[petlenz]

Closes #162. Kinematic half of #20.

What

materials/j2_kinematic_plasticity.h — J2 with Armstrong-Frederick kinematic hardening on top of the existing isotropic hardening_source, returned as one coupled system over {Δλ, Δβ} by vector_newton. The first shipped material on the coupled solver.

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

R_λ = q − 3G Δλ − σ_0 − H(κ_n + Δλ)          yield on the returned state
R_β = (1 + γ Δλ) β − β_n − (2/3) C Δλ N       Armstrong-Frederick, β = β_n + Δβ

Jacobian and the partial strain derivatives are derived (see the header). The consistent tangent comes through the solver's strain_derivative opt-in (#19) and the chain rule dσ/dε = C_e − 2G [N ⊗ dΔλ/dε + Δλ dN/dε].

γ = 0 is Prager: radial in the shifted space, closed form Δλ = (q_tr − σ_0 − H_n)/(3G + C + H'), one Newton step from the trial seed. γ > 0 rotates the flow direction during the step and the solve iterates. The tensor unknown is the increment Δβ, so the solver's zero seed is the trial state; with β itself as the unknown even Prager needed several steps once β_n ≠ 0.

vector_newton — two small additions the material needs:

• graph_driven (bool, default true). false leaves the unknowns without an update callback, so the solver runs only when a material calls solve() through a material_ref. This is the design plasticity.md §9 settled on for local_newton, applied here: a graph-driven solver would also run a cold-start solve at every point, elastic ones included, where the yield residual has a negative root. The material checks the flag and throws.
• reset(): zero the unknowns, mark not converged. The material's residual callback is still a property the engine fires once per update, at whatever the solver's slots hold, and it writes κ_n + Δλ into the history the hardening source reads. On an elastic step the previous point's converged Δλ would otherwise be added to this point's κ. Found by the Bauschinger test below; the elastic test now pins it.

Registered as j2_kinematic_plasticity; the conformance tests pick it up. Documented in plasticity.md (§2 interface, §9 the solver decision) and the README.

Tests (tests/test_j2_kinematic_plasticity.cpp, 8)

test	what it pins
PragerStepMatchesTheClosedForm	Δλ, β, κ and the shifted yield condition against the closed form, on a second step whose committed β_n is not parallel to the trial deviator
OnlyRecoveryNeedsMoreThanOneNewtonStep	Prager converges with max_iter = 1 on that step; Armstrong-Frederick throws
RecoveryReturnsOnTheShiftedSurfaceAndRotatesTheFlow	γ > 0: shifted surface, evolution law met by the final N, flow direction rotated away from η_tr
AnElasticStepDoesNotRunTheSolver	elastic stress and tangent, no solve, and after a committed plastic step an elastic one leaves κ, β and the unknowns clean
RejectsAGraphDrivenSolver	the graph_driven = true configuration throws
JacobianMatchesDirectionalFiniteDifferences	all four blocks as directional derivatives along symmetric directions, off the root
TangentMatchesDirectionalFiniteDifferencesOfTheStress	consistent tangent vs central differences of the stress, γ = 0 and γ = 40, non-proportional committed history
ReverseYieldIsShiftedByTheBackStress	Bauschinger: the reverse elastic limit sits at β_12 − Y/√3, probed from both sides

Directional finite differences rather than component loops: the Mandel space the solver works in sees IIsym as the identity, so perturbing one off-diagonal component alone compares against the wrong object.

Mutations, each caught:

mutation	caught by
dR_β/dε sign	tangent
J_βλ drops γ β	Jacobian, tangent
J_ββ coupling 1.5 → 2	Jacobian, tangent
tangent drops Δλ dN/dε	tangent
tangent ignores dβ/dε	tangent
residual evaluated at β_n instead of β	closed form, one-step, recovery, Jacobian, tangent
no reset() on the elastic branch	elastic, Bauschinger

Not in this PR

• back_stress as stress_like (STATEV: stress-like tensor history without the engineering-shear factor (#63) #91 / STATEV layout applies the engineering-shear factor to every symmetric tensor, which mis-scales stress-like history #63): the history is a plain tensor2, so statev_map exports it with the strain shear factor. Noted in the header and the docs.
• The two-surface half of Physics integration tests for vector_newton: two-surface plasticity and kinematic hardening #20.
• The elastic-branch options the issue listed: this takes option 3 (material_ref, conditional solve), the one §9 already settled for the scalar solver. Nothing was measured for the identity-system alternative because the material never poses one.