/NONLOCAL/MAT

Block Format Keyword Non-local regularization for elasto-plastic failure criteria (as in, dependent to plastic strain) and shells thickness variation.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/NONLOCAL/MAT/mat_ID/unit_ID
R len MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGSbGaamyzaiaad6gaaeqaaaaa@39C7@ L e MAX MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGLbaabeaakmaaCaaaleqabaGaamytaiaadgeacaWGybaa aaaa@3A89@

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID Unit identifier.

(Integer, maximum 10 digits)

 
R l e n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGSbGaamyzaiaad6gaaeqaaaaa@39C7@ Non-local internal length.

(Real)

[ m ]
L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGLbaabeaakmaaCaaaleqabaGaamytaiaadgeacaWGybaa aaaa@3A89@ Mesh convergence element length target.

(Real)

[ m ]

Comments

  1. The non-local regularization is used to get mesh independent results (size, orientation) in case of instabilities such as failure and/or thickness variation (for shells). The mesh independent results implies a mesh convergence for mesh sizes L e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaadwgaa8aabeaaaaa@3821@ less than or equal to the maximum value set by you, L e L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGHKjYOcaWG mbWdamaaDaaaleaapeGaamyzaaWdaeaapeGaamyBaiaadggacaWG4b aaaaaa@3EEB@ . This maximum mesh size L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGLbaabeaakmaaCaaaleqabaGaamytaiaadgeacaWGybaa aaaa@3A89@ is then the highest mesh size used for which results are mesh convergent.
    The non-local formulation is compatible with elasto-plastic material laws only. When activated, the computation of the attached failure criteria based on plastic strain and/or the shell thickness variation depends on a regularized nodal "non-local" plastic strain calculated on the entire mesh. The non-local plastic strain at nodes denoted ε p n l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaDaaaleaapeGaamiCaaWdaeaapeGaamOBaiaadYga aaaaaa@3AF7@ is computed accounting for its own gradient and its local counterpart ε p is computed at the Gauss points following the set of equations:(1)
      R l e n 2 Δ ε p n l γ ε ˙ p n l + ( ε p ε p n l ) = ζ ε ¨ p n l     ε p n l n = 0   o n o n         Ω Γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaauaabeqaceaaae aaqaaaaaaaaaWdbiaacckacaWGsbWdamaaDaaaleaapeGaamiBaiaa dwgacaWGUbaapaqaa8qacaaIYaaaaOGaeuiLdqKaeqyTdu2damaaDa aaleaapeGaamiCaaWdaeaapeGaamOBaiaadYgaaaGccqGHsislcqaH ZoWzcuaH1oqzgaGaa8aadaqhaaWcbaWdbiaadchaa8aabaWdbiaad6 gacaWGSbaaaOGaey4kaSYaaeWaa8aabaWdbiabew7aL9aadaWgaaWc baWdbiaadchaa8aabeaak8qacqGHsislcqaH1oqzpaWaa0baaSqaa8 qacaWGWbaapaqaa8qacaWGUbGaamiBaaaaaOGaayjkaiaawMcaaiab g2da9iabeA7a6jqbew7aLzaadaWdamaaDaaaleaapeGaamiCaaWdae aapeGaamOBaiaadYgaaaGccaGGGcGaaiiOaaWdaeaapeGafy4bIe9d ayaalaWdbiabew7aL9aadaqhaaWcbaWdbiaadchaa8aabaWdbiaad6 gacaWGSbaaaOGaeyyXICTabmOBa8aagaWca8qacqGH9aqpcaaIWaaa aiaacckapaqbaeqabiqaaaqaa8qacaWGVbGaamOBaaWdaeaapeGaam 4Baiaad6gaaaGaaiiOaiaacckacaGGGcGaaiiOa8aafaqabeGabaaa baWdbiabfM6axbWdaeaapeGaeu4KdCeaaaaa@788D@
    The parameters γ and ζ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOTdOhaaa@37C9@ are automatically set. You have to set the parameter R l e n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGSbGaamyzaiaad6gaaeqaaaaa@39C7@ (or L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGLbaabeaakmaaCaaaleqabaGaamytaiaadgeacaWGybaa aaaa@3A89@ , Comment 2) which defines a non-local "internal length" corresponding to a radius of influence in the non-local variable computation. This defines the size of the non-local regularization band L r = f ( R l e n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaadkhaa8aabeaak8qacqGH9aqpcaWG MbWaaeWaa8aabaWdbiaadkfapaWaaSbaaSqaa8qacaWGSbGaamyzai aad6gaa8aabeaaaOWdbiaawIcacaGLPaaaaaa@3FFA@ (Figure 1).


