Tabulated Failure Model /FAIL/TAB1

In Radioss, /FAIL/TAB1 is the most sophisticated failure model for ductile material. The plastic failure strain can be defined as a function of: stress triaxiality, strain rate, Lode angle, element size, temperature, and instability strain.

Damage is accumulated based on user-defined functions. The functionality of this failure model will be described starting with the most basic input to the most complex options.

Plastic Failure Strain

Similar to /FAIL/JOHNSON and /FAIL/BIQUAD, it is possible to define a curve that represents the plastic failure strain, ε f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamOzaaqabaaaaa@391D@ , as a function of stress triaxiality, σ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda ahaaWcbeqaaiaacQcaaaaaaa@38FD@ . Unlike /FAIL/JOHNSON and /FAIL/BIQUAD where the failure strain curve is defined using parameters in predefined equations, in /FAIL/TAB1 any number of discrete points can be entered to create an arbitrary function that represents the failure strain curve. This curve is defined using the /TABLE entity and is referenced in table_ID1. This method can be used for shell and solid elements.


Figure 1. Material Failure Curve Defined using Discrete Points with a Local Maximum
Example /TABLE, dimension=1
Input for failure plastic-strain versus triaxiality using table_ID1.
/TABLE/1/4711
failure plastic-strain vs triaxiality 
#dimension
         1
#        Triaxiality      Failure_Strain     
             -0.7000              0.3386
             -0.6000              0.3068
             -0.5000              0.2794
             -0.4000              0.2558
             -0.3333              0.2419
             -0.3000              0.2355
             -0.2000              0.2180
             -0.1000              0.2029
              0.0000              0.1900
              0.1000              0.1789
              0.2000              0.1693
              0.3000              0.1610
              0.3333              0.1585
              0.4000              0.1539
              0.5000              0.1478
              0.6000              0.1425
              0.7000              0.1380

Strain Rate Dependency

/FAIL/TAB1 can also include the influence of strain rate on material failure. For this case the /TABLE must be defined such that, the first dimension is the function ID for the failure curve and the second dimension is the strain rate where that failure curve is applied.

Example /TABLE, dimension=2
/TABLE/1/4711
failure plastic-strain vs triaxiality and strain rate
#dimension
         2
#   FCT_ID                   strain_rate  
      3000                          1E-4  
      3001                           0.1 
      3002                           1.0  
/FUNCT/3000
failure plastic-strain vs triaxiality 
#        Triaxiality      Failure_Strain     
             -0.7000              0.3386
             -0.6000              0.3068
             -0.5000              0.2794
             -0.4000              0.2558
             -0.3333              0.2419
             -0.3000              0.2355
             -0.2000              0.2180
             -0.1000              0.2029
              0.0000              0.1900
              0.1000              0.1789
              0.2000              0.1693
              0.3000              0.1610
              0.3333              0.1585
              0.4000              0.1539
              0.5000              0.1478
              0.6000              0.1425
              0.7000              0.1380
/FUNCT/3001
failure plastic-strain vs triaxiality 
#        Triaxiality      Failure_Strain     
                -0.7             0.27088
                -0.6             0.24544
                -0.5             0.22352
                -0.4             0.20464
             -0.3333            0.19352
                -0.3              0.1884
                -0.2              0.1744
                -0.1             0.16232
                   0                0.152
                 0.1             0.14312
                 0.2             0.13544
                 0.3              0.1288
             0.3333              0.1268
                 0.4             0.12312
                 0.5             0.11824
                 0.6               0.114
                 0.7              0.1104

Lode Angle with Solid Elements

For solid elements, the failure strain can also depend on the 3D stress state defined using the Lode angle.

This can be included by adding failure strain as a function of Lode angle parameter in the /TABLE entity referenced by table_ID1. For shells elements, it is only necessary to define the failure strain as a function of stress triaxiality. But for solid elements, it is more accurate to include the failure strain as a function of stress triaxiality and Lode angle.

In Radioss, the Lode angle is entered using a normalized and dimensionless Lode Angle parameter ξ and is defined here.

