ABSTRACT
The compressive strength behavior of a Polymer Bonded Explosives (PBX) analog was measured as a function of temperature (25°C to 95°C) and strain rate (100 to 700 s-1) using the split Hopkinson pressure bar (SHPB).The result exhibits the flow stress of the PBX analog is strong dependency on temperature and strain rate. According to the result, an improved Sargin constitutive model was used to describe the dynamic compressive behavior of the material, and the modeling curves fit well with the experimental.
Key words: PBX analog; Compressive strength; SHPB; Constitutive model
INTRODUCTION
Physically-based constitutive models are needed to predict the mechanical behavior, damage evolution, and performance of modern energetics for their safe application. Understanding and modeling the mechanical response of polymers and polymer-based composites is of great interest for defense and commercial applications related to (1) the need for predictive constitutive model descriptions for use in large-scale finite-element simulations of damage and deformation, and (2) focused emphasis on understanding the dynamics of localization phenomena and mechanical failure of polymeric composites. New continuum models, based on actual physical and chemical mechanisms, to describe complex loading processes must account for complex phenomenology, including temperature, strain rate, orientation effects like crosslinking and chain stretching, or texture (created by extrusion or directional formation), and aging effects on mechanical performance, if a predictive capability is to be achieved.
Conventional methods have been used to measure the mechanical properties of polymers and polymer composites at low strain rates. However, high strain rate methods using the split Hopkinson pressure bar (SHPB) must be modified to achieve stress equilibrium with these low sound speed materials; to obtain adequate pressure bar signal output at very low stress levels; and to minimize undesirable stress triaxiality caused by friction on specimen interfaces.
A number of previous studies have probed the constitutive response of a wide variety of plastic bonded explosives (PBX).[1-10] Some literatures investigated PBX at low strain rate.[2,5,7] PBX behave quite differently at high strain rates compared to low strain rate. Although many literatures gave the curves and trends about the mechanical properties of PBXs as function of temperature and strain rate,[6,8-10] there are few literatures showed the expressions to quantify them.
In the present investigation, a PBX analog instead of a true PBX was studied because of safety. Uniaxial compression tests were performed at strain rates from 100 to 700 s-1 and for temperatures from 25°C to 95°C on the PBX analog. From the data of experiments, a constitutive model was used to describe the dynamic compressive behavior of the PBX analog. The model curves can not only predict the raise part but also the decline part of experimental.
EXPERIMENTAL
The SHPB used for this study was equipped with 20mm diameter 7075A All-alloy, to improve the signal-to-noise output associated with low strength materials. The length of incident bar, transmission bar and striker bar were 2000mm, 1200mm and 200mm, respectively. The specimens geometry were 10mm diameter and 6mm long.
Specimens were tested at temperatures of 25°C, 50°C, 65°C, 80°C and 95°C. Controlled temperature variations between 25°C to 95°C were achieved putting the specimen in a small box which was heated using hot water. The water temperature was controlled to the tested temperature, and specimen was equilibrated at temperature for ~15 min prior to testing. After heated, the specimen was quickly taken out of the box and then installed on the bars. The gas gun was fired once the installation was finished. The whole time from take the specimen out of the box to finish loading was 2~3seconds. The temperature variety of the specimen during installation is show in figure 1. Because of the low constant of heat exchange of the specimen, it could been seen that the temperature variety of the specimen was less than 1°C in 3 seconds although at 95°C. The error of temperature less than ~2 percent proved the heated method is valid.
The surfaces of bars which connected the specimen during loading were lubricated with either a thin spray coating of boron nitride or a thin layer of molybdenum disulfide grease before the specimen installed, which could minimize undesirable stress triaxiality caused by friction on specimen interfaces.
