Extracting true stresses and strains from nominal stresses and strains in tensile testing

Seemingly a simple task, the extraction of the flow curve (true stress vs. true plastic strain) from nominal stresses and strains in standard tensile testing still has its unsolved points. This study addresses two of them: (i) in materials without yield point phenomenon (or generally in the region of homogeneous plastic deformation), the true stress is typically calculated assuming constant volume, ignoring the elastic volume changes. Here, we derive a set of exact analytical solutions for true stresses and strains with remarkable simplicity and beauty that fully account for the elastic volume changes. This set of exact solutions is cross-checked by finite element simulations as well as zeroth- and first-order approximations; perfect agreement has been found. (ii) In materials with a pronounced yield point phenomenon, a complicated three-dimensional stress state inevitably arises at the edge of the Luders bands, which masks the real (or inherent) material behaviour. To determine the real material behaviour in the Luders region, here we use a new macroscopic analytical approach characterised by a high true upper yield point, a typical strain hardening behaviour common for many materials, and the triaxiality of the stress state that inevitably develops at the edges of the Luders bands and that determines the stress level at the observed lower yield point. This approach is verified by experiments (including video observations as well as digital image correlation (DIC) strain distribution measurements) and finite element simulations with very good agreement.