MAT-06

True stress from engineering

σt = σe (1+εe) at constant volume.

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PlasticityISO 6892

Governing equation

\sigma_t=\sigma_e(1+\varepsilon_e),\quad \varepsilon_t=\ln(1+\varepsilon_e)

$\varepsilon_e \in \left[0, 0.8\right]$

where

\sigma_e: Engineering stress (MPa)
\varepsilon_e: Engineering strain (—)
\sigma_t: True stress (MPa)
\varepsilon_t: True strain (—)

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_t=\sigma_e(1+\varepsilon_e),\quad \varepsilon_t=\ln(1+\varepsilon_e)

$\varepsilon_e \in \left[0, 0.8\right]$

$\varepsilon_e = 0$ $\sigma_t = 350$

Along the curve

②

Domain of the variable — blue

\varepsilon_e_{\min} = 0,\quad \varepsilon_e_{\max} = 0.8

$\varepsilon_e \in \left[0, 0.8\right]$

$\varepsilon_e = 0$ $\varepsilon_e = 0$

Along the curve

③

Response curve — green

\sigma_t=\sigma_e(1+\varepsilon_e),\quad \varepsilon_t=\ln(1+\varepsilon_e)

$\varepsilon_e \in \left[0, 0.8\right]$

$\varepsilon_e = 0$ $\sigma_t = 350$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma_t}{\mathrm{d}\varepsilon_e}

$\varepsilon_e \in \left[0, 0.8\right]$

$\varepsilon_e = 0.005$ $\sigma_t^{\prime} = 350$

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-06 — True stress from engineering) is the form associated with ISO 6892. Working symbols: $\sigma_e$ , $\varepsilon_e$ $\rightarrow$ $\sigma_t$ , $\varepsilon_t$ . Constancy of volume in plastic flow converts instantaneous area to A0/(1+εe), hence true stress is engineering stress times (1+εe).

Purpose

Purpose: compute

\sigma_t

\varepsilon_t

from

\sigma_e

\varepsilon_e

in Materials via

\sigma_t=\sigma_e(1+\varepsilon_e),\quad \varepsilon_t=\ln(1+\varepsilon_e)

σt = σe (1+εe) at constant volume. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\sigma_e = 350.000\,\mathrm{MPa}

\varepsilon_e = 0.120\,\mathrm{—}

, the governing relation

\sigma_t=\sigma_e(1+\varepsilon_e),\quad \varepsilon_t=\ln(1+\varepsilon_e)

yields

\sigma_t = 392.000\,\mathrm{MPa}

\varepsilon_t = 0.1133\,\mathrm{—}

. Before necking, incompressible plasticity. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Engineering stress \sigma_e350 MPaEngineering strain \varepsilon_e0.12 —

Outputs

True stress \sigma_t392.000 MPa
True strain \varepsilon_t0.1133 —

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Narration of this film

Before necking, incompressible plasticity.

Constancy of volume in plastic flow converts instantaneous area to A0/(1+εe), hence true stress is engineering stress times (1+εe).

Reading speed

Watch on YouTube