarithmic interpolation base 10.
= 3
Logarithmic interpolation base n.

(Integer)

TAB_TEMP Tabulated yield stress – plastic strain – temperature identifier.

(Integer)

f u n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaaaaa@393B@ Uniaxial scale factor.

(Real > 0.0)

R Lankford coefficient.

Default = 1.0 (Real > 0.0)

f p s 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIXaaa aaaa@3A07@ First component of the plane strain scale factor.

(Real > 0.0)

f p s 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIYaaa aaaa@3A08@ Second component of the plane strain scale factor ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 1).

(Real > 0.0)

α p s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySde2damaaBaaaleaapeGaamiCaiaadohaa8aabeaaaaa@39EF@ Average factor to compute the second component of the plane strain scale factor ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 2).

Default = 0.5 (Real > 0.0)

f s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3933@ Shear scale factor.

(Real > 0.0)

R m i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaqhaaWcbaWdbiaad2gaa8aabaWdbiaadMgaaaaaaa@392B@ Maximum uniaxial engineering stress for the direction at i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaaaa@36F7@ degree with respect to the rolling direction (RD) ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 3).

(Real > 0.0)

A g i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaqhaaWcbaWdbiaadEgaa8aabaWdbiaadMgaaaaaaa@3914@ Maximum uniaxial uniform elongation in % for the direction at i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaaaa@36F7@ degree with respect to the rolling direction (RD) ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 3).

(Real > 0.0)

A g i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaqhaaWcbaWdbiaadEgaa8aabaWdbiaadMgaaaaaaa@3914@ Lankford coefficient for the direction at i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaaaa@36F7@ degree with respect to the rolling direction (RD) ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 3).

Default = 1.0 (Real > 0.0)

w p s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadchacaWGZbaapaqabaaaaa@394C@ Plane strain weight factor ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 4).

(Real > 0.0)

w s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3944@ Shear weight factor ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 4).

(Real > 0.0)

Examples

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/25
Local unit system
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW110/1/25
Steel: Icrit = 1, example with 3 angles (0°, 45° and 90° to the RD)
#        Init. dens.
             7.85E-9
#                  E                  nu
            194200.0                 0.3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#    Icrit   TAB_YLD          MAT_Xscale          MAT_Yscale                 fBI               rhoBI            
         1         0                 0.0                 0.0               1.004               0.889                         
#               YLD0               DSIGM                BETA               OMEGA                   n
               107.1               179.6                0.25                8.07                 1.0
#               EPS0                SIGS                 DG0               Deps0                   m
                 0.0                20.0                 800             3.61e-3                 1.0
#               TINI              C_HARD               F_CUT        VP   Ismooth  TAB_TEMP
               293.0                 0.0             10000.0         1         1         0
#          fUN_THETA             R_THETA          fPS1_THETA          fPS2_THETA           fSH_THETA 
               1.021                0.64               1.061              0.5305               0.560
               0.987                0.48               1.037              0.5185               0.640 
               1.009                0.76               1.048              0.5240               0.560 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW110/2/25
Steel :Icrit = 2, example with 3 angles (0°, 45° and 90° to the RD)
#        Init. dens.
             7.85E-9
#                  E                  nu
            194200.0                 0.3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#    Icrit   TAB_YLD          MAT_Xscale          MAT_Yscale                 fBI               rhoBI            
         2         0                 0.0                 0.0               1.004               0.889                         
#               YLD0               DSIGM                BETA               OMEGA                   n
               107.1               179.6                0.25                8.07                 1.0
#               EPS0                SIGS                 DG0               Deps0                   m
                 0.0                20.0                 800             3.61e-3                 1.0
#               TINI              C_HARD               F_CUT        VP   Ismooth  TAB_TEMP
               293.0                 0.0             10000.0         1         1         0
#          fUN_THETA             R_THETA          fPS1_THETA          ALPS_THETA           fSH_THETA 
               1.021                0.64               1.061                 0.5               0.560
               0.987                0.48               1.037                 0.5               0.640 
               1.009                0.76               1.048                 0.5               0.560 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW110/3/25
Steel
#        Init. dens.
             7.85E-9
#                  E                  nu
            194200.0                 0.3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#    Icrit   TAB_YLD          MAT_Xscale          MAT_Yscale                 fBI               rhoBI            
         3         0                 0.0                 0.0               1.004               0.889                         
#               YLD0               DSIGM                BETA               OMEGA                   n
               107.1               179.6                0.25                8.07                 1.0
#               EPS0                SIGS                 DG0               Deps0                   m
                 0.0                20.0                 800             3.61e-3                 1.0
#               TINI              C_HARD               F_CUT        VP   Ismooth  TAB_TEMP
               293.0                 0.0             10000.0         1         1         0
#               RM_0               RM_45               RM_90                AG_0               AG_45
               408.4               408.4               408.4                20.0                20.0
#              AG_90                 R_0                R_45                R_90
                20.0                0.64                0.48                0.76    
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW110/4/25
Steel :example with 3 angles (0°, 45° and 90° to the RD)
#        Init. dens.
             7.85E-9
#                  E                  nu
            194200.0                 0.3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#    Icrit   TAB_YLD          MAT_Xscale          MAT_Yscale                 fBI               rhoBI            
         4         0                 0.0                 0.0               1.004               0.889                         
#               YLD0               DSIGM                BETA               OMEGA                   n
               107.1               179.6                0.25                8.07                 1.0
#               EPS0                SIGS                 DG0               Deps0                   m
                 0.0                20.0                 800             3.61e-3                 1.0
#               TINI              C_HARD               F_CUT        VP   Ismooth  TAB_TEMP
               293.0                 0.0             10000.0         1         1         0
#          fUN_THETA             R_THETA                W_PS                W_SH
               1.021                0.64              0.4125                0.75
               0.987                0.48              0.4125                0.75
               1.009                0.76              0.4125                0.75 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

