Definitions of standard stress and standard heat flux in simple bodies, treated according to Mach and Painlevé

This work concerns the foundations of the thermodynamic theory of ordinary continuous media within classical physics; it constitutes the natural extension of earlier papers [A. Bressan and A. Montanaro, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl. 1 (1990), no. 3, 59–94; MR1082620 (92c:73007)(Part 2); Montanaro, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl. 1 (1990), no. 5, 123–146; MR1088046 (92b:73004)]. In our paper [A. Bressan and A. Montanaro, op. cit.] we defined contact forces, in the purely mechanical case for general simple bodies, in terms of forces at a distance and some kinematic notions. This was achieved in two steps: (i) by proving a strict uniqueness theorem for the (ordinary) response function for the stress of a hyperelastic body, which allows us to implicitly define contact forces for these bodies [A. Bressan and A. Montanaro, op. cit. (Part 1)]; and then (ii) by extending this uniqueness to the standard functional for the stress of any simple body by taking into account the possibility of suitable experiments, which roughly consist in cutting parts of the simple body and putting them in contact with some hyperelastic bodies [A. Bressan and A. Montanaro, op. cit. (Part 2)]. Our previous work [A. Montanaro, op. cit.] constitutes the thermodynamic extension of the work of Bressan and Montanaro [op. cit. (Part 1)], in that in it uniqueness theorems are proved for all the (ordinary) response functions of a thermoelastic body. The present paper extends the work of Montanaro [op. cit.] in the thermodynamic case, in the same way as the paper of Bressan and Montanaro [op. cit. (Part 2)] extends that of Bressan and Montanaro [op. cit. (Part 1)] in the purely mechanical case. Indeed, by defining the standard functionals and by postulating the possibility of suitable experiments of cutting the body and putting it in contact with some thermoelastic bodies, here we extend the uniqueness of the response functions of a thermoelastic body to the standard functionals for the stress and for the heat flux in a generic simple body. This allows us to define the corresponding notions of stress and heat flux for general simple bodies in thermodynamics, as well.