Memory-dependent and size-dependent effects on nonlinear electro-magneto-thermo-viscoelastic vibration analysis of a polymer spherical microshell with fractional order strain

The stress-strain prediction for viscoelastic materials using integer-order thermoviscoelastic models faces challenges due to the limitations of integer-order derivatives.With the miniaturization of devices, size-dependent effects on elastic deformation become more significant. Additionally, experiments and theories indicate that thermal conductivity in materials should not be treated as a constant in practical analyses. This study develops a fractional-order thermoviscoelastic model by integrating the fractional three-phase-lag heat conduction model, the fractional strain model and the nonlocal elasticity theory. The proposed model is then applied to analyze the nonlinear electro-magneto-thermo-viscoelastic response of a polymer spherical microshell under ramp-type heating and an external magnetic field. Taking into account the variable thermal conductivity, the nonlinear governing equations are derived. The Laplace and Kirchhoff transformations are employed to derive and solve the governing equations. The dimensionless temperature, induced magnetic field, displacement and stress are obtained and illustrated graphically. The results show that the nonlinear thermoviscoelastic response of the polymer spherical shell can be adjusted by the suitably modified parameters, which strongly depend on the size-dependent effect, memory-dependent effect and the variable thermal conductivity.