On the issue of classical heat transfer equation applicability to highly elastic materials at large strains

Аuthors

Korneyev V. S., Korneyev S. A.^*, Shalay V. V.

Omsk State Technical University, 11, prospect Mira, Omsk, 644050, Russia

*e-mail: korneyev@omgtu.ru

Abstract

A quantitative accuracy evaluation of the mathematical model describing the distribution of temperature fields and heat flux in structural elements from highly elastic materials with large strains based on the classical heat transfer equation, which does not contain expressions for mechanical work, was performed. The general approach of the nonlinear theory of thermoelasticity is applied with account for the fact that natural and artificial elastomers, widely applied in engineering (primarily rubber and rubber-cord composites), are capable of experiencing large elastic deformations, which contribution overlaps the mechanical effects associated with inelastic (viscoelastic) strain. A comparative analysis of direct consequences of the fundamental law of the internal energy change and experimental data obtained by Joule, James and Guth while elastomers testing on uniaxial adiabatic tension, revealed that large deformations (up to 100% and more) insignificantly affected the total thermal balance. If the term, containing the tensor of latent heats of the isothermal strain relative to the term, containing specific heat capacity at the constant strain, is neglected in the thermal balance equation, then maximum absolute error from this neglecting does not exceed 0.2°С. This evaluation of thermodynamically reversible temperature effect is equally small by the order of magnitude, as for the metals at small strains in the elastic region. Along with this, the temperature effect from the thermodynamically irreversible process of the internal forces of viscous resistance mechanical work turning into a heat at the moderately high deformation velocities is not negligiblysmall, which is accounted for in classical thermal conductivity equation as a correction for the internal heat sources power.

It was noted, that for the widely used model of the thermoelastic body with temperature-dependent heat capacity, the internal energy transfer equation splits into two independent equations, i.e. separately for the thermal energy and for the deformation energy. In the theory of elastic shells, the equation for the thermal energy, which coincides with the classical heat conduction equation, serves to determine the temperature and heat fluxes distribution, and the equation for deformation energy is used for obtaining the governing relationships that relate the linear forces and moments to the bending strains and membrane deformations of the shell. The results obtained are of practical interest for structural elements calculation from highly elastic materials, in particular, while the studying pneumatic elements with the rubber-cord shell.

Keywords:

heat conduction equation, elastomers, finite deformations

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