  1. If I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 1, the law uses the classic Vegter yield criterion which is defined by:
    (1)
    ϕ   = σ ¯ σ Y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dyMaaiiOaiabg2da9iqbeo8aZ9aagaqea8qacqGHsislcqaH dpWCpaWaaSbaaSqaa8qacaWGzbaapaqabaaaaa@3FDD@
    Where,
    ϕ
    Yield function.
    σ Y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamywaaWdaeqaaaaa@3904@
    Yield stress.
    σ ¯ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafq4Wdm3dayaaraaaaa@37F3@
    Interpolated Vegter equivalent stress.
    The Vegter yield locus, under plane stress conditions, is obtained by defining three second order Bezier interpolation curves in the principal stresses space (Figure 1).


    Figure 1. Yield criterion defined by the classic Vegter formulation in plane-stress (Icrit=1)
    These curves are used to link four reference points which are measured experimentally at different loading conditions: shear, uniaxial tension, plane strain, and equibiaxial tension. Between two reference points, the yield locus is then defined in:(2)
    σ =   σ 1 σ 2   =     σ ¯ f 1 f 2   =   σ 1 r 1 σ 2 r 1 + 2 μ σ 1 r 1 σ 1 h σ 2 r 1   σ 2 h + μ 2 σ 1 r 2 +   σ 1 r 1 2 σ 1 h σ 2 r 2 +   σ 2 r 1   2 σ 2 h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wdiabg2da9iaacckadaqadaWdaeaafaqabeGabaaabaWd biabeo8aZ9aadaWgaaWcbaWdbiaaigdaa8aabeaaaOqaa8qacqaHdp WCpaWaaSbaaSqaa8qacaaIYaaapaqabaaaaaGcpeGaayjkaiaawMca aiaacckacqGH9aqpcaGGGcGaaiiOaiqbeo8aZ9aagaqea8qadaqada WdaeaafaqabeGabaaabaWdbiaadAgapaWaaSbaaSqaa8qacaaIXaaa paqabaaakeaapeGaamOza8aadaWgaaWcbaWdbiaaikdaa8aabeaaaa aak8qacaGLOaGaayzkaaGaaiiOaiabg2da9iaacckadaqadaWdaeaa faqabeGabaaabaWdbiabeo8aZ9aadaqhaaWcbaWdbiaaigdaa8aaba WdbiaadkhacaaIXaaaaaGcpaqaa8qacqaHdpWCpaWaa0baaSqaa8qa caaIYaaapaqaa8qacaWGYbGaaGymaaaaaaaakiaawIcacaGLPaaacq GHRaWkcaaIYaGaeqiVd02aaeWaa8aabaqbaeqabiqaaaqaa8qacqaH dpWCpaWaa0baaSqaa8qacaaIXaaapaqaa8qacaWGYbGaaGymaaaaki abgkHiTiabeo8aZ9aadaqhaaWcbaWdbiaaigdaa8aabaWdbiaadIga aaaak8aabaWdbiabeo8aZ9aadaqhaaWcbaWdbiaaikdaa8aabaWdbi aadkhacaaIXaaaaOGaeyOeI0IaaiiOaiabeo8aZ9aadaqhaaWcbaWd biaaikdaa8aabaWdbiaadIgaaaaaaaGccaGLOaGaayzkaaGaey4kaS IaeqiVd02damaaCaaaleqabaWdbiaaikdaaaGcdaqadaWdaeaafaqa beGabaaabaWdbiabeo8aZ9aadaqhaaWcbaWdbiaaigdaa8aabaWdbi aadkhacaaIYaaaaOGaey4kaSIaaiiOaiabeo8aZ9aadaqhaaWcbaWd biaaigdaa8aabaWdbiaadkhacaaIXaaaaOGaeyOeI0IaaGOmaiabeo 8aZ9aadaqhaaWcbaWdbiaaigdaa8aabaWdbiaadIgaaaaak8aabaWd biabeo8aZ9aadaqhaaWcbaWdbiaaikdaa8aabaWdbiaadkhacaaIYa aaaOGaey4kaSIaaiiOaiabeo8aZ9aadaqhaaWcbaWdbiaaikdaa8aa baWdbiaadkhacaaIXaaaaOGaeyOeI0IaaiiOaiaaikdacqaHdpWCpa Waa0baaSqaa8qacaaIYaaapaqaa8qacaWGObaaaaaaaOGaayjkaiaa wMcaaaaa@9D6E@
    Where,
    σ r 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wd8aadaahaaWcbeqaa8qacaWGYbGaaGymaaaaaaa@395A@ , σ r 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wd8aadaahaaWcbeqaa8qacaWGYbGaaGOmaaaaaaa@395B@
    Two reference points.
    σ h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wd8aadaahaaWcbeqaa8qacaWGObaaaaaa@3895@
    Hinge point, which is computed automatically.
    μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiVd0gaaa@37BF@
    Parameter computed at each time step to determine the position on the yield locus.

