4. CAE modeling fundamentals for airbag tethers

An airbag tether is a thin, flexible fabric strap that constrains the inflated cushion geometry. In CAE terms it is a highly compliant, nearly inextensible membrane that carries in-plane tension and buckles immediately under compression. Getting the tether right in an explicit LS-DYNA model is not a matter of drawing a rectangle and assigning a material — the mesh topology, element formulation, and constitutive law all interact, and each choice has a measurable effect on the predicted cushion shape, tether load, and occupant kinematics.

Mesh Strategy: Topology Follows Load Path

Tethers are typically modeled with shell elements because the through-thickness stress is negligible and the dominant response is in-plane membrane tension. The first decision is element size. A tether that is 50 mm wide and 300 mm long, sewn at both ends, should be meshed with a structured grid aligned to the strap axis. A uniform element size of 5–8 mm is a practical starting point for a full-vehicle model; this gives roughly 6–10 elements across the width and 40–60 along the length. Fewer than four elements across the width will under-predict the transverse wrinkling that occurs when the tether is loaded off-axis.

Mesh alignment matters more than element count. The element edges should run parallel and perpendicular to the tether centerline. If the mesh is skewed relative to the load direction, the element coordinate system rotates, and the membrane stress state becomes contaminated by artificial shear. For a tether that wraps around a cushion fold, use a mapped mesh on the flat strap and transition to a free mesh only at the sewn attachment, where the geometry is irregular.

At the sewn ends, the tether is attached to the airbag panel. Two modeling approaches are common. The first is a shared-node connection, where the tether mesh and the panel mesh are coincident and merged. This is the stiffest and simplest option, but it forces the panel mesh to conform to the tether edge, which can degrade panel element quality. The second is a tied contact or constrained nodal rigid body, which allows independent meshes. Tied contact with a penalty formulation is more forgiving of mesh mismatch but introduces a small compliance at the joint. For design sensitivity studies, the shared-node approach is preferred because it removes contact stiffness as a variable.

A useful check: after meshing, run a free-edge check on the tether. Any free edge that is not a physical boundary indicates a mesh discontinuity that will cause a stress singularity and premature element deletion.

Membrane Elements and Formulation

In LS-DYNA, the tether is usually represented by *ELEMENT_SHELL with a membrane formulation, or by *ELEMENT_MEMBRANE if bending stiffness is explicitly unwanted. The distinction is important. A shell element with the Belytschko-Tsay formulation includes bending stiffness and transverse shear, which for a 0.3 mm thick fabric is physically negligible but numerically can stabilize the solution. A membrane element has no bending stiffness at all, so it cannot resist any out-of-plane load and will wrinkle freely. For tethers that are expected to go slack and fold, membrane elements are more realistic. For tethers that remain taut throughout deployment, shell elements with a small bending stiffness are more robust.

Shell thickness is a real input, not a placeholder. A typical polyester tether is 0.3–0.5 mm thick. If the thickness is set to 1.0 mm to improve stability, the axial stiffness will be over-predicted by a factor of two to three, and the tether will appear stronger than it is. Use the measured thickness and control stability through hourglass control and time-step scaling instead.

Hourglass control is critical for membrane elements. The default LS-DYNA hourglass coefficient of 0.1 is often too low for a thin, highly loaded tether. An IHQ value of 4 (stiffness form) with a coefficient of 0.05–0.1 is a reasonable starting point. Monitor the hourglass energy; if it exceeds 5% of the internal energy, the mesh is too coarse or the hourglass control is inadequate.

Material Constitutive Models

Woven polyester and nylon tethers are anisotropic and exhibit a nonlinear, stiffening stress–strain response. The most common constitutive model for this behavior in LS-DYNA is *MAT_FABRIC (MAT_034), which is specifically designed for woven fabrics. It requires the Young's moduli in the warp and weft directions, Poisson's ratios, and a shear modulus. For a typical polyester tether, the warp modulus is on the order of 8–10 GPa, the weft modulus is lower (4–6 GPa), and the shear modulus is 0.5–1.0 GPa. These are engineering estimates; the actual values should come from tensile tests on the specific webbing.

If the tether is modeled as a simple strap with a single dominant load direction, *MAT_ORTHOTROPIC_ELASTIC (MAT_002) is sufficient and computationally cheaper. However, it cannot capture the nonlinear stiffening that occurs as the weave straightens under load. For a tether that experiences large strain, MAT_034 with a nonlinear shear curve is more accurate.

