Size-dependent thermoelastic diffusion in micro-scale cylinders: Influence of thermal and diffusive length scales with phase delay effects

This work presents the first comprehensive framework to simultaneously incorporate spatial nonlocality and temporal phase-lag effects for coupled heat and mass transport in a cylindrical configuration. The model introduces two distinct nonlocal length scales, for thermal conduction and mass diffusion, respectively, alongside dual relaxation times, overcoming the scale limitations of classical continuum theories. A dual-phase-lag thermoelastic diffusion model is derived by generalizing the Lord-Shulman theory through nonlocal reformulations of Fourier’s and Fick’s laws. This formulation ensures finite signal speeds and captures the size-dependent response essential at micro- and nanoscales. The theory is applied to an infinitely long solid cylinder subjected to a transient surface load: a Gaussian-modulated cosine thermal pulse and an exponentially decaying chemical potential. Under axisymmetric conditions, all field quantities depend only on radial position and time. The governing equations are solved analytically via the Laplace transform, with solutions in the transformed domain expressed using modified Bessel functions. Time-domain results are recovered through numerical inversion based on a Fourier-series expansion. Results demonstrate that nonlocal and phase-lag parameters significantly dampen mechanical fields (displacement and stress) while amplifying and smoothing thermal and diffusive fields (temperature and concentration). The interplay between thermal and diffusive nonlocalities produces synergistic damping and enhanced penetration depths, effects unattainable with local or single-phase-lag models. This advanced framework provides a critical predictive tool for the design and analysis of micro- and nano-scale systems, including MEMS, semiconductor devices, energy-storage media, and biomedical implants, where coupled thermo-diffusive-mechanical interactions dictate performance and reliability.