Model
The plate is a Mindlin-Reissner plate meshed with MITC4 quadrilaterals on a uniform grid, which stays accurate from thin to moderately thick plates without shear locking. Fasteners prescribe the plate height h = d·(1 − squeeze) above the rigid base over their contact patch. The cord is a one-sided line foundation: at each point along the seal the local compression is c = (d − w)/d, and the line force is q(c), zero where the plate lifts off.
Cord stiffness
Lindley's relation for an elastomer cylinder between flat plates: q = E·d·(1.25·c^1.5 + 50·c^6) N/mm. The durometer estimate of E uses Gent's relation, which can be off by roughly 30% for a given compound. A measured curve is interpolated with a monotone cubic (PCHIP) and extended linearly past the last point.
Solution
Modified Newton-Raphson with a line search, reusing the factored tangent while convergence is fast. The banded Cholesky solve runs in a background worker. Each run checks equilibrium: the fastener loads must sum to the total seal force.
Limits
- Small deflection and linear isotropic plate. A warning appears when deflection exceeds half the thickness.
- The cord is a line load; its contact width and friction are ignored.
- Plate contact with the base plane is detected and flagged, not modeled.
- Fastener patches resolve to the mesh nodes inside each circle, so results shift by about 1–2% as the mesh changes. Stress at and next to a patch is approximate.
References
- P. B. Lindley, "Load-compression relationships of rubber units," J. Strain Analysis 2 (1967).
- A. N. Gent, "On the relation between indentation hardness and Young's modulus," Trans. IRI 34 (1958).
- K.-J. Bathe and E. Dvorkin, "A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation," IJNME 21 (1985).
- S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells; M. Hetényi, Beams on Elastic Foundation. Both supply the closed-form checks in the validation suite.