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DOI: 10.46698/v0156-6205-6438-l
Gradient Model of Non-Stationary Thermoelasticity for a Composite Rod
Nesterov, S. A.
Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 3.
Abstract: Modeling the behavior of micro-sized rod structures is of great practical importance for the design of thermoelectric devices. Previous studies did not take into account the combined effect of thermal and mechanical scaling parameters on the distribution of temperature and displacements of composite bodies. This led to large discrepancies between calculations and experimental data. We study the problem of refining the non-stationary thermal stress state of a composite micro-rod. One end of the rod is clamped and is at zero temperature, and a non-stationary heat flux is specified at the other end. A generalized (gradient) model of thermoelasticity for a composite micro-rod in a quasi-static formulation is obtained. We introduce a thermal scaling parameter into the defining relations of this model, and take into account second-order temperature gradients in the expression for the heat flux. The formulation of the gradient model includes heat conduction and equilibrium equations of a higher order compared to the classical theory as well as an extended list of boundary conditions. In this work, numerical-analytical and asymptotic solutions are constructed for the problem in the case of weak coupling. The problem is reduced to a sequential solution of the problems of gradient heat conductivity and gradient theory of elasticity in Laplace transforms. The inversion of transforms is carried out on the basis of the modified Durbin method. The solution of the singularly perturbed problem of heat conductivity is constructed using the asymptotic approach. Expressions for the temperature are presented as the sum of the classical solution and boundary layer functions. The following effects, not described earlier, are discovered: 1) with an increase in the scale parameter, the temperature values at small times increase; 2) the temperature changes more smoothly in the vicinity of the conjugation point. The accuracy of the asymptotic formulas is verified by comparing with the numerical-analytical solution. The limits of applicability of the asymptotic formulas are determined. The presented approach will allow more accurate calculations of the strength of composite micro-sized bodies, such as a microbeam and a microcylinder.
For citation: Nesterov, S. А. Gradient Model of Non-Stationary Thermoelasticity for a Composite Rod, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp. 91-107 (in Russian). DOI 10.46698/v0156-6205-6438-l
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