MAT-25

Considère necking strain

For Hollomon σ = K ε^n, necking starts at εt = n. Load maximum dP = 0.

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PlasticityConsidère 1885

Governing equation

\varepsilon_{\mathrm{u}}=n,\quad\sigma_{\mathrm{u}}=K n^n

$n \in \left[0.05, 0.5\right]$

where

K: Strength coefficient (MPa)
n: Strain-hardening n (—)
\varepsilon_u: Necking strain (—)
\sigma_u: True stress at neck (MPa)

The figure, part by part

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①

Governing law — red

\varepsilon_{\mathrm{u}}=n,\quad\sigma_{\mathrm{u}}=K n^n

$n \in \left[0.05, 0.5\right]$

$n = 0.05$ $\varepsilon_u = 0.05$

Along the curve

②

Domain of the variable — blue

n_{\min} = 0.05,\quad n_{\max} = 0.5

$n \in \left[0.05, 0.5\right]$

$n = 0.05$ $n = 0.05$

Along the curve

③

Response curve — green

\varepsilon_{\mathrm{u}}=n,\quad\sigma_{\mathrm{u}}=K n^n

$n \in \left[0.05, 0.5\right]$

$n = 0.05$ $\varepsilon_u = 0.05$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\varepsilon_u}{\mathrm{d}n}

$n \in \left[0.05, 0.5\right]$

$n = 0.053$ $\varepsilon_u^{\prime} = 1$

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-25 — Considère necking strain) is the form associated with Considère 1885. Working symbols: $K$ , $n$ $\rightarrow$ $\varepsilon_u$ , $\sigma_u$ . Considère: necking when dσ/dε = σ. Power-law hardening σ = K ε^n satisfies that identity at ε = n.

Purpose

Purpose: compute

\varepsilon_u

\sigma_u

from

K

n

in Materials via

\varepsilon_{\mathrm{u}}=n,\quad\sigma_{\mathrm{u}}=K n^n

For Hollomon σ = K ε^n, necking starts at εt = n. Load maximum dP = 0. 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}

n = 0.220\,\mathrm{—}

, the governing relation

\varepsilon_{\mathrm{u}}=n,\quad\sigma_{\mathrm{u}}=K n^n

yields

\varepsilon_u = 0.220\,\mathrm{—}

\sigma_u = 573.4\,\mathrm{MPa}

. A tensile curve, a load peak, a neck. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Strength coefficient K800 MPaStrain-hardening n n0.22 —

Outputs

Necking strain \varepsilon_u0.220 —
True stress at neck \sigma_u573.4 MPa

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Narration of this film

A tensile curve, a load peak, a neck.

Considère: necking when dσ/dε = σ. Power-law hardening σ = K ε^n satisfies that identity at ε = n.

Reading speed

Watch on YouTube