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An experimental investigation of dynamic crack propagation in a brittle material reinforced with a ductile layer
Optics and Lasers in Engineering 40 (2003) 289–306
An experimental investigation of dynamic crack propagation in a brittle material reinforced with a ductile layer Raman P. Singha,*, Venkitanarayanan Parameswaranb a
Department of Mechanical Engineering, Mechanics of Advanced Materials Laboratory, State University of New York, Stony Brook, NY 11794, USA b Department of Mechanical Engineering and Applied Mechanics, Dynamic Photomechanics Laboratory, University of Rhode Island, Kingston, RI 02881, USA
Abstract The fracture behavior of a dynamically loaded edge crack in a brittle-ductile layered material, as a function of applied loading rate, was experimentally investigated. Layered specimens were prepared by sandwiching a thin layer of ductile aluminum between two thick layers of brittle Homalite-100. The layers were bonded using Loctite Depend 330 adhesive, and a naturally sharp edge crack was introduced in one of the Homalite-100 layers. These single-edge notched specimens were loaded in dynamic three-point bending using a modified Hopkinson bar. The fracture process was imaged in real time using dynamic photoelasticity in conjunction with digital high-speed photography, and the applied load and load-point displacement histories were determined from the strain signals recorded at two locations on the Hopkinson bar. The results of this study indicated two distinct mechanisms of dynamic failure, depending on the applied loading rate. At lower loading rates, the starter crack arrested on reaching the aluminum layer and then caused delamination along the aluminum– Homalite interface. On the contrary, as the loading rate was increased, interfacial delamination was followed by crack re-initiation in the Homalite layer opposite to the initial starter crack. It was determined that the times required for crack initiation, delamination and crack re-initiation decreased as the loading rate was increased. However, it was also observed that the applied load values associated with each event increased with increasing loading rate. These observations indicate that both the dynamic failure process and plausibly the failure mode transition are affected by the rate-dependent properties of Homalite, aluminum and the interfacial bond. Finally, based on the measured peak loads and the observed failure mechanisms it was concluded that the incorporation of a thin ductile reinforcement layer can
*Corresponding author. Tel.: +1-631-632-4354; fax: +1-631-632-8544. E-mail address: [email protected] (R.P. Singh). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 8 9 - 1
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increase both the overall fracture toughness and strength of a nominally brittle material. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Brittle-ductile multilayered materials; Dynamic failure; Split Hopkinson pressure bar; Highspeed imaging
1. Introduction A highly attractive approach for improving the poor intrinsic toughness of ceramics and modern intermetallics involves the fabrication of multilayered laminated structures in which brittle layers alternate with tough metallic layers [1]. Introduction of the secondary tough (and usually ductile) phase introduces several extrinsic energy dissipation mechanisms that lead to significant improvement in overall fracture toughness. Furthermore, brittle-ductile multilayered structures can be readily fabricated with proper control over interfacial and constituent properties, even more so than fiber/particle-reinforced materials [2,3]. This control over the fabrication process enables the design of materials with optimally tailored fracture toughness and mechanical behavior. These factors have led to diverse applications of multilayered materials ranging from microelectronic components to ballistic armor. The increase in fracture toughness of these multilayered brittle-ductile materials originates from the deflection of the crack at various interfaces and from the plastic deformation of the ductile metallic layers [1]. However, the underlying fracture processes and the associated energy dissipation mechanisms are quite complex, even under relatively simple quasi-static loading conditions. In recent years, several ceramic–metal laminated systems have been developed [4–6]. Boundary value measurements have been used to determine the effective toughness and some effort has also been made towards experimentally characterizing strain fields in the vicinity of the crack tip [7,8]. Considerable analytical and numerical modeling has also been conducted to characterize and optimize various energy dissipation mechanisms during fracture in multilayered laminates [5,9–13]. However, these investigations have focused primarily on fracture processes resulting from quasi-static loading conditions. Nevertheless, it should be emphasized, that structures based on multilayered materials are frequently subjected to stress-wave dominated dynamic loading conditions either during service (e.g. impact, thermomechanical shock), or under accident conditions (e.g. bird hits, ballistic impact, turbine fan blade failure, etc.). Such adverse loading conditions significantly influence the fundamental fracture processes and material constitutive response. Furthermore, transient stresswave dominated loading results in even more complex failure behavior as compared to quasi-static situations. This necessitates the fundamental characterization of dynamic failure processes in multilayered materials as a function of applied loading rate. There are few reported investigations that focus on the fracture of brittle-ductile multilayered materials under dynamic loading conditions. The scarcity of dynamic
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experimental studies to identify the fundamental failure processes and energy dissipation mechanisms arises from the difficulty in performing these experiments and extracting relevant real time, full-field information with sufficient spatial and temporal resolution. Lack of experimental observations has also led to the paucity of appropriate numerical and/or analytical models for predicting material behavior. In the past few years, several investigations have focused on dynamic crack propagation along bimaterial interfaces [14–16], dynamic fracture in graded materials with discrete property variation [17], and dynamic failure of discontinuously reinforced composites [18]. In addition, Xu and Rosakis [19,20] have recently reported on experimental investigations of impact-induced failure in homogeneous and layered materials. While, these investigations have provided considerable insight into the complex issues governing the dynamic failure of layered materials, a fundamental mechanistic view of crack growth in multilayered materials, as a function of loading rate, is still lacking. In light of these observations, this paper presents a detailed experimental study to characterize dynamic crack propagation in a brittle material reinforced with a single ductile layer. The objective here is to conduct well controlled, model experiments that will examine the fundamental fracture behavior and failure mode transitions as a function of loading rate. The specific material geometry investigated consists of a brittle material reinforced with a single ductile layer. Since the fracture behavior of multilayered structures is intimately governed by interaction of cracks with the material interfaces, the investigated configuration represents a highly effective model for studying the fundamental characteristics of crack propagation, interfacial delamination and re-initiation. This investigation employs various experimental techniques, including high-speed imaging and modified Hopkinson bar setup with two-point strain measurements, to provide real time and quantitative observations of the dynamic failure process. The goal of these experiments is to establish the effect of loading rate on the fracture process and thus provide a foundation for developing appropriate analytical/numerical models that would be able to predict energy dissipation and overall fracture characteristics in dynamically loaded multilayered materials.
2. Specimen fabrication Fig. 1 shows a schematic of the brittle-ductile layered fracture specimen used in this investigation. It consists of two Homalite-100 layers bonded to a thin aluminum layer. Homalite-100 is a highly crosslinked polymer that is ideally suited for dynamic fracture studies of brittle materials. It also exhibits stress-induced birefringence, which enables the use of photoelasticity for imaging full-field stress distributions in real time. The thin ductile reinforcement layer consisted of 96.5% aluminum, which has a low yield stress and thus exhibits extensive plastic deformation. Homalite layers were machined in a high-speed diamond splitting saw. Water mist cooling was employed to minimize the generation of residual stresses and microcracks on the machined surface. All surfaces to be bonded were polished
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Brittle Layer (Homalite-100)
Ductile Layer (Aluminum)
38 mm
250 µm
Precrack
Brittle Layer (Homalite-100) 163 mm
Fig. 1. Schematic of the layered Homalite-100/aluminum single-edge notched fracture specimen.
Table 1 Material properties of Homalite-100 and aluminum Properties 5
Quasi-static modulus, Eð’e ¼ 10 =sÞ Dynamic modulus, Eð’e ¼ 102 =sÞ Poisson’s ratio, n Density, r Quasi-static fracture toughness, KI Yield strength, sY Longitudinal wave speed, c1 Shear wave speed, c2
Homalite-100
Aluminum
2.17 GPa 5.3 GPa 0.35 1230 kg/m3 0.45 MPa/m1/2 N/A 2630 m/s 1263 m/s
70 GPa 70 GPa 0.33 2700 kg/m3 — 54 MPa 6242 m/s 3144 m/s
and thoroughly cleaned with acetone. Loctite Depend 330 adhesive was then used to bond the Homalite-100 and aluminum layers as per the manufacturer’s recommended procedure. This bonding procedure results in a nominally ‘strong bond’ with quasi-static shear strength of 22.5 MPa and tensile strength of 17 MPa. The thickness of the adhesive layer was determined to be B5–10 mm, which is only 2% of the aluminum layer. Finally, a naturally sharp precrack was introduced in the specimen by first machining a notch with a low-speed diamond wafering saw and then by extending a sharp crack by gently tapping a fresh razor blade at the root of the machined notch. Table 1 lists the material properties and elastic wave speeds of the Homalite-100 and aluminum layers. For quasi-static loading conditions, the elastic mismatch between the two materials may be expressed in terms of Dundurs parameters [21] a¼
m1 ðk2 þ 1Þ m2 ðk1 þ 1Þ ; m1 ðk2 þ 1Þ þ m2 ðk1 þ 1Þ
ð1aÞ
b¼
m1 ðk2 1Þ m2 ðk1 1Þ ; m1 ðk2 þ 1Þ þ m2 ðk1 þ 1Þ
ð1bÞ
where kj ¼ ð3 nj Þ=ð1 þ nj Þ for the case of plane stress, nj are the Poisson’s ratios, and mj are the shear moduli, which relate to the Young’s moduli as mj ¼ Ej =2ð1 þ nj Þ: For the Homalite–aluminum interface a ¼ 0:940 and b ¼ 0:305: Under stress-wave dominated dynamic loading conditions, however, mismatch in material properties is governed by the differences in elastic wave speeds. As listed in Table 1, the elastic
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stress-wave speeds and thus the characteristic acoustic impedances of these two materials differ considerably.
