The attenuation of a plane shock wave in organopIastic was experimentally and numerically investigated during its interaction with an overtaking rarefaction ...
Combustion, Explosion, and Shock Waves, Vol. 31, No. 2, 1995
A T T E N U A T I O N O F A S H O C K W A V E IN O R G A N O P L A S T I C
S. A. Bordzilovskii, S. M. Karakhanov, L. A. Merzlfievskii, and A. D. Resnyanskii
UDC 539.3
The attenuation of a plane shock wave in organopIastic was experimentally and numerically investigated during its interaction with an overtaking rarefaction wave. Measurements were carried out with manganin gauges. An earlier formulation model of the dynamic deformation of composites was used in calculations. A comparison of calculated and experimental data has shown their good agreement.
Interest in investigations of shock-wave processes in condensed media is due not only to the quest for studying their features. Much more important is the fact that shock-wave loading allows extremely high thermodynamic parameters of the state of a material to be obtained. Finding the relationship between loading parameters and characteristics of the state provides the experimental information necessary for deriving governing equations (equations of state), closing the models, and verifying the calculations. In such investigations, the simplest way is to f'md experimentally the relation between the shock wave velocity and the mass velocity behind its front. The object of the measurements is to plot the shock adiabats (the Hugoniots) of a material. This helps us to estimate the parameters of the state behind the front of steady shock waves. As the experimental methods (laser interferometry, manganin gauges, etc.) are improved, it becomes possible to determine the space-time distributions (profiles) of characteristic parameters, namely the stress tensor components, pressure, and mass velocity. By analyzing the profiles experimentally obtained, we get more information on irreversible dissipative losses and on the relation between strength characteristics and the hydrostatic pressure, deformation, and strain rate. The majority of a great many studies in this area are devoted to the behavior and properties of homogeneous isotropic materials. Much attention is given to porous media, since these investigations allow us to substantially extend the range of thermodynamic parameters obtained in experiments. Composites, especially polymeric ones, are much less studied though they are coming into ever increasing use in modern constructions and technologies. This paper describes experiments in the evolution of short shock pulses propagating in organoplastic as they interact with an overtaking rarefaction wave. The process is also numerically analyzed on the basis of an earlier formulated and tested model which uses Maxwell representations of the viscoelastic behavior of the matrix and reinforcing material. A method of phenomenological averaging was used in constructing the model, which allowed us to exclude the hypothesis for structure periodicity of the material commonly used in model constructing.
EXPERIMENTS AND THEIR RESULTS The scheme of experiments in attenuation of a short shock pulse in organoplastic is shown in Fig. t. An aluminum projectile 1 (3 mm thick and 50 mm in diameter) was accelerated to velocity vo = 1.56 km/sec with the help of a specially constructed and spent charge. On impact of such a velocity, the shock-wave amplitude ~r = 14 GPa in the aluminum target and 8-9 GPa in the organoplastic. The target was composed of a screen 2 of aluminum 1.8 mm thick and 100 mm in diameter, and a set of four plates 4 made of the composite under study, each of which was 3 mm thick and 60 mm in diameter. The stresstime profiles in the specimen were registered with manganin gauges 3 placed between the plates. Lavrent'ev Institute of Hydrodynamics, Russian Academy of Sciences, Novosibirsk 630090. Translated from Fizika Goreniya i Vzryva, Vol 31, No. 2, pp. 125-130, March-April, 1995. Original article submitted August 29, 1994.
