Journal of Theoretical and Applied Mechanics, 2008, vol. 38, No. 4, pp. 67–82
DEVELOPMENT OF A NEW TESTING METHOD FOR POLYMER MATERIALS AT HIGH STRAIN RATE M. Khlif, N. Masmoudi, C. Bradai National School of Engineers of Sfax, Analysis Laboratory of Electro Mechanical Systems, BP.1173. W. 3038 Sfax, Tunisia, e-mail:
[email protected]
V. Grolleau, G. Rio University of Southern Bretagne, France, Mechanical laboratory of Engineering and Materials
[Received 02 January 2008. Accepted 09 June 2008] Abstract. The increasing use of polymeric materials in transport fields requires knowledge of their mechanical behaviour at high strain rate to optimize the structures. The particular behaviour of polymers, compared to metals, is characterized by low Young modulus, weak density and viscoplastic behaviour, which make conventional experimental test inoperative. The objective of this work is the development of a dynamic tensile test reaching from 100 to 500 s−1 (strain rate) based on the Charpy testing machine. The proposed test device is composed of an instrumented bar and a sensing block for wave strain measurement to determine the stress and strain on the tested material. The main objective consists in studying mechanical behaviour of polymer material at high strain rate. Key words: high strain rate, dynamic behaviour, shock, Hopkinson bar, sensing block.
I. Introduction Impact studies are very interesting techniques to characterize and study the mechanical behaviour of materials from both theoretical and experimental viewpoints. From a theoretical perspective, this kind of technique allows the characterization of the dynamic response of materials exposed to sudden loads, as well as the propagation of elastic, plastic and impact waves [1]. They also provide the means to study the dynamic behaviour of materials, which differs significantly from that under static or quasi-static load application, and the
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measurement of parameters such as dynamic elastic modulus, absorbed energy, deceleration peak, impact time, and maximum deformation, etc. [2]. From an experimental viewpoint, several configurations to perform impact tests have been described in literature. The Hopkinson pressure bar has been used for a variety of purposes such as ballistic determinations, longitudinal wave transmission and studies of contact relations at the impact point [1]. Many static and dynamic tests involving notched specimens have been carried out to assess certain properties of materials. These tests are, in general, not instrumented so that they do not provide stress strain behaviour data of materials, but rather compare their notch sensitivity and brittleness. Instrumented impact testers yield information on forces, strain and energies absorbed during the impact by measuring these signals continuously throughout the impact process. In addition, the introduction of transducers has provided the possibility of analyzing the impact process according to mechanical models [3]. Before the Charpy and Izod instruments, the US Board proposed falling weight equipment for testing iron and steel in 1881 [4]. However, due to the difficulties of measuring the energies involved, this system was less attractive than that using Charpy and Izod methods [3]. Recently, the improvements in the field of instrumentation and better knowledge of materials behaviour have led to the use of instrumented impact testing machines to evaluate the resistance of the tested materials to shock. In summary, many experimental tests using Hopkinson bars are available in the literature especially for metallic materials, however, there is a lack of works for polymer materials. Indeed, to study the mechanical behaviour of polymers, Hopkinson bars must be made from low impedance materials in order to get a correct measure of displacements and forces. On the other hand, when using polymer Hopkinson bars there are two problems: the fixing mode of the specimen with the bars and the created interfaces between two different materials (fixing elements and polymer material). The above problems diminish the efficiency of the Hopkinson bars for polymer materials. For this reason, the LASEM and LG2M have developed a new instrumented falling pendulum tester based on Charpy, Hopkinson theories associated with the use of sensing block technique in order to characterize the dynamical behaviour of a polymeric material at high strain rate. II. Experimental techniques design Dynamic tensile test can be implemented by any testing machine which applies a uniaxial tensile load to the polymer specimen with specified high
Development of a New Testing Method. . .
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strain rate [5]. The instrumentation of the machine is very important in order to get valuable information about strain, load and displacement. II.1. Test device description The test machine is divided into three parts. The first part shown in Fig. 1 represents the shock generating system composed of a rotating pendulum which excites the end part of the input bar. In order to change the intensity of the shock, a range of variable masses of the pendulum are used. A pneumatic device commands the shock generating system and allows only one impact. The velocity of the excitation shock is a function of the pendulum altitude position and can be determined by using the following formulae: p v = 2gh.
