Lagging Behavior

Abstract

All microstructural interactions require finite periods of time to accomplish, which are in the range from femtoseconds (10 -15 s, phonon-electron interactions in metals) to seconds or even longer (slow conducting materials with subscale constituents). Effect of such finite times may not be noticeable as the process time is much greater, exemplified by the nanosecond (10 -9 s) transient as compared to the picosecond (10 -12 s) delay in support of sufficient collisions between electrons and phonons as they approach thermal equilibrium. As the process time enters the same domain of time for the microstructural interactions to take place, however, individual (and yet statistically meaningful) behaviors of the energy carriers would result in new physical responses that are not observed in macroscale. This chapter is dedicated to the physical foundation and mathematical methods describing the lagging response in times comparable to the phase lags characterizing the microstructural interactions.Accommodating the finite times required for completing the physical interactions in microor nanoscale, the lagging response describes the heat flux vector and the temperature gradient that occur at different instants of time in the heat-transfer process. If the heat flux precedes the temperature gradient in the time history, the heat flux is the cause and the temperature gradient is the effect of heat flow. If the temperature gradient precedes the heat flux, on the other hand, the temperature gradient becomes the cause and the heat flux becomes the effect. This concept of precedence does not exist in the classical theory of diffusion because the heat flux vector and the temperature gradient are assumed to occur simultaneously. This chapter establishes the theoretical foundation for the lagging response in times comparable to the characteristic times in micro/nanoscale heat transfer. It results in a new type of energy equation in heat transport, capturing the classical theories of diffusion (macroscopic in both space and time), thermal waves (macroscopic in space but microscopic in time), phonon scattering, and phononelectron interaction (microscopic in both space and time) in the same framework of thermal lagging. The resulting model employing the two phase lags in describing the transient process is called the dual-phase-lag model. Universality of the dual-phase-lag model facilitates a consistent approach describing the intrinsic transition from one type of behavior (diffusion,

The vibration analysis of a nanobeam due to a ramp-type heating under Moore-Gibson-Thompson theory of thermoelasticity

“…where 𝑖 is an imaginary number unit and Re is the real part. For faster convergence, many numerical experiments have concluded that the value 𝜅 satisfies the relation 𝜅𝑡 ≈ 4.7 Tzou [22,23].…”