Examination of the stress-strain curve of mild steel from a statistical point of view

Abstract

If a long bar of mild steel is cut up into a large number of short bars, and the stress-strain curve of each of these bars is measured, considerable fluctuations from bar to bar will often occur. An examination of the character of these fluctuations on an ordinary round bar about 12 m long and of 10 mm diameter is reported. The main result of the examination is that the lower yield point σ y l and the difference between the upper and the lower yield point σ y u − σ y l are independent random variables, and further that σ y l is normally distributed with parameters (μ, σ), and that σ y u − σ y l is exponentially distributed with parameter λ. On the basis of the measured results and by applying the principles of the theory of probability, an asymptotic expression is given for the stress-strain curve of a long steel bar, imagined to have been taken from the same batch as the one used in our examination. Finally it is found that the fluctuations occurring in the stress-strain curve during plastic strain must theoretically develop with greater and greater amplitudes and in such a way that the minimum points in the greater part of the plastic strain interval are normally distributed with parameters (μ + λσ 2, σ), while the maximum points pass along an increasing curve. This curve is the aforesaid asymptotic stress-strain curve.