The stress state at a point, P, could be expressed with principle stresses ( σ 1 , σ 2 , σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeo8aZ9aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGa eq4Wdm3damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacYcacqaHdp WCpaWaaSbaaSqaa8qacaaIZaaapaqabaaaaa@409E@ ) or could also be expressed using stress invariants ( I 1 , J 2 , J 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadMeapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaa dQeapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaaiilaiaadQeapa WaaSbaaSqaa8qacaaIZaaapaqabaaaaa@3DC1@ ). The advantage of using stress invariants is that they are constant and do not depend on the orientation of the coordinate system. In Figure 2, to correctly describe stress state of point P σ 1 , σ 2 , σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeo8aZ9aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGa eq4Wdm3damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacYcacqaHdp WCpaWaaSbaaSqaa8qacaaIZaaapaqabaaaaa@409E@ using stress invariants, the magnitude of O O ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHpbGaaC 4taiaacEcaaaa@38BA@ as:(1) 3 σ m = 3 3 I 1
Where,
σ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamyBaaqabaaaaa@3940@
Mean stress
I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadMeapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@3862@
First stress invariant I 1 = σ 1 + σ 2 + σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadMeapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyypa0Ja ae4Wd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHRaWkcaqGdp WdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabgUcaRiaabo8apaWa aSbaaSqaa8qacaaIZaaapaqabaaaaa@4297@
O O ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHpbGaaC 4taiaacEcaaaa@38BA@ is in the hydrostatic axis, which means the principle stress in this axis is the same ( σ 1 = σ 2 = σ 3 ). | O O ' | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaabdaqaai aah+eacaWHpbGaai4jaaGaay5bSlaawIa7aaaa@3BDC@ is the hydrostatic pressure.


Figure 2. Stress State of P
The magnitude of O ' P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHpbGaai 4jaiaahcfaaaa@38BB@ is:(2) 2 J 2 = 2 3 σ V M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaaqaai aaikdacaWGkbWaaSbaaSqaaiaaikdaaeqaaaqabaGccqGH9aqpdaGc aaqaamaalaaabaGaaGOmaaqaaiaaiodaaaaaleqaaOGaeq4Wdm3aaS baaSqaaiaadAfacaWGnbaabeaaaaa@3F3B@

Here, J 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbWaaS baaSqaaiaaikdaaeqaaaaa@3815@ is the second invariant of deviatoric stress s ( s = σ - p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHZbWaae WaaeaacaWHZbGaaCypaiaaho8acaWHTaGaaCiCaaGaayjkaiaawMca aaaa@3DA3@ with J 2 = 1 2 ( S 1 2 + S 2 2 + S 3 2 ) = 1 2 [ ( σ 1 σ 2 ) 2 + ( σ 2 σ 3 ) 2 + ( σ 3 σ 1 ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOm aaaacaGGOaGaam4uamaaBaaaleaacaaIXaaabeaakmaaCaaaleqaba GaaGOmaaaakiabgUcaRiaadofadaWgaaWcbaGaaGOmaaqabaGcdaah aaWcbeqaaiaaikdaaaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodaae qaaOWaaWbaaSqabeaacaaIYaaaaOGaaiykaiabg2da9maalaaabaGa aGymaaqaaiaaikdaaaWaamWaaeaadaqadaqaaiabeo8aZnaaBaaale aacaaIXaaabeaakiabgkHiTiabeo8aZnaaBaaaleaacaaIYaaabeaa aOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabm aabaGaeq4Wdm3aaSbaaSqaaiaaikdaaeqaaOGaeyOeI0Iaeq4Wdm3a aSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaaca aIYaaaaOGaey4kaSYaaeWaaeaacqaHdpWCdaWgaaWcbaGaaG4maaqa baGccqGHsislcqaHdpWCdaWgaaWcbaGaaGymaaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaaakiaawUfacaGLDbaaaaa@66BC@ .