Ù â
Ú
1 – pendulum, 2 – pneumatic device, 3 – input bar Fig. 1. Shock generating system
The second part, the shock receiving system, is illustrated in Fig. 2. It is composed of the input bar which receives the shock excitation and transmits the corresponding strain wave to the specimen located between the input bar and the sensing block. The input bar is used to measure the force. The choice of a long bar enables us to solve the problem of interference of the waves (incident and reflected waves) and to measure the loading force for a dynamic test. Moreover, this bar is used to load the specimen by transmitting the
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incident wave. The wave transmitted by the specimen is finally received by the sensing block divided into a sensing receiver and a base block. A strain gage located at the sensing receiver measures the corresponding transmitted stress wave. At last, the stress wave moving from the sensing receiver is dissipated in the base block which has an important relative large mass after some reflections between its upper surfaces. The technique described above has the following advantages compared with the Hopkinson bar system [2, 6]: The sensing block system is a compact structure. Due to the important mass of the base block, the reflection disturbance is negligible and the short length of the sensing receiver allows a long measuring time.
1 – input bar, 2 – specimen, 3 – sensing block, 3a – sensing receiver, 3b – base block Fig. 2. Receiving shock system
The last part is the data acquisition system which analyses the signals measured by the two strain gages. These gages were used to measure the strain wave propagation. The first one is located in the middle of the input bar, the second is mounted on the sensing block. A data acquisition card “NI-PCI-6250” is used to collect the signals issued from the gages which were reliably amplified. The visualization and analysis of the results are done by a Lab-view software, Fig. 3.
Development of a New Testing Method. . .
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1 – dynamic analyser, 2 – amplificator, 3 – computer equipped by data acquisition card Fig. 3. Data acquisition system
II.2. Test stand preparations To get a uniform signal, the different parts of the reserving shock system have similar impedance. To this effect, the diameters and the materials of the bar and the sensing receiver are chosen identical to avoid a substantial change in the amplitude of the travelling signals (Table. 1). Both connections between the specimen and the bars were implemented with steel rings screwed onto both the bar and the specimen. Their diameter and position were especially designed so that their contribution to the impedance was negligible (Fig. 4).
Strike
Strain gage G1
Input bar
Specimen
Strain gage G2
30 L = 3m
Fig. 4. Scheme of the complete set-up
Sensing block
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M. Khlif, V. Grolleau, N. Masmoudi, C. Bradai, G. Rio Table 1. Material properties of the input bar and the sensing block Characteristics Young’s modulus E Mass density ρ Poisson’s ratio υ
Value 71 GPa 2,83 103 kg/m3 0,3
III. Wave propagation analysis A tensile test is considered in this study. For this effect, a specimen is placed between the input bar and the sensing block system and the pendulum is launched on the free part which causes a propagation of an elastic wave through the bar. At the interface between the input bar and the specimen, a part of this wave propagates in the specimen while another part is reflected [7–8]. The transmitted wave in the specimen encounters the sensing block. The elastic strain wave propagation generates a dynamic loading on the specimen. The corresponding deformation and force applied on the specimen can be recovered by the difference of two acquired signals in the gages placed on the bar and the sensing block. The equations for the analysis of the compression split Hopkinson bar assume that stresses and velocities at the end of the specimen are propagated down the bars in a random way. By considering that the wave-transit time in the short specimen is small compared to the total time of the test, many stress and strain wave reflections can take place uniformly back and forth along the specimen. The stress and strain are then assumed to be uniform along the specimen. By using the same material and cross-section of the input and the sensing block, a set of relatively simple relations can be derived for the stress, strain, and strain rate in the specimen. Figure 5 shows the incident reflected and transmitted strain pulses ε i , εr and εt on the specimen, the input bar and the sensing block. Subscripts 1 and 2 are used to represent the two strain pulses at the ends of the specimen.
Fig. 5. Schematic of specimen and strain pulses
Development of a New Testing Method. . .