3. Experimental configuration for dynamic fracture A modified Hopkinson bar setup was employed to subject the precracked Homalite/aluminum specimens to three-point bending at high loading rates. Meanwhile, the two-point strain measurement technique was used in conjunction with high-speed digital imaging and dynamic photoelasticity to obtain comprehensive qualitative and quantitative characterization of the dynamic failure process. Experiments were conducted at various loading rates to examine rate dependence on the fracture process and identify any associated failure-mode transitions. Details of the experimental setup are provided as follows. 3.1. Modified Hopkinson bar setup Dynamic failure characteristics of single-edge notched Homalite/aluminum specimens were investigated using the experimental configuration schematically illustrated in Fig. 2. Specimens were loaded under dynamic three-point bend loading using the modified Hopkinson bar setup [22]. In this adaptation of the conventional split Hopkinson pressure bar, a single incident bar is employed to dynamically load a specimen under three-point bending, as shown in Fig. 2. The incident bar, made of maraging steel, had a length of 1239.8 mm, a diameter of 12.7 mm and a bar wave speed of 4809 m/s. The end of the bar, which contacted the specimen, was rounded Flash Lamps Diffuser Sheet
Striker Bar
Gas Gun
Specimen
Incident Bar
A
Pulse Shaper
B
Data Acquisition
Circular Polarizer Rigid Block
Circular Polarizer Monochromator High-Speed Digital Camera
Side View of
Trigger Pulse
Specimen
Fig. 2. Modified Hopkinson bar configuration for dynamic fracture experiments, also showing high-speed digital imaging and dynamic photoelasticity setups.
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with a radius of 6.35 mm. Dispersion of the stress-wave pulse introduced by rounding the bar end results in inadmissible errors only when the duration of the event under investigation is of the order of L=c0 ; where L is the length of the rounded section and c0 is the longitudinal wave speed in the bar. For the current configuration L=c0 ¼ 1:32 ms, which is much smaller than the time duration of the fracture events under investigation and thus, the rounded end of the loading bar did not introduce appreciable errors due to pulse dispersion. A steel projectile of 12.7 mm diameter was launched using a single-stage gas gun to impact the incident bar and generate a compressive stress-wave pulse. The length of the projectile was selected to result in a loading pulse duration of B500 ms. As will be seen later, this pulse duration was sufficient to ensure that the fracture events of interest occur during loading by the first input stress-wave pulse. A pulse shaper was used at the projectile impact location to control the rise time of the stress-wave pulse. Experiments were conducted at various loading rates by systematically varying the impact velocity from 2.3 to 7.6 m/s. 3.2. Determination of dynamic load-point displacement and force Strain gages (EA-13-060LZ-120, Micro Measurements, Inc.) were installed at two locations, A and B; on the loading bar, as shown in Fig. 2. Two diametrically opposed gages were employed at each location to eliminate bending strains. The strain gage signals were demodulated using Ectron 563F signal conditioners and then recorded on a LeCroy 8025 data acquisition system at a sampling rate of 1 point/ms. Subsequently, the two-point strain measurement technique was applied to determine dynamic variations of the load-point displacement and the load-point force at the loading end of the incident bar [22,23]. Based on the theory of one-dimensional wave propagation the normal forces at locations xA and xB are given in terms of the recorded strains eA and eB as NA ðtÞ ¼ AEeA ðtÞ;
ð2aÞ
NB ðtÞ ¼ AEeB ðtÞ;
ð2bÞ
where A and E are the cross-sectional area and elastic modulus of the incident bar, respectively. Then, using the method of characteristics, the particle velocity, vA ðtÞ; at gage location A can be expressed as [22,23] 1 vA ðtÞ ¼ vA ðtp Þ þ ½NA ðtÞ NA ðtp Þ þ 2NB ðt TBA Þ; ð3Þ Z where Z is the acoustic impedance of the incident bar, tp ¼ t 2TBA ; and TBA ¼ ðxB xA Þ=c0 : Similarly, the particle velocity, vE ðtÞ; and normal force NE ðtÞ; at the loading end of the bar, xE ; is given as 1 1 vE ðtÞ ¼ ½vA ðt þ TEA Þ þ vA ðt TEA Þ þ ½NA ðt þ TEA Þ NA ðt TEA Þ; ð4Þ 2 2Z 1 Z NE ðtÞ ¼ ½NA ðt þ TEA Þ þ NA ðt TEA Þ þ ½vA ðt þ TEA Þ vA ðt TEA Þ; 2 2
ð5Þ
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where TEA ¼ ðxE xA Þ=c0 : Using Eqs. (4) and (5), the load-point displacement, uðtÞ; and the load-point force, F ðtÞ; at the loading end of the bar is given as Z t uðtÞ ¼ vE ðtÞ dt; ð6Þ 0
F ðtÞ ¼ NE ðtÞ
ð7Þ
3.3. High-speed, real-time digital imaging of fracture processes High-speed digital imaging was employed along with dynamic photoelasticity to obtain real time, full-field quantification of the dynamic failure process. The experimental configuration shown in Fig. 2 illustrates the use of these techniques in conjunction with the modified Hopkinson bar setup. Two crossed circular polarizers were placed on either side of the specimen to form a light-field circular polariscope. Meanwhile, Power Light 2500DR xenon flash lamps (Photogenic Inc.) were used as light sources to illuminate the specimen in transmission. Since Homalite-100 exhibits stress-induced birefringence, this arrangement results in the formation of isochromatic fringe patterns during the failure process. Images of the isochromatic fringes were captured at discrete, preset instances of time using an Imacon 200 ultra-highspeed digital camera (DRS Hadland). This CCD based camera provides 16 independently programmable digital images of dynamic events up to a maximum framing rate of 200 106 frames/s. The resolution of each image is 1280 1024 pixels. The camera allows for minimum exposure duration of 5 ns, however, a 100 ns exposure was sufficient for the current experiments. Finally, the camera field-of-view was selected to image only the central portion of the specimen, which was the area of interest for investigating the failure process. The xenon flash lamps are a broadband source of light and thus, a monochromatic filter was placed just before the camera to ensure that the imaged isochromatic fringe patterns correspond to a single wavelength of light. Then, the isochromatic fringes represent contours of constant maximum shear stress and their generation is governed by the stress-optic law [24] s 1 s2 ¼
Nfs ; h
ð8Þ
where s1 and s2 are the in-plane principal stresses, fs is the material fringe value, N is the isochromatic fringe order and h is the specimen thickness. A trigger signal was generated by the strain gage at location A: This signal first triggered the data acquisition system for recording strain pulses, and also fired the xenon flash lamps, which require a finite warm-up time. Simultaneously, the digital camera was also primed to start the framing sequence after a preset delay. In this manner, it was possible to directly relate the loading stress-wave pulse to specific fracture events imaged by the high-speed camera.
