Reflection of thermo-elastic wave in semiconductor nanostructures nonlocal porous medium
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Reflection of thermo-elastic wave in semiconductor nanostructures nonlocal porous medium HASHMAT Ali1, ADNAN Jahangir2, AFTAB Khan1 1. Department of Mathematics, COMSATS University Islamabad, Islamabad Campus 44000, Pakistan; 2. Department of Mathematics, COMSATS University Islamabad, Wah Campus 47040, Pakistan © Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract: The current work is an extension of the nonlocal elasticity theory to fractional order thermo-elasticity in semiconducting nanostructure medium with voids. The analysis is made on the reflection phenomena in context of three-phase-lag thermo-elastic model. It is observed that, four-coupled longitudinal waves and an independent shear vertical wave exist in the medium which is dispersive in nature. It is seen that longitudinal waves are damped, and shear wave is un-damped when angular frequency is less than the cut-off frequency. The voids, thermal and non-local parameter affect the dilatational waves whereas shear wave is only depending upon non-local parameter. It is found that reflection coefficients are affected by nonlocal and fractional order parameters. Reflection coefficients are calculated analytically and computed numerically for a material, silicon and discussed graphically in details. The results for local (classical) theory are obtained as a special case. The study may be useful in semiconductor nanostructure, geology and seismology in addition to semiconductor nanostructure devices. Key words: three-phase lag model; semiconductor; fractional order time derivative; non-local theory; nanostructure; voids; reflection Cite this article as: HASHMAT Ali, ADNAN Jahangir, AFTAB Khan. Reflection of thermo-elastic wave in semiconductor nanostructures nonlocal porous medium [J]. Journal of Central South University, 2020, 27. DOI: https://doi.org/10.1007/s11771-020-4472-1.
1 Introduction Eringen’s [1] nonlocal elasticity theory is the most widely used theory and different nonclassical thermoelasticity theories have been developed with some extra atomistic features based on Eringen’s [1] theory. The nonlocal theory [1] supposed that the involvement of strain field at the neighboring points about any local point is also very significant in the study of stress applied at that particular point in the solid continuum. Therefore, the nonlocal elasticity theory has area of study for atoms or molecules containing long-range forces, thus, an internal length scale parameter is needed to formulate the problem.
ALTAN [2] proved the uniqueness theorem of linear nonlocal elasticity theory. CHIRITA [3] solved some boundary value problems using the idea of nonlocal theory of elasticity. CRACIUN [4] presented his model under the effect of nonlocal thermoelasticity. The nonlocal elasticity models characterized by the presence of nonlocality residuals of fields (like body force, mass, entropy, internal energy, etc.) have been proposed by EDELEN et al [5, 6]. ERINGEN et al [7−10] used the non-local theory to different oth
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