Abstract
Two simple and fully analytical models are presented for a SMA bar or wire of circular cross section subjected to fully-reversed cyclic torsional loading, by taking into consideration the reorientation of the martensitic variants occurring during unloading and reverse loading. The process is assumed to take place at constant temperature between the start temperatures of the martensitic and austenitic transformations. In the first part of this work, the same shear modulus is taken both for Martensite and Austenite. In the second part, different elastic shear moduli are considered for the two phases. The volume fractions of both positive and negative twisted Martensite are assumed to evolve linearly with the shear stress. The bar is initially in a state of Austenite. As the applied torque is increased the martensitic transformation starts from the outer surface and then it extends inwards. If the maximum applied torque is large enough, then the complete Martensitic transformation takes place in the outer region of the cross section. During unloading and subsequent reverse loading the martensitic reorientation process may occur starting from the boundary between the fully martensitic outer region and the intermediate transforming region. Particular attention is focused on modeling the unloading and reverse loading processes. At each stage, the radial distributions of shear stress and Martensite variant are calculated analytically. A closed form relation between the applied torque and the angle of twist is presented for the entire process in the case of equal shear moduli, and only for the loading and elastic unloading stages in the case of different shear moduli. The approach is then validated against analytical, numerical and experimental results available in the literature for the direct loading–unloading process. Application to the seismic response of dissipative systems based on SMA helical springs is also envisaged.
Highlights
The problem of cyclic torsion of a shape memory alloy (SMA) bar of circular cross section is the fundamental basis for modeling the superelastic (SE) and the shape memory (SM) effects of SMA helical springs, which are finding ever more innovative applications in structural and civil engineering
In a SMA spring cyclically loaded at low temperatures, a large hysteresis loop similar to that exhibited by ductile metals is observed, which is due to the reorientation process between martensitic variants rather than to phase transformation or dislocation glide mechanisms
The present study aims to develop two exact analytical models, which allow for obtaining closedform solutions for the shear stress and Martensite distributions in a SMA bar with circular cross section during each step of the loading-unloading processes with opposite sign at constant temperature T lower than As
Summary
The problem of cyclic torsion of a shape memory alloy (SMA) bar of circular cross section is the fundamental basis for modeling the superelastic (SE) and the shape memory (SM) effects of SMA helical springs, which are finding ever more innovative applications in structural and civil engineering. Viet et al (2019) worked out an analytical solution for the problem of SMA circular shaft subject to a SE loading-unloading cycle in torsion, which required the solution of a cubic equation for the shear stress in the phase transforming region during loading These authors validated their model against numerical and finite element results and used it for simulating the SE behavior of SMA helical springs (Mirzaeifar et al, 2011; Heidari, et al 2016; Viet et al, 2020). The present study aims to develop two exact analytical models, which allow for obtaining closedform solutions for the shear stress and Martensite distributions in a SMA bar with circular cross section during each step of the loading-unloading processes with opposite sign at constant temperature T lower than As. The 1D constitutive models adopted here is derived from a phenomenological 3D model based on the von Mises effective stress and strain.
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