(1)
73
The displacement at the ends of the specimen is given by: Z t u1 = C0 ε1 dt, 0
(2)
u2 =
Z
t
C0 ε2 dt,
C0 =
0
s
E , ρ
where C0 is the elastic-wave velocity in the bar and the sensing block, E is the Young modulus, ρ is the material density. In terms of incident and reflected pulses the displacement u1 becomes: Z t (3) u 1 = C0 (εi − εr )dt. 0
Tanimura [9] showed that the rising time of the load applied to the sensing block must be higher than 25 µs for the relative error from the sensing block measurement to remain below 3%. This condition is satisfied as one can see in Figs 7 and 11. Under this condition, the sensing receiver can be considered as in equilibrium. Thus, the displacement u 2 reads: (4)
u2 = −lsp F2 /(E A),
where lsp is the length of the sensing receiver, A is the cross-sectional area of the input bar and sensing receiver. The average strain in the specimen can be expressed by: (5)
l ε s = u1 − u2 ,
where l is the length of the specimen. During tests, the sensing receiver strain remains lower than 10 −3 and its length is about 30 mm. Thus, u2 remains lower than 0.03 mm while l εs is expected higher than 1 mm. Thus, the displacement u 2 can be neglected in equation (5) and the stain of the specimen reads: Z C0 (6) εs = (εi − εr )dt. l The forces at the two ends of the specimen are obtained by the following equations: (7)
F1 = E A (εi + εr ),
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M. Khlif, V. Grolleau, N. Masmoudi, C. Bradai, G. Rio F2 = E A εt .
(8)
From equations (7) and (8), the average force can be computed by: (80 )
Fav =
EA (εi + εr + εt ). 2
If it is assumed that the forces are equal at both ends of the specimen (F1 = F2 ), then combining equations (7) and (8) yields to: (9)
ε i + εr = εt .
Applying the equilibrium condition of the specimen, the stress and strain rate become: (10)
σs = E
A εt , As
where As is the cross-sectional area of the specimen. It is important to note that the stress, strain, and strain rate are average values, and that they are computed from a uniaxial stress-state assumption. To achieve tests with cyclic loading of specimens and to record the loading cycles, it is necessary to eliminate reflected strain pulses ε r from the
Fig. 6. Lagrangian diagram of pulses in the input bar and the sensing block during the registration of one cycle of loading
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back end-face of the sensing block. To this effect the sensing block is designed to be a massive block in order to trap the s transmitted wave from the specimen. Figure 6 represents the Lagrangian diagram of distribution of deformation along the system for one cycle of loading. IV. Analysis and processing of gage measurements The incident strain wave εi from the impact point is measured in a first time by the input bar gage G1, this wave encounters the specimen so a part of it will be reflected εr and measured again by the same gage G1. The length of the input bar (3 m) and the placement of the gage G1 (in the middle of the input bar) allow a correct time separation between the incident and the reflected wave. The force applied on the first extremity of the specimen and its velocity can be determined by: Finput (t) = Ai Ei (εi (t) + εr (t)),
Vinput (t) = − Ci (εi (t) − εr (t)). A part of the incident strain wave εt is transmitted through the specimen to the sensing receiver and measured by the strain gage G2. The corresponding force applied on the second extremity of the specimen is determined by: Foutput (t) = At Et εt (t), where Ai , Ei , and Ci are the input bar cross-sectional area, the Young modulus and the elastic wave velocity, respectively. A t and Et are the sensing receiver cross-sectional area and the Young modulus, respectively. V. Results and discussion V.1. The experimental technique confirmation The objective of this study is to observe the evolution of the strain wave in the input bar and the sensing receiver during its propagation. In the first experience, the specimen is not mounted. The steel made pendulum with 23 kg weight applies one impact compressive load at 5 m/s on the end part of the input bar which is in a direct contact with the sensing receiver. Figure 7 shows the evolution of the measured signals from gages G1 and G2. The signal measured by gage G1 represents the amplitude evolution of the incident and reflected strain wave in the middle of the input bar versus
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Fig. 7. Measured signals of the strain wave on the bar and the sensing receiver from gages G1 and G2
time after some reflections. This signal is an alternating incident-reflected wave strain. It is noted that its amplitude decreases in a slow way as the time increases due to the structure damping effect. The interval of time from 0 to t1 represents the time delay for the strain wave to reach the measuring point (gage G1).The measured value of t 1 ≈ 0.3 ms corresponds exactly to the theoretical calculated time: L t1 =
2
C0
= 0.3 ms.
