The Forming Quality Analyser takes a minimal material card and a measured strain state, then classifies forming quality as PASS / MARGINAL / FAIL.

1. Open the dashboard at /.
2. Set the Material card sliders: strength coefficient K (MPa), hardening exponent n, and Lankford r-values r₀, r₄₅, r₉₀.
3. Set the Nominal thickness (mm).
4. Enter the Major strain ε₁ and Minor strain ε₂ at the point of interest (from DIC, strain gauges, or simulation output).
5. Optionally enter a Measured thickness to compute thinning directly rather than deriving it from the strain state.
6. Click Assess. The API at POST /api/assess returns the verdict in <100 ms.
7. Read the Verdict panel: PASS (green) / MARGINAL (amber) / FAIL (red), FLD safety margin, thinning ratio, and any active flags.
8. The FLD plot and Hardening-curve plot update live as you move sliders — no re-submit needed. ^[3]

Data flows left-to-right: the material card feeds physics functions that compute the FLD safety margin and thinning ratio; the classifier combines them into a verdict. ^{[3] [4]}

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flowchart LR
  mc["Material Card\nK · n · r₀ · r₄₅ · r₉₀ · t₀"] --> flc["fld0_keeler_brazier\nforming_limit_curve"]
  mc --> hard["hollomon_flow_stress\nHardening plot"]
  mc --> hill["hill48_equivalent_stress\nYield surface"]
  ss["Strain state\nε₁ · ε₂"] --> margin["safety_margin_to_fld"]
  flc --> margin
  ss --> thin["thinning_ratio"]
  mc --> thin
  margin --> cls["classify_quality"]
  thin --> cls
  cls --> vrd["QualityResult\nPASS / MARGINAL / FAIL"]
  hard --> hplot["Hardening-curve tile\n/plot/hardening"]
  flc --> fplot["FLD tile\n/plot/fld"]
  ss --> fplot

The plot tiles update via postMessage from the outer page — sliders do not reload the iframe; they push new curve data over GET /api/preview.

Verdict

PASS (green) — FLD safety margin > 0.10 AND thinning < 20 %. The part meets all forming-quality criteria.
MARGINAL (amber) — margin in (0, 0.10] OR thinning in [20 %, 25 %). The part is near the forming limit — review recommended.
FAIL (red) — margin ≤ 0 (strain state above the FLC) OR thinning ≥ 25 %. Do not ship. ^[3]

FLD Safety Margin

A margin of 0.10 means the measured major strain sits 10 % below the forming-limit curve at the same minor strain. The FLD tile marks the operating point as a filled circle against the copper-coloured FLC. The closer the circle is to the curve, the higher the forming risk. A circle above the curve is a FAIL. ^{[3] [4]}

Thinning Ratio

A thinning ratio of 0.25 (25 %) is the industry rule-of-thumb red line for forming-grade steels and aluminiums. If measured_thickness_mm is omitted, thinning is estimated from the strain state via volume conservation: $\theta \approx 1 - e^{-(\varepsilon_1+\varepsilon_2)}$.

Flags

The flags array in the API response carries human-readable reasons — e.g. ABOVE FLC — margin -0.042 or THINNING 26.3% >= 25% limit.

Typical DP800 deep-draw with a moderate strain state:

POST /api/assess
{
  "name": "DP800",
  "thickness_mm": 1.5,
  "K_mpa": 1310,
  "n": 0.16,
  "r0": 0.83,  "r45": 0.94,  "r90": 0.97,
  "major_strain": 0.15,
  "minor_strain": -0.05
}

Expected response: verdict = "PASS", fld_safety_margin ≈ 0.31, thinning_ratio ≈ 0.09. The operating point sits well below the Keeler-Brazier FLC on the drawing side (ε₂ < 0). ^[3]

HSLA350 blank with a strain state above the FLC (biaxial stretching corner):

POST /api/assess
{
  "name": "HSLA350",
  "thickness_mm": 2.0,
  "K_mpa": 650,
  "n": 0.18,
  "r0": 1.1,  "r45": 0.9,  "r90": 1.0,
  "major_strain": 0.42,
  "minor_strain": 0.05
}

Expected response: verdict = "FAIL", FLD margin < 0 (operating point above the FLC stretching-side arm). Remediation: increase blank-holder force or reduce punch travel to keep ε₁ below FLD₀ + 0.6 ε₂. ^{[3] [4]}

The dashboard (GET /) provides the complete single-page analysis workflow: material-card sliders, strain-state inputs, an Assess button, a quality-verdict panel, and two live Plotly iframe tiles (FLD and hardening curve).

Sliders call GET /api/preview on every change and push updated curve data into the plot iframes via window.postMessage, keeping both plots in sync without a full page reload.

