delta-theory 10.3.1

Unified materials strength and fatigue prediction based on geometric first principles

δ-Theory: Unified Materials Strength & Fatigue Framework

"Nature is Geometry" — Predicting material properties from Geometric Structural Principles

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🎯 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?

Module	Predicts	Method	Accuracy
v10.3	Creep Module	Q_self = k_B × T_m × Q_base(struct) × g_ssoc(pattern)	3.5% (20 metals)
v10.2	AM Alloy Fatigue Module	N = min( N_init + N_prop, N_static )	3.2% (25 metals)
v10.0	Yield stress σ_y	SSOC f_de (δ_L-free)	3.2% (25 metals)
v4.1	τ/σ, R_comp	α-coefficient geometry	Cu-calibrated
v6.10	Fatigue life N	r_th (BCC=0.65, FCC=0.02, HCP=0.20)	4-7%
v8.1	Forming Limit Curve FLC	7-mode discrete, 1-point calibration	4.7%
DBT	Ductile-Brittle Transition	Grain size, segregation	—
Lindemann	Melting point prediction	Iizumi-Lindemann Law	—

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📦 Installation

pip install delta-theory

From Source

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

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🚀 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!

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📦 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

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🔬 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

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

Symbol	Meaning	Source
8√5/(5π)	≈ 1.1384, geometric coefficient	Derived
M	= 3.0, unified Taylor factor (all structures)	Geometric
Z	Bulk coordination number	Crystal structure
α₀	Packing fraction (BCC=0.289, FCC=0.250, HCP=0.250)	Crystal geometry
(b/d)²	= 3/2, universal geometric ratio	Crystal geometry
f_de	SSOC electronic factor (structure-selective)	This work
√(E·kT)	Core energy term (δ_L-free)	Thermodynamic
V_act	= b³, activation volume	Crystal geometry
HP	= 1 - T/T_m, homologous fraction	Temperature

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

Structure	Channel	X (input)	g_d (gate)	f_de formula
FCC	PCC	μ (shear modulus)	{0, 1} discrete	f_μ × f_shell × f_core × f_lanthanide
BCC	SCC	ΔE_P (Peierls)	d⁴ JT anomaly	f_JT × f_5d × f_lat × f_complex (or f_sp)
HCP	PCC	R (CRSS ratio)	sigmoid	f_elec × f_aniso × f_ca × f_5d × f_lanthanide × f_sp_cov

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

v10.1 Gate Extensions

Gate	Physics	Targets
f_lanthanide (FCC/HCP)	4f crystal field: f = 1 + 0.423×n_f_eff, + 5d¹ contribution	Ce, Nd
f_complex (BCC)	Complex unit cell: (N_atoms/2)^0.25	Mn (58 at/cell)
f_sp (BCC)	Unified sp branch: pure sp=0.10, p-block d¹⁰=0.80	Li, Na, Sn
f_elec d¹ gate (HCP)	Period-dependent directionality: ≤4→3.0, ≥5→1.5	Sc, Y
f_sp_cov (HCP)	sp³ covalent bonding: f=1.905	Be
fcc_gate p-block (FCC)	d¹⁰ + p-block → g_d=0	In

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)

Structure	Metals	MAE	Key Results
BCC (11)	Fe, W, V, Cr, Nb, Mo, Ta, Li, Na, Mn, Sn	2.0%	W d⁴ JT: 744 vs 750 MPa
FCC (12)	Cu, Ni, Al, Au, Ag, Pt, Pd, Ir, Rh, Pb, Ce, In	10.6%	Ce 4f: 65.0 vs 65 MPa
HCP (14)	Ti, Mg, Zn, Zr, Hf, Re, Cd, Ru, Co, Be, Sc, Y, Nd, Bi	4.0%	Be sp_cov: 300 vs 300 MPa
All 37	—	5.5%	Cd, In excluded: ~2.6%
<5% error	32/37	<10%	35/37

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)

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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