236
0010-5082195/3102-0236512.50
9
Plenum Publishing Corporation
I
iv ~
i
jz
Fig. 1. Experiment setup: 1) projectile; 2) aluminum plates; 3) plates of studied composite; 4) manganin gauges. 6, GPa
0
O,,e
1,2
7,8
2,4 t, gsec
Fig. 2. Loading pulse. To minimize the acoustic impedance mismatch of the test material and the inserts insulating the gauges, mylar sheets with a thickness of 100/~m and a density of 1.39 g/cm 3 were used as an insulator in the experiments with composites, and PTFE sheets of the same thickness and a of density 2.18 g/cm 3 were used in the experiments with aluminum. A large diameter of the screen was required to cut off the detonation products, to protect the electric leads of the gauges, and to level off the loading pulse front. The gauges were made of manganin wire, which was flattened to a strip - 15 #m thick and 0.5 mm wide and soldered to copper leads 20/zm thick and 1.5 mm wide. The maximum thickness of the gauge in a contact zone did not exceed 40 #m, and the electric resistance of the active element did not exceed 0.25 f~. The stress-time profiles were recorded with two digital oscillographs $9-27 with a digitizing rate of 50 MHz. The impedance mismatch of the screen and the material under study results in a discontinuity decay when a shock wave arrives at the contact surface, and a weak rarefaction wave originates and propagates in the screen towards the loading pulse. The interaction between these meeting waves complicates the profile of the loading pulse, which makes it difficult to interpret the data registered by the first gauge and to compare the characteristic parameters of the loading pulse with similar measurements made by other researchers and with calculated data. To clear up the trouble, further experiments were run with the plates 4 replaced by aluminum ones. The obtained characteristics of the loading pulse were used as reference characteristics. The experimentally recorded loading-pulse shape is shown in Fig. 2. A unidirectional aramid-fibrous composite with an epoxy matrix was investigated. The reinforcing aramid fibers were 15 #m thick. The specimens were placed so that the reinforcing fibers were loaded transversely. The dependences a(t) experimentally obtained at different distances from the loaded surface are shown in Fig. 3. The initial density of the organoplastic specimens was - 1.14 g/cm3, which points to a low homogeneity degree and the presence of air bubbles inside. This affects the results of the experiments. In particular, it explains the noise in the first experimental stress-time profile (see Fig. 3), which attain 50% of the total amplitude of the signal. Furthermore, the maximal amplitudes in this set of experiments could be distorted by stress jumps in passage through the measuring stations. The dependence of the peak amplitude Crmax on the distance covered by a shock wave are shown in Fig. 4 (points). Data obtained in further calculations are shown here too (curve).
N U M E R I C A L SIMULATION OF SHOCK-WAVE PROCESSES
To model numerically the shock wave processes in organoplastic we use an earlier elaborated model of a composite [1, 2].
237
~, !GPa 7'o-
e~max,~GPa
7,5-
i
i•'
5,0 2,5
|,,
s t, gsec
2 Fig. 3
0
~
1oJ x, mm
Fig.4
Fig. 3. Shock-wave profiles at various distances from loaded surface. Fig. 4. Variation of maximum pulse amplitude with propagation. We assume that the volumetric concentrations of the composite components and all the necessary mechanical characteristics of each composite component are known. The following relations (~ is the component's number) are valid for the composite components
pc,Ou__~ ~ Oa~ _ O, O~ Oxj
0~ 1 (Ou~ Ou~ a~ - (1/3)(a~' 1 + a~'2 + a~'a)61j 0"-'-~- 2 \ Ox~ + -~-x/] = 2#~.v '~
(1)
Here pC, is the density; ui~ is the velocity; (rUe is the stress tensor; e l f is the tensor of elastic strains; ~a is the relaxation time of shear stresses; and t~C'is the shear modulus. Stresses and strains are connected by Hooke's law
(2) and can be calculated from the elastic potential E ~ using the Murnaghan formulas
,,OE '~
aG=P
(3)
Oe~i
To calculate characteristics of the state of a composite that are averaged over an elementary volume we use the mixture rule. In the case of fibrous composites, for average quantities from (1) and a number of hypothesis we obtain [1, 2]
Oui POt
O~j Ot
Oaij Oxj = O,
l ( Ou~
Ouj,I
OA
2 k~xj + Oxi] = -qa/./,
.0t" = -~b,
(4)
where a is the difference in elastic strains of the composite components along the fibers (the parameter which characterizes nonequilibrium of the elastic strains); r and r are the relaxation functions; and eij are the elastic macrostrains of the composite measured from the relieved state. The detailed structure of Eqs. (4) is explained in [1, 2]. In the case of a laminated composite the number of parameters characterizing nonequilibrium grows to three, and the system of corresponding equations takes the form
Oui
Oaij _ 0,
0~1 Ot
~O ~ j -- l { O u i
OAz = -r
238
=
(5)
0A23
O~ = - r '
A1 ----"(A22 q- A33)/2,
Ou~
AS
O---i- = - i'2s,
~--" ( A 2 2
--
A33)/2.
~, GPa
10-
_o.oo5 ~, k b a r 2
:
5-
- ~'2, 5
-2~,00 o
O,8
~,8
Fig. 5
I
q
2
J
~, k~sec
Fig. 6
Fig. 5. Calculated shock-wave profiles in model copper-epoxy composite. Fig. 6. Calculated shock-wave profiles in organoplastic.