    The four measured reference points are referred to as: f s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3933@ , f u n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaaaaa@393B@ , f p s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa8Nza8aadaWgaaWcbaWdbiaadchacaWGZbaapaqabaaaaa@3943@ , and ƒ b i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby 6aaSbaaSqaaiaadkgacaWGPbaabeaaaaa@39BC@ . The coordinates of these reference points are provided by you.

    Where, f s h 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadohacaWGObaapaqaa8qacaaIYaaa aaaa@3A00@ = f s h 1 =   f s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeyOeI0IaamOza8aadaqhaaWcbaWdbiaadohacaWGObaapaqaa8qa caaIXaaaaOGaeyypa0dcbmGaa8hOaiabgkHiTiaadAgapaWaaSbaaS qaa8qacaWGZbGaamiAaaWdaeqaaaaa@4140@ , f u n 2 = 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadwhacaWGUbaapaqaa8qacaaIYaaa aOGaeyypa0JaaGimaaaa@3BD2@ , f b i 2 =   f b i 1 = f b i   MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadkgacaWGPbaapaqaa8qacaaIYaaa aOGaeyypa0dcbmGaa8hOaiaadAgapaWaa0baaSqaa8qacaWGIbGaam yAaaWdaeaapeGaaGymaaaakiabg2da9iaadAgapaWaaSbaaSqaa8qa caWGIbGaamyAaaWdaeqaaOWdbiaa=bkaaaa@4578@ . For a given direction with respect to the rolling direction, only one component must be set for the shear f s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3933@ , uniaxial tension f u n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaaaaa@393B@ , and equibiaxial tension reference point ƒ b i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby 6aaSbaaSqaaiaadkgacaWGPbaabeaaaaa@39BC@ . For the plane-strain reference point, the two components f p s 1   MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIXaaa aOGaaiiOaaaa@3B35@ and f p s 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIYaaa aaaa@3A08@ must be provided by you.

    There is a freedom on the choice of the second coordinate f p s 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIYaaa aaaa@3A08@ . If it is not set, its value will be taken as an average of the second coordinates of the two neighboring hinge points.