POST /api/assess — primary quality-assessment endpoint. Accepts a JSON body matching the MaterialCardInput schema and returns a QualityResult object.

Request fields

Field	Range	Default	Description
name	—	DP800	Grade designation (label only)
thickness_mm	0.5–6.0 mm	1.5	Nominal blank thickness
K_mpa	100–3000 MPa	1310	Hollomon strength coefficient [2]
n	0.01–0.60	0.16	Hollomon strain-hardening exponent [2]
r0	0.1–4.0	0.83	Lankford r₀ at rolling direction [1]
r45	0.1–4.0	0.94	Lankford r₄₅ at 45° [1]
r90	0.1–4.0	0.97	Lankford r₉₀ at transverse direction [1]
major_strain	0–1	0.15	Major engineering strain ε₁
minor_strain	−0.5–0.5	−0.05	Minor engineering strain ε₂
measured_thickness_mm	0.5–6.0 | null	null	Direct thickness measurement; if omitted, thinning derived from strains

Response fields (QualityResult)

Field	Type	Description
verdict	str	"PASS", "MARGINAL", or "FAIL"
score_pct	float [0, 100]	Normalised quality score (convenience indicator)
fld_safety_margin	float	Fractional distance below FLC; negative = above limit
thinning_ratio	float	(t₀ − t) / t₀; ≥ 0.25 triggers FAIL
message	str	Human-readable verdict summary
flags	list[str]	Active failure / warning reasons

GET /api/preview — lightweight endpoint for live slider updates. Returns FLC trace data (200 points) and Hollomon hardening-curve data (200 points) as plain Python lists.

The outer-page JavaScript posts this data into the plot iframes via window.postMessage, triggering a Plotly.restyle call inside each iframe without reloading the iframe source. This keeps the UI responsive at slider-drag frequency.

Parameters: same as MaterialCardInput passed as query-string arguments.

Response shape:

{
  "fld": {"eps2": [...], "eps1": [...]},
  "hardening": {"eps": [...], "sigma": [...]},
  "operating_point": {"major": float, "minor": float}
}

GET /plot/fld — standalone axes-only Forming Limit Diagram tile rendered via Plotly. Draws the FLC (copper line) and the current operating point (filled circle).

Designed as an iframe tile: the outer page loads it once and then pushes live updates via postMessage without reloading the iframe src. Accepts the full material-card and strain parameters as query-string arguments for the initial render.

The FLC is computed by forming_limit_curve() using the Keeler-Brazier intercept. ^[3]

GET /plot/hardening — standalone axes-only Hollomon hardening-curve tile. Shows flow stress σ (MPa) vs. plastic strain εₚ over [0.0001, 0.50]. Same postMessage live-update pattern as the FLD tile. ^[2]

Useful for checking whether the K and n values entered are physically reasonable for the grade — compare visually against typical tensile-test curves for the same grade.

The FLD safety margin quantifies how far the measured strain state lies below the Forming Limit Curve at the same minor strain. ^{[3] [4]}

$$m = \frac{\varepsilon_1^{\text{limit}}(\varepsilon_2) - \varepsilon_1^{\text{meas}}}{\varepsilon_1^{\text{limit}}(\varepsilon_2)}$$

where $\varepsilon_1^{\text{limit}}(\varepsilon_2)$ is the FLC ordinate at the measured minor strain (from forming_limit_curve()), and $\varepsilon_1^{\text{meas}}$ is the measured major strain.

• $m > 0.10$ → PASS
• $0 < m \leq 0.10$ → MARGINAL
• $m \leq 0$ → FAIL (operating point above FLC)

The thinning ratio measures the fractional thickness loss at the point of interest.

$$\theta = \frac{t_0 - t_{\text{meas}}}{t_0}$$

where $t_0$ is the nominal blank thickness and $t_{\text{meas}}$ is either the directly measured thickness or the value derived from volume conservation under plane stress:

$$t_{\text{meas}} = t_0 \exp(-(\varepsilon_1 + \varepsilon_2))$$

Industry rule-of-thumb limits for forming-grade steel and aluminium:

• $\theta < 0.20$ (20 %) → PASS
• $0.20 \leq \theta < 0.25$ → MARGINAL
• $\theta \geq 0.25$ (25 %) → FAIL

The classifier applies two independent checks and combines them to a three-bucket verdict. A single-metric FAIL overrides the overall verdict regardless of the other metric. ^[3]

Verdict	FLD margin condition	Thinning condition
PASS	$m > 0.10$	$\theta < 0.20$
MARGINAL	$0 < m \leq 0.10$	$0.20 \leq \theta < 0.25$
FAIL	$m \leq 0$	$\theta \geq 0.25$

Rule precedence: FAIL is checked first, then MARGINAL, then PASS.

