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

Three-panel von Mises stress composite for Fe16Cr0 at peak indentation load: overview, near-field contact patch, and close-up contact zone
Von Mises stress at peak load (h = 0.404 µm) — Fe16Cr0 reference case. Three-panel composite: overview ∣ near-field (full contact patch, a = 1.128 µm) ∣ close-up (sub-contact detail). Colorbar capped at 20,000 MPa (~1.9× mean contact pressure) to suppress penalty contact edge singularities and reveal the physical stress bulb.
4 compositions · 0–100% Cr DFT → ML → FEM pipeline Eᵣ within ±12% of Hill VRH

A three-stage first-principles-to-FEM pipeline for Fe-Cr binary alloys, targeting the composition-dependent reduced indentation modulus Eᵣ across the full BCC solid-solution range. Stage 1 computes C11, C12, and C44 for 17 compositions using the strain–stress method in Quantum ESPRESSO on a GPU-accelerated cloud node. Stage 2 trains Gaussian-Process and MLP surrogates to interpolate the elastic tensor across composition space, generating FEM-ready material cards. Stage 3 runs axisymmetric conical-indentation simulations in CalculiX and extracts Eᵣ via the Oliver–Pharr method.

The central methodological challenge was penalty-contact validity. An earlier model generation used a fixed penalty stiffness that silently saturated — producing a spurious flat Eᵣ ≈ 40 GPa across all compositions with no solver warnings. The validated model sets K = 5 × C11 per composition, confirmed by checking observed contact pressure against the K × helement ceiling. The resulting Eᵣ values track the analytical Hill VRH reference within −12% to +7%, consistent with the elastic-only constitutive model, idealised conical geometry, and lower-bound penalty stiffness.

Post-processing is staged across all three pipeline steps. DFT output undergoes a five-tier convergence audit that assigns quality weights carried into ML training. The ML stage selects models per elastic constant via leave-one-out cross-validation and an fe02cr14 ablation study. FEM output is corrected for the CAX 2° wedge (×180 force scaling) and zero-contact unloading points before Oliver–Pharr fitting. Field visualisation in ParaView 6.1.1 uses per-composition colorbar caps derived from the mean contact pressure to suppress penalty singularity nodes and reveal the physical stress and strain fields.

GitHub

Pipeline schematic: Stage 1 DFT → Stage 2 ML → Stage 3 FEM, each with a post-processing step below
Three-stage pipeline with explicit post-processing at each step. Stage 1 produces a five-tier elastic-constant dataset. Stage 2 selects the best GP/MLP surrogate per constant and emits CalculiX and ABAQUS material cards. Stage 3 corrects for wedge scaling and zero-contact exclusion before Oliver–Pharr fitting and ParaView field visualisation.
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 × helement ceiling.
Fe16Cr0 von Mises stress animation — full load-unload cycle
Fe16Cr0 — 0% Cr  ·  Eᵣ = 196.9 GPa (−12.4% vs VRH)
 Fe8Cr8 von Mises stress animation — full load-unload cycle
Fe8Cr8 — 50% Cr  ·  Eᵣ = 220.5 GPa (−1.1% vs VRH)
Fe4Cr12 von Mises stress animation — full load-unload cycle
Fe4Cr12 — 75% Cr  ·  Eᵣ = 247.7 GPa (−3.7% vs VRH)
 Fe0Cr16 von Mises stress animation — full load-unload cycle
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).