    The hinge point located between two reference points is computed using the reference points coordinates, but also the normal to the yield surface in each reference point. Using the normality rule, the normal to the yield surface can be expressed using the strain rate tensor component:(3)
    n =   n 1 n 2 =   ε ˙ 1 ε ˙ 2 =     1 ε ˙ 2 ε ˙ 1 =     1 ρ r MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa8NBaiabg2da9iaacckadaqadaWdaeaafaqabeGabaaabaWd biaad6gapaWaaSbaaSqaa8qacaaIXaaapaqabaaakeaapeGaamOBa8 aadaWgaaWcbaWdbiaaikdaa8aabeaaaaaak8qacaGLOaGaayzkaaGa eyypa0JaaiiOamaabmaapaqaauaabeqaceaaaeaapeGafqyTdu2day aacaWaaSbaaSqaa8qacaaIXaaapaqabaaakeaapeGafqyTdu2dayaa caWaaSbaaSqaa8qacaaIYaaapaqabaaaaaGcpeGaayjkaiaawMcaai abg2da9iaacckacaGGGcWaaeWaa8aabaqbaeqabiqaaaqaa8qacaaI Xaaapaqaa8qadaWcaaWdaeaapeGafqyTdu2dayaacaWaaSbaaSqaa8 qacaaIYaaapaqabaaakeaapeGafqyTdu2dayaacaWaaSbaaSqaa8qa caaIXaaapaqabaaaaaaaaOWdbiaawIcacaGLPaaacqGH9aqpcaGGGc GaaiiOamaabmaapaqaauaabeqaceaaaeaapeGaaGymaaWdaeaapeGa eqyWdi3damaaBaaaleaapeGaamOCaaWdaeqaaaaaaOWdbiaawIcaca GLPaaaaaa@5D99@
    Where, ρ r MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaCaaaleqabaWdbiaadkhaaaaaaa@390C@ is the strain rate ratio in a given reference point. For the shear reference point, ρ s h = 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaam4CaiaadIgaa8aabeaakiabg2da 9iabgkHiTiaaigdaaaa@3CC0@ . For the plane-strain reference point, ρ p s = 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaamiCaiaadohaa8aabeaakiabg2da 9iaaicdaaaa@3BDA@ . For the uniaxial tension reference point, this ratio can be computed from the Lankford coefficient denoted R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOuaaaa@36E0@ :(4)
    ρ u n =   R R + 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaamyDaiaad6gaa8aabeaak8qacqGH 9aqpcaGGGcWaaSaaa8aabaWdbiabgkHiTiaadkfaa8aabaWdbiaadk facqGHRaWkcaaIXaaaaaaa@40DA@

    The Lankford coefficient must be provided by you. Finally, the strain rate ratio in the equibiaxial tension ƒ b i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby 6aaSbaaSqaaiaadkgacaWGPbaabeaaaaa@39BC@ must also be set.

