Energy balance in dynamic fracture, investigated by a ... · PDF fileEnergy balance in dynamic fracture, investigated by a potential drop technique J.A. HAUCH and M.P. MARDER Center

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International Journal of Fracture90: 133–151, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

Energy balance in dynamic fracture, investigated by apotential drop technique

J.A. HAUCH and M.P. MARDERCenter for Nonlinear Dynamics and Department of Physics, University of Texas at Austin, Austin, Texas 78712,U.S.A.

Received 23 January 1997; accepted in revised form 8 January 1998

Abstract. A puzzling question in dynamic fracture has been why cracks in amorphous brittle materials alwaystravel at velocities smaller than the Rayleigh wave speed. The answer is that the energy per length needed for thecrack to propagate depends strongly on velocity. As the energy flux to the crack tip increases, the crack choosesnew modes of dissipation such as micro-cracking and the creation of subsurface damage zones to dissipate thisenergy. In this paper we use a potential drop technique to measure length and velocity of a crack with highspatial precision and time resolution so as to investigate the modes of dissipation in Homalite-100 and makequalitative comparisons with PMMA. The technique is capable of resolving crack initiation, run, and arrest. Usingthis technique we search for a ‘forbidden band’ of velocities in PMMA, Homalite-100, and glass, and we showthat no such velocity gap exists in these amorphous materials at room temperature.

Key words: Energy balance, dissipation, fracture energy, velocity gap, potential drop technique.

1. Introduction

Ever since Griffith broke glass rods in his laboratory in England and realized that flaw growthis only possible if the energy released by crack advance is larger than the energy needed tocreate the surface (Griffith, 1920), energy and how it is dissipated has been at the heart ofquantitative fracture mechanics. In 1947 Mott realized that the inclusion of a kinetic energyterm in Griffith’s framework can extend this energy approach to encompass dynamic fracture(Mott, 1947). He found that the velocity of a crack should asymptotically approach a terminalvelocity, which Stroh proposed should be the Rayleigh wave speed in the material (Stroh,1957). In fact this result was already implicit in the results of Yoffe (Yoffe, 1951). As correctedby Dulaney and Brace in 1960 (Dulaney and Brace, 1960) Mott’s scaling argument remainedessentially unscathed, despite tremendous increases in the mathematical sophistication of dy-namic fracture mechanics over the next years (Freund, 1990). However there was a difficulty.Cracks in brittle amorphous materials never were observed to reach the Rayleigh wave speed(Kobayashi et al., 1974; Irwin et al., 1979).

This difficulty was always more apparent than real. On the one hand, cracks on the cleavageplanes of brittle crystals, or along weak interfaces did achieve velocities close to the Rayleighwave speed (Gilman et al., 1958; Hull and Beardmore, 1966; Cotterell, 1965; Field, 1971).On the other, no theory of dynamic fracture was able to make any predictions about thevelocity of a crack without making a presumption about the energy needed per unit lengthto propagate a crack. Most dynamical equations assumed this quantity to be a constant (Hall,1953). However, the experimental results can easily be explained by dropping this assumptionand allowing fracture energy to increase with velocity. For a long time it has been known that

134 J.A. Hauch and Marder

cracks in brittle polymers at high stress intensity factors attempt to branch, with the branchessubsequently dying out leaving behind a series of short micro-cracks at an angle to the maincrack (Irwin et al., 1979; Kobayashi and Mall, 1978; Kobayashi and Dally, 1977; Dally, 1979;Ravi-Chandar and Knauss, 1984a). In the brittle plastics PMMA and Homalite-100 Doyle(1983) and Ravi-Chandar and Knauss (1984) showed that the increase in fracture energywas connected to the generation of micro-cracks underneath the fracture surface. The centralquestion in dynamic fracture mechanics therefore was reduced to determining why energyconsumption should rise so abruptly past a critical threshold, or putting it more properly, whywhen energy flux to the crack tip exceeded a threshold, the mean velocity stopped increasing.

A tentative answer to this question has been given in the context of lattice calculations.In analytically solvable models as well as extensive computer simulations, it has been shownthat when energy flux to a crack tip passes a critical threshold, the tip becomes unstable,sometimes to micro-cracks and sometimes to dislocations (Liu and Marder, 1991; Marder andLiu, 1993; Marder, 1993; Marder and Gross, 1994; Abraham et al., 1994; Zhou et al., 1994).Some of the images produced by these simulations bear great visual resemblance to imagesof PMMA past its threshold of instability (Sharon et al., 1995; Sharon et al., 1996). Theatomic scale calculations make additional predictions as well. The most striking is that thereshould be a range of velocities, roughly between 0 percent and 20 percent of the Rayleighwave speed, at which steady crack motion would be impossible (Marder, 1993; Marder etal., 1994), a process also known as lattice trapping (Thomson et al., 1971; Sinclair and Lawn,1972; Gumbsch, 1995). All the calculations are really appropriate for comparison with crystalsat low temperature, not glass at one third of its melting temperature. Still one might easilywonder if in brittle amorphous solids such a velocity gap exists.

