Shear properties of epoxy under high strain rate loading.
Polymer matrix composites are finding increasing uses in high performance applications. During their service life, composite structures undergo different loading conditions. High velocity impact loading is one of the critical loading conditions. Typical examples are: collision, crash landing, and rigid body impact on to a structure. The overall behavior of composites is governed by the polymer matrix along with the reinforcing material. Interlaminar properties of the composites are significantly lower than the inplane properties along the fiber direction. Inter-laminar properties are mainly governed by the properties of polymer matrix. The mechanical properties of composites and polymer matrices under quasi-static loading conditions are well documented. Studies on shear behavior of polymer matrices under high strain rate loading are limited.
There are different test methods used for the evaluation of mechanical properties based primarily on strain rate required (1). Torsional Split Hopkinson Bar (TSHB) testing is the most widely used technique to determine shear properties of materials under high strain rate loading. Split Hopkinson Pressure Bar (SHPB) testing is also used with single-lap and double-lap shear specimens. Drop weight impact testing technique is used for shear characterization of materials under intermediate strain rates.
Sayers and Harris (2) used drop weight impact test technique for determining the interlaminar shear properties of carbon/epoxy composites. The specimen used was similar to the standard interlaminar shear strength (ILSS) test specimen. During testing, the authors observed other failure modes including bending. Further, they reported that the ILSS decreases with increasing strain rate. Werner and Dharan (3) carried out interlaminar shear tests on woven graphite/epoxy using compressive SHPB apparatus. They used short-beam shear specimen, and they observed considerable scatter in the data. Even though there was a slight decrease in ILSS at high strain rate compared with that at low strain rate, they concluded that the interlaminar shear stress response is relatively constant for all the strain rates.
Harding and Li (4) and Harding and Dong (5) used double-lap specimens made of plain weave carbon/epoxy and plain weave glass/epoxy composites. They observed an increase in the ILSS with increasing strain rate in both the materials. However, based on finite element analysis of the specimens, they reported that the shear stress distribution along the overlap zone was not uniform. They also observed a large shear and translaminar normal stress concentrations at both the ends of the failure plane. They concluded that the stress concentration is the main reason for the initiation of failure.
Single-lap specimens were used for determining the interlaminar shear properties of composites (6-9). The major advantage of single-lap specimen is that the central interlaminar plane is on the axis of the externally applied loading. This leads to a very low normal stress in the plane of shear failure. Also this leads to almost constant shear stress along the shear failure plane. However, there can be stress concentration at the interface between the specimen and the loading bars on to which the specimen is attached. This is because of elastic property mismatch between the specimen and the loading bars. This possible stress concentration can lead to premature failure in the adhesive bond between the specimen and the loading bars even before the shear failure of the specimen.
Torsional tests are the most accurate test techniques for the evaluation of shear properties. Typical studies based on the torsional tests are presented in Refs. (10-12), In these studies, short, circular thin-walled tubular specimens were used. These studies were on metallic specimens. Modified geometry of the specimen was used for interlaminar shear properties of composites by Leber and Lifshitz (13). The studies were carried out on TSHB apparatus. They used thin-walled tubular specimens with gauge length of 8-10 mm, total length of 20 mm, internal diameter of 14.5 mm, and external diameter of 22 mm. Three different specimen wall thicknesses of 1, 1.5, and 2 mm were used. The material used for the study was plain weave E-glass/epoxy. From the numerical analysis of the fractured surfaces, they concluded that fracture in both static and dynamic tests was approximately at the mid plane of the specimen. The studies were carried out at strain rate in the range of 300-500 per sec. They observed that the stress-strain behavior was nonlinear. Their study indicated that the dynamic fracture stress increased by about 50%, the initial shear modulus increased by about 60%, and the failure strain increased by about 25%. Further, they reported that stress uniformity along the axis of the specimen cannot be taken granted. This is because the specimen was not compact, i.e., was having a larger aspect ratio. They also studied the response of fiber reinforced polymers to high strain rate loading in combined tension and shear (14). The studies were carried out on SHPB apparatus. Gillespie et al. (15) studied ILSS of composites made using plain weave S-glass subjected to out of plain high strain rate compressive loading. The studies were carried out on SHPB apparatus.