A small cylinder made of brass as a shaper was installed on the impacted-surface of incident bar, to change the shape of incident wave. The existence of shaper enlarged the time of incident wave rose to maximum, which made it is more easier to achieve stress equilibrium on the both surfaces of specimen and constant strain rate loading. The typical wave of loading process is showed in figure 2. As is shown in figure, the stress equilibrium is clearly achieved between the two sides of the specimen since the transmitted stress wave is nearly identical to the sum of the incident and reflected waves during the whole loading period.
Fig. 1 Variety of temperature of specimen during testing
Fig. 2 The stress histories in the bars. in: input; tr: transmitted; re: reflected
RESULTS
The stress-strain curves of specimens under different temperatures and strain rates are shown in figure 3.
Fig. 3 Curves of engineering stress-strain at different temperatures and strain rates
These data show that the flow stress peaks consistently between 2% and 3.5% strain before slowly decaying with further strain.
The temperature dependence of elastic modulus, flow stress and failure strain at high strain rate is summarized in Figure 4. The points in the Figure 4 are the experiment results. These plots show the elastic modulus, flow stress increase with increasing strain rate, and the higher strain rate is, the higher increasing speed becomes. The elastic modulus are approximately independent of temperature, because at the nearly same strain rate, elastic modulus at different temperatures present confusion. The flow stress decreases with increasing temperature except 50°C, which performs nearly the same value of 25°C. The result could be analyzed as the stronger force connect crystal and bonder at ~50°C, so the flow stress maintained. As the temperature went on increasing, the effect of the temperature-soften presented and the flow stress decreased. Failure strain of the specimen is independent of strain rate and temperature, and presents between 2% and 3.5% (Figure 4 (c)).
Fig. 4 Varieties of mechanic behaviour at different temperatures and strain rates
Constitutive models of PBX now available usually can only describe the climb part of the stress-stain curve depend on temperature and strain rate, because of the complex structure of PBX. And the other part of curve that the flow stress coming down with further strain takes an important part in understanding the remain-strength of material. Because of the similar shape of stress-stain curve between the specimen and concrete, the Sargin model which described concrete behavior originally is modified to describe the behavior of the specimen.[11] The Sargin model describe as:
Where
is the secant modulus at peak compressive strength, parameter ft affects decline part of stress- strain curve.
From analysis above, the mechanical behavior affected by temperature and strain rate were explained as the following equations:
Where A, B, C and D are the parameters controlled by experimental results, B, C and D mean strain rate effect, temperature effect, both strain rate and temperature effect, respectively.
From the experiment results, take s0=138s"1, room temperature T= 25°C. a0= 19.29Mpa, e0=2.98% and £,0=0.802Gpa are flow stress, failure strain and elastic modulus at strain rate £c and temperature Tr respectively. {1=0.5.
Constitutive parameters fitting by experimental results used the inheritance arithmetic are shown in table 1. Table 1 Modify Sargin model parameters_
|
A |
B |
C |
D |
|
|
E |
1.173 |
2.83E-03 |
-0.1163 |
5.41E-04 |
|
a |
1.104 |
4.70E-03 |
-0.1573 |
0 |
|
s |
0.934 |
1.91E-03 |
-0.0202 |
-8.77E-04 |
The solid lines in figure 4 (a), (b), and (c) are the fitting results by using equations (2), (3), and (4) and the parameters in table 1. And the model curves fitting by equation (1) are compared to experimental result in figure 5.
Fig. 5 Comparison of stress-strain curves with modeling to experimental
CONCLUSION
Split Hopkinson pressure bar tests were made on a PBX analog at different temperatures. The following conclusions can be drawn: 1) the compressive flow strength of specimen is strongly dependent on both strain rate and temperature. The elastic modulus is only mildly dependent on strain rates. The failure strain is not evidently temperature and strain rates dependent. 2) the specimen has an invariant strain to failure of approximately 2~3 percent. 3) The modify Sargin constitutive models could describe and predict the stress-strain curve of the specimen as a function of temperature and strain rate not only the climbing part, but also the declining part.
