A reflected shear-wave technique for determining dynamic rock strength

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 14, pp. 247-259. Pergamon Press 1977. Printed in Great Britain

A Reflected Shear-wave Technique for Determining Dynamic Rock Strength C. YOUNG* O. D U B U G N O N t The states of stress that may be obtained by conventional plane-shock-wave loading of rocks are not adequate to determine dynamic strengths at pressures of practical interest. Although the desired states of stress can be obtained with spherically or cylindrically diverain0 shock waves, the high-explosive or high-energy-discharge systems required make it difficult to obtain laboratory quality data. Since the controlled reflection of a planar shock wave can also produce the desired states of stress and since electromagnetic gauges can be used to make high quality in-material measurements, a wave reflection technique has been developed and used to study the dynamic failure of Solenhofen limestone. A conventional gas gun impact facility, capable of launchin# 50 mm dia projectiles at up to 500 m s-1, is used to generate a primary longitudinal shock wave in the limestone targets. The targets are constructed with a reflection interface inclined to the incident lonoitudinai wave so as to generate a large amplitude reflected shear wave. Electromaonetic particlevelocity and shear-stress gauges are placed within the targets to measure the amplitudes and wave velocities of the incident lonoitudinal and the reflected shear and longitudinal waves. With these data the states of stress realized in the taroet with the superposition of various waves is calculated and used to deduce the dynamic strength of the target material. For Solenhofen limestone the dynamic shear strength, at loading rates up to 106 s-i, does not exceed statically determined values by more than 25%.

INTRODUCTION Evaluation of novel rapid excavation and drilling techniques and further optimization of conventional rock breakage by explosives both require knowledge of the strength and failure characteristics of rocks at high rates of loading. Loading rates up to 103 s - t may be obtained with split-Hopkinson bar devices [I-3] but only a quasi-uniaxial state of stress may be obtained unless the experimentally complicated provision is made to subject the sample to a confining pressure prior to its dynamic loading [4, 5]. High explosives and exploding wires have been used to generate spherical 16"1 and cylindrical [7"1 shock waves in materials, but the dominating effects of the geometrical wave divergence make data from such experiments difficult to analyze. Plane-shock-wave techniques utilizing either planar impact or explosive lenses [8] can be used to obtain loading rates to at least 10e s-1 and have provided valuable data on the dynamic failure of many materials, especially ductile metals. For materials with * Institut CERAC SA, CH-1024 Ecublens, Switzerland; present Department,Colorado State University, Fort CoUin& CO 80523, U.~Jk. t Institut CERAC SA, CH-1024 Ecublens, Switzerland.

address: Civil Engineering

a strongly pressure-dependent yield behavior, of which rocks and ceramics are the best examples, the states of stress resulting from the uniaxial strain of plane waves will not effect failure until pressures generally higher than characteristic of novel and explosive rock fragmentation are obtained. There is thus a need for new experimental techniques which can be used to study the dynamic failure of rocks, and other brittle materials, in stress- and strain-rate regimes of practical interest. Any experimental loading technique can only be as valuable as the diagnostic techniques which can be employed with it. Both in-material and external gauge techniques have been successfully developed and applied to shock-wave loading characteristic of the above experimental techniques [8]. The in-material techniques, which include various piezoresistive pressure gauges, optical techniques giving in-material accelerations and particle velocities for transparent materials, and electromagnetic techniques giving particle velocity and stress in non-conducting materials, have the advantage that the gauge emplacement does not perturb wave propagation. The external gauge techniques, such as rear surface piezoelectric crystals, mirrors permitting rear surface velocity measurements and pres-

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C. Young and O. Dubugnon

sure bars, have the strong limitation that wave reflections from the gauge material or the free surface perturb subsequent deformation and wave propagation in the target material. For shock-wave measurements in materials that are both opaque and non-conducting, the electromagnetic gauge techniques have several desirable characteristies [8, 9]. As the output voltage generated by the electromagnetic gauges can be directly converted by Faraday's law to the in-material particle velocity, no gauge calibration is necessary and both the precision and accuracy of the measured signals are higher than for piezoresistive pressure gauges. In materials like rocks and composities which can be highly heterogeneous the electromagnetic gauges have the further advantage that perturbations of the shock front caused by this heterogeneity do not severely perturb the measured signal. In contrast the small-scale deformations imposed upon piezoresistive gauges in heterogeneous media can significantly alter the resistance and thus the pressure measured by the gauge. Finally, because of their extreme simplicity, the electromagnetic gauges may be emplaced in the target materials of interest by any of a number of techniques. The simplicity and reliability of electromagnetic gauge techniques permitted us to consider developing novel shock-wave loading techniques for determining the dynamic shear strength of rocks and other brittle materials. As electromagnetic gauges can be used to directly measure the in-material particle velocities and even the stresses associated with large amplitude shear waves [10] consideration was given to experimental techniques for shear-wave generation. The states of stress associated with large amplitude shear waves could be used to directly deduce the dynamic shear strength of the material of interest. If large amplitude shear waves could also be generated in materials already subjected to static or dynamic (longitudinal wave) compressive states of stress, then dynamic material behavior in stress ranges of practical interest could be investigated at will. SHEAR-WAVE GENERATION

Numerous experimental techniques have been used to generate large-amplitude shear waves in materials. One practical technique simply involves detonation, from a line source, of a sheet of high explosive placed on a planar surface. Depending upon the detonation velocity relative to wave speeds in the material, such loading is referred to as superseismic (detonation velocity, cd, greater than longitudinal-wave velocity, cp), subseismic (cd less than shear-wave velocity, c,), or transseismic (c, < cd < cp). The superseismic case with the generation of distinct longitudinal and shear-wave shock fronts is the easiest to analyze. The principal limitations of the use of this technique derive from the hazards of high explosives and the difficulty of making good diagnostic measurements in an explosive environment.