Dai et al. (16), (17) used a specimen with gauge length and wall thickness in the range of 1.9-2.1 and 0.5-0.7 mm, respectively. They used TSHB apparatus for their studies. Their studies were on unidirectional [C.sub.f]/A356.0 metal matrix composite and A356.0 aluminum alloy. The fibers were aligned with the direction of torsional axis. They observed multi-stage failure process and a multi-scale zigzag fracture at all the surfaces. Their studies indicated that the carbon fibers did not improve the shear strength of aluminum matrix.
High strain rate behavior of polycarbonate and poly-methyl methacrylate was studied by Fleck et al. (18) using TSHB apparatus. The yield and fracture behaviors were investigated. Jelf and Fleck (19) studied behavior of unidirectional carbon-fiber epoxy laminates under combined compression and shear loading. They observed that failure was by plastic microbuckling. Hu and Feng (20) studied the transient large strain response of polymer melts at high shear rates. They studied the effect of temperature in the range of 150-210[degrees]C. The studies were carried out on Kolsky torsion bar.
Gilat et al. (21-23) studied high strain response of two types of epoxy under shear loading. They used resins E 862 and PR 520. Tubular specimens of gauge length 2.54 mm were used. Thickness of the wall was 0.63 mm and 1.27 mm. The studies were carried out on TSHB in the shear strain rate range of 400-700 per sec. For comparison, studies were carried out at quasi-static strain rate also. It was observed that there was significant enhancement of shear strength at high strain rate loading when compared with that at quasi-static loading. The ultimate strain decreased at high strain rate loading compared with that at quasi-static loading.
There are only limited studies on high strain rate behavior of resins under shear loading. Epoxy resin is one of the widely used matrices along with high performance carbon fibers for high technology applications.
The objective of this investigation is to study the effect of strain rate on epoxy resin LY 556 with hardener HY 951 under shear loading using TSHB. Studies are carried out in the shear strain rate range of 385-880 per sec. The shear behavior is characterized using one-dimensional wave propagation theory in elastic bars.
The stress-strain plots under high strain rate shear loading for the specimens can be obtained by the analysis of shear stress waves in the elastic bars placed on either side of the specimen. The portion of the incident wave that is transmitted through the specimen provides a measure of the shear stress in the specimen, whereas the remaining portion of the incident wave that is reflected provides a measure of strain rate and strain (12), (24).
The shear strain rate in the specimen is given by
[[gamma].sub.S](t) = [2C[D.sub.S]/[L.sub.S]D][[gamma].sub.R](t) (1)
The shear strain in the specimen is given by
[[gamma].sub.S](t) = [2C[D.sub.S]/[L.sub.S]D] [t.integral 0] [[gamma].sub.R](t) dt (2)
The shear stress in the specimen is give by
[[tau].sub.S](t) = [G[D.sup.3]/[8[D.sub.S.sup.2][t.sub.S]]] [[gamma].sub.T](t) (3)
where C is torsional wave velocity in the bars, [D.sub.s] is diameter of the bars, [D.sub.s] is mean diameter of the specimen, [L.sub.s] is specimen gauge length, [t.sub.s] is tubular specimen wall thickness, G is shear modulus of the bars, [[gamma].sub.T] is transmitted shear strain pulse, and [[gamma].sub.R] is reflected shear strain pulse.
The photograph of the TSHB apparatus used for the studies is presented in Fig. 1. The construction and working of TSHB is presented in Appendix A.
[FIGURE 1 OMITTED]
Schematic arrangement of a short, circular thin-walled tubular specimen used for this study is given in Fig. 2a. Specimens with wall thickness ([t.sub.s]) of 2 mm and gauge length ([L.sub.s]) of 3.5 mm were used. Other dimensions used were: [D.sub.i] = 10 mm and [D.sub.o] = 22 mm. The overall length of the specimen was 7.7 mm. Specimens were made from epoxy LY 556 with hardener HY 951. The gauge length used in this study is in between the gauge lengths used by Leber and Lifshitz (13) and Dai et al. (16).
