Issue 39

A. Risitano et alii, Frattura ed Integrità Strutturale, 39 (2017) 202-215; DOI: 10.3221/IGF-ESIS.39.20 205 for the internal variables. 2. The specific heat C   (for the constants  and  ) is assumed to be independent respect to the internal state of the material and hardening. 3. The conduction is linear isotropic, and thus, the Fourier law q k     is valid. 4. The total derivatives with respect to time are reducible at partial derivatives; therefore, it is possible to neglect the convective terms, and the transformation velocities can be considered negligible. 5. The external temperature   is assumed to be time/space constant; therefore, the external flux r e is considered to be independent of time. 6. The initial temperature   (x,y,z,0) of any point can be considered to be coincident with the external temperature    To understand the logical process, we have to start following points presented in [20], where Maugin connected the mathematical theory of plasticity and fracture to the field of thermomechanical for metallic material. Starting from the second law of thermodynamics and the principle of continuity of mass, the inequality equation of Clausius-Duhem can be written as follows: r t int 0       Intrinsic and thermal dissipation are respectively: v e p r t A q int : :                        where   v e p : :        is the inelastic component and   A   is the hardening dissipation that the material uses to determine its own internal characteristics. By the theorem of energy in the local form e e D r q :       , using the previous assumptions and the temporal derivatives of the Helmholtz’s potential (  ), it is possible write the following equation of heat diffusion: e r e e C k r 2 , int : :                             Assuming that the coupling terms between temperature and internal variables is negligible : 0         , t he diffusion equation can be write: sm e C k r 2 ,            with e sm r e int :             (1) where:    , ,        is the Helmholtz free energy;  r e is the external heat flux;   int r is the intrinsic dissipation per unit of volume of irreversible phenomena;    is the internal temperature;   sm   includes all terms that yield internal dissipations and is called the thermoelastic or isentropic term. Focusing the attention on the first part of static tensile test, it is possible to individuate the thermoelastic zone where transformations are reversible. Focusing on the first part of the true curve while maintaining a constant external temperature during testing, visco-plastic phenomena can be neglected, and it is possible to set all the terms of intrinsic