Stress–Strain Curves

This simulation shows the engineering stress–strain curve for a tensile test of a metal specimen. Pick a material from the menu. The Young's modulus E, strength coefficient K, hardening exponent n, and fracture strain ε_f sliders are inactive for these materials; select "Custom" to change them with the sliders (they start from the values of the material that was selected).

Set the applied strain with the strain slider, press Pull to run the test, or click and drag on the plot. The specimen on the right stretches, necks, and fractures as the strain increases.

Check true stress–strain to add the true curve (shown up to the onset of necking), 0.2% offset to find the yield strength, areas to shade the modulus of resilience (green) and toughness (blue), and unload to see the elastic recovery and the permanent strain. Zoom to elastic region shows strains up to 0.01.

The gray dashed line (0.2% offset line) starts at a strain of 0.002 and is parallel to the elastic part of the curve (slope E). The yield strength σ_y is the stress where this line crosses the stress–strain curve.

The red dashed line (unloading) shows what happens if the load is removed at the current strain. The specimen recovers elastically along a line parallel to the elastic region, and the point where the line reaches zero stress is the permanent strain ε_p that remains in the specimen.

At full scale both dashed lines look nearly vertical because the elastic strains are very small (less than 0.01). Check zoom to elastic region to see them clearly.

In the elastic region, stress is proportional to strain (Hooke's law):

\[\sigma = E\,\varepsilon\]

where \(\sigma\) is engineering stress (force per original area, MPa), \(\varepsilon\) is engineering strain (\(\Delta L/L_0\)), and \(E\) is Young's modulus.

True stress and true strain use the instantaneous area and length. Assuming constant volume during uniform deformation:

\[\sigma_T = \sigma\,(1+\varepsilon), \qquad \varepsilon_T = \ln(1+\varepsilon)\]

Plastic deformation is described by the Hollomon equation:

\[\sigma_T = K\,\varepsilon_T^{\,n}\]

so the engineering stress in the uniform plastic region is

\[\sigma = \frac{K\,[\ln(1+\varepsilon)]^{n}}{1+\varepsilon}\]

The elastic line and the Hollomon curve meet at the proportional limit. Necking begins at the maximum of the engineering curve (Considère criterion), where \(d\sigma_T/d\varepsilon_T = \sigma_T\), which gives

\[\varepsilon_{T,u} = n, \qquad \varepsilon_u = e^{n}-1, \qquad \sigma_{UTS} = K\,n^{n}e^{-n}\]

After necking begins, the engineering stress decreases until fracture at \(\varepsilon_f\). This part of the curve uses an empirical quadratic that drops to 75% of the UTS at fracture:

\[\sigma = \sigma_{UTS} - 0.25\,\sigma_{UTS}\left(\frac{\varepsilon-\varepsilon_u}{\varepsilon_f-\varepsilon_u}\right)^{2}\]

The 0.2% offset yield strength \(\sigma_y\) is where the curve crosses a line of slope \(E\) that starts at \(\varepsilon = 0.002\):

\[\sigma(\varepsilon) = E\,(\varepsilon - 0.002)\]

The modulus of resilience \(U_r\) is the elastic energy per volume absorbed up to yielding, and the toughness \(U_t\) is the total area under the engineering curve:

\[U_r = \frac{\sigma_y^{2}}{2E}, \qquad U_t = \int_0^{\varepsilon_f}\sigma\,d\varepsilon\]

1 MPa × (mm/mm) = 1 MJ/m³. When a specimen is unloaded from a plastic strain \(\varepsilon\), it recovers elastically along a line of slope \(E\), leaving a permanent strain

\[\varepsilon_p = \varepsilon - \sigma/E\]

A brittle material (gray cast iron here) fractures with little or no plastic deformation, so it has no necking and may have no 0.2% offset yield strength. Material parameters are approximate values from standard materials-science tables.

Definitions

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.