We have studied experimentally stress profiles upon propagation of a shock wave in a unidirectional composite in the case where the normal to the surface of ...
Combustion, Explosion, and Shock Waves, Vol. 33, No. 3, 1997
SHOCK-WAVE STRUCTURE
IN A UNIDIRECTIONAL
COMPOSITE WITH DIFFERENTLY
ORIENTED
FIBERS
S. A. Bordzilovskii, S. M. Karakhanov, and L. A. Merzhievskii
UDC 539.3
We have studied experimentally stress profiles upon propagation of a shock wave in a unidirectional composite in the case where the normal to the surface of the wave front is directed at angle 0 to a reinforcing fiber. For 0 = 5 and 15~ the elastic precursor behind which the shock wave propagates was registered. For the case 0 = 45 ~ the elastic precursor becomes a plastic wave with a smeared front, and, for 0 = 90 ~ a single shock wave was recorded. Measurement results show that the stress at the yield point depends on the orientation of the fiber and on the direction of the shock-wave motion.
The prediction of the behavior of fiber-reinforced polymeric composites under dynamic loading is required in designing composite structures to be used under extreme conditions. In the last decade, similar problems have arisen more often owing to the ever increasing use of composite materials. To solve them successfully, it is necessary to develop models of composites that describe adequately their behavior over a wide range of loading parameters (stresses, strains, and strain rates). A model of calculating the effective characteristics of a composite as a whole was proposed and tested by Merzhievskii et al. [1]. The model is based on the Maxwell representations of the mechanisms of irreversible deformation of the composite components and also on the method of phenomenological averaging for calculation of the effective characteristics of the composite as a whole. The calculation results obtained by this model were compared with the dynamic [2] and shock-wave [3] experiments in which specimens were loaded along the principal directions of the model or structure material. The materials studied were characterized by the anisotropy of mechanical properties, which is most pronounced in unidirectional composites. A numerical experiment in which the shock wave (SW) propagated along reinforcing fibers was performed in [4]. The authors also determined the effect of SW splitting, which is caused by the difference in the velocities of stress pulses over materials of the components. At present, the relevant experimental data are not available. The goal of the present paper is to record wave profiles in a unidirectional composite during the SW motion along reinforcing fibers or at an angle to them. Apart from the determination of the wave structure, our measurements were aimed at establishing the relationship between the wave velocity and the angle of the loading direction with the fiber. EXPERIMENT The experiment is shown schematically in Fig. 1. A plane-wave lens and a booster T N T charge generated a SW in a paraffin pad. The SW then passed through a copper plate to a specimen backed with an additional plate. The specimen and the back plate were made of organoplastic to be examined. The plane-wave lens and the TNT charge were 75 mm in diameter, and the booster charge was 60 mm high. The paraffin pad and the copper plate were 20 and 10 m m thick, respectively, to ensure a measurement time of ~-5 #sec before the
Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Fizika Goreniya i Vzryva, Vol. 33, No. 3, pp. 132-138, May-June, 1997. Original article submitted February 4, 1997. 354
0010-5082/97/3303-0354 $18.00 Q 1998 Plenum Publishing Corporation
,11
O', GPa 8
11
6
2 3
,
6
"'
0
0
016
Fig. 1
112
118
214 t,/Jsec
Fig. 2
Fig. 1. Scheme of the experiment: 1) TNT charge; 2) paraffin; 3) copper; 4) specimen; 5) back plate; 6 and 7) leads of the manganin gauges. Fig. 2. Profiles
~ri(t)
at the top (1) and back (2) surfaces of the specimen for 0 = 5~
TABLE 1 h, mm 7.62 7.75 7.33 7.9 9.5
p,g/cm 3
0, deg
1.27 1.27 1.27 1.14 1.14 1.14 1.14
5 15 15 45 45 9O 9O
ariEL I a~x,11 GPa 0.2 5.4 0.15 4.8 0.15 4.4 0.5 3.8 0.5 4.0 --
a~x,2
mm/~sec 2.8 2.4 2.3 1.3 2.0
5
3
5
3
5.9 6.0 6.1 3.2 3.4 --