The signal measured by gage G2 represents the amplitude evolution of the transmitted strain wave in the sensing receiver versus time after some reflections. The maximum amplitude of the signal is observed at the first propagation through the sensing receiver (transmitted wave), this can be explained by the fact that the wave is totally transmitted by the input bar to the sensing receiver. Add to that, the amplitude of the signal is largely decreased with the time. The rapid attenuation of this wave is due to the massive base block which traps and dissipates the receiving wave. The time interval from 0 to t 2 represents the time delay for the transmitted strain wave to reach the measurement point located on the sensing receiver (gage G2). The incident and the transmitted waves at the first propagation have approximately identical amplitudes (Fig. 7). The last remark shows that the wave loss in the interface between the input bar and the sensing receiver is negligible.
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Development of a New Testing Method. . .
V.2. Effect of the impact velocity on the strain wave In this section, the influence of the impact velocity on the strain wave propagation is studied. To this effect, different altitudes of the pendulum are chosen to set four impact velocities: 2 m/s, 3 m/s, 4 m/s and 5 m/s. Figure 8 represents the propagation of the incident and reflected strain waves in the input bar for the four velocities, measured always in the middle of the input bar by gage G1. 0,15
Incident Wave strain
Recorded signal [V]
0,1
0,05
Time [ms]
0 0
0,2
0,4
0,6
0,8
1,2
1,4
1,6
1,8
2 5m/s
-0,05
-0,1
1
4m/s
Reflected Wave strain
3m/S 2m/S
-0,15
Fig. 8. Shock velocity influence on the wave propagation in the input bar
It is noted that the strain amplitude increases as the impact velocity increases. The loading time remains constant. Figure 9 represents the transmitted strain wave in the sensing block system which has the same amplitude of the incident wave for all the velocities. It is also observed here that the time delays are conserved. V.3. High strain rate tensile tests In this section, a polypropylene specimen is introduced between the input bar and the sensing block as shown in Fig. 2. This specimen is chosen according to the standard ISO 8652 [10] and used for the dynamic tensile tests. The mechanical characteristics of the tested material are obtained by a quasi static experimental tensile test and presented in Table 2. The geometric specifications of the specimen are presented in Fig. 10. It has a thickness (W = 5 mm) and a gauge length (l = 10 mm). The objective of the tests is the determination of the mechanical behaviour of the polypropylene in presence of high strain rate (the case of shock).
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M. Khlif, V. Grolleau, N. Masmoudi, C. Bradai, G. Rio 0,14
Recorded signal [V]
0,12 0,1
Transmitted Wave strain
0,06
5m/s 4m/s 3m/s
0,04
2m/s
0,08
0,02 0 -0,02 0
0,5
1
1,5
2
Time
[ms] 2,5
-0,04
Fig. 9. Shock velocity influence on the wave propagation in the sensing block system Table 2. Mechanical characteristics of polypropylene Characteristics Elastic deformation limit Young’s modulus Mass density Poisson’s ratio
Rp0,2 E ρ υ
Value 22 MPa 1400 MPa 750 kg/m3 0,4
Fig. 10. Specimen geometry used for the dynamic tensile test
If we denote by V the loading velocity, the strain rate (ε) ˙ can be estiv mated by the following formula: ε˙ = . l A tensile impact at velocity of 5 m/s is generated by the pendulum at the end part of the input bar. The corresponding wave propagates from the bar to the sensing block through the specimen. Gages G1 and G2 measure the
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Development of a New Testing Method. . .