Some of the problems of using high explosives could be avoided by using an inclined-impact technique to provide a swept surface loading. A conventional gas gun facility could readily be used to launch flyer plates against approximately inclined target blocks of materials of interest. In this case the sweep rate of the pressure loading on the interface would be controlled by the projectile velocity and the angle between the target and the flyer plate. As for sheet explosive loading it would be possible to develop a superseismic, swept pressure loading which would generate the distinct wave structures desired for material studies. A third technique by which large-amplitude shear waves might be generated would involve the instantaneous impact of an inclined flyer plate against an equally inclined target face. Conservation of momentum on the impact interface would result in the generation of a longitudinal wave associated with that component of impact velocity perpendicular to the interface and a shear wave associated with the component of velocity parallel to the interface. Measurements of the longitudinal and shear waves as they propagated into the target material would provide data on the dynamic behavior and strength of the target material. By modifying the angle of impact and the impact velocity the relative and absolute magnitudes of the longitudinal and shear waves can be experimentally controlled. Some experiments to evaluate the feasibility of this technique have been carried out [11-13], but two important experimental problems will make extensive application of the technique difficult. In order to obtain the planarity of impact required for good shock-wave generation and measurement, projectile rotation must be limited to a few milliradians. Adequate control of projectile rotation can only be achieved if the launch tube is slotted or otherwise modified[12, 13]. The second, and probably more important, experimental limitation arises from the need to have a high vacuum in the launch tube between the projectile and the inclined target. Any gas in the launch tube whether residual or generated by blow-by around the projectile would be swept ahead of the inclined impactor and could in large part be trapped in the space between impactor and target. Since the impact surfaces must be ground i'elatively smooth for good shock-wave generation, even a small amount of air in the impact plane would provide an air cushion which could decouple the transverse (shear-wave) motion of the impactor from the target block. Slotting of the launch tube would increase projectile sealing and blow-b~ problems. A fourth technique for generating large amplitude shear waves would involve the reflection of conventional longitudinal shock waves off of an appropriately inclined and impedance mismatched interface. With the realization that it is possible to get very large (in some cases 100%) energy conversion from the incident longitudinal (P) wave into a reflected shear (S) wave it was decided to investigate the experimental and analytical aspects of a reflected wave technique in detail. A wave reflection technique could be utilized to give relative

Technique for Determining Dynamic Rock Strength amplitudes and orientations between incident and re-. fleeted P- and S-waves which could be utilized to deduce the dynamic material properties. Since the technique was foreseen to be used principally for studying the dynamic behavior of rocks in the intermediate stress region, the rock response would be essentially linearly elastic prior to failure and linear elastic wave theory could be used to predict the relative wave forms 114]. In the first approximation the experimental measurement of non-linear behavior could be used as an indication of material failure under the loading conditions. The orientation of wave fronts and particle velocities for the case of linear elastic wave reflection from a free surface is shown in Fig. 1. In the case when material failure or other non-linear effects occur the wave structure will of course be much more complicated. As will be discussed in more detail later, the state of stress leading to failure can be restricted to the reflected shear wave, with the incident and reflected longitudinal waves propagating as elastic waves. In this case the wave structure shown in Fig. 1 can be modified to a good approximation by simply introducing a shear failure wave propagating into the material behind an elastic shear wave. For the linearly elastic free surface reflection case the relative amplitudes of the incident and reflected waves is determined solely by the reflection angle and the Poisson's ratio of the material [14]. As illustrated in Fig. 2 there exists for every elastic solid at least one, and possibly two, critical reflection angles at which the amplitude of the reflected P-wave will be a minimum. In this case the majority if not all of the energy

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in the incident P-wave will go into the reflected shear wave. These critical angles depend upon the Poisson's ratio of the material. For a Poisson's ratio of greater than 0.25, as is shown in Fig. 2 for Solenhofen limestone, there will exist one minimum of reflected P-wave amplitude as a function of angle and in no case will the reflected P-wave amplitude equal or be less than zero. For a Poisson's ratio less than 0.25 there exists a range of angles where the reflected P-wave is compressive rather than tensile and there are two angles of reflection at which the P-wave amplitude is exactly zero. By appropriately orienting a reflection face and choosing the wave loading conditions so that most of the wave propagation takes place under essentially elastic conditions, the measurement and analysis of the characteristics of shear wave propagation could be done with reasonable facility and accuracy. The states of stress associated with various waves active during experiments with a reflected technique and how their superposition can lead to the controlled failure of a pressure dependent material can be most clearly represented in Mohr-stress space. The Mohr circles, representing the stress states for various combinations of the incident and reflected waves are shown in Fig. 3, along with a representative failure envelope. It is clear from the circle for the incident P-wave, indicated P~ in Fig. 3, that the stress levels within this wave fall well below the elastic limit of the material. Upon reflection at the angle which gives minimum reflected P-wave amplitudes (--63 ° for v = 0.3), both reflected Failure envelope

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longitudinal (Pr) and reflected shear (S,) waves will be generated. The effect of the reflected P-wave will be to reduce the principal stresses of the incident P-wave, thus, moving the Mohr circle to the left (P~ + P,) while the shear wave, which cannot change the mean pressure, serves to expand the Mohr circle, possibly to the point that it intersects the failure envelope. Measurement of the amplitudes of each of these waves of interest serves to establish the degree to which the material fails under a given loading condition. For example, if the material were weaker than shown in Fig. 4, the reflected shear wave would have to be lower in amplitude, representing a smaller expansion of the Pi + P, Mohr circle and the reflected P-wave would have to be larger in amplitude so as to properly satisfy the conservation of momentum and energy conditions at the interface. Given the theoretical possibility of generating large amplitude shear waves through wave reflection and the potential of using electromagnetic velocity gauges for measuring and interpreting these waves, considerable effort has been devoted to developing and evaluating an experimental wave-reflection technique. Although the dimensional limitations of a standard laboratory gas gun would make the generation of the required incident P-wave over a large surface area impossible, it was felt that the laboratory environment of a gas gun facility would be far preferable to that which could be obtained utilizing high explosives. The geometry of the wave-reflection technique that has been developed and utilized for most experiments to date is shown schematically in Fig. 4. The technique involves essentially: (1) the generation by conventional impact of a longitudinal shock wave whose amplitude is below the elastic limit of the material being studied; (2) the reflection of this wave from an inclined surface or interface such that the state of stress created by the superposition of the incident and reflected waves just exceeds the elastic limit of the material to be studied; and (3) finally, the measurement of the amplitude of the reflected Pand S-waves so that this state of stress may be determined. One important feature of the wave-reflection technique, as illustrated in Fig. 4, is that the reflected

