On the onset of necking in the tensile test

A very small inhomogeneity in macroscopic material properties is considered for the investigation of necking of a (polycrystalline) bar in the tensile test. A simple, one-dimensional model is adopted (i.e. spatial variation only along the barʹs length), corresponding to which the evolving nonuniformity of cross-sectional area with increasing load, up to the maximum, also is small. The inhomogeneity is represented by a single modulus varying slightly with position, but with the same dimensionless form of the stress–strain curve throughout. A transcendental equation is derived that relates the strains at the weakest and strongest sections to their respective material strengths. It is shown that, at the well known Considère strain (corresponding to the maximum load), the decrease in area at the minimum section is only a little greater than at the strongest, but its rate of change with the latter strain is infinite. The analysis thus gives an idealized representation of the rapid increase in necking that typically ensues from the maximum load in experiments. In an extended Appendix the traditional case of a bar of theoretically homogeneous material is reviewed and reanalyzed, including consideration of rate-dependence.