Mechanical behaviour of acrylic under high strain rate tensile loading

Transcription

1 ARTICLE IN PRESS Mechanical behaviour of acrylic under high strain rate tensile loading N.K. Naik, Yernamma Perla Aerospace Engineering Department, Indian Institute of Technology Bombay, Powai, Mumbai 476, India Abstract An investigation of the effect of strain rate on tensile properties of acrylic is presented. Experimental studies were carried out on tensile split Hopkinson pressure bar (SHPB) apparatus in the strain rate range of 2 33 s 1. It is observed that the tensile strength and Young s modulus are enhanced and the ultimate strain is decreased at high strain rate loading compared with those at quasi-static loading. Studies were carried out with end-threaded cylindrical specimens. During tensile SHPB testing of the specimens, it was observed that the peak force obtained from the strain gauge mounted on the transmitter bar is lower than the peak force obtained from the strain gauge mounted on the incident bar. Further, an analytical method is presented based on a variable rate power law for the prediction of high strain rate tensile strength of acrylic. Using the analytical method, high strain rate tensile stress strain behaviour is presented up to a strain rate of 2 s 1. Keywords: Strain rate effect; Acrylic; Tensile properties; Split Hopkinson pressure bar; Power law 1. Introduction Mechanical response of materials is sensitive to the rate at which they are loaded. Even though there are many studies on the high strain rate behaviour of metals/alloys, only limited information is available on high strain behaviour of polymeric materials. Polymeric materials and their composites are finding increasing applications in high technology as well as traditional applications. This is primarily because of their low density and production cost. For the effective use of polymeric materials in structural applications, their behaviour under high strain rate loading should be studied. The objective of the present study is to evaluate the mechanical behaviour of acrylic under tensile high strain rate loading. Whilst Izod/Charpy and drop weight tests are used for the evaluation of mechanical properties at intermediate strain rates, split Hopkinson pressure bar (SHPB) or Kolsky apparatus is widely used for the study of mechanical behaviour of materials at high strain rates. SHPB apparatus works based on one-dimensional wave propagation theory in elastic bars. Historically, John Hopkinson originated the concept of testing materials under dynamic loads in 1872 with stress wave experiments in iron wires

2 ARTICLE IN PRESS 55 [1,2], followed by the experiments by his son Bertram Hopkinson [3]. The first use of a long thin bar to measure the pulse shape induced by an impact is considered due to Bertram Hopkinson [4]. This study was presented in Although Bertram Hopkinson s observations were qualitative, he was able to correctly establish that the velocity of impact was the most important cause of failure. An important modification was made by Kolsky in 1949 [5]. He presented the concept of split bars to determine dynamic compressive stress strain behaviour of different materials and for one-dimensional pressure bar data analysis. He also presented a complete experimental procedure. The SHPB is sometimes called the Kolsky bar. After Kolsky [5] introduced the SHPB technique for dynamic testing of specimens, SHPB has become a widely used experimental technique to test the materials at high strain rates. The details regarding SHPB technique are presented in Refs. [6 8]. The tensile SHPB was originally introduced by Harding et al. [9] and later by Lindholm et al. [1] and Nicholas [11]. Since then, there are typical studies on tensile high strain rate behaviour of different metals and alloys. There are also typical studies on high strain rate behaviour of polymeric materials under tensile loading [12 18]. Summary of the observations made on mechanical behaviour in these papers is presented in Table 1. The studies are presented up to a strain rate of 4 s 1. Even though the polymeric materials used are different for different studies and the strain rate used is different, the consolidated information given in Table 1 gives an overview regarding the behaviour of polymeric materials under high strain rate tensile loading. It can be generally observed that the tensile strength increases at high strain rate loading compared with that at quasi-static loading. Although there are studies on the tensile behaviour of several polymeric materials under high strain rate loading, investigations do not appear to be available on acrylic. The dynamic behaviour of acrylic should be fully understood for its effective use. The objective of the present investigation is to determine high strain rate behaviour of acrylic under tensile loading. Tensile strength, Young s modulus and ultimate strain are evaluated. Experimental studies were carried out on tensile SHPB apparatus. Force versus time plots based on strain gauge signals obtained from incident bar and transmitter bar were derived and compared. Further, an analytical method is presented for the Table 1 High strain rate mechanical behaviour of polymeric materials: tensile loading (from literature) Material Strain rate, e (s 1 ) High strain rate properties Reference s ult (MPa) E (GPa) e ult (%) PS-PPE blends [12] CARILON [13] 828/T [14] PMMA Epoxy E [15] Poly Propylene [16] PP/POE (9:1) PP/POE (7:3) PP/POE (65:35) TPO 4 32 [18] HDPE 4 72 PC/ABS 4 78 PP/glass 4 24 PS PPE: Polystyrene poly phenylene ether; PP/POE: polypropylene/octene ethylene; HDPE: high density poly ethylene; TPO: thermoplastic elastomer olefinic; PC/ACB: polycarbonate/acrylonitrile butadiene styrene terpolymer; PMMA: poly methyl methacrylate.