To identify the point P, the angle in the circular plane must be calculated. This angle is called the Lode Angle θ :(3) cos ( 3 θ ) = 27 2 J 3 σ V M 3 = 3 3 2 J 3 J 2 3 / 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaabogacaqGVbGaae4Camaabmaapaqaa8qacaaIZaGaeqiUdeha caGLOaGaayzkaaGaeyypa0ZaaSaaa8aabaWdbiaaikdacaaI3aaapa qaa8qacaaIYaaaamaalaaapaqaa8qacaWGkbWdamaaBaaaleaapeGa aG4maaWdaeqaaaGcbaWdbiabeo8aZ9aadaqhaaWcbaWdbiaadAfaca WGnbaapaqaa8qacaaIZaaaaaaakiabg2da9maalaaapaqaa8qacaaI ZaWaaOaaa8aabaWdbiaaiodaaSqabaaak8aabaWdbiaaikdaaaWaaS aaa8aabaWdbiaadQeapaWaaSbaaSqaa8qacaaIZaaapaqabaaakeaa peGaamOsa8aadaqhaaWcbaWdbiaaikdaa8aabaWdbiaaiodacaGGVa GaaGOmaaaaaaaaaa@51E5@
with 0 θ π 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaaicdacqGHKjYOcqaH4oqCcqGHKjYOdaWcaaWdaeaapeGaeqiW dahapaqaa8qacaaIZaaaaaaa@3F21@ and J 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadQeapaWaaSbaaSqaa8qacaaIZaaapaqabaaaaa@3865@ is the third invariant of deviatoric stress calculated as:(4) J 3 = S 1 S 2 S 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaadQeapaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeGaeyypa0Ja ae4ua8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaqGtbWdamaaBa aaleaapeGaaGOmaaWdaeqaaOWdbiaabofapaWaaSbaaSqaa8qacaaI Zaaapaqabaaaaa@3F7D@
In /FAIL/TAB1 a normalized and dimensionless Lode Angle parameter ξ with the range ( 1 ξ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbmaabmaapaqaa8qacqGHsislcaaIXaGaeyizIm6daiabe67a49qa cqGHKjYOcaaIXaaacaGLOaGaayzkaaaaaa@3FD6@ is used and defined as:(5) ξ = cos ( 3 θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabe67a4jabg2da9iaabogacaqGVbGaae4Camaabmaapaqaa8qa caaIZaGaeqiUdehacaGLOaGaayzkaaaaaa@4031@


Figure 3. Stress State with Different Lode Angle
The special characters for Lode Angle θ and Load Angle Parameter ξ are:
Lode Angle Parameter ξ Lode Angle θ Stress State
1 0 Uniaxial tension + hydrostatic pressure (triaxial tension or axisymmetric tension)
0 30 Pure shear + hydrostatic pressure

(plane strain)

-1 60 Uniaxial compression + hydrostatic pressure

(axisymmetric compression)

A failure strain surface can be created from the stress triaxiality and Lode angle failure data.


Figure 4. 3D Failure Surface
The material failure surface could be created using the following material tests.


Figure 5. Stress State and Lode Angle for Various Tests

Example /TABLE, dimension=3

Input for failure plastic-strain versus triaxiality, strain rate, and Lode angle using table_ID1
/TABLE/1/4711
failure plastic-strain vs triaxiality and strain rate
#dimension
         3
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#   FCT_ID                   strain_rate          Lode_angle  
      3000                          1E-4                  -1
      3001                           0.1                   0
      3002                           1.0                   1
....


Figure 6. Failure Surface when table_ID1 references a /TABLE with dimension=3

Scaling Failure Strain

Material failure based on temperature and element size can be considering in /FAIL/TAB1 by including functions that scale the failure strain depending on the element size and/or temperature using:(6) ε f = X s c a l e 1 f ( σ * , ε ˙ , ξ ) f a c t o r e l f a c t o r T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamOzaaqabaGccqGH9aqpcaWGybGaam4CaiaadogacaWG HbGaamiBaiaadwgacaaIXaGaeyyXICTaamOzaiaacIcacqaHdpWCda ahaaWcbeqaaiaacQcaaaGccaGGSaGafqyTduMbaiaacaGGSaGaeqOV dGNaaiykaiabgwSixlaadAgacaWGHbGaam4yaiaadshacaWGVbGaam OCamaaBaaaleaacaWGLbGaamiBaaqabaGccqGHflY1caWGMbGaamyy aiaadogacaWG0bGaam4BaiaadkhadaWgaaWcbaGaamivaaqabaaaaa@5F52@
Where,
X s c a l e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGybGaam 4CaiaadogacaWGHbGaamiBaiaadwgacaaIXaaaaa@3C97@
General scaling factor
f a c t o r e l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaam yyaiaadogacaWG0bGaam4BaiaadkhadaWgaaWcbaGaamyzaiaadYga aeqaaaaa@3E02@
Scale factor based on element size
f a c t o r T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaam yyaiaadogacaWG0bGaam4BaiaadkhadaWgaaWcbaGaamivaaqabaaa aa@3D00@
Scale factor based on temperature

