>[!warning]
>This content has not been peer reviewed.
# Tensile Test — Results
Output of **[[Tensile Test - Code]]** (`rst_tensile_test.py`).
## 1. Corrected model
The field equation is the **inverted fidelity** form:
$\sigma(\varepsilon) = E\,\varepsilon \times \mu\!\left(\frac{N}{\varepsilon},\, n\right)$
This gives Hooke's law (σ = Eε) at small strain and saturates at σ → EN at large strain. See [[Tensile Test (RST)]] §2 for the derivation and the gravity ↔ materials duality.
---
## 2. Discovery fit — 17 pure metals (annealed, 293 K)
The `--discovery` mode fits RST parameters (N, n) and Ramberg-Osgood parameters (K, m) to known yield and ultimate strength for 17 ductile metals. Both models have 2 free parameters.
### Key result: n clusters by crystal structure
*Left: N vs homologous temperature (no clear correlation across materials). Center: n vs crystal structure — FCC clusters low (≈0.35), BCC clusters high (≈1.0). Right: RST vs Ramberg-Osgood fit quality (both achieve near-zero loss).*
### Discovery table
| Metal | Crystal | T/Tₘ | E (GPa) | σ_y | σ_u | N_fit | n_fit |
|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|
| Al | FCC | 0.314 | 70 | 35 | 90 | 0.00198 | 0.465 |
| Cu | FCC | 0.216 | 130 | 70 | 220 | 0.00343 | 0.409 |
| Ni | FCC | 0.170 | 200 | 60 | 320 | 0.02305 | 0.228 |
| Ag | FCC | 0.237 | 83 | 55 | 170 | 0.00384 | 0.443 |
| Au | FCC | 0.219 | 78 | 30 | 130 | 0.00841 | 0.289 |
| Pd | FCC | 0.160 | 121 | 40 | 190 | 0.01218 | 0.258 |
| Fe | BCC | 0.162 | 211 | 170 | 350 | 0.00198 | 0.654 |
| Cr | BCC | 0.134 | 279 | 210 | 280 | 0.00102 | 1.020 |
| V | BCC | 0.134 | 128 | 180 | 260 | 0.00208 | 1.153 |
| W | BCC | 0.079 | 411 | 750 | 980 | 0.00240 | 1.485 |
| Mo | BCC | 0.101 | 329 | 325 | 500 | 0.00159 | 0.933 |
| Nb | BCC | 0.107 | 105 | 105 | 195 | 0.00207 | 0.772 |
| Ti | HCP | 0.151 | 116 | 140 | 240 | 0.00221 | 0.895 |
| Zr | HCP | 0.138 | 68 | 230 | 350 | 0.00525 | 1.535 |
| Mg | HCP | 0.317 | 45 | 21 | 90 | 0.00892 | 0.309 |
| Zn | HCP | 0.423 | 108 | 30 | 50 | 0.00053 | 0.567 |
| Co | HCP | 0.166 | 209 | 345 | 500 | 0.00244 | 1.222 |
### Crystal-type averages
| Structure | n (mean ± std) | N (mean) | Count |
|:---|:---|:---|:---|
| **FCC** | 0.349 ± 0.093 | 0.00881 | 6 |
| **BCC** | 1.003 ± 0.269 | 0.00185 | 6 |
| **HCP** | 0.906 ± 0.440 | 0.00387 | 5 |
Spearman correlation N vs T/Tₘ: ρ = 0.23, p = 0.38 (not significant across materials).
### Interpretation
- **n** measures the **effective lattice connectivity** (relational path redundancy). FCC metals with 12 slip systems and low Peierls barriers have many deformation pathways → gradual elastic-plastic transition → low n. BCC metals with higher barriers have fewer easy pathways → sharper yield → high n.
- **N** measures the **substrate rendering capacity** (saturation strain). It is material-specific and approximately equals σ_u/E for metals with moderate-to-high n; for low-n metals (FCC), N is larger because slow saturation at finite ε_max requires compensation.
- **Ramberg-Osgood** achieves identical fit quality (loss ≈ 0), confirming both are 2-parameter models fitting 2 targets. The difference is physical meaning: R-O's (K, m) are empirical; RST's (N, n) map to substrate capacity and lattice connectivity.
---
## 3. Temperature check — Cu (FCC) and Fe (BCC)
The `--temperature` mode fits N and n at 6 temperatures (77–673 K) for a single material.
*Top: N vs T (substrate capacity decreases with temperature — thermal softening). Bottom: n vs T (topology is not strictly constant but crystal-type ordering is preserved).*
### Copper (FCC, Tₘ = 1358 K)
| T (K) | T/Tₘ | σ_y | σ_u | N_fit | n_fit | σ_y/σ_u |
|:---:|:---:|:---:|:---:|:---:|:---:|:---:|
| 77 | 0.06 | 100 | 350 | 0.00576 | 0.412 | 0.286 |
| 200 | 0.15 | 85 | 280 | 0.00439 | 0.415 | 0.304 |
| 293 | 0.22 | 70 | 220 | 0.00343 | 0.409 | 0.318 |
| 400 | 0.29 | 55 | 180 | 0.00337 | 0.375 | 0.306 |
| 500 | 0.37 | 40 | 150 | 0.00445 | 0.313 | 0.267 |
| 673 | 0.50 | 25 | 100 | 0.00467 | 0.268 | 0.250 |
n mean = 0.365 ± 0.056. Range [0.268, 0.415].
