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

Abstract

An analytical model is developed for a prismatic SMA beam with rectangular cross section subjected to alternating bending at temperature below the austenitic transformations. The loading path consists in a loading-unloading cycle under bending and then under reversed bending. Two opposite martensitic variants take place, whose volume fractions evolve linearly with the axial stress. Different Young’s moduli are taken for the austenitic and martensitic phases. As the bending moment is increased, the martensitic transformation starts from the top and bottom and then it extends inwards. If the maximum applied bending moment is large enough, then the complete Martensitic transformation takes place at the upper and lower parts of the cross section. During unloading and the following reversed bending, reorientation of the Martensite variant into the opposite one takes place starting from the boundary between the fully martensitic region and the intermediate transforming region. Special attention is devoted to calculate analytically the axial stress and Martensite variant distributions within the cross section at each stage of the process. A closed form moment-curvature relation is provided for loading and elastic unloading and in integral form for the rest of the process. The approach is then validated by comparison with analytical results available in the literature.

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

“…However, the reorientation of Martensite variants under torsional loading is taken into consideration here analytically for the first time. Recently, Radi (2021) performed an analytical investigation of the SM effect in a SMA beam under direct and reversed bending loading by adopting a 1D SMA constitutive model that incorporates two opposite Martensite variants. The reorientation process between multiple martensitic variants was also considered in a previous work on the response of SMA thick-walled cylinders under internal pressure (Radi, 2018).…”

Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys

“…Beams manufactured from shape memory alloys have been intensively investigated due to their superb actuation and energy harvesting capabilities [42][43][44]. Although analytical models can be very useful for quick concept validation and preliminary design studies [45][46][47], full-scale FEM simulations are often indispensable in detailed studies [42,44,48,49], since they are not constrained by assumptions on a particular geometry, material response, boundary conditions, etc. The simulated problem mimics a simple beam with a rectangular cross-section (with its width being half of the height) loaded at both ends and supported in the middle so that bending is invoked.…”

Effect of Tension-Compression Asymmetry Response on the Bending of Prismatic Martensitic SMA Beams: Analytical and Experimental Study

“… Comparison of analytical results considering symmetric material parameters and finite element for a rectangular SMA beam subjected to a moment of 80 N.mm at −40 °C, [ 28 , 35 ]. …”

“…Several theoretical works were also studied for bending of laminated composite PE beams considering the constitutive equations with different material behavior in tension and compression [ 31 , 32 , 33 ]. More recently, assuming the effect of tension–compression asymmetry, a constitutive model to predict the response of PE and SME beams under a three-point bending test and pure bending is proposed in [ 34 , 35 ], respectively.…”