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4. Experimental results and discussion of fracture behavior Dynamic fracture experiments were conducted at various loading rates using projectile impact velocities of 2.3, 4.4, 5.6, 6.6 and 7.6 m/s. Typically, three tests were performed at each impact velocity to obtain confidence in experimental repeatability. The resulting failure processes were imaged using photoelasticity in conjunction with high-speed digital photography, and the applied load and load-point displacement were determined using the two-point strain measurement technique. The following sub-sections discuss the experimental observations. 4.1. Dynamic fracture behavior for low velocity impact Fig. 3 shows a sequence of photoelastic fringe patterns obtained for a Homalite– aluminum single-edge-notched specimen subjected to dynamic three-point bend loading. The projectile impact velocity was set to 2.3 m/s, which represented the lower limit of the loading rate regime investigated. Also, note that only nine frames (out of a total of sixteen) are shown in Fig. 3. Furthermore, the field of view of the camera was adjusted so that only the central part of the specimen was photographed, thus enabling the imaging of the main crack, the interface and the impact loading point with sufficient resolution. From the digital images it was observed that the main crack initiated 173 ms after arrival of loading pulse at specimen, and subsequently propagated under predominantly mode-I (opening) conditions till it reached the Homalite–aluminum interface. The crack then arrested at the interface while the dynamic loading on the specimen continued to increase further. The delamination crack initiated at 233 ms after the arrival of the loading pulse at the specimen, and then started to propagate along the Homalite–aluminum interface. This interface crack arrested as the loading pulse passed through the specimen and no further crack propagation was observed for the remaining duration of the experiment. The purely anti-symmetric isochromatic fringe patterns around the interface crack indicate that interface delamination occurred under mode-II (shear) dominated conditions. Fig. 4 shows a plot of the main crack and the interface delamination crack positions as a function of time elapsed after the arrival of the loading pulse at the specimen. This figure highlights the instances of initiation, propagation and arrest of both the main crack and the interface delamination crack. From the time history of crack extensions, the average propagation speed of the main crack was determined to be 150 m/s and that of the interface delamination was 21 m/s. An observation of the specimen, at the end of the test, showed that delamination occurred only on the Homalite–aluminum interface that was on the same side as the main crack. While the delamination crack grew further with repeated loading (due to multiple pulse reflections), no other fracture was observed and the final specimen was recovered physically intact. The strains recorded during the dynamic fracture event were analyzed to determine the load-point displacement and the load-point force, as discussed earlier. The dynamic variation of these quantities is plotted in Fig. 5. It was observed that
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Fig. 3. Sequence of photoelastic fringe patterns obtained for dynamic three-point bend loading of a Homalite–aluminum single-edge notched specimen subjected to low velocity impact.
the main crack initiated during the rising portion of the load pulse. However, the interfacial delamination initiated at the tailing end of the loading pulse. For this experiment, the maximum load sustained by the specimen was 550 N. 4.2. Dynamic fracture behavior for high velocity impact Similar experiments, as those discussed above, were carried out for higher impact velocities of 4.4, 5.6, 6.6 and 7.6 m/s. Fig. 6 shows a sequence of isochromatic fringe
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Crack Extension (mm)
5 Main Crack Delamination
4 3 2 1 0 0
50
100
150
200
250
300
350
400
450
Time (µs)
Fig. 4. Extension of main crack and interfacial delamination for low velocity impact.
500
1.0
Force
300 200
Displacement
Delamination Initiation
400
Main Crack Initiation
Load-point Force, F(t), (N)
Experiment: ML025 Impact Velocity: 2.3 m/s
0.8 0.6 0.4
100
0.2
0
0.0 0
50
100
150
200
250
300
350
400
Load-point Displacement, uE (mm)
1.2
600
450
Time (µs) Fig. 5. Variation of dynamic load-point displacement and load-point force for low velocity impact.
patterns obtained for a Homalite–aluminum fracture specimen subjected to dynamic three-point bend loading with an impact velocity of 4.4 m/s. As before, only select frames are shown and the field of view is centered on the region of interest. In this case, the main crack initiated 120 ms after the arrival of the loading pulse, and then propagated under mode-I (opening) conditions till it arrested at the Homalite– aluminum interface. The delamination crack initiated 165 ms after the arrival of the loading pulse, and started to propagate along the interface. However, shortly afterwards, the delamination crack arrested and two cracks re-initiated in the Homalite layer across the aluminum layer. The re-initiated cracks originated at the same location as the interfacial delamination crack and continued to propagate,
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Fig. 6. Sequence of photoelastic fringe patterns obtained for dynamic three-point bend loading of a Homalite–aluminum single-edge notch specimen subjected to high velocity impact.
while the interface did not delaminate any further. The aluminum layer was intact during this entire failure process and provided closing tractions as shown in frames 11 and 14 of Fig. 6. The time histories of propagation for the main crack, the delamination crack and the re-initiated cracks are plotted in Fig. 7. As before, average crack speeds were obtained from the crack extension versus time data. The average speeds for the main crack, the interfacial delamination, and the two re-initiation cracks were determined to be 250, 40 and 325 m/s, respectively. Dynamic variations of the load-point
300
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Crack Extension (mm)
Experiment: ML023 Impact Velocity: 4.4 m/s
12
Main Crack Delamination Crack Re-initiation
9
6
3
0 0
50
100
150
200
250
300
350
Time (µs)
Fig. 7. Extension of main crack, interfacial delamination and re-initiated cracks for high velocity impact.
600 400 200
1.0 0.8 0.6 0.4 Crack Re-initiation
800
Displacement Delamination Initiation
1000
Main Crack Initiation
Load-point Force, F(t), (N)
Experiment: ML023 Impact Velocity: 4.4 m/s
0 0
50
100
150
200
0.2 Force
0.0 250
300
Load-point Displacement, uE (mm)
1.2
1200
350
Time (µs) Fig. 8. Variation of dynamic load-point displacement and load-point force for high velocity impact.
displacement and the load-point force were determined from the measured strain signals, and are plotted in Fig. 8. It was observed that the main crack initiated during the rising portion of the load pulse, while the interfacial delamination initiated near the peak load. For this experiment, the maximum load sustained by the specimen was 1090 N. Experiments conducted at impact velocities of 5.5, 6.6 and 7.6 m/s resulted in similar observations. The dynamic fracture process, for all these loading rates, resulted in limited interfacial delamination followed by re-initiation of cracks across
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the aluminum layer. The times associated with initiation of main crack, interfacial delamination and crack re-initiation were determined from the high-speed images of the fracture process. The average crack speeds were also calculated as for other experiments. In addition, the applied load and load-point displacement were determined from the recorded strain signals. It was observed that the times associated with various fracture events and the maximum load sustained by the specimen were all functions of the impact velocity and hence the loading rate. 4.3. Effect of loading rate on the dynamic failure process Based on the experimental observations it is possible to identify two different modes of dynamic failure. For an impact velocity of 2.3 m/s, only interfacial delamination was observed and the specimen was recovered intact. Meanwhile, for all higher impact velocities, 4.4–7.6 m/s, limited interfacial delamination followed by crack re-initiation in the second Homalite half across the aluminum layer was observed. These two modes of failure are schematically illustrated in Fig. 9. The fundamental issue of interest is the identification of loading conditions that will lead to crack re-initiation across the aluminum layer reinforcement. It is believed that this transition is a function of the rate-dependent fracture toughness of Homalite-100 and also the adhesive layer. The effect of loading rate on the yield and post-yield hardening characteristics of the aluminum layer is also expected to contribute to this failure mode transition. Fig. 10 shows the times associated with the initiation of the main crack, interfacial delamination and crack re-initiation. As the impact velocity, and thus the loading rate, was increased the time associated with each failure event decreased. This is inline with the fact that at higher impact velocity, the load applied to the specimen increases at a faster rate and thus crack initiation can be expected to occur earlier during the dynamic loading process. However, it was also observed that the applied loads associated with the initiation of the main crack and interfacial delamination also increased, as shown in Fig. 11. These observations indicate that both the dynamic failure process and plausibly the failure mode transition are affected by the rate-dependent properties of Homalite, aluminum and the interfacial bond. Finally,
Re-initiation
Delamination
(a)
(b)
Fig. 9. Two dynamic failure modes of Homalite–aluminum layered specimens: (a) interfacial delamination, and (b) interfacial delamination followed by crack re-initiation.
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300
Time, t, (µs)
250 200 150 100 50 0
2
3
4
5
6
7
8
Impact Velocity (m/s)
Fig. 10. Time associated with the initiation of the main crack, interfacial delamination and re-initiated cracks, as a function of impact velocity.
