# ⚛  L1 Principle — Continuum Damage Mechanics — scalar/tensor damage

**ID:** `L1-231` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Structural Mechanics — *Kachanov-Lemaitre continuum damage*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** damage field d
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554024

---

## 🧠 1. Introduction

**Continuum Damage Mechanics — scalar/tensor damage** is a **nonlinear inverse problem** whose unknown lives in **damage field d** space, within the **Kachanov-Lemaitre continuum damage** sub-domain of **Structural Mechanics**.

Measurements consist of mechanical vibrations or strain signals via a **dic ultrasonic** sensing mechanism.

The forward operator applies, in order: L · effective stress operator; L · damage evolution operator; L · thermodynamic force operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · effective stress → L · damage evolution → L · thermodynamic force → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.thermodynamic_force` `L.damage_evolution` `L.effective_stress` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.effective_stress` | L · effective stress operator |
| `L.damage_evolution` | L · damage evolution operator |
| `L.thermodynamic_force` | L · thermodynamic force operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Structural Mechanics |
| Sub domain | Kachanov-Lemaitre continuum damage |
| Carrier | mechanical |
| Problem class | nonlinear_inverse |
| Solution space | damage_field_d |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `damage` · **Forward operator:** `damage_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N dof | — | 10000 |
| N inc | — | 200 |
| S err | — | 0 |
| Snr db | dB | 30 |
| Evolution law error | — | 0 |
| Damage threshold error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N dof | — | 100 – 1000000 |
| N inc | — | 10 – 10000 |
| S err | — | 0.0 – 0.3 |
| Snr db | dB | 0 – 40 |
| Evolution law error | — | 0.0 – 0.3 |
| Damage threshold error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xef6df6f3891114a724edf582ef9b0d60aa69be32561eb0d8fed9001d668cad8a`
- **Chain tx hash:** `0xb136ad71119e152d1cf2a696f1a8cf069b9b1b0813f864894806087d5b2a79f8`
- **Chain block:** `41554024`

---

## File Mapping

This bundle consists of: `L1-231.md`, `L1-231.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-231.md` | Source of truth — edit this | Human or LLM |
| `L1-231.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-231`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._