    Figure 1. Non-local regularization principle illustration
    The failure criterion damage variable is then computed using the non-local plastic strain.(2)
    D = t = 0 Δ ε p n l ε f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9maaqahabaWaaSaaaeaacqqHuoarcqaH1oqzdaqhaaWcbaGaamiC aaqaaiaad6gacaWGSbaaaaGcbaGaeqyTdu2aaSbaaSqaaiaadAgaae qaaaaaaeaacaWG0bGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5 aaaa@4712@

    Where, ε f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadAgaaeqaaaaa@38B5@ is the plastic strain at failure depending on the failure criterion formulation.

  2. To set the non-local length parameter R l e n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGSbGaamyzaiaad6gaaeqaaaaa@39C7@ , you can select:
    • Directly input the value of R l e n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGSbGaamyzaiaad6gaaeqaaaaa@39C7@ in the input card, if a direct control on this parameter is wanted. In this case, the parameter L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGLbaabeaakmaaCaaaleqabaGaamytaiaadgeacaWGybaa aaaa@3A89@ must be ignored.
    • Input the maximum mesh size L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGLbaabeaakmaaCaaaleqabaGaamytaiaadgeacaWGybaa aaaa@3A89@ for which results are mesh convergent. The non-local regularization will then be effective for all mesh sizes L e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaadwgaa8aabeaaaaa@3821@ such as L e L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGHKjYOcaWG mbWdamaaDaaaleaapeGaamyzaaWdaeaapeGaamyBaiaadggacaWG4b aaaaaa@3EEB@ . In this case, an automatic set of R l e n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGSbGaamyzaiaad6gaaeqaaaaa@39C7@ is realized according to the value of L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGLbaabeaakmaaCaaaleqabaGaamytaiaadgeacaWGybaa aaaa@3A89@ , and the input value of R l e n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGSbGaamyzaiaad6gaaeqaaaaa@39C7@ is ignored.

      For instance, if you want to get converged and mesh-independent results for a mesh size of 5mm, L e M A X = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaqhaaWcbaWdbiaadwgaa8aabaWdbiaad2gacaWGHbGa amiEaaaak8aacqGH9aqpcaaI1aaaaa@3CE5@ mm. In this case, the results will be converged, mesh-size and mesh orientation independent for L e 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaadwgaa8aabeaak8qacqGHKjYOcaaI 1aaaaa@3AAF@ mm.