4000 Load at Main Crack Initiation Load at Delamination Initiation Peak load
Dynamic Load, NE, (N)
3500 3000 2500 2000 1500 1000 500 0 2
3
4
5
6
7
8
Impact Velocity (m/s)
Fig. 11. Applied load associated with initiation of main crack and delamination, as a function of impact velocity.
it was also observed that the peak load for any given experiment was consistently higher than the load required to initiate the main crack. These observations imply that the incorporation of a thin ductile reinforcement layer can increase both the overall fracture toughness and strength of a nominally brittle material. In order to quantify the conditions governing the dynamic failure process and failure mode transition, it is necessary to examine the histories of crack-tip driving tractions and energy release rate as functions of applied loading rate. These parameters are normally quantified by analyzing the experimentally obtained isochromatic fringe patterns around the crack tip, in conjunction with the stressoptic law, given by Eq. (8), and appropriate asymptotic solutions that describe the
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stress fields surrounding a crack propagating dynamically along the interface of a ductile layer in a brittle material. The elastodynamic stress fields, in the vicinity of a crack propagating along a bimaterial interface, are given in terms of the complex stress intensity factor, K d ðtÞ; as [25] 1 1 sij ¼ pffiffiffiffiffiffiffiRe½K d ðtÞrieðvÞ s* 1ij ðy; vÞ þ pffiffiffiffiffiffiffiIm½K d ðtÞrieðvÞ s* 2ij ðy; vÞ; 2pr 2pr
ð9Þ
where r is the radial distance from the crack tip, s* 1ij and s* 2ij are known functions of angle y and crack velocity v; eðvÞ is the oscillatory index, andK d ðtÞ ¼ K1d ðtÞ þ iK2d ðtÞ: Note that a non-zero oscillatory index results in a coupled and inherently mixedmode stress field that is typical for cracks along bimaterial interfaces. The existence of K-dominant crack tip fields, such as those given by Eq. (9), for interfacial crack propagation in multilayered specimen geometries has been examined only for the case of quasi-static crack propagation along the interface of an elastic reinforcing layer in a brittle material. Based on linear-elastic finite element analyses it was demonstrated that the region of K-dominance near the crack tip is very limited, extending merely to B1/10 of the height of the reinforcing layer [26]. The dynamic loading situation is even more complex and is expected to result in further limitations on the spatial and temporal validity of analytical expressions of the crack tip stress fields. It has been shown that the behavior of a dynamically loaded three-point bend fracture specimen can be characterized by a short-time response dominated by discrete waves, and a long-time response that is essentially quasi-static (yet still rate dependent) [27]. Short-time response can be distinguished from long-time behavior in terms of a transition time, tT ; as the time at which the kinetic energy in the specimen is approximately equal to the deformation energy. The effects of material inertia dominate prior to the transition time, but the deformation energy dominates at times significantly greater than tT : Thus, if failure events occur at tbtT ; then inertial effects can be neglected and quasi-static models can be applied to interpret experimental observations. The transition time is given as [27] H tT ¼ DS ; ð10Þ c0 where S is a dimensionless shape factor, H is the width of the specimen, c0 is the longitudinal bar wave speed in the specimen and D is a dimensionless coefficient that depends implicitly on the impact conditions ’ tDðtÞ ; D¼ ð11Þ DðtÞ tT
where D and D’ are the displacement and velocity at the load-point and t is the elapsed time from the instant of impact. For the current-loading geometry the transition time was estimated to be typically 150–200 ms, while the typical time to failure was 150–300 ms. Since the time to failure is very close to the characteristic transition time, the relevant near-tip fields cannot be inferred from quasi-static models and the stress-wave dominated inertial effects must be taken into account.
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Based on the above discussion it is evident that analytical solutions for transient crack propagation along the interface of a thin layer embedded in a brittle material would be required to be able to quantify the dynamic failure process based on experimental observations. Furthermore, these analytical solutions would also have to account for the constrained plastic deformation expected to occur in the ductile reinforcing layer. The development of appropriate analytical solutions of this nature is difficult, and the only resort would be to conduct dynamic finite element simulations for modeling the fracture process. Such a simulation would then enable the extraction of the crack-tip driving tractions and energy release rate as functions of applied loading rate. Such simulations are currently in progress and will be published in the near future [28].
5. Summary An experimental investigation was conducted to characterize the dynamic failure behavior of a brittle material reinforced with a single ductile layer, as a function of loading rate. Model specimens were prepared by sandwiching a thin layer of ductile aluminum between two thick layers of brittle Homalite-100 using Loctite Depend 330 adhesive. A naturally sharp edge crack, introduced in one of the Homalite-100 layers, served as the starter precrack. The specimens were subjected to dynamic three-point bending in a modified Hopkinson bar at projectile impact velocities of 2.3, 4.4, 5.6, 6.6 and 7.6 m/s. The entire dynamic failure phenomena of crack initiation, interlayer delamination and crack re-initiation were observed using dynamic photoelasticity combined with high-speed digital photography. In addition, strains recorded at two locations on the Hopkinson bar were analyzed using the twopoint strain measurement technique to determine the time histories of the load and load-point displacement were obtained. Two distinct mechanisms of dynamic failure were identified, as a function of loading rate. At lower loading rates (impact velocity of 2.3 m/s), the starter crack arrested on reaching the aluminum layer and then caused delamination along the aluminum–Homalite interface. This delamination arrested as the loading pulse passed through the specimen and no further fracture was observed. On the contrary, as the loading rate was increased, interfacial delamination was followed by crack reinitiation in the Homalite layer opposite to the initial starter crack. The re-initiated cracks originated at the same location as the interfacial delamination crack and continued to propagate, while the interface did not delaminate any further. It was determined that the times required for crack initiation, delamination and crack re-initiation decreased, as the loading rate was increased. However, it was also observed that the applied load values associated with each event increased with increasing loading rate. These observations indicate that both the dynamic failure process and plausibly the failure mode transition are affected by the rate-dependent properties of Homalite, aluminum and the interfacial bond. Finally, based on the measured peak loads and the observed failure mechanisms it was concluded that the incorporation of a thin ductile reinforcement layer can increase both the overall
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fracture toughness and strength of a nominally brittle material. Quantification of the crack-tip driving tractions and energy release rate, as functions of applied loading rate, is currently in progress.
Acknowledgements The authors are indebted to Prof. Arun Shukla at the University of Rhode Island for providing the experimental facilities that made this investigation possible. Notably, we acknowledge the availability of the Imacon 200 high-speed digital camera (supported by the National Science Foundation Grant No. CMS 9870947). Raman P. Singh also thanks Prof. Toshio Nakamura at the State University of New York, Stony Brook for the several discussions regarding this research effort.
References [1] Cao HC, Evans AG. On crack extension in ductile/brittle laminates. Acta Metall Mater 1991;39:2997–3005. [2] Evans AG, Lu MC, Schmauder S, Ruhle M. Some aspects of the mechanical strength of ceramic/ metal bonded systems. Acta Metall 1986;34:1643–4. [3] Dalgeish BJ, Trumble KP, Evans AG. The strength and fracture of alumina bonded with aluminum alloys. Acta Metall 1989;37:1923–31. [4] Flinn BD, Ruhle M, Evans AG. Toughening in composites of Al2O3 reinforced with Al. Acta Metallurgica 1989;37:3001–6. [5] Chen Z, Mecholsky Jr JJ. Control of strength and toughness of ceramic/metal laminates using interface design. J Mater Res 1993;8(9):2362–9. [6] Shaw MC, Marshall DB, Dadkhah MS, Evans AG. Cracking and damage mechanisms in ceramic/ metal multilayers. Acta Metall Mater 1993;41:3311–20. [7] Ma Q, Shaw MC, He MY, Dalgleish BJ, Clarke DR, Evans AG. Stress redistribution in ceramic/ metal multilayers containing cracks. Acta Metall Mater 1995;43(6):2137–42. [8] Shaw MC, Clyne TW, Cocks ACF, Fleck NA, Pateras SK. Cracking patterns in metal-ceramic laminates: effects of plasticity. J Mech Phys Solids 1996;44(5):801–21. [9] He MY, Heredia FE, Wissuchek DJ, Shaw MC, Evans AG. Mechanics of crack growth in layered materials. Acta Metall Mater 1993;41(4):1223–8. [10] Huang Y, Zhang HW, Wu F. Multiple cracking in metal-ceramic laminates. Int J Solids Struct 1994;31:2753. [11] Ballarini R, Charalambides PG, Islam S. Near-tip dual length scale mechanics of mode-I cracking in laminated brittle matrix composites. Int J Fract 1995;70:275–304. [12] Jha M, Charalambides PG. A finite element analysis of fracture initiation in ductile/brittle periodically layered composites. Int J Fract 1998;90:299–323. [13] Hwu KL, Derby B. Fracture of metal/ceramic laminates-II. Crack growth resistance and toughness. Acta Metall Mater 1999;47(2):545–63. [14] Lambros J, Rosakis AJ. Dynamic decohesion of bimaterials: experimental observations and failure criteria. Int J Solids Struct 1995;32(17/18):2677–702. [15] Singh RP, Shukla A. Subsonic and transonic crack growth along a bimaterial interface. J Appl Mech 1996;63:919–24. [16] Kavaturu M, Shukla A. Dynamic fracture criteria for crack growth along bimaterial interfaces. J Appl Mech 1998;56(2):293–9.
