*2.2. Modeling Results*

Equations from (1) to (5) were solved numerically using an explicit difference scheme and a fixed grid method [22] implemented in Delphi 7.0.

The physical properties for an A356 alloy (Al-7% Si), listed in Table 1, were used in solving the problem.


**Table 1.** Physical quantities used in calculation.

The ranges of process parameters in the calculation were as follows:


The calculation showed that the temperature profile weakly depends on vibration within the given range of amplitude and frequency. Figure 2a shows the temperature in its center *T*(0,*t*) without and with vibration at a frequency of 50 and 80 Hz. The movement of the solidification front *rs*<sup>1</sup> (that is, corresponding to the liquidus temperature *Tm*1) is nonlinear (Figure 2b).

**Figure 2.** (**a**) Time dependence of the temperature in the center of liquid metal volume for different conditions: 0—without vibration, 50—vibration of *f* = 50 Hz, 80–vibration of *f* = 80 Hz; *Ts* = 630 K, *T*<sup>0</sup> = 900 K.; (**b**) Time dependence of the movement of the solidification front from the start point *tcol* for two thermal modes: 1—*Ts* = 630 K, *T*<sup>0</sup> = 900 K, and 2—*Ts* = 430 K, *T*<sup>0</sup> = 1050 K under *A* = 0.5 mm, *f* = 50 Hz.

First of all, the thermal solution allows one to define the kinetics of the process. Figure 2a shows the time-dependence of the temperature in the center of the melt volume *T*(0,*t*). Vibration accelerates the process of solidification: the "plateau" on the temperature profile corresponding to the solidification becomes shorter with the vibration.

The calculation shows that the duration of alloy solidification as well as the cooling time (combined cooling to the temperature of liquidus and then to the solidus), depends on the initial temperature difference between the melt and the chill mold, and is less dependent on frequency (Figure 3a). The solidification time slightly shortens with the increasing frequency (Figure 3b). As one can see, the solidification time is short in all cases, a few seconds. However, it is during this brief period of time that the vibration has an effect on the solidification and structure formation.

**Figure 3.** Dependencies of the cooling time (**a**) and the solidification time (**b**) on the vibration frequency: 1—*Ts* = 630 K, *T*<sup>0</sup> = 900 K, and 2—*Ts* = 430 K, *T*<sup>0</sup> = 1050 K; *A* = 0.5 mm.

Figure 4 shows the dependence of the cooling time *tcol* on the initial mold temperature, as well as the dependence of the solidification time on the initial melt temperature. The higher the initial temperature of the metal and that of the chill mold, the more time needed for cooling; which is the expected result.

**Figure 4.** Dependence of (**a**) the cooling time before solidification and (**b**) the solidification time on the initial temperature of the chill mold *Ts* for different starting melt temperatures *T*0.

When analyzing the stress amplitude imposed on the semi-solid alloys, one should note that the strength of semi-solid alloys can be significantly lower than that of the solid alloy, i.e., the tensile strength of an A356 alloy decreases from 157 MPa in the solid state [27] to 4.98 MPa at 860 K and to ~ 0.01 MPa at 880 K [24]. Nevertheless, a certain critical amplitude of mechanical stress *σmin* that is higher than the tensile strength of the semi-solid alloy, is required to effect the fracture of growing crystals, and as a consequence, the refinement of the crystalline grains in the casting. Not only the value of *σmin* is important but also the operating time of these stresses during the alloy solidification (number of oscillations during solidification). A combination of those can be characterized by the value *Zσ* (Equation (6)).

Figure 5a shows the dependence of the specific stress integral *Zσ* on the vibration frequency for the two following conditions: *T*<sup>s</sup> = 630 K, *T*<sup>0</sup> = 900 K and *T*<sup>s</sup> = 430 K, *T*<sup>0</sup> = 1050 K. The calculation demonstrates that the effectiveness of the vibration grows nonlinearly with increasing the vibration frequency and saturates at a certain frequency. This effect can be explained by the reduction of the solidification time during which the vibration affects the growing crystals (see Figure 3b). It can be assumed that the fracture of growing crystals is possible only under conditions when a relative value *Z* exceeds a certain critical value: Z = *Z*σ/*Zs* = *Z*σ/*c*1ρ<sup>1</sup> > *Zcr*.

The dependence of this integral stress characteristic on amplitude is, however, linear and the values become substantial at the frequencies above 50 Hz (Figure 5b).

**Figure 5.** (**a**) The dependence of the specific stress integral on the vibration frequency for two thermal conditions: 1—*Ts* = 630 K, *T*<sup>0</sup> = 900 K, and 2—*Ts* = 430 K, *T*<sup>0</sup> = 1050 K; *A* = 0.5 mm. (**b**) The dependence of the specific stress integral on the vibration amplitude for the four values of frequency.

Thus, according to the proposed approach, the vibration of liquid and solidifying metal causes the following effects:


Considering that the vibration stresses operate on growing crystals in a particular short period of solidification, the time integral characteristic of mechanical stresses can be suggested as a measure of the effectiveness of vibration. The higher this value, the greater the effect of vibration stress on crystals. The integral characteristic of mechanical stresses increases linearly with rising amplitude but grows nonlinearly with rising vibration frequency. It is necessary to consider the time and mechanical conditions under vibration treatment of an alloy during solidification as:

