Combustion, Explosion, and Shock Waves, Vol. 32, No. 3, 1996

NUMERICAL

SIMULATION OF FRACTURE

OF PLATES BY CONFINED EXPLOSIVE CHARGES UDC 539.3

A. V . G e r a s i m o v

A numerical simulation of the interaction between an HE charge confined within a shell and an elastoplastic plate was performed. The cases of a fized charge and a charge with a convez bottom moving at a given velocity were considered. The main features of plate deformation and fracture and the difference of the obtained results from those for an unconfined charge resting on a plate were revealed. It was shown that for the moving charge an optimum (from the point of view of spalIation in the plate) size of the contact spot ezists at the moment of detonation-wave arrival at the interface between the plate and the charge.

Study of the interaction between an HE charge and a deformable target (plate) is of interest from the viewpoint of improvement in the technology of explosive treatment of materials and evaluation of possible consequences of accidental situations (fall of an HE charge, an accidental detonation, etc.). A great body of experimental data from studies of the stress-strain state and fracture of metal plates under the action of surface cylindrical HE charges can be found in [1]. The deformation of a plate and expansion of detonation products (DP) during explosion of an HE charge has been studied theoretically [2]. The process of fracture of a plate under loading of this type was simulated numerically by Belov et al. [3, 4]. All the papers mentioned were concerned with unconfined charges placed on the target surface. In the present work, we study the influence of the charge shell and impact velocity on the stress-strain state and fracture of thick elastoplastic plates under explosive load. The interaction between a deformable target and a confined HE charge moving with velocity V is a complex multistep process. It includes the following stages: the fall and impact of the system (shell + HE) as an inert deformable body on a deformable target, the initiation of detonation, detonation-wave (DW) propagation through the HE, the interaction between the DW and the shell, shell fracture and scattering, shock-wave (SW) formation in the plate, and plate fracture under the action of intense rarefaction waves. The process is complicated from a physical point of view and from the viewpoint of mathematical formulation of the problem and numerical simulation. The cases of a fixed charge with and without a shell are particular cases of the above problem. Figure I shows three cases of interaction between HE charges and a plate considered in this paper. The bottom of the moving charge has the shape of an elliptical segment. In this case, it is interesting to estimate how the size of the contact spot M D influences the degree of spall fracture of the plate at the moment of DW arrival. In all three cases, initiation occurs at point F or at the top of the charge F K . A damaged (porous) medium model is employed to describe the fracture of the shell and plate. The basic relations describing three-dimensional axisymmetric motion of a strong, porous, compressible elastoplastic medium are based on the conservation laws of mass, momentum, and energy and are written as [1,5]

l dp_ p dt

{Ou Ov v , ~ Oz + -~r + r } '

du OSzz OSrz Srz P dt - 3 ~ + ---~-r + - r-

Op OZ '

Institute of Apphed Mathematics and Mechanics, Tomsk 634050. Translated from Fizika Goreniya i Vzryva, Vol. 32, No. 3, pp. 126-133, May-Jtme, 1996. Original article submitted August 30, 1995. 0010-5082/96/3203-0347 $15.00 (~) 1996 Plenum Publishing Corporation

347

t/

1 1 1 Fig. 1. Variants of interaction between HE charge and plate: unconfined charge (a), confined charge (b), moving confined charge (c) and computation scheme (d).

dv O S r ~ OS~z ST~ - Soo P dt - O----r-+---~-z + r dE dt

Op Or'

(1)

p2 ~ + ~ &~-g; + s - N + soo~r + &. -g;z + N

The physical relations (Prandtl-Rice equations) are

DSz,

(Ou

DS,,

ldp~

(v + i< 3p dt / ' + sL + s L

lclp'~

(2)

D Sr~ ( Ou Ov ) D---~ +.~Sr. = # ~r + -~z •

d---'[-+ ISoo = 21~ The Mises yield condition is

(Ov

D----~+ ASrT = 2# ~rr + 3p dt ]'

D---i-+~&z =2~ ~ + 3pdt]'

l = a

+

The equation of state is P = P(P, E).