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[17] Parameswaran V, Shukla A. Dynamic fracture of a functionally gradient material having discrete property variation. J Mater Sci 1998;33(13):3303–11. [18] Irfan MA, Prakash V. Dynamic deformation and fracture behavior of novel damage tolerant discontinuously reinforced aluminum composites. Int J Solids Struct 2000;37(33):4477–507. [19] Xu LR, Rosakis AJ. Impact failure characteristics in sandwich structures; Part I: basic failure mode selection. Int J Solids Struct, 2002, in press. [20] Xu LR, Rosakis AJ. Impact failure characteristics in sandwich structures; Part II: effects of impact speed and interfacial strength. Int J Solids Struct, 2002, submitted for publication. [21] Dundurs J. Mathematical theory of dislocations. New York: American Society of Mechanical Engineering, 1969. p. 70. [22] Bacon C, F.arm J, Lataillade JL. Dynamic fracture toughness determined from load-point displacement. Exp Mech 1994;34:217–33. [23] Lundberg B, Carlsson J, Sundin KG. Analysis of elastic waves in non-uniform rods from two-point strain measurement. J Sound Vib 1990;137(3):483–93. [24] Dally JW. Dynamic photoelastic studies of fracture. Exp Mech 1979;19:349–61. [25] Yang W, Suo Z, Shih CF. Mechanics of dynamic debonding. Proc R Soc London 1991;A433:679–97. [26] Becker TL, McNaney JM, Cannon RM, Ritchie RO. Limitations on the use of the mixed-mode delaminating beam test specimen: effects of the size of the region of K-dominance. Mech Mater 1997;25(4):291–308. [27] Nakamura T, Shih CF, Freund LB. Analysis of a dynamically loaded three-point-bend ductile fracture specimen. Eng Fract Mech 1986;25(3):323–39. [28] Kumar BG, Parameswaran V, Singh RP, Nakamura T. Energy dissipation during dynamic failure of a brittle-ductile layered material. Int J Fract, submitted for publication.
An experimental investigation of dynamic crack propagation in a brittle material reinforced with a ductile layer Raman P. Singha,*, Venkitanarayanan Parameswaranb a
Department of Mechanical Engineering, Mechanics of Advanced Materials Laboratory, State University of New York, Stony Brook, NY 11794, USA b Department of Mechanical Engineering and Applied Mechanics, Dynamic Photomechanics Laboratory, University of Rhode Island, Kingston, RI 02881, USA
Abstract The fracture behavior of a dynamically loaded edge crack in a brittle-ductile layered material, as a function of applied loading rate, was experimentally investigated. Layered specimens were prepared by sandwiching a thin layer of ductile aluminum between two thick layers of brittle Homalite-100. The layers were bonded using Loctite Depend 330 adhesive, and a naturally sharp edge crack was introduced in one of the Homalite-100 layers. These single-edge notched specimens were loaded in dynamic three-point bending using a modified Hopkinson bar. The fracture process was imaged in real time using dynamic photoelasticity in conjunction with digital high-speed photography, and the applied load and load-point displacement histories were determined from the strain signals recorded at two locations on the Hopkinson bar. The results of this study indicated two distinct mechanisms of dynamic failure, depending on the applied loading rate. At lower loading rates, the starter crack arrested on reaching the aluminum layer and then caused delamination along the aluminum– Homalite interface. On the contrary, as the loading rate was increased, interfacial delamination was followed by crack re-initiation in the Homalite layer opposite to the initial starter crack. It was determined that the times required for crack initiation, delamination and crack re-initiation decreased as the loading rate was increased. However, it was also observed that the applied load values associated with each event increased with increasing loading rate. These observations indicate that both the dynamic failure process and plausibly the failure mode transition are affected by the rate-dependent properties of Homalite, aluminum and the interfacial bond. Finally, based on the measured peak loads and the observed failure mechanisms it was concluded that the incorporation of a thin ductile reinforcement layer can
*Corresponding author. Tel.: +1-631-632-4354; fax: +1-631-632-8544. E-mail address: [email protected] (R.P. Singh). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 8 9 - 1
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increase both the overall fracture toughness and strength of a nominally brittle material. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Brittle-ductile multilayered materials; Dynamic failure; Split Hopkinson pressure bar; Highspeed imaging
1. Introduction A highly attractive approach for improving the poor intrinsic toughness of ceramics and modern intermetallics involves the fabrication of multilayered laminated structures in which brittle layers alternate with tough metallic layers [1]. Introduction of the secondary tough (and usually ductile) phase introduces several extrinsic energy dissipation mechanisms that lead to significant improvement in overall fracture toughness. Furthermore, brittle-ductile multilayered structures can be readily fabricated with proper control over interfacial and constituent properties, even more so than fiber/particle-reinforced materials [2,3]. This control over the fabrication process enables the design of materials with optimally tailored fracture toughness and mechanical behavior. These factors have led to diverse applications of multilayered materials ranging from microelectronic components to ballistic armor. The increase in fracture toughness of these multilayered brittle-ductile materials originates from the deflection of the crack at various interfaces and from the plastic deformation of the ductile metallic layers [1]. However, the underlying fracture processes and the associated energy dissipation mechanisms are quite complex, even under relatively simple quasi-static loading conditions. In recent years, several ceramic–metal laminated systems have been developed [4–6]. Boundary value measurements have been used to determine the effective toughness and some effort has also been made towards experimentally characterizing strain fields in the vicinity of the crack tip [7,8]. Considerable analytical and numerical modeling has also been conducted to characterize and optimize various energy dissipation mechanisms during fracture in multilayered laminates [5,9–13]. However, these investigations have focused primarily on fracture processes resulting from quasi-static loading conditions. Nevertheless, it should be emphasized, that structures based on multilayered materials are frequently subjected to stress-wave dominated dynamic loading conditions either during service (e.g. impact, thermomechanical shock), or under accident conditions (e.g. bird hits, ballistic impact, turbine fan blade failure, etc.). Such adverse loading conditions significantly influence the fundamental fracture processes and material constitutive response. Furthermore, transient stresswave dominated loading results in even more complex failure behavior as compared to quasi-static situations. This necessitates the fundamental characterization of dynamic failure processes in multilayered materials as a function of applied loading rate. There are few reported investigations that focus on the fracture of brittle-ductile multilayered materials under dynamic loading conditions. The scarcity of dynamic
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experimental studies to identify the fundamental failure processes and energy dissipation mechanisms arises from the difficulty in performing these experiments and extracting relevant real time, full-field information with sufficient spatial and temporal resolution. Lack of experimental observations has also led to the paucity of appropriate numerical and/or analytical models for predicting material behavior. In the past few years, several investigations have focused on dynamic crack propagation along bimaterial interfaces [14–16], dynamic fracture in graded materials with discrete property variation [17], and dynamic failure of discontinuously reinforced composites [18]. In addition, Xu and Rosakis [19,20] have recently reported on experimental investigations of impact-induced failure in homogeneous and layered materials. While, these investigations have provided considerable insight into the complex issues governing the dynamic failure of layered materials, a fundamental mechanistic view of crack growth in multilayered materials, as a function of loading rate, is still lacking. In light of these observations, this paper presents a detailed experimental study to characterize dynamic crack propagation in a brittle material reinforced with a single ductile layer. The objective here is to conduct well controlled, model experiments that will examine the fundamental fracture behavior and failure mode transitions as a function of loading rate. The specific material geometry investigated consists of a brittle material reinforced with a single ductile layer. Since the fracture behavior of multilayered structures is intimately governed by interaction of cracks with the material interfaces, the investigated configuration represents a highly effective model for studying the fundamental characteristics of crack propagation, interfacial delamination and re-initiation. This investigation employs various experimental techniques, including high-speed imaging and modified Hopkinson bar setup with two-point strain measurements, to provide real time and quantitative observations of the dynamic failure process. The goal of these experiments is to establish the effect of loading rate on the fracture process and thus provide a foundation for developing appropriate analytical/numerical models that would be able to predict energy dissipation and overall fracture characteristics in dynamically loaded multilayered materials.
