>[!warning] >This content has not been peer reviewed. # Tensile Test — RST application **Pillar:** The uniaxial tensile test (stress σ vs strain ε) is mapped onto the Resource Triangle using the **inverted fidelity** identity. The same μ function used in the gravity sector governs the elastic-to-plastic transition, but with the argument flipped: gravity uses μ(signal/threshold) while materials use μ(threshold/signal). This duality derives plasticity as **substrate saturation** — the lattice's rendering capacity is exceeded. --- ## 1. Standard view (tensile test) - **Elastic regime:** σ = E ε (Hooke's law); E = Young's modulus. - **Yield point:** strain at which plastic deformation begins; stress may plateau or drop. - **Ultimate strength:** maximum stress before necking/failure. - **Temperature:** higher T typically lowers yield and modulus (more disorder, more thermal activation). - **Lattice / structure:** crystal structure and defects set the sharpness of the elastic–plastic transition. --- ## 2. The Fidelity Inversion (Gravity ↔ Materials) In the gravity sector, the RST field equation is: $g_N = \frac{g_{obs}^2}{(g_{obs}^n + a_0^n)^{1/n}}$ The noise floor $a_0$ **boosts** a weak gravitational signal — this is the MOND-like effect that replaces dark matter. At weak acceleration ($g_{obs} \ll a_0$), the substrate amplifies the observed gravity above the Newtonian prediction. In the materials sector, the **same fidelity function** produces the opposite effect. When the deformation request (strain ε) exceeds the substrate's rendering capacity (N), the stress response **saturates** below the elastic prediction: $\sigma(\varepsilon) = E\,\varepsilon \times \mu\!\left(\frac{N}{\varepsilon},\, n\right)$ | Regime | Gravity sector | Materials sector | |:---|:---|:---| | **Strong signal** | Newtonian (μ → 1) | **Hooke's law** (σ = Eε) | | **Weak signal / high load** | MOND boost (g > g_N) | **Yield / saturation** (σ < Eε) | | **Noise role** | Boosts weak signal | Limits strong deformation | | **Fidelity argument** | μ(g/a₀, n) | μ(N/ε, n) | The inversion is mandatory: a finite-bandwidth substrate that **assists** weak signals must also **lag** when overloaded. These are two sides of the same $L^n$ resource budget. --- ## 3. RST mapping (corrected model) | Tensile concept | RST concept | |:---|:---| | Imposed strain ε | **Signal** (deformation request to the substrate). | | Stress response σ | **Rendered output** = Eε × μ(N/ε, n). | | Young's modulus E | Scale factor linking substrate fidelity to physical stress. | | Temperature / disorder | Modulates **N** (substrate capacity) and **n** (effective connectivity). | | Crystal structure | Sets the **base topology** n_lattice (FCC < HCP < BCC). | **Corrected field equation:** $\sigma(\varepsilon) = E\,\varepsilon \times \mu\!\left(\frac{N}{\varepsilon},\, n\right) = \frac{E \cdot N}{\left(1 + \left(\frac{N}{\varepsilon}\right)^n\right)^{1/n}}$ - **ε ≪ N** (elastic): μ → 1, so σ → Eε (Hooke's law). - **ε ≫ N** (saturated): μ → N/ε, so σ → EN (ultimate stress plateau). - **ε ≈ N** (transition): the elastic–plastic "knee," whose sharpness is controlled by n. --- ## 4. Parameters and their physical meaning ### N — Substrate rendering capacity N sets the strain scale at which the substrate can no longer maintain elastic fidelity. The saturation stress is approximately E·N, so N ≈ σ_u / E. - **Cross-material:** N is material-specific. It does NOT correlate with homologous temperature T/Tₘ across different metals. - **Single material, varying T:** N decreases monotonically with temperature (verified for Fe). Higher T → lower rendering capacity → lower ultimate strength. ### n — Effective lattice connectivity (Active Morphism Paths) n measures the **relational path redundancy** of the lattice at the operating temperature: how many deformation pathways are available to distribute the workload. **Discovery fit (17 pure metals, 293 K):** | Crystal structure | n (mean ± std) | Interpretation | |:---|:---|:---| | **FCC** (coord 12) | **0.35 ± 0.09** | High redundancy — 12 slip systems, many pathways, gradual transition | | **BCC** (coord 8) | **1.00 ± 0.27** | Low redundancy — higher Peierls barriers, sharper yield | | **HCP** (coord 12) | **0.91 ± 0.44** | Variable — depends on c/a ratio and available twinning modes | n is **not** a static topology constant. It is the **effective connectivity** at a given temperature: $n_{eff}(T) = n_{lattice} \times g(T/T_m)$ where $n_{lattice}$ is the 0 K base topology and $g$ captures thermal activation of additional slip systems. As T increases, more pathways become available, effective redundancy rises, and n decreases (softer transition). The crystal-type ordering (FCC < BCC) is preserved across the entire temperature range tested (77–673 K). --- ## 5. Tensile n derivation from slip-system geometry (Tier 2.5) **Goal:** Replace "FCC → 0.35 fitted" with a structural derivation from slip-system topology. See [[expanded theory/The Spectrum of Relational Topologies]] §8 for the connectivity heuristic (approximation for computational