  2. If I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 2, the standard Vegter criterion is used. It is the same criterion as the classical Vegter, with a different input card. In this card, the second coordinate of the plane strain point is computed as a weighted average of the second coordinate of the two neighboring hinge points σ h 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamiAaiaaigdaaeqaaaaa@3AFC@ and σ h 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamiAaiaaikdaaeqaaaaa@3AFD@ :
    (5)
    f p s 2 = σ h 1 2 + α p s σ h 2 2   σ h 1 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadchacaWGZbaapaqaa8qacaaIYaaa aOGaeyypa0Jaeq4Wdm3damaaDaaaleaapeGaamiAaiaaigdaa8aaba WdbiaaikdaaaGccqGHRaWkcqaHXoqypaWaaSbaaSqaa8qacaWGWbGa am4CaaWdaeqaaOWdbmaabmaapaqaa8qacqaHdpWCpaWaa0baaSqaa8 qacaWGObGaaGOmaaWdaeaapeGaaGOmaaaakiabgkHiTiaacckacqaH dpWCpaWaa0baaSqaa8qacaWGObGaaGymaaWdaeaapeGaaGOmaaaaaO GaayjkaiaawMcaaaaa@5188@
  3. If I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 3, the Vegter 2017 yield criterion is used. It is the same criterion as the classical Vegter, with a different input card. In this card, all parameters are determined from the maximum uniaxial engineering stress R m i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaqhaaWcbaWdbiaad2gaa8aabaWdbiaadMgaaaaaaa@392B@ , the maximum uniform elongation A g i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyqa8aadaqhaaWcbaWdbiaadEgaa8aabaWdbiaadMgaaaaaaa@3914@ , and the Lankford coefficient R i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaahaaWcbeqaa8qacaWGPbaaaaaa@381A@ . These three parameters must be given for three directions at an angle i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaaaa@36F7@ degrees with respect to the rolling direction RD: 0 degrees, 45 degrees, and 90 degrees. For this criterion, the number of angles is locked at 3.
  4. If I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 4, the law uses the simplified Vegter Lite yield criterion which is defined by:
    (6)
    ϕ   = σ ¯ σ Y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dyMaaiiOaiabg2da9iqbeo8aZ9aagaqea8qacqGHsislcqaH dpWCpaWaaSbaaSqaa8qacaWGzbaapaqabaaaaa@3FDD@
    Where,
    ϕ
    Yield function.
    σ Y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamywaaWdaeqaaaaa@3904@
    Yield stress.
    σ ¯ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafq4Wdm3dayaaraaaaa@37F3@
    Interpolated Vegter equivalent stress.
    The Vegter yield locus, under plane stress conditions, is obtained by defining two second order Nurbs interpolation curves in the principal stresses space. These curves are used to link three reference points which are measured experimentally at different loading conditions: uniaxial compression, uniaxial tension, and equibiaxial tension. Between two reference points, the yield locus is then defined in:(7)
    σ =   ( σ 1 σ 2 )   =     σ ¯ ( f 1 f 2 )   = ( 1 μ ) 2 ( σ 1 r 1 σ 2 r 1 ) + 2 μ ( 1 μ ) w h ( σ 1 h σ 2 h ) + μ 2 ( σ 1 r 2 σ 2 r 2 ) ( 1 μ ) 2 + 2 μ ( 1 μ ) w h +   μ 2   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wdiabg2da9iaacckadaqadaWdaeaafaqabeGabaaabaWd biabeo8aZ9aadaWgaaWcbaWdbiaaigdaa8aabeaaaOqaa8qacqaHdp WCpaWaaSbaaSqaa8qacaaIYaaapaqabaaaaaGcpeGaayjkaiaawMca aiaacckacqGH9aqpcaGGGcGaaiiOaiqbeo8aZ9aagaqea8qadaqada WdaeaafaqabeGabaaabaWdbiaadAgapaWaaSbaaSqaa8qacaaIXaaa paqabaaakeaapeGaamOza8aadaWgaaWcbaWdbiaaikdaa8aabeaaaa aak8qacaGLOaGaayzkaaGaaiiOaiabg2da9maalaaapaqaa8qadaqa daWdaeaapeGaaGymaiabgkHiTiabeY7aTbGaayjkaiaawMcaa8aada ahaaWcbeqaa8qacaaIYaaaaOWaaeWaa8aabaqbaeqabiqaaaqaa8qa cqaHdpWCpaWaa0baaSqaa8qacaaIXaaapaqaa8qacaWGYbGaaGymaa aaaOWdaeaapeGaeq4Wdm3damaaDaaaleaapeGaaGOmaaWdaeaapeGa amOCaiaaigdaaaaaaaGccaGLOaGaayzkaaGaey4kaSIaaGOmaiabeY 7aTnaabmaapaqaa8qacaaIXaGaeyOeI0IaeqiVd0gacaGLOaGaayzk aaGaam4Da8aadaWgaaWcbaWdbiaadIgaa8aabeaak8qadaqadaWdae aafaqabeGabaaabaWdbiabeo8aZ9aadaqhaaWcbaWdbiaaigdaa8aa baWdbiaadIgaaaaak8aabaWdbiabeo8aZ9aadaqhaaWcbaWdbiaaik daa8aabaWdbiaadIgaaaaaaaGccaGLOaGaayzkaaGaey4kaSIaeqiV d02damaaCaaaleqabaWdbiaaikdaaaGcdaqadaWdaeaafaqabeGaba aabaWdbiabeo8aZ9aadaqhaaWcbaWdbiaaigdaa8aabaWdbiaadkha caaIYaaaaaGcpaqaa8qacqaHdpWCpaWaa0baaSqaa8qacaaIYaaapa qaa8qacaWGYbGaaGOmaaaaaaaakiaawIcacaGLPaaaa8aabaWdbmaa bmaapaqaa8qacaaIXaGaeyOeI0IaeqiVd0gacaGLOaGaayzkaaWdam aaCaaaleqabaWdbiaaikdaaaGccqGHRaWkcaaIYaGaeqiVd02aaeWa a8aabaWdbiaaigdacqGHsislcqaH8oqBaiaawIcacaGLPaaacaWG3b WdamaaBaaaleaapeGaamiAaaWdaeqaaOWdbiabgUcaRiaacckacqaH 8oqBpaWaaWbaaSqabeaapeGaaGOmaaaakiaacckaaaaaaa@9B00@
    (8)
    0 μ 1 ,    w h 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaGimaiabgsMiJkabeY7aTjabgsMiJkaaigdacaGGSaGaaeiiaiaa bccacaWG3bWdamaaBaaaleaapeGaamiAaaWdaeqaaOGaeyyzImRaae imaaaa@435D@
    Where,
    σ r 1 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wd8aadaahaaWcbeqaa8qacaWGYbGaaGymaaaaaaa@395A@ , σ r 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wd8aadaahaaWcbeqaa8qacaWGYbGaaGOmaaaaaaa@395B@
    Two reference points.
    σ h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wd8aadaahaaWcbeqaa8qacaWGObaaaaaa@3895@
    Hinge point, which is computed automatically.
    μ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiVd0gaaa@37BF@
    Parameter computed at each time step to determine the position on the yield locus.
    w h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadIgaa8aabeaaaaa@384C@
    Weight factor associated with the hinge point.


    Figure 2. Yield criterion defined by the classic Vegter formulation in plane-stress (Icrit=2)

    In Figure 2, yield criterion is defined by the classic Vegter formulation in plane-stress ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 2).

    As for the classical Vegter formulation, you must set the parameters f u n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaaaaa@393B@ , ƒ b i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby 6aaSbaaSqaaiaadkgacaWGPbaabeaaaaa@39BC@ , R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOuaaaa@36E0@ , and ρ b i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaamOyaiaadMgaa8aabeaaaaa@39F8@ . In this simplified formulation, the two hinge points are in pure shear and plane strain condition. The two associated weight factors w s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3944@ and w p s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadchacaWGZbaapaqabaaaaa@394C@ must also be defined.