Element Length Dependency

In numerical simulations, the element size will affect the material failure. Using the same failure parameters, a coarse mesh will fail earlier than a fine mesh.


Figure 7. Influence of Element Mesh Size on Material Failure in Numerical Simulation
To account for the variation in results based on mesh size, the element size scale factor can be defined to scale the failure strain based on mesh element size. The scale factor defined in Equation 6 is:(7) f a c t o r e l = F s c a l e e l f e l ( S i z e e l E l _ r e f ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaam yyaiaadogacaWG0bGaam4BaiaadkhadaWgaaWcbaGaamyzaiaadYga aeqaaOGaeyypa0JaamOraiaadohacaWGJbGaamyyaiaadYgacaWGLb WaaSbaaSqaaiaadwgacaWGSbaabeaakiabgwSixlGacAgadaWgaaWc baGaamyzaiaadYgaaeqaaOWaaeWaaeaadaWcaaqaaiaadofacaWGPb GaamOEaiaadwgadaWgaaWcbaGaamyzaiaadYgaaeqaaaGcbaGaamyr aiaadYgacaGGFbGaamOCaiaadwgacaWGMbaaaaGaayjkaiaawMcaaa aa@5899@
Where,
F s c a l e e l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaam 4CaiaadogacaWGHbGaamiBaiaadwgadaWgaaWcbaGaamyzaiaadYga aeqaaaaa@3DD1@
Element size function scale factor
f e l ( S i z e e l E l _ r e f ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiaadwgacaWGSbaabeaakmaabmaabaWaaSaaaeaacaWGtbGa amyAaiaadQhacaWGLbWaaSbaaSqaaiaadwgacaWGSbaabeaaaOqaai aadweacaWGSbGaai4xaiaadkhacaWGLbGaamOzaaaaaiaawIcacaGL Paaaaaa@461E@
failure strain scale factor as a function of the normalized element size (defined via fct_IDel), and E l _ r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbGaam iBaiaac+facaWGYbGaamyzaiaadAgaaaa@3BC8@ is the reference element size used to normalize the element size
For example, a mesh size of 2mm was used in a material validation simulation. After the initial validation, the same simulation is reran using E l _ r e f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbGaam iBaiaac+facaWGYbGaamyzaiaadAgaaaa@3BC8@ =2 with different element sizes to determine the correct scaling factor for each element size. Using the results of the second validation, a failure strain scale factor function is constructed as (Figure 8):


Figure 8. Example of Element Size Scale Factor Function fct_IDel in /FAIL/TAB1

Temperature Dependency

To account for how the temperature affects material failure, failure strain values defined in Equation 6 can be scaled using a temperature scale factor function:(8) f a c t o r T = F s c a l e T f T ( T s t a r t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaam yyaiaadogacaWG0bGaam4BaiaadkhadaWgaaWcbaGaamivaaqabaGc cqGH9aqpcaWGgbGaam4CaiaadogacaWGHbGaamiBaiaadwgadaWgaa WcbaGaamivaaqabaGccqGHflY1ciGGMbWaaSbaaSqaaiaadsfaaeqa aOWaaeWaaeaacaWGubWaaSbaaSqaaiaadohacaWG0bGaamyyaiaadk hacaWG0baabeaaaOGaayjkaiaawMcaaaaa@502F@
Where,
F s c a l e T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaam 4CaiaadogacaWGHbGaamiBaiaadwgadaWgaaWcbaGaamivaaqabaaa aa@3CCF@
Temperature function scale factor
f T ( T s t a r t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiaadsfaaeqaaOWaaeWaaeaacaWGubWaaSbaaSqaaiaadoha caWG0bGaamyyaiaadkhacaWG0baabeaaaOGaayjkaiaawMcaaaaa@3FB8@
Failure strain scale factor as a function temperature defined via fct_IDT.