efficiency, not part of core RST derivation). ### 5.1 Slip-system enumeration - **FCC:** 12 slip systems — 4 {111} planes × 3 ⟨110⟩ directions per plane (Hull & Bacon 2011; standard metallurgy). Easy glide; many redundant pathways. - **BCC:** 48 slip systems before restriction (4 {110} × 6 ⟨111⟩ + 6 {211} × 2 ⟨111⟩ + …); at yield, **pencil glide** (⟨111⟩ screw dislocation) restricts to effectively **1D** path — one Burgers vector dominates. - **HCP:** Depends on c/a; basal, prismatic, pyramidal slip; twinning modes add complexity. Variable connectivity. **Reference:** Hull, D. & Bacon, D. J. (2011). *Introduction to Dislocations*, 5th ed. Butterworth-Heinemann. ### 5.2 Active systems at yield At the elastic–plastic transition, only a **subset** of slip systems are thermally activated. For FCC, typically **~3** of the 12 systems are simultaneously active (Schmid factor, stress orientation). For BCC, pencil glide gives **1** effective path. ### 5.3 Derivation: $n = 1/n_{\text{active}}$ When workload is **distributed** across $k$ redundant parallel paths, $n \approx 1/k$ (Spectrum §8, "Reciprocal paths" heuristic). | Crystal | $n_{\text{active}}$ | $n_{\text{theory}} = 1/n_{\text{active}}$ | $n_{\text{fitted}}$ | Match | |:---|:---|:---|:---| | FCC | ~3 | 1/3 ≈ 0.33 | 0.35 ± 0.09 | 95% | | BCC | 1 (pencil glide) | 1.0 | 1.00 ± 0.27 | 100% | **Epistemological status:** Tier 2.5 (post-hoc derivation with independent structural input). Full Tier 3 requires predicting $n$ for an **unseen** metal before any fit. ### 5.4 Testable prediction A **new FCC metal** (not in the 17-metal discovery fit) should have $n \in [0.25, 0.45]$ from full-curve fitting. A new **BCC** metal: $n \in [0.7, 1.3]$. Violation (e.g. FCC with $n > 1.0$) would refute the connectivity rule. See [[expanded theory/Fidelity Inversion — Gravity and Materials]] §7 (falsifiability). ### 5.5 Optional script `rst_tensile_n_derivation.py` — compute $n_{\text{theory}}$ from slip-system counts; compare with fitted values; output table. See [[expanded theory/The Spectrum of Relational Topologies]] for the derivation engine. --- ## 6. Results See **[[Tensile Test Results]]** for: - Discovery fit table (17 metals, RST vs Ramberg-Osgood) - Crystal-structure correlation plot - Temperature check (Cu FCC, Fe BCC) - Shape comparison (RST vs R-O curve shapes for 7 metals) - **Experimental validation** against full stress-strain curves from three open-access datasets: - Mendeley Al 6061-T651 (9 room-temp specimens from 9 lots) [^1] - KupferDigital CuZn38As brass (18 specimens) [^2] - NIST Al 6061-T6 dynamic compression (4 specimens) [^3] **Key validation result:** RST fits full experimental Al 6061-T651 stress-strain curves to R² = 0.987 (mean over 9 specimens), compared to R² = 0.990 for Ramberg-Osgood. N is consistent across lots (0.00401 ± 0.00021), confirming it is a genuine material property. [^1]: Aakash, B.S. et al. *Mendeley Data*, V2, 2019. DOI: [10.17632/rd6jm9tyb6.2](https://doi.org/10.17632/rd6jm9tyb6.2). [^2]: BAM. *Zenodo*, 2024. DOI: [10.5281/zenodo.10820299](https://doi.org/10.5281/zenodo.10820299). [^3]: Mates, S.P. et al. *NIST PDR*, 2024. DOI: [10.18434/mds2-3090](https://doi.org/10.18434/mds2-3090). ## 7. Limitations and open questions 1. **Two targets, two parameters (discovery mode):** The discovery fit matches yield and ultimate exactly — any 2-parameter model can do this. The signal is in the **clustering of n by crystal structure**, not the fit quality. 2. **Full-curve validation performed but limited:** Experimental validation (§5 of Results) uses Al 6061-T651 data from Mendeley [^1]. RST achieves R² = 0.987 — competitive with R-O (0.990) but with a systematic residual in the post-yield work-hardening region. 3. **Work-hardening gap:** The RST saturation model (σ → E·N) does not capture strain-hardening above yield. Materials with large σ_u/σ_y gaps (brass: 0.69) show larger residuals than those with small gaps (Al 6061: 0.89). 4. **Validation data is an alloy:** Al 6061-T651 is precipitation-hardened, not pure Al. Direct comparison with the pure-metal discovery fit requires caution. 5. **No necking/failure:** The corrected model is monotonically increasing (no stress maximum). Experimental curves are truncated at ultimate before comparison. 6. **Literature data for discovery:** Yield/ultimate values are handbook approximations for annealed pure metals. Real materials vary with processing, grain size, and purity. 7. **ε_max dependence:** The fitted N depends on the choice of maximum strain (currently 10%). A different ε_max shifts N but not n significantly. ## 8. Links - **Code:** [[Tensile Test - Code]] - **Results:** [[Tensile Test Results]] - **Resource Triangle, fidelity:** [[expanded theory/Resource Triangle]], [[expanded theory/Relational Substrate Theory (RST)]] - **Fidelity inversion (gravity ↔ materials duality):** [[expanded theory/Fidelity Inversion — Gravity and Materials]] - **Electronic transport (n_el vs n_tensile comparison):** [[../Electronic Transport/Electronic Transport (RST)]] - **Roadmap:** [[../../Applications Roadmap]]