[FIGURE 2 OMITTED]
Thin-walled tubular specimens with configuration as shown in Fig. 2a were machined using a standard carbide cutting tool. It was observed that the surfaces of all the specimens were free from damages. For bonding the specimens to the incident and transmitter bars, room temperature curing araldite adhesive was used. Figure 2b presents the photograph of a fractured specimen. The fractured surface is indicated by an arrow.
In this study, the shear strain rate was varied in the range of 385-880 per sec. Details regarding control of shear strain rate and pulse duration during testing are given in Appendix B.
Calibration of TSHB apparatus was carried out by bonding together the incident and transmitter bars. With this the incident and transmitter bars together can be treated as a single bar. For bonding the incident and transmitter bars together, room temperature curing araldite adhesive was used. Strain gauge signals on oscilloscope during calibration are presented in Fig. 3. Channel 1 shows 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.sub.1][a.sub.2]. whereas T is the transmitted pulse with pulse duration equal to [b.sub.1][b.sub.2]. During calibration, reflected pulse (R) is not present. The magnitudes and durations of incident and transmitted pulses are nearly the same.
[FIGURE 3 OMITTED]
The transmitted pulse (T) proceeds further and reaches to the free end of the transmitter bar. After reflection, this pulse travels back in the transmitter bar. As this pulse reaches to the strain gauge mounted on the transmitter bar, the pulse is captured as indicated by [b.sub.3][b.sub.4]. As the pulse further proceeds, the strain gauge mounted on the incident bar captures the pulse as indicated by [a.sub.3][a.sub.4].
The pulse durations [a.sub.1][a.sub.2], [b.sub.1][b.sub.2], [b.sub.3][b.sub.4], and [a.sub.3][a.sub.4] are nearly equal (550 [micro] sec). During calibration, the distance between the clamp and the rotary actuator and the distances between the strain gauges and interface between the bars were maintained the same. Hence, [a.sub.2] and [b.sub.1] correspond to the same time interval. It may be noted that [b.sub.1][b.sub.3] is more than [b.sub.1][b.sub.2]. This is because the distance between the strain gauge on the transmitter bar and the interface between the bars is less than the distance between the strain gauge and the free end of the transmitter bar. Further, it may be noted that [a.sub.3] and [b.sub.4] correspond to the same time interval.
Figure 4 presents comparison of torque versus time behavior derived from strain gauge signals during calibration on TSHB apparatus. Here, [T.sub.1] represents torque at the interface between the bars and calculated based on incident strain gauge signal I, whereas [T.sub.2] represents torque at the interface between the bars and calculated based on transmitted strain gauge signal T. From the figure, it can be seen that torque versus time plots based on [T.sub.1] and [T.sub.2] are matching well. This indicates that the TSHB apparatus is perfectly aligned and friction free and ready for use for further experimentation. The torque induced is calculated as
[FIGURE 4 OMITTED]
[T.sub.1] = [GJ/r]([[gamma].sub.I] + [[gamma].sub.R]) (4)
[T.sub.2] = [GJ/r] ([[gamma].sub.T]) (5)
Here, G is shear modulus of the bars, J is polar moment of inertia of the bars, r is radius of the bars, [[gamma].sub.1] is incident shear strain pulse, [[gamma].sub.T] is transmitted shear strain pulse, and [[gamma].sub.R] is reflected shear strain pulse. During calibration, [[gamma].sub.R] = 0.
A typical oscilloscope plot for epoxy LY 556 under high strain rate torsional loading is shown in Fig. 5. Channel 1 shows 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. The plot shows incident (I), reflected (R), and transmitted (T) pulses. Point P indicates the end of rise time. Using the shear wave pulses and the expressions given in theory section, shear strain rate, shear strain, and shear stress are derived as a function of time. From this, shear stress-shear strain plots arc obtained.
[FIGURE 5 OMITTED]
As the stored torque is released instantaneously by breaking the notched clamping bolt, the part of the incident bar between the clamp and the rotary actuator, where the torque was stored gets unloaded whereas the remaining part of the incident bar, specimen, and the transmitter bar get loaded.
RESULTS AND DISCUSSION
A typical specimen fractured during TSHB testing is shown in Fig. 2b. The fractured surface is indicated by an arrow. It was observed that the specimen fractured at the mid-span of the gauge length. The fracture was confined to a small region on a plain parallel to the faces of the incident and transmitter bars. This indicated that the failure was under pure shear. Buckling of the thin-walled tubular specimen was not observed.
