Modelling of hysteretic loops in cyclic-loading Nb₃Sn wire: kinematic and mixed hardening

Abstract The Nb 3 Sn composite wire may undergo cyclic loading of Lorentz forces in an extremely strong magnetic field. The study of stress–strain relationships under these cyclical conditions, particularly the formation of hysteresis loops, is essential for assessing the operational performance of these wires. We concentrate on the role of component copper in determining the hysteresis curves, since copper is the main ingredient and exhibits characteristic elastic-plastic stress–strain relation. When employing the Johnson-Cook model, which excludes the effect of kinematic hardening, we find that the stress–strain curve shows no hysteresis loop; however, hysteresis loops appear when the Chaboche model is used. To identify the reason for the hysteresis phenomenon, we calculated the hysteresis curves using elementary nonlinear isotropic and kinematic hardening models. Our findings suggest that the hysteresis loops are attributable to the kinematic hardening in copper. This property effectively increases the reversed yield limit of copper as the initial unloading stress increases. Hysteresis loops show up once this yield limit exceeds exactly zero.