r-Value (Plastic Anisotropy / Lankford Coefficient)

1. Definition

The r-value (Lankford coefficient, plastic strain ratio) is the ratio of true width strain to true thickness strain at a chosen longitudinal strain in a tensile test. It quantifies plastic anisotropy—how preferentially the sheet deforms in-plane versus through-thickness.

Higher r means the sheet resists thinning more strongly for a given length strain, which improves many deep-drawing situations.

2. Formula and Test Strain

r = ε_width / ε_thickness (true strains). Strains of about 15%–20% are commonly used for low-carbon sheet. Like n, the reported ratio depends on the reference strain chosen. If uniform elongation is below that window (typical of many AHSS grades), measure r before necking per ASTM E517 / ISO 10113 and report the strain.

Test methods: ASTM E517 and ISO 10113.

3. r0, r45, r90 and r-bar

Values are measured at 0°, 45°, and 90° to the rolling direction. The normal (average) anisotropy is often:

r̄ = (r₀ + 2 r₄₅ + r₉₀) / 4

In many cold-rolled drawing steels, r₉₀ is highest and r₄₅ lowest—do not design only on r₄₅.

4. Planar Anisotropy and Earing

Planar anisotropy is commonly expressed as:

Δr = (r₀ − 2 r₄₅ + r₉₀) / 2

Δr near zero favors non-earing behavior in cylindrical draws. Large |Δr| produces ears that must be trimmed.

5. Deep Drawing and LDR

Higher r-bar generally improves limiting draw ratio (LDR) by favoring in-plane flow over thinning. Friction and die-corner radius also matter, but AHSS guidance notes r-value as a major material lever for drawability.

6. HSS / AHSS Behavior

Hot-rolled steels and high-strength steels with tensile strength greater than about 450 MPa often have r-bar near 1. Higher-strength AHSS (tensile ≥800 MPa) can fall below 1, hurting deep-draw corners—one reason open-ended draw designs are preferred for those grades.

7. Relevance to Roll Forming

• Longitudinal roll-form bends: r-value is secondary to elongation and inside bend radius-to-thickness (R/t).
• Wide stretch / drawn features on hybrid lines: r matters.
• Hole expansion and flangeability: related anisotropy effects appear in advanced formability metrics.

8. Boundaries

• r ≠ n.
• r-value is not elastic (Young's modulus) anisotropy; it is measured in the plastic regime.
• High r does not excuse an undersized bend radius on low-elongation coil.

Buyer / Engineer FAQ

Do I need r-value on a roofing panel RFQ?

Usually not. Thickness, coating, yield, and elongation dominate. Ask for r when deep-draw or hole-expansion performance is specified.

What is a “good” r-bar?

Values greater than 1 indicate resistance to thinning and generally help cup drawing. Many HSS/AHSS sit near 1; some tensile >800 MPa grades can be below 1.

Why measure at 0°, 45°, 90°?

Rolled sheet is anisotropic. Directional r values feed average r-bar and planar anisotropy delta-r.

Is delta-r = 0 ideal?

For cylindrical deep drawing / can making, delta-r near 0 reduces earing and trimming waste.

r vs n—which for stretch failures?

Stretch/FLC concerns lean on n and thickness; draw/thinning concerns lean on r. Many stampings need both.

Can roll forming ignore r completely?

For simple longitudinal bends, often yes operationally. For wide stretch flanges, embosses, or drawn ends on hybrid lines, no.

Plant Practice Notes

Lead RFQ conversations with measurable acceptance criteria, not adjectives. Document whether a number comes from a mill certificate, a plant trial, or a published standard table.

When two heats of the same grade behave differently, pull certificates and process logs before rebuilding tooling. Most mystery forming issues are heat-to-heat property or lubrication shifts.

• Keep units consistent (MPa vs ksi; mm vs in).
• Record test methods when citing formability indices.
• Attach heat numbers to scrap photos and CAPA files.
• Separate design minimums from actual forming windows.
• Update SOPs when a new grade or coating family is introduced.
• Train receiving to stop on out-of-window certificates.

1. Define the decision this page supports (buy / form / inspect / redesign).
2. List the three certificate or process fields that matter most.
3. List shop symptoms that should trigger a re-read of this page.
4. Name the next specialist encyclopedia page to open.
5. File the lesson learned after each coil-related incident.

Auditors respond better to traceable certificate fields than to verbal grade nicknames. Keep EN 10204 type and heat mapping visible at receiving.

After any coil change that moves yield by a large step, expect springback and load changes even when thickness is identical.

Do not invent universal machine kW or m/min values on materials encyclopedia pages.

Extended Notes

Specify r̄ (r-bar / r_m) on the purchase order together with the test method (ASTM E517 or ISO 10113) and the longitudinal strain used. A single unlabeled “r” is not comparable across heats if direction or strain is missing.

When earing, ear-trim scrap, or planar flow is specified, request r₀, r₄₅, and r₉₀—not only r-bar—so Δr can be checked.

• Do not confuse Lankford r-value with inside bend radius / thickness (R/t) used on roll-form flowers.
• For grades whose uniform elongation is below the common 15%–20% low-carbon window, measure r before necking at the strain the standard allows; report the strain (for example r₁₀).
• r-value is usually omitted on roofing and structural coil certificates; require it when deep draw, hole expansion, or other thinning-limited modes are in the spec.