Exact solutions for isothermal cyclic torsional loading of a circular SMA bar exploiting the shape memory effect

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 of the process, 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.

Analytical modeling of the shape memory effect in SMA beams with rectangular cross section under reversed pure bending

“…respectively, where ξ 0 is the effective volume fraction of Martensite at the beginning of each stage. Equations (2.7) and (2.8) agree with the 1D constitutive relations for the multivariant Martensite volume fractions considered by Govindjee and Kasper (1997, 1999), De la Flor et al (2011), Rizzoni et al (2013) and Radi (2021).…”

“…This occurrence makes the residual beam curvature after complete unloading as well as the area enclosed by each cycle a little bit smaller than those obtained under the simplifying assumption of equal Young’s moduli of the two phases (Ostadrahimi et al, 2015). Therefore, such a simplifying assumption overestimates the shape memory effect and the energy dissipation capabilities of SMA, as already observed under torsional loading (Radi, 2021).…”

“…However, no complete and detailed analytical study of a SMA beam under reversed bending has been performed up to now. Recently, Radi (2021) performed an analytical investigation of the shape memory effect in SMA circular rod under cyclic torsional loading by adopting a 3D SMA constitutive model that incorporates two opposite Martensite variants. He also analyzed the transformation and reorientation processes into multiple martensitic variants in a previous work on the response of SMA thick-walled cylinders under internal pressure (Radi, 2018).…”

“…However, they necessarily require the adoption of numerical procedures, also for simulating simple mechanical problems. The constitutive model developed here originates from those proposed for multi-variants Martensite (Brinson, 1993; Brinson and Huang, 1996; De la Flor et al, 2011; Fahimi et al, 2019; Govindjee and Kasper, 1997, 1999; Marfia and Rizzoni, 2013; Rizzoni and Marfia, 2015; Rizzoni et al, 2013) and recently adopted by Radi (2021) for modeling cyclic torsional loading. It is able to simulate the Martensite reorientation process occurring when a transition from tensile to compressive stress, or vice versa, takes place.…”