Our purposes in this paper are threefold. First we intend to relate details of the dynamicpotential drop method we have employed for several years to investigate crack dynamics. Themethod is comparatively simple and inexpensive, and our discussion may enable other groupsto utilize it. Second we investigate whether micro-cracks in Homalite-100 provide the samequalitative explanation for the rise in fracture energy that Sharon et al. have demonstrated inPMMA. Finally we ask whether a velocity gap exists in glass, PMMA, and Homalite. Despiteprevious misleading indications the answer is no.

2. Experimental technique

In dynamic fracture mechanics crucial data allowing quantitative and qualitative comparisonwith theory are the length and the velocity of the crack as a function of time or each other, andthe stress state of the sample. These quantities cannot be obtained through measurement witha single localized probe. Several measurement techniques have been developed to measurethese quantities. The techniques are variations on four major themes.

(1) One is high-speed photography, from which measurement of crack velocities have beenobtained for over 50 years (Schardin et al., 1955; Dally, 1979; Ravi-Chandar and Knauss,1984a, 1984b, 1984c, 1984d; Knauss and Ravi-Chandar, 1985, Field, 1971). This techniquehas two limitations in measuring fast fracture. First, cameras cannot be triggered to capturecrack motion unless the time of initiation is known with some certainty (Wenner and Rogers,1977). This fact dictates rather violent loading conditions for such experiments, since thesecan be guaranteed to produce crack motion in a short window of time. Second, it has beenimpossible to gather more than 128 frames in a single experiment (Ravi-Chandar and Knauss,

Energy balance in dynamic fracture135

1982), and therefore the time series for crack motion cannot be known in great detail. Anadvantage of this technique however is that in a variety of birefringent materials it can alsoyield information about the dynamic stress intensity factor at the crack tip in addition to thecrack length.

(2) A second is the resistive grid technique, in which a series of resistive wires is laid downon the sample, and is broken by a running crack (Kobayashi et al., 1974; Wenner and Rogers,1977; Cotterell, 1965). Sample preparation for this method is usually difficult and resolutionis limited by the experimenters’ ability to lay down thin wires and interface with them.

(3) A third method that has also been applied widely is acoustic modulation of the cracktip. In this type of experiment the crack tip is deflected by ultra-sound waves introduced intothe specimen, leaving behind a set of periodic markings that can then be measured and reflectthe velocity of the crack tip (Field, 1971; Kerkhof, 1973). This type of measurement yieldslength and velocity data; however, it is intrinsically intrusive and may modify the dynamicsof the crack tip, particularly in the presence of instabilities. Another disadvantage is that thetechnique is limited to materials that produce smooth fracture surfaces, and have low acousticattenuation.

(4) Potential drop techniques measure crack length through changes in the resistance pro-duced by crack propagation in a specimen. The method has widely been used to measurequasi-static crack growth for conductive specimens (Wenner and Rogers, 1977; Buck, 1989).It can be adapted to measure the surface velocity in insulators by applying conductive coatings(Stalder et al., 1983; Béguelin et al., 1983; Stalder and Kausch, 1985; Stalder et al., 1989;Cudré-Mauroux et al., 1991; Wang and Kim, 1993). The resolution of this technique is limitedonly by deviation of the crack from the straight cut assumed in analysis, and by the speed of theacquisition equipment. Crack velocity can therefore be obtained at higher rates, with greateraccuracy, and greater density of measurements than other techniques. In addition, the methodcan be applied to a wide range of materials and fracture processes ranging from polymers tosingle crystals, and from fast fracture and crack arrest to interface fracture in virtually everyspecimen geometry.

In this paper we use a variation of the potential drop technique that has been developed at theCenter for Nonlinear Dynamics. The technique yields measurements of the crack length witha resolution of 0.2 mm and the crack velocity with a resolution of 10 m/s or better at a temporalresolution of 5 MHz. With this method data sets of 64000 points for both the length and thevelocity are obtained on both sides of the specimen simultaneously, allowing full resolutionof initiation, run and arrest for fracture processes lasting up to 3.2 ms. We use the techniqueto investigate the dynamics of cracks in PMMA, Homalite-100 and glass.

2.1. COATING

The main ingredient of potential drop techniques is a conductor that is ruptured during thefracture process. As the fracture proceeds, the resistance of the conductor changes, and pro-vides information on the length and velocity of the crack. The conductor may be the fracturesample itself, or if the specimen is an insulator, it can be a thin metallic coating that is appliedto the fracture sample. All experiments described in this paper concern insulating samplescoated with aluminum, since aluminum is brittle and a good conductor. The coating is evapo-rated onto the samples in a vacuum of 10−6 Torr. To assure that the coating does not affect thephysical properties of the sample its thickness must be very small in comparison to the samplethickness. In our experiments the coating thickness is varied with sample size but is between


Figure 1. The figure shows the coating geometry and electrode placement on the fracture sample. Plotted above theplate is the current density in the coating obtained by solving Laplace’s equation. The graph shows the resistanceas a function of crack length, simulation and calibration. The inlay is an enlargement of the calibration data. Itdemonstrates that there are no small scale variations within the resolution of the measurement.

