Combustion, Explosion, and Shock Waves, Vol. 41, No. 4, pp. 474–480, 2005
Effect of Strength and Plasticity of the Material and Particle Size of a Porous Medium on Shock-Wave Deformation V. A. Ogorodnikov,1 M. V. Zhernokletov,1 S. V. Mikhailov,1 S. V. Erunov,1 and V. V. Komissarov1
UDC 532.529+539.374
Translated from Fizika Goreniya i Vzryva, Vol. 41, No. 4, pp. 124–131, July–August, 2005. Original article submitted May 31, 2004.
The process of dynamic deformation of shock-loaded cylindrical “porous” samples of lead, tungsten, and the 95% W + 3.5% Ni + 1.5% Fe alloy consisting of particles 0.1–2.5 mm in size is considered. The shock-wave intensity was slightly lower than the values corresponding to complete compaction of the material. The influence of the particle size and material strength and plasticity on the processes considered is examined. Key words: fragmented medium, compaction, strength, plasticity, particle size, shock waves.
In solving a number of applied problems associated, e.g., with acceleration of cylindrical shells by the energy of an explosion, one has to predict the characteristics of this process. The problem becomes much more complicated if the shell material loses its compactness during acceleration. The reasons for the loss of compactness can be spalling of the shell, its fragmentation by shear stresses, or the loss of stability in the course of shell implosion [1–3]. To correctly calculate the shell motion in such a state, one has to know the behavior of the fragmented material under dynamic compression. A medium with discontinuities is called a damaged, fragmented, or porous medium. Examples of such media are sand, crushed stone, chips, sintered powders, various foam-based materials, and ceramics. Investigation of their behavior under dynamic compression, especially in the range of pressures corresponding to their complete compaction, is of independent interest. It should be noted that development of reliable models of deformation of such materials is hindered by the lack of experimental data on their behavior under dynamic compression. This particularly refers to materials with a wide range of characteristic particle sizes (from tens of micrometers to several millimeters). 1
Institute of Experimental Physics, Russian Federal Nuclear Center, Sarov 607188;
[email protected].
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In the range of comparatively low pressures, particles of materials under consideration under dynamic or shock-wave loading are first loaded by the shock wave (SW) and then are unloaded to surrounding pores. Because of the finite volume of the “porous” space, the particles experience several circulations of compression and rarefaction waves. This leads to formation of a transitional zone between the SW front and the region of final states; the width of this zone is determined by the time of decay of compression and rarefaction waves circulating in the particles or the time of pore implosion and by the thermal relaxation of particles [4]. When the oscillations decay, the SW propagates in a steady mode; dissipation of kinetic energy of the particle material in the shock-transition zone results in an increase in the thermal component of internal energy and in enhanced shock-induced heating of the porous material. In addition to these features, sintered powders, foam-based materials, and ceramics whose particles are connected by a skeleton have additional peculiarities caused by skeleton compression or breakdown. For this reason, the behavior of a porous or fragmented material under shock-wave compression can be characterized by propagation of complicated multi-wave structures consisting of one or several elastic precursors and a wave of irreversible compression [5–10]. If the particles of a porous
c 2005 Springer Science + Business Media, Inc. 0010-5082/05/4104-0474
Strength and Plasticity of the Material and Particle Size of a Porous Medium TABLE 1 Material
W–Ni–Fe
Tungsten Lead
∆, mm
ρ00 , g/cm3
k
0.25–0.5
9.8 ± 0.4
1.84 ± 0.08
0.5–0.7
8.9 ± 0.4
2.02 ± 0.08
1.6–2.5
10.4 ± 0.1
1.73 ± 0.02
0.1
6.7 ± 0.1
2.68 ± 0.1
2.0–2.5
7.3 ± 0.1
1.55 ± 0.05
