true stress and strain
Imagine pulling a piece of taffy: as it thins in the middle, the same pull is now squeezed through a skinnier neck, so that spot is worked far harder than the fat ends. Engineering stress ignores this — it always divides by the original width. True stress and true strain are the honest bookkeeping that uses the ACTUAL, shrinking cross-section and the running length at each instant.
True stress is sigma_T = F / A_i, force over the instantaneous (current) area A_i, not the original A0. True strain is epsilon_T = ln(L_i / L0), a sum of tiny fractional stretches rather than one big fraction. Because a stretched bar gets thinner (its volume stays about constant during plastic flow), the current area is smaller than the original, so true stress is always higher than engineering stress once plastic stretching begins. For small strains the two agree closely: at 0.2 percent strain they differ by a fraction of a percent.
This matters because the engineering stress-strain curve turns DOWNWARD after the peak (it looks like the material is getting weaker as it necks), but the material is actually still hardening — the true-stress curve keeps rising until fracture. Engineers quote engineering values because they map directly to a load on a part of known original size, while metallurgists and anyone modelling forming (rolling, forging, deep drawing) use true stress and strain because that is what the metal really experiences.
At the neck of a failing steel bar the engineering stress might read 500 MPa (using the original area) while the true stress is over 800 MPa, because the neck has shrunk to a fraction of its starting width.
The downturn on the engineering curve is an artefact of dividing by the original area — the true curve keeps climbing.
The apparent softening after the tensile peak is not real weakening; it is the original-area convention. The material keeps work-hardening right up to fracture, as the true-stress curve shows.