### Iron (BCC, Tₘ = 1811 K)
| T (K) | T/Tₘ | σ_y | σ_u | N_fit | n_fit | σ_y/σ_u |
|:---:|:---:|:---:|:---:|:---:|:---:|:---:|
| 77 | 0.04 | 300 | 500 | 0.00240 | 0.963 | 0.600 |
| 200 | 0.11 | 230 | 420 | 0.00215 | 0.803 | 0.548 |
| 293 | 0.16 | 170 | 350 | 0.00198 | 0.652 | 0.486 |
| 400 | 0.22 | 140 | 300 | 0.00186 | 0.601 | 0.467 |
| 500 | 0.28 | 110 | 250 | 0.00175 | 0.538 | 0.440 |
| 673 | 0.37 | 80 | 190 | 0.00159 | 0.486 | 0.421 |
n mean = 0.674 ± 0.163. Range [0.486, 0.963].
### Interpretation
- **N for Fe** decreases monotonically with T (factor 1.5×), tracking σ_u. Thermal softening = reduced substrate capacity.
- **n** is NOT strictly constant: it decreases with T for both materials. The cause is that σ_y/σ_u changes with temperature (especially for BCC Fe, where the Peierls barrier is strongly T-dependent). Higher T activates more deformation pathways → higher effective connectivity → lower n.
- **Crystal-type ordering preserved:** Fe's n (0.49 at 673 K) remains above Cu's mean (0.37) across the entire range. The base topology sets the floor; temperature modulates within that range.
- n is best understood as a **dynamic degrees-of-freedom count**, not a static geometry parameter.
---
## 4. Shape comparison — RST vs Ramberg-Osgood curve shapes
The `--shapes` mode generates both RST and R-O curves using parameters calibrated to the same yield and ultimate, then measures where the full curve shapes diverge.
| Metal | Crystal | n_rst | R² (RST vs R-O) | RMSE (MPa) | Max deviation |
|:---:|:---:|:---:|:---:|:---:|:---:|
| Al | FCC | 0.47 | 0.845 | 5.8 | −36% at ε ≈ 0.02% |
| Cu | FCC | 0.41 | 0.890 | 13.5 | −33% at ε ≈ 0.02% |
| Ni | FCC | 0.23 | 0.965 | 13.0 | −14% at ε ≈ 0.1% |
| Fe | BCC | 0.65 | 0.751 | 24.7 | −30% at ε ≈ 0.1% |
| W | BCC | 1.49 | 0.686 | 51.9 | −20% at ε ≈ 0.2% |
| Ti | HCP | 0.90 | 0.673 | 16.9 | −26% at ε ≈ 0.1% |
| Co | HCP | 1.22 | 0.651 | 31.3 | −22% at ε ≈ 0.1% |
All deviations are **negative** (RST < R-O in the transition zone), meaning RST enters plasticity earlier than R-O. The maximum divergence is concentrated at the **elastic-to-plastic knee** (ε ≈ 0.01–0.2%). This is the strain range where experimental data would most effectively discriminate between the two models.
*Top panels: overlaid RST (colored) and R-O (gray) curves for 7 metals. Bottom panels: residual (RST − R-O) in MPa. Red line marks the strain of maximum relative deviation.*
---
## 5. Experimental validation — full-curve fit
The `--validate` mode fits RST (N, n) and Ramberg-Osgood (K, m) to **complete experimental stress-strain curves** — not just two summary statistics. Both models have 2 free parameters; the optimizer minimizes MSE across all data points.
### Data sources
| Dataset | Material | Type | Specimens | Source |
|:---|:---|:---|:---:|:---|
| Mendeley | Al 6061-T651 | Uniaxial tension, 20 °C, DIC strain | 19 (9 lots) | Aakash et al. (2019) [^1] |
| KupferDigital | CuZn38As (brass) | Uniaxial tension, 26 °C, extensometer | 20 | BAM (2024) [^2] |
| NIST | Al 6061-T6 | Dynamic compression (Kolsky bar, ~2500/s), 23 °C | 4 | Mates et al. (2024) [^3] |
[^1]: Aakash, B.S., Connors, J.P., Shields, M.D. "Stress-strain data for aluminum 6061-T651 from 9 lots at 6 temperatures under uniaxial and plain strain tension." *Mendeley Data*, V2, 2019. DOI: [10.17632/rd6jm9tyb6.2](https://doi.org/10.17632/rd6jm9tyb6.2). License: CC BY 4.0.
[^2]: Bundesanstalt für Materialforschung und -prüfung (BAM). "KupferDigital mechanical testing datasets: Brinell hardness, Vickers hardness, and tensile tests." *Zenodo*, 2024. DOI: [10.5281/zenodo.10820299](https://doi.org/10.5281/zenodo.10820299). License: CC BY 4.0.
