4. Material models for crashworthiness

In a full-vehicle PAM-CRASH model, the accuracy of the predicted occupant compartment intrusion, belt force, and airbag firing time depends far more on the constitutive response of the sheet-steel and aluminum closures than on the mesh density of the body-in-white. A crashworthiness material card must therefore reproduce four distinct physical behaviors simultaneously: elastic stiffness, plastic hardening, strain-rate sensitivity, and ductile fracture. This section explains how those behaviors are represented in PAM-CRASH, how the required parameters are obtained from coupon and component tests, and how to judge whether a calibrated card is fit for sign-off.

Elastoplasticity: The Backbone of the Card

For metals, the standard choice is an isotropic elastoplastic model (in PAM-CRASH typically material law 1 or 2, or the more capable law 103/104 for shells). The elastic branch is defined by Young's modulus E, Poisson's ratio ν, and density ρ. The plastic branch is defined by a true stress–true plastic strain curve, entered as a table of pairs. Two conventions must be respected: the curve must be true stress versus true plastic strain, not engineering stress–strain, and the first point must correspond to the initial yield stress at zero plastic strain.

Conversion from the tensile test output is straightforward but frequently done incorrectly. Given engineering stress σ_eng and engineering strain ε_eng:

σ_true = σ_eng * (1 + ε_eng)
ε_true = ln(1 + ε_eng)
ε_pl   = ε_true - σ_true / E

For a typical DP590 sheet, the initial yield is around 380 MPa, the ultimate true stress reaches roughly 950 MPa at a true plastic strain near 0.15, and the curve is extrapolated to the failure strain using a power-law or linear extension. The extrapolation matters: shell elements in a crash model routinely exceed the uniform elongation of the coupon, so an unconservative extrapolation will under-predict tearing.

Strain-Rate Dependence

Automotive steels are strain-rate sensitive. At the strain rates typical of a 50 km/h frontal impact (10²–10³ s⁻¹), the flow stress of a mild steel can rise by 50–80% relative to quasi-static, while a high-strength steel may rise by only 20–30%. PAM-CRASH captures this through a rate-dependent multiplier on the yield surface, most commonly the Cowper–Symonds model:

σ_dynamic / σ_static = 1 + (ε̇ / D)^(1/p)

where D and p are calibrated constants. Alternatively, a tabulated rate-scaling factor versus strain rate can be supplied directly, which is preferable when the test matrix covers several decades of rate. The calibration data come from split-Hopkinson pressure bar (SHPB) tests at 10²–10³ s⁻¹ and from intermediate-rate servo-hydraulic tests at 1–100 s⁻¹, anchored to quasi-static values at 10⁻³ s⁻¹.

Failure Criteria

Ductile fracture in crash-loaded sheet metal is driven by stress triaxiality and plastic strain accumulation. Three criteria are common in PAM-CRASH:

• Equivalent plastic strain limit — simplest, but ignores stress state and tends to be mesh-sensitive.
• Johnson–Cook damage — accumulates damage D = Σ Δε_pl / ε_f(η, ε̇) where the fracture strain depends on triaxiality η and rate. Failure occurs when D = 1.
• GISSMO — a phenomenological damage model that regularizes the softening branch and is well suited to shell meshes. It requires a fracture locus (equivalent plastic strain at failure versus triaxiality) plus a damage exponent.

For sign-off work, GISSMO is preferred for closures and rails because it handles the post-necking softening without pathological mesh dependence. The fracture locus is built from a set of coupon geometries that sample different triaxialities: a smooth tensile specimen (η ≈ 0.33), a notched tensile specimen (η ≈ 0.5–0.7), and a shear specimen (η ≈ 0).

Calibration Workflow: Coupon to Component

Calibration proceeds in three tiers, each validating the previous one before moving on.

1. Coupon level. Fit the hardening curve to tensile data, fit D and p to SHPB data, and fit the fracture locus to the notched and shear specimens. Acceptance: predicted force–displacement curves within 5% of test peak force and within 10% of displacement at fracture.
2. Component level. Validate on a hat-section axial crush or a three-point bend of a closed section. This exposes mesh-size sensitivity and the interaction between rate effects and failure. Acceptance: crush force within 8% and fold pattern matching the test photograph.
3. Sub-system level. A bumper beam or front rail section under sled impact. This is the last checkpoint before the full-vehicle model.

The following SVG summarizes how the four material behaviors map onto the calibration tiers and what observable output validates each tier.

Practical Calibration Example: DP590 Front Rail

Consider a DP590 front rail section calibrated for a 50 km/h frontal sled. The quasi-static tensile test gives E = 210 GPa, ν = 0.3, initial yield 380 MPa, and a true stress of 950 MPa at ε_pl = 0.15. SHPB tests at 500 s⁻¹ and 1500 s⁻¹ give flow-stress multipliers of 1.28 and 1.42 respectively, which fit Cowper–Symonds with D = 6500 s⁻¹ and p = 3.4. The fracture locus from notched and shear coupons yields ε_f = 0.55 at η = 0.33, ε_f = 0.38 at η = 0.6, and ε_f = 0.72 at η = 0.05.

After running the hat-section crush at 5 m/s, the predicted peak force is 62 kN against a measured 58 kN — a 6.9% over-prediction, within the 8% acceptance band. However, the simulated fold initiates 15 mm earlier than the test. The typical cause is an over-stiff rate multiplier at the intermediate strain rate; reducing D to 5500 s⁻¹ shifts the fold initiation downstream and brings the force–displacement curve into alignment. This is the kind of iteration that must be completed before the card is released to the full-vehicle model.

Sign-Off Checklist for a Material Card

The card is only released to the full-vehicle model once all five rows are satisfied. Any subsequent change to the hardening curve or fracture locus invalidates the component validation and requires re-running Tier 2. This discipline is what separates a material card that predicts occupant injury metrics reliably from one that merely produces plausible-looking deformation plots.