Elastic Constants of Fe–Cr Alloys — DFT → ML → CALPHAD → FEM

Homogenized Young's modulus over the CALPHAD equilibrium grid, 298–1200 K by 0–100% Cr, with the Curie/Néel curve overlaid and four FEM runs marked; adjacent line-cut panel showing modulus versus temperature for four compositions

Homogenized Young's modulus over the CALPHAD equilibrium grid (298–1200 K, 0–100% Cr), with the TDB-computed Curie/Néel curve overlaid and the four CALPHAD-linked FEM runs marked. Right: line cuts for four compositions — each flat segment is a single stable phase, each step a phase transition. The surrogate carries no temperature dependence of its own; every change in modulus with temperature comes from CALPHAD.

17 compositions · 0–100% Cr DFT → ML → CALPHAD → FEM Eᵣ within ±12% of Hill VRH

What is the indentation modulus of an alloy that has unmixed into two phases? Single-crystal elasticity cannot answer it — there is no one crystal to describe.

DFT elastic constants for 17 compositions (Quantum ESPRESSO, strain–stress) feed a surrogate model; CALPHAD equilibrium (pycalphad) supplies which phases are stable and what each phase is made of; Voigt-Reuss-Hill and Hashin-Shtrikman homogenisation produce isotropic material cards; CalculiX runs axisymmetric conical nanoindentation and Eᵣ follows by Oliver–Pharr.

The coupling is the point. The surrogate is queried at each phase's own composition rather than the nominal alloy composition, so it needs no structural or thermal feature — the thermodynamics supplies that context. At 50% Cr and room temperature this matters: equilibrium splits the alloy into two BCC phases, one nearly pure iron and one nearly pure chromium.

Direct interpolation over the DFT points reproduces the surrogate to 1.85% mean deviation, so the elastic constants are DFT-derived rather than ML-approximated. In the single-phase regime Eᵣ tracks the analytical Hill VRH reference within −12% to +7%.

The methodological core was penalty-contact validity. An earlier model generation used a fixed penalty stiffness that saturated silently — a spurious flat Eᵣ ≈ 40 GPa across every composition, clean convergence, no solver warning. It surfaced only by checking observed contact pressure against the K × h_element ceiling. Penalty convergence is not physical correctness.

Extension in progress: first-principles elastic constants for the σ phase, which currently inherits BCC-structure values at its own composition.

GitHub

Four-stage pipeline, DFT through CALPHAD-coupled FEM, with explicit post-processing at each step. CALPHAD supplies phase fractions and each phase's own composition, so the surrogate is queried at that composition directly — no structural or thermal feature is added to the model.

Er vs Cr% plot showing K-saturated v16 flat at ~40 GPa versus validated v19 tracking Hill VRH within ±12% — K-dependency diagnostic: the v16 model (fixed K = 10,000 MPa/µm) produces a spurious flat Eᵣ ≈ 40 GPa regardless of composition — the substrate elastic constants never enter the force response. The v19 model (K = 5 × C11) tracks the analytical Hill VRH curve within −12% to +7%. Both models converged cleanly; the saturation is detectable only by checking contact pressure against the K × h_element ceiling.

Fe16Cr0 — 0% Cr · Eᵣ = 196.9 GPa (−12.4% vs VRH)	Fe8Cr8 — 50% Cr · Eᵣ = 220.5 GPa (−1.1% vs VRH)
Fe4Cr12 — 75% Cr · Eᵣ = 247.7 GPa (−3.7% vs VRH)	Fe0Cr16 — 100% Cr · Eᵣ = 274.0 GPa (+6.8% vs VRH) ⚠️ AFM

Von Mises stress — full load–unload cycle for all four compositions. Three-panel composite: overview (indenter motion) ∣ near-field (contact patch decay) ∣ close-up (sub-contact detail). Fe0Cr16 shows a distinctly broader stress field driven by its anomalously low C12 (66 GPa, AFM ground state) and low Poisson ratio (ν ≈ 0.22).