Scaling the failure strain based on temperature works with any material that has /HEAT/MAT defined or materials that include thermos plasticity such as /MAT/LAW2 (PLAS_JOHNS).

In fct_IDT, the temperature is defined relative to the melting and initial temperature.(9) T = T T i n i T m e l t T i n i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaW baaSqabeaacqGHxiIkaaGccqGH9aqpdaWcaaqaaiaadsfacqGHsisl caWGubWaaSbaaSqaaiaadMgacaWGUbGaamyAaaqabaaakeaacaWGub WaaSbaaSqaaiaad2gacaWGLbGaamiBaiaadshaaeqaaOGaeyOeI0Ia amivamaaBaaaleaacaWGPbGaamOBaiaadMgaaeqaaaaaaaa@48AD@


Figure 9. Influence of Temperature on Material Failure with fct_IDT in /FAIL/TAB1

Element Failure Treatment

In /FAIL/TAB1 an accumulative damage model is used. The damage can be output for contour plotting using the Engine options, /ANIM/SHELL/DAMA or /ANIM/BRICK/DAMA.

The accumulated damage is calculated by first calculating a damage increment:(10) Δ D = Δ ε p ε f n D p ( 1 1 n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WGebGaeyypa0ZaaSaaaeaacqqHuoarcqaH1oqzdaWgaaWcbaGaamiC aaqabaaakeaacqaH1oqzdaWgaaWcbaGaamOzaaqabaaaaOGaeyyXIC TaamOBaiabgwSixlaadseadaWgaaWcbaGaamiCaaqabaGcdaahaaWc beqaamaabmaabaGaaGymaiabgkHiTmaalaaabaGaaGymaaqaaiaad6 gaaaaacaGLOaGaayzkaaaaaaaa@4D3A@
Where,
Δ ε p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGuoGaeq yTdu2aaSbaaSqaaiaadchaaeqaaaaa@3A40@
The change in plastic strain of the integration point
ε f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamOzaaqabaaaaa@391C@
Plastic failure strain for the current stress triaxiality
D p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadchaaeqaaaaa@3848@ and n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3751@
Damage parameters
The accumulated damage is:(11) D = Δ D D c r i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey ypa0ZaaSaaaeaadaaeabqaaiabfs5aejaadseaaSqabeqaniabggHi LdaakeaacaWGebWaaSbaaSqaaiaadogacaWGYbGaamyAaiaadshaae qaaaaaaaa@4146@
Where, D c r i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadogacaWGYbGaamyAaiaadshaaeqaaaaa@3B1A@ is defined in /FAIL/TAB1 with a recommended between 0 and 1. The elements fail when D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey yzImRaaGymaaaa@39A9@ which means Δ D D c r i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaaeabqaai abfs5aejaadseaaSqabeqaniabggHiLdGccqGHLjYScaWGebWaaSba aSqaaiaadogacaWGYbGaamyAaiaadshaaeqaaaaa@412D@ .


Figure 10. Influence of D c r i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadogacaWGYbGaamyAaiaadshaaeqaaaaa@3B1A@ in /FAIL/TAB1

It is also interesting to understand the influence of the damage accumulation parameter, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3751@ in Equation 10.

When n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3751@ =1 damage is linear, otherwise the damage curve is nonlinear.


Figure 11. Influence of Damage Parameter n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3751@

Material Instability (Diffuse Necking)

In a tensile test, a material will reach a maximum engineering stress and then the stress will decrease or sometimes called soften. This maximum engineering stress point is called the necking point. After necking, the true stress actually increases, due to the decrease in the cross sectional area. This is called diffuse necking. In sheet metal, thickness thinning or localized necking in diffuse area may appear if the material continues to be loaded in the tensile direction. The diffuse necking normally appears in stress triaxiality ranges of 0 < σ * 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa WdbiaaicdacqGH8aapcqaHdpWCpaWaaWbaaSqabeaapeGaaiOkaaaa k8aacqGHKjYOdaWcaaqaaiaaikdaaeaacaaIZaaaaaaa@3E51@ and localized necking in the range of 1 3 σ * 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaai aaigdaaeaacaaIZaaaaabaaaaaaaaapeGaeyizImQaeq4Wdm3damaa CaaaleqabaWdbiaacQcaaaGcpaGaeyizIm6aaSaaaeaacaaIYaaaba GaaG4maaaaaaa@3FD0@ .


