4. Material models for restraint fabrics and membranes

Airbag cushions and seat-belt webbing are not rigid bodies. They are thin, flexible, highly anisotropic structures that fold, unfold, stretch, and vent during a crash event. In MADYMO, the fidelity of a restraint simulation depends heavily on how accurately the fabric material is characterized. This section covers the constitutive models available for restraint fabrics and membranes: orthotropic elasticity, viscoelasticity, and failure criteria, with practical guidance on parameter identification and validation.

Why Fabric Is Not Isotropic

Woven airbag fabric (typically nylon 6.6, 420–840 denier) and seat-belt webbing (polyester or nylon) derive their mechanical behavior from interlaced warp and weft yarns. The yarns carry load primarily along their own direction, so the stiffness in the warp direction differs from the weft direction. This is the definition of orthotropic behavior. Additionally, the crimp interchange between warp and weft during loading produces a nonlinear, progressively stiffening response that neither a simple linear elastic model nor an isotropic model can capture.

In MADYMO, the fabric material is assigned to membrane elements (usually triangular or quadrilateral shell elements with zero bending stiffness). The material model must therefore describe in-plane behavior: two principal stiffness directions, shear coupling, and the transition from low-stiffness crimp decrimping to high-stiffness yarn extension.

Orthotropic Elasticity

The baseline model for airbag fabric in MADYMO is an orthotropic elastic material. The stress–strain relationship in the material coordinate system (1 = warp, 2 = weft) is defined by four independent constants: E1, E2, ν12, and G12. The constitutive matrix for plane stress is:

[σ₁]   [ E₁/(1-ν₁₂ν₂₁)    ν₂₁E₁/(1-ν₁₂ν₂₁)    0      ] [ε₁]
[σ₂] = [ ν₁₂E₂/(1-ν₁₂ν₂₁)  E₂/(1-ν₁₂ν₂₁)      0      ] [ε₂]
[τ₁₂]  [ 0                  0                    G₁₂    ] [γ₁₂]

For a typical 420-denier nylon airbag fabric, representative values used in MADYMO examples are E1 ≈ 2.5–4.0 GPa, E2 ≈ 2.0–3.5 GPa, ν12 ≈ 0.15–0.25, and G12 ≈ 0.05–0.15 GPa. Note that G12 is deliberately low because the fabric offers little resistance to in-plane shear until yarns lock. These values are illustrative starting points; actual parameters must be identified from biaxial tensile tests or derived from yarn-level models.

The orthotropic elastic model is computationally efficient and works well for the early inflation phase, where the fabric is mostly unfolding and membrane stresses are moderate. However, it cannot capture the strain-rate sensitivity or the permanent set observed in real fabrics after high-speed deployment.

Viscoelasticity

During a deployment event, strain rates in the fabric can exceed 100 s-1. Nylon and polyester both exhibit viscoelastic behavior: the stress at a given strain depends on the loading rate, and the material creeps or relaxes over time. MADYMO supports viscoelastic material definitions through a Prony series representation of the shear relaxation modulus:

G(t) = G∞ + Σ Gᵢ · exp(-t / τᵢ)

where G∞ is the long-term modulus, and each term (Gᵢ, τᵢ) represents a relaxation mode. In practice, a two- or three-term Prony series is sufficient for airbag fabric. The parameters are fitted to stress-relaxation or dynamic mechanical analysis (DMA) data. For example, a three-term fit for nylon fabric might yield τ₁ = 0.01 s, τ₂ = 0.1 s, τ₃ = 1.0 s, with corresponding modulus fractions of 0.3, 0.4, and 0.2 of the instantaneous modulus.

When viscoelasticity is active, the material response becomes history-dependent. This is important for simulating the post-deployment behavior of a belt that has already been loaded once, or for airbag fabrics that experience multiple pressure pulses during a rollover event.

Practical note: Viscoelastic parameters are often calibrated by matching the simulated pressure–time curve of a tank test or a static deployment test. Start with the elastic model, then add viscoelastic terms only if the simulated peak pressure or rebound behavior deviates significantly from the test data.