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S-wave is roughly perpendicular to the incident P-wave. This relative orientation of the two waves, such that their particle velocities are nearly parallel, permits the accurate measurements needed for quantitative analyses. EXPERIMENTAL TECHNIQUE The experiments to develop and evaluate the shearwave reflection technique have been carried out on a 50-mm dia precision impact facility. The integral part of the impact facility is a 50-mm bore gas gun which, with a 3.5 m long launch tube and a large-orifice, fastacting valve connecting the breech of the gun to a 101. 3 x 107Pa (300 bar) reservoir, can be used to launch projectiles to velocities as high as 500 m s-1. To conserve floor space and to permit the utilization of an optical target alignment system the gun is mounted vertically with the muzzle end exiting in a target chamber located slightly above floor level. The target chamber which is constructed entirely of aluminum and nonmagnetic stainless steel is shown in Fig. 5. The two large Helmholtz coils, which are powered by up to 100 A of current obtained from a bank of 10 conventional car batteries, can generate a very uniform magnetic field greater than 0.1 tesla (1 kG) in the impact region located at the center of the cylindrical target chamber. For all experiments, the projectile velocity is measured by a simple fibre-optic-reflecting technique employing a strong light source directed into one leg of a Y-shaped fibre-optic bundle and a photomultiplier connected to the other leg of the fibre-optic bundle. The common end of the bundle is carefully positioned in the barrel of the gas gun so that reflecting bands on a projectile will provide a strong light return as the projectile passes in front of the bundle. Signals obtained from two reflecting bands spaced at a known distance on the projectile can be recorded on an oscilloscope or used to trigger a time-interval counter, thus providing an accurate measure of projectile velocity. Because of the vertical orientation of the impact facility, target alignment may be carried out rapidly and precisely. Once the target block is firmly attached to the support table (Fig 5) a precision front-surface mirror 25 mm in diameter is placed on the target at the center of the 50 mm dia impact area. With the upper, breech end of the gun opened, prior to the placement of the projectile and fast-firing valve assembly, a low power telescope in a special fixture is placed in the breech. Alignment is rapidly effected by adjusting three precision screws on the target support table until the reflected image of the telescope, the circumference of the precision mirror and the exit circumference of the gun are all seen as concentrically aligned by the telescope. With this technique, alignment is typically realized by two persons in 10-15 rain and tilts measured during impact experiments have been of the order of 0.3 mrad. Planarity of impact, or tilt, has been measured in all experiments by either of two techniques. Initially,

Technique for Determining Dynamic Rock Strength

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Fig. 5. Aluminium target chamber of 50 mm gas gun with Helmholtz coils for electromagnetic measurements.

a short-circuiting signal and trigger generating technique was used in which three pairs of wires are placed and lapped into the target surface at the perimeter of the impact area. One of each pair of wires is held at +9 V and the other is temporarily held at ground through a system of low resistance voltage dividers which eventually lead to the recording oscilloscopes. At the moment of impact a very thin conducting film, vaporplated onto the periphery of the front face of the impactor (flyer plate), short circuits each of the three pairs of wires. Because the voltage dividing circuit causes each pair to generate a characteristic voltage the sequence of closure and consequently the orientation and degree of tilt can be measured on a single oscilloscope trace. A second and more recently utilized tilt measuring technique involves the placement of two or more electromagnetic conducting loops on the impact surface of the target. As these loops must not perturb the impact conditions they are made either by vaporplating thin conducting films on the target face or by spraying a conducting paint into very shallow, 0.0lmm deep, grooves machined into the target face with a precision engraving machine. Once these grooves are filled with paint the target face is lightly ground so as to remove any paint protruding beyond the impact surface. The electromagnetic tilt measuring loops are placed in a V configuration so that one of the elements has a longer effective length (higher final generated voltage) than the other. The rise time of the signals generated by these two loops provides a measure of the tilt along each loop and the relative displacements of their two signals in time gives a measure of tilt between the two loops. In all experiments the tilt signal obtained by a.,',l.st.S. 1 4 - - 5 : 6 - - C

either of the two above techniques is used, by means of the "gate-out" signal from an internally triggered tilt measuring oscilloscope, to trigger all the oscilloscopes recording the electromagnetic particle velocity and stress gauge signals. All signals of interest are recorded on both Tektronix 7000 series oscilloscopes and a Bell & Howell model VR-3700B high-frequency (2 MHz) 14-channel tape recorder. For the tilt measurements, where only a temporary record is needed, Tektronic model 7623 storage oscilloscopes are utilized. For the various electromagnetic gauge signals Tektronix model 7403N and 7603 oscilloscopes with 7A13 or 7A18 plug-ins are utilized in conjunction with model C59 cameras. Although the maximum frequency response of the Bell & Howell recorder does not permit a quantitative treatment of the recorded data, the long-time base signals obtained with this instrument have been valuable for studying the long-time deformation of the target, determining the long-time performance of various gauge and tilt measuring systems, and providing a valuable source of data when triggering or other instrumental difficulties caused oscilloscope records to be lost. In using a relatively small diameter gas gun facility to generate the incident longitudinal shock wave required for the reflection technique, critical evaluation needs to be made of the planarity of the incident and reflected waves and of any perturbing effects arising from the limited dimensions of the target. Only if perturbing effects arising from the corner between the impact face and the reflecting face (Fig. 4), and only if the generated shear wave has good planar structure, can a simple and direct analysis of wave structure be made. In order to further establish the validity of the