3 56 ARTICLE IN PRESS prediction of high strain rate strength of acrylic under tensile loading. The method is based on a variable rate power law. 2. Tensile split Hopkinson pressure bar apparatus 2.1. Experimental technique and data acquisition Different configurations of tensile SHPB apparatus have been used over the years [6 8,19]. Schematic arrangement of the tensile SHPB apparatus and a photograph of the apparatus used for the present study are given in Fig. 1. The main parts of the apparatus are: incident bar, transmitter bar, high-pressure propelling mechanism, specimen holders and the support stand. The high-pressure propelling mechanism consists of a cylinder, piston, hollow shaft, springs and the front disc. As the highpressure air supply is provided, the piston along with the front disc is accelerated instantaneously. As the front disc of the propelling mechanism impacts on to the disc mounted on the incident bar, an elastic stress pulse is generated and travels along the incident bar. When the pulse reaches the specimen, which is sandwiched between the incident and transmitter bars, part of the stress pulse is reflected and the remaining part is transmitted through the specimen to the transmitter bar. The strain gauges mounted at the centres of incident and transmitter bars provide the time resolved measure of the signals. The strain gauge mounted on the incident bar measures incident and reflected pulses whereas the strain gauge mounted on the transmitter bar measures the transmitted pulse. The strain gauges are installed mid-way on the incident and transmitter bars to avoid overlapping of the signals. During loading, the specimen undergoes dynamic elastic plastic deformation. From the reflected pulse, the strain rate applied and the strain in the specimen are estimated, whilst the transmitted pulse provides a measure of the stress. The reflection of the pulse from the interface of the bar and specimen is due to the impedance mismatch between the bar and the specimen [2]. The diameter of the incident and transmitter bars is 16 mm and the length is 2 mm.the bars are made of SUS44C martensite stainless steel with Young s modulus of 23 GPa and density of 7667 kg m 3. The entire strain/deformation history within the specimen is obtained by taking measurements along Fig. 1. Split Hopkinson pressure bar (SHPB) apparatus tensile loading: (a) schematic arrangement and (b) photograph.

4 ARTICLE IN PRESS 57 the incident and transmitter bars from the strain gauges with the help of amplifier and oscilloscope. From these signals and using one-dimensional wave propagation theory, strain rate versus time, strain versus time, stress versus time, and stress versus strain plots in the specimen are obtained. Further, force history at the interface between the specimen and the incident bar (F 1 ) and the specimen and the transmitter bar (F 2 ) is obtained Specimen configuration Arrangement of specimens and holders is presented in Fig. 2. For the experimental studies, endthreaded cylindrical specimens were used. Fig. 2a presents a calibration specimen with holders attached to the incident and the transmitter bars. A typical end-threaded cylindrical specimen is shown in Fig. 2b. The gauge length is 12 mm and the diameter is 5.5 mm. Right-handed and lefthanded threads were provided on either side of the specimens for easy mounting of the specimen onto the holder. Fig. 2c presents an end-threaded cylindrical specimen held in the holders. 3. Theory The design of SHPB is based on one-dimensional wave propagation in elastic bars, which deals with the motion of particles in the longitudinal direction. The one-dimensional system can ideally be considered to be of infinite length and negligible diameter. Since this is not possible in practise, the theory is adopted with certain approximations. The analytical relations to calculate strain rate, strain and stress as a function of time in the specimen in SHPB testing are: [7,21] Strain rate; SðtÞ ¼ð2C =l S Þ R ðtþ, (1) Average strain; S ðtþ ¼ð2C =l S Þ Z t R ðtþ dt, (2) Stress; s S ðtþ ¼E A B T ðtþ, (3) A S where C is elastic wave velocity in the bars, l S is specimen gauge length, A B is cross-sectional area of the bars, A S is cross-sectional area of the specimen, E is Young s modulus of the bars, e R is reflected strain pulse, e T is transmitted strain pulse and t is time duration. 4. Calibration of tensile SHPB apparatus For commissioning and assessing the accuracy of SHPB apparatus, calibration was first carried out. During calibration, incident and transmitter bars were first joined together using an end-threaded cylindrical calibration specimen made of the same material as that of the bars. For the next set of calibration experiments, an end-threaded cylindrical calibration specimen was mounted onto a set of Calibration specimen, 2. Specimen, 3. Holder, 4. Incident bar, Transmitter bar Fig. 2. Specimens and holders: (a) end-threaded cylindrical calibration specimen with holders, (b) end-threaded cylindrical specimen (dimensions in mm) and (c) end-threaded cylindrical specimen with holders.

