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Third Medium Contact – Material Modeling for Contact Problems
Software Lab 2025 | TUM School of Engineering and Design
Self-contacting geometries in the IGAeasy framework produce non-physical deformation behavior under excessive loads. To address this, I implemented a Third Medium Contact (TMC) material model — a continuum-based approach that introduces a virtual "third medium" between potentially contacting surfaces, avoiding the computational overhead of traditional discrete contact detection algorithms.
Formulation
The strain energy density is decomposed into isotropic and anisotropic contributions:
Isotropic part — a hyperelastic formulation in terms of the principal invariants of the right Cauchy-Green tensor, with a stress-free reference configuration enforced analytically.
Anisotropic part — provides directional stiffness aligned with the contact normal, governed by a pseudo-invariant J₄ = n · C · n and an exponent controlling sensitivity.
Dynamic stiffness activation — the key feature of the formulation. A compression-triggered stiffness cM activates only when det(F) falls below a prescribed threshold, following a power law cM = a₅ [det(F)]ⁱ. This creates a compressible cushion layer that offers negligible resistance under tension but rapidly stiffens under contact compression.
The full pipeline — strain energy, Second Piola-Kirchhoff stress, and material tangent stiffness — was derived and implemented in both 2D and 3D, with the tangent stored in Voigt notation (3×3 for plane problems, 6×6 for 3D) for direct use in element stiffness assembly.
Implementation
The model was implemented as a Python class ThirdContactMaterial within the IGAeasy framework, exposing:
strain_energy(F, n)— strain energy density evaluationcompute_stress_2D / 3D(F, n)— Second Piola-Kirchhoff stressget_C_voigt_cart_2D / 3D(F, n)— material tangent in Voigt notation
If no fiber direction is provided, the model defaults to purely isotropic behavior.
Results
A parameter sweep over det(F) confirmed the compression-triggered activation: stiffness remains zero above the threshold and follows the prescribed power law below it, verifying the intended behavior.
A 2D membrane example, a rectangular domain compressed against a fixed edge, demonstrated large deformations prior to stiffness activation, consistent with the expected soft-layer response.
Convergence challenges were encountered near the activation threshold, where the sudden stiffness jump ill-conditions the Newton-Raphson tangent matrix. Load stepping produced partial results through the pre-contact phase, but full convergence through activation remains an open problem with the current solver.
Limitations & Future Work
- No tangential contact behavior — friction at the interface is not captured
- Nonlinear solver improvements needed: adaptive load stepping near the threshold would help
- A systematic parameter sensitivity study (a₁–a₅, exponent i, tolerance) would provide practical design guidelines