A third option is *MAT_PIECEWISE_LINEAR_PLASTICITY (MAT_024) with a high yield stress and a linear hardening curve. This treats the tether as an isotropic elastic–plastic material. It is a poor physical representation but is sometimes used as a placeholder when fabric data is unavailable. The danger is that it will not capture the directional dependence, and the tether may appear to carry load in directions where it should be compliant.

Tether CAE Model: Mesh, Element, and Material Decision Path Mesh Strategy • Element size: 5–8 mm • ≥ 4 elements across width • Align edges to load axis • Mapped mesh on flat strap • Shared nodes at sewn ends • Free-edge check before run Element Formulation • Shell (Belytschko-Tsay) – includes bending stiffness – robust for taut tether • Membrane (no bending) – realistic for slack/fold • Thickness = measured value • Hourglass control: IHQ=4, coef 0.05–0.1 Constitutive Model • MAT_FABRIC (MAT_034) – warp/weft moduli, shear – nonlinear stiffening • MAT_ORTHOTROPIC_ELASTIC – linear, cheaper • MAT_024 (placeholder only) – isotropic, not physical Tether Load Path and Boundary Conditions Tether strap (shell/membrane elements) Sewn anchor Sewn anchor Inflation pressure load (out-of-plane) Tension Tension Coordinate convention: X = tether axis, Y = width, Z = out-of-plane (cushion normal)

Putting It Together: A Reproducible Modeling Sequence

Consider a driver-side airbag tether, 300 mm long, 50 mm wide, 0.35 mm thick, sewn at both ends to the cushion panel. The modeling sequence is as follows.

  1. Geometry preparation. Create a mid-surface representation of the tether. Do not model the sew lines as solid geometry; represent them as a shared node line or a tied contact surface.
  2. Meshing. Use a mapped mesh with 6 elements across the width (8.3 mm each) and 50 elements along the length (6 mm each). This gives 300 shell elements. Check aspect ratio: keep it below 3:1.
  3. Element formulation. Use *SECTION_SHELL with ELFORM=2 (Belytschko-Tsay) and NIP=2 through-thickness integration points. Set thickness to 0.35 mm.
  4. Material. Assign *MAT_FABRIC with the following example parameters: EA=9.0 GPa, EB=5.0 GPa, PRBA=0.3, GAB=0.8 GPa. These are representative values for polyester webbing and should be replaced with test data.
  5. Hourglass control. In *CONTROL_HOURGLASS, set IHQ=4 and QH=0.08.
  6. Contact. Define tied contact between the tether ends and the cushion panel using *CONTACT_TIED_SHELL_EDGE_TO_SURFACE with a penalty scale factor of 0.1.
  7. Output. Request *DATABASE_ASCII for element stresses and *DATABASE_BINARY_D3PLOT for animation. Monitor hourglass energy in glstat.

After the run, check three things. First, the tether axial force should rise smoothly as the cushion inflates and then plateau. A sudden drop indicates element deletion or contact failure. Second, the hourglass energy should remain below 5% of internal energy. Third, the tether should not exhibit excessive wrinkling unless the physical tether is expected to go slack. If the tether wrinkles while still under tension, the mesh is too coarse or the material model is too compliant.

A common failure mode is using a shell element with a default thickness of 1.0 mm and a linear elastic material. This over-predicts tether stiffness by a factor of three, which delays cushion deployment and under-predicts occupant excursion. Always verify thickness and material data against physical measurements.

Practical Calibration and Validation

Material data for tethers is rarely available in the required form. A tensile test on a 50 mm wide strap at 100 mm/min gives a force–displacement curve. Convert this to engineering stress–strain using the original cross-sectional area (width × thickness). The initial modulus is the slope of the first linear region; the stiffening region is captured by a piecewise linear curve in MAT_034. If the tether is woven, test both warp and weft directions. For shear, a bias tensile test or a picture-frame test is needed; if unavailable, use a shear modulus of 10% of the lower axial modulus as a first estimate.

Validation is done at the component level before full-vehicle simulation. Build a simple model of the tether alone, apply a prescribed displacement at one end, and compare the reaction force to the test curve. Adjust the material parameters until the initial stiffness and the onset of stiffening match. Only then insert the tether into the full airbag model. This stepwise approach prevents compensating errors between the tether and the cushion.

The final check is a mesh convergence study. Refine the tether mesh from 6 elements across the width to 10 and then to 14. If the peak tether force changes by less than 5%, the mesh is adequate. If it changes by more than 10%, the coarser mesh is under-resolving the stress distribution and should not be used for design decisions.

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