The well established continuum theory of lattice defects is usually formulated in a three-dimensional material space. The defects are represented by differential geometric properties of the material manifold. Moving defects lead to differential geometric quantities changing with time. This motivates to augment the three space-like coordinates in the material space by a time-like coordinate to a four-dimensional manifold. The lattice vectors of a crystalline solid represent three space-like vectors in the material manifold. They can be completed to a tetrad field by a fourth time-like vector. The additional components of the tetrad are related to temperature and heat flux. The derivatives of the Lagrangian density for a hyper-elastic solid with respect to the components of the tetrad can be arranged in a four-dimensional second-order tensor in which entropy density and current are coupled to material momentum. The representation of entropy production leads to a Cattaneo type constitutive equation for heat transfer.
On the Thermodynamic Coupling in the Continuum Theory of Defects
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Schnepp, J. "On the Thermodynamic Coupling in the Continuum Theory of Defects." Proceedings of the ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. Volume 1: Advanced Computational Mechanics; Advanced Simulation-Based Engineering Sciences; Virtual and Augmented Reality; Applied Solid Mechanics and Material Processing; Dynamical Systems and Control. Nantes, France. July 2–4, 2012. pp. 535-542. ASME. https://doi.org/10.1115/ESDA2012-82821
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