2. Specimen fabrication Fig. 1 shows a schematic of the brittle-ductile layered fracture specimen used in this investigation. It consists of two Homalite-100 layers bonded to a thin aluminum layer. Homalite-100 is a highly crosslinked polymer that is ideally suited for dynamic fracture studies of brittle materials. It also exhibits stress-induced birefringence, which enables the use of photoelasticity for imaging full-field stress distributions in real time. The thin ductile reinforcement layer consisted of 96.5% aluminum, which has a low yield stress and thus exhibits extensive plastic deformation. Homalite layers were machined in a high-speed diamond splitting saw. Water mist cooling was employed to minimize the generation of residual stresses and microcracks on the machined surface. All surfaces to be bonded were polished
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292
Brittle Layer (Homalite-100)
Ductile Layer (Aluminum)
38 mm
250 µm
Precrack
Brittle Layer (Homalite-100) 163 mm
Fig. 1. Schematic of the layered Homalite-100/aluminum single-edge notched fracture specimen.
Table 1 Material properties of Homalite-100 and aluminum Properties 5
Quasi-static modulus, Eð’e ¼ 10 =sÞ Dynamic modulus, Eð’e ¼ 102 =sÞ Poisson’s ratio, n Density, r Quasi-static fracture toughness, KI Yield strength, sY Longitudinal wave speed, c1 Shear wave speed, c2
Homalite-100
Aluminum
2.17 GPa 5.3 GPa 0.35 1230 kg/m3 0.45 MPa/m1/2 N/A 2630 m/s 1263 m/s
70 GPa 70 GPa 0.33 2700 kg/m3 — 54 MPa 6242 m/s 3144 m/s
and thoroughly cleaned with acetone. Loctite Depend 330 adhesive was then used to bond the Homalite-100 and aluminum layers as per the manufacturer’s recommended procedure. This bonding procedure results in a nominally ‘strong bond’ with quasi-static shear strength of 22.5 MPa and tensile strength of 17 MPa. The thickness of the adhesive layer was determined to be B5–10 mm, which is only 2% of the aluminum layer. Finally, a naturally sharp precrack was introduced in the specimen by first machining a notch with a low-speed diamond wafering saw and then by extending a sharp crack by gently tapping a fresh razor blade at the root of the machined notch. Table 1 lists the material properties and elastic wave speeds of the Homalite-100 and aluminum layers. For quasi-static loading conditions, the elastic mismatch between the two materials may be expressed in terms of Dundurs parameters [21] a¼
m1 ðk2 þ 1Þ m2 ðk1 þ 1Þ ; m1 ðk2 þ 1Þ þ m2 ðk1 þ 1Þ
ð1aÞ
b¼
m1 ðk2 1Þ m2 ðk1 1Þ ; m1 ðk2 þ 1Þ þ m2 ðk1 þ 1Þ
ð1bÞ
where kj ¼ ð3 nj Þ=ð1 þ nj Þ for the case of plane stress, nj are the Poisson’s ratios, and mj are the shear moduli, which relate to the Young’s moduli as mj ¼ Ej =2ð1 þ nj Þ: For the Homalite–aluminum interface a ¼ 0:940 and b ¼ 0:305: Under stress-wave dominated dynamic loading conditions, however, mismatch in material properties is governed by the differences in elastic wave speeds. As listed in Table 1, the elastic
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stress-wave speeds and thus the characteristic acoustic impedances of these two materials differ considerably.
3. Experimental configuration for dynamic fracture A modified Hopkinson bar setup was employed to subject the precracked Homalite/aluminum specimens to three-point bending at high loading rates. Meanwhile, the two-point strain measurement technique was used in conjunction with high-speed digital imaging and dynamic photoelasticity to obtain comprehensive qualitative and quantitative characterization of the dynamic failure process. Experiments were conducted at various loading rates to examine rate dependence on the fracture process and identify any associated failure-mode transitions. Details of the experimental setup are provided as follows. 3.1. Modified Hopkinson bar setup Dynamic failure characteristics of single-edge notched Homalite/aluminum specimens were investigated using the experimental configuration schematically illustrated in Fig. 2. Specimens were loaded under dynamic three-point bend loading using the modified Hopkinson bar setup [22]. In this adaptation of the conventional split Hopkinson pressure bar, a single incident bar is employed to dynamically load a specimen under three-point bending, as shown in Fig. 2. The incident bar, made of maraging steel, had a length of 1239.8 mm, a diameter of 12.7 mm and a bar wave speed of 4809 m/s. The end of the bar, which contacted the specimen, was rounded Flash Lamps Diffuser Sheet
Striker Bar
Gas Gun
Specimen
Incident Bar
A
Pulse Shaper
B
Data Acquisition
Circular Polarizer Rigid Block
Circular Polarizer Monochromator High-Speed Digital Camera
Side View of
Trigger Pulse
Specimen
Fig. 2. Modified Hopkinson bar configuration for dynamic fracture experiments, also showing high-speed digital imaging and dynamic photoelasticity setups.
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with a radius of 6.35 mm. Dispersion of the stress-wave pulse introduced by rounding the bar end results in inadmissible errors only when the duration of the event under investigation is of the order of L=c0 ; where L is the length of the rounded section and c0 is the longitudinal wave speed in the bar. For the current configuration L=c0 ¼ 1:32 ms, which is much smaller than the time duration of the fracture events under investigation and thus, the rounded end of the loading bar did not introduce appreciable errors due to pulse dispersion. A steel projectile of 12.7 mm diameter was launched using a single-stage gas gun to impact the incident bar and generate a compressive stress-wave pulse. The length of the projectile was selected to result in a loading pulse duration of B500 ms. As will be seen later, this pulse duration was sufficient to ensure that the fracture events of interest occur during loading by the first input stress-wave pulse. A pulse shaper was used at the projectile impact location to control the rise time of the stress-wave pulse. Experiments were conducted at various loading rates by systematically varying the impact velocity from 2.3 to 7.6 m/s. 3.2. Determination of dynamic load-point displacement and force Strain gages (EA-13-060LZ-120, Micro Measurements, Inc.) were installed at two locations, A and B; on the loading bar, as shown in Fig. 2. Two diametrically opposed gages were employed at each location to eliminate bending strains. The strain gage signals were demodulated using Ectron 563F signal conditioners and then recorded on a LeCroy 8025 data acquisition system at a sampling rate of 1 point/ms. Subsequently, the two-point strain measurement technique was applied to determine dynamic variations of the load-point displacement and the load-point force at the loading end of the incident bar [22,23]. Based on the theory of one-dimensional wave propagation the normal forces at locations xA and xB are given in terms of the recorded strains eA and eB as NA ðtÞ ¼ AEeA ðtÞ;
ð2aÞ
NB ðtÞ ¼ AEeB ðtÞ;
ð2bÞ
where A and E are the cross-sectional area and elastic modulus of the incident bar, respectively. Then, using the method of characteristics, the particle velocity, vA ðtÞ; at gage location A can be expressed as [22,23] 1 vA ðtÞ ¼ vA ðtp Þ þ ½NA ðtÞ NA ðtp Þ þ 2NB ðt TBA Þ; ð3Þ Z where Z is the acoustic impedance of the incident bar, tp ¼ t 2TBA ; and TBA ¼ ðxB xA Þ=c0 : Similarly, the particle velocity, vE ðtÞ; and normal force NE ðtÞ; at the loading end of the bar, xE ; is given as 1 1 vE ðtÞ ¼ ½vA ðt þ TEA Þ þ vA ðt TEA Þ þ ½NA ðt þ TEA Þ NA ðt TEA Þ; ð4Þ 2 2Z 1 Z NE ðtÞ ¼ ½NA ðt þ TEA Þ þ NA ðt TEA Þ þ ½vA ðt þ TEA Þ vA ðt TEA Þ; 2 2
ð5Þ
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where TEA ¼ ðxE xA Þ=c0 : Using Eqs. (4) and (5), the load-point displacement, uðtÞ; and the load-point force, F ðtÞ; at the loading end of the bar is given as Z t uðtÞ ¼ vE ðtÞ dt; ð6Þ 0
F ðtÞ ¼ NE ðtÞ
ð7Þ
3.3. High-speed, real-time digital imaging of fracture processes High-speed digital imaging was employed along with dynamic photoelasticity to obtain real time, full-field quantification of the dynamic failure process. The experimental configuration shown in Fig. 2 illustrates the use of these techniques in conjunction with the modified Hopkinson bar setup. Two crossed circular polarizers were placed on either side of the specimen to form a light-field circular polariscope. Meanwhile, Power Light 2500DR xenon flash lamps (Photogenic Inc.) were used as light sources to illuminate the specimen in transmission. Since Homalite-100 exhibits stress-induced birefringence, this arrangement results in the formation of isochromatic fringe patterns during the failure process. Images of the isochromatic fringes were captured at discrete, preset instances of time using an Imacon 200 ultra-highspeed digital camera (DRS Hadland). This CCD based camera provides 16 independently programmable digital images of dynamic events up to a maximum framing rate of 200 106 frames/s. The resolution of each image is 1280 1024 pixels. The camera allows for minimum exposure duration of 5 ns, however, a 100 ns exposure was sufficient for the current experiments. Finally, the camera field-of-view was selected to image only the central portion of the specimen, which was the area of interest for investigating the failure process. The xenon flash lamps are a broadband source of light and thus, a monochromatic filter was placed just before the camera to ensure that the imaged isochromatic fringe patterns correspond to a single wavelength of light. Then, the isochromatic fringes represent contours of constant maximum shear stress and their generation is governed by the stress-optic law [24] s 1 s2 ¼
Nfs ; h
ð8Þ
where s1 and s2 are the in-plane principal stresses, fs is the material fringe value, N is the isochromatic fringe order and h is the specimen thickness. A trigger signal was generated by the strain gage at location A: This signal first triggered the data acquisition system for recording strain pulses, and also fired the xenon flash lamps, which require a finite warm-up time. Simultaneously, the digital camera was also primed to start the framing sequence after a preset delay. In this manner, it was possible to directly relate the loading stress-wave pulse to specific fracture events imaged by the high-speed camera.
