On standard dissipative gradient models
Tóm tắt
A general presentation on gradient models for solids in thermo-mechanics is given in this paper. Firstly, it is shown that the introduction of the gradients of the internal variables can be conveniently done under the formalism of the generalized standard materials. The derivation of the governing equations and the associated boundary conditions of a solid can be derived in a straightforward manner from the basic assumption of existence of an energy potential and a dissipation potential. Secondly, the attention is focussed on the introduction of the temperature gradient. Two descriptions, respectively by Forest et al. (Thermoelasticity of second-grade media. Kluwer, Dordrecht, 2000) and by Nguyen and Andrieux (C R Mecanique 333:139–145, 2005) are discussed. The last one is based upon an interesting definition of the entropy and of the internal energy by Legendre transform of the free energy with respect to the temperature and the temperature gradient. The obtained results are justified in a simple example of micro-macro modeling of thermal conduction in a rigid composite. In particular, it is shown that the associated thermal equation removes the paradox of instantaneous propagation.
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