δ-Theory: Unified Materials Strength & Fatigue Framework

🎯 Overview

δ-Theory is a unified framework that predicts material properties from crystal structure geometry. Unlike traditional empirical fitting approaches, it derives material behavior from Geometric Structural Principles

Core Equation

$$\Lambda = \frac{K}{|V|_{\text{eff}}}$$

• K: Destructive energy density (stress, thermal, electromagnetic, etc.)
• |V|_eff: Effective cohesive energy density (bond strength)
• Λ = 1: Critical condition (fracture / phase transition)

What Can δ-Theory Predict?

📦 Installation

pip install delta-theory

From Source

git clone https://github.com/miosync-inc/delta-theory.git
cd delta-theory
pip install -e .

🚀 Quick Start

Yield Stress (v10.1 SSOC)

from delta_theory import calc_sigma_y, MATERIALS, calc_f_de, sigma_base_v10

# Full yield stress with strengthening mechanisms
mat = MATERIALS['Fe']
result = calc_sigma_y(mat, T_K=300)
print(f"σ_y = {result['sigma_y']:.1f} MPa")
print(f"f_de = {result['f_de']:.4f}")
print(f"branch = {result['sigma_base_branch']}")

# Direct SSOC calculation
print(f"W f_de = {calc_f_de(MATERIALS['W']):.3f}")   # → 2.993 (d⁴ JT anomaly)
print(f"Fe σ_base = {sigma_base_v10(MATERIALS['Fe']):.1f} MPa")

Fatigue Life

from delta_theory import fatigue_life_const_amp, MATERIALS

result = fatigue_life_const_amp(
    MATERIALS['Fe'],
    sigma_a_MPa=150,
    sigma_y_tension_MPa=200,
    A_ext=2.5e-4,
)
print(f"N_fail = {result['N_fail']:.2e} cycles")

FLC Prediction (v8.1 — 7-Mode Discrete!)

from delta_theory import FLCPredictor, predict_flc

# Quick prediction
eps1 = predict_flc('Cu', 'Plane Strain')  # → 0.346

# Full usage
flc = FLCPredictor()
for mode in ['Uniaxial', 'Plane Strain', 'Equi-biaxial']:
    print(f"{mode}: {flc.predict('Cu', mode):.3f}")

# Add new material from v6.9 parameters
flc.add_from_v69('MySteel', flc0=0.28, base_element='Fe')
betas, eps1s = flc.predict_curve('MySteel')  # All 7 modes!

📦 Repository Structure

delta-theory/
├── delta_theory/                       # 🔧 Main package
│   ├── material.py                     # Data layer — 37 metals + SSOC params
│   ├── creep.py                        # Diffusion Creep Module
│   ├── ssoc.py                         # Calculation layer — f_de (NEW!)
│   ├── am_fatigue.py                   # AM fatigue SNCurve
│   ├── unified_yield_fatigue_v10.py    # Application layer — σ_y, τ/σ, S-N
│   ├── unified_yield_fatigue_v6_9.py   # ← backward compat shim (re-exports v10)
│   ├── unified_flc_v8_1.py             # FLC 7-mode discrete
│   ├── dbt_unified.py                  # DBT/DBTT prediction model
│   ├── lindemann.py                    # Iizumi-Lindemann melting law
│   ├── banners.py                      # ASCII art banners
│   └── fatigue_redis_api.py            # FatigueData-AM2022 API
│
├── apps/                              # 🖥️ Applications
│   └── delta_fatigue_app.py           # Streamlit Web App
│
├── examples/                          # 📚 Usage examples
└── tests/                             # 🧪 Tests

🔬 Core Modules

1. Yield Stress — v10.0 SSOC (Structure-Selective Orbital Coupling)

δ_L-free unified yield stress from crystal geometry

Architecture (Multi-Layer Design)

material.py                    → Data layer      (25 metals, SSOC + crystal params)
ssoc.py                        → σ_y calculation  (f_de: PCC/SCC 3-factor model)
unified_yield_fatigue_v10.py    → Application layer (σ_y + τ/σ + R_comp → S-N, FLC)

Unified Equation

$$\sigma_y = \frac{8\sqrt{5}}{5\pi M Z} \cdot \alpha_0 \cdot \left(\frac{b}{d}\right)^2 \cdot f_{de} \cdot \frac{\sqrt{E_{\text{coh}} \cdot k_B \cdot T_m}}{V_{\text{act}}} \cdot HP$$

Key Insight: δ_L ∝ √(k_B·T_m / E_coh), so E_bond × δ_L ∝ √(E_coh · k_B · T_m). v10.0 uses this relationship directly, eliminating δ_L dependence.