  5. To take into account the anisotropy, you can define a set of parameters for several N a n g l e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadggacaWGUbGaam4zaiaadYgacaWG Lbaapaqabaaaaa@3BD6@ directions at different angle ϕ with respect to the rolling direction RD. These directions must be equally distributed between 0 and π 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiabec8aWbWdaeaapeGaaGOmaaaaaaa@38D0@ . For all loading directions that are located between the given directions, the Vegter model proposes to use a Fourier series interpolation.
    Not all parameters must be defined for each direction. The parameter ƒ b i MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby 6aaSbaaSqaaiaadkgacaWGPbaabeaaaaa@39BC@ is uniform among all the directions. The strain rate ratio ρ b i MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaamOyaiaadMgaa8aabeaaaaa@39F8@ in the direction 0 (rolling direction) is sufficient to determine this ratio for all other directions:(9)
    ρ b i θ =   ρ b i + 1 + ρ b i 1 cos 2 θ ρ b i + 1 ρ b i 1 cos 2 θ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaamOyaiaadMgaa8aabeaak8qadaqa daWdaeaapeGaeqiUdehacaGLOaGaayzkaaGaeyypa0JaaiiOamaala aapaqaa8qadaqadaWdaeaapeGaeqyWdi3damaaBaaaleaapeGaamOy aiaadMgaa8aabeaak8qacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaey 4kaSYaaeWaa8aabaWdbiabeg8aY9aadaWgaaWcbaWdbiaadkgacaWG PbaapaqabaGcpeGaeyOeI0IaaGymaaGaayjkaiaawMcaaiGacogaca GGVbGaai4CaiaaikdacqaH4oqCa8aabaWdbmaabmaapaqaa8qacqaH bpGCpaWaaSbaaSqaa8qacaWGIbGaamyAaaWdaeqaaOWdbiabgUcaRi aaigdaaiaawIcacaGLPaaacqGHsisldaqadaWdaeaapeGaeqyWdi3d amaaBaaaleaapeGaamOyaiaadMgaa8aabeaak8qacqGHsislcaaIXa aacaGLOaGaayzkaaGaci4yaiaac+gacaGGZbGaaGOmaiabeI7aXbaa aaa@698F@
    For all other parameters, the following Fourier interpolation is used:(10)
    f u n θ = m = 0 N a n g l e 1 ψ un m   cos 2 m θ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadwhacaWGUbaapaqabaGcpeWaaeWa a8aabaWdbiabeI7aXbGaayjkaiaawMcaaiabg2da9maawahabeWcpa qaa8qacaWGTbGaeyypa0JaaGimaaWdaeaapeGaamOta8aadaWgaaad baWdbiaadggacaWGUbGaam4zaiaadYgacaWGLbaapaqabaWcpeGaey OeI0IaaGymaaqdpaqaa8qacqGHris5aaGccaqGipWdamaaDaaaleaa peGaaeyDaiaab6gaa8aabaWdbiaad2gaaaGccaGGGcGaae4yaiaab+ gacaqGZbWaaeWaa8aabaWdbiaaikdacaWGTbGaeqiUdehacaGLOaGa ayzkaaaaaa@5849@
    In the case of the classic, standard, and 2017 Vegter formulation ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 1, 2, 3), a similar operation is used. (11)
    f s h 2 θ = f s h 1   π 2 θ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaqhaaWcbaWdbiaadohacaWGObaapaqaa8qacaaIYaaa aOWaaeWaa8aabaWdbiabeI7aXbGaayjkaiaawMcaaiabg2da9iabgk HiTiaadAgapaWaa0baaSqaa8qacaWGZbGaamiAaaWdaeaapeGaaGym aaaakiaacckadaqadaWdaeaapeWaaSaaa8aabaWdbiabec8aWbWdae aapeGaaGOmaaaacqGHsislcqaH4oqCaiaawIcacaGLPaaaaaa@4B91@
    In the case of the simplified Vegter Lite formulation ( I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 4), a similar interpolation is used on R MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOuaaaa@36E0@ , w s h MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaaaaa@3944@ , and w p s MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadchacaWGZbaapaqabaaaaa@394C@ . In this case:(12)
    w s h θ =   w s h π 2 θ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Da8aadaWgaaWcbaWdbiaadohacaWGObaapaqabaGcpeWaaeWa a8aabaWdbiabeI7aXbGaayjkaiaawMcaaiabg2da9iaacckacaWG3b WdamaaBaaaleaapeGaam4CaiaadIgaa8aabeaak8qadaqadaWdaeaa peWaaSaaa8aabaWdbiabec8aWbWdaeaapeGaaGOmaaaacqGHsislcq aH4oqCaiaawIcacaGLPaaaaaa@494D@
