Skip to content

P92 tensor admission: a research gate, not a material card

Source and decision

The version-2 PDF of Zhang et al., Metals 10 (2020) 307 was visually rechecked on printed pages 7-10. Table 3 and equations 9-13 use scalar stress, backstress and inelastic strain; Table 4 supplies E, but not a Poisson ratio. Its SHA-256 remains c00a71167969edaab6a1b52641140cb5a3f241e82a90b74b464f293f46fcd604. The existing published-figure screen stays unchanged.

A three-dimensional extension therefore needs explicit assumptions: isotropic elasticity with a separately supplied Poisson ratio, pressure-independent J2 flow, deviatoric backstresses and the stress-norm convention below. Those assumptions are not additional experimental facts from the source.

Candidate normalization

With s = dev(sigma), eta = s - X1 - X2, define q = sqrt(3/2 * eta:eta), v = (<q-R-k>/Z)^n, and ep_dot = (3/2) * v * eta/q, with zero flow on the inactive branch. Use sigma_dot = C:(strain_rate-ep_dot) and

Xi_dot = (2/3) Ci ai ep_dot - Ci v Xi - gamma(p) J(Xi)^(m-1) Xi
J(Xi) = sqrt(3/2 * Xi:Xi)
R_dot = b(Q-R)v + gamma_r(p)|Qr-R|^(mr-1)(Qr-R) + H v
p_dot = v

The source's scalar chi maps to diag(2 chi/3, -chi/3, -chi/3), not diag(chi, 0, 0). Consequently J(Xi)=abs(chi) and multiplying the axial backstress equation by 3/2 recovers the source equation exactly. Substituting the raw Frobenius norm would multiply the static-recovery coefficient by (2/3)^((m-1)/2), a substantial change at the published exponents.

NEML's Chaboche documentation provides a useful implementation comparison, but explicitly uses an opposite hardening sign convention and its own norm factors. Copying its parameters without a convention map would not reproduce this source model.

Numerical implementation and evidence

First verify rates: exact tension/compression/inactive reduction to the existing scalar reference, hydrostatic independence, traceless inelastic/backstress rates, and covariance under a constant change of Cartesian basis. The latter is small-strain tensor covariance, not finite-rotation objectivity.

Then solve the complete coupled backward-Euler local system in Mandel stress coordinates and an orthonormal five-component deviatoric backstress basis. The unknown vector has 18 entries: stress(6), X1(5), X2(5), R, p. Its algorithmic tangent must differentiate the converged update at fixed old state: B = solve(I-dt*df/dy, [C; 0]), taking the first six rows. Independent strain differences and the stress-free lateral condensation must recover the existing uniaxial tangent. A last-substep tangent is not a complete substepped tangent.

All experiments stay in tools/, outside the constitutive catalogue. Effective stress power eta:ep_dot=q*v >= 0 is checked, but is not claimed to be the full thermodynamic dissipation of a softening/recovery model. Stored energy and general first-law closure are not invented.

The coupled local update is now implemented in tools/p92_tensor_normalization.py. Fifty numerical tests cover the rate reduction, analytic 18-by-18 Jacobian, complete six-direction strain tangent, stress-free lateral condensation, stress-unit scaling, constant-basis rotation covariance, and time refinement against independent Radau integration.

tools/p92_tensor_material_reference.py is a research-only local provider for the bounded physical-time consumer (PR #150). It declares its supplied Poisson ratio, fixed published temperature, source parameters and named state. It does not contain a global nonlinear solver. Mandel shear scaling and component permutation are explicitly converted to the public tensor-Voigt convention. Temperature fields and other external fields are rejected rather than ignored.

Ten FEniCSx/provider tests verify the public tangent checker with all shear components, rejection of unsupported fields, 600/650 Celsius homogeneous structures with free lateral contraction against the independent scalar backward-Euler history, restart at the start of a hold, and all reversal/hold endpoints of a tension-hold-compression-hold-unload cycle. The expanded suite including scalar references, physical-time and atomic-restart regressions, and CI routing passed 171 tests, with one dedicated two-rank admission test skipped in the serial run (already run separately for PR #150). This is numerical composition evidence, not a multiaxial experimental validation or a complete fatigue simulation.

Structural refinement, nonproportional material data, temperature interpolation, cyclic-life prediction and a production material catalogue entry remain separate. The source does not supply a unique multiaxial model or Poisson ratio; no claim of calibrated three-dimensional P92 behavior follows from this gate.

Integration follow-up

PR #150 is merged. This research branch is now based on that verified main, including the independent scalar figure screen and co-located finite-strain history consumer; the latter is not automatically available on this small-strain route. CI changes to the upstream scalar reference now also select the tensor and structural tests, preventing an unnoticed dependency regression.

At both published temperatures, structural runs with 16, 32 and 64 increments approach an independently integrated continuous-time scalar Radau reference. Each refinement reduces final axial-stress error by at least 30% in the tested ramp/hold path. This is time-discretization evidence for a homogeneous patch, not spatial convergence or experimental multiaxial validation. Invalid complex, boolean, array-valued and out-of-range Poisson assumptions are rejected at construction.

Force-controlled 300 MPa ramp/hold structures at both temperatures also show continued inelastic accumulation during the hold. The accumulated strain approaches an independently integrated stress-controlled scalar Radau history under 16/32/64 increments, with at least 25% error reduction on each refinement. This complements the displacement-controlled relaxation checks; it is not experimental creep-life validation.

The integrated local regression is 245 passed, 1 skipped (the serial run skips the dedicated two-rank admission check). Generated documentation and targeted lint checks pass. No new release or material maturity promotion is made by this integration.

Next gate: retain fixed published parameters and test additional full cyclic experimental histories with explicit digitization uncertainty. Separately, extend co-located accepted histories to the small-strain consumer before using structural paths as life-assessment inputs. Temperature interpolation and multiaxial calibration need independent evidence, not inferred constants.