incident, the transmitted and the reflected strain waves. The corresponding signals are presented in Fig. 11. 0,4
Recorded signals [v]
0,35
Sensing block Bar
0,3 0,25 0,2 0,15 0,1 0,05 0 -0,05 0
1
2
3
4
5
6
7
8
9
10
Time [ms]
-0,1
Fig. 11. Signals recorded in the input bar (gage G1) and the sensing block (gage G2)
Using the recorded incident εi (t) and transmitted εt (t) strain signals in the bar and the sensing block, the strain ε s (t) and the stress σs (t) can be determined by using equations (6) and (10). They are presented in Fig. 13 for different strain rates. 50
25 100s-1
45
200s-1
100s-1
40
300s-1
20
500s-1
200s-1 300s-1
35
Stress [MPa]
Strain %
500s-1
15
10
30 25 20 15 10
5 Time [ms]
5
Time [ms] 0
0 0
0,1
0,2
0,3
0,4
(a) Strain-time curves
0,5
0
0,1
0,2
0,3
0,4
0,5
(b) Stress-time curves
Fig. 12. Strain-time and stress-time curves for different strain rates
Using the results presented in Fig. 12(a, b), the evolution of the stress versus the strain on the specimen can be determined σ s (εs ), (Fig. 13). Figure 13 represents the dynamic behaviour of the tested polypropylene
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M. Khlif, V. Grolleau, N. Masmoudi, C. Bradai, G. Rio
Fig. 13. Tensile stress-strain curves for different strain rates
material. For a given strain rate value, the curve can be divided into three zones: -I- an elastic zone: It corresponds to a low strain, the material shows an elastic behaviour with a slight plastic strain hardening; -II- a viscoelastic zone: The strain increases slowly with the stress, a viscoelastic behaviour of the material is shown with a plastic strain hardening. On the other hand, for a given value of the strain, it is shown that the stress increases with the strain rate; -III- a plastic zone: In this zone, a change of the direction of the curves is observed, the stress increases quickly with the strain. The above results are in agreement with many existing studies in the literature that use two Hopkinson bar systems [11, 12, 13]. The above figures describe the dynamic behaviour of the tested polypropylene material. At low strain, the material shows an elastic behaviour with a slight plastic strain hardening. For a given value of the strain, the stresses on the specimen increase with the strain rate values.
Development of a New Testing Method. . .
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VI. Conclusions In this paper, an experimental method of determining the dynamic tensile behaviour of polymer material is proposed. A mixed system composed of an instrumented input bar and a sensing block is used to capture the incident, reflected and transmitted wave signals. In this technique, the out bar of Hopkinson bar system is replaced by a sensing block with great inertia in order to tap the transmitted strain wave. The equations of the travelling waves are developed. At first, validation tests without specimens using compressive loads are achieved. The obtained results are in agreement with the wave propagation theory. The shock velocity influence on the strain wave propagation is studied. It was noted that the wave strain amplitude increases with the shock velocity while the time loading remains unchangeable. Dynamic tensile tests of polypropylene material specimens are realized in the second part of this study. The stress-strain curves presenting the dynamical tensile behaviour at different strain rates are presented. The results show that there are three dynamical behaviour zones: an elastic, a viscoelastic and a plastic zone. The mentioned results are in agreement with the literature. We can note that this new experimental method using a reduced bulkiness device is adaptable with both tensile and compressive tests. Its particularity is to associate Hopkinson technique and sensing block technique in an only one.
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[8] Zhao, H., G. Gary. A Three-dimensional Longitudinal Wave Propagation in an Infinite Linear Viscoelastic Cylindrical Bar. Application to Experimental Techniques. J. Mech. Phys. Solides, 43 (1995), 1335–1348. [9] Tanimura, S. et al . Evaluation of Accuracy in Measurement of Dynamic Load by Using Load Sensing Block Method, Proceedings of 4th International Symposium on Impact Engineering ISIE4, Kumamoto, Japan, 2001, 77–82. [10] Krawczak, P. Essais m´ecanique des plastiques, Techniques de l’Ing´enieur, AM, 3, 1997, 510. [11] Gomez-del Rio, T., E. Barbero, R. Zaera, C. Navarro. Dynamic Tensile Behaviour a Low Temperature of CFRP Using a Split Hopkinson Pressure Bar. J. Composites Science and Technology, 65 (2005), 61–71. [12] Yuan, J. M., V. P. W. Shim. Tensile Response of Ductile α-Titanium at Moderately High Strain Rates. International Journal of Solids and Structures, 39 (2002), 213–224. [13] Chen, W., F. Lu, M. Cheng. Tension and Compression Tests of Two Polymers under Quasi-Static and Dynamic Loading. J. Polymer Testing, 21 (2002), 113–121.