5 58 ARTICLE IN PRESS threaded holders. The incident and transmitter bars were joined together using the set of threaded holders with end-threaded cylindrical calibration specimen (Fig. 2a). The diameter of the holders and calibration specimen was equal to those of the incident and transmitter bars. The set of holders was made using the same material as that of the incident and transmitter bars. With this arrangement, incident and transmitter bars along with the set of holders and the end-threaded cylindrical calibration specimen can be treated as a single continuous bar. Strain gauge signals on the oscilloscope during calibration with the set of holders and calibration specimen are presented in Fig. 3. Channel 1 indicates the output of the strain gauge mounted on the incident bar whereas channel 2 indicates the output of the strain gauge mounted on the transmitter bar. Here, I is the incident pulse with pulse duration equal to a 1 a 2 whereas T is the transmitted pulse with pulse duration equal to b 1 b 2. During calibration, a reflected pulse (R) is not present. The amplitude and duration of incident and transmitted pulses are nearly the same. The strain gauge signals obtained either with the calibration specimen mounted on to the set of holders and then the set of holders mounted on to the elastic bars, or with calibration specimen mounted onto the elastic bars directly were nearly identical. Similar observation was made by Chocron Benoulo et al. [22]. Force versus time plots were obtained from the strain gauge signals and are presented in Fig. 4. The force history obtained based on the strain gauge mounted on the incident bar is indicated as F 1 whereas the force history obtained based on the strain gauge mounted on the transmitter bar is indicated as F 2. It may be noted that the forces F 1 and F 2 match very well. This indicates that the stress states within the incident bar and transmitter bar are exactly the same. This ensures that the tensile SHPB apparatus is perfectly aligned and friction free. The apparatus is then ready for further investigations. 5. Experimental studies: acrylic Studies were carried out in the strain rate range of 2 33 s 1. Studies were also carried out at quasistatic loading for comparison. Strain rates used and tensile properties are given in Table 2 and Figs Strain gauge signals obtained on an oscilloscope during testing are presented in Fig. 5. The durations of incident and reflected signals are represented by a 1 a 2 and a 3 a 4, respectively. In the present case, a 1 a 2 ¼ 452 ms. It may be noted that a 1 a 2 and a 3 a 4 are nearly the same. Force versus time plots are presented in Fig. 6. Force history on the incident bar is plotted based on strain gauge signal I+R, Force, F (KN) F1 F2 F1 F I Fig. 4. Comparison of force versus time behaviour, derived from strain gauge signals obtained during calibration. 1-> a 1 a 2 volts 2-> T b 1 b 2 Table 2 Strain rate effect on tensile properties of acrylic (present experimental study) Strain rate, e (s 1 ) Strength, s ult (MPa) Young s modulus, E (GPa) Ultimate strain, e ult (%) Time Fig. 3. Strain gauge signals on oscilloscope during calibration on SHPB-tensile loading. Quasi-static (4, 5) (6, 3) (3, 2)

6 ARTICLE IN PRESS 59 Volts 1-> a 1 I a 2 a 3 a 4 R C > Time Fig. 5. Strain gauge signals from high strain rate tensile test on SHPB, d s ¼ 5.5 mm, l s ¼ 12 mm; incident, reflected and transmitted signals, acrylic. T Strain, ε (%) Force, F (KN) F1 F Fig. 6. Comparison of force versus time behaviour, derived from strain gauge signals, d s ¼ 5.5 mm, l s ¼ 12 mm, acrylic. whereas force history on the transmitter bar is plotted based on strain gauge signal T. The force history obtained based on signal I+R is referred to as F 1 and the force history obtained based on signal T is referred to as F 2 for further discussion. Force F 1 would be acting on the interface between the incident bar and the