Figure 5 presents a typical oscilloscope plot for epoxy LY 556 with hardener HY 951 under high strain rate torsional loading. The duration of incident pulse, [a.sub.1][a.sub.2] = 510 [micro] sec. Here, [a.sub.1][a.sub.2] and [a.sub.3][a.sub.4] are nearly the same. Figure 6 presents comparison of torque versus time behavior derived from strain gauge signals obtained during testing. Peak torque [T.sub.1] is more than peak torque [T.sub.2]. From Fig. 6, it can be observed that peak torque [T.sub.1] is attained at a time interval of 135 [micro] sec. During this period, micro damage initiation and propagation and macro damage formation, propagation and fracture of the specimen take place. During the evolution of damage at the fracture surface, the shear stress wave that is transiting within the specimen would encounter the damage being formed. This would lead to incident pulse transmission and reflection at the fracture surface. Further, this would lead to stress wave attenuation. The stress at the interface between the incident bar and the specimen would be more than that at the interface between the transmitter bar and the specimen. In other words, the magnitude of I + R would be more than the magnitude of T. Hence [T.sub.1] is more than [T.sub.2].
[FIGURE 6 OMITTED]
High strain rate shear test results are presented in Figs. 7 and 8 and Table 1. The results presented in Table 1 are the averages of five test results. Time versus shear strain rate plot, time versus shear strain plot, and time versus shear stress plot are generated using the equations given in theory section and the strain gauge signals (Fig. 5) and presented in Fig. 7. Here, point P indicates end of rise time and point A represents peak shear stress. For this case, shear strain rate is 500 per sec, shear strength is 52 MPa, and the corresponding shear strain is 4.58%. The total duration of the incident pulse is about 510 [micro] sec, and the rise time is about 55 [micro] sec. During the initial stage of loading, the strain rate increases. Then, the strain rate marginally decreases. Further, it increases to the peak value. The fracture of the specimen takes place at time duration of 152 [micro] sec.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
TABLE 1. Shear properties of epoxy LY 556 under high strain rate loading. Shear strain Shear strength Ultimate shear Shear modulus rate (per sec) (MPa) strain (%) (GPa) Quasi-static 37(+1.2, -2.6) 5.2(+0.05, -0.37) 1.8(+0.93, -0.54) 385 (+40, -10) 50(+4.1, -3.3) 4.67(+0.12, -0.05) 2.1(+0.22, -0.75) 500 (+20, -17) 52.0(+3.1, -4.2) 4.58(+0.18, -0.22) 2.1(+0.21, -0.45) 880 (+13, -21) 53.5(+4.8. -3.2) 4.68(+0.15, -0.34) 2.2(-0.29, -0.55) Thin-walled tubular shear specimen: wall thickness. [t.sub.s] = 2 mm: gauge length. [L.sub.s] = 3.5 mm.
Time versus shear stress plot (Fig. 7c) can be subdivided into region I and region II. Region I represents the behavior of the material until the shear strength is reached. Region II represents post failure behavior of the material. The shear stress-shear strain plot, as given in Fig. 8, is generated using shear strain versus time and shear stress versus time plots, as given in Fig. 7.
The rise time is about 55 [micro] sec. During this period, which is the initial stage of loading, the strain rate is changing. Hence, the shear modulus obtained from the test would not be exact. The end of rise time is represented by the point P. The shear stress versus shear strain plot, as given in Fig. 8, would represent the actual behavior of the material beyond point P. Shear modulus can be determined, as a first approximation, by joining point P to the origin. The shear modulus obtained this way would indicate the lower bound. The shear modulus can also be found by extrapolating the shear stress-shear strain curve in region I from point P to the origin (Fig. 8). The shear modulus obtained this way can be taken as the actual shear modulus.
The shear properties at quasi-static loading were obtained for epoxy LY 556 using short circular thin-walled tubular specimens. The specimens were tested on Tinius Olsen Torsion Testing Machine. The shear stress-shear strain plot obtained is presented in Fig. 9.