3.6 3.1 3.3 3.0 2.9 3.2 3.5
to, #sec 0.8 1.2 1.0 0.25 0.35
arrival of the second wave reflected from the detonation product-paraffin interface. Because of the mismatch of the dynamic impedances of paraffin and copper, the SW amplitude in the specimen decreased to 3.8-5.4 GPa, depending on the specimen thickness (Table 1). Specimens of diameter 48 mm and thickness 7-8 mm were made of a unidirectional aramid-fibrous composite with an epoxy matrix. The reinforcing fibers were 15 #m thick. The angle of slope of the fibers t9 was reckoned from the normal to the wave-front surface (Fig. 1). The stress in the specimens was recorded by manganin gauges, whose design and main characteristics are given in [3]. The gauges were placed in the upper and lower planes of the specimen and recorded the normal stress-time profiles cri(t), where i = 1 and 2 refers to the first and second profiles, respectively. One gauge was positioned at the boundary between the copper plate and the specimen (this gauge registered the input stress pulses), while the other was located between the specimen surface and the back plate. The amplitude of the recorded elastic precursor appeared to be small and to reduce the level of noise, and the gauges were shielded by a thin aluminum foil. Experiments were performed with specimens in which the angles of slope of the fibers were ~ = 5, 15, 45, and 90 ~
355
0"2. GPa 4
c,D, krn/sec
:l
I
I
i
l
I
I
o
1
2
3
4
Fig. 3
so~ \\./I 4
.O__B______
2
I
t, #sec
2/
t 0
~,
i
0
----"
D
i
i 45
i
t i 0, deg
90
Fig. 4
Fig. 3. Stress-time profiles at the back surface of the specimen for 0 = 15~ (1), 45 ~ (2), and 90 ~ (3). Fig. 4. Velocities of the elastic precursor (1) and of the plastic wave (2) vs. the angle of slope of the reinforcing fibers. EXPERIMENTAL RESULTS The SW with the falling-off pressure profile originated from the copper plate and propagated toward the organoplastic specimen. The profile al(t) had a steep leading edge whose duration did not exceed the resolution time of the measuring system, which was equal to ___50 nsec (Fig. 2). In the leading edge, we sometimes observed a stress peak whose amplitude was 5-20% larger than the level of the following signal. This peak occurred owing to the higher acoustic impedance of the dacron insulation by which the piezoresistive element of the gauge was protected. The duration of the stress peak was ~50 nsec, which corresponds to the reverberation time of the acoustic wave in the gauge. To avoid this effect, we determined the maximum stress in the specimen by extrapolating the stress profile to a point corresponding to the moment of arrival of the SW at the specimen. We distinguished three types of profiles cr2(t), depending on the mutual orientation of the SW motion and of the reinforcing fibers. For 0 = 5 and 15~ we observed a distinct elastic precursor and the subsequent main stress jump with a steep front (curve 1 in Fig. 3); for 0 = 45 ~ the elastic precursor became gradually a smeared plastic wave (curve 2 in Fig. 3); and at 0 = 90 ~ we observed the SW with an _~50-nsec steep front (curve 3 in Fig. 3). The elastic precursor was distinctly observed at 0 = 5 and 15~ Its velocity was --_6 mm/#sec and was 1.7-1.8 times larger than the velocity of the main stress jump. For this reason, for the specimen 7.5 mm thick, the elastic precursor was ahead of the plastic-wave front by approximately 3 mm. The precursor had a complicated structure: its amplitude was first constant for ___0.55/tsec and then increased smoothly for ~0.45 #sec. In signal records (curve I in Fig. 3), one can separate the yield point or the Hugoniot elastic limit, and, for 0 = 5 and 15 ~ the duration of the front of the main stress jump is not larger than the resolution time of the recording system. From the qualitative viewpoint, this corresponds to the results of [4]. The elastic precursor, which is approximately three times smaller than in the previous case, was also observed for 0 = 45 ~ but its velocity decreased, precisely as the velocity of the fundamental wave (curve 2 in Fig. 3). The latter velocity fell off by 28%, which exceeds both the measurement accuracy equal to ---5% and the scatter of the results from shot to shot, which was "10%. Under such loading conditions, the duration of its front increased up to ___160 nsec. In experiments with 0 = 90 ~ the elastic precursor was not observed, and the duration of the front of the second profile was approximately equal to the resolution time of the system (curve 3 in Fig. 3). The experimental conditions and the measurement results are given in Table 1. We used the following
