4. Material modeling for airbag fabrics

An airbag is not a rigid pressure vessel. It is a sewn, woven, coated or uncoated nylon or polyester membrane that inflates in tens of milliseconds, stretches biaxially, leaks gas through its own pores and seams, and then vents through dedicated exhaust holes. If the fabric is modeled as a simple isotropic shell with a single Young's modulus, the simulation may still run, but the predicted bag shape, internal pressure history, and occupant contact force will drift away from physical reality. This chapter explains how to select and calibrate the constitutive law for airbag fabric, how to represent gas permeability, and how to account for strain-rate dependence in PAM-CRASH.

Why the Constitutive Law Matters

Airbag fabric is a plain-weave textile. Warp and weft yarns interlace, so the in-plane stiffness along the warp direction differs from the weft direction, and both differ from the off-axis response. The coating (typically silicone or neoprene) controls permeability and friction but contributes little to load carrying. During deployment, yarns straighten and crimp interchange occurs before the fabric itself extends, producing a nonlinear, progressively stiffening stress–strain curve. Three modeling levels are commonly used in PAM-CRASH, and the correct choice depends on what the simulation must predict.

Linear Elastic and Orthotropic Formulations

The linear elastic shell is defined by Young's modulus E, Poisson's ratio ν, density ρ, and thickness t. For a 470 dtex silicone-coated nylon 6.6 at low strain, a representative elastic modulus is on the order of 200–400 MPa with ν ≈ 0.3. This is adequate only when the bag is treated as a kinematic surface and pressure is prescribed rather than computed.

The orthotropic law introduces independent moduli along the material axes. In PAM-CRASH the fabric reference direction is defined by the element's local coordinate system, so the warp direction must be assigned consistently across the mesh. A typical calibration set for the same coated nylon is:

MAT_ORTHOTROPIC
  E11 = 1200 MPa   (warp)
  E22 = 1100 MPa   (weft)
  E33 =  150 MPa   (through-thickness, shell)
  G12 =  120 MPa   (in-plane shear)
  G23 =   60 MPa
  G31 =   60 MPa
  nu12 = 0.25
  rho  = 1.15e-9 tonne/mm^3
  t    = 0.35 mm

These values are illustrative and must be replaced by coupon test data (ASTM D5035 strip tensile or ISO 13934-1) for the actual fabric. The critical practical point is that E11 and E22 should be extracted from the initial linear region of the warp and weft curves, not from the secant modulus at failure, otherwise the bag will appear far too stiff during the early inflation phase.

Viscoelastic and Strain-Rate Effects

Airbag deployment imposes strain rates from roughly 10 s⁻¹ to over 500 s⁻¹. Nylon and polyester both stiffen with increasing strain rate, and the effect is not negligible: the same fabric can show a 30–60 % increase in tensile stress at 100 s⁻¹ compared with quasi-static loading. A viscoelastic model represents this through a Prony series or a rate-dependent yield surface.

In PAM-CRASH, a common approach is to combine an elastic or orthotropic backbone with a rate multiplier. The stress is written as a convolution of the relaxation modulus with strain rate, or, in simplified form, as:

sigma(t) = E_inf * eps + sum_i  E_i * integral( exp(-(t-tau)/tau_i) * d(eps)/dtau ) dtau

where E_inf is the long-term modulus and E_i, tau_i are Prony pairs. For a first calibration, two or three Prony terms are usually sufficient. The relaxation times should span the deployment window: one short term (0.1–1 ms) to capture the initial stiff response, and one longer term (10–100 ms) to capture the post-inflation relaxation as the bag settles against the occupant.

Permeability Modeling

Permeability is the second mechanism by which the bag loses pressure, and it is often the dominant one for uncoated fabrics. Two leakage paths exist: through the fabric pores (fabric permeability) and through the seams and vent holes (orifice flow). PAM-CRASH handles these differently.

Fabric permeability is defined as a mass flow rate per unit area per unit pressure difference, typically in units of l/(dm²·min·bar) or converted to SI. For a coated fabric the value may be below 1; for an uncoated fabric it can exceed 50. The flow is modeled as a porous media or as a distributed leak across the shell elements. The governing relation is often written as:

m_dot = C_perm * A * (P_in - P_out)^n

where C_perm is the permeability coefficient, A the element area, and n typically between 1 and 2 depending on whether the flow is laminar or turbulent. Vent holes and seam leaks are modeled separately as concentrated orifices using a discharge coefficient Cd (commonly 0.6–0.8) and the orifice area.

Calibration Workflow and Verification

A practical calibration sequence for a driver-side airbag module is:

1. Run quasi-static strip tensile tests on warp, weft, and 45° bias coupons. Extract E11, E22, and G12 from the initial linear region.
2. Run dynamic tensile tests at 1, 10, and 100 s⁻¹. Fit Prony terms or build a rate-dependent table.
3. Measure fabric permeability with a static pressure-decay test on a clamped fabric sample. Record the pressure–time curve and fit C_perm and n.
4. Build a single-element or small-panel model in PAM-CRASH and reproduce the coupon curves. Verify that the simulated stress–strain response matches the test within 10 % up to 5 % strain.
5. Run a full bag inflation simulation with a prescribed mass flow inlet. Compare the internal pressure history and bag envelope against a tank test or high-speed video.

Typical failure symptoms and their causes: if the bag inflates too slowly, the permeability or vent area is too high; if the bag is too stiff and rebounds sharply, the elastic modulus is overestimated or the rate multiplier is too aggressive; if the bag shape is asymmetric, the warp/weft orientation in the mesh is inconsistent. Each of these can be diagnosed by isolating the material, permeability, and orifice contributions in separate verification runs.

The figure summarizes the complete modeling chain: coupon and permeability tests feed the orthotropic–viscoelastic constitutive law and the two leakage mechanisms, which together drive the simulated bag envelope, pressure history, and occupant loading. The dashed feedback path is the calibration loop that every airbag model must pass through before it can be used for restraint system validation.