Failure Criteria

Restraint fabrics fail by yarn rupture, seam failure, or excessive leakage through pores. MADYMO provides several failure criteria that can be applied to membrane elements:

  • Maximum principal stress: Element is deleted when σ₁ exceeds a user-defined ultimate tensile strength (e.g., 500–700 MPa for nylon 6.6).
  • Maximum principal strain: Element is deleted when ε₁ exceeds a critical value (e.g., 0.20–0.30 for airbag fabric).
  • Tsai-Wu or Hashin: Interactive criteria that account for combined stress states, more appropriate for orthotropic materials under biaxial loading.
  • Leakage-based failure: Not a material deletion criterion, but a porosity model that increases venting as the fabric stretches.

Element deletion must be used with caution. Deleting elements creates holes that can cause numerical instability and non-physical pressure loss. A common practice is to define a failure strain slightly above the maximum expected strain in the simulation, so that failure only occurs in extreme cases, and to use a separate leakage model for normal venting.

Parameter Identification Workflow

The following workflow is typical for characterizing a restraint fabric in MADYMO:

  1. Biaxial tensile testing: Apply loads along warp and weft simultaneously at ratios of 1:1, 2:1, and 1:2. Record stress–strain curves.
  2. Curve fitting: Fit the orthotropic elastic constants to the initial linear region. Fit viscoelastic Prony series to the rate-dependent data.
  3. Single-element validation: Apply a prescribed displacement to a single membrane element in MADYMO and compare the reaction force to the test data.
  4. Component-level validation: Simulate a simple airbag deployment (e.g., a static deployment into free air) and compare the pressure–time history and bag shape to high-speed video.

If the simulated pressure rises too quickly, the fabric is too stiff; reduce E1 and E2. If the bag rebounds too slowly, increase the viscoelastic damping terms. If the bag bursts prematurely, check the failure strain against the maximum principal strain in the simulation.

Fabric Material Model Selection and Calibration Flow Biaxial Tensile Test Warp/Weft ratios Strain rates: 0.01–100 s⁻¹ Model Selection Orthotropic elastic + Viscoelastic (Prony) + Failure criteria Parameter Fitting E₁, E₂, ν₁₂, G₁₂ Gᵢ, τᵢ (Prony terms) σ_ult, ε_ult Single-Element Check Prescribed displacement Compare reaction force Component Validation Tank test / free-air deploy Pressure–time comparison Match? Yes → Finalize No → Adjust params Iterate: adjust E, G, or failure strain Key Parameters: Test input Model choice Fitted constants Element check Component test Decision / iterate

Practical Considerations for Belt Webbing

Seat-belt webbing is typically polyester, which has a higher strain-rate sensitivity than nylon. The same orthotropic elastic and viscoelastic models apply, but the failure criterion is usually based on ultimate tensile strength of the webbing (e.g., 25–30 kN for a 48 mm wide belt). In MADYMO, belt elements are often modeled as 1D seat-belt elements with a user-defined force–displacement curve rather than a full membrane material. However, when the belt is modeled as a 2D membrane (for example, to capture folding and contact with the occupant), the orthotropic elastic model with a low shear modulus is appropriate.

Common pitfall: Using isotropic elastic properties for airbag fabric. This overestimates shear stiffness and can lead to unrealistic bag shapes and pressure histories. Always define at least E₁, E₂, and G₁₂ separately.

Summary of Model Selection

Model When to Use Key Parameters Limitations
Orthotropic elastic Initial inflation, moderate strain rates E₁, E₂, ν₁₂, G₁₂ No rate effects, no permanent set
Viscoelastic (Prony) High-speed deployment, multiple pulses G∞, Gᵢ, τᵢ Requires DMA or relaxation data
Failure criteria Burst prediction, seam failure σ_ult, ε_ult, Tsai-Wu coefficients Element deletion can cause instability

In practice, a calibrated orthotropic elastic model with a simple maximum strain failure criterion is sufficient for many airbag deployment simulations. Viscoelasticity becomes necessary when the simulation must capture the full pressure–time history including the rebound phase, or when the fabric experiences multiple loading cycles. The calibration workflow shown above ensures that the material model is not just numerically stable but physically representative of the actual restraint fabric.

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