356
TABLE 2 Loading conditions Statics (v -- 0.45-0.75) SW (v = 0.30)
Material Fiber Matrix Composite Composite
E, GPa 110-160 2 78-95 33
Y, MPa 2400-4200 130-160 280-310 80-110
r, MPa
20-44 15
notation: h is the thickness of the specimen, p is its density, O'HEL is the Hugoniot elastic limit, amax,1 and Crm~x,2 are the maximum stress amplitudes for the first and second profiles, respectively, c is the velocity of the elastic precursor; u is the front velocity of the fundamental wave or the velocity of the lower stress level in the plastic wave, and to is the duration of the elastic precursor. The last two rows of Table 1 give the SW velocity from [3] in which the SW attenuation in the unidirectional organoplastic was measured, with specimens loaded transversely. According to the data of Table 1, we plotted curves of the velocities of the elastic precursor and of the fundamental wave vs. the angle 0 (Fig. 4). Figure 4 shows the marked effect of SW splitting within the range ~ = 5-45 ~ DISCUSSION OF RESULTS
The material under study is transversal-isotropic [1] and is characterized by five independent moduli of elasticity. The specific features of irreversible (plastic) deformation of such a material are more complicated than the yielding conditions for isotropic bodies. For a quantitative analysis of the results obtained for the case of a uniaxial stress state, we shall use the simplest model. We first consider the experimental results for 0 = 5 and 15~ The material is assumed to be conditionally isotropic with some volume-averaged mechanical properties. In addition, all directions in a plane perpendicular to the fibers are assumed to be of equal strength. The velocity of a longitudinal elastic wave can be represented in the following form [5]: c2 = Z(1 - #)/p(1 + #)(1 - 2#),
(1)
where E is the equivalent modulus of elasticity of the composite in the direction of reinforcing, # is the equivalent Poisson ratio that characterizes the deformation over the thickness during specimen loading in the direction of reinforcing, and p is the mean density of the composite. Provided that the Poisson ratios for the fiber and for the matrix are practically the same, the equivalent modulus of elasticity of the composite can be calculated by the following mixture rule [6]: E = v E I + (1 - v ) E m = v E I,
(2)
where E l and Em are the moduli of the fiber and of the epoxy matrix, respectively, and v is the volume fraction of the fiber. Because of a lack of information on static tests of the material used in the experiments, we compared the mechanical properties of the components of aramid composites subjected to shock loading with the corresponding characteristics obtained after static tests on the basis of the data of [7, 8]. The results are given in Table 2. Table 2 shows values of the modulus of elasticity of the composite that were calculated using Eqs. (1) and (2) and the velocity of the elastic precursor for 0 = 5 ~ In comparison with the shock-wave loading results, it is necessary to take into account the fact that in the static tests, the volumetric content of the fiber was larger. For E, the correction of our results with allowance for this factor yields the 50-82 GPa range, which is overlapped by the static-modulus-of-elasticity range for the aramid composite. This allows one to conclude that the velocity of the elastic precursor depends on the properties of the fiber, in particular, on the large value of the Young's modulus and on the volumetric fiber content. One more experimentally measured quantity was the amplitude of the elastic wave or the Hugoniot elastic limit. Within the framework of the
357
Fig. 5. Vector diagram of the sound velocity: points (• refer to the data of [10] and points (,) refer to the data of Table 1; c* = c(O)/c(O).