medium are connected by a skeleton, a three-wave configuration propagates over such a medium [5–7, 9]. The first wave is elastic, the second wave is generated by skeleton breakdown, and the third wave is caused by splitting of fragments and a decrease in the degree of their anisotropy, i.e., is a wave of irreversible compression. If the particles of a porous medium are not connected by a skeleton, a two-wave configuration propagates in such a medium [8, 10]. We will call it a fragmented medium. It follows from an analysis of data described in the literature that for a wide range of porous materials, beginning from foam plastics and ending by metal ceramics, the pressure that ensures a multi-wave profile of the compression wave is within 5–7 GPa. It is this range of pressures that is of greatest interest for research. Note, the porous materials in [5–7, 10] had the characteristic particle size smaller than 0.1 mm. A question arises whether the features observed are valid for porous material with a characteristic particle size up to several millimeters. The present paper describes the research of dynamic compression of tungsten powder with a particle size ∆ ≈ 0.1 mm, composite material 95% W + 3.5% Ni + 1.5% Fe (below W–Ni–Fe) with ∆ = 0.25–2.5 mm, and lead with ∆ = 2–2.5 mm. The particles of the W–Ni–Fe alloy with an initial density ρ0 = 18.0 g/cm3 had the form of scales and were obtained by grinding chips with subsequent separation into fractions with particle sizes of 0.25–0.5, 0.5–0.7, and 1.6–2.5 mm. Lead particles had a spherical form. These material with substantially different strength and plasticity were used to clarify the influence of these characteristics on the behavior of the curves of compression of porous materials. In addition, the effect of the particle size on compression in samples with close values of porosity was examined. The particles were densely packed into a cartridge, and the real density of the porous material ρ00 and the corresponding porosity k = ρ0 /ρ00 were determined by weighting. Some characteristics of the examined materials are listed in Table 1. The experiments were performed in a BUT-96 ballistic shock tube [11], which allows studying dynamic compressibility of materials
475
in steady shock waves with impactor velocities up to W0 ≈ 500 m/sec. High velocities of impactors were obtained in explosive experiments. The layout of the experiments is shown in Fig. 1. In experiments performed with the use of the scheme shown in Fig. 1a, the particles of the examined material were packed into a cartridge made of steel 10 mm thick with inner and outer diameters equal to 40 and 95 mm, respectively, and were shielded by copper screens 2 mm thick on two sides. As the experimental assembly was located at the seat of the BUT-76 evacuated shaft with a residual air pressure of ≈2 kPa, the pressure of air in the space between the particles was the same. The targets were loaded by impactors made of steel 22 mm thick and 75 mm in diameter, which made it possible to generate a steady SW with a duration of ≈10 µsec in the sample. The velocity of the impactors was measured by electric-contact gauges within 0.5%. The thickness of the copper screens was chosen to eliminate microcumulation associated with implosion of voids in the material and to ensure deviations of the end faces of the target from a plane within ≈0.01 mm, which allows reduction of skew in impactor–target collisions to 10−3 rad. In each experiment, a capacitance gauge [12] 10 mm in diameter measured the velocity profile of the free surface W (t) of the copper screen averaged over the diameter. The electric-contact gauge mounted on the impactor–target collision surface was used to obtain time tags corresponding to the beginning of SW propagation in the target in oscillograms. After that, based on the known thicknesses of the sample and screens and the measured time interval between the impactor–target collision (by the electric-contact gauge) and SW passage to the free surface (by the capacitance gauge), the wave velocity in the sample was determined. On the basis of the results of calculations and experiments where the porous material was replaced by a reference sample made of polyethylene or AMg-6 aluminum-based alloy, it was found that such a formulation allows correct determination of the compression-wave profile p(t) in the porous material without any distortions. The velocity profiles of the free surface of copper screens registered in studying shock compression of samples made of the fragmented W–Ni–Fe alloy with a particle size of 0.5–0.7 mm are plotted in Fig. 2a. Figure 3 shows the photographs of the longitudinal section of lead samples before and after shock-wave loading with different impactor velocities W0 corresponding to different degrees of material compaction and the photograph of the W–Ni–Fe sample. An analysis of the W (t) profiles reveals a complicated character of shock-wave-induced deformation
476
Ogorodnikov, Zhernokletov, Mikhailov, et al.
