Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating

• Published in: Mathematics (Basel) 2023-03, Vol.11 (5), p.1099
• Main Authors: Fayik, Mohsen, Alhazmi, Sharifah E., Abdou, Mohamed A., Awad, Emad
• Format: Article
• Language: English
• Subjects: Constraint modelling; Elastic plates; Exact solutions; finite-speed thermal waves; Hankel transform; HDPL thermoelasticity; Heat conductivity; Heat transfer; Heating; Laplace transform; Lord-Shulman thermoelasticity; Mathematical models; MHDPL thermoelasticity; Phase lag; Stress distribution; Thermal stress; Thermalization (energy absorption); Thermoelasticity; Time domain analysis; Velocity; Wave fronts; Wave reflection
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11051099

The present study provides a theoretical estimate for the thermal stress distribution and the displacement vector inside a nano-thick infinite plate due to an exponentially temporal decaying boundary heating on the front surface of the elastic plate. The surface heating is in the form of a circular ring; therefore, the axisymmetric formulation is adopted. Three different hyperbolic models of thermal transport are considered: the Maxwell-Cattaneo-Vernotte (MCV), hyperbolic Dual-Phase-Lag (HDPL) and modified hyperbolic Dual-Phase-Lag (MHDPL), which coincides with the two-step model under certain constraints. A focus is directed to the main features of the corresponding hyperbolic thermoelastic models, e.g., finite-speed thermal waves, singular surfaces (wave fronts) and wave reflection on the rear surface of the plate. Explicit expressions for the thermal and mechanical wave speeds are derived and discussed. Exact solution for the temperature in the short-time domain is derived when the thermalization time on the front surface is very long. The temperature, hydrostatic stress and displacement vector are represented in the space-time domain, with concentrating attention on the thermal reflection phenomenon on the thermally insulated rear surface. We find that the mechanical wave speeds are approximately equal for the considered models, while the thermal wave speeds are entirely different such that the modified hyperbolic dual-phase-lag thermoelasticity has the faster thermal wave speed and the Lord-Shulman thermoelasticity has the slower thermal wave speed.