  6. Parameters must define a convex yield locus, otherwise the simulation might be unstable. That is why a data check is present in the Radioss Starter to check the yield criterion convexity.
  7. When only one direction is defined, N a n g l e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadggacaWGUbGaam4zaiaadYgacaWG Lbaapaqabaaaaa@3BD6@ = 1, the material law is isotropic and must be used with /PROP/TYPE1 (isotropic shell). If more than 1 direction is defined, N a n g l e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOta8aadaWgaaWcbaWdbiaadggacaWGUbGaam4zaiaadYgacaWG Lbaapaqabaaaaa@3BD6@ > 1 or I c r i t MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGJbGaamOCaiaadMgacaWG0baabeaaaaa@3AB3@ = 3, the material law becomes anisotropic and must be used with /PROP/TYPE9 (orthotropic shells).
  8. The yield stress σ Y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamywaaWdaeqaaaaa@3904@ is defined by the following expression:
    (13)
    σ Y = σ 0 + Δ σ m β ε p + ε 0 + 1 e Ω ε p + ε 0 n + σ 0 1 + k T Δ G 0 ln ε ˙ ε 0 ˙ m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamywaaWdaeqaaOWdbiabg2da9iab eo8aZ9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGHRaWkcaqGuo Gaeq4Wdm3damaaBaaaleaapeGaamyBaaWdaeqaaOWdbmaadmaapaqa a8qacqaHYoGydaqadaWdaeaapeGaeqyTdu2damaaBaaaleaapeGaam iCaaWdaeqaaOWdbiabgUcaRiabew7aL9aadaWgaaWcbaWdbiaaicda a8aabeaaaOWdbiaawIcacaGLPaaacqGHRaWkdaqadaWdaeaapeGaaG ymaiabgkHiTiaadwgapaWaaWbaaSqabeaapeGaeyOeI0IaaeyQdmaa bmaapaqaa8qacqaH1oqzpaWaaSbaaWqaa8qacaWGWbaapaqabaWcpe Gaey4kaSIaeqyTdu2damaaBaaameaapeGaaGimaaWdaeqaaaWcpeGa ayjkaiaawMcaaaaaaOGaayjkaiaawMcaa8aadaahaaWcbeqaa8qaca WGUbaaaaGccaGLBbGaayzxaaGaey4kaSIaeq4Wdm3damaaDaaaleaa peGaaGimaaWdaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5 wzaGqba8qacqWFgls5aaGcdaWadaWdaeaapeGaaGymaiabgUcaRmaa laaapaqaa8qacaWGRbGaamivaaWdaeaapeGaaeiLdiaadEeapaWaaS baaSqaa8qacaaIWaaapaqabaaaaOWdbiGacYgacaGGUbWaaeWaa8aa baWdbmaalaaapaqaa8qacuaH1oqzpaGbaiaaaeaadaWfGaqaa8qacq aH1oqzpaWaaSbaaSqaa8qacaaIWaaapaqabaaabeqaa8qacaGGzlaa aaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faa8aadaahaaWcbeqaa8 qacaWGTbaaaaaa@8368@
    Where,
    σ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38E0@
    Initial yield stress.
    Δ   σ m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaacckacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaaa aa@3B56@
    Hardening stress increment.
    β
    Large strain hardening parameter.
    ε 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38C4@
    Initial plastic strain.
    Ω MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdaaa@3738@
    Small strain hardening parameter.
    η
    Hardening exponent.
    σ 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaDaaaleaapeGaaGimaaWdaeaatuuDJXwAK1uy0Hwm aeHbfv3ySLgzG0uy0Hgip5wzaGqba8qacqWFgls5aaaaaa@44CC@
    Limit dynamic flow stress.
    k MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aaaaa@36F9@
    Boltzmann constant.
    T MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@36E2@
    Temperature.
    Δ G 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaadEeapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3903@
    Maximum activation enthalpy.
    ε ˙ 0
    Inviscid limit strain rate.
    m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyBaaaa@36FB@
    Strain rate dependence exponent.
    The parameter, Ω MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdaaa@3738@ is called small strain hardening parameter, as its effect has a strong influence on the hardening curve at small strain (Figure 3a). Similarly, the large strain hardening parameter has a bigger influence at large strain (Figure 3b).


