The Monotonic Flow Curve

The monotonic flow curve is often the primary source of information on a material’s mechanical properties. It is typically derived from a tensile test, where the force and extension of the gauge section of a cylindrical specimen are recorded. The strain rate is chosen to be low enough that it does not influence the result (i.e., quasi-static loading). The resulting stress–strain curve supplies both input data and allowable values for structural analyses.

In this course we illustrate the true stress–strain behaviour of a typical low-strength structural steel (S235). For material behaviour in FEA we generally use true stress–strain curves rather than the engineering versions. “True” refers to using the instantaneous geometric quantities: stress is the load divided by the instantaneous cross-sectional area, and strain is the logarithmic strain based on the instantaneous gauge length. “Engineering” values use the original area and length measured on the unloaded specimen.

When we speak about hardening in monotonic stress–strain behaviour, we mean the increase in stress after first yield under continuously increasing load. This should be distinguished from cyclic hardening/softening, which will be treated in the next section.

For non-linear FEA, a simple and widely used option to represent the monotonic flow curve is a rate-independent bilinear elastoplastic model, usually employed within a quasi-static analysis. Often this is the best practical choice because it is simple, and can be built from the limited data typically available (yield stress, and elongation at break).

Although solvers often step the load using an internal “time” parameter, this is pseudo-time used for load incrementation; it is not physical time unless a genuinely time-dependent material or dynamic procedure is adopted.

A linear-elastic, perfectly (ideally) plastic material can be approximated by a bilinear model with a very small hardening modulus. This idealisation underpinned many classical closed-form solutions developed before non-linear FEA became commonplace, and it remains a useful, inherently conservative benchmark.