Coupled Thermomechanical Analyses of Shape Memory Alloy Structures undergoing Large Deformation during Static and Dynamic Loading

Abstract

Shape Memory Alloys (SMAs) exhibit large displacement and force capabilities due to the stress and temperature-dependent diffusionless martensitic phase transformation. During the same, the variation of material properties, e.g., modulus of elasticity, thermal expansion coefficient, and specific heat during phase transition, also significantly affect the temperature of the system, which in turn dictates the extent of phase transformation in SMA behaviour and hence its response. Also, SMA-based structures undergo minor hysteretic loops and significant deformation (up to 8%) during thermomechanical loading. So, for effective modelling of SMA-based structures, a coupled thermomechanical analysis is required, considering the effect of finite deformation and partial phase transformation. In this study, coupled thermomechanical nonlinear finite element (FE) formulations have been developed to address material and geometric non-linearity arising out of the SMA behaviour. Both the stress and thermal equilibrium equations are solved simultaneously in an incremental-iterative FE framework, simulating the coupled SMA response. In addition, the effect of the inner loop arising out of partial transformation has been taken into account, and the simulated responses are compared with experimental observations, as reported in the literature. To account for the effect of large deformation, a Total Lagrangian (TL) formulation has been developed considering Green-Lagrange strain and Second Piola-Kirchhoff stress measures, following Lagoudas et al.. Next, an Updated Lagrangian (UL) formulation has been developed considering different objective stress rates, e.g., Jaumann, Green-Naghdi, and Logarithmic, by extending the infinitesimal strain-based constitutive model for finite deformation. In addition, incremental objectivity has been maintained to preclude the generation of any factitious stresses during superposed rigid body motion. The efficacy and robustness of the developed FE models are corroborated through various practical applications of SMA-based members, e.g., SMA ring, SMA-actuated beam, morphing of corrugated airfoil, compliant gripper, undergoing large displacement and rotation (LDR) while subjected to a variety of thermomechanical loading conditions. Finally, an FE formulation has been developed to simulate the dynamic response of SMA-based structures, considering the effect of transformation-induced latent heat, large deformation and inner hysteretic subloops. Several problems related to the transient and forced vibration responses of SMA-based structures are simulated using the developed framework for validation purpose. The combined effect of inertia and transformation-induced latent heat increases the amplitude of the displacement response than that of the uncoupled case. This is due to the temperature rise in the stress-concentrated zone, which stabilizes austenite and reduces energy dissipation. On the contrary, the effect of finite strain yields a decrement in the amplitude of vibration as compared to material nonlinear-only analysis. Moreover, the effect of inner subloops entails lesser dissipation of energy, rendering an increment in displacement amplitude.