The quality score maps the two forming metrics to a single scalar in [0, 100] for display on the gauge tile. It is a convenience indicator only — the shipping decision is made by classify_quality, not by this score.

$$Q = \operatorname{clip}\!\left(50(1+m) + 50 \cdot \frac{0.25 - \theta}{0.25},\; 0,\; 100\right)$$

where $m$ is the FLD safety margin and $\theta$ is the thinning ratio. A score of 100 corresponds to a 100 % safety margin and zero thinning. At the FAIL boundary (margin = 0, thinning = 0.25), the score resolves to approximately 50.

The normal anisotropy $\bar{r}$ summarises directional plastic resistance to thinning. It is used internally to parametrise the Hill-1948 yield criterion. ^[1]

$$\bar{r} = \frac{r_0 + 2r_{45} + r_{90}}{4}$$

A value $\bar{r} > 1$ indicates good resistance to thinning — the material deforms laterally rather than through the thickness. Typical values: DP800 ≈ 0.9, IF steel ≈ 1.8, AA6014 ≈ 0.6.

The Hill-1948 quadratic yield criterion computes the equivalent stress for anisotropic sheet-metal under plane-stress conditions. ^[1]

$$\bar{\sigma} = \sqrt{(G+H)\sigma_{xx}^2 - 2H\sigma_{xx}\sigma_{yy} + (F+H)\sigma_{yy}^2 + 2N\sigma_{xy}^2}$$

The anisotropy coefficients are parametrised by the Lankford r-values via the standard substitution (Hill 1948, eq. 5.7, p. 286):

$$G = \frac{1}{1+r_0}, \quad H = \frac{r_0}{1+r_0}, \quad F = \frac{r_0}{r_{90}(1+r_0)}, \quad N = \frac{(r_0+r_{90})(1+2r_{45})}{2r_{90}(1+r_0)}$$

The quadratic form is positive semi-definite for positive r-values, so $\bar{\sigma} \geq 0$ by construction. A QualityGateError is raised if the computed equivalent stress is non-finite.

Hollomon's power law relates flow stress to equivalent plastic strain for metals in the work-hardening regime. ^[2]

$$\sigma = K \varepsilon_p^n$$

where $K$ (MPa) is the strength coefficient and $n$ is the strain-hardening exponent (Hollomon 1945, eq. 3, p. 271). Valid domain: $\varepsilon_p > 0$, $n \in (0, 1)$. Typical values for DP800: $K \approx 1310$ MPa, $n \approx 0.16$. The exponent $n$ also controls the Keeler-Brazier FLD₀ intercept — higher $n$ raises the forming limit. ^[3]

A CalibrationError is raised for negative strain inputs (Hollomon is undefined for $\varepsilon_p < 0$).

The Keeler-Brazier formula gives the plane-strain intercept FLD₀ of the forming-limit curve as a function of sheet thickness and hardening exponent. ^[3]

$$\text{FLD}_0 = \frac{23.3 + 14.13\,t}{100} \cdot \frac{n}{0.21}$$

where $t$ is the sheet thickness in mm and $n$ is the Hollomon exponent (Keeler & Brazier 1977, eq. 4, p. 521). The factor $(n/0.21)$ normalises the original Keeler curve (fit on $n = 0.21$ mild steel) to the material at hand.

Validity window: $0.5 \leq t \leq 3$ mm. The app saturates $t$ at 3 mm for thicker blanks to avoid extrapolating outside the empirically calibrated range. ^[4]

The full FLC is constructed by a two-slope linearisation anchored at the plane-strain intercept FLD₀ (Keeler-Brazier ^[3], conceptual basis M-K ^[4]):

$$\varepsilon_1^{\text{limit}}(\varepsilon_2) = \begin{cases}\text{FLD}_0 - \varepsilon_2 & \varepsilon_2 \leq 0 \;\text{(drawing side, slope} -1\text{)} \\\text{FLD}_0 + 0.6\,\varepsilon_2 & \varepsilon_2 > 0 \;\text{(stretching side, slope} +0.6\text{)}\end{cases}$$

The drawing-side slope of $-1$ reflects that deep-drawing is constrained by the same total major strain regardless of minor strain. The stretching-side slope of $+0.6$ is the standard M-K-derived envelope value used by most press-shop FLD overlays. ^[4]

The classify_quality() function applies a two-variable rule table to issue the three-bucket verdict. Thresholds are industry-typical values for forming-grade sheet steels and aluminium alloys. ^[3]

Rule (evaluated in order — first match wins):

1. FAIL if $m \leq 0$ or $\theta \geq 0.25$.
2. MARGINAL if $m \leq 0.10$ or $\theta \geq 0.20$.
3. PASS otherwise.

A hard FAIL on either metric overrides the other metric's result — there is no weighted averaging. A part that clears the FLD check but exceeds the thinning limit is a FAIL.