A return map at the closed secondary nodes on the mortar slip in its rate form (w_A = B_A dx,
B_A = [int Phi_A N_i P, -int Phi_A N_j P]): stick within mu p_A, slip back to the cone, static
coefficient for a node that stuck in the last increment and kinetic for one that slipped; the
friction does the virtual work t . B_A dx, so a slipping interface carries mu N exactly. The
tangent is symmetrised (stick and slip stiffness kept, the coupling to the pressure and the
change of B left out; the residual is exact). Tractions and slip states are committed with the
converged increment. After a chattering cutback, nodes open by less than 1e-3 of their
element's thickness keep their penalty in the tangent for the rest of the step. Results carry
the shear traction, the slip and stick/slip status.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
Frictionless contact pairs enforced by an augmented Lagrangian: nodal pressures from the
mortar weighted gaps, Uzawa updates after each converged increment until the gaps are within
1e-5 of the element thickness, and the consistent tangent (outer products of the weighted
gap gradients and the pressure-weighted Hessians from the automatic differentiation of the
mortar integrals). Finite sliding searches and integrates on the current configuration every
evaluation; without nonlinear geometry the pairs and their mortar integrals are fixed in the
reference configuration (small sliding, linear and exact). The tangent's pattern takes the
face pairs' blocks; the step cuts back when the active set chatters. Initial overclosure is
offset or refused; general contact, 2D models and arc length are refused by name; the linear
analyses refuse contact by name instead of the mesh.
With contact the convergence force scale comes from the converged states, the loads and the
current state. The loads-and-reactions balance check takes the nodal reactions into its scale
(no spurious warning for a part squeezed between held faces).
Tests: the contact patch test on non-matching hex8/tet4 meshes (exact with small sliding,
to 1e-4 with finite sliding), Hertz cylinder on a flat (peak pressure within 3%, half-width
bracketed) with quadratic Newton convergence, overclosure, thread-count determinism and the
refusals; Hertz sphere on a flat (heavy).
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