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

Symbol	Formula	Meaning
α_t	(1/Z) Σ max(b·d, 0)	Tensile projection of bond vectors
α_s	(1/Z) Σ |b·n| |b·s|	Shear projection onto slip system (n, s)
C_class	(τ/σ)_Cu / (α_s/α_t)_FCC	Cu torsion 1-point calibration
T_twin	0.6 (Mg) ~ 1.0 (most)	Twinning correction factor
A_texture	1.0 (default)	Texture adjustment slot

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

Structure	Tensile ref	Slip system	α_s/α_t
BCC	[100]	{112}⟨111⟩	0.4714
FCC	[110]	{111}⟨110⟩	0.4082
HCP	[100]	(0001)⟨1000⟩	0.4330

Compression Ratio R = σ_c / σ_t

Driven by twinning asymmetry in HCP metals:

Metal	R_comp	Mechanism
BCC, FCC	1.0	Symmetric (no twinning effect)
Ti	1.0	Slip-dominated HCP
Mg	0.6	Twin-dominated (strong tension-compression asymmetry)
Zn	1.2	Reverse twinning effect

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")

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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):

Structure	r_th	n	τ/σ †	R_comp †	Fatigue Limit
BCC	0.65	10	0.565	1.0	✅ Clear
FCC	0.02	7	0.565	1.0	❌ None
HCP	0.20	9	0.327*	0.6*	△ Intermediate

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

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4. unified_flc_v8_1.py — FLC 7-Mode Discrete

Core Equation

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

Component	Formula	Description
|V|_eff	Calibrated from FLC₀	Material forming capacity
C_j	1 + 0.75β + 0.48β²	Localization correction (frozen)
R_j	w_σ + w_τ/(τ/σ) + w_c/R_comp	Mixed resistance

7 Standard Forming Modes

Mode	β	C_j	Physical Meaning
Uniaxial	-0.370	0.788	Deep drawing (tension + compression)
Deep Draw	-0.306	0.815	Drawing dominant
Draw-Plane	-0.169	0.887	Transition region
Plane Strain	0.000	1.000	FLC₀ reference
Plane-Stretch	+0.133	1.108	Transition to biaxial
Stretch	+0.247	1.214	Stretching dominant
Equi-biaxial	+0.430	1.411	Balanced biaxial tension

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

Material	Structure	MAE	Data Points
Cu	FCC	3.4%	7
Ti	HCP	4.8%	7
SPCC	BCC	4.2%	7
Al5052	FCC	6.8%	7
SUS304	FCC	3.9%	7
DP590	BCC	4.2%	7
Mg_AZ31	HCP	5.6%	7
Overall	—	4.7%	49

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5. dbt_unified.py — Ductile-Brittle Transition

View	Fixed Axis	Solve For	Use Case
View 1	Temperature T	Grain size d*	Ductile window detection
View 2	Grain size d	Temperature T*	DBTT prediction
View 3	d, T	Time t	Segregation evolution

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")

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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)

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⌨️ 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

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🧪 Testing

pytest tests/ -v

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📖 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

Version	Feature
v4.1	τ/σ and R_comp from α-coefficient geometry (Cu 1-point calibration)
v5.0	Yield stress from δ-theory
v6.9b	Unified yield + fatigue with multiaxial (τ/σ, R_comp)
v6.10	Universal fatigue validation (2472 points)
v7.0	Geometric factorization: f_d → (b/d)² × f_d_elec
v7.2	FLC from free volume consumption
v8.1	FLC 7-mode discrete formulation
v8.2	v6.9 integration + CLI commands
v10.0	SSOC: δ_L-free unified yield stress (25 metals, 3.2% MAE)

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💡 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

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📄 License

MIT License (Code) — See LICENSE

Data sources (FatigueData-AM2022): CC BY 4.0

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👥 Authors

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

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📚 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}
}

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"Nature is Geometry" 🔬

From yield stress to fatigue life to forming limits — all from crystal structure