/FAIL/GURSON

Block Format Keyword A Gurson-Nahshon-Hutchinson failure model describing the damage in terms of void nucleation and growth in metal plasticity.

The modified Gurson formulation adds additional damage accumulation terms for shear dominated loads, specific treatment under compressive loading, and elastic stiffness loss with damage.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/FAIL/GURSON/mat_ID/unit_ID
q 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3817@ q 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3817@ Iloc
ε n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaad6gaaeqaaaaa@38BD@ As Kw
f c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3839@ f F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3839@ f I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3839@
Rlen Hchi L e MAX MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeadaqhaa WcbaGaamyzaaqaaiaad2eacaWGbbGaamiwaaaaaaa@3A49@
Optional line
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
fail_ID        

Definition

Field Contents SI Unit Example
mat_ID Material identifier.

(Integer, maximum 10 digits)

unit_ID (Optional) Unit Identifier.

(Integer, maximum 10 digits)

q 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3817@ First Gurson damage coefficient.

Default = 1.5 (Real)

q 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3817@ Second Gurson damage coefficient, maximum value = 1.02.

Default = 1.0 (Real)

Iloc Damage variable accumulation method flag.
= 0
Set to 1.
= 1 (Default)
Local damage formulation.
= 2
Non-local damage regularization using the Micromorphic method.
= 3
Non-local damage regularization using the Peerlings method.

(Integer)

ε n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaad6gaaeqaaaaa@38BD@ Equivalent plastic strain at void nucleation.

(Real)

As Linear void nucleation slope.

(Real)

Kw Shear damage growth coefficient.

(Real)

f c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3839@ Critical void volume fraction at void coalescence.

(Real)

f F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3839@ Void volume fraction at ductile failure.

(Real)

f I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaGaaGimaaqabaaaaa@37EC@ Initial void volume fraction.

(Real)

Rlen Radius of non-local variable influence (Iloc > 1).

(Real)

[ m ]
Hchi Non-local penalty parameter (Micromorphic method only, Iloc = 2).

(Real)

[ Pa ]
L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeadaqhaa WcbaGaamyzaaqaaiaad2eacaWGbbGaamiwaaaaaaa@3A49@ Mesh convergence element length target. 5

(Real)

[ m ]
fail_ID (Optional) Failure criteria identifier.

(Integer, maximum 10 digits)