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4. Experimental results and discussion of fracture behavior Dynamic fracture experiments were conducted at various loading rates using projectile impact velocities of 2.3, 4.4, 5.6, 6.6 and 7.6 m/s. Typically, three tests were performed at each impact velocity to obtain confidence in experimental repeatability. The resulting failure processes were imaged using photoelasticity in conjunction with high-speed digital photography, and the applied load and load-point displacement were determined using the two-point strain measurement technique. The following sub-sections discuss the experimental observations. 4.1. Dynamic fracture behavior for low velocity impact Fig. 3 shows a sequence of photoelastic fringe patterns obtained for a Homalite– aluminum single-edge-notched specimen subjected to dynamic three-point bend loading. The projectile impact velocity was set to 2.3 m/s, which represented the lower limit of the loading rate regime investigated. Also, note that only nine frames (out of a total of sixteen) are shown in Fig. 3. Furthermore, the field of view of the camera was adjusted so that only the central part of the specimen was photographed, thus enabling the imaging of the main crack, the interface and the impact loading point with sufficient resolution. From the digital images it was observed that the main crack initiated 173 ms after arrival of loading pulse at specimen, and subsequently propagated under predominantly mode-I (opening) conditions till it reached the Homalite–aluminum interface. The crack then arrested at the interface while the dynamic loading on the specimen continued to increase further. The delamination crack initiated at 233 ms after the arrival of the loading pulse at the specimen, and then started to propagate along the Homalite–aluminum interface. This interface crack arrested as the loading pulse passed through the specimen and no further crack propagation was observed for the remaining duration of the experiment. The purely anti-symmetric isochromatic fringe patterns around the interface crack indicate that interface delamination occurred under mode-II (shear) dominated conditions. Fig. 4 shows a plot of the main crack and the interface delamination crack positions as a function of time elapsed after the arrival of the loading pulse at the specimen. This figure highlights the instances of initiation, propagation and arrest of both the main crack and the interface delamination crack. From the time history of crack extensions, the average propagation speed of the main crack was determined to be 150 m/s and that of the interface delamination was 21 m/s. An observation of the specimen, at the end of the test, showed that delamination occurred only on the Homalite–aluminum interface that was on the same side as the main crack. While the delamination crack grew further with repeated loading (due to multiple pulse reflections), no other fracture was observed and the final specimen was recovered physically intact. The strains recorded during the dynamic fracture event were analyzed to determine the load-point displacement and the load-point force, as discussed earlier. The dynamic variation of these quantities is plotted in Fig. 5. It was observed that
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Fig. 3. Sequence of photoelastic fringe patterns obtained for dynamic three-point bend loading of a Homalite–aluminum single-edge notched specimen subjected to low velocity impact.
the main crack initiated during the rising portion of the load pulse. However, the interfacial delamination initiated at the tailing end of the loading pulse. For this experiment, the maximum load sustained by the specimen was 550 N. 4.2. Dynamic fracture behavior for high velocity impact Similar experiments, as those discussed above, were carried out for higher impact velocities of 4.4, 5.6, 6.6 and 7.6 m/s. Fig. 6 shows a sequence of isochromatic fringe
298
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Crack Extension (mm)
5 Main Crack Delamination
4 3 2 1 0 0
50
100
150
200
250
300
350
400
450
Time (µs)
Fig. 4. Extension of main crack and interfacial delamination for low velocity impact.
500
1.0
Force
300 200
Displacement
Delamination Initiation
400
Main Crack Initiation
Load-point Force, F(t), (N)
Experiment: ML025 Impact Velocity: 2.3 m/s
0.8 0.6 0.4
100
0.2
0
0.0 0
50
100
150
200
250
300
350
400
Load-point Displacement, uE (mm)
1.2
600
450
Time (µs) Fig. 5. Variation of dynamic load-point displacement and load-point force for low velocity impact.
patterns obtained for a Homalite–aluminum fracture specimen subjected to dynamic three-point bend loading with an impact velocity of 4.4 m/s. As before, only select frames are shown and the field of view is centered on the region of interest. In this case, the main crack initiated 120 ms after the arrival of the loading pulse, and then propagated under mode-I (opening) conditions till it arrested at the Homalite– aluminum interface. The delamination crack initiated 165 ms after the arrival of the loading pulse, and started to propagate along the interface. However, shortly afterwards, the delamination crack arrested and two cracks re-initiated in the Homalite layer across the aluminum layer. The re-initiated cracks originated at the same location as the interfacial delamination crack and continued to propagate,
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Fig. 6. Sequence of photoelastic fringe patterns obtained for dynamic three-point bend loading of a Homalite–aluminum single-edge notch specimen subjected to high velocity impact.
while the interface did not delaminate any further. The aluminum layer was intact during this entire failure process and provided closing tractions as shown in frames 11 and 14 of Fig. 6. The time histories of propagation for the main crack, the delamination crack and the re-initiated cracks are plotted in Fig. 7. As before, average crack speeds were obtained from the crack extension versus time data. The average speeds for the main crack, the interfacial delamination, and the two re-initiation cracks were determined to be 250, 40 and 325 m/s, respectively. Dynamic variations of the load-point
300
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Crack Extension (mm)
Experiment: ML023 Impact Velocity: 4.4 m/s
12
Main Crack Delamination Crack Re-initiation
9
6
3
0 0
50
100
150
200
250
300
350
Time (µs)
Fig. 7. Extension of main crack, interfacial delamination and re-initiated cracks for high velocity impact.
600 400 200
1.0 0.8 0.6 0.4 Crack Re-initiation
800
Displacement Delamination Initiation
1000
Main Crack Initiation
Load-point Force, F(t), (N)
Experiment: ML023 Impact Velocity: 4.4 m/s
0 0
50
100
150
200
0.2 Force
0.0 250
300
Load-point Displacement, uE (mm)
1.2
1200
350
Time (µs) Fig. 8. Variation of dynamic load-point displacement and load-point force for high velocity impact.