Figure 12. Sketch of Diffuse Necking and Localized

In /FAIL/TAB1 it is possible to account for the material instability (diffuse necking) with options table_ID2, or Inst_start and Fad_exp.

The material will start to weaken due to diffuse necking when the instability strain is reached. The reduced stress in the material is:(12) σ r e d u c e d = σ ( 1 ( D i n s t a b i l i t y i n s t _ s t a r t 1 i n s t _ s t a r t ) F a d _ exp ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda WgaaWcbaGaamOCaiaadwgacaWGKbGaamyDaiaadogacaWGLbGaamiz aaqabaGccqGH9aqpcqaHdpWCcqGHflY1daqadaqaaiaaigdacqGHsi sldaqadaqaamaalaaabaGaamiramaaBaaaleaacaWGPbGaamOBaiaa dohacaWG0bGaamyyaiaadkgacaWGPbGaamiBaiaadMgacaWG0bGaam yEaaqabaGccqGHsislcaWGPbGaamOBaiaadohacaWG0bGaai4xaiaa dohacaWG0bGaamyyaiaadkhacaWG0baabaGaaGymaiabgkHiTiaadM gacaWGUbGaam4CaiaadshacaGGFbGaam4CaiaadshacaWGHbGaamOC aiaadshaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGgbGaamyyai aadsgacaGGFbGaciyzaiaacIhacaGGWbaaaaGccaGLOaGaayzkaaaa aa@7031@
Where,(13) D i n s t a b i l i t y = Δ ε p ε f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadMgacaWGUbGaam4CaiaadshacaWGHbGaamOyaiaadMga caWGSbGaamyAaiaadshacaWG5baabeaakiabg2da9maaqaeabaWaaS aaaeaacqqHuoarcqaH1oqzdaWgaaWcbaGaamiCaaqabaaakeaacqaH 1oqzdaWgaaWcbaGaamOzaaqabaaaaaqabeqaniabggHiLdaaaa@4BD6@

with ε f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamOzaaqabaaaaa@391D@ being the diffuse necking strain based on the current stress triaxiality.

The strain at which instability starts could be either input with a curve (blue curve in Figure 13) using table_ID2 or input as a constrain strain using the option Inst_start.

In a uniaxial tensile test σ * = 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiabeo8aZ9aadaahaaWcbeqaa8qacaGGQaaaaOWdaiabg2da9maa laaabaGaaGymaaqaaiaaiodaaaaaaa@3BE3@ .
  • If material instability is not included then damage is calculated using the red failure curve in Figure 13
  • If material instability is included and modeled using the curve input (table_ID2) then:
    • Damage due to diffuse necking starts when the plastic strain defined by the blue curve in Figure 13 is reached.
    • Damage due to diffuse necking is linear if Fad_exp=1 is used. Increasing the Fad_exp leads to more energy dissipated during damage. Figure 13 shows the influence of Fad_exp from 1 to 10 in stress-strain curve. It is recommend to use a Fad_exp value of 5 to 10.
    • Once the strain reaches the red curve the element fails.


      Figure 13. Influence of Parameter Fad_exp and Material Instability Region
  • If only Inst_start is used without curve input in table_ID2, then the diffuse necking plastic strain is the constant value, Inst_start, for all stress triaxiality.


    Figure 14. Constant Inst_start Describing Diffuse Necking

Currently diffuse necking (material instability) in /FAIL/TAB1 can only be used with material law numbers > 28.

1 Wierzbicki, Tomasz, "Addendum to the Research Proposal on Fracture of Advanced High Strength Steels", page 19, January 2007.
2 Wierzbicki, Tomasz. "Fracture of AHSS Sheets–Addendum to the Research Proposal on Fracture of Advanced High Strength Steels." Impact and Crashworthiness Laboratory, Massachusetts Institute of Technology (2007).