The systems (4) and (5) are complemented with the relationships for the generalized elastic potential, the Murnaghan formulas, and the functions for the relaxation time of shear stresses for every composite component [3]. A specially developed modification of the discontinuity decay method was used in a numerical solution of the specific problems of shock wave deformation. Let us consider a solution of the problem of propagation and evolution of a plane shock wave in a fibrous composite when the wave propagates along the fibers directed along the spatial x-axis. The appropriate system of equations results from (4) under the assumption that all functions depend only on time and one spatial variable. Figure 5 shows the calculated shock-wave profiles in a model copper-epoxy composite [4] in (a, x)-coordinates and the profiles of dependences of the incompatibility characteristic 2~ of elastic strains of the reinforcing fiber materials and the matrix for various instants of time. Calculations were performed for initial nonequilibrium 2x0 = 0.005, fiber concentration c = 0.5, and mass velocity u 1 = 200 m/sec. The peculiarities of the shock wave structure observed in calculations are in full agreement with the nature of the compression diagram established in [2]. Three regions of the a - - e diagram of a composite point to the existence of three types of disturbances in a material propagating at different velocities, namely, the elastic disturbances, when the matrix is deformed elastically and the reinforcing fiber is deformed plastically, and the plastic disturbances. This specifies the shock wave splitting in a composite into three subsequent drops. The elastic properties of a composite are determined, first of all, by the elastic properties of the fibers, and the effective elastic modulus is close to Young's modulus for the fiber. Consequently, the main plastic change of the form occurs in the second wave, and its propagation velocity differs little from the velocity of a plastic wave. Since the relation between the elastic characteristics of the components is not the same, the above peculiarities of the structure of a shock wave front are less pronounced in the experiments. Now consider a problem which is consistent with the above-described experiments on a shock wave's attenuation during its interaction with an overtaking rarefaction wave. The experimental conditions were completely reproduced in the calculations. Control calculations for aluminum plates instead of the composite ones verified the program and the accuracy of the loading pulse reproduction by comparison with the experimental data. Figure 6 shows the calculated profiles of the attenuating shock pulse. A comparison with the corresponding experimental profiles presented in Fig. 3 shows a good qualitative similarity and an agreement of quantitative characteristics in all cases except for the first profiles. Here the duration of the pulses is the same and the maximum value of the amplitude in the experiment is close to the average value of the calculated "shelf" (9.1 and 10.5 GPa) but the pulses are qualitatively different. In our opinion, this difference may be related to some specific feature of the experiment procedure not discussed here or to the structure nonuniformity of the specimens in the experiments. The calculated change in the shock-wave amplitude, as it propagates, is shown in Fig. 4 with a solid line.
239
CONCLUSION Attenuation of shock waves in organoplastic was experimentally investigated in interaction with an overtaking rarefaction wave. Data obtained in the experiments give new information which characterizes not only the properties of a material itself. It forms a complementary basis for model constructing and testing. It is pertinent to note that similar researches on metals made radical changes in the concepts of mechanisms of a shock wave deformation and formed the basis for a variety of new models [5]. Our results also show that the model of [1, 2] for a dynamic deformation of composites covers well the investigated shock-wave process. This confirms the acceptability of the concepts and hypotheses used in model constructing and shows that the approach being developed is promising for the description of the behavior of composite materials and structures under shock-wave loading. The work was supported by the International Science Foundation (Grant RCC000).
REFERENCES .
2. 3. 4. 5.
240
E. I. Romensky, A. D. Resnyansky, and L. A. Merzhievsky, "The model of viscoelastic composite," J. Physique IV, Colloq. C3, Vol. 1, No. 10, 923-930 (1991). L. A. Merzhievskii, A. D. Resnyanskii, and E. I. Romenskii, "A model of the dynamic deformation of unidirectional composites," Dokl. Akad. Nank SSSR, 327, No. 1, 48-54 (1992). L. A. Merzhievskii and S. A. Shamonin, "Construction of the dependence of the relaxation time of shear stresses on parameters of the medium state," Zh. Prild. Mekh. Tekh. Fiz., No. 5, 170-179 (1980). L. A. Merzhievskii and O. A. Nizhnikov, "Dynamic compression of a model unidirectional composite," Fiz. Goren. Vzryva, 29, No. 5, 76-80 (1993). L. A. Merzhievskii and A. D. Resnyanskii, "Selecting a model to describe the attenuation of shock waves in metals," Fiz. Goren. Vzryva, 19, No. 1, 99-105 (1983).