above assumptions, the value of O'HEL is related to the yield strength of the material by the relation [9] 1 - 2# Y --
1 -
~t e T H E L "
(3)
The value of Y was estimated under shock loading at angle 0 = 5~ using Eq. (3) and the data of Table 1. Comparison with the static mechanical properties of [7, 8] shows that this quantity is the closest to the strength of the binder - - epoxy - - remaining, however, 1.5 times smaller than the static strength of the latter. For 0 = 15~ the elastic-precursor profile is of a complicated structure owing to the increase in its amplitude from ___150 to ___300 MPa during the second half of its duration (see curve 1 in Fig. 3). Such a structure makes it possible to distinguish two levels of stresses, which correspond to the onset of transition or to the complete transition of the matrix to a yield state. Note that the second level almost coincides with the mean strength of the organocomposite upon static compression along the fibers [7]. We emphasize the fact that, for 0 = 5 and 15~ c/u ~_ 1.8, which is close in magnitude to the ratio of the velocities of the elastic longitudinal and transverse waves in an isotropic material: Cel/Ctr = x/r3 [5]. At the same time, if the main stress jump is a plastic wave, i.e., the material in the wave is in a yield state, the ratio of the elastic-wave velocity to the plastic-wave velocity in an isotropic material is Cel/CO ~- 1.3 [9]. The considerable deviation of the ratio c/u from the latter value is likely to be caused by a more complicated mechanism of deformation in the front of the main stress jump than plastic shear. It is natural to assume that the observed effect is caused by the physical inhomogeneity of the material and also by the larger difference between the strength properties of the fiber and of the matrix material. For example, according to various data, the strength of aramid fibers varies from 2.4 to 4.2 GPa [9], which is substantially larger than the strength of the epoxy matrix. Therefore, at stresses that are larger than the elastic limit of the epoxy binder in the matrix material, plastic shear occurs, while at stresses that are smaller than ___2-3 GPa, the fiber behaves elastically. Precisely this range of stresses, from 0.3 to 5.5 GPa, is typical of the second wave in our tests. One can assume that the plastic strain of the matrix material causes transverse disturbances of the fiber that propagate with the velocity of elastic transverse waves and "drive" the plastic wave over the matrix material. Upon compression of the specimen at an angle of 45 ~ to the direction of reinforcing, the maximum shear stresses occur in planes coinciding with the direction of the fibers. Therefore, if the shear stresses exceed the interlaminar strength, slipping arises in these planes. In the experiments with 0 = 45 ~ the amplitude of the elastic precursor decreased by a factor of 3 compared with the case of specimen compression along the fibers and reached values that are smaller by one order of magnitude than the epoxy strength in static tests. 358
In Table 2, the maximum shear stresses r in the planes at 0 = 45 ~ which are reached in the elastic-precursor front, are compared with the data on the shear strength of the aramid composite in the static tests of [8]. The comparison enables us to conclude that the strength of the interlaminar shear 7 is a dominating quantity in the mechanism of specimen fracture at 0 = 45 ~ One should note that the data obtained are in qualitative agreement with the results of [10], in which Rose et al. studied the features of sound propagation in unidirectional graphite-epoxy composites. Figure 5 shows a comparison of the results obtained by Rose et al. with those of Table 1. In this figure, the vectors of the velocities of sound oscillations, which are referred to the sound velocity in a direction coinciding with the direction of the fibers, are reckoned from the origin. In our experiments, the velocity of sound for 0 = 90~ is chosen equal to that in the matrix material. The solid curve in the figure indicates the approximation of the experimental data by means of an ellipse with axes 1 and c (90~ respectively. Thus, the data obtained show that the strength of an organic composite (stress at the point of transition to a yield state) during shock-wave loading depends on the orientation of the fiber and on the direction of the SW motion. The results of SW structure experiments allows us to conclude that the strength is determined by the minimum value from the complete set of strength characteristics of particular composites. For example, the strength is determined by a transition of the epoxy matrix to a yield state at small angles between the direction of reinforcing and that of SW propagation and is determined by the interlaminar shear in the aramid composite at a 45 ~ angle of loading. This work was supported by the International Science Foundation and the Government of the Russian Federation (Grant No. RCC300). REFERENCES 1o
2. 3. 4. 5. 6. 7. 8. 9. 10.
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