Geodesic
Thermodynamically consistent gradient elasticity with an internal variable
Theoretical and applied mechanics, Tome 47 (2020) no. 1.

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The role of thermodynamics in continuum mechanics and the derivation of proper constitutive relations is a topic discussed in Rational Mechanics. The classical literature did not use the accumulated knowledge of thermostatics and was very critical of the heuristic methods of irreversible thermodynamics. In this paper, a small strain gradient elasticity theory is constructed with memory effects and dissipation. The method is nonequilibrium thermodynamics with internal variables; therefore, the constitutive relations are compatible with thermodynamics by construction. The thermostatic Gibbs relation is introduced for elastic bodies with a single tensorial internal variable. The thermodynamic potentials are first-order weakly nonlocal, and the entropy production is calculated. The constitutive functions and the evolution equation of the internal variable are then constructed. The second law analysis has shown a contribution of gradient terms to the stress, also without dissipation.
Keywords: nonequilibrium thermodynamics, generalised continua, gradient elasticity 
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     author = {Peter V\'an},
     title = {Thermodynamically consistent gradient elasticity with an internal variable},
     journal = {Theoretical and applied mechanics},
     pages = {1 - 17},
     publisher = {mathdoc},
     volume = {47},
     number = {1},
     year = {2020},
     url = {http://geodesic.mathdoc.fr/item/TAM_2020_47_1_a2/}
}
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Peter Ván. Thermodynamically consistent gradient elasticity with an internal variable. Theoretical and applied mechanics, Tome 47 (2020) no. 1. http://geodesic.mathdoc.fr/item/TAM_2020_47_1_a2/