20–50 nm in all cases. The resistance as a function of crack length is sensitive to the geometryof the coating, and placement of the electrodes on the sample. Coating geometry and electrodeplacement used in our experiments are shown in Figure 1. This particular configuration waschosen for the linearity of resistance as a function of crack length, which can be obtainedby solving Laplace’s equation subject to the appropriate boundary conditions. A surface plotof the current density in the coating is depicted in Figure 1. To ensure good agreement withsimulation the functional form of the resistance was checked in PMMA by cutting the coatingand measuring the resistance. Figure 1 compares the two measurements, showing only slightdiscrepancies in the beginning and the end that can be attributed to the evaporation processby which the aluminum was deposited on the sample. The evaporation geometry results in aslightly thicker coating in the middle of the sample. To alleviate this problem it is possible totake the samples after the experiment and measure the functional form of the resistance on asample by sample basis in detail. This was not done in our experiments since the variationswere found to be less than the resolution of the digitizers used. The average resistance gradienthowever was measured on each sample by bridging the crack with a piece of copper in orderto account for variations in the average thickness of the coating between samples, which couldbe as high as 15 percent.

To gain confidence in the method several issues need to be addressed. When the samplesare loaded there will be a change in the cross sectional area of the coating that will affect itsresistance. The change in area is determined by sample strainε and the Poisson ratioν of thecoating material, and is given by:δA/A = νε2 ' 0.345×0.12 = 3.45×10−5. This will resultin a maximum shift in resistance of:δR = δA/A×Rmax ' 50�×3.45×10−5 ' 2×10−3�,which is an order of magnitude smaller than the resolution of the instrumentation.

Another concern is electrical discharge across the crack. The potential drop between theelectrodes increases with the length of the crack, but never exceeds 3 V over the useful rangeof the measurements. The largest possible potential drop is then across the crack at the positionof the electrodes. To exceed a critical electric field of 106 V/m for sparking in air this entirevoltage drop must occur over a distance of 3µm. The separation of the crack faces at theposition of the electrodes will be on the order of the total strain of the plate or about 50µmin the polymers, which makes discharge impossible. In glass, however, the separation of the


crack faces is much smaller, only about 2–3µm, and electrical discharge has been observed inour experiments. The process lasts about 1µs and is clearly distinguishable from the fracturedynamics, since it is characterized by an apparent jump to a negative velocity followed by arapid apparent jump to a large positive velocity that quickly decays.

Small scale variations of the coating thickness could also affect the data. The method willbe most sensitive to variations in the coating at the position of the peak in the current densityin front of the crack tip (Figure 1). The full width at half maximum of this peak should thusdetermine a length-scale on which variations in the coating thickness will cause problems.From the simulations the full width at half maximum is found to be greater than 1 percent ofthe plate length which is on the order of millimeters. The length-scale on which the coatingthickness is expected to vary due to the vapor deposition process can be estimated by the vapordroplet size of the aluminum which is on the order of 1 nm. No statistical variations can be ex-pected for length-scales on the order of 1000 droplets, about 1µm. Thus thickness variationsin the coating pose no problem since the method is inherently an averaging process involvinga large area of the coating. In addition it can be seen from the inset of Figure 1 that there are nomeasurable small scale variations that are within the resolution of the measurement method.

Generation of static charge on the crack faces could also interfere with the measurements.We can estimate an upper bound for the currents due to charge generation if we assume oneelectron per bond will end up on one of the crack faces, and a bond density of 5× 1012 bondsper square millimeter. A crack traveling at 300 m/s a 3.2 mm thick plate will break bonds at arate of 5× 1018 bonds/s. If all the electrons end up on the same crack face this could lead toa current of 800 mA. To preserve charge neutrality, this charge might generate an additionalcurrent loop around the crack tip, but due to the principle of superposition it could have noeffect on the measurements at the position of the electrodes.

2.2. INSTRUMENTATION

To instrument the samples electrodes are attached on either side of the seed crack using aconductive paint that is made from a suspension of silver particles. The electrodes are partof a Wheatstone bridge powered by batteries for noise reduction. The output of the bridge isdigitized at a rate of 10 MHz by a Burr-Brown ZDP1002 12 bit Analog to Digital converter. Inaddition the output of the bridge is also fed into an analog differentiator that outputs a voltageproportional to the time rate of change of the input signal, and is then digitized at 20 MHz bya Markenrich WaagII 8 bit Analog to Digital converter. Details of the electronics are shown inFigure 2.

The bridge voltage time series can be inverted to find the length of the plate using theexpression

RPl = Rb

[Vbat(R1 + Ra)

RaVbat − VBr(R1 + Ra)− 1

], (1)

whereRPl is the plate resistance,Vbat is the input voltage to the bridge, andVBr is the outputvoltage of the bridge. The resistancesR1, Ra, andRb are the other three resistances in thebridge as defined in Figure 2. Using a lookup table generated by the solution of Laplace’sequation, or by direct measurement on the sampleRPl can then be converted to crack length. Asimpler procedure is to consider only the data in the linear resistance regime and approximate


Figure 2. The diagram shows the details of the instrumentation used in the experiments. The values ofRa = 500�

andRb = 250� were chosen to minimize the power dissipated in the plate.R1 is chosen individually in eachexperiment to match the initial bridge output to the range of the digitizers, and ranges from 30 to 70�.

