Thermo-viscoelastic dynamic response of nano-plate based on the non-singular kernel fractional order derivatives

In this paper, a new temporal-spatial fractional order model is proposed to study the dynamic behavior of thermo-viscoelastic nanoplates. Traditional singular kernel in Caputo fractional order differentiation is replaced by the non-singular kernel and thus leads to a new generalized fractional order differential model with the integer order differential models as a special case. This improved model can more flexibly describe small-scale mechanical behavior and time-dependent heat conduction behavior and provides a clear physical explanation for the fractional order parameters. Spatial nonlocal effects are described in terms of nonlocal strain gradient elasticity and spatial fractional order derivatives, while the time-dependent effects are described in terms of non-Fourier heat conduction, viscoelasticity, and time fractional order derivatives. In addition, it is the first time that the nonlocal characteristic lengths and the memory characteristic times are introduced as two new small-scale parameters in the fractional order derivatives of non-singular kernels to focus on the short-range nonlocal behaviors and the short-term memory behaviors. Numerical examples of the free vibration and the forced vibration under step loading are given, and the effects of the spatial fractional order parameter and the temporal fractional order parameter are both discussed.