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Elastic impact of a bar on a half-space
Journal o f Sound and Vibration (1975) 41(3), 335-346
ELASTIC IMPACT OF A BAR ON A HALF-SPACE W. JANACH Institut CERA C, Chemin des Larges Pi~ces, CH-1024 Ecublens, Switzerland (Received I0 December 1974, and in revised form 31 January 1975) When a semi-infinite bar impacts on a half-space, the pressure and particle velocity at the contact surface jump to a value which is given by the impedances of the two materials. "The presence of the half-space is gradually felt by the bar as an increase of effective impedance. As a result, the contact pressure starts to increase while the velocity keeps decreasing. Whert the contact surface eventually comes to rest, the pressure reaches a final value. This transition between the initial jump and the final state is investigated. A simple model is derived for small impact velocities and linear elastic material behaviour based on the assumption of a linear pressure-penetration relationship. Impact experiments of an aluminium and a plexiglas bar on a plexiglas block with a velocity of 1 to 2 m/s show good agreement with results from the model. 1. INTRODUCTION When two elastic bodies come into contact with a finite velocity, elastic waves start propagating from the region of contact. The pressures developed will depend on impact velocity, impedance and shape of the two bodies. As soon as the pressure exceeds the material strength, plastic flow or brittle fracture will take place locally. Here only the elastic domain will be considered. The particular geometry investigated is a semi-infinite fiat-ended bar impacting on a half-space (Figure 1). A system of dispersing wave s is set up in the half-space, whereas the waves are not attenuated as they propagate through the bar. The interaction of these two different behaviours results in a characteristic impact process. This process is fundamental to m a n y rock breakage methods such as impact by a hammer, drill rod, impactor or projectile. All ofthese cases are of more complex nature than the process that is investigated here mainly because of more complicated geometry and material failure. When the fiat end of the bar impacts on the half-space, the pressure and particle velocity at the interface j u m p to a value which is given by the impedances of the two materials. The conditions are the same as if the bar were impacting on a second bar. The initial transition from a uniaxial strain to a uniaxial stress situation in the bar is not considered. As the interface moves into the half-space, the increasing effective impedance of the half-space begins to be felt by the bar and as a consequence the contact pressure starts to increase while the velocity
vI Ip,
Figure I. Contact region of semi-infinite bar impacting on half-space. 335
336
w. JANACH
keeI3'S decreasing. In the limit the interface will come to rest, corresponding to an effective impedance of infinity for the half-space. The pressure and penetration will then correspond to the static loading of a half-space by a cylinder. The average final pressure will be the same as if the hat had impacted on a rigid half-space. At the instant of impact, the average pressure a{ the interface between the bar and the half-space jumps to a value that is the same as if the half-space were replaced by a second bar of equal diameter and then increases gradually with time to the final value corresponding to the impact on a rigid half-space. The dynamic response of an elastic half-space to various kinds of loading has been exten9sively investigated by previous workers [l, 2, 3]. Pekeris [1] treated the case of a suddenly applied point load at the surface for an elastic material with equal Lam6 constants: i.e., a Poisson ratio of 0-25. He derived closed form expressions for the horizontal and vertical displacements at the surface. The case of the load being uniformly distributed over a circular area was investigated by Eason [2], who derived and evaluated expressions for the displacement inside the half-space, again for a Poisson ratio of 0.25. Boucher and Kolsky [3] have treated the problem ofthe reflection of elastic stress pulses at the interface between a bar and a half-space already in contact. They computed the reflected pulse from the Fourier components of the incident pulse using the dynamic response of the half-space in the frequency domain and compared it to experimental results. The problem of flat impact of a bar on a half-space could be treated in the same way as done by Boucher and Kolsky. This would require considerable computation in order to cover a wide range of impedance ratios. Here, however, a simplified approach is used that takes into account the main features of the process. The resulting expressions are of closed form and offer tlle advantage of showing immediately the influence of the impedance ratio. The main purpose of the model is to estimate the rise-time of the pressure for the transition from its initial bar on bar to its final bar on rigid half-space value. For the particular case of a steel cylinder impacting on rock the impedance ratio is typically 3. In this case, the average impact pressure will build up to a final value of 4 times the initial value if no failure takes place and reflections from the back end of the cylinder have not reached the interface.