RPl by a linear function with an offsetR0 due to contact resistance and a slope determined bythe derivative ofRPl

RPl = R0 + dRPl

dll, (2)

in which case an expression for the lengthl of the crack is obtained

l = RPl − R0dRPl

dl

= dl

dRPl

[Rb

(Vbat(R1 + Ra)

RaVbat − VBr(R1 + Ra)− 1

)− R0

]. (3)

The time rate of change of the output of the Wheatstone bridge,VBr is determined by

dVBr

dt= −Vbat

Rb

(Rb + RPl)2

dRPl

dt= −Vbat

Rb

(Rb + RPl)

dRPl

dl

dl

dt. (4)

Crack velocity can now be extracted using the calibrated response of the analog differentiators

Vdif = KdVBr

dt+ Vo, (5)

whereVdif is the differentiator output voltage andVo is an arbitrary offset voltage, and com-bining it with the voltage time series from the Wheatstone bridge to obtain

dl

dt= (Vo − Vdif)

KVbat

(Rb + RPl)2

Rb

dl

dRPl

. (6)


Figure 3. Crack length and velocity data from a full thin strip experiment in Homalite-100. The stress intensityfactor for this experiment wasKI = 0.52 MPa

√m which is in region where the crack velocity is highly sensitive

to KI . The linear resistance approximation of (2) was used to convert the data. The data clearly show that thisapproximation starts to fail for crack lengths greater than 23 cm for the 28 cm plate, which is to be expected fromthe calibration. In all later analysis data that falls into the nonlinear region is ignored.

Every quantity in this expression is known either from direct measurement or deduction. Theprecision of the measurements is usually limited by internal noise of the electronics, but canbe manipulated by adjusting the input voltage of the bridge and the coating thickness whichdetermines the magnitude of dRPl/dl in (2). Typical resolution in the experiments performedis 0.2mm for the crack length and 10 m/s for the crack velocity at a bandwidth of 5 MHz. Afull data set for both length and velocity for a thin strip experiment in Homalite-100 is shownin Figure 3.

To ensure consistency in the experiments these measurements are done on both sides ofthe sample simultaneously, resulting in two data sets for each experiment. The ZPD1002has a 256 Kbyte buffer for two channels, resulting in a 64 K sample, 6.4 ms long data set at12 bit resolution for each side of the plate. Similarly the WaagII has a 128 Kbyte buffer for 2channels, yielding a 64 K sample, 3.2 ms long data set at 8 bits for each side. Synchronizationis achieved by triggering both cards simultaneously with a comparator when the outout voltageof the Wheatstone bridge passes through 0 V, coinciding approximately with the middle of theplate. Both cards are set up such that they store pre-trigger information in half of their buffercapacity, and post-trigger information in the other half. The setup yields information for theentire fracture process in a typical experiment with duration between 1–3 ms.

2.3. THIN STRIP LOADING CONFIGURATION

One set of experiments was done in a thin strip configuration with fixed displacement bound-ary conditions identical to the loading used in (Fineberg et al., 1991, 1992; Gross et al., 1993;Marder and Gross, 1995; Sharon et al., 1995, 1996). Thin strip geometry refers to experimentswith a large aspect ratio L/W (Figure 4, i.e. specimen that are narrow and long). The advantageof this loading geometry is that it makes the task of determining energy flux to the crack tiptrivial. Once a fracture achieves steady-state conditions one has necessarily that the fractureenergyG and the stress intensity factorKI are determined by

G = K2I

E= σ 2

2EW = Eδ2

2W, (7)


Figure 4. Loading configurations: (a) The thin strip configuration was used in constant fracture energy experi-ments. The plate is loaded by displacing the edges by a constant distanceδ. (b) The Double Cantilever Beamconfiguration was used in crack arrest experiments. The plate is loaded by displacing the circular discs at a constantrateδ. During fracture the conditions are assumed to be quasi-static.

whereE is Young’s modulus,σ is the stress far from the crack tip andδ andW are definedin Figure 4. Steady-state here means that the crack has come into dynamic equilibrium withwaves reflecting from the upper and lower boundaries, but is sufficiently far from beginningand end of the sample. Sample dimensions in these experiments varied with an aspect ratioL/W between 3.3 and 3.6 in all experiments, but most experiments were done on specimenthat measured 20 cm× 5.5 cm with a thickness of 0.32 mm. Using these sample dimensionsand the two conditions for steady-state shows that the data for which the expression is validare acquired in the time interval from 60µs to 400µs. Since the fracture energy is fixed,constant velocity experiments (see Figures 3 and 5) can be realized in this configuration. As aresult, observations can be performed on comparatively large areas of fracture surface createdat the same fracture energy. In early experiments the thin strip boundary conditions wererealized by attaching oil hardened precision ground steel strips to the edges of the sampleswith Permabond 910 industrial grade adhesive. With this method in Homalite-100 the stressvariation along the length of the plate was less than 3 percent of the total stress. In later exper-iments a clamp type jaw was used and the samples were glued directly to the jaws, resulting inimproved boundary conditions. The edges were then displaced in our home built screw typetensile testing machine by a constant distance in a sequence of steps, such that the stress inthe sample increased by about 1 percent of the anticipated failure stress during each step. Iffailure did not occur within one minute, the process was repeated. In a successful experimentthe failure occurred 10–60 s after the stepping process so that effects due to the stepping of thejaws could be neglected. In order for the boundary conditions to remain constant throughout


the experiment, jaw motion due to machine compliance during the fracture process must alsobe kept to a minimum. The jaw motion can be estimated with the following expression