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wave-reflection technique, optical measurements with transparent targets have been used to complement the electromagnetic gauge experiments using opaque targets. The most successful optical technique involves illuminating the target block at pre-selected instants of time with red and green filtered flashes from two Nanolight spark tubes (lmpulsphysik model 8N18D). The two transmitted light images thus created are then recorded on high-speed color film (Ektachrome EH-135) in a 35 mm reflex camera with a 400 mm, f5.6 lens. After film development the red and green images are separated by means of appropriate filters to yield unusually high quality black and white images of the wave propagation phenomena at two distinct times within one experiment. Two Nanolight photographs at 2.91 and 4.03 ms after impact on a glass target are shown in Fig. 6. In these photographs a well-developed shear wave (S, in Fig. 6) nearly orthogonal to the incident P-wave (P~) may be seen. Neither the planarity nor the amplitude (as indicated by the shadow intensity) of the shear wave are perturbed by either the impact corner at the upper left or the gauge plane (GP). In the 4.03 Ms image the P-wave may be seen to have been both reflected and transmitted at the gauge plane. The reflection of the P-wave at the gauge plane was expected since the two glass blocks utilized to make the target were intentionally separated by the 20/~m thickness of the gauge element. The excellent wave structure seen in this and comparabk optical experiments confirms the validity and quahty of the shearwave phenomena measured with electromagnetic gauges in opaque targets. Two basically different types of electromagnetic gauge have been developed and successfully utilized with the reflected-shear-wave technique. One of these

is the well-known electromagnetic particle-velocity gauge [8] placed in target blocks so as to measure the particle velocity of both longitudinal and shear wa~es at various distances from the impact face and the reflection face. The other is a recently developed electromagnetic shear-stress gauge [10], which functions in a fashion analogous to the earlier developed electromagnetic stress gauge for longitudinal waves [9]. It has been possible to develop a relatively simple gauge configuration which permits the simultaneous measurement of the particle velocities of both P- and S-wa,,es and the shear-stress amplitude of shear waves created by the reflection technique. Tile success of this gauge derives from the fact that the reflection technique permits the orientation of the reflected shear wave to be roughly perpendicular to the incident longitudinal wave. As shown in Fig. 7 this gauge (which because of the conP-wove front /~ (11 to x-z plane )

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Technique for Determining Dynamic Rock Strength

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figuration of the active elements will be referred to as used to glue a 0.1 mm dia copper wire into the approa Z-gauge) must be positioned as precisely as possible priate grooves and the surface of the target block was parallel to the wave front of the incident P-wave. By lightly ground so as to remove any excess glue and orienting the imposed magnetic field parallel to the im- portions of the wires which might extend above the pact plane and perpendicular to the intersection of this reference surface. These finished portions of the target plane with the reflection interface (roughly perpendicu- blocks were then carefully positioned, pressed and lar to the shear-wave front) pick-up leads from the glued (but only at their edges) to the adjacent portions active elements of the Z-gauge exiting from the target of the target blocks to which they had already been in a direction perpendicular to the intersection of the mated. impact and reflection planes, will generate no signal. It is important to note that the target block pieces Consequently the two active elements for particle vel- should not be completely glued together. During the ocity measurements, at ho and hi in Fig. 7, will generate dynamic loading and deformation of the target a glue characteristic voltages proportional to the amplitudes interface would certainly prove to be weaker than the of the particle velocities in both the P- and S-waves rock and would thus provide a "lubricated" surface as these waves sweep across the Z-gauge. As discussed leading to internal slip and perturbation of the desired in detail by Young and Dubugnon [10] the inclined wave structure in the target. By having the pieces of shear-stress element in the Z-gauge will generate a volt- the target blocks in intimate dry contact as described age whose amplitude is directly proportional to the above, the pressure of the incident P-wave would submomentum of the shear wave which has swept past ject the gauge plane interface to high normal pressures reference point ho in Fig. 7. Since any time-dependent which could in principle preclude slippage during the change in shear momentum between ho and the wave propagation of the reflected shear wave. The validity front can only be effected by a shear stress acting at of this conclusion may be illustrated by referring to ho, the time derivative of this momentum (gauge-output Fig. 3, where the states of stress associated with the voltage) provides a direct measure of the shear stress various waves are depicted in Mohr-stress space. The acting at ho. point marked GP on the circle, Pi + P, + S,, which By appropriately placing one or more Z-shaped just satisfies the failure conditions for the material, corgauges in a non-conducting target material of interest, responds to the state of stress on a gauge plane oriented high quality measurements can be made of the shear- parallel to the incident P-wave front. In order for slipwave stress and the particle velocities of the incident page to occur on this gauge plane the angle of sliding P-wave and the reflected S-wave. friction (qY) of the limestone would have to be less than Installation of electromagnetic gauges with the 25 °. All experimental data available on Solenhofen Z-shaped geometry shown in Fig. 7 has been carried limestone and many other rocks [15] indicate that the out in both polymethylmethacrylate (PMMA) and angle of friction for sliding on precut surfaces is typiSolenhofen limestone by slicing the target material on cally greater than 30°. Thus slippage would not be a plane parallel to the incident P-wave front. In expected on a gauge plane with this orientation. PMMA, gauges were fabricated by gluing a 20-pm In order to avoid the fabrication effort needed for thick foil of copper on one of the two faces of the gauge emplacement on perfectly mated surfaces parallel gauge parting planes. A Z-gauge element with three to the P-wave front and the possibilities for slippage active elements and four pick-up leads was then cut on this interface, a second gauge installation technique from the foil and the undesired foil peeled from the has been developed for targets of Solenhofen limestone. surface of PMMA. Using an acrylic cement (Tensol No. As illustrated in Fig. 8 the target blocks for this tech7) the target block slices were then glued together. Finally, the wave-reflection faces were ground flat at the proper angles, gauge elements to measure tilt and Side block trigger oscilloscopes were constructed on the impact faces with conducting paint, and coaxial cables for Slice .L o.z ~ /, / cabling all gauges to oscilloscopes were attached. As for the PMMA target block the gauges in Solen~/~ L , \ \ , ~ Laser drilled hofen limestone were installed on parting planes -\ .... : i ~ " ~ hole, for . parallel to the incident P-wave front. Because the blocks of target material must be brought into intimate ,°c. I / \\",",3 ", contact across the gauge plane when the target is reasW", ", \ ',,",, ', sembled, it is necessary that the conducting electromag\ \ \, '~ \\ ~,\ \\ / i f - \,", netic gauge elements be embedded in the surface of y ~ ", ,, \ \ \ the gauge plane. Grooves 0.1 mm wide by 0.1 mm deep were machined into one of the gauge plane surfaces I/ ~ tPick-ull wlril by means of a precision engraving machine, such as y ~\ i. i - . , pl,.. utilized in the watch industry. Once these grooves were %1 % machined with the desired pattern a pressure sensitive Fi t. 8, Schematic of target block sliced perpendicular to a 1 for installation of electromagnetic gauges. cyanoacrylate adhesive (Loctite Ltd. type I.S.-150) was