Fig. 1. Schematic of the experiment: 1) impactor; 2, 5) copper screen; 3) cartridge; 4) material under study; 6) electric-contact gauge; 7) capacitance gauge; 8) layer of a high explosive; 9) air gap; 10) manganin pressure gauges.
of the fragmented material under dynamic compression. For W0 ≈ 40–100 m/sec, “extended” profiles of the compression wave are observed. As the impactor velocity or the pressure amplitude increases, the duration of the leading front decreases. The W (t) profiles of the fragmented W–Ni–Fe sample with a particle size of 0.25–0.7 mm are close to each other, and their configuration approaches the two-wave configuration at W0 < 100 m/sec. For W–Ni–Fe samples with a particle size of 1.6–2.5 mm, the above-mentioned feature is less expressed. This agrees with the conclusions drawn by Kiselev [8]: the width of the compression-wave front in a porous material is determined by the time needed to fill the pores; the greater the characteristic size of pores or grains, the greater this time. Note, a similar structure consisting of two compression waves was observed, e.g., in studying dynamic compression of porous aluminum [8] and pressed copper powder samples [10] with a porosity k ≈ 1.2–1.5 and close velocities of the impactor. With increasing loading intensity, the second wave catches up with the first wave, the W (t) profiles are transformed, and the two-wave configuration is no longer observed for W0 ≈ 300 m/sec. The results of processing experimental data are listed in Table 2, where W0 is the impactor velocity, ρ00 is the density of the samples, D1,por and D2,por are the measured velocities of the first and second compression waves in the fragmented material, u1,por and u2,por are their mass velocities, p1,por and p2,por are their pressures, kσ , ρ, and v are the compression factor, density, and volume of the fragmented material behind the front of the second compression wave, kσ = D2,por /(D2,por − u2,por ), ρ = kσ ρ00 , and v = 1/ρ. The state in the fragmented material in the second compression wave was determined with the use of the reflection method [13] by the formulas
u2,por =
W0 2(ρc)Fe (ρc)Cu , [(ρc)Fe + (ρc)Cu ][(ρ00 D2 )por + (ρc)Cu ] p2,por = (ρ00 D2 )por u2,por
derived in the acoustic approximation from the law of conservation of momentum in considering the Riemann problem at the contact boundary between the copper screen and the porous material (c is the velocity of sound). The state in the first compression wave was estimated by the formulas u1,por = uCu
(ρc)Cu + (ρD)por , (ρ00 D1 )por + (ρD)por
p1,por = (ρ00 D1 )por u1,por derived in the acoustic approximation from the law of conservation of momentum in considering the Riemann problem at the contact boundary between the porous material and the copper screen. The value of (ρD)por = (pCu − p2,por )/(u2,por − uCu ) was determined from the combination of experimental data obtained by the method of reflection and by registering the W (t) profiles. Note, the value of (ρD)por depends on the state of the fragmented material compressed behind the SW front and changes from (ρ00 D)por for the examined loading intensity to ρ0 D for the continuous material. In studying dynamic compression by the scheme shown in Fig. 1b, we registered the pressure profiles by manganin pressure gauges [14] manufactured in the form of bifilar spirals 4.5 mm in diameter and 0.03 mm thick. The initial resistance of the gauge was 1.5 Ω. Pulsed feeding of the gauges followed the potentiometer scheme; it was switched on 10 µsec before the SW front arrival. Each gauge had four identical outputs: one pair served to feed the voltage gauge, and the other pair served to register the signal at the inputs