(3)

Here p is the current density; r and z are the radial and axial coordinates; u and v are the radial and axial velocity components; Srr, Szz, See, and Srz are the components of the stress tensor deviator; p is the pressure; E is the internal specific energy; # is the shear modulus; a is the yield strength; D / D t is Yanman's derivative. All quantities in Eqs. (1)-(3) refer to the porous medium. These relations are supplemented by a kinetic equation that describes expansion and compression of spherical pores [6]: da

dt -

(a0

--

1 ) 2 / a a ( a - 1)l/aAp sign(p),

77

a -- -V p -+

vs

vs

'

Ap = IPl _ ds In ~ a a--l'

(4)

where a0, ds, and 77 are material constants; vs is the specific volume of the continuous component of the porous medium; and vv is the specific volume of the pores. The pressure in the porous medium is described by the equation of state for the continuous

Ks (1 - (1/2) F~ )

component [6]:

p = ps(vs, E)/a,

(1 -

ps =

cg)2

+ psrE,

where subscript s refers to the matrix; P is the Gr/ineisen parameter; c and Ks are material constants, = 1 - pos/ps; and p0 is the initid density. The strength characteristics of the porous material were given by [3, 6]

= ~s/a,

# = #s(1 - ¢ ) ( 1 \

348

6Ks + 12#, (I)~ 9K~ + 8~s ,'

6 Z, EXfl

t

r, cm

-4

-2

0

Fig. 2. Test calculations.

where ~ = (a - 1 ) / a is the porosity; Ks and #s are matrix material constants. For the limiting value of porosity ¢ . = 0.3, the material at a given point is considered fractured. The set of equations describing motion of DP as that of a nonviscous nonthermoconductive gas is derived from relations (1)-(3) for an elastoplastic medium by setting the strength parameters equal to zero: 1 dp _ (Ou Ov v~ du Op dv Op dE p dp p at ,-8-; + + ;,' PTi = Oz' P--gi = - o-7' : dr' P = p(p' E). (S) In the simulation of detonation of an HE charge, the method of [7] is used. If the HE in the calculation cell is compressed to a critical value p., the equation of state for a solid is replaced by the equation of state for detonation products, which is written as p = (1 - ~ ) f l ( p ) +/~f2(p), f l ( p ) = A[(p/po) n - B], f2(p) = CP k, (6) where f~ = 0 for p < p.. For detonation of the main HE charge a detonator is required, which is simulated by a few cells with an elevated HE density, and, hence, with an elevated DP pressure. The Landau-Stanyukovich equation [the third in system (6)], which describes satisfactorily the DP behavior at high pressures, gives an erroneous velocity for gas expansion into vacuum [5]. Therefore, this equation, which is applicable for charges in a thick-walled shell, is replaced by a two-term equation of state that describes adequately the DP expansion into vacuum [5]:

(7)

f2(P) -- FPk + GPrn'q'l.

This equation is used to solve the problem of the interaction between a plate and an unconfined HE charge. In formulas (6) and (7), A, n, B, k, C, F , G, and m are constants. System (1)-(6) is supplemented by initial and boundary conditions. At the initial moment all points of the shell and of the HE have axial velocity u = v, whose sign is taken into account, and the plate is assumed to be in an unperturbed state. The boundary conditions are specified in the following way (see Figs. 1 and 2). The boundary A B C M N L G is assumed to be free of stresses: crn = zn = 0. The condition of slip along the rigid wall is specified on the symmetry axis A D E F G : rn = 0 and vn = 0. Displacements along the z-axis at point B of the plate base are absent: u = 0. The condition of perfect slip of one material over another along the tangent and the condition of nonpenetration through the normal to the contact surface are specified at the varying contact section D M between the plate and the shell: crnl = an2, Vnl = vn2, and "rnl = 7"n2 ----O. Since the contact spot D M varies in the process of interaction between the charge and the target, its current dimensions must be determined during solution of the problem for each time. For the contact interface E J K F between the HE and the shell the specified conditions are analogous to the previous ones: Crnl

