import type { SpringbackInput, SpringbackResult } from './types.js';
/**
 * Sheet-metal springback: how much a bend elastically opens up after the tool is
 * released, and how far to overbend so the part lands on the target angle.
 *
 * The elastic core of a bent sheet recovers on unloading, so the final radius is
 * always larger (and the final angle shallower) than the tool's. The recovery
 * grows with the R/t ratio and with the yield-strength-to-stiffness ratio Y/E —
 * which is why aluminium springs back far more than steel at the same geometry.
 *
 * @formula x  = Y · R_i / (E · T)                (dimensionless, model domain x < 0.5)
 * @formula Ks = R_i / R_f = 4x³ − 3x + 1         (springback factor, 0 < Ks ≤ 1 on the domain)
 * @formula R_f = R_i / Ks · θ_final = θ_tool · Ks ⇒ θ_overbend = θ_target / Ks
 * @reference Kalpakjian & Schmid, "Manufacturing Engineering and Technology",
 *            springback equation for pure bending (also in ASM Metals Handbook
 *            Vol. 14B, Sheet Forming). Valid for R_i > 2T, where the neutral axis
 *            sits at mid-thickness; θ scales with Ks because the neutral-axis arc
 *            length is conserved through unloading.
 *
 *            Model domain: the cubic factors as Ks = (x + 1)(2x − 1)², a double root
 *            at x = 0.5 — which is exactly the elastic limit: x ≥ 0.5 ⇔ the maximum
 *            bending strain T/2R_i stays at or below the yield strain Y/E, so the
 *            sheet never yields and springs back completely (no permanent set).
 *            Past the root the polynomial climbs again (x = 1 → Ks = 2, i.e. negative
 *            springback), which is a mathematical artifact with no physical meaning,
 *            so inputs with x ≥ 0.5 are rejected rather than evaluated (ISSUE-20260714:
 *            the unguarded cubic returned Infinity at x = 0.5 and shrinking radii beyond).
 * @units thickness/bendRadius mm · bendAngle degrees · yieldStrength MPa · elasticModulus GPa
 * @validation thickness > 0, bendRadius > 0, 0 < bendAngle < 180; for material
 *             'custom', yieldStrength > 0 and elasticModulus > 0 are required;
 *             x = Y·R_i/(E·T) < 0.5 (below the elastic limit, see @reference).
 * Even inside the domain the required overbend can pass 180° (thin sheet, large radius,
 * target angle near 180°) — a valid model prediction that no single press-brake stroke
 * can execute. The result discloses this via `overbendExceeds180` instead of throwing:
 * the number itself still tells the fabricator the part cannot be made in one hit.
 *
 * Tight bends (R_i ≤ 2T) are common in practice but sit below the model's stated
 * validity (the neutral axis shifts off mid-thickness), so rather than rejecting them
 * the result discloses the accuracy caveat via `radiusBelow2T`.
 *
 * @param input - Geometry plus material (preset or custom Y/E)
 * @returns Springback factor, final radius, springback angle, overbend angle, and
 *          disclosure flags: `overbendExceeds180` when the overbend is not achievable
 *          in a single bend, `radiusBelow2T` when the geometry is below the model's
 *          stated validity (R_i > 2T)
 * @throws {RangeError} thickness, bendRadius or bendAngle out of range
 * @throws {RangeError} material is 'custom' but yieldStrength/elasticModulus missing or non-positive
 * @throws {RangeError} x = Y·R_i/(E·T) ≥ 0.5 — the bend stays fully elastic (no permanent
 *                      set), outside the springback model's domain
 */
export declare function springback(input: SpringbackInput): SpringbackResult;
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