Mathematical Model of the Specimen Shape During the Necking Stage in the Uniaxial Tensile Test

The mathematical model of the necking shape is built based on the analysis of the shape characteristics during the necking stage in the uniaxial tensile test. It is that the radius r of the section vertical to the central axis is the function about the distance z(z≥0) from this section to the minimal section. This function is r=${r_n} + \frac{{{r_c} - {r_n}}}{{1 + {{(\frac{z}{{{z_1}}})}^{{p_1}}} + {{(\frac{z}{{{z_2}}})}^{{p_2}}}}}$. The function of the necking sample's surface is √x²+y²= ${r_n} + \frac{{{r_c} - {r_n}}}{{1 + {{(\frac{z}{{{z_1}}})}^{{p_1}}} + {{(\frac{z}{{{z_2}}})}^{{p_2}}}}}$ in the coordinate system with the origin at the center of the minimal sec[JP4]tion of the necking specimen and the central axis as the z-axis. The necking shape can be attributed with the six characteristic parameters of r_c、r_n、z₁、z₂、p₁、p₂ in this mathematical model. The availability of the necking shape's mathematical model is verified by the curves' fitting with the test data of the low alloy steel and the simulation data with three sets of the stress-strain relationship's parameters.