Comments

  1. The Gurson damage model may only be used with the elasto-plastic material /MAT/LAW104. The yield surface definition of the material law is modified by adding the damage evolution terms: (1)
    ϕ = σ e q 2 σ y l d 2 1 + 2 q 1 f * c o s h ( η t   q 2 T r ( σ ) 2 σ y l d ) ( q 1 f * ) 2 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dyMaeyypa0ZaaSaaa8aabaWdbiabeo8aZ9aadaqhaaWcbaWd biaadwgacaWGXbaapaqaa8qacaaIYaaaaaGcpaqaa8qacqaHdpWCpa Waa0baaSqaa8qacaWG5bGaamiBaiaadsgaa8aabaWdbiaaikdaaaaa aOGaeyOeI0IaaGymaiabgUcaRiaaikdacaWGXbWdamaaBaaaleaape GaaGymaaWdaeqaaOWdbiaadAgapaWaaWbaaSqabeaapeGaaiOkaaaa kiaadogacaWGVbGaam4CaiaadIgadaqadaWdaeaapeWaaSaaa8aaba WdbiabeE7aO9aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGGGcGa amyCa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaWGubGaamOCam aabmaapaqaaGqad8qacaWFdpaacaGLOaGaayzkaaaapaqaa8qacaaI YaGaeq4Wdm3damaaBaaaleaapeGaamyEaiaadYgacaWGKbaapaqaba aaaaGcpeGaayjkaiaawMcaaiabgkHiTmaabmaapaqaa8qacaWGXbWd amaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaadAgapaWaaWbaaSqabe aapeGaaiOkaaaaaOGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaI YaaaaOGaeyypa0JaaGimaaaa@6B4D@
    Where,
    q 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3817@ , q 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaa@3817@
    Two Gurson-Tveergard-Needleman parameters.
    f * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaahaaWcbeqaa8qacaGGQaaaaaaa@37F1@
    Effective damage
    η t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdG2damaaBaaaleaapeGaamiDaaWdaeqaaaaa@390B@
    Factor defined as:
    η t = { 0   ,     f t =   0     and     T r ( σ ) < 0 1 ,  otherwise MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdG2damaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabg2da9maa ceaapaabaeqabaWdbiaaicdacaGGGcGaaeilaiaacckacaGGGcGaam Oza8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGH9aqpcaGGGcGa aGimaiaacckacaGGGcGaaeyyaiaab6gacaqGKbGaaiiOaiaacckaca WGubGaamOCamaabmaapaqaa8qacqaHdpWCaiaawIcacaGLPaaacqGH 8aapcaaIWaaabaGaaGymaiaacYcacaqGGaGaae4BaiaabshacaqGOb GaaeyzaiaabkhacaqG3bGaaeyAaiaabohacaqGLbaaaiaawUhaaaaa @5E43@
    f t   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGGGcaaaa@3988@
    Total void volume fraction that is computed incrementally.
    d f t = d f n + d f g + d f s h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadAgapaWaaSbaaSqaa8qacaWG0baapaqabaGcpeGaeyyp a0JaamizaiaadAgapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeGaey 4kaSIaamizaiaadAgapaWaaSbaaSqaa8qacaWGNbaapaqabaGcpeGa ey4kaSIaamizaiaadAgapaWaaSbaaSqaa8qacaWGZbGaamiAaaWdae qaaaaa@4699@
    The kinetic equations of the damage factor increments are:
    • Void nucleation (creation of microcavities), decreasing at low triaxiality.(2)
      d f n = { A s   d ε p ,     ε p ε n     and     τ 0 A s   ( 1 + 3 τ ) d ε p ,   ε p ε n     and   1 3 τ < 0 0 ,   ε p < ε n     and     τ <   1 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadAgapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeGaeyyp a0Zaaiqaa8aabaqbaeqabmqaaaqaa8qacaWGbbWdamaaBaaaleaape Gaam4CaaWdaeqaaOWdbiaacckacaWGKbGaeqyTdu2damaaBaaaleaa peGaamiCaaWdaeqaaOWdbiaacYcacaGGGcGaaeiiaiabew7aL9aada WgaaWcbaWdbiaadchaa8aabeaak8qacqGHLjYScqaH1oqzpaWaaSba aSqaa8qacaWGUbaapaqabaGcpeGaaiiOaiaacckacaqGHbGaaeOBai aabsgacaGGGcGaaiiOaiabes8a0jabgwMiZkaaicdaa8aabaWdbiaa dgeapaWaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaaiiOamaabmaapa qaa8qacaaIXaGaey4kaSIaaG4maiabes8a0bGaayjkaiaawMcaaiaa dsgacqaH1oqzpaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeGaaiilai aabccacqaH1oqzpaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeGaeyyz ImRaeqyTdu2damaaBaaaleaapeGaamOBaaWdaeqaaOWdbiaacckaca GGGcGaaeyyaiaab6gacaqGKbGaaiiOaiabgkHiTmaalaaapaqaa8qa caaIXaaapaqaa8qacaaIZaaaaiabgsMiJkabes8a0jabgYda8iaaic daa8aabaWdbiaaicdacaGGSaGaaeiiaiabew7aL9aadaWgaaWcbaWd biaadchaa8aabeaak8qacqGH8aapcqaH1oqzpaWaaSbaaSqaa8qaca WGUbaapaqabaGcpeGaaiiOaiaacckacaqGHbGaaeOBaiaabsgacaGG GcGaaiiOaiabes8a0jabgYda8iaacckacqGHsisldaWcaaWdaeaape GaaGymaaWdaeaapeGaaG4maaaaaaaacaGL7baaaaa@9488@
      Where, τ is the stress triaxiality defined as:(3)
      τ =   T r ( σ ) 3   σ e q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdqNaeyypa0JaaiiOamaalaaapaqaa8qacaWGubGaamOCamaa bmaapaqaaGqad8qacaWFdpaacaGLOaGaayzkaaaapaqaa8qacaaIZa GaaiiOaiabeo8aZ9aadaWgaaWcbaWdbiaadwgacaWGXbaapaqabaaa aaaa@44F2@


