# Mathematical Formulation of the Problem

The mathematical model of the dynamic behavior of a solid in two-dimensional axi- symmetrical formulation is constructed on the basis of a system of equations written in general Lagrange formulation [55-60]. The use of three cylindrical coordinate systems is assumed: Euler (or spatial) with the actual configuration of the examined solids, initial Lagrangian (material) which coincides at the initial moment with the coordinates of Euler, and finally, computed Lagrangian system (also material) in which the regular initial calculation configuration of the examined solid, obtained from its real initial configuration by a means of some homeomorphous transformation is specified. The coordinates of the points in these systems are denoted (z, r, and *ф)* and *(z, r,* and *(p*), respectively. The equations of motion used in this case have the form [32,59,60]:

where p„ is the initial density of the material, v, are the components of the velocity of the material point; *D„* are components of the first Piola-Kirchhoff stress tensor, and

The accepted model of the elastoplastic solid (Mises’s model) includes geometrical relations:

where {*e*are the components of the strain rate tensor. Equations for the deviator component of the Cauchy stress tensor

and also the Mises condition of ideal plasticity are used (10.66). In the above equation, *s,j* are the components of the stress deviator, and <5,y is the Kronecker symbol.

The components of the Cauchy stress tensor <7,, are computed according to (1.5). The hydrodynamic pressure *p* and temperature *T* are computed using the wide-range equations of state (1.10) [44,56]:

Generally speaking, described equations resemble the equations of Section 10.2. Some results of calculations are presented in Figure 10.26. The evolution of cold, thermal, and total pressures in one of the evaporated cells are showm. To provide more comprehensive information, the evolution is shown in relation to a monotonically

FIGURE 10.26 Dependence of the cold (1), thermal (2), and total (3) pressure on the relative volume.

decreasing quantity, i.e., relative volume, and not in relation to time. At the initial moment, the thermal compression pressure forms as a result of irradiation, where the material is undeformed and, consequently, cold pressure is equal to zero. The total positive pressure causes an expansion of the material with an increasing of its rate. The expansion is accompanied by an increase of the cold tensile pressure which partially compensates the gradually decreasing thermal pressure. However, the cold pressure has the maximum value P_{f}._{max} at some relative volume. When the latter exceeds certain critical level V_{max}, the cold pressure starts to decrease and the material continues to expand without obstacles; that is, it is evaporated.

It should be mentioned that if the initial thermal pressure was lower than P_{c>max}, then the tendency for expansion of the material would be retained only up to the moment of reaching the relative volume at which the cold and thermal pressures are equal to each other. In this case, the matter would be in the state of thermal equilibrium characterized by the absence of stresses in the material deformed in relation to its initial state.

To examine the material fracture, we used the criterion of instantaneous local fracture (Section 10.4). It was assumed that the distribution of energy as a result of instantaneous absorption of the emission pulse by the plate material can be represented in the following form:

where E is the specific internal energy, which is absorbed symmetrically relatively in the center of the loaded spot; *E _{s}* is the specific energy on the surface of the plate in the center of the loading spot;

*f(z)*and

*g(r)*are functions which depend only on

*z*and

*r,*respectively.

The formulated problem is solved numerically using the explicit Lagrangian finite-difference schema [60].

There were two aims for calculations: the first w'as to verify the numerical algorithms, and the second was to examine the dynamics of failure of solids by pulsed radiation when the absorption of energy increases. It w'as attempted to examine the possibility of changing the fracture mechanisms of DSO during its approach to the Earth where the power of effect on the DSO of every pulse in a series increases. In particular, it was attempted to examine the fracture of DSO by pulses when the power of every pulse in a series is increased.

# Calculation Results and Comparison with Experiments

Verification of the model. The absorption of radiation in the depth of the material is described by the Bouguer-Lambert law (10.44):

where *L* is the coefficient of attenuation of radiation in the material which depends on the mechanical properties of the absorbed material and on the wavelength of radiation.

However, the calculations based on Eq. (10.74) require a highly detailed numerical mesh. It is necessary to use from seven to ten cells to describe the exponent. Therefore, a simplified approach w'as used in which the absorption of radiation along the depth reduces linearly.

The variation of the density of energy release along the radius was described using the Gaussian distribution [49]:

where *R,* is the effective radius of the irradiation spot. Quantity *R, =* 1.0 mm was taken from Ref. [61]. The pulse energy was founded using the following relationship:

where the initial distribution of specific energy *E(z„ r)* is taken from (10.73). Since the law' of attenuation of radiation in the material w'as not specified in the calculations carried out in Ref. [61], here the attenuation coefficient *L* value typical for all metals is 10^{5}nr'.