  3. When the non-local regularization is used for shell elements, an additional regularization is made on the thickness variation computation avoiding an additional localization issue. In the common local case (Figure 2), the compatibility of thickness between shell elements is not ensured, due to the lack of kinematic equations in the z-direction, and the thickness variation is locally computed at Gauss points. By introducing the non-local plastic strain in the "in-thickness" strain increment, the compatibility is restored (Figure 3).(3)
    Δ ε z z = ν 1 ν   ( Δ ε x x   Δ λ n l   n x x + Δ ε y y   Δ λ n l   n y y ) +   Δ λ n l   n z z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiabew7aL9aadaWgaaWcbaWdbiaadQhacaWG6baapaqabaGc peGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiabe27aUbWdaeaapeGaaG ymaiabgkHiTiabe27aUbaacaGGGcWaaeWaa8aabaWdbiabfs5aejab ew7aL9aadaWgaaWcbaWdbiaadIhacaWG4baapaqabaGcpeGaaiiOai abgkHiTiabfs5aejabeU7aS9aadaWgaaWcbaWdbiaad6gacaWGSbaa paqabaGcpeGaaiiOaiaad6gapaWaaSbaaSqaa8qacaWG4bGaamiEaa WdaeqaaOWdbiabgUcaRiabfs5aejabew7aL9aadaWgaaWcbaWdbiaa dMhacaWG5baapaqabaGcpeGaaiiOaiabgkHiTiabfs5aejabeU7aS9 aadaWgaaWcbaWdbiaad6gacaWGSbaapaqabaGcpeGaaiiOaiaad6ga paWaaSbaaSqaa8qacaWG5bGaamyEaaWdaeqaaaGcpeGaayjkaiaawM caaiabgUcaRiaacckacqqHuoarcqaH7oaBpaWaaSbaaSqaa8qacaWG UbGaamiBaaWdaeqaaOWdbiaacckacaWGUbWdamaaBaaaleaapeGaam OEaiaadQhaa8aabeaaaaa@7537@
    Where, Δ λ n l = f ( ε p n l ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiabeU7aS9aadaWgaaWcbaWdbiaad6gacaWGSbaapaqabaGc peGaeyypa0JaamOzamaabmaapaqaa8qacqaH1oqzpaWaa0baaSqaa8 qacaWGWbaapaqaa8qacaWGUbGaamiBaaaaaOGaayjkaiaawMcaaaaa @43C0@ is the non-local plastic multiplier.


    Figure 2. Transverse strain incompatibility (local)


    Figure 3. Transverse strain compatibility (non-local)
    Note: This last point implies that the identified parameters can be used on solid and shells, as results will be identical within the same range of stress triaxiality 2 3 η 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS GaaeaacaaIYaaabaGaaG4maaaacqGHKjYOcqaH3oaAcqGHKjYOdaWc caqaaiaaikdaaeaacaaIZaaaaaaa@3F10@ .
  4. List of compatible material laws for shells thickness variation regularization:
    • /MAT/LAW2 (PLAS_JOHNS)
    • /MAT/LAW22 (DAMA)
    • /MAT/LAW27 (PLAS_BRIT)
    • /MAT/LAW32 (HILL)
    • /MAT/LAW36 (PLAS_TAB)
    • /MAT/LAW43 (HILL_TAB)
    • /MAT/LAW44 (COWPER)
    • /MAT/LAW48 (ZHAO)
    • /MAT/LAW57 (BARLAT3)
    • /MAT/LAW60 (PLAS_T3)
    • /MAT/LAW63 (HANSEL)
    • /MAT/LAW64 (UGINE_ALZ)
    • /MAT/LAW72 (HILL_MMC)
    • /MAT/LAW76 (SAMP)
    • /MAT/LAW78
    • /MAT/LAW87 (BARLAT2000)
    • /MAT/LAW93 (ORTH_HILL) (CONVERSE)
    • /MAT/LAW104 (JOHNS_VOCE_DRUCKER)
    • /MAT/LAW109
    • /MAT/LAW110 (VEGTER)
  5. List of elasto-plastic failure model and coupled damage model compatible with non-local regularization:
    • MMC damage model in /MAT/LAW72
    • Damage model in /MAT/LAW76
    • /FAIL/BIQUAD
    • /FAIL/COCKROFT
    • /FAIL/EMC
    • /FAIL/HC_DSSE (for shells)
    • /FAIL/JOHNSON
    • /FAIL/ORTHBIQUAD
    • /FAIL/RTCL
    • /FAIL/SPALLING
    • /FAIL/TAB1
    • /FAIL/USERi
    • /FAIL/WIERZBICKI
    • /FAIL/WILKINS
  6. List of material laws with non-local regularized temperature computation:
    • /MAT/LAW104 (JOHNS_VOCE_DRUCKER)
    • /MAT/LAW109