Analytical model for a superelastic Timoshenko shape memory alloy beam subjected to a loading–unloading cycle

Abstract

We propose a new analytical model for a superelastic shape memory alloy prismatic cantilever beam subjected to a concentrated force at the tip. The force is gradually increased and then removed and the corresponding distribution of phase transformation fields in the beam is determined, analytically, in both the transverse and longitudinal directions. Analytical moment–curvature and shear force–shear strain relations are also derived during loading and unloading of the beam. The proposed model is validated against an exact numerical beam model as well as a three-dimensional finite element analysis model for the same beam, with very good agreement in each case. Moreover, an experiment is proposed and carried out to characterize the load–deflection response of a shape memory alloy beam under the same boundary conditions as those considered in deriving the model. The obtained response is in good agreement with the analytical model as well as three-dimensional finite element analysis simulations. The analytical method provides a direct mathematical way for describing the material and structural properties of the beam and the distribution of the different solid phase regions as they change under the influence of an applied load and allows the determination of details such as the boundaries of solid phase regions immediately and accurately using equations. The same would require postprocessing at possibly significant computational cost and personal effort if finite element analysis or similar numerical methods are used.

A Visco-Hyperelastic Constitutive Model to Characterize the Stress-Softening Behavior of Ethylene Propylene Diene Monomer Rubber

“…In addition to the traditional hyper-elastic constitutive models, some new hyper-elastic constitutive models have also been developed recently. For example, Yao et al [ 19 ] proposed a constitutive model to explain the cyclic response of hyper-elastic NiTi shape memory alloy, and Nguyen et al [ 20 ] put forward an analytical model for the addition and removal cycles of hyper-elastic memory alloy beam. To model the hyper-elastic fracture and fatigue behavior of memory alloys, Simoes et al [ 21 ] offered a native model; Viet et al [ 22 ] presented a new hyper-elastic cyclic model for the behavior of large refractive index hyper-elastic shape memory alloy coil springs.…”

Stress-dependent generalized Prandtl–Ishlinskii hysteresis model of a NiTi wire with superelastic behavior

“…Various constitutive hysteresis models of the stressdependent thermomechanical property of SMAs have been introduced, such as the Tanaka model (Tanaka and Nagaki, 1982;Tanaka and Iwasaki, 1985;Xiao et al, 2019), the Mu¨ller model (Mu¨ller, 1979;Mu¨ller and Wilmanski, 1980), the Boyd-Lagoudas model (Boyd and Lagoudas, 1994;De Blonk and Lagoudas, 1998;Li et al, 2019), the Ivshin-Pence model (Ivshin and Pence, 1994), the Brinson model (Cheng et al, 2017), and the Zaki-Moumni model (Van Viet et al, 2018. In these models, the stress dependency is treated mainly as a shift in the martensite volume fraction with respect to the transition temperature and applied stress.…”