2. Material failure and damage fundamentals in PAM-CRASH

Ductile fracture in metals is not a single event but a progressive process: voids nucleate, grow, and finally coalesce into a crack. PAM-CRASH captures this process through two complementary layers — a fracture criterion that decides when a material point fails, and a damage evolution variable that tracks how much of the load-carrying capacity has been consumed before that point. Both layers depend on the local stress state, which is why stress triaxiality and the Lode parameter appear in almost every modern failure card.

Stress triaxiality: the hydrostatic driver

Stress triaxiality is the dimensionless ratio of mean (hydrostatic) stress to the von Mises equivalent stress:

eta = sigma_m / sigma_vm
sigma_m  = (sigma_1 + sigma_2 + sigma_3) / 3
sigma_vm = sqrt( 0.5 * [ (s1-s2)^2 + (s2-s3)^2 + (s3-s1)^2 ] )

Here sigma_1, sigma_2, sigma_3 are the principal stresses and s1, s2, s3 the principal deviatoric stresses. The sign convention matters: tension is positive, so a uniaxial tensile test gives eta = 1/3, pure shear gives eta = 0, and a notched or pre-cracked specimen can reach eta = 0.6–1.0 or higher. High positive triaxiality opens voids; negative triaxiality (compression) closes them and suppresses fracture. This is why a smooth tensile coupon and a notched coupon of the same steel fail at very different plastic strains.

The Lode parameter: the third invariant

Triaxiality alone cannot distinguish between axisymmetric tension and pure shear, because both can share the same eta. The Lode parameter resolves the shape of the stress state:

L = (2*sigma_2 - sigma_1 - sigma_3) / (sigma_1 - sigma_3)   [-1, +1]

L = +1 corresponds to axisymmetric tension, L = 0 to generalized shear, and L = -1 to axisymmetric compression. Many PAM-CRASH failure models (e.g., extended Mohr–Coulomb or Bai–Wierzbicki style criteria) use the pair (eta, L) — or equivalently (eta, theta_bar) where theta_bar is the normalized Lode angle — to define a fracture locus. Ignoring L is a common source of over-prediction in shear-dominated crash modes such as spot-weld pull-out or shear-dominated bending.

Fracture strain and the fracture locus

The fracture strain epsilon_f is the equivalent plastic strain at the onset of fracture. In a calibrated model it is not a constant but a function of the stress state:

epsilon_f = f(eta, L)   or   epsilon_f = f(eta, theta_bar)

A typical calibration uses at least three specimen geometries — smooth round bar (eta ≈ 0.33), notched round bar (eta ≈ 0.6–0.8), and a shear or flat-grooved specimen (eta ≈ 0, L ≈ 0) — to fit the locus. In PAM-CRASH this is entered through material cards such as /MAT/PLAST with a failure option, or dedicated damage cards like /MAT/DAMAGE / /FAIL/... depending on the release. The plastic hardening curve (sigma_y vs. epsilon_p) is defined separately; the failure card only supplies the fracture criterion.

Damage evolution variables

Instead of an instantaneous on/off switch, PAM-CRASH uses a scalar damage variable D that accumulates with plastic straining:

D = integral( d(epsilon_p) / epsilon_f(eta, L) )
Fracture initiates when D reaches 1.0

Once D = 1, the element does not vanish instantly. A damage evolution law degrades the stiffness and strength over a characteristic length, typically using a fracture energy G_f to regularize mesh dependence:

sigma = (1 - D) * sigma_undamaged
G_f = integral( sigma d(epsilon_p) )   [energy per unit area]

This softening phase is what makes the difference between a mesh-sensitive brittle deletion and a physically meaningful crack path. In PAM-CRASH the softening slope is often controlled by G_f and the element characteristic length L_c, giving a displacement at failure u_f = 2*G_f / sigma_y.

How the variables interact — a visual summary

The diagram below shows how the local stress state feeds the fracture locus, how plastic strain accumulates damage, and how the damage variable drives stiffness degradation until element deletion.

Calibration workflow in practice

A reproducible calibration sequence for a typical automotive steel in PAM-CRASH looks like this:

1. Collect base data. Tensile tests at multiple strain rates to fit the hardening curve; notched and shear specimens to populate the fracture locus.
2. Extract eta and L histories. Run each specimen as a small PAM-CRASH model with the hardening curve only (no failure), and post-process eta and L at the critical element up to the experimental fracture point.
3. Fit epsilon_f(eta, L). Plot the fracture strain against the average eta and L at fracture for each specimen and fit a surface or piecewise curve.
4. Set G_f. Choose a fracture energy consistent with the material class (e.g., 20–80 kJ/m² for mild steel) and verify that the softening response is mesh-insensitive by running the same coupon at two element sizes.
5. Validate. Re-run the full coupon set with the failure card active and compare force–displacement curves and fracture location against the experiments.

What to check in the results

After a simulation, inspect the damage variable D and the equivalent plastic strain epsilon_p in the critical region. A well-calibrated model shows D rising smoothly to 1.0 at the experimentally observed fracture location, with the crack path following the expected shear or tensile mode. If D jumps from 0 to 1 in a single increment, the fracture locus is too steep or G_f is too small. If elements survive well beyond the experimental failure strain, the locus is too high or the triaxiality range was not covered during calibration.

Understanding these four quantities — eta, L, epsilon_f, and D — is the prerequisite for every advanced failure card in PAM-CRASH. The next step is to map them onto the specific material models and card syntax available in your solver release.