Application of hysteresis functions in vibration problems

In the paper, the notion of the hysteresis function introduced by the author is developed further and applied in some vibration problems for hysteretic materials. The stress–strain relation is built on the basis of well-known extended Masing's rules in which the scaled backbone curve used in the second rule is replaced by a more general function called the hysteresis function. This modification allows us to regulate the dependence of the damping ratio on the strain amplitude in the process of cycle deforming of a material with further extending to arbitrary deformation processes. The three cases of hysteretic systems are considered: the system with limited stress when strain increases without bound, the system with linear backbone curve, and the system with the backbone curve having increasing stiffness for increasing strain. For a number of dynamic examples, a comparison of different hysteresis functions is carried out.