specimen, whereas force F 2 would be acting on the interface between the transmitter bar and the specimen. It may be noted that the forces F 1 and F 2 are not matching (Fig. 6). The magnitude of peak force F 2 is significantly lower than the magnitude of peak force F 1. As the stress wave propagates and encounters a boundary between two materials, it would partly transmit and partly reflect. The boundary can be either because of impedance mismatch or area mismatch. Because of transmission and reflection of the incident stress wave at the boundaries, attenuation of the incident stress wave would take place. For the end-threaded cylindrical specimens used for the investigations, the minor diameter at F1 F2 Stress, σ (MPa) 7 5 A O Fig. 7. High strain rate tensile test results for acrylic, d s ¼ 5.5 mm, l s ¼ 12 mm: (a) time versus strain rate plot, (b) time versus strain plot and (c) time versus stress plot. Stress, σ (MPa) o A Strain, ε (%) Fig. 8. Stress versus strain plot from high strain rate tensile test on SHPB for acrylic, d s ¼ 5.5 mm, l s ¼ 12 mm, e ¼ 33 s 1. Tensile strength, X t (MPa) Fig. 9. Tensile strength versus strain rate plot for acrylic, d s ¼ 5.5 mm, l s ¼ 12 mm.

7 51 ARTICLE IN PRESS the threaded portion was more than the diameter of the specimen gauge section. This would lead to area mismatch at either side of the gauge length of the specimen. This would lead to the attenuation of the stress wave as it passes the specimen. Further, complete equilibrium within the specimen may not be possible once flow stress state is reached. Dynamic stress equilibrium cannot be achieved during damage initiation and propagation stages. It may be noted that the peak force is attained after a time of 133 ms in the case of F 1 and 98 ms in the case F 2. During this period, 61 transits take place one transit equals the time required for a pulse to travel from one end of the specimen to the other end. Since F 1 and F 2 are not equal, the specimen would not be under uniform stress during the loading process. The tensile properties reported further are based on F 2. This is to obtain a conservative estimate of tensile properties. Time versus strain rate, strain and stress plots are given in Fig. 7. Here, point C indicates the first peak strain rate. A typical true stress true strain plot is presented in Fig. 8, which can be subdivided into two regions. Region one represents the behaviour of the material until the tensile strength is reached. Region two represents the post-failure behaviour. In the case of acrylic, necking was not observed. Point A indicates the ultimate tensile failure. The specimen separated into two pieces during loading at a strain as indicated by point A. Hence, the plot presented in Fig. 8 beyond point A does not represent the behaviour of acrylic. The strain corresponding to point A is 2.1%. For acrylic, ultimate strain presented in Table 2 is with respect to the ultimate tensile strength. Figs. 5 and 7 show a rise time of 11 ms. During this period, which is the initial stage of loading, the strain rate is not constant. Hence, Young s modulus obtained based on strain gauge data during this period would not be exact. As a first approximation, Young s modulus is presented based on the initial linear portion of the stress strain curve. Tensile properties at different strain rates are given in Table 2 and Fig. 9. It can be observed that the tensile strength is enhanced at high strain rate loading compared with that at quasi-static loading. Further, in the strain rate range, there is an increase in the tensile strength as the strain rate is increased. Enhancement in Young s modulus at high strain rate tensile loading is marginal, but there is a significant decrease in the ultimate tensile strain at high strain rate loading compared with that at quasi-static loading. For comparison, tensile properties at quasi-static loading are presented in Table 2. A property