      Figure 1. Nucleation of cavities
    • Void growth at high triaxiality:(4)
      d f g = ( 1 f t )   T r ( ε p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadAgapaWaaSbaaSqaa8qacaWGNbaapaqabaGcpeGaeyyp a0ZaaeWaa8aabaWdbiaaigdacqGHsislcaWGMbWdamaaBaaaleaape GaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaiaacckacaWGubGaamOC amaabmaapaqaaGqad8qacaWF1oWdamaaBaaaleaapeGaamiCaaWdae qaaaGcpeGaayjkaiaawMcaaaaa@4738@


      Figure 2. Growth of cavities at high triaxiality
    • Additional shear void growth at low triaxiality which is shear dominated: (5)
      d f s h = K w f t   w ( θ )   S   : ε p σ e q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadAgapaWaaSbaaSqaa8qacaWGZbGaamiAaaWdaeqaaOWd biabg2da9iaadUeapaWaaSbaaSqaa8qacaWG3baapaqabaGcpeGaam Oza8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGGGcGaam4Damaa bmaapaqaa8qacqaH4oqCaiaawIcacaGLPaaacaGGGcWaaSaaa8aaba acbmWdbiaa=nfacaWFGcGaaiOoaiaa=v7apaWaaSbaaSqaa8qacaWG WbaapaqabaaakeaapeGaeq4Wdm3damaaBaaaleaapeGaamyzaiaadg haa8aabeaaaaaaaa@5006@
      Where, w ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Damaabmaapaqaa8qacqaH4oqCaiaawIcacaGLPaaaaaa@3A66@ is a weight function depending on the Lode angle: (6)
      w ( θ ) = 1 ( cos ( 3 θ ) ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Damaabmaapaqaa8qacqaH4oqCaiaawIcacaGLPaaacqGH9aqp caaIXaGaeyOeI0YaaeWaa8aabaWdbiaabogacaqGVbGaae4Camaabm aapaqaa8qacaaIZaGaeqiUdehacaGLOaGaayzkaaaacaGLOaGaayzk aaWdamaaCaaaleqabaWdbiaaikdaaaaaaa@46AD@


      Figure 3. Growth of cavities at low triaxiality
      To represent the cavities coalescence when a critical void volume fraction f c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3839@ is reached by f t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadogaa8aabeaaaaa@3839@ , the effective damage (which has an influence on the stress computation) f * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaahaaWcbeqaa8qacaGGQaaaaaaa@37F1@ is introduced in the model and its expression depends on f t   MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGGGcaaaa@3988@ : (7)
      f * = f ( f t ) = { f t ,     f t < f c f c + ( 1 q 1 f c ) ( f t f c ) ( f R f c ) ,     f t f c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaahaaWcbeqaa8qacaGGQaaaaOGaeyypa0JaamOzamaa bmaapaqaa8qacaWGMbWdamaaBaaaleaapeGaamiDaaWdaeqaaaGcpe GaayjkaiaawMcaaiabg2da9maaceaapaqaauaabeqaceaaaeaapeGa amOza8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGGSaGaaiiOai aacckacaWGMbWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabgYda 8iaadAgapaWaaSbaaSqaa8qacaWGJbaapaqabaaakeaapeGaamOza8 aadaWgaaWcbaWdbiaadogaa8aabeaak8qacqGHRaWkdaqadaWdaeaa peWaaSGaa8aabaWdbiaaigdaa8aabaWdbiaadghapaWaaSbaaSqaa8 qacaaIXaaapaqabaaaaOWdbiabgkHiTiaadAgapaWaaSbaaSqaa8qa caWGJbaapaqabaaak8qacaGLOaGaayzkaaWaaSaaa8aabaWdbmaabm aapaqaa8qacaWGMbWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiab gkHiTiaadAgapaWaaSbaaSqaa8qacaWGJbaapaqabaaak8qacaGLOa Gaayzkaaaapaqaa8qadaqadaWdaeaapeGaamOza8aadaWgaaWcbaWd biaadkfaa8aabeaak8qacqGHsislcaWGMbWdamaaBaaaleaapeGaam 4yaaWdaeqaaaGcpeGaayjkaiaawMcaaaaacaGGSaGaaiiOaiaaccka caWGMbWdamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabgwMiZkaadA gapaWaaSbaaSqaa8qacaWGJbaapaqabaaaaaGcpeGaay5Eaaaaaa@6E51@
      Where, f R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadkfaa8aabeaaaaa@3828@ is the total void volume fraction at rupture for which f * = 1 / q 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaahaaWcbeqaa8qacaGGQaaaaOGaeyypa0JaaGymaiaa c+cacaWGXbWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa@3C7A@ .


