True Stress/Strain vs Engineering Stress/Strain

True Stress and Strain

If the deformation of a bar in tension becomes significant, its cross-sectional area will change. The traditional engineering definitions for stress and strain are no longer accurate, and new measures, namely true stress and true strain, are introduced. Alternative names for these quantities are Cauchy stress, logarithmic strain, and natural strain.

The true stress is ${\sigma }_T=\frac{F}{a}$ , where $a$ is the final deformed cross-sectional area.

The true strain is ${ϵ}_T=\ln \left(\frac{l}{L}\right)$ , where $l$ is the final length and $L$ is the initial undeformed length of the bar.

Engineering Stress and Strain

The engineering stress (or nominal stress) is ${\sigma }_E=\frac{F}{A}$ , where $A$ is the initial undeformed cross-sectional area.

The engineering strain (or nominal strain) is ${ϵ}_E=\frac{\mathrm{\Delta }l}{L}$ , where $\mathrm{\Delta }l$ is the change of the bar's length and $L$ is the undeformed bar's lengh.

Stresses are related to the strains through constitutive laws, which are governed by material properties. The simplest of the stress/strain laws is Hooke’s law, which is given by $\sigma =\mathrm{{\rm E}}ϵ$ .