--~

--p,

Vnl

----

Vn2,

rnl

=

Tn2

=

O,

349

where crn and ~'n are the normal and tangential components of the stress vector; vn is the normal velocity component at the point of contact; subscripts 1 and 2 refer to the media in contact. The finite-difference scheme of the "cross" type discussed in great detail in [2] was used in the solution. Division of the calculated domain into individual subdomalns eliminates empty cells, i.e., cells that are not used in calculations, and allows one to choose the required computation grid for each subdomain (a fine or coarse grid as necessary). This technique allows one to take all advantages of the Lagrange method of describing the motion of a continuous medium, particularly, an acceptable accuracy of keeping track of shock waves, of contact and free boundaries. Also, it allows one to vary the geometry of the calculated domains and to analyze inhomogeneous rotation bodies. A combined (parabolic plus linear) artificial viscosity was introduced into the numerical scheme to eliminate nonphysical oscillations behind the shock front. A tensor viscosity, which was realized in triangular cells adjacent to the calculated grid node [2], was introduced to eliminate "hourglass" instability. To calculate the contact interfaces, we developed a special procedure for analyzing slip between grids of different regions (metal-metal and metal-gas) under slip boundary conditions. The procedure is based on the assumption of nonpenetration through the normal and of perfect slip along the tangent to the contact surface. In this case, the main and auxiliary groups of cells form at the contact. The positions of these groups are changed either in each step or after a certain number of steps in time. In the absence of slip, the procedure automatically describes the condition of attachment. The problems of discontinuity decay at the contact interface [5] and the fracture of a thick plate [1] were treated as test problems. Calculation of discontinuity decay at the plate-DP interface for copper and TNT/HMX 36/64 composition gave the following values of pressure and mass velocity: p = 44.9 GPa and u = 933 m/sec (in the analytical solution of [5], p = 42.01 GPa and u = 920 m/sec). A great body of experimental data on the problem of an unconfined HE charge on the surface of a thick plate can be found in [1]. The calculation results were compared with the experimental data for systems with geometric parameters as follows: copper-plate thickness of 3.8 cm, diameter of 15.2 cm, charge height of 5.1 cm, and charge diameter of 2.54 cm. The pattern of plate fracture at the end of the interaction process, as is shown in Fig. 2, is in good agreement with the experimental data of [1]. The zones in the plate with the limiting value of porosity ¢ . = 0.3 are shown by hatching. The material in these zones is considered completely fractured. As one can see from Fig. 2, fracture occurred in the contact zone between the HE charge and the plate, in the vicinity of the symmetry axis, and a spa~ formed close to the rear surface of the plate. The thickness of the detached spall plate was 0.33 cm [1]. This is shown by the dashed curve in Fig. 2. The current configurations of the metal-DP systems, the porous zones (hatched), and the fractured zones of the target (shaded) are shown in Fig. 3 for three times. The plate thickness was 3 cm, and the radius was 12 cm; the height of the HE-charge was 4.4 cm, and the radius was 3 cm. The plate material was copper (p0 = 8.9 g/cm 3, # = 46 GPa, and a = 0.2 GPa). The parameters for Eq. (4) were chosen as in [6]. The HE charge (TNT/RDX 36/64 composition) had density p0 = 1.717 g/cm 3 and detonation velocity D = 8,000 m/sec. The equation of state (7) was used in the form proposed in [5]. In the case of a confined charge, the charge and plate dimensions were the same. The thickness of the copper-shell walls was as follows: the bottom was 0.5 cm thick, the top, 0.6 cm, and the side wall, 0.5 cm. The copper shell enables calculations to be performed without considering discontinuities in the shell upon its stretching to a relative radius of 3-4 [8]. The HE was initiated over the surface F K . More intense rarefaction of the plate material in the case of the unconfined charge (see Fig. 3a) results in a smaller crater, earlier formation of a spall, and a larger volume of the fractured zone as compared with the effect of the confined charge (Fig. 3b). In the latter case, the shell leads to a smoother drop in the detonationproduct pressure along the charge radius, and, hence, to the formation of larger porous zones in the plate material in the former case. Note, however, that the value of this additional porosity is small. The features of shell deformation manifest themselves in formation of two zones of intense plastic flow of material and in relatively less deformed elements of the top, and bottom, and also of the side wall (see Fig. 3b). The calculated picture of plate damage by the unconfined HE charge is in good agreement with the experimental data of [4]. In this case, a crater and a funnel-shaped, fractured, and detached spall plate are 350

~,cm ~

13

r'~

13

t=15

t,,=11pseI

10

psec

v t =20psec 16

f

1)

I

......

13 10

7 7

4 4 r, cm - 9

-6

-3

0

1

-12

-9

•,cm t=11~sec

-6

-15

-;

-12

-9

-6

2pcm

b

z, c m

1 )

t=15psec

1

10

t=20/Jsec f ~

7 ~

4

~.