δjaw(t) = 1

6

F

M

t3

tbreak. (8)

Whereδjaw is the displacement of the jaw,F is the total load applied,M is the jaw mass,t is time andtbreak is the total duration of the fracture process. It should be noted thatδjaw

is independent of the compliance of the tensile testing machine, and solely depends on theapplied load, mass of the jaw and experiment duration. The total variation in stress due tothis effect is on the order of 2 percent, but it is proportional to the time cubed which makesit negligible for the initial stages of the experiment. The loading configuration is shown inFigure 4(a).

2.4. DOUBLE CANTILEVER BEAM LOADING CONFIGURATION

Our second loading configuration is a double cantilever beam type loading configuration usedfor crack arrest experiments. In this configuration circular steel discs were attached to thesamples on either side of the seed crack (Figure 4). The samples were then loaded by dis-placing the discs at a constant rate. Loading of this type is stable by energy considerations(Freund, 1990) and can lead to crack arrest (Kalthoff et al., 1977). Displacement rates inthese experiments were 5× 10−4 m/s. During the fracture process conditions can be assumedto be quasi-static. Crack extension will thus unload the crack, meaning that the availablefracture energy is slowly decreased until crack extension is no longer possible, and the crackarrests. Samples of PMMA, Homalite-100, and glass were loaded in this way. Crack arrestwas achieved in PMMA, and Homalite-100, but not in glass. The reason for this is most likelythat the tensile testing machine, with a compliance of 6.7 × 104 lb/in, is ‘soft’ compared tothe glass sample, resulting in boundary conditions that are closer to constant force than toconstant displacement rate leading to unstable fracture.

3. Results

3.1. DISSIPATION PROCESSES INHOMALITE -100

It is well-known that Homalite-100 as well as many other brittle materials exhibits a sharprise in fracture energy once the crack velocity surpasses a certain threshold. As a result of thisrise in fracture energy cracks in most brittle materials never reach the theoretically predictedsound speed. Since energy is not used to increase crack velocity it must be dissipated in otherways. This dissipation process should leave behind traces that make it possible to accountfor the energy introduced into the system. In PMMA it was shown by Sharon et al. that thelarge increase in fracture energy could be explained by a crack tip instability that sets in ata critical velocity (Sharon et al., 1995). Above the critical velocity the crack attempted tobranch off to the side with the side branches subsequently dying out, resulting in a series ofmicro-cracks. The combined surface area of these micro-cracks in addition to the area of themain crack accounts for nearly all the energy dissipated (Sharon et al., 1996). Here we wantto determine whether such micro-cracking, which is also well-known to exist in Homalite-100 (Irwin et al., 1979; Kobayashi and Mall, 1978; Kobayashi and Dally, 1977; Dally, 1979;


Figure 5. Velocity profile for three thin strip experiments in Homalite-100. At the top is a detailed view of thecrack initiation. The average acceleration during this stage is 2.7 × 108 m/s2.

Figure 6. (a) Average Velocity vs.KI for all thin strip experiments in Homalite-100. (b) Fracture energyG andnormalized fracture energyG/Go vs. Average Velocity for the thin strip experiments in Homalite-100. The dottedlines indicate the values reported by Dally for (1) minimum fracture energy, (2) onset of micro-cracking (3)successful branching.

Ravi-Chandar and Knauss, 1984), can provide the same explanation for the increase in dis-sipation in Homalite-100. Our findings show that this is not entirely the case, as there arelarge increases in dissipation even before the onset of micro-cracking. We believe that thisdissipation occurs below the fracture surface where the material structure is modified by thecrack without leaving traces on the crack surface. We find some evidence for this; however,thickness of the subsurface damage zone does not increase monotonically with energy flux aswe would expect.