      Figure 4. Coalescence of cavities
      To take into account the effect of the stiffness loss, a damage variable D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiraaaa@36D5@ is computed as: (8)
      D = q 1   f * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGebGaeyypa0JaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacaGGGcGaamOza8aadaahaaWcbeqaa8qacaGGQaaaaaaa@3D14@
      The effective damage f * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaahaaWcbeqaa8qacaGGQaaaaaaa@37F1@ is normalized by its rupture value 1 / q 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaGymaiaac+cacaWGXbWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa @3985@ which gives 0     D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaGimaiaacckacqGHKjYOcaGGGcGaamiraiabgsMiJkaaigdaaaa@3DFC@ . The stress tensor is then computed as: (9)
      σ = ( 1 D )   C ˜   ε e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGaa83Wdiabg2da9maabmaapaqaa8qacaaIXaGaeyOeI0Iaamir aaGaayjkaiaawMcaaiaacckaceWFdbWdayaaiaWdbiaacckacaWF1o WdamaaBaaaleaapeGaa8xzaaWdaeqaaaaa@4231@

      Where, C ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqadabaaaaaaa aapeGab83qa8aagaacaaaa@36FA@ is the elastic stiffness matrix.

      The material fails when the cumulated total damage factor reaches the limit value f R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOza8aadaWgaaWcbaWdbiaadkfaa8aabeaaaaa@3828@ . The element is then deleted.

