Description
The semi-continuous casting of aluminum ingots, upstream of the production of sheets by rolling, is shown in figure 1.2. It is a typical process used in industry which shows well, first, the complexity of the phenomena occurring, and second, the different scales that can be distinguished. At the macroscopic scale of the process (fig. l.2(a)), liquid metal is injected through a nozzle (forced convection), cools on contact with the mold (heat exchange) and solidifies (phase transformation). The temperature gradients induce convection movements in the liquid (natural convection) and stress in the solid (deformation). The convection, in tum, modifies the temperature distribution in the liquid, as the deformation of the solid changes the thermal contact with the mold. One sees then that the problems of mass, heat and momentum exchange are completely coupled. For alloys, one must consider also the redistribution of the different constituents during solidification.
It is apparent, even at this scale, that the problem is far from simple. And yet, the macroscopic scale directly influences the microscopic, namely the formation of microstructures during solidification. As shown in figure l .2(b ), grains of aluminum form in the liquid, grow and can be entrained by convection. Strong convection can also break up the dendrites that compose a grain, giving birth to new grains. This coupling between macroscopic and microscopic scales works in both directions, as the formation of crystals liberates heat, rejects solute, and slows the convection in the liquid and thus affects the processes at the largest scale.
It should be noticed that the environment considered in figure 1.2 is not homogeneous as it is composed of at least two phases, namely solid and liquid. The conservation equations for multiphase environments are presented later, in chapter 5. Even though very complex, the phenomena occurring at the macroscopic scale of the continuous casting process (fig. l.2(a)) or those that control the formation of microstructures at the microscopic scale can be described with four conservation equations (mass, momentum, energy and solute)! These equations need to be matched with the appropriate boundary and initial conditions and the constitutive equations controlling the behavior of the material in question in its different phases. By behavior we mean the relation between cause and effect such as between stress and strain for elastic deformation, stress and strain rate in fluid mechanics, or temperature-gradient and heat-flux, etc. The type of behavior (elastic, plastic, viscous, etc.) and also the values of thermo-mechanical properties of the material (specific mass, viscosity, elastic modulus, strain-hardening coefficient, thermal conductivity, etc.) enter into the constitutive equations for the different phases of the material. Notice that the formalism of the conservation equations allows an approach to problems as diverse as the injection of polymers in a mold (fig. l.3(a)), the diffusion of water during the drying of concrete (fig. l.3(b)), the strain of a test piece under tension (fig. l.3(c)) or the diffusion that occurs in a sintered ceramic (fig. l.3(d)).
The goal of this chapter is to lay out a presentation, if not exhaustively, at least in overview, of the formalism of the mechanics of continuous media, while attempting to show its generality, describing equally well a process such as continuous casting, a mechanical test, or the evolution of the microstructure in a material.