The first step in the investigation was to determine the form of velocity as a function ofthe stress intensity factorKI . For this purpose a series of samples was loaded in the stripconfiguration. The values ofKI and the fracture energyG were then obtained from (7), and theaverage velocity was calculated using only the parts of the velocity records where the linear


resistance approximation in (2) holds. Typical velocity records for three different fractureenergies are shown in Figure 5. The results of all thin strip experiments in Homalite-100 aresummarized in Figure 6(a) and 6(b). Figure 6(a) shows the crack velocity as a function of stressintensity factor. Our experimental results are compared to data obtained in a large series ofexperiments done in the late 1970’s (Irwin et al., 1979; Kobayashi and Mall, 1978; Kobayashiand Dally, 1977). In a summary of these experiments by Dally (1979) effects due to specimengeometry were averaged to produce the solid curve shown in Figure 6(a) and our data arewithin the limits of this geometrical variation. Thus Figure 6(a) demonstrates the validity of(7) for the thin strip geometry. Also indicated in the figure are the minimum stress intensityfactor for which propagation is possible, the stress intensity factor where attempted crackbranching is first observed, and the value ofKI where successful crack branching occurred inthe specimen geometries used. An interesting feature of the thin strip geometry is that due tothe equilibrium with the side boundaries the instability is suppressed by reflected stress wavesand catastrophic branching prevented. This allows an extension of theKI vs. velocity curve tomuch higher values of the stress intensity factor, in our case up to 1.78 MPa

√m.

Figure 6(b) recasts the same data in terms of fracture energy as a function of velocity. Againthe same three regions that were indicated in Figure 6(a) are indicated. The plot clearly showsthere are two transition regions. The first transition occurs approximately atG = 160 J/m2

and another one occurs atG = 500 J/m2. For fracture energies between 80 J/m2 and 160J/m2

the fracture energy increases linearly with velocity at a rate of approximately 0.3 J s/m3; i.e.small changes in fracture energy produce large changes in crack velocity. In the range from160 J/m2 to 500 J/m2 the fracture energy still increases linearly with velocity but now at arate of 4 J s/m3, a factor of 13 larger, but changes inG still produce significant changes invelocity. Beyond energies of 500 J/m2 the slope is infinite and even large changes in energyflux have no effect on the crack velocity up to fracture energies of about 1000 J/m2 at whichpoint increases in fracture energy affect the crack velocity again. The second transition point at500 J/m2 coincides with the previously reported onset of the micro cracks. Beyond that pointall extra energy deposited in the system is dissipated by increasing the density and length ofthe micro-cracks.

Figure 7 shows a series of transmission microscopy pictures of the crack at several differentvalues of the fracture energy. The pictures have a depth of field ranging from 20µm to 100µmand thus show only a slice through the sample. Any particular micro-crack extends only afraction of the specimen thickness through the sample, such that the distribution of the micro-cracks is highly three dimensional, with different layers of cracks emerging as the microscopeis focused at different depths. The images obtained by this method show no micro-cracks forenergies less than 500 J/m2 in accordance with the findings by Dally et al. By this point theenergy used to create new surface has increased by 600 percent overG0, the minimum energyat which crack propagation is possible. Observation of the fracture surface by eye revealsthat between the first and the second transition the fracture surface changes uniformly from amirror surface to a surface of uniform roughness. Detailed measurements of surface roughnessin PMMA however have shown that the extra surface created due to this mirror-mist transitioncannot account for the large increases in energy observed. Ravi-Chandar and Knauss explainthe extra dissipation in terms of parabolic surface markings. From our observations at highfracture energies these surface markings are the surface signature of micro-cracks.

The first evidence of these parabolic markings was found atKI = 0.93 MPa√

m at anenergy 300 percent above the minimum, indicating that micro-cracking does exist for stressintensity factors below the previously reported values. These micro-cracks are not observable


Figure 7. Development of micro-cracking with increasing fracture energy (a)KI = 0.93 MPa√

m, G = 360 J/m2

(b) KI = 1.11 MPa√

m, G = 513 J/m2 (c) KI = 1.23 MPa√

m, G = 630 J/m2 (d) KI = 1.33 MPa√

m,G = 737 J/m2 (e) KI = 1.72 MPa

√m, G = 1233 J/m2. The pictures show a top view of the crack which

propagated from left to right. To accommodate the rapidly increasing length of the micro-cracks the magnificationis decreased. There is no indication of micro-cracking in (a) from this view.

by transmission microscopy for two resons. First their total length is only on the order of 15–25µm (Figure 8(c)) with an extension into the sample of only a fracture of the length, whichmakes them invisible due to diffraction of the light around the edge of the main crack, and dueto warping of the fracture surface which is larger than the extension of the micro-cracks intothe sample. Second their opening is very small, such that the total amount of light reflected bythem is negligible, making them invisible. Nevertheless the density of these parabolic surfacemarkings atKI = 0.93 MPa

√m is quite low, making it unlikely that these are responsible for a

factor of 4.5 increase in dissipation. This indicates that other subsurface mechanisms besidesmicro-cracking must be at play. A mechanism that appears to be important at these lowerenergies is subsurface flaw growth as was proposed by Knauss and Ravi-Chandar (1984). Alarge increase in subsurface flaw size is noticeable as the stress intensity factor increases from0.62 MPa

√m to 0.79 MPa

√m (Figures 8(a,b)). Using confocal microscopy it was determined

that these flaws exist only within a range of approximately 20µm of the surface, indicatingthat the fracture process is responsible for their creation. The growth of these flaws also con-stitutes a creation of new surface which will dissipate significant amounts of energy. As thestress intensity factor increases further the density of these flaws decreases again as the surfacestarts to break up due to river formations, in the vicinity of which the density of subsurfaceflaws is noticeably less.