Strength and Plasticity of the Material and Particle Size of a Porous Medium
477
TABLE 2 Material
∆, mm
0.25–0.5
W–Ni–Fe
0.5–0.7
1.6–2.5
Tungsten
Lead
0.1
2.0–2.5
W0 , m/sec
ρ00 , g/cm3
D1,por , D2,por , u1,por , u2,por , p1,por , p2,por , m/sec m/sec m/sec m/sec MPa MPa
kσ
ρ, g/cm3
v, cm3 /g
35
9.8
364
304
11
34
35
101
1.13
11.00
0.091
56
9.7
356
323
9
54
33
169
1.20
11.60
0.086
60
9.7
376
357
8
57
31
197
1.19
11.50
0.087
82
9.5
—
369
—
78
—
273
1.27
12.05
0.083
88
9.6
—
92
10.2
—
367
—
84
—
296
1.30
12.50
0.080
392
—
86
—
344
1.28
13.10
0.076
199
10.1
—
466
—
183
—
861
1.64
16.50
0.060
200
9.4
—
450
—
186
—
787
1.68
16.00
0.063
291
9.4
—
618
—
261
—
1 516
1.73
16.30
0.061
306
10.0
—
655
—
270
—
1 768
1.70
17.00
0.059
333
9.6
—
676
—
294
—
1 908
1.77
17.00
0.059
43
8.8
292
242
4
42
9
89
1.21
10.60
0.094
66
8.9
327
296
13
64
40
169
1.28
11.40
0.088
70
9.3
386
343
9
67
31
214
1.24
11.50
0.087
85
8.8
347
307
2
82
7
221
1.36
12.00
0.083
208
8.6
—
456
—
192
—
753
1.73
14.80
0.067
296
8.8
—
623
—
267
—
1 464
1.75
15.40
0.065
88
10.5
703
542
3
79
21
450
1.17
12.30
0.081
179
10.4
708
564
4
160
37
938
1.39
14.50
0.069
190
10.3
692
567
3
170
23
993
1.43
14.70
0.068
189
10.3
696
569
4
172
19
1 008
1.43
14.7
0.068
266
10.4
—
775
—
226
—
1 821
1.41
14.70
0.068
74
6.5
306
256
9
73
18
121
1.39
9.09
0.109
155
6.5
—
278
—
154
—
278
2.24
14.57
0.069
180
6.7
—
302
—
178
—
360
2.44
16.32
0.061
275
6.9
—
426
—
267
—
785
2.67
17.89
0.056
80
7.3
434
365
9
78
29
208
1.27
9.29
0.108
155
7.4
—
476
—
147
—
517
1.44
10.72
0.093
176
7.4
—
509
—
169
—
637
1.49
11.08
0.090
211
7.2
—
551
—
202
—
801
1.57
11.37
0.088
478
Ogorodnikov, Zhernokletov, Mikhailov, et al. TABLE 3 ∆, mm
0.25–0.5
0.5–0.7
ρ00 , g/cm3
9.7
9.7
D2,por , km/sec
u2,por , m/sec
p2,por , GPa
kσ
ρ, g/cm3
v, cm3 /g
1.160
459
5.17
1.66
16.05
0.062
0.200
477
5.56
1.66
16.10
0.062
1.270
507
6.27
1.66
16.14
0.062
1.370
550
7.30
1.67
16.22
0.062
1.670
678
10.99
1.68
16.33
0.061
0.990
387
3.72
1.64
15.93
0.063
1.094
430
4.56
1.64
15.97
0.063
1.260
498
6.09
1.65
16.05
0.062
1.269
502
6.18
1.65
16.05
0.062
1.345
534
6.97
1.66
16.09
0.062
1.476
590
8.44
1.67
16.15
0.062
of the HP54645D and TDS3052 oscillographs. Mica 0.1 mm thick and epoxy glue were used for insulating the gauges. The total thickness of the gauges was 0.25 mm. The shock-loading pressure was determined by the measured electric resistance R of the gauges in a compressed state. To pass from the resistance ratio R/R0 to the pressure p, we used the dependence p = p(R/R0 ) from [15]. As is seen from Fig. 1b, such a formulation of experiments allows the first gauge to register the pressure profiles p(t) in the copper screen in the incident SW (pscr ) and in the rarefaction wave (pback ) emanating from the material examined and allows the second gauge to measure the pressure profile pback at the screen–sample interface. The pressure amplitude was changed by varying the power and thickness of HE charges or the material of the layered screens. The space between the particles of the examined material was filled by air under standard conditions (p = 100 kPa). The profiles p(t) recorded in different experiments with different intensities of the incident SW are plotted in Fig. 2b, c. As was already noted, the first gauge measured the pressure in the screen and in the rarefaction wave reflected from the W–Ni–Fe sample with fragmented particles (pscr and pback in Fig. 2b). The second gauge was mounted at the screen–sample interface and registered the pressure in insulating mica gaskets and the pressure established in the W–Ni–Fe sample (pins and pback in Fig. 2c). The error of pressure measurement was ±5%. The position of the experimental point in the p–u coordinates was determined from the dependence 5 ppor [16, 17] u2por = u20 + (k − 1), where u0 and upor are 6 ρ0