  2. By default I l o c = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadYgacaWGVbGaam4yaaWdaeqaaOWd biabg2da9iaaigdaaaa@3BDC@ , the damage variable is calculated step by step using the local plastic strain values at each integration point. However, one may want to use non-local regularization which offers mesh size and the mesh orientation independent results (mesh convergence) for all meshes using the mesh size L e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaWgaaWcbaWdbiaadwgaa8aabeaaaaa@3821@ less than 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@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamita8aadaqhaaWcbaWdbiaadwgaa8aabaWdbiaad2gacaWGHbGa amiEaaaaaaa@3B07@ is then the highest mesh size used for which results are mesh convergent.
    If one of the non-local formulations is used, ( I l o c > 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiikaiaadMeapaWaaSbaaSqaa8qacaWGSbGaam4Baiaadogaa8aa beaak8qacqGH+aGpcaaIXaGaaiykaaaa@3D37@ , the damage increments depend 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 computed at the Gauss points following the set of equations: (10)
        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 aaqaaaaaaaaaWdbiaacckacaGGGcGaamOua8aadaqhaaWcbaWdbiaa dYgacaWGLbGaamOBaaWdaeaapeGaaGOmaaaakiaacckacaGGGcGaeu iLdqKaeqyTdu2damaaDaaaleaapeGaamiCaaWdaeaapeGaamOBaiaa dYgaaaGccaGGGcGaeyOeI0IaaiiOaiabeo7aNjaacckapaWaaCbiae aapeGaeqyTdu2damaaDaaaleaapeGaamiCaaWdaeaapeGaamOBaiaa dYgaaaaapaqabeaapeGaaiy2caaakiaacckacqGHRaWkcaGGGcWaae Waa8aabaWdbiabew7aL9aadaWgaaWcbaWdbiaadchaa8aabeaak8qa cqGHsislcqaH1oqzpaWaa0baaSqaa8qacaWGWbaapaqaa8qacaWGUb GaamiBaaaaaOGaayjkaiaawMcaaiabg2da9iaacckacqaH2oGEpaWa aCbiaeaapeGaeqyTdu2damaaDaaaleaapeGaamiCaaWdaeaapeGaam OBaiaadYgaaaaapaqabeaapeGaaiiQaaaakiaacckacaGGGcaapaqa a8qacuGHhis0paGbaSaapeGaaiiOaiabew7aL9aadaqhaaWcbaWdbi aadchaa8aabaWdbiaad6gacaWGSbaaaOGaaiiOaiaac6cacaGGGcGa bmOBa8aagaWca8qacqGH9aqpcaaIWaaaaiaacckapaqbaeqabiqaaa qaa8qacaWGVbGaamOBaaWdaeaapeGaam4Baiaad6gaaaGaaiiOaiaa cckacaGGGcGaaiiOa8aafaqabeGabaaabaWdbiabfM6axbWdaeaape Gaeu4KdCeaaaaa@8815@
    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 Rlen (or L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeadaqhaa WcbaGaamyzaaqaaiaad2eacaWGbbGaamiwaaaaaaa@3A49@ - Comment 5) which defines a non-local “internal length” which corresponds 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 5).


    Figure 5. Non-local regularization principle
    To help choose a value for the parameter Rlen, one may follow the following expression:(11)
    R l e n   3   L e m a x   π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaadYgacaWGLbGaamOBaaWdaeqaaOWd biabgIKi7kaacckadaWcaaWdaeaapeGaaG4maiaacckacaWGmbWdam aaDaaaleaapeGaamyzaaWdaeaapeGaamyBaiaadggacaWG4baaaOGa aiiOaaWdaeaapeWaaOaaa8aabaWdbiabec8aWbWcbeaaaaaaaa@4749@
  3. If I l o c = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadYgacaWGVbGaam4yaaWdaeqaaOWd biabg2da9iaaikdaaaa@3BDD@ , the non-local Micromorphic method will be used. For this method, the parameter is required, Hchi. This parameter and the non-local plastic strain ε p n l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaDaaaleaapeGaamiCaaWdaeaapeGaamOBaiaadYga aaaaaa@3AF7@ are introduced in the constitutive equation as: (12)
    R c h i ( ε p , ε p n l )   =     R ( ε p )   H c h i   ( ε p n l   ε p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaadogacaWGObGaamyAaaWdaeqaaOWd bmaabmaapaqaa8qacqaH1oqzpaWaaSbaaSqaa8qacaWGWbaapaqaba GcpeGaaiilaiabew7aL9aadaqhaaWcbaWdbiaadchaa8aabaWdbiaa d6gacaWGSbaaaaGccaGLOaGaayzkaaGaaiiOaiabg2da9iaacckaca GGGcGaamOuamaabmaapaqaa8qacqaH1oqzpaWaaSbaaSqaa8qacaWG Wbaapaqabaaak8qacaGLOaGaayzkaaGaaiiOaiabgkHiTiaadIeapa WaaSbaaSqaa8qacaWGJbGaamiAaiaadMgaa8aabeaak8qacaGGGcWa aeWaa8aabaWdbiabew7aL9aadaqhaaWcbaWdbiaadchaa8aabaWdbi aad6gacaWGSbaaaOGaeyOeI0IaaiiOaiabew7aL9aadaWgaaWcbaWd biaadchaa8aabeaaaOWdbiaawIcacaGLPaaaaaa@616F@

