[petlenz]

First half of #20, which is #15's acceptance criterion deferred by #17: J2 with a tensor back-stress, unknowns {Δγ (scalar), β (symmetric tensor)} solved as one coupled 7×7.

Until now vector_newton had only been driven through synthetic systems with closed-form roots. Nothing had put it through a return map — which matters, because the review round that produced #54 showed how much of its behaviour was unexercised.

The system

With η = s_tr − dev(β), q = σ_eq(η), N = (3/2)η/q:

R_γ = q − 3G·Δγ − σ_y            yield on the RETURNED state, because
                                 σ_eq(s − β) collapses to q − 3G·Δγ exactly
R_β = β − β_n − (2/3)·H·Δγ·N     Prager evolution

Nothing pre-substitutes the radial-return simplification — N is taken from the current iterate, so it genuinely has to be solved coupled rather than decoupling into a scalar equation. The Jacobian is derived rather than differenced:

dq/dB = −N        dN/dB = −((3/2)IIdev − N⊗N)/q

the same shape as j2's existing flow_normal_stress_derivative.

Four tests, and why each exists

Against the closed form, not against itself. For linear kinematic hardening the return is radial in the shifted space, so Δγ = (q_tr − σ_y)/(3G + H) exactly — an independent oracle. The returned state must also sit on the shifted surface.

The elastic branch is not optional, and I found that the hard way. Without it R_γ = q − 3G·Δγ − σ_y still has a root for any trial state, and the solver finds it. Measured on a step with q_tr = 2.10 against σ_y = 50: it converged to Δγ = −0.0148 with a spurious back stress of |β| = 36.3. Negative plastic flow, reported as success. That was a real bug in my model, hit while writing it.

The Bauschinger effect, because it's what separates kinematic from isotropic hardening: in pure shear the elastic span between forward and reverse yield stays 2σ_y/√3 however far the forward loading went — the surface translates instead of expanding.

A finite-difference check of the Jacobian, which exists because the first three cannot catch a Jacobian error. I verified that rather than assuming: flipping the sign of jacobian_g_B leaves all three passing, because Newton converges to the root of the residual and a wrong Jacobian changes only the path. Mutation-verified across every block:

mutation	caught
jacobian_g_g value	yes
jacobian_g_B sign	yes
jacobian_B_g sign	yes
jacobian_B_B 1.5 → 2.0	yes

Still open on #20

The consistent tangent through tangent_checker (needs the strain_derivative wiring — dR/dε for both blocks), and the two-surface model. This PR is the first half; #20 stays open.

306/306 pass.

[petlenz]

Review findings (design, tmech expressions instead of component loops, modern C++, comments, tests, correctness). Severity: must = wrong result, hang, build break or unreachable feature; should = design, duplication, accuracy; nit = style.

kinematic hardening — needs changes

• [must] tests/CMakeLists.txt:9 conflicts with main (macro now wraps numsim_add_test with -Werror + ASan twin); rebase, re-run.
• [should] IIsym/IIdev rebuilt every residual → plasticity_detail::cached_IIdev; back_stress must become stress_like once STATEV: stress-like tensor history without the engineering-shear factor (#63) #91 lands; FD Jacobian loops → tmech::num_diff_central; 19-line class docblock; 1e-30 magic twice; trial_deviator() thrice per call.

Conflicts and landing order (open PRs on main)

• tests/CMakeLists.txt: core: reject malformed models instead of running them (#56, #57, #60, #64, #69) #79 and core: fail loudly when a material reaches for something unwired (#59, #62) #84 both insert after test_property_graph; solvers: report a Runge-Kutta step that did not converge (#58, #61) #83 vs main's foreach; tests: drive vector_newton through a real return map (kinematic hardening) #55 vs main.
• core/material_interface.h: core: fail loudly when a material reaches for something unwired (#59, #62) #84 and io: name the parameter, the material and the entry when a deck is wrong (#66, #67) #85 overlap.
• Order: core: fail loudly when a material reaches for something unwired (#59, #62) #84 → io: name the parameter, the material and the entry when a deck is wrong (#66, #67) #85 (rebased on it) → core: reject malformed models instead of running them (#56, #57, #60, #64, #69) #79 → solvers: report a Runge-Kutta step that did not converge (#58, #61) #83; solvers: drop the relative-residual guard, it fired on rounding noise (#21) #53 merges clean; tests: drive vector_newton through a real return map (kinematic hardening) #55/cmake: make the installed package compile for a consumer (#70, #71) #81/build: clean the warning set and enforce it in CI (#76) #86 rebase onto 7bcba93 and shrink to their non-CMake content.
• STATEV: stress-like tensor history without the engineering-shear factor (#63) #91 ↔ tests: drive vector_newton through a real return map (kinematic hardening) #55: back_stress becomes stress_like.

[petlenz]

Closing in favour of #162, a shipped J2 material with Armstrong-Frederick kinematic hardening on vector_newton.

The review finding that decided it: linear Prager hardening is radial in the shifted space, and from the cold start the first Newton step is the radial return. With max_iter = 1 all four tests here still pass, and the coupling term in J_ββ is multiplied by Δγ = 0 in that step, so removing it changes nothing in any solve (3 residual evaluations either way; only the FD test notices). The solver-side coverage this adds already exists in test_vector_newton.

What carries over to #162: the closed-form Prager test and the dN/dβ derivation. What does not: the Bauschinger loop, the elastic-branch test and the component-wise FD loop, replaced by a directional check.