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C. Young and O. Dubugnon

nique are cut so as to yield a thin slice oriented perpendicular to both the impact face and the reflection face. The orientation of these cuts is the most desirable that can be obtained because for any combination of incident and reflected waves the faces will be subjected always to the intermediate principal stress, a2, and thus will never be subjected to shear stresses. Interfaces with this orientation have the lowest probability of interfering with wave propagation and measurements. Thb conductive gauge elements are emplaced in the intermediate slice by first drilling small 0.2 mm dia holes through the slice as illustrated in Fig. 8 with a pulsed ruby laser drill, as used in the manufacture of jewels in the watch industry. Because the pulsed laser can only drill to a limited depth it is necessary to drill with the laser from both sides of the 5-7-mm thick intermediate slice and then to finish the holes to a clean cylindrical form with a 0.2-mm high-speed steel drill. The gauge pick-up leads may be oriented anywhere on the faces of the intermediate slice since the orientation of these faces, in the plane defined by the magnetic field and all possible particle motions, precludes the generation of any spurious signals. Typically, a precision engraving machine is used to cut the fine grooves necessary to install the pick-up leads from the laser drilled holes to a free surface where coaxial cables may be attached. Subsequently, 0.1-mm dia copper wire or 0.2 mm aluminum wire is threaded through the holes and glued into the pick-up lead grooves. To preclude spurious signals being generated by either the collapse of the slightly oversized holes or the oscillations of the 0.I mm copper wires in the 0.2mm holes, these holes are filled as much as possible with 0.05-mm dia glass fibres and then injected with a low-viscosity epoxy resin. Finally, the two faces of the intermediate slice are lightly ground and the target block is glued together using a high impedance cement made with a 50/50 (by weight) mixture of silicon carbide power (800 mesh) and epoxy resin. Almost all of the above discussion on shear-wave generation and measurement has been addressed to a reflection interface which is a free surface. As may be seen graphically in Fig. 3, wave reflection from a free surface cannot provide all of the states of stress that would be desirable to investigate. The Mohr circle corresponding to the sum of the incident and all reflected waves must by definition pass through the origin of the shear stress and normal stress axes. Such a Mohr circle would touch the failure envelopes expected to be encountered at very low pressures, corresponding very closely to the states of stress that would be realized with uniaxial stress loading. An obvious and simple method to increase the mean pressure at which expanding Mohr circles would intersect the failure surface, would be to replace the free reflection surface with an impedance mismatched interface. Depending upon the impedance difference between the target material and adjacent interface material residual stresses of various amplitudes will be maintained on the interface after reflection takes place. These residual stresses will cause

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the least principle stress in the target material to remain in compression and thus permit the sampling of target material failure at higher mean pressures. Using linear wave theory the relative amplitudes of the reflected waves in the target material and the waves transmitted into the interface material have been calculated. In these calculations the interface was treated as either a sliding, no tractional stress, interface or as a locked interface. The amplitudes of the reflected and transmitted waves relative to the incident P-wave amplitude are shown in Figs. 9 and l0 for a target material of Solenhofen limestone with adjacent material of PMMA and silica glass, respectively. For both sets of calculations the interface was treated as a sliding frictionless surface. These relative wave amplitudes as a function of interface angle should be compared to those of Fig. 2 where wave reflection was from a free surface. Because of the relatively low impedance of PMMA as compared to Solenhofen limestone, the reflected P-wave as shown in Fig. 9 is always a relief wave which serves to reduce the mean pressure active in the limestone prior to the arrival of the reflected shear wave. As shown in Fig. 10, however, the higher impedance of fused silica which is comparable to that of Solenhofen limestone, causes the reflected P-wave to be a compressive wave over a broad range of reflection angles. Such a compressive reflected P-wave will cause the mean pressure active in the limestone to increase prior to the arrival of the reflected shear wave. By varying the reflection angle, the impedance of the

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radlans

Fig. 10. Absolute relative amplitude of reflected and transmitted w a v e s for Solenhofen limestone interfaced with silica glass.

Technique for Determining Dynamic Rock Strength adjacent material and finally the pressure of the incident P-wave, it is possible to subject Solenhofen limestone and any comparable material to dynamic stresses of interest in the intermediate pressure region. In view of the fact that analysis of data obtained with the reflected shear-wave technique requires the assumption of plane-wave propagation, it is worth mentioning two experimental techniques that have been developed to reduce significantly the generation of spurious, undesirable waves at the corner between the impact face and the reflection interface. The simpler of these techniques is utilized for free-face reflection experiments when the impact flyer has a different impedance than the target material. For most of the free-face reflection experiments with Solenhofen limestone an impactor of PMMA has been utilized. By using a low impedance PMMA. impactor it was possible to obtain the low pressures (2-6 kbars) desired for the incident P-wave without having to go to very low impact velocities which would complicate both the quality of tilt at impact and the reproducibility of impact velocity, Because of the relative impedances, the particle velocity of the incident P-wave in the Solenhofen limestone is typically 20-25~ of the PMMA flyer impact velocity. Reference to Fig. 2 in the range of free-face reflection angles utilized shows that the shear-wave amplitude (particle velocity) is roughly equal to that of the incident P-wave. Thus, for the corner to have the minimum perturbing effect upon wave generation and propagation, it should move after impact at a velocity between 40 to 50~ of the flyer impact velocity. By placing a rectangular block of PMMA adjacent to the reflection corner of the target such that one face of the block is perfectly parallel to the impact plane of the target while the adjacent face just touches the target at the reflection corner the portion of the flyer which does not impact the target will, through its symmetrical impact on PMMA, have a particle velocity exactly onehalf of the impact velocity. In experiments made with and without a PMMA block adjacent to a free face reflection target, it was observed that the mixed impact condition served to significantly reduce the amplitudes of perturbing waves generated at the corner. The second and more complicated experimental technique for reducing corner effects has been developed for targets which have an adjacent impedance mismatched material at the reflection interface. For this experimental configuration the objective is to have a wave structure in both the target and the adjacent material which is as similar as possible to that which would be realized for the reflection of incident plane waves from an infinite interface. As seen in Fig. 9 the transmitted P-wave has an amplitude which is significantly greater than all other transmitted and reflected waves and is significantly greater than the transmitted shear wave at all angles of impact. It is this large amplitude transmitted P-wave which should be simulated as much as possible by modifying the impact conditions. As illustrated in Fig. 11, the introduction of the proper tilt into the impact face of the adjacent material will

I

255

Projectile

o,

.

