In this paper, the memory-dependent nonlocal magnetothermoelasticity theory is used for the reflection problem in a magnetized electrically conducting thermo-triclinic solid half-space. The velocity equation is derived by formulating and solving the governing equations for a triclinic magnetothermoelastic medium according to the memory-dependent derivative nonlocalized thermoelasticity. Three quasi-plane waves, namely, quasi-longitudinal displacement (qP), quasi-thermal (qT), and quasi-shear vertical (qSV) waves, propagate in the medium according to the plane-wave solution. The wave velocities are calculated. For the incidence of a coupled quasi-plane wave, the equations for the reflection coefficient and energy ratio are derived. These characteristics are presented graphically.
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References
A. C. Eringen, Nonlocal polar elastic continua, Int. J. Eng. Sci., 10, 1–16 (1972).
A. C. Eringen, Theory of nonlocal thermoelasticity, Int. J. Eng. Sci., 12, 1063–1077 (1974).
A. C. Eringen, Nonlocal Continuum Theories, Springer, New York (2002).
Y. J. Yu, X. G. Tian, and X. R. Liu, Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity, Eur. J. Mech. A/Sol., 60, 238–253 (2016).
K. Wang and B. Wang, Vibration modelling of carbon-nanotube-based biosensors incorporating thermal and nonlocal effects, J. Vib. Control, 22, No. 5, 1405–1414 (2016).
C. Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation, C. R. Acad. Sci. Paris, 247, 431–433 (1958).
K. Diethelm, Analysis of Fractional Differential Equation. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin, Heidelberg (2010).
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010).
J. L. Wang and H. F. Li, Surpassing the fractional derivative: Concept of the memory dependent derivative, Comput. Math. Appl., 62, No. 3, 1562–1567 (2011); doi: https://doi.org/10.1016/j.camwa.2011.04.028.
M. A. Ezzat, A. S. El-Karamany, and A. A. El-Bary, Generalized thermo-viscoelasticity with memory-dependent derivatives, Int. J. Mech. Sci., 89, 470–475 (2014).
M. A. Ezzat, A. S. El-Karamany, and A. A. El-Bary, A novel magneto-thermoelasticity theory with memory-dependent derivative, J. Electromagn. Waves Appl., 29, No. 8, 1018–1031 (2015).
M. A. Ezzat, A. S. El-Karamany, and A. A. El-Bary, Modeling of memory-dependent derivatives in generalized thermoelasticity, Eur. Phys. J. Plus, 131, Article ID 372 (2016).
R. S. Dhaliwal and H. H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math., 33, 1–8 (1980).
H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, 15, 299–309 (1967).
J. Wang and R. S. Dhaliwal, Uniqueness in generalized nonlocal thermoelasticity, J. Therm. Stresses, 16, 71–77 (1993).
B. Singh and A. K. Yadav, Plane waves in a rotating monoclinic magnetothermoelastic medium, J. Eng. Phys. Thermophys., 89, 428–440 (2016).
M. Zilmer, D. Gajewski, and B. M. Kashtan, Reflection coefficients for weak anisotropic media, Geophys. J. Int., 129, Issue 2, 389–398 (1997).
T. Mensch and P. Rasolofosaon, Elastic wave velocities in anisotropic media of arbitrary symmetry — Anisotropy generalization of Thomsen's parameters ε, δ and γ, Geophys. J. Int., 128, Issue 1, 43–63 (1997).
A. Chattopadhyay, Wave reflection and refraction in triclinic crystalline media, Arch. Appl. Mech., 73, 568–579 (2004).
A. Chattopadhyay, Wave reflection in triclinic crystalline media, Arch. Appl. Mech., 76, 65–74 (2006).
A. Chattopadhyay, Reflection for three-dimensional plane waves in triclinic crystalline medium, Appl. Math. Mech., 28, No. 10, 1309–1318 (2007).
B. Singh and A. K. Yadav, Plane waves in a transversely isotropic rotating magnetothermoelastic medium, J. Eng. Phys. Thermophys., 85, 1226–1232 (2012).
B. Singh and A. K. Yadav, Reflection of plane waves in a rotating transversely isotropic magneto-thermoelastic solid half-space, J. Theor. Appl. Mech., 42, No. 3, 33–60 (2012).
A. K. Yadav, Magnetothermoelastic waves in a rotating orthotropic medium with diffusion, J. Eng. Phys. Thermophys., 94, 1663–1672 (2021).
P. Zhang and T. He, A generalized thermoelastic problem with nonlocal effect and memory-dependent derivative when subjected to a moving heat source, Waves Random Complex Media, 30, Issue 1, 142–156 (2020); doi: https://doi.org/10.1080/17455030.2018.1490043.
A. K. Yadav, Reflection of plane waves in a fraction-order generalized magneto-thermoelasticity in a rotating triclinic solid half-space, Mech. Adv. Mater. Struct., 1–18 (2021); doi: https://doi.org/10.1080/15376494.2021.1926017.
S. Biswas, Modeling of memory-dependent d erivatives in orthotropic medium with three-phase-lag model under the effect of magnetic field, Mech. Based Des. Struct. Mach., 47, No. 3, 302–318 (2019); doi: https://doi.org/10.1080/15397734.2018.1548968.
M. Bachher and N. Sarkar, Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer, Waves Random Complex Media, 29, Issue 4, 595–613 (2019); doi: https://doi.org/10.1080/17455030. 2018.1457230.
Iqbal Kaur, Parveen Lata, and Kulvinder Singh, Reflection and refraction of plane wave in piezo-thermoelastic diffusive half spaces with three phase lag memory dependent derivative and two-temperature, Waves Random Complex Media, 32, Issue 5, 2499–2532 (2022); doi:https://doi.org/10.1080/17455030.2020.1856451.
N. Sarkar and S. De, Reflection of thermoelastic plane waves at a stress-free insulated solid boundary with memorydependent derivative, Indian J. Phys., 95, No. 6, 1203–1211 (2021); https://doi.org/https://doi.org/10.1007/s12648-020-01788-2.
Nihar Sarkar, Soumen De, Narayan Das, and Nantu Sarkar, Reflection of thermoelastic waves from the insulated surface of a solid half-space with time-delay, J. Heat Transf., 142, No. 9 (2020); doi:https://doi.org/10.1115/1.4046924.
J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Company, Amsterdam (1973).
B. A. Auld, Acoustic Fields and Waves in Solids, Vol. 1, Krieger Publishing Company, Malabar, Florida, USA (1990).
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Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 96, No. 6, pp. 1669–1684, November–December, 2023
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Yadav, A.K., Schnack, E. Plane Wave Reflection in a Memory-Dependent Nonlocal Magnetothermoelastic Electrically Conducting Triclinic Solid Half-Space. J Eng Phys Thermophy 96, 1658–1673 (2023). https://doi.org/10.1007/s10891-023-02836-4
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DOI: https://doi.org/10.1007/s10891-023-02836-4