displacement and the load-point force were determined from the measured strain signals, and are plotted in Fig. 8. It was observed that the main crack initiated during the rising portion of the load pulse, while the interfacial delamination initiated near the peak load. For this experiment, the maximum load sustained by the specimen was 1090 N. Experiments conducted at impact velocities of 5.5, 6.6 and 7.6 m/s resulted in similar observations. The dynamic fracture process, for all these loading rates, resulted in limited interfacial delamination followed by re-initiation of cracks across
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the aluminum layer. The times associated with initiation of main crack, interfacial delamination and crack re-initiation were determined from the high-speed images of the fracture process. The average crack speeds were also calculated as for other experiments. In addition, the applied load and load-point displacement were determined from the recorded strain signals. It was observed that the times associated with various fracture events and the maximum load sustained by the specimen were all functions of the impact velocity and hence the loading rate. 4.3. Effect of loading rate on the dynamic failure process Based on the experimental observations it is possible to identify two different modes of dynamic failure. For an impact velocity of 2.3 m/s, only interfacial delamination was observed and the specimen was recovered intact. Meanwhile, for all higher impact velocities, 4.4–7.6 m/s, limited interfacial delamination followed by crack re-initiation in the second Homalite half across the aluminum layer was observed. These two modes of failure are schematically illustrated in Fig. 9. The fundamental issue of interest is the identification of loading conditions that will lead to crack re-initiation across the aluminum layer reinforcement. It is believed that this transition is a function of the rate-dependent fracture toughness of Homalite-100 and also the adhesive layer. The effect of loading rate on the yield and post-yield hardening characteristics of the aluminum layer is also expected to contribute to this failure mode transition. Fig. 10 shows the times associated with the initiation of the main crack, interfacial delamination and crack re-initiation. As the impact velocity, and thus the loading rate, was increased the time associated with each failure event decreased. This is inline with the fact that at higher impact velocity, the load applied to the specimen increases at a faster rate and thus crack initiation can be expected to occur earlier during the dynamic loading process. However, it was also observed that the applied loads associated with the initiation of the main crack and interfacial delamination also increased, as shown in Fig. 11. These observations indicate that both the dynamic failure process and plausibly the failure mode transition are affected by the rate-dependent properties of Homalite, aluminum and the interfacial bond. Finally,
Re-initiation
Delamination
(a)
(b)
Fig. 9. Two dynamic failure modes of Homalite–aluminum layered specimens: (a) interfacial delamination, and (b) interfacial delamination followed by crack re-initiation.
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300
Time, t, (µs)
250 200 150 100 50 0
2
3
4
5
6
7
8
Impact Velocity (m/s)
Fig. 10. Time associated with the initiation of the main crack, interfacial delamination and re-initiated cracks, as a function of impact velocity.
4000 Load at Main Crack Initiation Load at Delamination Initiation Peak load
Dynamic Load, NE, (N)
3500 3000 2500 2000 1500 1000 500 0 2
3
4
5
6
7
8
Impact Velocity (m/s)
Fig. 11. Applied load associated with initiation of main crack and delamination, as a function of impact velocity.
it was also observed that the peak load for any given experiment was consistently higher than the load required to initiate the main crack. These observations imply that the incorporation of a thin ductile reinforcement layer can increase both the overall fracture toughness and strength of a nominally brittle material. In order to quantify the conditions governing the dynamic failure process and failure mode transition, it is necessary to examine the histories of crack-tip driving tractions and energy release rate as functions of applied loading rate. These parameters are normally quantified by analyzing the experimentally obtained isochromatic fringe patterns around the crack tip, in conjunction with the stressoptic law, given by Eq. (8), and appropriate asymptotic solutions that describe the
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stress fields surrounding a crack propagating dynamically along the interface of a ductile layer in a brittle material. The elastodynamic stress fields, in the vicinity of a crack propagating along a bimaterial interface, are given in terms of the complex stress intensity factor, K d ðtÞ; as [25] 1 1 sij ¼ pffiffiffiffiffiffiffiRe½K d ðtÞrieðvÞ s* 1ij ðy; vÞ þ pffiffiffiffiffiffiffiIm½K d ðtÞrieðvÞ s* 2ij ðy; vÞ; 2pr 2pr
ð9Þ
where r is the radial distance from the crack tip, s* 1ij and s* 2ij are known functions of angle y and crack velocity v; eðvÞ is the oscillatory index, andK d ðtÞ ¼ K1d ðtÞ þ iK2d ðtÞ: Note that a non-zero oscillatory index results in a coupled and inherently mixedmode stress field that is typical for cracks along bimaterial interfaces. The existence of K-dominant crack tip fields, such as those given by Eq. (9), for interfacial crack propagation in multilayered specimen geometries has been examined only for the case of quasi-static crack propagation along the interface of an elastic reinforcing layer in a brittle material. Based on linear-elastic finite element analyses it was demonstrated that the region of K-dominance near the crack tip is very limited, extending merely to B1/10 of the height of the reinforcing layer [26]. The dynamic loading situation is even more complex and is expected to result in further limitations on the spatial and temporal validity of analytical expressions of the crack tip stress fields. It has been shown that the behavior of a dynamically loaded three-point bend fracture specimen can be characterized by a short-time response dominated by discrete waves, and a long-time response that is essentially quasi-static (yet still rate dependent) [27]. Short-time response can be distinguished from long-time behavior in terms of a transition time, tT ; as the time at which the kinetic energy in the specimen is approximately equal to the deformation energy. The effects of material inertia dominate prior to the transition time, but the deformation energy dominates at times significantly greater than tT : Thus, if failure events occur at tbtT ; then inertial effects can be neglected and quasi-static models can be applied to interpret experimental observations. The transition time is given as [27] H tT ¼ DS ; ð10Þ c0 where S is a dimensionless shape factor, H is the width of the specimen, c0 is the longitudinal bar wave speed in the specimen and D is a dimensionless coefficient that depends implicitly on the impact conditions ’ tDðtÞ ; D¼ ð11Þ DðtÞ tT
where D and D’ are the displacement and velocity at the load-point and t is the elapsed time from the instant of impact. For the current-loading geometry the transition time was estimated to be typically 150–200 ms, while the typical time to failure was 150–300 ms. Since the time to failure is very close to the characteristic transition time, the relevant near-tip fields cannot be inferred from quasi-static models and the stress-wave dominated inertial effects must be taken into account.
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Based on the above discussion it is evident that analytical solutions for transient crack propagation along the interface of a thin layer embedded in a brittle material would be required to be able to quantify the dynamic failure process based on experimental observations. Furthermore, these analytical solutions would also have to account for the constrained plastic deformation expected to occur in the ductile reinforcing layer. The development of appropriate analytical solutions of this nature is difficult, and the only resort would be to conduct dynamic finite element simulations for modeling the fracture process. Such a simulation would then enable the extraction of the crack-tip driving tractions and energy release rate as functions of applied loading rate. Such simulations are currently in progress and will be published in the near future [28].
5. Summary An experimental investigation was conducted to characterize the dynamic failure behavior of a brittle material reinforced with a single ductile layer, as a function of loading rate. Model specimens were prepared by sandwiching a thin layer of ductile aluminum between two thick layers of brittle Homalite-100 using Loctite Depend 330 adhesive. A naturally sharp edge crack, introduced in one of the Homalite-100 layers, served as the starter precrack. The specimens were subjected to dynamic three-point bending in a modified Hopkinson bar at projectile impact velocities of 2.3, 4.4, 5.6, 6.6 and 7.6 m/s. The entire dynamic failure phenomena of crack initiation, interlayer delamination and crack re-initiation were observed using dynamic photoelasticity combined with high-speed digital photography. In addition, strains recorded at two locations on the Hopkinson bar were analyzed using the twopoint strain measurement technique to determine the time histories of the load and load-point displacement were obtained. Two distinct mechanisms of dynamic failure were identified, as a function of loading rate. At lower loading rates (impact velocity of 2.3 m/s), the starter crack arrested on reaching the aluminum layer and then caused delamination along the aluminum–Homalite interface. This delamination arrested as the loading pulse passed through the specimen and no further fracture was observed. On the contrary, as the loading rate was increased, interfacial delamination was followed by crack reinitiation in the Homalite layer opposite to the initial starter crack. The re-initiated cracks originated at the same location as the interfacial delamination crack and continued to propagate, while the interface did not delaminate any further. It was determined that the times required for crack initiation, delamination and crack re-initiation decreased, as the loading rate was increased. However, it was also observed that the applied load values associated with each event increased with increasing loading rate. These observations indicate that both the dynamic failure process and plausibly the failure mode transition are affected by the rate-dependent properties of Homalite, aluminum and the interfacial bond. Finally, based on the measured peak loads and the observed failure mechanisms it was concluded that the incorporation of a thin ductile reinforcement layer can increase both the overall
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fracture toughness and strength of a nominally brittle material. Quantification of the crack-tip driving tractions and energy release rate, as functions of applied loading rate, is currently in progress.
Acknowledgements The authors are indebted to Prof. Arun Shukla at the University of Rhode Island for providing the experimental facilities that made this investigation possible. Notably, we acknowledge the availability of the Imacon 200 high-speed digital camera (supported by the National Science Foundation Grant No. CMS 9870947). Raman P. Singh also thanks Prof. Toshio Nakamura at the State University of New York, Stony Brook for the several discussions regarding this research effort.
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