I

"./.p;~\

Adjacent

~,/~,

material

~

arget

Fig. 11. Schematic of tilted impact face on material adjacent to main target for reducing corner effects.

result in the adjacent material being loaded along the impact face much as if plane-wave reflection had been taking place on an infinite interface. The angle of impact tilt (ct) of the adjacent material is calculated so that the phase velocity, c~, of impact on the impact face is equal to the phase velocity of the transmitted P-wave along the impact face. Tilted impact faces on the adjacent material have been experimentally observed to reduce significantly the generation of perturbing waves. The reflected shear-wave technique, coupled with the electromagnetic particle-velocity and shear-stress gauges and the just discussed techniques for avoiding undesirable corner effects, offers an experimental tool which can facilitate the study of the dynamic shear strength and failure of rocks and similar materials. In the following section, data obtained on Solenhofen limestone are used to demonstrate the applicability of the technique. DATA AND ANALYSIS Using the reflected shear-wave technique and the associated electromagnetic gauges as discussed in the preceding section, fifteen instrumented experiments have been carried out using Solenhofen limestone as the target material. Experiments have been done with freesurface reflection and with both glass and PMMA used as an impedance mismatched material at the reflection interface. In order to provide various desired loading conditions, the inclination of the reflection interface has been varied between 45 and 72° to the plane of impact. All experiments were done with a PMMA flyer and with impact velocities ranging from 115 to 360 m s- t. With various combinations of impact velocity, reflection-interface angle and reflection-surface conditions, it has been possible to subject Solenhofen limestone targets to both elastic and inelastic loading conditions with maximum principal stresses ranging up to I0 kbars. Depending upon the target fabrication technique utilized and the nature of data desired, the number of gauges in the limestone targets ranged from 2 to 5 electromagnetic particle-velocity or shear-stress gauges. For the great majority of experiments electromagnetic

C. Young and O. Dubugnon

._6 e ~ - I mpact face

~,r

/ /~pr

J

;;,.A

Or

V~

o

/~oooe,

/~

"~''°"

4-Smm

d2

~

.I

,,, . . . . , , .

".._../°..e .,,,,o..~

-I° c~ pi

d3

.

d4

~

/

/

--4ram

__

/

1

Fig. 12. Typical placement of electromagnetic velocity gauges in targets of Solenhofen limestone.

particle-velocity gauges installed in laser drilled holes in transverse slices (Fig. 8) were utilized. In general the 5 gauge locations utilized included 3 gauges at the same distance from the reflection interface but at different depths into the target from the impact face and 2 additional gauges at the same depth from the impact face but at increasing distances from the reflection surface. As illustrated in Fig. 12 the latter two gauges were typically placed at the same depth as the intermediate gauge near the reflection surface. The distance of gauges from the reflection surface ranged from 2.2 to 4.8 mm and the depth of gauges from the impact interface were in multiples of 4-5 ram. The three gauges located along the reflection interface served to provide data which established the self-similarity of the reflection conditions, while the two gauges located deeper within the target from the reflection interface served to provide data on the stability and linearity of shearwave propagation. Oscilloscope traces obtained from 4 of the 5 gauges utilized as discussed above in one experiment (shot No.

il

4 •

t I I

$2V2) are shown in Fig. 13. All 4 traces arc at 10 mV and 1 /2s/div (except gauge 5 which has a time base of 1.1 /~s/div). In all traces the distinctive longitudinal and shear-wave structure characteristic of these experiments can be seen. The less than desirable limitations on target size imposed by the 50 mm impact dia can be seen in these traces where all guages show a drop in signal amplitude at about 4.5 #s, corresponding to the arrival of the principal shock at the gauge leads on the rear of the target. The sharp drop in signal amplitude arriving between 8 and 9 its corresponds to the relief wave arriving from the rear surface of the 12-mm thick PMMA flyer. The nearly identical amplitudes and wave forms of the incident longitudinal and reflected shear waves for the three gauges located at the same 2.25 mm from the reflection interface attest to the validity of the reflection technique and the assumption of plane waves utilized for data analysis. The distinct shear wave seen by gauge 4 located at 6.6 mm from the reflection interface provides a measure of the ability of Solenhofen limestone to transmit a very large amplitude shear wave. The modification of shear-wave structure between gauges 2 and 4 is utilized to determine the dynamic mechanical strength of Solenhofen limestone. The most direct method of reducing the data obtained from the shear-wave reflection experiments has been to graphically represent the states of stress associated with the various waves and their interactions in Mohr-stress space. The series of Mohr circles corresponding to the superposition of the various stress waves as calculated from the experimental data of Fig. 13, are shown in Fig. 14. The experimentally determined values of particle velocity and wave velocity for the longitudinal and shear waves coupled with the previously determined density of this Solenhofen limestone (2600 kg m-3) can be used to directly calculate the normal and shear stresses active in longitudinal and shear waves, respectively. If the reflected shear wave is highly dispersive then the shear stress is calculated on an incremental basis, over the variations in wave velocity, to arrive at a more exact determination of the state of stress. In order to plot the Mohr circles for longitudinal waves it is necessary to know also the principal stress acting parallel to the wave front. For an elastic "Z static

©oml~l~km

S

I-.

= 3

o

•t ~

2

//

,e

m 1

0

2

4 Normal

Fig. 13. Oscilloscope records of four electromagnetic gauges in shearwave experiment No. S'V2.

6 stress.

8 On(kbar)

Fig. 14. Mohr circles for the superposition of incident and reflected waves as measured by gauges 1-5 in experiment No. S'V2.

