Hall current effect in double poro-thermoelastic material with fractional-order Moore–Gibson–Thompson heat equation subjected to Eringen's nonlocal theory

Abstract

The present study focuses on the analysis of one-dimensional thermoelastic interaction in an infinite medium consisting of double poro-thermoelastic material with voids (DPTMWV) in the presence of Hall current by developing the fundamental relations in light of Eringen's nonlocal elasticity theory. A fractional-order Moore–Gibson–Thompson (MGT) heat conduction equation is introduced to the model in the context of a continuous heat source. The Caputo fractional derivative is applied in the domain of Laplace transform. The displacement, temperature, volume fractions, and stresses are determined with the help of the eigenvalue approach and the numerical inversions of these physical variables are obtained by using Zakian's algorithm. The effects of several parameters, such as Hall current, fractional order, nonlocality, and time, are graphically executed. Furthermore, the nonlocal models under different generalized thermoelasticity theories are compared to visualize the variations in the distributions corresponding to the prior mentioned variables.

Keywords:

• Eringen's nonlocal elasticity theory
• double poro-thermoelastic material with voids
• Hall current
• fractional-order Moore–Gibson–Thompson equation
• eigenvalue approach
• Zakian's algorithm

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix

$\begin{array}{rl}{Q}_1& =M_1+M_{10}+M_{15}-M_3{M}_{12}+M_4{M}_{16}+M_5-M_2{M}_8,\\ Q_2& =M_1{M}_{10}-M_{11}{M}_{14}+M_1{M}_{15}+M_{10}{M}_{15}-M_3{M}_{12}{M}_{15}\\ & +M_3{M}_{11}{M}_{16}-M_4{M}_{12}{M}_{14}+M_4{M}_{10}{M}_{16}\\ & +M_1{M}_5+M_5{M}_{10}+M_5{M}_{15}-M_3{M}_5{M}_{12}+M_4{M}_5{M}_{16}+M_2{M}_6{M}_{12}-M_7{M}_{13}\\ & +M_2{M}_7{M}_{16}-M_2{M}_8{M}_{10}-M_2{M}_8{M}_{15}-M_4{M}_8{M}_{13}-M_6{M}_9+M_3{M}_8{M}_9,\\ Q_3& =M_1{M}_{10}{M}_{15}-M_1{M}_{11}{M}_{14}+M_1{M}_{10}{M}_5-M_5{M}_{11}{M}_{14}+M_1{M}_5{M}_{15}\\ & +M_5{M}_{10}{M}_{15}-M_3{M}_5{M}_{12}{M}_{15}+M_3{M}_5{M}_{11}{M}_{16}-M_4{M}_5{M}_{12}{M}_{14}+M_4{M}_5{M}_{10}{M}_{16}\\ & +M_6{M}_{11}{M}_{13}+M_2{M}_6{M}_{12}{M}_{15}-M_2{M}_6{M}_{11}{M}_{16}-M_4{M}_6{M}_{12}{M}_{13}-M_1{M}_7{M}_{13}\\ & -M_7{M}_{10}{M}_{13}-M_2{M}_7{M}_{12}{M}_{14}+M_2{M}_7{M}_{10}{M}_{16}+M_3{M}_7{M}_{12}{M}_{13}\\ & +M_2{M}_8{M}_{11}{M}_{14}-M_2{M}_8{M}_{10}{M}_{15}-M_3{M}_8{M}_{11}{M}_{13}-M_4{M}_8{M}_{10}{M}_{13}\\ & -M_1{M}_6{M}_9-M_6{M}_9{M}_{15}-M_4{M}_6{M}_9{M}_{16}+M_7{M}_9{M}_{14}\\ & -M_3{M}_7{M}_9{M}_{16}+M_3{M}_8{M}_9{M}_{15}+M_4{M}_8{M}_9{M}_{14},\\ Q_4& =M_1{M}_5{M}_{10}{M}_{15}-M_1{M}_5{M}_{11}{M}_{14}+M_1{M}_6{M}_{11}{M}_{13}\\ & -M_1{M}_7{M}_{10}{M}_{13}-M_1{M}_6{M}_9{M}_{15}+M_1{M}_7{M}_9{M}_{14},\\ L_{11}& =(-M_{15}{M}_3{M}_5-M_{14}{M}_4{M}_5+M_{15}{M}_2{M}_6+M_{13}{M}_4{M}_6-M_{14}{M}_2{M}_7+M_{13}{M}_3{M}_7)M_{16},\\ L_{12}& =[M_{14}{M}_4+M_3(M_{15}+M_5)-M_2{M}_6]M_{16},L_{13}=M_3{M}_{16},\\ L_{21}& =M_1{M}_{16}(M_{15}{M}_6-M_{14}{M}_7),\\ L_{22}& =M_{16}[M_1{M}_6+M_{15}{M}_6+M_{16}{M}_4{M}_6-M_{14}{M}_7+M_{16}{M}_3{M}_7-(M_{15}{M}_3+M_{14}{M}_4)M_8],\\ L_{23}& =(M_6-M_3{M}_8)M_{16},L_{31}=M_1{M}_{13}{M}_{16}{M}_7-M_1{M}_{15}{M}_{16}{M}_5,\\ L_{32}& =(M_1{M}_{15}+M_1{M}_5+M_{15}{M}_5+M_{16}{M}_4{M}_5-M_{13}{M}_7\\ & +M_{16}{M}_2{M}_7-M_{15}{M}_2{M}_8-M_{13}{M}_4{M}_8)M_{16},\\ L_{33}& =(M_2{M}_8-M_1-M_{15}-M_{16}{M}_4-M_5)M_{16},L_{41}=M_1{M}_{14}{M}_{16}{M}_5-M_1{M}_{13}{M}_{16}{M}_6,\\ L_{42}& =(M_{14}{M}_2{M}_8-M_1{M}_{14}-M_{14}{M}_5+M_{16}{M}_3{M}_5+M_{13}{M}_6-M_{16}{M}_2{M}_6-M_{13}{M}_3{M}_8)M_{16},\\ L_{43}& =(M_{14}-M_{16}{M}_3)M_{16}.\end{array}$

Additional information

Funding

The authors convey their sincere thanks to the Council of Scientific and Industrial Research for providing financial support to carry out this research work through the project [25(0296)/19/EMR-II] entitled ‘Mathematical Modeling of Elastic Waves in Fractional-Order Thermoelastic Solids with Micro-configurations and Initially Stressed Media’ for providing financial support for this research work.