Figure 8. Development of the fracture surface with increasing stress intensity factor, fracture energy (a)KI = 0.62 MPa

√m,G = 160 J/m2 (b)KI = 0.79 MPa

√m,G = 263 J/m2 (c)KI = 0.93 MPa

√m,G = 360 J/m2

(d) KI = 1.11 MPa√

m, G = 513 J/m2 (e) KI = 1.23 MPa√

m, G = 630 J/m2 (f) KI = 1.72 MPa√

m,G = 1233 J/m2. The parabolic markings become first visible in (c) and increase in size and density as the fractureenergy increases. They appear to be the surface signature of the micro-cracks in Homalite-100.

Figure 9. The figure shows the areas of the samples that are displayed in Figures 10 and 11. The detail on theleft is the sample edge that is shown in Figure 10, and the detail on the right is the sample corner that is shown inFigure 11. The three important surfaces are labeled: (A) fracture surface, (B) original surface of the sample and(C) surface created by subsequent fracture to reveal the structure inside.


Figure 10. Edge view of the fracture samples taken with SEM for increasing stress intensity factor, fracture energy(a) KI = 0.51 MPa

√m, G = 108 J/m2 (b) KI = 0.62 MPa

√m, G = 160 J/m2 (c) KI = 0.79 MPa

√m,

G = 263 J/m2 (d) KI = 0.93 MPa√

m, G = 360 J/m2 (e) KI = 1.23 MPa√

m, G = 630 J/m2 (f)KI = 1.72 MPa

√m, GI = 1233 J/m2. Figure 9 shows a sketch of the part of the sample that is imaged. In all

pictures the lighter surface on the top is the fracture surface (marked A in (a)) and the dark surface on the bottomis the top surface (marked B in (a)) of the sample. All pictures indicate some form of subsurface activity; howeverno clear trend is visible.

Figure 11. Three surface SEM view for two values of the fracture energy (a)KI = 0.51 MPa√

m, G = 108 J/m2

(b) KI = 0.79 MPa√

m, G = 263 J/m2. Figure 9 shows the part of the sample that is imaged. The surfaces aremarked in the following way: (A) fracture surface, (B) original surface of the sample and (C) surface created bysecond fracture to reveal structure inside.

To investigate the subsurface structure further a series of Scanning Electron Microscopepictures were taken of the edge of the fracture samples. The pictures shown in Figure 10reveal that there is distinguishable structure below the fracture surface at all fracture energies.The typical thickness of this damage zone is between 5–25µm. To determine if the observedprocess zone exists everywhere underneath the fracture surface the samples were carefullyfractured one more time, this time perpendicular to the original fracture surface, to revealthe structure below. This approach is intrinsically intrusive; nevertheless in several cases itrevealed structure underneath the fracture surface throughout the whole sample thickness.Figure 11 shows SEM pictures of the edge where the three surfaces meet. Underneath thefracture surface a filament structure exists that has a well defined boundary.


Creation of this damage zone should use up considerable amounts of fracture energy, butit is unclear how one could quantify its contribution to dissipation. A first attempt would beto measure the thickness of the damage zone as a function of fracture energy. However fromFigure 10 it is obvious that the thickness is not monotonically increasing with fracture energyas one would expect. Furthermore the appearance of this damage zone changes significantlywith fracture energy. Thus dissipation in the damage zone may not be directly proportionalto the zone thickness. The pictures lead to the conclusion that the main source of dissipationat all fracture energies is below the surface. However, we were puzzled to find that the widthof this damage zone does not appear to increase monotonically with energy until the onset ofmicro-cracking and we cannot account for this observation.

3.2. THE VELOCITY GAP

In the final part of this paper we want to turn our attention to the forbidden band of velocitiesthat is predicted by the analytical solution of the lattice models (Marder, 1993; Marder andGross, 1995). The prediction is that there is a forbidden band of velocities ranging from 0 m/sup to about 20 percent of the Rayleigh wave speed. The only way for a crack velocity to passthrough this range is in the course of large acceleration or deceleration. Previous experimentsin PMMA in the thin strip configuration have shown that fracture initiation is always accom-panied by a rapid acceleration stage up to about 20 percent of the Rayleigh wave speed. Theresults shown in Figure 5 similarly show an initial rapid acceleration stage for Homalite-100,in most cases all the way up to the steady state velocity. The experiments provide sufficienttime resolution to investigate crack initiation. The data in both materials clearly show thatthere is an acceleration in excess of 2× 10−8m/s2 visible in the surface velocity at initation.It is difficult however to attribute this behavior to any fundamental property since it may bedue to other fast processes that occur at the moment of initation. One possible explanation isthe three dimensionality of the plate, causing fracture to initiate preferentially in the middleof the plate and subsequently growing outwards causing a sharp jump in the surface velocityof the crack. Surface markings that grow spherically outwards at the beginning of the cracksupport this picture in Homalite-100. Another possible explanation is crack tip blunting dueto plastic or visco-elastic flow in the vicinity of the crack tip. If fracture initiates from a bluntseed-crack there must be a fast transition from a blunt crack to a sharp traveling crack. Thissharpening of the crack could also lead to a large acceleration visible in the surface velocity.

