MAT-36

Hollomon hardening

σ = K ε^n. Power-law true stress–strain in the plastic range.

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PlasticityHollomon 1945

Governing equation

\sigma=K\varepsilon^n

$\varepsilon \in \left[0.001, 0.6\right]$

where

K: Strength coefficient (MPa)
\varepsilon: True strain (—)
n: Hardening n (—)
\sigma: True stress (MPa)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\sigma=K\varepsilon^n

$\varepsilon \in \left[0.001, 0.6\right]$

$\varepsilon = 0.001$ $\sigma = 201$

Along the curve

②

Domain of the variable — blue

\varepsilon_{\min} = 0.001,\quad \varepsilon_{\max} = 0.6

$\varepsilon \in \left[0.001, 0.6\right]$

$\varepsilon = 0.001$ $\varepsilon = 0.001$

Along the curve

③

Response curve — green

\sigma=K\varepsilon^n

$\varepsilon \in \left[0.001, 0.6\right]$

$\varepsilon = 0.001$ $\sigma = 201$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma}{\mathrm{d}\varepsilon}

$\varepsilon \in \left[0.001, 0.6\right]$

$\varepsilon = 0.005$ $\sigma^{\prime} = 14280.8$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-36 — Hollomon hardening) is the form associated with Hollomon 1945. Working symbols: $K$ , $\varepsilon$ , $n$ $\rightarrow$ $\sigma$ . Hollomon fitted the plastic branch of many metals to a power law. n is the strain-hardening exponent, K the strength coefficient.

Purpose

Purpose: compute

\sigma

from

K

\varepsilon

n

in Materials via

\sigma=K\varepsilon^n

σ = K ε^n. Power-law true stress–strain in the plastic range. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

K = 800.000\,\mathrm{MPa}

\varepsilon = 0.080\,\mathrm{—}

n = 0.200\,\mathrm{—}

, the governing relation

\sigma=K\varepsilon^n

yields

\sigma = 482.7\,\mathrm{MPa}

. A log–log σ–ε line, an n slope. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Strength coefficient K800 MPaTrue strain \varepsilon0.08 —Hardening n n0.2 —

Outputs

True stress \sigma482.7 MPa

Reading speed

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Narration of this film

A log–log σ–ε line, an n slope.

Hollomon fitted the plastic branch of many metals to a power law. n is the strain-hardening exponent, K the strength coefficient.

Reading speed

Watch on YouTube