In this chapter, we review the fundamental concepts and formulation of the thermomechanics of continuous media. First, we revise the expressions of the two first laws of thermodynamics for thermomechanical processes, that is, those with changes in temperature and strains as state-independent variables. These equations constitute the generalization of the energy balance equation that was presented, particularized for the isothermal case, in chapter “ Basic Equations of Continuum Mechanics” . Next, we introduce different thermodynamic potentials such as the internal energy density, the Helmholtz free energy density, and the dissipation density. This latter is expressed in terms of additional independent internal state variables that take into account history-dependent changes in the material’s internal microstructure. The associated thermodynamic fluxes are then defined as derivatives of the dissipation density with respect to the corresponding thermodynamic drivers (internal variables). The next section introduces the fundamental principles for simple (local, nongraded) materials. These allow establishing the general constitutive equations for the rest of the state- dependent variables, stress and entropy, from the expression of the chosen thermodynamic potential and the fulfillment of the second law of thermodynamics. These expressions are finally applied to several examples: two types of non-dissipative materials, such as ideal fluids and elastic solids in thermomechanical processes, and two dissipative cases: damage mechanics and nonlinear viscoelastic solids. We finish with the formulation and results of a complex thermomechanical process, the extrusion forming process of an aluminum billet under high temperature and viscoplastic behavior.
- 2.Ball J (1977) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 61:317–401Google Scholar
- 10.Merodio J, Ogden RW (2019) Basic equations of continuum mechanics. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). SpringerGoogle Scholar
- 12.Montagut E, Kachanov M (1988) On modeling a microcracked zone by weakened elastic material and on statistical aspects of crack-microcrack interaction. Int J Fract 37:R55–R62Google Scholar
- 13.Mullins L (1947) Effect of stretching on the properties of rubber. J Rubber Res 16:275–289Google Scholar
- 17.Serrin J (1959) Mathematical principles of classical fluid mechanics. In: Flügge S (ed) Encyclopedia of physics, vol VIII/1. Springer, Berlin, Heidelberg, pp 125–263Google Scholar
- 22.Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S (ed) Handbuch der Physik, III(1). Springer, Berlin, Heidelberg, pp 225–793Google Scholar
- 23.Zhou J, Li L, Duszczyk J (2003) 3D FEM simulation of the whole cycle of aluminium extrusion throughout the transient state and the steady state using the updated Lagrangian approach. J Mater Process Technol 134:282–297Google Scholar