Technique for Determining Dynamic Rock Strength wave the ratio between the maximum and minimum principal stresses acting in a longitudinal wave is given uniquely by Poisson's ratio. For a Poisson's ratio of 0.28 the ratio between the stresses is 0.39. The correct Poisson's ratio to utilize for the large amplitude waves encountered in these experiments can be obtained directly from the relative velocities of the longitudinal and shear waves. Unfortunately, it is not possible to know the value of the Poisson's ratio effective at all times during a wave-reflection experiment. It is these unknowns in the value of Poisson's ratio that provide the greatest source of error in the Mohr-stress space analysis. In Fig. 14 it is seen that some of the Mohr circles associated with the state of stress realized by the incident and reflected longitudinal waves and the reflected

257

shear waves exceed what would be a logical extension of the statically determined strength of Solenhofen limestone [16]. The stress circles 1, 2 and 5 correspond to gauges located at just 2.2 mm from the reflection interface. At this distance the rates of loading due to the generation of a large amplitude shear wave at the interface were quite high (on the order of 106 s- 1). The stress circles corresponding to gauges 3 and 4 at 4.4 and 6.6 mm from the reflection interface, however, show a reduction in shear-stress amplitude that would be logically expected for the lower loading rates realized at these locations. It is important to note that the state of stress realized at gauges 3 and 4, where strain rates associated with the shear-wave loading were of the order of 104 s- 1 coincide closely with that which would be predicted from the static data of Handin et al. [16].

TABLE |. STRESS STATES CALCULATED FROM ELECTROMAGNETICVELOCITY GAUGES IN SOLENHOFEN LIMESTONE

Gauge

up

No.

(m/s)

Shot

(see

No.

Fig. 12)

ai 10-' MPa, (kbar)

tq (m,,'s)

z 102 MPa (kbar)

•1

:c2

~3

deg

deg

deg Adjacent material

(see Fig. 1)

SS 1

2 3

39 36

5.6 5.2

20 27

1.6 2.1

70

40

39

Vacuum

SS2

2 3

39 37

5,6 5,3

14 24

1.1 1.9

70

40

39

Vacuum

SS3

1 2 4

21 2t 21

3.0 3.0 3.0

19 17 16

1.9 1.3 1.2

62

56

32.7

Vacuum

SS4

1 2 3 4

35 35 31 32

4.9 4.9 4.4 4.6

29 20 27 18

2.3 1.6 2.1 1.4

64.5

51

34.5

Vacuum

SS5

1 2

29 26

4.2 3.8

27 22

2.1 1.8

63

54

33.4

Vacuum

SS6

l 2 3 4

64 68 49 57

9.2 9.7 7.1 8.1

46 27 46 31

3.6 2.1 3.6 2.4

61.5

57

32.3

Vacuum

S2VI

1 2 3 4

33 34 32 32

4.7 4.8 4.6 4.6

23 22 24 22

1.8 1.7 1.9 1.7

60

60

31

PMMA

S-'V2

1 2 3 4 5

53 52 52 52 52

7.6 7.4 7.4 7.4 7.4

30 27 21 15 29

2.4 2.1 1.7 1.2 2.3

63.3

53.5

33.6

PMMA

$2V4

1 2 3 4 5

60 50 55 50 47

8.6 7.2 7.9 7.2 6.7

18 18 14 11 19

1.4 1.4 1.1 0.9 1.5

62.5

55

33

Glass

S 2V5

1 2 3 4 5

35 37 33 35 33

5.0 5.2 4.8 5.0 4.8

18 18 14 15 18

2.3 2.3 1.8 1.9 2.3

52.5

75

26,5

PMMA

$2V6

1 2 3 4 5

32 32 33 33 32

4.6 4.6 4.7 4.7 4.6

22 23 22 21 21

1.8 1.8 1.8 1.7 1.7

63

54

33.4

PMMA

$2V9

1 2 3 4 5

65 63 66 70 --

9.3 8.7 9.0 9.8 --

36 27 20 16 27

2.7 2.0 1.6 1.3 2.0

62.5

55

33

PMMA

(see Figs. 13 and 14)

258

C. Young and O. Dubugnon

In order to estimate a maximum dynamic strength value for the limestone, the shear-wave amplitudes from gauges at various distances were extrapolated back to the reflection interface. For the reduced experimental data shown in Fig. 14 (shot No. SZV2) the Mohr circle corresponding to the extrapolated state of stress at the interface is noted "S". This state of stress is not much different than that actually measured at gauge 1, and consequently the maximum strength can be taken as that measured by gauges adequately close to the reflection interface. One advantage of using PMMA as a reflection interface material is illustrated in both Fig. 9 and Fig. 14. When PMMA is used against Solenhofen limestone as an interface material the amplitude of the reflected P-wave into the limestone is small, being on the order of 8~o of the incident P-wave amplitude for the reflection angles utilized. As precision measurements of the amplitude of the reflected P-wave could not be done in these experiments, it was advantageous to have the P-wave amplitude as small as possible. In this case it was possible to carry out the analysis in Mohr-stress space with the minimum possibility that the states of stress are unknowingly perturbed by the reflected P-wave. The reflected P-wave was usually oriented so that its particle velocity had a component capable of being detected by the gauges utilized in these experiments. Measurement of the approximate amplitude of the reflected P-wave confirmed that it was not significantly different than that predicted from the curves of Figs. 2, 9 and 10. The reduced data obtained from all of the electromagnetic gauges used in experiments on Solenhofen limestone to date are summarized in Table 1. Using the most accurate measure of wave speeds, the orientation angles ~2 and ~3 (see Fig. 1) for the reflected waves were calculated and subsequently used to calculate the states of stress realized with various wave superpositions. For all the experiments conducted to date, longitudinal and shear-wave velocities had average values of cp = 5.50 km s- 1 and c~ = 3.05 km s- t, respectively. Gauges 1 and 3 in experiment SS6 illustrate the very large shear-wave amplitudes that have been realized with the wave reflection technique. Dynamic strength envelopes enclosing the maximum and minimum strength Mohr circles for all experiments on Solenhofen limestone are shown in Fig. 15. The Mohr circles giving the maximum strength envelope were all determined from the data of gauges located closest to the reflection interfaces where strain rates, as calculated from stress wave rise times, were as high as 106 s-1. The minimum strength envelope is derived from gauges located furthest from the reflection interface and includes shear waves whose rise times gave strain rates as low as 104s-1. That the lower strain rate envelope falls below the statically determined failure envelope for combined compression plus torsion loading [16], and that the maximum strength envelope does not greatly exceed the static envelope, indicates that the dynamic failure of this rock is not markedly differ-

