In this study, an analytical solution is presented to simulate the mechanical response of thermo-viscoelastic shape memory polymer materials considering the effects of time, temperature, loading rate, etc. Based on the proposed formulation and based on the curved-bar-torsion theory, the behavior of smart helical springs with variable stiffness concerning time and temperature is studied. Moreover, numerical solutions based on the finite element method are used to validate the proposed formulation. The presented formulations are evaluated for different mechanical loading and temperature conditions, and simulated the behavior of both helical compression and extension springs for force recovery or shape recovery processes of shape memory polymer materials. The presentation of multiple shape memory properties along with the effects of parameters such as temperature and rate of temperature have also been investigated. The proposed method is much faster than the numerical solution and the matching of the responses of the two methods at different loading conditions shows the accuracy of the presented method. Therefore, using this propound method, design or optimization processes can be used with much lower computational cost than numerical solutions.
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