enhancement factor is obtained as the ratio of the tensile strength at high strain rate loading to the tensile strength at quasi-static loading. It is found to vary from 1.14 to 1.73, corresponding to strain rate varying from 2 to 33 s Analytical studies Tensile stress strain behaviour for acrylic at a strain rate of 33 s 1 is presented in Fig. 8. The region of the stress strain curve up to point A represents the actual behaviour of the material. It can be seen that this region is not linear. The stress strain behaviour can be represented based on strain rate dependent nonlinear parameters in the form of a variable rate power law as follows: s ¼ k 1 ðþ k 2. (4) Here, k 1 and k 2 are strain rate dependant nonlinear parameters. A power law was used in Refs. [23,24] for the characterisation of high strain rate behaviour of materials. Strain rate dependent nonlinear parameters k 1 and k 2 are presented as a function of strain rate in Fig. 1. It can be noted that k 1 and k 1 are nonlinear parameters. Hence, k 1 and k 2 are further represented in the form of a power law as follows: k 1 ¼ k 3 ð Þ k 4, (5) k 2 ¼ k 5 ð Þ k 6. (6) Here, k 3, k 4, k 5 and k 6 are material properties. For acrylic, k 3 ¼ 13.48, k 4 ¼.162, k 5 ¼.599 and k 6 ¼.19. For acrylic, tensile stress strain relation as a function of strain rate can be represented as follows: s ¼ 13:48 ð_þ :162 ðþ ð:599ð_þ :19Þ. (7) 6.1. Prediction of tensile stress strain behaviour at high strain rate Using Eq. (7), tensile stress strain behaviour can be obtained at different strain rates for acrylic. From the stress strain relation, local modulus and rate of change of local modulus can be obtained as a function of strain. Fig. 11 presents rate of change of

8 511 k 1 (MPa) k 1 = (έ).162 tensile strengths at different strain rates. There is a good match between the predicted and experimental tensile strength values (Table 3). Overall, as the strain rate increases, the tensile strength increases although the rate of increase decreases Conclusions High strain rate behaviour under tensile loading was studied for acrylic. General observations are: k k 2 =.599 (έ) Fig. 1. Strain rate dependent nonlinear parameters for acrylic, tensile loading: (a) k 1 versus e and (b) k 2 versus e. Rate of change of modulus έ = 5 per sec έ = 1 per sec Strain, ε (%) Fig. 11. Rate of change of local modulus as a function of strain for acrylic, tensile loading. local modulus as a function of strain for different strain rates. In the range of strain from 2.5% to 3.5%, the rate of change of local modulus is a very small quantity. This indicates that the ultimate tensile strength is reached. It would be a fair approximation to assume that the ultimate tensile strain is in the region of %. Rate of change of local modulus as a function of strain is presented in Fig. 11 for strain rates of 5 and 1 s 1. The curves almost merge with each other. This indicates that the range of ultimate tensile strain can be taken as % for different strain rates. Using Eq. (7), tensile stress strain plots are predicted at various strain rates and are presented in Fig. 12. Fig. 13 and Table 3 present the predicted The peak force obtained based on a strain gauge mounted on the transmitter bar is lower than the peak force obtained based on a strain gauge mounted on the incident bar. There is an increase in tensile strength and Young s modulus at high strain rate loading compared with those at quasi-static loading, although the increase in Young s modulus is marginal. There is a decrease in ultimate tensile strain at high strain rate loading compared with that at quasi-static loading. Stress, σ (MPa) per sec 1 per sec 15 per sec 2 per sec 3 per sec strain, ε (%) Fig. 12. Predicted stress versus strain plots under high strain rate tensile loading for acrylic. Tensile strength, X t (MPa) ε = 3.5 % ε = 2.5 % Fig. 13. Predicted tensile strength versus strain rate plots for acrylic.

