Modeling of Fracture, Melting, Vaporization, and Phase Transition
So far we have considered waves of heat, stresses, and destruction arising from the action of relatively weak radiation pulses. Next, we take into account the possibility of the plasma appearance. In particular, two models are introduced for consideration: The first mode uses the wide-ranged equations of state (see Section 1.1), and the second model uses the separation of arising phases.
For the first (continuum) model, we continue to use the Eqs. (10.22)-(10.34). At the same time, the formulation of the problem changes because we replace the equations of state (10.35)—(10.37) by more general equations, which take into account the possibility of the material evaporation (the appearance of the plasma phase of material). When the second model is used, the surface of evaporation is determined at each step of the problem solution. The method of the previous part is used where material retains the strength characteristics. In the vapor zones, the equations of high-temperature gas are used. On the evaporation surface, the Knudsen conditions are written.
1. The continuum model. The possibility of the material evaporation is taken into account. Additionally, here we use the term dQ/dz from (10.24). This term describes the radiation energy. The generalized Bouguer-Lambert law is written for this energy:
Eq. (10.44) describes the dependence of the radiation attenuation coefficient L(V,T) on the density and temperature of the material. If L(V,T) = const, then
To close the above system of differential equations, we use wide-range equations of state (see Eqs. (1.10) and (2.50)) [44]:
The process of the transition of the solid to the liquid and gaseous states without explicit defining the phase boundary can be expressed using the above equations. Here, we assumed that the material is vaporized if the thermal component of pressure exceeds the greatest (negative) pressure of the cold component [44]:
In this case,
For example, here m = 6, n = 4 for iron and aluminum. It is assumed that the substance is in the gaseous state in those regions, i.e., those finite- difference cells, in which V > V,.
2. The Knudsen model (conditions). When the second approach is used, the vaporization surface is identified in explicit form. Knudsen conditions are written on this surface, these being the boundary conditions of the problem. The part of the material, which has retained its strength properties, is described by (10.22)—(10.39), (10.44), and (10.45).
The motion of the gases (vaporization products) is described by equations that are valid for a high-temperature gas [18]:
where z is the Lagrangian coordinate, and N is the number of moles of gas.
The surface of contact of the solid and the gas is modeled by relations characterizing the kinetics of the phase transformation [45] (the Knudsen conditions):
where M = v(2RT)05, and erfc is the complementary error function. The quantities without indices correspond to the thermodynamic parameters for the gas at the discontinuity; the subscript s denotes the values of these quantities on the surface of the solid. It should be noted that the formulas from (10.50) for Cp/Cv = 5/3 are identical to those used in Ref. [46]. Thus, they constitute a special case of the equations for the Knudsen discontinuity [45]. Eqs. (10.50) correspond to the Hugoniot relations for a shock-wave discontinuity except for the fact that these equations presume a two-velocity gas model. These equations are analogous to the Hugoniot relations for an ideal gas when Cp/Cv = 5/3.
The presence of a nonequilibrium Knudsen layer is due to the flow of particle returned to the surface of discontinuity from the gaseous medium. The magnitude of this reverse flow depends on the conditions of gasdy- namic dispersal, characterized by the Mach number (M). The latter determines the quantity of mass vaporized and the rebound pressure [47]. The maximum value of M is unity. Here, the dynamics of processes taking place in the solid and gas can be examined independently of one another.
The Knudsen model, in accordance with [45], is augmented by the equation of energy balance at the discontinuity:
and the mass conservation law is
where p, is the density of the substance ahead of the discontinuity, L is the specific heat of sublimation, and 2 is the velocity of the vaporization front. Eq. (10.51) describes the action of the external load, which is characterized by the specific power density associated with the incoming energy q(t).
Thus, at Л/ = 1 the second approach reduces to the solution of Eqs.
(10.22)—(10.39) with the boundary conditions (10.50)—(10.52).
Boundary conditions [1,6,7]. Boundary conditions are set to zero for stresses and heat influx due to thermal conductivity:
where H is thickness of the loading plate.
1. Distributed load. On the loading surface,
2. Instant load. In this case, Qo(0,t) = 0 in (10.53).
Ideal contact conditions are assumed for a multilayer material on the surfaces of contacts of layers.
Initial conditions [1,6,7]. We assume that
- 1. Distributed load. In this case, all unknown values are zero at t = 0.
- 2. Instant load. In this case, the temperature is given in the form of an initial distributed front:
The initial thermal stresses and the initial distribution of thermal energy arise in the material in this case. The energy flow Q(z„t = 0) = 0.