    Where, R ( ε p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOuamaabmaapaqaa8qacqaH1oqzpaWaaSbaaSqaa8qacaWGWbaa paqabaaak8qacaGLOaGaayzkaaaaaa@3B9B@ is the classical work-hardening function. This newly defined micromorphic work-hardening function Rchi is introduced in the flow stress computation σ y l d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdm3damaaBaaaleaapeGaamyEaiaadYgacaWGKbaapaqabaaa aa@3B01@ . The parameter Hchi becomes a penalty parameter and if H c h i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaOcaeaaaaaa aaa8qacaWGibWdamaaBaaaleaapeGaam4yaiaadIgacaWGPbaapaqa baGcpeGaeyOKH4QaeyOhIukaaa@3E19@ , then ε p ε p n l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaamiCaaWdaeqaaOWdbiabgkziUkab ew7aL9aadaqhaaWcbaWdbiaadchaa8aabaWdbiaad6gacaWGSbaaaa aa@3FF4@ and ε p n l ε p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaDaaaleaapeGaamiCaaWdaeaapeGaamOBaiaadYga aaGccqGHsgIRcqaH1oqzpaWaaSbaaSqaa8qacaWGWbaapaqabaaaaa@3FE4@ and so ε p ε p n l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTdu2damaaBaaaleaapeGaamiCaaWdaeqaaOWdbiabgIKi7kab ew7aL9aadaqhaaWcbaWdbiaadchaa8aabaWdbiaad6gacaWGSbaaaa aa@3FB8@ . This method is thermo-dynamically well defined. However, it is hard to identify the input values and it changes the plastic behavior of the model. This is why it is recommended to use the Peerlings method I l o c = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadYgacaWGVbGaam4yaaWdaeqaaOWd biabg2da9iaaiodaaaa@3BDE@ .

