4. Calibrate Abdominal Soft Tissue Material Properties Against Belt Penetration Tests

Why Belt-Penetration Tests Are the Right Calibration Anchor

An abdominal soft-tissue model used for submarining assessment must reproduce two things simultaneously: how far the lap belt sinks into the abdomen (penetration depth) and how much force the abdomen pushes back at that depth (force–displacement response). Belt-penetration tests — a rigid or padded indenter pressed into an isolated abdomen specimen at a controlled rate — give exactly this pair of curves. Calibrating against them is the most direct way to make the belt–abdomen interaction in a full sled model physically credible, because the lap belt's ability to stay on the iliac crest depends on how much the abdominal wall yields before it loads the pelvis.

The task in this case is to adjust a hyperelastic (nonlinear elastic) matrix plus a viscoelastic relaxation spectrum so that simulated penetration and reaction force track the published test corridors. The model is a hybrid: an Ogden-type hyperelastic ground substance for the quasi-static response, and a Prony-series viscoelastic overlay for rate dependence. Calibration is a two-stage problem — fit the quasi-static curve first, then fit the rate-dependent scaling — because the two parameter sets are nearly separable when the test rates are well chosen.

Reference Data and Target Corridors

Published belt-penetration datasets typically report force versus indenter displacement at several rates (e.g., 1 mm/s quasi-static, 100 mm/s, and 1 m/s) for a defined indenter geometry and boundary condition. Before touching material parameters, fix the test configuration in the model so the comparison is apples-to-apples:

• Indenter: 50 mm radius hemispherical or 100 mm × 40 mm belt-section pad, rigid, with a defined friction coefficient (0.3 is a common belt–skin assumption).
• Specimen: abdominal wall slab or full abdomen block with the same thickness and backing condition as the test (rigid back plate vs. compliant spine).
• Outputs: indenter reaction force RF3 and penetration depth measured along the indentation axis, both filtered per the test's channel class.

Build the target as a corridor, not a single curve: upper and lower bounds from specimen-to-specimen scatter. A calibration that lands inside the corridor at all three rates is defensible; one that matches a mean curve but exits the corridor at 1 m/s is not.

Two-Stage Parameter Identification

Stage 1 — hyperelastic fit at quasi-static rate. Use the 1 mm/s curve, where viscous effects are negligible. Start from literature values for abdominal adipose/muscle composites and fit the Ogden strain-energy parameters. A practical starting set for a soft abdominal surrogate is:

*MAT_OGDEN_RUBBER  (or equivalent hyperelastic card)
  MU1 = 0.012  MPa    ALPHA1 = 6.0
  MU2 = 0.004  MPa    ALPHA2 = -2.0
  D1  = 0.0           (nearly incompressible, nu ≈ 0.495)

Fit by minimizing the area between simulated and target force–displacement curves over the penetration range, weighting the toe region (0–15 mm) and the stiffening region (15–40 mm) separately so the fit does not sacrifice the initial compliance to chase the plateau. The toe region controls how early the belt "bites"; the stiffening region controls peak force transmitted toward the pelvis.

Stage 2 — viscoelastic fit across rates. With the hyperelastic response locked, add a Prony series and fit the shear relaxation moduli G_i and relaxation times tau_i to the 100 mm/s and 1 m/s curves. Two or three terms spanning the test time scales are usually sufficient:

*MAT_VISCOELASTIC (overlay)
  G1 = 0.35   TAU1 = 0.002 s
  G2 = 0.25   TAU2 = 0.020 s
  G3 = 0.10   TAU3 = 0.200 s
  BULK = 1.0  (elastic bulk, no volumetric relaxation)

These are normalized shear moduli relative to the instantaneous response; the long-term modulus must remain consistent with the Stage 1 fit. If the 1 m/s curve overshoots while 100 mm/s is on target, shorten TAU1 or reduce G1; if both high-rate curves are too soft, increase the sum of G_i.

Running the Calibration and Reading the Result

Run each rate as a separate load case with the indenter displacement prescribed and reaction force requested as history output. Compare against the corridor at matched penetration depths, not matched times, since the test and model may reach a given depth at different instants. A defensible calibration satisfies three checks:

1. Corridor containment: the simulated curve stays between the upper and lower bounds across the full penetration range at all three rates.
2. Toe-region fidelity: the initial slope (0–10 mm) matches within roughly 15%, because this governs when the belt begins to load the abdomen rather than the pelvis.
3. Peak-force agreement: the maximum reaction force at 40 mm penetration is within ±10% of the test mean at each rate.

If the model passes at 1 mm/s but drifts out of the corridor at 1 m/s, the problem is almost always in the viscoelastic spectrum, not the hyperelastic fit — revisit TAU1 and G1 before touching the Ogden terms. If the toe region is too stiff at every rate, reduce MU1 or lower ALPHA1; if the plateau force is too high, the long-term modulus is over-set and the bulk response needs review.

Calibration Record (Deliverable)

Capture the following in a short calibration record so the fit is reproducible and auditable:

With the abdomen calibrated to the penetration corridor, the lap-belt interaction in the full model now has a physically grounded compliance, which is the prerequisite for trusting any downstream submarining flag or belt-routing sensitivity result.