Abstract
An integral formula is used to average a coupled problem of thermoelasticity for a nonuniform rod of variable cross section. Effective characteristics are found. It is shown that, in addition to the expected effective coefficients, there appear five independent coefficients characterizing the temperature change rate effect on the stresses in the rod, on the longitudinal heat flux, and on the entropy distribution along the length of the rod. A feature of these new coefficients is that they become equal to zero in the case of a uniform rod. The homogenization of the thermoelasticity equations for nonuniform rods allows one to propose a new theory of heat conduction in rods. This new theory differs from the classical one by the fact that some new terms are added to the Duhamel–Neumann law, to the Fourier heat conduction law, and to the entropy expression. These new terms are proportional to the temperature change rate with time. It is also shown that, in the new theory of heat conduction, the propagation velocity of harmonic heat perturbations is dependent on the oscillation frequency and is finite when the frequency tends to infinity.
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References
B. E. Pobedrya, Numerical Methods in the Theory of Elasticity and Plasticity (Mosk. Gos. Univ., Moscow, 1995) [in Russian].
V. I. Gorbachev, “Integral Formulae in the Coupled Problem of the Thermoelasticity of an Inhomogeneous Body. Application in the Mechanics of Composite Materials,” Prikl. Mat. Mekh. 78 (2), 277–299 (2014) [J. Appl. Math. Mech. 78 (2), 192–208 (2014)].
B. E. Pobedrya, Mechanics of Composite Materials (Mosk. Gos. Univ., Moscow, 1984) [in Russian].
D. I. Bardzokas and A. I. Zobnin, Mathematical Modeling of Physical Processes in Composite Materials of Periodic Structure (Editorial, Moscow, 2003) [in Russian].
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Original Russian Text© V.I. Gorbachev, 2017, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2017, Vol. 72, No. 2, pp. 48–54.
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Gorbachev, V.I. Heat propagation in a nonuniform rod of variable cross section. Moscow Univ. Mech. Bull. 72, 48–53 (2017). https://doi.org/10.3103/S0027133017020042
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DOI: https://doi.org/10.3103/S0027133017020042