  4. If I l o c = 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamysa8aadaWgaaWcbaWdbiaadYgacaWGVbGaam4yaaWdaeqaaOWd biabg2da9iaaiodaaaa@3BDE@ , the non-local Peerlings method will be used. For this method, the parameter Hchi is used. Only the non-local length Rlen is used. This method is simpler than the Micromorphic one. It introduces the non-local plastic strain in the softening variable kinetic equation (damage and temperature if thermal effects are considered): (13)
    f ˙ t = A  ε p nl ˙ Void nucleation + ( 1 f t )Tr( ε p nl ˙ ) Void growth ( high triaxiality )   +   K w f t w( θ ) S : ε p nl ˙ σ eq Shear nucleation (low triaxility) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GabmOza8aagaGaamaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabg2da 98aadaagaaqaa8qacaWGbbGaaiiOa8aadaWfGaqaa8qacqaH1oqzpa Waa0baaSqaa8qacaWGWbaapaqaa8qacaWGUbGaamiBaaaaa8aabeqa a8qacaGGzlaaaaWdaeaapeGaaeOvaiaab+gacaqGPbGaaeizaiaabc kacaqGUbGaaeyDaiaabogacaqGSbGaaeyzaiaabggacaqG0bGaaeyA aiaab+gacaqGUbaak8aacaGL44papeGaey4kaSYaaGbaaeaadaqada WdaeaapeGaaGymaiabgkHiTiaadAgapaWaaSbaaSqaa8qacaWG0baa paqabaaak8qacaGLOaGaayzkaaGaamivaiaadkhadaqadaWdaeaada WfGaqaaGqad8qacaWF1oWdamaaDaaaleaapeGaa8hCaaWdaeaapeGa a8NBaiaa=XgaaaaapaqabeaapeGaaiy2caaaaOGaayjkaiaawMcaaa WcbaWdauaabeqaceaaaeaapeGaaeOvaiaab+gacaqGPbGaaeizaiaa bckacaqGNbGaaeOCaiaab+gacaqG3bGaaeiDaiaabIgaa8aabaWdbm aabmaapaqaa8qacaqGObGaaeyAaiaabEgacaqGObGaaeiOaiaabsha caqGYbGaaeyAaiaabggacaqG4bGaaeyAaiaabggacaqGSbGaaeyAai aabshacaqG5baacaGLOaGaayzkaaaaaaGccaGL44pacaGGGcGaaeiO aiaabUcacaqGGcGaaeiOa8aadaagaaqaa8qacaWGlbWdamaaBaaale aapeGaam4DaaWdaeqaaOWdbiaadAgapaWaaSbaaSqaa8qacaWG0baa paqabaGcpeGaam4Damaabmaapaqaa8qacqaH4oqCaiaawIcacaGLPa aadaWcaaWdaeaapeGaa83uaiaacckacaGG6aWdamaaxacabaWdbiaa =v7apaWaa0baaSqaa8qacaWFWbaapaqaa8qacaWFUbGaa8hBaaaaa8 aabeqaa8qacaGGzlaaaaGcpaqaa8qacqaHdpWCpaWaaSbaaSqaa8qa caWGLbGaamyCaaWdaeqaaaaaaqaabeqaa8qacaqGtbGaaeiAaiaabw gacaqGHbGaaeOCaiaabckacaqGUbGaaeyDaiaabogacaqGSbGaaeyz aiaabggacaqG0bGaaeyAaiaab+gacaqGUbaabaGaaeikaiaabYgaca qGVbGaae4DaiaabccacaqG0bGaaeOCaiaabMgacaqGHbGaaeiEaiaa bMgacaqGSbGaaeyAaiaabshacaqG5bGaaeykaaaak8aacaGL44paaa a@BB48@
    (14)
    T ˙ =ω( ε p nl ˙ ) η ρ C p   σ ¯ ¯ : ε p nl ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabmiva8aagaGaa8qacqGH9aqpcqaHjpWDdaqadaWdaeaadaWfGaqa a8qacqaH1oqzpaWaa0baaSqaa8qacaWGWbaapaqaa8qacaWGUbGaam iBaaaaa8aabeqaa8qacaGGzlaaaaGccaGLOaGaayzkaaWaaSaaa8aa baWdbiabeE7aObWdaeaapeGaeqyWdiNaam4qa8aadaWgaaWcbaWdbi aadchaa8aabeaaaaGcpeGaaiiOa8aadaqdaaqaa8qacuaHdpWCpaGb aebaaaWdbiaacQdapaWaaCbiaeaaieWapeGaa8xTd8aadaqhaaWcba Wdbiaa=bhaa8aabaWdbiaa=5gacaWFSbaaaaWdaeqabaWdbiaacMTa aaaaaa@5288@

    This method is recommended since it is simple to identify the input parameters and does not modify the plastic behavior of the material.

  5. To set the non-local length parameter Rlen, you can choose:
    • Directly input the value of Rlen 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@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeadaqhaa WcbaGaamyzaaqaaiaad2eacaWGbbGaamiwaaaaaaa@3A49@ must be ignored and set to none.
    • Input the maximum mesh size L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeadaqhaa WcbaGaamyzaaqaaiaad2eacaWGbbGaamiwaaaaaaa@3A49@ for which results have reached mesh convergence. 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 Gaamita8aadaqhaaWcbaWdbiaadwgaa8aabaaaaaaa@3822@ 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 Rlen is calculated automatically according to the value of L e M A X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeadaqhaa WcbaGaamyzaaqaaiaad2eacaWGbbGaamiwaaaaaaa@3A49@ , and the input value of Rlen is ignored. For instance, if you want to get converged and mesh-independent results for a mesh size of 5 mm, 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.
  6. When a 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 6), 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 7). (15)
    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 6. Transverse strain incompatibility (local)


    Figure 7. 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@ .
  7. To create a specific damage output DAMA in ANIM and H3D files, the total damage is normalized by its rupture value: (16)
    D = f t f R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiraiabg2da9maalaaapaqaa8qacaWGMbWdamaaBaaaleaapeGa amiDaaWdaeqaaaGcbaWdbiaadAgapaWaaSbaaSqaa8qacaWGsbaapa qabaaaaaaa@3C7E@