[FIGURE 9 OMITTED]
The shear properties of epoxy LY 556 under different strain rates are presented in Table 1. It can be observed that the shear strength is enhanced significantly at high strain rate loading compared with quasi-static loading. Shear modulus is also increased at high strain rate loading compared with quasi-static loading. Ultimate shear strain is less at high strain rate loading compared with that at quasi-static loading. The shear strength enhancement at shear strain rate of 880 per sec is 45% compared with that at quasi-static loading, whereas the shear modulus is enhanced by 22%. In the range of parameters considered, the change in shear properties with the change in shear strain rate is not significant.
The qualitative behavior of epoxies E 862 and PR 520 studied by Gilat et al. (21-23) is similar to the observations of this study for epoxy LY556 under high strain rate shear loading. The qualitative behavior obtained by this study is also similar to the qualitative behavior obtained for plain weave E-glass/epoxy and plain weave carbon/epoxy (24).
Generally, the shear strength is higher at high strain rate compared with that at quasi-static strain rate. This can be attributed to the fact that at lower strain rates, the damage propagates slowly utilizing most of the applied energy. However, at higher strain rates, there is no sufficient time for the damage to initiate and propagate. Under such conditions, more work needs to be carried out for the damage initiation and propagation. This would lead to enhanced shear strength at high shear strain rates. The viscoelastic nature of the resin used is also responsible for the enhancement of shear strength.
Shear properties of epoxy LY 556 with hardener HY 951 are generated under high strain rate loading. The experimental studies were carried out on a TSHB apparatus.
It was observed that the fracture was at the mid-span of the gauge length of the specimen. The fracture was confined to a small region on a plain parallel to the faces of the incident and transmitter bars. It was further observed that the shear strength is enhanced under high strain rate loading compared with quasi-static loading. The shear strength enhancement at shear strain rate of 880 per sec is 45% compared with that at quasi-static loading.
The shear modulus is enhanced under high strain rate loading compared with quasi-static loading. The shear modulus enhancement at shear strain rate of 880 per sec is 22% compared with that at quasi-static loading. In the range of parameters considered, the change in shear properties with the change in shear strain rate is not significant.
Torsional Split Hopkinson Bar Apparatus
The photograph of the TSHB apparatus used for the studies is presented in Fig. 1. The main components are: incident and transmitter elastic bars, torque pulley/rotary actuator, clamping mechanism, linear actuator, support blocks with rotary bearings, and stand. The instruments used are: fatigue resistant strain gauges for dynamic measurement, balancing bridge, dynamic strain meter/amplifier, oscilloscope, and computer. Incident and transmitter bars are made of aluminum alloy 6061-T6 with diameter of 25.4 mm and length of 1.9 m each.
The strain gauges on the incident bar are mounted in such a way that the distance between the strain gauges and the specimen is more than the distance between the clamp and the torque pulley. The strain gauges on the transmitter bar are mounted in such a way that the distance between the strain gauges and the specimen as well as the distance between strain gauges and rear end of the transmitter bar are more than the distance between the clamp and torque pulley. This ensures that overlap of pulses docs not take place. Generally, the distance between the strain gauges on the incident bar and the specimen and the distance between the strain gauges on the transmitter bar and the specimen are the same.
Clamping mechanism is an important component of the apparatus. Schematic arrangement of clamping mechanism is shown in Fig. A1. Figure Ala shows the initial position of the clamping mechanism. The jaws of the vise are held together at the top by a notched clamping bolt as shown in Fig. Alb. Further, the incident bar is clamped using the linear actuators placed at the lower portion of the vise as shown in Fig. Alc. Clamping by linear actuators and pretightening of the clamping bolt are adjusted in such a way that the incident bar is rigidly clamped and perfectly aligned with the transmitter bar. The pretightening in the bolt is such that it is very near to the breaking point. At this stage, torque is stored in the incident bar between the clamp and the torque pulley through a rotary actuator. On further motion of linear actuators, notched clamping bolt would break instantaneously. This is shown schematically in Fig. A1d. As the bolt breaks, the incident bar is unclamped instantaneously leading to release of stored torque. The clamping bolt material must exhibit minimum ductility, but must not be so brittle as to fracture before the clamp is tight enough to hold the desired torque. The notched clamping bolts were made using 6061-T6 and 2040-T6 aluminum alloys and EN24 steel. The photograph of clamping mechanism is given in Fig. A2.
