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Towards an analytical mechanics of dissipative materials. (English) Zbl 1097.74005

The author considers a Hamiltonian-Lagrangian density in non-linear continuum theory of thermoelastic conductor. The free energy is dependent on the deformation gradient, on the temperature defined as the time derivative of a scalar field, the so called thermacy, as well as on the the gradient of the thermacy. In the Lagrangian density the basic fields are the placement (motion) and the thermacy (but the Lagrangian is not an explicit function of them), both being functions of the space-time parametrization. The field equations are Euler-Lagrange equations associated with the motion and the thermacy, and Noether theorem for the Lagrangian with respect to the space-time parametrization lead to the balance equations, as well as to the equations of the entropy and of the energy. The free energy is the potential for the first Piola-Kirchhoff stress, with respect to the deformation gradient, and for the entropy flux in terms of the gradient of the thermacy, while the linear momentum and the entropy density are derived as a generalized rate of basical fields. The material force of the true inhomogeneities and thermal forces of quasi-inhomogeneity are involved in the representations, as well as two Legendre transformations, related to mechanical and to thermal fields, respectively. The model holds for the non-zero variantion in time of the thermacy. The contribution to the gradient of the temperature is separately put into evidence, and the attention is focused on the appropriate heat- propagation equations. An analitical mechanical model is attached to certain finite continuum models. In order to account for anelasticity the free energy is apriori dependent on a new set of variables, which can be related to certain dissipative processes. Here the material forces of quasi-inhomogeneities, due to a non-uniform temperature field and a non-uniform internal variables of state, play a fundametal role.

MSC:

74A15 Thermodynamics in solid mechanics
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
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