Effect of Impedance Boundary on the Reflection of Plane Waves in Fraction-Order Thermoelasticity in an Initially Stresse
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Effect of Impedance Boundary on the Reflection of Plane Waves in Fraction‑Order Thermoelasticity in an Initially Stressed Rotating Half‑Space with a Magnetic Field Anand Kumar Yadav1,2 Received: 12 July 2020 / Accepted: 7 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this research article, the propagation of the plane waves in an initially stressed rotating magneto-thermoelastic solid half-space in the context of fractional-order derivative thermoelasticity is studied. The governing equations in the x–z plane are formulated and solved to obtain a cubic velocity equation that indicates the existence of three coupled plane waves. A reflection phenomenon for the incidence of a coupled plane wave for thermally insulated/isothermal surface is studied. The plane surface of the half-space is subjected to impedance boundary conditions, where normal and tangential tractions are proportional to the product of normal and tangential displacement components and frequency, respectively. The reflection coefficients and energy ratios of various reflected waves are computed numerically for a particular material and the effects of rotation, initial stress, magnetic field, fractional-order, and impedance parameters on the reflection coefficients and energy ratios are shown graphically. Keywords Energy ratio · Fractional-order derivative · Impedance boundary · Initial stress · Magnetic field · Reflection coefficients · Rotation · Thermoelasticity Abbreviations B Magnetic induction, NA−1·m−1 λ, µ Lame′s constants, Nm−2 CE Specific heat at constant strain, Jkg−1·deg−1 E Electric field strength, Vm−1 H Magnetic field strength, Am−1 h Perturbation of magnetic field strength, Am−1 * Anand Kumar Yadav [email protected] 1
Shishu Niketan Model Senior Secondary School, Sector 22‑D, Chandigarh, India
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Department of Mathematics, Maharishi Markandeshwar (Deemed To Be University), Mullana, Ambala, India
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International Journal of Thermophysics
(2021) 42:3
J Current Am−2 n Unit vector Ω Angular velocity, Hz u Displacement vector, m u, w Components of the displacement vector βT Thermal coefficient, Nm−2·deg−1 𝛽T = (3𝜆 + 2𝜇)𝛼 0 , and α0 The coefficient of linear thermal expansion deg−1 K Thermal conductivities Wm−1·deg−1 τ0 The thermal relaxation time, s P Initial pressure, Nm−2 T Change in temperature variable, deg T0 The uniform temperature, deg t Time, s v Wave speed, m/s µe Magnetic permeability, H m−2 ρ Density, kg·m−3 ω Circular frequency, Hz σ The electric conductivity of the medium, S·m−1 θ0 The angle of propagation measured from normal to the half-space, deg, x, y, z are cartesian coordinates eij Components of the strain tensor σij Components of the stress tensor δij Kronecker delta
1 Introduction The dynamical theory of thermoelasticity is the study of the interaction between thermal and mechanical fields in solid bodies and is of considerable importance in various engineering fields, soil dynam
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