1 r, cm - 9

-6

-3

0

-12

-9

-6

-3

4

0

-12

-9

-6

Fig. 3. Explosions of unconfined charge (a) and confined charge (b).

also observed. This is due to the relatively small thickness of the plate and the large radius of the HE charge as compared with those in Fig. 2. The investigation performed confirms the applicability of the proposed method for simulation of the interaction between plates and HE charges that are fixed in front of the target. It would be interesting to extend the proposed approach to of the interaction between a plate and a confined charge moving with velocity V = 200 m/sec. The bottom has the shape of an ellipsoid segment, and the contact-spot size increases with time during collision. The results of calculation of spallation for various contact-spot sizes at the moment of DW arrival at the bottom of the shell are given in Fig. 4. The plate radius is 11 cm, and the thickness is 2 cm. The plate material is steel with the following physicomechanical properties: p0 = 7.86 g/cm 3, # = 81.4 GPa, and cr = 0.64 GPa. The parameters for Eq. (4) were taken from [3]. The shell material is copper with the parameters given above. The HE charge has the following parameters: p0 = 1.6 g/cm 3 and D = 6,000 m/sec. The shell length is 14 cm, its external radius is 3.5 cm, the internal radius is 2.5 cm, the charge length is 11.6 cm, the thickness of the bottom is 0.4 cm, and the thickness of the top is 2 cm. The porosity and fracture zones of the plate material are given in Fig. 4a for the case where the DW reaches the bottom of the shell at the same moment as the shell hits the plate surface. The calculation results for t = 32/~sec after the beginning of initiation at the upper end F K are shown. For Fig. 4b the flattening of the bottom lasts for 10/~sec up to DW arrival; the data obtained 32 #sec after the beginning of the process are also shown. The longest interaction process between the shell and the HE charge, which are treated as inert bodies, (27 ~tsec) and the effect of the interaction on spallation are shown in Fig. 4c. These data correspond to a time of 40 ~tsec. The time of interaction between the plate and DW was about 14 #sec in all three cases. As one can see from Fig. 4, there is an optimum size of the contact spot between the shell and the plate at the moment of DW arrival, which gives the greatest spallation in the plate material. The formation of the spallation zone, which depends on the initial size of the contact spot in the shellplate system, is due to the complex nature of interaction between the rarefaction waves from the front and

351

f

zcr.

Gm

b

g , C!m

¢

f L f

((

I! I:

9

9 |

6

| 1", c m

15

=

6

.l

"" -9

-6

-3

0

-12

-9

-6

-3

0

-12

-9

-6

-3

0

Fig. 4. The interaction between plate and moving confined charge.

rear surfaces of the plate. In addition, the pressure variation on the front surface is affected by the subsequent radial flow of the material of the charge-shell system after complete flattening of the bottom. The obtained results show a significant influence of a charge shell on the deformation and fracture of the plate as compared with an unconfined charge. This difference is both qualitative and quantitative. A delay in spall formation, spall growth in the radial direction, and an increase in the porous zone are observed. Shell expansion is accompanied by intense plastic flow at the joints between the bottom and the walls, but less-deformed zones also exist. The most interesting result is that there is an optimum size of the contact spot which provides for maximum spaUation in the plate for a moving charge with a convex bottom. If the spot size is larger or smaller than the optimum size at the moment of detonation-wave arrival at the charge-target interface, spallation is decreased. The relationship between the porosity level and the contact-spot size allows one to decrease or increase the effectiveness of impact by varying the collision velocity, physicomechanical characteristics, or dimensions. REFERENCES

1°

2. 3. 4. 5. 6.

7. 8.

352

J. Reinhart and J. Pearson, Behavior of Materials Under Impulse Loading [Russian translation], Izd. Inostr. Lit., Moscow (1958). M. L. Wilkins, "Analysis of elastoplastic flows," in: Calculations Methods in Hydrodynamics [Russian translation], Mir, Moscow (1967), pp. 212-263. N. N. Below, A. I. Korneev, and A. P. Nikolaev, "Numerical analysis of plate failures under pulse loads," PrikI. Mekh. Tekh. Fiz., No. 3, 132-136 (1985). S. G. Sugak, G. I. Kanel', V. E. Fortov, et al., "Numerical simulation of the action of an explosion on an iron stab," Fiz. Goreniya Vzryva, 19, No. 2, 121-128 (1983). K. e. Stanyukovich (ed.), Physics of Explosion [in Russian], Nanka, Moscow (1975). J. N. Johnson, "Dynamic fracture and spallation in ductile solids," J. Appl. Phys., 52, No. 4, 28122825 (1981). V. Ya. Gol'din, N. N. Kalitkin, Yu. L. Levitan, and B. L. Rozhdestvenskii, "Analysis of twodimensional flows with detonation," Zh. Vychis. Mat. Mat. Fiz., 12, No. 6, 1606-1611 (1972). E. F. Gryaznov, V. A. Odintsov, and V. V. Selivanov, "Smooth circular spallations," Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 6, 148-153 (1976).