4. Implementing Johnson-Cook damage in PAM-CRASH

The Johnson-Cook (JC) damage model is one of the most widely used ductile failure criteria in explicit crash simulation. In PAM-CRASH it is implemented through material law 32 (elastic-plastic with JC hardening) combined with a damage card that activates element erosion once the accumulated damage variable reaches unity. This section walks through the three practical pillars of a working implementation: identifying the five JC damage parameters from coupon data, wiring them into the correct PAM-CRASH cards, and controlling element deletion so that erosion does not introduce non-physical behavior.

1. The Johnson-Cook damage formulation in PAM-CRASH

The JC damage model defines a scalar damage indicator D that accumulates plastic strain weighted by the current stress triaxiality and strain rate:

D = Σ ( Δε_p / ε_f )

ε_f = [ D1 + D2·exp(D3·σ*) ] · [ 1 + D4·ln(ε̇*) ] · [ 1 + D5·T* ]

where:
  σ*  = σ_m / σ_eq        (stress triaxiality)
  ε̇*  = ε̇_p / ε̇_0        (dimensionless plastic strain rate)
  T*  = (T − T_room)/(T_melt − T_room)
  D1..D5 = damage parameters

Failure occurs when D reaches 1.0. In PAM-CRASH the damage evolution is tracked per integration point; when the criterion is met, the element is flagged for deletion at the end of the current time step. The five parameters D1–D5 are not independent of the plasticity constants A, B, n, C, and m used by material law 32, so identification must be done in a consistent sequence.

2. Parameter identification workflow

Parameter identification starts from the same set of tests used for JC plasticity, but the focus shifts to the fracture strain at different stress states. A practical sequence is:

  1. Smooth tensile specimen — gives the reference fracture strain at low triaxiality (σ* ≈ 0.33). This anchors D1.
  2. Notched tensile specimens (e.g., R = 1.0, 2.0, 4.0 mm) — provide fracture strains at increasing triaxiality. These constrain D2 and D3.
  3. Split-Hopkinson or high-rate tensile tests — supply fracture strain at elevated strain rates, fixing D4.
  4. Elevated-temperature tests — if thermal softening matters, D5 is fitted from the reduction in fracture strain with temperature.

A common simplification for crash applications at room temperature is to set D5 = 0 and fit D1–D4 only. For a typical automotive steel (e.g., DP780), a reasonable starting set is D1 = 0.12, D2 = 1.65, D3 = −1.80, D4 = −0.015, D5 = 0. These values are illustrative and must be re-identified for the actual material grade.

Always verify that the identified D1–D5 reproduce the experimental fracture strain within ±10% before proceeding to full-vehicle models. A mismatch here propagates directly into energy absorption predictions.

3. PAM-CRASH card setup

In the PAM-CRASH input deck, the JC damage model is activated through the material definition and a companion damage card. The essential cards are:

/MAT/32/1
  RO = 7.85e-9        $ density (t/mm^3)
  E  = 210000         $ Young's modulus (MPa)
  NU = 0.30           $ Poisson's ratio
  A  = 450            $ yield stress (MPa)
  B  = 780            $ hardening modulus (MPa)
  N  = 0.45           $ hardening exponent
  C  = 0.014          $ strain-rate coefficient
  M  = 1.05           $ thermal softening exponent
  TM = 1500           $ melting temperature (K)
  TR = 293            $ room temperature (K)

/DAM/1/1
  D1 = 0.12
  D2 = 1.65
  D3 = -1.80
  D4 = -0.015
  D5 = 0.0
  EPS0 = 1.0e-3       $ reference strain rate (1/s)

The /DAM card references the material ID and must be linked to the same part set. In the part definition, the damage flag is enabled with IDAM = 1. If the damage card is omitted, material law 32 behaves as pure JC plasticity with no erosion.

4. Element deletion criteria and controls

Element deletion in PAM-CRASH is not automatic when D = 1. Several controls govern when and how elements are removed:

ControlCard / keywordTypical valueEffect
Damage threshold/DAM DCRIT1.0D at which erosion triggers
Minimum time step/DT/NODA1e-7 sPrevents instability after deletion
Element deletion flag/PART IDEL1Enables deletion for the part
Shell thickness criterion/MAT/32 THIN0.1Deletes shells below 10% thickness

Two practical pitfalls deserve attention. First, deleting elements removes mass and load path; if too many elements erode in a load-bearing region, the structure may appear artificially compliant. Second, deletion can create free surfaces that expose interior elements to contact, sometimes causing penetration. A common mitigation is to set DCRIT slightly above 1.0 (e.g., 1.05) or to use a damage-based stiffness degradation before deletion.

Never enable element deletion without checking the energy balance. A sudden drop in internal energy or a spike in hourglass energy after erosion indicates that the deletion criteria are too aggressive or that the mesh is too coarse in the failure zone.

5. Verification and troubleshooting

After running a single-element or coupon-level simulation, inspect the following:

  • Force–displacement curve — should match the experimental curve up to fracture, then drop sharply. A gradual drop suggests damage is spreading too slowly (DCRIT too high or mesh too coarse).
  • Damage contour — D should localize in the necking region, not diffuse across the specimen.
  • Energy balance — internal energy should decrease monotonically after fracture; hourglass energy should remain below 5% of internal energy.
  • Time step — after deletion, the critical time step may drop; check that it does not fall below the minimum specified in /DT/NODA.

If the simulation terminates prematurely, reduce the time step or increase the minimum element size. If fracture occurs too early, re-examine the identified D1–D5 against the experimental fracture strain. If fracture occurs too late, check that the damage card is actually linked to the part and that IDAM is set.

Johnson-Cook Damage Implementation Flow in PAM-CRASH 1. Parameter Identification Smooth tensile → D1 Notched tensile → D2, D3 High-rate tests → D4 Elevated temp → D5 2. Card Setup /MAT/32 (A,B,n,C,m) /DAM (D1–D5, EPS0) /PART IDAM=1, IDEL=1 /DT/NODA min time step 3. Damage Accumulation D = Σ Δε_p / ε_f ε_f = f(σ*, ε̇*, T*) Track per integration point D ≥ DCRIT → flag element 4. Element Deletion Remove element at end of step Update mass & load path Check energy balance Monitor hourglass energy 5. Verification Force–disp. match, D contour, energy balance, time step Re-identify if mismatch > 10% Key Parameters at a Glance D1: reference fracture strain D2: triaxiality sensitivity D3: triaxiality curvature D4: strain-rate sensitivity D5: temperature sensitivity DCRIT: deletion threshold (≈1.0) EPS0: reference strain rate IDAM: damage activation flag IDEL: deletion activation flag

6. Practical checklist for a new material

When introducing a new material into a PAM-CRASH crash model, follow this sequence to avoid the most common errors:

  1. Fit JC plasticity constants A, B, n, C, m from tensile tests at multiple rates and temperatures.
  2. Fit D1–D5 from fracture strains at varying triaxiality and rate.
  3. Write the /MAT/32 and /DAM cards with consistent units (mm, t, s, MPa).
  4. Set IDAM = 1 and IDEL = 1 on the relevant part cards.
  5. Run a single-element or coupon simulation and compare force–displacement and fracture strain.
  6. Check energy balance and hourglass energy; adjust DCRIT or mesh size if needed.
  7. Scale up to component level, then full vehicle, re-verifying at each stage.

The JC damage model is powerful but sensitive to parameter quality and mesh resolution. A well-identified parameter set combined with disciplined card setup and deletion controls will produce robust, physically meaningful crash simulations. The next step after implementation is calibration against full-scale test data, which is covered in the following chapter.

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