4 static

~

static eLtenlion /q ~

maximum

Dynamic strength

3 2

.c

1

Normal

stress, On (kbar)

Fig. 15. Maximumand minimumstrengthenvelopesof Mohr circles for all experiments conductedon Solenhofenlimestoneas compared to previouslydetermined[16] static failure envelopes. ent from its static failure. Certainly, a very large increase in dynamic strength as might have been predicted by an extrapolation of split-Hopkinson bar data [17] is not observed. The sharp increase in apparent strength at loading rates around 103 s - l observed by Green and Perkins [17] probably represents a transition to uniaxial strain conditions as suggested by Brace and Jones [18]. This transition to a state of uniaxial strain from the uniaxial stress condition usually assumed for split-Hopkinson bar loading has been interpreted as a direct result of dilatancy or bulking by Janach [19]. CONCLUSIONS The more important results of the research efforts may be divided into two distinct areas; one related to experimental technique and the other related to the dynamic strength of rocks. Although the development of technique comprised much the greater portion of expended effort, the observations made on the dynamic strength of rock have important implications. In terms of experimental technique the results demonstrate that large amplitude, well-defined shear waves can be systematically generated by a wave-reflection technique even though the impact area utilized to generate an initial longitudinal wave and the reflection surface area are of finite dimensions. Both optical measurements in transparent targets and gauge measurements in limestone and PMMA targets show that the reflected shear-wave structure is amenable to planar wave analyses. The basic reflection technique has been well complemented by the development of in-material diagnostic techniques enabling the unambiguous interpretation of wave structure. The electromagnetic-gauge techniques have been refined to permit a direct measure of shear-wave stress [10] and of both longitudinal and shear-wave particle velocities. Gauge installation techniques have been developed which offer the minimum probability of perturbing wave propagation. In applying the experimental techniques to a study of the dynamic failure of Solenhofen limestone it was revealed that the dynamic strength of this rock is not greatly different from statically determined values. The

Technique for Determining Dynamic Rock Strength

259

best measure of the strain-rate dependence of strength Rock Mech. (Edited by Cording E. J.), pp. 797-823. American Society of Civil Engineers, Washington, 12(2. (1972). is obtained from the attenuation of shear-wave ampli7. Fyfe I. Plane-strain plastic wave propagation in a dynamically tude with distance from the reflection surface. The loaded hollow cylinder. In Mechanical Behavior of Materials strain-rate dependence thus obtained, over a range of under Dynamic Loads (Edited by Lindholm U. S.), pp. 314-328. Springer, New York (1968). strain rates from 106 s-1 to 104 s-1, is comparable to 8. Fowles G. R. Experimental technique and instrumentation. Dynathat given by Brace and Jones [18] for uniaxial deformic Response of Materials to lntense Impulsive Loadin# (Edited mation. The results indicate that for some rocks, at by Chou P. C. & Hopkins A. K.), Chap. 8, pp. 405-480. Air Force Materials Laboratory Wright Patterson Air Force Base, least, statically determined rock strengths may be used Ohio (1972). to first approximation for dynamic analyses and con- 9. Young C., Fowles R. & Swift R. P. An electromagnetic stress trolling factors, other than an intrinsic rate dependence,. gage. In Shock Wares and the Mechanical Properties of Solids (Edited by Burke J. J. & Weiss V.), pp. 203-223. Syracuse Univermust be found to explain unusually high rock strengths sity Press, New York (1971). measured in split-Hopkinson bar experiments. 10. Young C. & Dubugnon O. An electromagnetic shear stress gage Received 31 May 1977.

REFERENCES 1. Davis D. H. & Hunter S. C. The dynamic compression testing of solids by the method of the split-Hopkinson bar. J. Mech. Phys. Solids It, 115-179 (1963). 2. Hakalehto K. O. Brittle fracture of rocks under impulsive loads. Int. J. Fract. Mech. 6, 249 (1970). 3. Lundberg B. A split Hopkinson bar study of energy absorption in dynamic rock fragmentation. Int. d. Rock Mech. Min. Sci. Geomech. Abstr. 13, 187--197 (1976). 4. Christensen R. J., Swanson S. R. & Brown W. S. Split-Hopkinson bar tests on rock under confining pressure. Expl Mech. 12, 508-513 (1972). 5. Lindholm U. S., Yeakley L. M. & Nagy A. The dynamic strength and fracture properties of Dresser basalt. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 11, 181-191 (1974). 6. Swift R. P. & McKay M. W. Dynamic behavior of rocks to spherical stress waves. In Stability of Rock Slopes, 13th Syrup.

for large amplitude shear waves. Expl Mech., in press. 11. McKay M. W., Shadel R.. Swift R. P. & Young C. Experimental studies of stress-wave propagation in earth media. DASA 2525. Defense Atomic Support Agency, Washington. DC (1970). 12. Abou-Sayed A. S, Clifton R. J. & Hermann L. The oblique-plate impact experiment. Expl Mech. 16, 127-132 (1976). 13. Gupta Y. M. Shear measurements in shock-loaded solids. Appl. Phys. Lett. 29, 694-697 (1976). 14. Kolsky H. Stress Wares in Solids. Dover, New York (1963). 15. Handin J. On the Coulomb-Mohr failure criterion. J. geophys. Res. 74, 5343-5348 0969). 16. Handin J., Heard H. C. & Magouirk J. N. Effects of the intermediate principal stress on the failure of limestone, dolomite, and glass at different temperatures and strain rates. J. geophys. Res. 72, 611--640 (1967). 17. Green S. J. & Perkins R. D. Uniaxial compression tests at strain rates from 10-4/s to 10~/s on three geologic materials. Proc. lOth Syrup. Rock Mechanics (Edited by Gray K. E.). A.LM.E. (1968). 18. Brace W. F. & Jones A. H. Comparison of uniaxial deformation in shock and static loading of three rocks. J. geophys. Res. 76, 4913--4921 (1971). 19. Janach W. The role of bulking in brittle failure of rocks under rapid compression. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 13, 177-186 (1976).