Experimental investigations on the mechanical deformations of thermo-induced shape memory polymers (TSMPs) at equilibrium temperatures and shape memory effects at non-equilibrium temperatures are conducted at different strain rates. It is found that the loading history conditions change the glass transition temperature, which is the nature of the change of shape memory effect. Meanwhile, temperature hysteresis is observed in stress freezing and strain recovery stages due to the varied temperature rate. The effect of the loading histories on the shape memory effect is reflected by introducing the temperature rate, strain rate, and deformation-dependent delay factors into the rubbery phase volume fraction. Based on the phase transition theory, a logarithmic rate-based viscoelastic-viscoplastic model is established at finite deformation. In this model, the viscoelastic rubbery phase is parallel with the viscoplastic glassy phase. Then, a frozen phase is introduced into the parallel model by changing its frozen volume fraction to reflect the storage/release of the deformation. Finally, the proposed model is validated by comparing the simulated rate-dependent mechanical deformation at equilibrium temperatures and shape memory effect at non-equilibrium temperatures with corresponding experimental results.