    Figure 3. Effect of hardening parameter on the yield stress evaluation
  9. If Tini is defined, the temperature will be constant at the Tini value over the simulation. This will enable a constant strain rate dependency (Figure 4a).
    If you want to compute a temperature evolution to take into account the thermal activation effect (slight increase of strain rate dependency following the increase of the temperature), the option /HEAT/MAT must be defined (Figure 4a).


    Figure 4. Effect of temperature setting on the strain rate dependency
  10. The strain rate ε ˙ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqyTdu2dayaacaaaaa@37C8@ computation depends on the value of the flag V P MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOvaiaadcfaaaa@37B9@ :
    • If V P MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOvaiaadcfaaaa@37B9@ = 1: the plastic strain rate is used.
    • If V P MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOvaiaadcfaaaa@37B9@ = 2: the total strain rate is used.
    • If V P MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOvaiaadcfaaaa@37B9@ = 3: the deviatoric strain rate is used.
    In all cases the strain rate computation includes a filtering, using the cutting off frequency F c u t MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOra8aadaWgaaWcbaWdbiaadogacaWG1bGaamiDaaWdaeqaaaaa @3A09@ , which is user-defined, such that:(14)
    ε ˙ f = α   ε ˙ n + ( 1 α ) ε ˙ n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqyTdu2dayaacaWaaSbaaSqaa8qacaWGMbaapaqabaGcpeGaeyyp a0JaeqySdeMaaiiOaiqbew7aL9aagaGaamaaCaaaleqabaWdbiaad6 gaaaGccqGHRaWkdaqadaWdaeaapeGaaGymaiabgkHiTiabeg7aHbGa ayjkaiaawMcaaiqbew7aL9aagaGaamaaCaaaleqabaWdbiaad6gacq GHsislcaaIXaaaaaaa@4A45@

    With α =   2 π   F c u t   Δ t MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdeMaeyypa0JaaiiOaiaaikdacqaHapaCcaGGGcGaamOra8aa daWgaaWcbaWdbiaadogacaWG1bGaamiDaaWdaeqaaOWdbiaacckaca qGuoGaamiDaaaa@44C0@ .

  11. If you want to use a tabulated hardening yield stress, the ID of the tabulated function TAB_YLD must be defined. This table can be used to define several yield stress evolutions with the plastic strain, at several strain rates. Two scale factor can be also defined in the X and Y directions. In this case, the hardening parameters σ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38E0@ , Δ   σ m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaacckacqaHdpWCpaWaaSbaaSqaa8qacaWGTbaapaqabaaa aa@3B56@ , β , Ω MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdaaa@3738@ , and η are ignored, and the yield stress becomes:(15)
    σ Y = f Y ε p , ε ˙ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamywaaWdaeqaaOWdbiabg2da9iaa bAgapaWaaSbaaSqaa8qacaqGzbaapaqabaGcpeWaaeWaa8aabaWdbi abew7aL9aadaWgaaWcbaWdbiaabchaa8aabeaak8qacaGGSaGafqyT duMbaiaaaiaawIcacaGLPaaaaaa@4373@
    Furthermore, you can also define a tabulated yield stress with plastic strain and temperature TAB_TEMP, to consider the thermal softening in adiabatic condition. In this case, the yield stress becomes:(16)
    σ Y = f Y ε p , ε ˙ f temp ε p , T f temp ε p , T ini MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamywaaWdaeqaaOWdbiabg2da9iaa bAgapaWaaSbaaSqaa8qacaqGzbaapaqabaGcpeWaaeWaa8aabaWdbi abew7aL9aadaWgaaWcbaWdbiaabchaa8aabeaak8qacaGGSaGafqyT duMbaiaaaiaawIcacaGLPaaadaWcaaWdaeaapeGaaeOza8aadaWgaa WcbaWdbiaabshacaqGLbGaaeyBaiaabchaa8aabeaak8qadaqadaWd aeaapeGaeqyTdu2damaaBaaaleaapeGaaeiCaaWdaeqaaOWdbiaacY cacaqGubaacaGLOaGaayzkaaaapaqaa8qacaqGMbWdamaaBaaaleaa peGaaeiDaiaabwgacaqGTbGaaeiCaaWdaeqaaOWdbmaabmaapaqaa8 qacqaH1oqzpaWaaSbaaSqaa8qacaqGWbaapaqabaGcpeGaaiilaiaa bsfapaWaaSbaaSqaa8qacaqGPbGaaeOBaiaabMgaa8aabeaaaOWdbi aawIcacaGLPaaaaaaaaa@5DB6@
    Where,
    Tini
    Reference temperature.
    T MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@36E2@
    Actual temperature computed using the /HEAT/MAT option.
  12. You might also want to use a kinematic hardening by setting coefficient Chard:
    • If Chard = 0: the isotropic hardening is used.
    • If Chard = 1: the Prager-Ziegler kinematic hardening is used.
    • If 0Chard1: the model is interpolated between the isotropic and the kinematic hardening.


      Figure 5.