A Mathematical Study of Magneto-Thermoelasticity: Theory, Wave Propagation, and Boundary Value Problems

Prasad S. Patil

doi:10.36948/ijfmr.2026.v08i04.86376

A Mathematical Study of Magneto-Thermoelasticity: Theory, Wave Propagation, and Boundary Value Problems

Author(s)	Prasad S. Patil
Country	India
Abstract	This paper presents a unified mathematical treatment of magneto-thermoelasticity, synthesizing the classical coupled theory, its principal generalized (finite-speed) extensions, and their application to plane-wave dispersion and boundary value problems in a perfectly conducting elastic half-space. Beginning from the Duhamel–Neumann stress–temperature relation and the Lorentz body force induced by a primary magnetic field, the coupled momentum equation is reduced, under the perfectly conducting approximation, to a magnetically modified Navier–Lamé equation, and coupled in turn with the Lord–Shulman, Green–Lindsay, and Green–Naghdi (Types I–III) heat-conduction laws. All three generalized theories are situated within a single non-dimensional parameter family governed by a magnetic pressure number and a thermoelastic coupling constant, so that the resulting dispersion relation and boundary value problem can be compared across theories in one consistent notation. Two complementary boundary value problems are solved: a plane-wave dispersion analysis for the unbounded medium, yielding a single relation, quadratic in the squared wavenumber, valid across three generalized theories; and a Laplace-transform solution of a half-space subjected to a sudden thermal shock at a traction-free boundary. The results are verified two ways — analytically, through uniqueness and reciprocity arguments for the coupled field equations, and numerically, through a from-first-principles computation of dilatational-wave phase velocity, thermal-mode attenuation, and the transient temperature and stress fields, the last obtained by Stehfest numerical Laplace inversion and checked against the imposed boundary data to within 10⁻⁷ in non-dimensional units. The computations confirm that the magnetic field acts as a monotonic stiffening mechanism on the dilatational wave speed under all three generalized theories, that the Green–Naghdi Type II theory alone propagates without spatial attenuation, and that the half-space thermal shock produces a temperature front trailed by a sign-changing stress response consistent with a mechanical precursor outrunning the thermal disturbance. The paper closes by situating these results against the literature on rotation, initial stress, and Hall-current extensions of the theory, and identifies their full combination, together with cylindrical and spherical boundary geometries, as the most direct extensions of the present formulation.
Keywords	magneto-thermoelasticity; generalized thermoelasticity; thermal relaxation; wave propagation; Laplace transform; boundary value problems
Published In	Volume 8, Issue 4, July-August 2026
Published On	2026-08-21
DOI	https://doi.org/10.36948/ijfmr.2026.v08i04.86376

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A Mathematical Study of Magneto-Thermoelasticity: Theory, Wave Propagation, and Boundary Value Problems

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