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Materials Engineering DISCUSSION
In a tensile test on a mild steel specimen, the stress rises to a maximum and then falls before fracture. My textbook says the material keeps strain hardening all the way to failure, which seems to contradict a falling curve. It also shows a "true" stress-strain curve that never comes down.
How do I convert my test data from engineering to true values, where does the conversion stop being valid, and if true stress is the real one, why is engineering stress used at all?
Engineering stress divides the load by the original cross-section, which stays on paper as a constant. The real cross-section shrinks as the bar stretches. Up to the maximum load, strain hardening outweighs the loss of area and the load keeps rising. At the ultimate point the two balance, a neck forms, and from then on the area in the neck falls faster than the material hardens. The load drops, so load divided by the original area drops too.
The metal in the neck is carrying a higher true stress than ever; only the bookkeeping makes the curve fall.
While the deformation is uniform the volume is constant, which gives two simple conversions: true stress σ_t = σ_e × (1 + ε_e) and true strain ε_t = ln(1 + ε_e).
Example: an engineering stress of 400 MPa at an engineering strain of 0.15 becomes σ_t = 400 × 1.15 = 460 MPa and ε_t = ln(1.15) = 0.140. At small strains the two sets of values are almost identical: at 0.2 percent strain the difference in stress is 0.2 percent, which is why it does not matter for yield strength.
Those formulas are valid only up to the onset of necking, because after that the strain is no longer uniform along the gauge length and the extensometer reading is an average over a non-uniform region. Beyond the ultimate point you need the actual minimum diameter of the neck: true stress is load divided by the current neck area, and true strain is ln(A0 / A). The stress state in the neck is also triaxial, so an accurate curve needs a correction such as Bridgman's.
A useful result: if the flow curve follows σ = K × ε^n, necking starts when the true strain equals n. A steel with n = 0.20 therefore necks at a true strain of 0.20, which is an engineering strain of e^0.20 - 1 = 0.22.
Engineering stress survives because it matches how parts are designed and tested. A designer knows the original dimensions and the applied load, and wants to know the load at which the part yields or reaches its maximum. Yield strength and UTS as engineering values answer that directly, they are simple to measure, and material specifications are written around them.
True stress-strain data matter when the deformation is large: metal forming, crash analysis and any nonlinear finite element model with plasticity. Most solvers expect true stress against true (logarithmic) plastic strain, and feeding them engineering values is a common source of wrong results.