the mass velocity of the continuous and porous material, respectively, ppor = pback is the pressure in the porous material, and ρ0 is the density of the continuous material. This dependence was obtained from the condition that, the porous material being compressed by the shock wave, the air in pores is compressed in accordance with the Hugoniot relation, and the ultimate compression is reached: h = γ + 1/(γ − 1) (γ = 1.4 for air). In calculations for the continuous W–Ni–Fe material with a density ρ0 = 18 g/cm3 , we used the relation D = 3.925 + 1.317u [18]. The results of processing the experimental data by examples of W–Ni–Fe samples with particle sizes ∆ = 0.25–0.5 and 0.5–0.7 mm are listed in Table 3. The results of experiments performed by the schemes shown in Fig. 1 are plotted in Fig. 4. An analysis of results (see Tables 2 and 3 and Fig. 4) obtained in studying the dynamic behavior of fragmented materials with different particle sizes (∆ = 0.25–2.5 mm) indicate that the measured parameters have a nonmonotonic dependence on ∆. Thus, the velocity of the compression wave in the fragmented W–Ni–Fe depends weakly on the particle size in the range ∆ = 0.25–0.7 mm, but the compression-wave velocity noticeably increases as the particle size increases to 1.6–2.5 mm. This seems to be related to the dependence of the density of initial packing of the fragmented material on the size and degree of asymmetry of its fragments (see the difference in initial densities depending on the particle size in Table 2). Note, the velocities of the first compression waves in the examined materials reach several hundred meters per second. A possible reason is the absence of rigid connection between the particles of the fragmented material.
Strength and Plasticity of the Material and Particle Size of a Porous Medium
479
Fig. 3. Photographs of the longitudinal section of the samples: lead sample before shock-wave loading (a), sample after loading with different impactor velocities (b and c), and W–Ni–Fe sample (d).
Fig. 2. Velocity profiles of the free surface (a) and pressures in the copper screen (b) and at the screen–sample interface (c) recorded by the capacitance and manganin gauges, respectively.
In addition, as the particle size increases, other conditions being identical, compression decreases, and the compression curves in the p–v coordinates move away from the curve for the continuous material. It can also be noted that the segments of the p(v) dependences obtained with the use of different loading schemes and measurement techniques (see Fig. 1) do not contradict each other. The results of experiments with the lead sample show that a decrease in strength and an increase in plasticity of the particle material lead to a decrease in pressure of complete compaction (see Fig. 4). Thus, with allowance for the previously obtained results on the shock-wave behavior of modern dampers of thermal and mechanical loads, such as zirconium dioxide ceramics (k 1.2) and chamotte (k 2) [9, 19], it could be proved that three-wave and two-wave struc-
480
Fig. 4. Compression curves of the materials examined: the open and filled points refer to loading by schemes in Fig. 1a and 1b, respectively.
tures of the compression wave are observed in the range of pressures below the pressure of complete compaction in porous materials with a characteristic particle size ∆ ≈ 0.2–2.0 mm connected (ceramics) or not connected (fragmented medium) by a skeleton. A similar behavior was observed in studying pressed [5] and sintered [10] powders with a characteristic particle size ∆ 0.1 mm.
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