SSOC f_de — Structure-Selective Orbital Coupling (v10.1)

$$f_{de}^{(s)} = \left(\frac{X_s}{X_{\text{ref}}}\right)^{2/3 \cdot g_d} \times f_{\text{aux}}^{(s)}$$ P_DIM = 2/3 — Universal geometric exponent: surface (2D) → volume (3D) dimension transformation

• PCC (Perturbative-Coupled Channel): Input field and response separable (FCC, HCP)
• SCC (Self-Consistent Channel): Field and response inseparable (BCC)

v10.1 Gate Extensions

BCC d⁴ Jahn-Teller Anomaly

t₂g⁴: one orbital doubly occupied → Oh→D₄h symmetry breaking
→ Maximum SCC self-generation of Peierls barrier
→ W (d⁴, 5d): f_JT=1.9 × f_5d=1.5 × f_lat=1.05 → f_de ≈ 2.99

Validation (37 Metals, T=300K)

Zero fitting parameters — All predictions from crystal geometry + thermodynamic data.

from delta_theory import calc_f_de, sigma_base_v10, MATERIALS

# Inspect SSOC factors
from delta_theory import calc_f_de_detail
detail = calc_f_de_detail(MATERIALS['W'])
# → {'f_jt': 1.9, 'f_5d': 1.5, 'f_lat': 1.05, 'f_de': 2.993}

# Inverse: experimental σ → back-calculate f_de
from delta_theory import inverse_f_de
f = inverse_f_de(MATERIALS['Fe'], sigma_exp_MPa=150.0)

2. Multiaxial Ratios — v4.1 α-Coefficient Theory

τ/σ (shear-to-tensile ratio) and R_comp (compression-to-tensile ratio) from crystal geometry

These ratios feed directly into fatigue (r_th presets) and FLC (R_j resistance).

Core Equation

$$\frac{\tau}{\sigma} = \frac{\alpha_s}{\alpha_t} \times C_{\text{class}} \times T_{\text{twin}} \times A_{\text{texture}}$$

Design Principles (v4.1)

• Cu torsion = anchor (Quality A data): C_class calibrated from Cu τ/σ = 0.565
• HCP: C_class NOT applied — avoids propagating uncertainty from estimated data
• BCC slip mixing: w110 parameter for {110}/{112} system weighting (default: w110=0, pure {112})

Geometric α Values by Structure

Compression Ratio R = σ_c / σ_t

Driven by twinning asymmetry in HCP metals:

from delta_theory import tau_over_sigma, sigma_c_over_sigma_t, C_CLASS_DEFAULT, MATERIALS

# τ/σ prediction (uses α-coefficients internally)
fe = MATERIALS['Fe']
print(f"Fe τ/σ = {tau_over_sigma(fe):.4f}")       # → 0.565
print(f"Fe R_comp = {sigma_c_over_sigma_t(fe)}")   # → 1.0
print(f"C_class = {C_CLASS_DEFAULT:.4f}")           # Cu-calibrated

# Yield by mode
from delta_theory import yield_by_mode
sigma_shear, info = yield_by_mode(fe, sigma_y_tension_MPa=150.0, mode='shear')
print(f"Fe τ_y = {sigma_shear:.1f} MPa")

3. Fatigue Life (v6.8/6.10)

$$\frac{dD}{dN} = \begin{cases} 0 & (r \leq r_{th}) \ A_{\text{eff}} \cdot (r - r_{th})^n & (r > r_{th}) \end{cases}$$

Structure Presets (No Fitting Required):

*HCP values depend on T_twin (twinning factor) †τ/σ and R_comp derived from v4.1 α-coefficient theory (Section 2)

