5.3: True and Nominal Stresses and Strains

It is common during uniaxial (tensile or compressive) testing to equate the stress to the force divided by the original sectional area and the strain to the change in length (along the loading direction) divided by the original length. In fact, these are “engineering” or “nominal” values. The true stress acting on the material is the force divided by the current sectional area. After a finite (plastic) strain, under tensile loading, this area is less than the original area, as a result of the lateral contraction needed to conserve volume, so that the true stress is greater than the nominal stress. Conversely, under compressive loading, the true stress is less than the nominal stress.

Consider a sample of initial length L0, with an initial sectional area A0. For an applied force F and a current sectional area A, conserving volume, the true stress can be written

$\sigma_{\mathrm{T}}=\frac{F}{A}=\frac{F L}{A_{0} L_{0}}=\frac{F}{A_{0}}\left(1+\varepsilon_{\mathrm{N}}\right)=\sigma_{\mathrm{N}}\left(1+\varepsilon_{\mathrm{N}}\right)$

where $$\sigma_n$$ is the nominal stress and $$\varepsilon_{\mathrm{N}}$$ is the nominal strain. Similarly, the true strain can be written

$\varepsilon_{\mathrm{T}}=\int_{L_{0}}^{L} \frac{\mathrm{d} L}{L}=\ln \left(\frac{L}{L_{0}}\right)=\ln \left(1+\varepsilon_{\mathrm{N}}\right)$

The true strain is therefore less than the nominal strain under tensile loading, but has a larger magnitude in compression. While nominal stress and strain values are sometimes plotted for uniaxial loading, it is essential to use true stress and true strain values throughout when treating more general and complex loading situations. Unless otherwise stated, the stresses and strains referred to in all of the following are true (von Mises) values.

The simulation below refers to a material exhibiting linear work hardening behaviour, so that the (plasticity) stress-strain relationship may be written

$\sigma=\sigma_{\mathrm{Y}}+K \varepsilon$

where σY is the yield stress and K is the work hardening coefficient. The sliders on the left are first set to selected σY and K values. The applied force, F, is then progressively raised via the third slider. The graph on the right then shows true stress-true strain plots, and nominal stress-nominal strain plots, while the schematic on the left shows the changing shape of the sample (viewed from one side).

Note that the elastic strains are not shown on this plot, so nothing happens until the applied stress reaches the yield stress. Since a typical Young's modulus of a metal is of the order of 100 GPa, and a typical yield stress of the order of 100 MPa, the elastic strain at yielding is of the order of 0.001 (0.1%). Neglecting this has only a small effect on the appearance of most stress-strain curves.

Simulation 2: Nominal and True Stresses and Strains