To fulfill the first task, numerical modeling w'as carried out of the formation and development of experimentally observed [61] thin-wall cupolas on the rear side of an aluminum barrier under pulsed laser radiation. The typical forms of the cross section of the specimen are shown in Figures 10.27 and 10.28. The form of the cupola

FIGURE 10.27 (a-c) A scheme of laser shock-induced fragmentation of aluminum target.

FIGURE 10.28 Form of the cupola on the rear (free) surface of the plate loaded by laser radiation.

shown in Figure 10.28 corresponds to the variant of irradiation of an aluminum round plate constrained at the edges. The thickness of the plate was *h* = 0.65 mm, radius *R =* 1.0 mm.

Formation of the cupola-like forms on the rear surface of the irradiated plates is a phenomenon which is utilized here, on the one hand, as a test but, on the other hand, presents a special subject for theoretical examination.

For the calculations, the material constants for aluminum were taken mainly from [44,56,48,61,62]. We will call this version of the calculations as the version C.

The number of calculation cells in the plate along the axis of symmetry was selected equal to 20 and in the radial directions 16.

Analysis of the formation of cupola-shaped convex areas on the rear surface of the irradiated plate indicates that the process consists of two stages. In the first short- time stage (the wave stage), a disk-shaped crack forms in the vicinity of the rear surface of the plate. The crack separates the sheet of material from the main volume. The thickness of this sheet is considerably smaller than that of the plate. The second long-time stage (conventionally referred to as the deformation) consists of the inertia movement of the disk-shaped sheet with its deceleration as a result of energy losses for plastic deformation. It should be mentioned that during the development of dynamic fracture of sheet, there is a time period characterized by simultaneous rapid occurrence of both wave movements and plastic deformations.

Let us examine these stages of fracture. Comparison of the spallation depth (0.065 mm in calculations and 0.064mm in experiments) and of the radius of the disk-shaped crack (0.5 mm in calculations and 0.4 mm in experiments) indicate that they are in satisfactory agreement.

Figure 10.29a shows the configuration of the difference mesh 0.2 x 10~^{6} second after the start of irradiation. This moment corresponds to the initial stage of development of the spherical cupola.

Irradiation takes place in the direction from top to bottom. The upper part of the figure shows a depression formed in the area of the cells in which the material evaporated. As in the experiments, the thickness of the spallation cupola is heterogeneous along the radius.

To explain the mechanism of formation of rear crack (spallation), preceding the development of a spherical cupola, it is necessary to take into account the evolution of the stressed state in the target (see Figures 10.29b [32] and 10.31). During irradiation of the top surface of the target, a disk-shaped zone of heating and compression forms in the vicinity of the surface. A stress wave, consisting of two halfwaves, propagates from this zone into the bulk of the material. The first halfwave (compression) repeats the form of the initial temperature distribution in the solid. The second halfwave is the tensile part of the wave. These halfwaves determine the steep of the stresses from the maximum compression to the maximum tension.

Thus, according to Figure 10.29, the impact of radiation can lead to the formation of a strong compression wave in the surface layer of the material. The interaction of this wave with the free surface of the target causes the development of a stretching wave propagating into the material behind the compression wave. The tensile stress may exceed the dynamic tensile strength of the material, which will cause the spalling of the front layer of the target (Figure 10.29b). Over time, that layer can break up into fragments (drops, cavitation), and the compression wave front can reach the back side of the target and cause the spallation of the back surface of the target (see, also, Figures 10.27, 10.28 ,and 10.31(right)).

FIGURE 10.29 Configuration of the calculation mesh in the zone of formation of a cupola *(t* = 0.203 x 10'^{6} second) (a). A scheme of the processes occurring during irradiation of the target by a laser radiation (b and c).

However, if pulsed radiation is accompanied by intensive evaporation, the tensile stresses are restricted and the pulse loses its central symmetry relatively compression and tension. This was shown in [37] using unidimensional calculations (see Sections 10.2 and 10.3). The same result is in two-dimensional calculations (Figure 10.30). The restriction and suppression of the tensile stresses takes place as a result of the recoil of the vapors accelerating from the top surface.

Generally speaking, during propagation of the compression wave, this wave can attenuate quite rapidly. However, reaching the rear surface, the wave can be reflected as the tensile wave with the amplitude sufficient for formation of rear spallation (Figure 10.30). For comparison, the dotted and dashed line 5 in Figure 10.30 shows the profile stresses obtained when fracture of the material was not taken into account.

The propagating flat compression wave has the form of a circle constantly narrowing as a result of lateral unloading. Since the diameter of the radiation spot was greater than the thickness of the target, the stress state in the center of the narrowing circle remains to be planar. In accordance with intuitive expectation, the radius of the disk-shaped crack (region in which the stress remains planar) is approximately equal to the difference of the spot radius and the plate thickness (see, also, Figures 10.23a and 10.24a).