[^3]: Mates, S.P. et al. "Dynamic compressive stress-strain curves of aluminum alloy 6061-T6 with rapid heating." *NIST Public Data Repository*, 2024. DOI: [10.18434/mds2-3090](https://doi.org/10.18434/mds2-3090).
### Mendeley Al 6061-T651 (the primary validation)
Room-temperature uniaxial tension, 9 specimens from different commercial lots. Strain measured by DIC (2-inch gauge length). Curves truncated at ultimate (pre-necking).
| Specimen | N_rst | n_rst | R²_rst | RMSE_rst (MPa) | R²_ro | RMSE_ro (MPa) |
|:---|:---:|:---:|:---:|:---:|:---:|:---:|
| T_020_A_1 | 0.00389 | 5.42 | 0.9885 | 8.1 | 0.9928 | 6.4 |
| T_020_A_3 | 0.00394 | 6.49 | 0.9736 | 11.7 | 0.9786 | 10.6 |
| T_020_B_2 | 0.00374 | 3.56 | 0.9912 | 7.7 | 0.9959 | 5.2 |
| T_020_C_1 | 0.00437 | 3.23 | 0.9930 | 6.4 | 0.9959 | 4.9 |
| T_020_D_2 | 0.00407 | 4.23 | 0.9842 | 10.2 | 0.9921 | 7.2 |
| T_020_E_1 | 0.00387 | 3.63 | 0.9916 | 7.0 | 0.9971 | 4.2 |
| T_020_E_3 | 0.00383 | 3.21 | 0.9859 | 7.6 | 0.9959 | 4.1 |
| T_020_G_1 | 0.00432 | 2.87 | 0.9929 | 6.3 | 0.9843 | 9.3 |
| T_020_I_1 | 0.00410 | 3.20 | 0.9802 | 10.4 | 0.9743 | 11.8 |
| **Mean** | **0.00401** | **3.98** | **0.9868** | **8.4** | **0.9896** | **7.0** |
| Std | 0.00021 | 1.14 | 0.006 | 1.9 | 0.008 | 2.9 |
**RST wins 2 of 9** specimens (G_1 and I_1, both with lower σ_y/σ_u ratio). Overall RST R² = 0.987, R-O R² = 0.990.
### KupferDigital CuZn38As (brass)
Extensometer data: only the monotonically-increasing elastic portion is usable (extensometer detachment at ~1% strain). With data restricted to the pre-yield region, both models achieve R² > 0.99 by fitting Hooke's law. This dataset does not test the elastic-to-plastic transition and is not discriminating.
Summary: RST R² = 0.995, R-O R² = 0.998 (18 specimens). Strain range too limited for shape comparison.
### NIST Al 6061-T6 (dynamic compression)
This data is **dynamic compression** at strain rates of 2400–2700/s (Kolsky bar), not quasi-static tension. The data starts at ~3% true strain (post-yield). Neither the RST nor R-O model was designed for this loading regime. Results shown for completeness:
Summary: RST R² = 0.622, R-O R² = 0.745 (4 specimens). R-O handles the flat post-yield plateau better.
### Validation figure
*Panels: experimental data (dots) overlaid with RST fit (solid) and R-O fit (dashed). Bottom: residual (model − experiment) in MPa. Mendeley Al 6061 specimens are the primary validation target.*
### Interpretation
- **RST matches real experimental curves to R² = 0.987** with only 2 physically-motivated parameters (N = substrate capacity, n = lattice connectivity). This is 0.3% below R-O's 0.990, which uses 2 empirical parameters (K, m).
- **N is highly consistent across lots:** 0.00401 ± 0.00021 across 9 specimens from different commercial sources, confirming N is a genuine material property (not an artifact of the fitting).
- **Full-curve n differs from 2-point n:** when fitting the entire curve shape (not just yield + ultimate), n increases from ~0.35 (discovery mode) to ~4.0. The full curve demands a sharper transition, indicating the discovery-mode n reflects a different aspect of the model (matching two points vs capturing the knee shape).
- **The 0.3% gap:** RST's residual is largest in the post-yield work-hardening region (ε > 2%), where the model saturates toward E·N while real materials continue to strain-harden. R-O's power-law form inherently captures gradual hardening.
---
## 6. Limitations
1. **Mendeley data is an alloy** (6061-T651), not pure aluminum. The precipitation-hardened microstructure complicates direct comparison with the pure-metal discovery fit.
2. **Kupfer data is elastic-only** (extensometer limitation). No constraint on the transition shape.
3. **NIST data is compression at high strain rate.** Not the regime the RST tensile model targets.
4. **No necking or failure modeled.** Curves are truncated at ultimate.
5. **Work-hardening gap:** RST saturates at σ ≈ E·N, while real metals continue to harden. Materials with large σ_u/σ_y ratios (e.g., annealed brass, σ_y/σ_u ≈ 0.69) will show larger residuals than materials with small ratios (e.g., Al 6061, σ_y/σ_u ≈ 0.89).
6. **Temperature data** (§3) uses handbook approximations, not measured curves.