[FIGURE 1A OMITTED]
[FIGURE 2A OMITTED]
Controlling Shear Strain Rate During TSHB Testing
The shear strain rate in the specimen is given by Eq. 3,
[[gamma].sub.S](t) = [2C[D.sub.S]/[L.sub.S]D] [[gamma].sub.R](t)
The shear strain rate depends on diameter of incident and transmitter bars, shear wave velocity in the bars, mean diameter and gauge length of the thin-walled tubular specimen, and amplitude of reflected wave pulse.
Generally, the configuration of TSHB and the materials for different components, especially for incident and transmitter bars, are worked out based on the range of applications visualized. On a specific TSHB apparatus, in order to achieve higher strain rates, either larger amplitude incident pulses or shorter specimen gauge lengths can be used. The former, i.e., the amplitude of the incident pulse, is limited by the yield strength of the incident bar or the amount of torque that can be generated by the particular experimental apparatus. The later, i.e., the length of the specimen is limited by the minimum specimen gauge length permitted on TSHB apparatus. The governing equation for the TSHB apparatus is based on the assumption that the plastic deformation is confined to the thin-walled portion of the tube, i.e., the gauge length of the specimen and does not extend into the collars of the specimen. It is also based on the assumption that the deformation is uniform along the gauge length. For the specimens having very short gauge lengths, these assumptions may not be valid. For the specimens having larger gauge lengths, as the specimen is no longer compact, i.e., having a larger aspect ratio, stress uniformity along the axis can no longer be taken for granted. Local buckling of thin-walled tubular specimens is a possibility.
Another possibility for increasing the strain rate is to increase the mean diameter of the tubular specimen. The maximum diameter of the collar of the specimen can be equal to the diameter of the bars. Considering the geometrical constrains and the bond strength required between the specimen and the bars, the mean diameter of the tubular specimen at gauge length can be determined. The possibility of controlling strain rate by varying the gauge length and mean diameter of the specimen is marginal.
The best way to control the shear strain rate is by governing the amount of torque applied on to the incident bar. As it can be seen from Eq. 3, the strain rate is directly proportional to the amplitude of reflected pulse. It may be noted that the amplitude of reflected pulse is related to the amplitude of incident pulse, which in turn, is related to the amount of torque applied on to the incident bar. By varying the torque applied on to the incident bar, the amplitude of reflected pulse, and hence the shear strain rate on the specimen can be controlled.
At lower strain rates, the specimen needs more time to reach higher strains. In other words, higher strains can be obtained by increasing the duration of the pulse. The procedure for controlling the duration of the pulse is given next.
Controlling Pulse Duration During TSHB Testing
The pulse duration is directly proportional to the distance between the clamp and the torque pulley. It is the time required for the pulse to travel twice the distance between the clamp and the torque pulley in the incident bar.
The duration of the pulse is,
t = [2[L.sub.1]/C]
where t is the pulse duration, [L.sub.1] is the distance between the torque pulley and the clamp and C is torsional wave velocity in the incident bar.
Precaution has to be taken on the maximum limit of the distance between the clamp and the torque pulley. This distance should be less than the distance between the strain gauges and the specimen. This ensures that the overlap of incident and reflected pulses does not take place. At lower shear strain rates, higher pulse duration is required for loading the specimen up to failure. The overall shear strain in the specimen can be obtained based on the shear strain rate and the duration of pulse. Higher pulse duration can be obtained by increasing the distance between the clamp and the pulley. Based on the above two considerations, the optimum distance between the clamp and the torque pulley can be worked out.
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Niranjan K. Naik, Ravikumar Gadipatri, Narasimha Moorthy Thoram, Venkateswara Rao Kavala, Veerraju Ch.
Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, Maharashtra, India
Correspondence to: N.K. Naik; e-mail: firstname.lastname@example.org
Contract grant sponsor: Structures Panel, Aeronautics Research & Development Board, Ministry of Defense, Government of India; contract grant number: DARO/08/1051204/M/I.
Published online in Wiley InterScience (www.interscience.wiley.com).
[C]2010 Society of Plastics Engineers