9 512 ARTICLE IN PRESS Table 3 Prediction of tensile strength at high strain rates, acrylic Strain rate, e (per s) Predicted strength, X t (MPa), at strain, e (%) Experimental strength, X t (MPa) (%) at e ¼ at e ¼ at e ¼ at e ¼ at e ¼ at e ¼ There is a good match between the predicted and experimental tensile strengths. Acknowledgement This work was partially supported by the Structures Panel, Aeronautics Research and Development Board, Ministry of Defence, Government of India, Grant no. DARO/8/15124/M/I. References [1] J. Hopkinson, in: B. Hopkinson (Ed.), On the Rupture of Iron Wire by a Blow (1872), Article 38, Original Papers by the Late John Hopkinson, Vol. II, Scientific Papers, Cambridge University Press, 191, pp [2] J. Hopkinson, in: B. Hopkinson (Ed.), Further Experiments on the Rupture of Iron Wire by a Blow (1872), Article 39, Original Papers by the Late John Hopkinson, Vol. II, Scientific Papers, Cambridge University Press, 191, pp [3] B. Hopkinson, The effects of momentary stress in metals, Proc. Phys. Soc. London A74 (194/195) [4] B. Hopkinson, A method of measuring the pressure produced in the detonation of high explosives by the impact of bullets, Philos. Trans. R. Soc. London A213 (1914) [5] H. Kolsky, An investigation of the mechanical properties of materials at very high rates of loading, Proc. Phys. Soc. London B62 (1949) [6] R.L. Sierakowski, S.K. Chaturvedi, Dynamic Loading and Characterization of Fiber-Reinforced Composites, Wiley- Interscience Publication, New York, 1997, pp [7] H. Kuhn, D. Medlinm (Eds.), Mechanical Testing and Evaluation. ASM Hand Book, vol. 8, ASM International, Materials Park, OH, 2, pp [8] B.A. Gama, S.L. Lopatnikov, J.W. Gillespie Jr., Hopkinson bar experimental technique, a critical review, Appl. Mech. Rev. 57 (24) [9] J. Harding, E.O. Wood, J.D. Campbell, Tensile testing of materials at impact rates of strain, J. Mech. Eng. Sci. 2 (196) [1] U.S. Lindholm, R.L. Bessey, G.V. Smith, Effect of strain rate on yield strength, tensile strength, and elongation of three aluminium alloys, J. Mater. 6 (1971) [11] T. Nicholas, Tensile testing of materials at high rates of strain, Exp. Mech. 21 (1981) [12] M.C.M.V. Sanden, H.E.H. Meijer, Deformation and toughness of polymeric systems: 4. Influence of strain rate and temperature, Polym. 35 (1994) [13] L.E. Govaert, P.J.D. Vries, P.J. Fennis, W.F. Nijenhuis, J.P. Keustermans, Influence of strain rate, temperature and humidity on the tensile yield behaviour of aliphatic polyketone, Polym. 41 (2) [14] W. Chen, F. Lu, M. Cheng, Tension and compression tests of two polymers under quasi-static and dynamic loading, Polym. Test. 21 (22) [15] A. Gilat, R.K. Goldberg, G.D. Roberts, High strain rate response of epoxy in tensile and shear loading, J. Phys. IV 11 (23) [16] J. Yang, Y. Zhang, Y. Zhang, Brittel-ductile transition of PP/POE blends in both impact and high-speed tensile tests, Polym. 44 (23) [17] A. Gilat, R.K. Goldberg, G.D. Roberts, Strain rate sensitivity of epoxy resin in tensile and shear loading, NASA/TM [18] X. Xiao, Dynamic tensile testing of plastic materials, Polym. Test. 27 (28) [19] G.H. Staab, A. Gilat, A direct-tension split Hopkinson bar for high strain rate testing, Exp. Mech. 31 (1991) [2] M.A. Meyers, Dynamic Behavior of Materials, Wiley Publications, New York, 1994, pp [21] S. Ellwood, L.J. Griffiths, D.J. Parry, A tensile technique for materials testing at high strain rates, J. Phys. 15 (1982) [22] I.S. Chocron Benloulo, J. Rodriguez, M.A. Martinez, V. Sanchez Galvez, Dynamic tensile testing of aramid and polyethylene fiber composites, Int. J. Impact Eng. 19 (1997) [23] C.A. Weeks, C.T. Sun, Modeling non-linear rate-dependent behaviour in fibre-reinforced composites, Compos. Sci. Technol. 58 (1998) [24] J. Tsai, C.T. Sun, Constitutive model for high strain rate response of polymeric composites, Compos. Sci. Technol. 62 (22)