A way to alleviate both of these difficulties is to measure instead the arrest dynamics ofa crack, since a moving crack has a sharp crack front with an established three dimensionalprofile. The crack arrest experiments were implemented in a Double Cantilever Beam config-uration. In this configuration the unloading of the crack is slow compared to the time scalesavailable to the crack dynamics, allowing the crack always to seek out a steady state. In thevelocity gap picture presented by the molecular models a crack in such a configuration shoulddecelerate smoothly until some velocity on the order of 20 percent of the Rayleigh wavespeed and then rapidly arrest. Figures 12(a) and 12(b) show the results of such experimentsin PMMA and Homalite-100 respectively. Both experiments clearly show the characteristicvelocity jump at initiation that was also observed in the thin strip experiments. For the de-celeration stage that begins almost immediately after the initial velocity jump the molecularmodels predict some hysteresis, i.e. a slow deceleration down to a certain velocity lower thanthe initial velocity jump, followed by a very rapid deceleration down to zero. In both, PMMAand Homalite-100, the crack initally decelerates slowly with the deceleration growing at an


Figure 12. Velocity records for experiments in the Double Cantilever Beam configuration. In PMMA and Homa-lite-100 crack arrest was achieved. In both materials the crack appears to decelerate smoothly but with increasingdeceleration until it arrests. In glass crack arrest was not achieved in this configuration. However there is no signof an initial velocity jump. This can be attributed to the sharp seed cracks that can be generated in glass.

increasing rate as the velocity approaches zero. Nevertheless there is no clear transition pointbetween slow and rapid deceleration as would be expected from theory. The data clearly showsthat a forbidden band of velocities does not exist in PMMA and Homalite-100. Figure 12(c)shows the result of an experiment in the same loading configuration in soda-lime glass. Dueto its low thermal shock resistance it is relatively easy to introduce sharp seed cracks withan established crack-front into glass. Therefore glass does not suffer from the same problemsas the polymers during initiation. However crack arrest experiments in glass are extremelydifficult. No crack was successfully arrested in the DCB loading in glass, owing to the com-pliance of the tensile testing machine used. The Rayleigh wave speed in soda-lime glass is onthe order of 3000 m/s. The forbidden band of velocities in glass should extend from zero up toabout 600 m/s. The data in Figure 12(c) shows smooth acceleration up to a velocity of 60 m/s.This means that the forbidden velocity band picture also does not fit for glass. Thus, theseexperiments indicate that the theoretical picture obtained in brittle crystals at low temperaturedoes not apply to amorphous materials at room temperature.

4. Conclusion

In our investigation we used a variation of the potential drop method for measuring the lengthand velocity of running cracks. Instrumentation and data reduction were discussed in detail.The advantage of this technique is that it is capable of yielding a direct measurement ofcrack velocity that is sustainable at very high acquisition rates. Synchronization of the dataacquisition with initiation from a blunt seed crack is achieved with relative ease. Moreoverrecords of the full fracture process, including initiation, run, and arrest are attained at temporaland spatial resolution that yield useful information about the dynamics of running cracks.Motivated by the good qualitative agreement between the fracture of PMMA and the theoryof fracture of crystals at low temperatures, i.e. both display a micro-cracking instability which


constitutes the major mode of dissipation, we have investigated the fracture of Homalite-100to determine if this same picture is applicable. In previous work it was found that micro-cracking in Homalite-100 sets in at fracture energies which were a factor of 7 higher than theminimum fracture energy at which crack propagation is possible. Although we have foundparabolic markings that indicate the existence of micro-cracks at fracture energies signifi-cantly lower than that, a factor of 4.5 above the minimum fracture energy, it is clear thatin Homalite-100 other modes of dissipation must be significant. Post mortem observationson the fracture samples with a Scanning Electron Microscope show that at all energies thereexists a damage zone with a thickness between 10–25µm underneath the fracture surface. Thematerial structure in this zone is modified by filamentation and void growth that at low fractureenergies remain hidden below a perfectly smooth fracture surface. At low fracture energies thisdamage zone is probably the main source of the dissipation, while at higher energies a complexthree dimensional set of micro-cracks evolves and becomes the main source of dissipation.We are puzzled by the observation that the thickness of the damage zone does not increasemonotonically with fracture energy, and it is unclear how it evolves until the onset of micro-cracking. It appears that the fracture processes of Homalite-100 are much more complex anddifficult to describe than the processes in PMMA. We also investigated the existence of aforbidden band of velocities that exists in the molecular models. In PMMA and Homalite-100we observe a velocity jump during crack initition that we attributed to crack-tip blunting orthree dimensional effects. However our experiments showed that no such forbidden band ofvelocities exists in amorphous materials at room temperature.

Acknowledgements

This project was supported by a TARP (Texas Advanced Research Program) grant issuedfor the investigation of dynamic fracture. Special thanks to the Cell Research Institute of theUniversity of Texas at Austin for the use of their Scanning Electron Microscope, and JohnMendenhall, Barbara Goettgens for their time.

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