4. unified_flc_v8_1.py — FLC 7-Mode Discrete

Core Equation

$$\varepsilon_{1,j} = |V|_{\text{eff}} \times \frac{C_j}{R_j}$$

7 Standard Forming Modes

1-Point Calibration

Measure only FLC₀ (Plane Strain) → Predict all 7 modes automatically!

from delta_theory import FLCPredictor

flc = FLCPredictor()
flc.add_from_v69('MySteel', flc0=0.28, base_element='Fe')

results = flc.predict_all_modes('MySteel')
for mode, eps1 in results.items():
    print(f"{mode}: {eps1:.3f}")

FLC Validation

5. dbt_unified.py — Ductile-Brittle Transition

from delta_theory import DBTUnified

model = DBTUnified()
result = model.temp_view.find_DBTT(d=30e-6, c=0.005)
print(f"DBTT = {result['T_star']:.0f} K")

6. lindemann.py — Iizumi-Lindemann Law

Melting point prediction from crystal geometry:

from delta_theory import iizumi_lindemann, predict_delta_L

# Predict Lindemann parameter from crystal structure
delta_L = predict_delta_L('Fe')

# Full validation
from delta_theory import validate_all, print_validation_report
results = validate_all()
print_validation_report(results)

⌨️ CLI Reference

Yield & Fatigue (v10.0)

# Single point calculation (SSOC)
python -m delta_theory.unified_yield_fatigue_v10 point --metal Fe --sigma_a 150

# Generate S-N curve
python -m delta_theory.unified_yield_fatigue_v10 sn --metal Fe --sigma_min 100 --sigma_max 300

# Calibrate A_ext
python -m delta_theory.unified_yield_fatigue_v10 calibrate --metal Fe --sigma_a 244 --N_fail 7.25e7

FLC

# Quick FLC₀ prediction
python -m delta_theory flc Cu

# All 7 modes
python -m delta_theory flc SPCC all

# List available materials
python -m delta_theory flc --list

DBT

# Single point calculation
python -m delta_theory.dbt_unified point --d 30 --c 0.5 --T 300

# Temperature axis analysis (DBTT)
python -m delta_theory.dbt_unified T_axis --d 30 --c 0.5

# Grain size axis analysis (ductile window)
python -m delta_theory.dbt_unified d_axis --T 300 --c 0.5 --find_c_crit

🧪 Testing

pytest tests/ -v

📖 Theory Background

Why "δ-Theory"?

δ_L (Lindemann Parameter) — The critical ratio of atomic displacement at melting point. This purely geometric parameter unifies explanations from material strength to fatigue limits. In v10.0, the explicit δ_L dependence is eliminated through SSOC, but the geometric spirit remains.

Key Insights

1. Materials = Highly Viscous Fluids — Deformation is "flow", not "fracture"
2. Fatigue Limits = Geometric Consequence of Crystal Structure — BCC/FCC/HCP differences emerge naturally
3. Forming Limit = Geometry + Localization — C_j captures strain path, R_j captures crystal resistance
4. SSOC — Electronic effects (d-orbitals, Jahn-Teller, relativistic) are captured through structure-selective channels, not fitting parameters
5. Fitting Parameters ≈ 0 — Derived from crystal geometry, not curve fitting

Version History

💡 Forming-Fatigue (Simple Rule)

"Forming makes it weak" — Stretched lattice = Nearly broken bonds

Before: ●──●──●──●  (r₀)
After:  ●───●───●───●  (r > r₀, about to break!)

Simple formula:

eta = eps_formed / eps_FLC      # How much capacity used
r_th_eff = r_th * (1 - eta)     # Remaining fatigue threshold

📄 License

MIT License (Code) — See LICENSE

Data sources (FatigueData-AM2022): CC BY 4.0

👥 Authors

• Masamichi Iizumi — Miosync, Inc. CEO
• Tamaki — Sentient Digital Partner

📚 Citation

@software{delta_theory_2025,
  author = {Iizumi, Masamichi and Tamaki},
  title = {δ-Theory: Unified Materials Strength, Fatigue, and Forming Framework},
  version = {10.0.0},
  year = {2025},
  url = {https://github.com/miosync-inc/delta-theory},
  doi = {10.5281/zenodo.18457897}
}