# Extreme Waves Excited by Radiation Impact

- 1. This chapter contains the results published in the USSR at the end of the last century. It developed the approach in which the separation of the problem into a gas-dynamic and strength parts in a wide range of variation of the impact energy, frequencies, and time of action of radiation was not used from the very beginning [1,6,7].
- 2. The behavior of the material on which the radiation pulse falls is largely determined by the wavelength and the radiation power. Figure 10.1 shows the regions of influence of the most important physical and mechanical processes manifested during the action of radiation [6]. It follows from the figure that in the very wide temperature range (10
^{2}—10^{6}K), there is no effect of strength and material can be in the plasmous (lOMO^{6}K), gaseous (10^{4}—10^{5}K), and condensed (10^{2}-10^{3}K) states [21-31].

Here, we study waves of pressure, stresses, and heat arising from the pulsed heating of material surface from 10^{2} to 10^{5} K. At low temperatures, when there is no melting, there are the processes of expansion of the nearsurface layers of the material and their destruction in the form of a surface spalling. The presence of melting and evaporation at higher temperatures leads to the complication of the formation of stress and pressure waves, which are determined both by the reactive recoil of the evaporation products and by the expansion of the unevaporated part of the solid.

The contribution of these two mechanisms to the formation of the waves depends on the intensity of the radiation incident on material. The effect

FIGURE 10.1 Fields of influence of some physical and mechanical processes: 1 - absent of strength, 2 - ionization and formation of plasma. 3 - evaporation, 4 - melting, 5 - phase transitions of the crystal lattice, and 6 - strength of the material.

of reactive recoil can be increased until the manifestation of the effect of screening radiation by flying pairs of material begins.

3. Certain additional results of Russian studies of the action of the radiation pulse on materials can be found in [21-32].

## Impulsive Deformation and Destruction of Bodies at Temperatures below the Melting Point

In this section are considered thermoelastic waves caused by impact radiation. Results published in [6,33,34] are used. Destruction occurs in rarefaction waves in the form of a surface spalling. The considered models relate to the action of longwave (thermal, laser) and short-wave radiation, acting during time less than 1 ns and more 10 ns.

### Thermoelastic Waves Excited by Long-Wave Radiation

The long-wavelength radiation is absorbed in a very thin near-surface layer of material of approximately 1 O'^{8} m. Therefore, in the mathematical formulation of the problem, this layer can be assumed to be infinite thin, and the energy source can be introduced into the boundary conditions for the temperature.

In [6,33], the effect of the energy flux on metallic targets coated with transparent liquid dielectrics was studied. The formula for the shock-wave pressure is derived under the condition that the amplitudes of the shock wave in the liquid and the amplitude of the compression wave in the metal are equal:

where *pi* and p, are the densities of the liquid and the metal, respectively, Д and Д shock-wave velocities, *у* the adiabatic constant, and *Q* the absorbed radiation flux.

The next follows from a comparison of the pressure calculated according to (22.1) and the experiment (Figure 10.2) is discussed next. First, the theory and the experiment are in satisfactory agreement with the fact that the bulk of the energy is absorbed. Second, the energy of the recoil for copper and aluminum targets at about 0.5 J/m^{2} is approximately of an order of magnitude, which is higher than that in the case of irradiation of the pure surface of a metal at the same densities of the radiation flux.

### Thermoelastic Waves Excited by Short-Wave Radiation

A purely thermomechanical problem is being studied. A pulsed heat source is introduced in the thermal part of the problem, and then, the propagation of the thermoelastic wave is studied. Short pulse radiation is absorbed in a sufficiently thick layer of matter. This effect may be simulated assuming the heat sources appearing in the material. The study of the formation and propagation of heat waves and stress waves generated by the action of this radiation on copper mirrors was performed

FIGURE 10.2 Dependence of pressure in a shock wave on the density of the incident radiation flux: (a) experiment and (b) theory (here ВТ is watt (W)), Гпа = GPa.

in [1,6,34]. The heat propagation in these mirrors was described by the relaxation model of the heat flux (the modified Fourier law) (1.13). Therefore, heat sources and the inertial term was introduced in the classical heat equation. As a result, we have

where *C _{v}* is the heat capacity per unit volume, k(T) is the thermal conductivity, r is the relaxation time of the heat flow, and

*Q(z.,t)*is the density of volumetric heat sources. Moreover,

where *Q _{u}(t)* is the heat release at the surface of a solid body (at

*z =*0), which is conveniently expressed in terms of the intensity of radiation

*W(z„t),*and

*ц*is a constant of the material. We assume that

where *L* is the coefficient of absorption of radiation. The field of dynamic stresses in a half-space is determined by the equations for thermoelastic waves:

where /3 is the coefficient of thermal expansion, *E* is the modulus of elasticity, *v* is the Poisson’s ratio, a is a constant, and *c* is the propagation velocity of the stress wave, which is given by

Initial and boundary conditions for temperature and stress are

The problem can be solved by using analytical methods [34].

A case of a stepwise loading of the body surface was considered. It is assumed that the material breaks down when the stress reaches the yield point. It was shown that the change in the temperature field in a metal occurs both as a result of the volume heat sources and due to the heat wave moving into the material with a constant velocity. During the propagation, the amplitude of the thermal wave decreases exponentially.

In the material, there are two stress waves: a wave of compressive and tensile stresses. The last do not move with a velocity of elastic waves, but with the velocity of the thermal wave. The compression waves move ahead of thermal wave.

According to [34], the threshold of plastic deformations for the pulses longer than 0.1 ns is approximately from 20 to 50 times smaller than that for melting of the mirror. For pulse durations less than 0.1 ns, this discrepancy increased. In the last case, the difference between the thresholds was more than 100 times due to dynamic effects (Figure 10.3).

Figure 10.3a shows the threshold of fracture of copper mirrors by melting of the surface (1-4) and plastic deformation 5 [34]. Experimental points are shown for wavelengths of the radiation of 1.06 x 10^{_6}m (1), 3 x 10~^{6}m (2), and 10.6 x 10~^{6}m (3). Curves 4 and 5 are the results of calculations: four takes into account the melting, and five appreciates the influence of the strength of the material.

It must be borne in mind for understanding Figure 10.3a that the theoretical thresholds of destruction by melting by the pulsed radiation are substantially higher than those obtained in the experiment (see points noted as 1, 2, and 3). Therefore, one should expect a similar decrease of the real destructive thermoelastic stresses in comparison with the theoretical ones.

FIGURE 10.3 Threshold of destruction (a). Impulse duration effect (b).

In Ref. [34], the influence of two dynamic effects on the allowable thermal load of mirrors was studied: the effect of the velocities of the propagation of the thermal and mechanical waves in the metal and the overheating of the electrons with respect to the crystal lattice. The first effect has a significant effect on the mechanism of mirror destruction and the second on the level of thermal stresses in the material (Figure 10.3). Figure 10.3b shows the effect of the duration of the laser pulse on the electron (curve 2) and lattice (curve 1) temperatures on the surface of a copper mirror at a radiation power density *W(t =* 0) = 3 x 10^{12} W/m^{2} (10.4). It is shown in Figure 10.3 that there is an equalization of the temperatures of the electronic and lattice subsystems with an increase in the time.