Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating
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...
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MDPI AG
2023-02-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/5/1099 |
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| author | Mohsen Fayik Sharifah E. Alhazmi Mohamed A. Abdou Emad Awad |
| author_facet | Mohsen Fayik Sharifah E. Alhazmi Mohamed A. Abdou Emad Awad |
| author_sort | Mohsen Fayik |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T07:18:45Z |
| format | Article |
| id | doaj.art-a723d2d50590434e9dcf2e5d71bd7ea4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T07:18:45Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a723d2d50590434e9dcf2e5d71bd7ea42023-11-17T08:08:11ZengMDPI AGMathematics2227-73902023-02-01115109910.3390/math11051099Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring HeatingMohsen Fayik0Sharifah E. Alhazmi1Mohamed A. Abdou2Emad Awad3Department of Mathematics, Faculty of Education, Alexandria University, Souter St. El-Shatby, Alexandria 21526, EgyptMathematics Department, Al-Qunfudhah University College, Umm Al-Qura University, Al-Qunfudhah 28821, Mecca, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Alexandria University, Souter St. El-Shatby, Alexandria 21526, EgyptDepartment of Mathematics, Faculty of Education, Alexandria University, Souter St. El-Shatby, Alexandria 21526, EgyptThe 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.https://www.mdpi.com/2227-7390/11/5/1099finite-speed thermal wavesLord-Shulman thermoelasticityHDPL thermoelasticityMHDPL thermoelasticityLaplace transformHankel transform |
| spellingShingle | Mohsen Fayik Sharifah E. Alhazmi Mohamed A. Abdou Emad Awad Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating Mathematics finite-speed thermal waves Lord-Shulman thermoelasticity HDPL thermoelasticity MHDPL thermoelasticity Laplace transform Hankel transform |
| title | Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating |
| title_full | Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating |
| title_fullStr | Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating |
| title_full_unstemmed | Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating |
| title_short | Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating |
| title_sort | transient finite speed heat transfer influence on deformation of a nanoplate with ultrafast circular ring heating |
| topic | finite-speed thermal waves Lord-Shulman thermoelasticity HDPL thermoelasticity MHDPL thermoelasticity Laplace transform Hankel transform |
| url | https://www.mdpi.com/2227-7390/11/5/1099 |
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