2. SIMPLE MODEL The model is based on one-dimensional wave motion in the bar and a linear functional relationship between average impact pressure, p, and penetration, st (see Figure 1). The following assumptions are made: linear elastic bar and half-space; one-dimensional treatment of the bar; bar impedance based on bar wave velocity, V ~ ; impact velocity small in comparison to wave velocity in bar, no failure; contact surface remains flat. Immediately after impact the pressure, p, and the velocity, v, at the interface will jump to the initial valuesp~ and v~ given by the impedances Pl Cl of the bar and P2 c2 of the half-space. The impedance of the bar is based on the bar velocity for wave propagation, whereas in the half-space it is based on the propagation velocity of longitudinal waves in an infinite medium. For an impact velocity V and under the stated assumptions the initial pressure and interface velocity are Pt = V Pl cl P2 c2
(1)
pt ct + P2 c2' and v~ = V,
Pl Cl
(2)
Pl cl + P2 c2" As the stress waves in the semi-infinite bar travel in only one direction, one has a simple wave
337
ELASTIC IMPACT OF A BAR ON A HALF-SPACE
region w'i'th parallel characteristics. This gives a simple relation between average pressure, p, and velocity, v, at the interface: (3)
P = Pl c l ( V - v).
The interface velocity, v, gradually decreases to zero. When the interface has come to rest, the half-space will have deformed to the same extent as if it had been statically loaded to the same average pressure with a flat ended cylinder. Not considered here is the actual pressure distribution over the contact surface, which is singular at the edge. Sneddon [4] gives the expression for the static displacement, us, at the interface with a rigid cylinder of radius R under the total load P as
P0. + 2t,). us = 8nt~0. + t0'
(4)
2 and p are the Lam6 constants. The displacement for a rigid cylinder is taken to be also representative for the average displacement with an elastic cylinder, where the interface will not remain fiat. It should be noted that the displacement us is proportional to the load P. Using the following relations between the elastic constants 2, p, E, v and the longitudiiial wave propagation velocity, c, in an infinite medium, 2pv
)" =
E
1 -- 2v'
P = 2 ( 1 + v)
~fp E0 c=
v)
(l
one can rewrite equation (4) in terms of e2, P2 and the Poisson ratio, v, of the half-space. After introducing the average final pressure, p:, over the circular surface of radius R, one obtains ~zRps(1 - v) z us = 2c2ZPz (1 - 2v)" (5) The final pressure, p:, is obtained from equation (3) as Pj = Pl cx V,
(6)
and corresponds to the case when the bar would impact on a rigid body of infinite impedance. P
Radiated wave energy
3= Us
U
Figure 2. Assumed pressure-penetration relationship.
338
w. JANACH
It is'possible to establish the evolution of the pressure, p, over time if one states how the pressure changes with penetration, tt. As in the static relation (4) the force is proportional to displacement, the most plausible assumption for the dynamic case is a linear relationship between pressure and penetration. As shown in Figure 2 one assumes that the pressure, p, increases linearly' with penetration, u, from Pt to Ps and obtains u
P =P~ + (Ps - P i ) - . u,
(7)
For the case of low half-space impedance, pi becomes small and e'xpression (7) approaches the static pressure-penetration relationship. If one considers that (8)
v = dxldt
the time functions of pressure and velocity at the interface can be derived. Taking equation (8) with equations (7) and (3) one obtains u~pl cl do
(P~"-Pt)
dt '
which, by using equations (1), (5) and (6), can be expressed as d/3 V=-T--
dt'
with T=-1+ 2c2 P2 c2 ] 1 - - ~ v ' the solution of which is
o o exp(
PI CI
vi
~
pzC2-3
7" p
a,T Figure 3. Velocity and average pressure at the interface between bar and half-space.
(9)
339
ELASTIC I M P A C T OF A BAR ON A H A L F - S P A C E I00
I
I
i
I
I
I
I
50
20
'~
5
~0"
2
I
OI
I
0-2
!
0-5
I
I
I
2
I
5
I
I
I0
I
gO
50
p~c~
Pzcz
Figure 4. Dimensionlesstime constant. The pressure is obtained from equation (3) with equations (9) and (2) as
Pl ci -I--P2 c2 exp
(lO) - -
.
Equations ~9) and (10) represent the velocity and pressure at the interface after impact and are plotted in Figure 3 for an impedance ratio of 3. The remarkable property of this solution is that the time constant, T, does not depend on impact velocity, V. The time constant is made dimensionless with the longitudinal wave velocity c2 in the half-space and the bar radius R: c.r
n=i
1
(ID
p c2}I- "
Figure 4 shows plots of the dimensionless time constant, c2 T/R, for various Poisson ratios of the half-space. It may be noted that it has a finite lower bound for an impedance ratio, Pl c~/p2c2, going towards zero. In this case the initial pressure, p~, approaches the final pressure, p:, and the initial interface velocity, v~, is almost zero. The exponential terms in equations "(9) and (10) become negligible and their time constant meaningless. Figure 5 shows in dimensionless representation the pressure as a function of time for various impedance ratios and a Poisson ratio of 0.3.
p/p, f
5
I0
15
20
czt/R
Figure 5. Pressure at interface for various impedance ratios.
340
w. JANACtt 3. ENERGY PARTITION
It has been seen how the impedance ratio of bar to'half-space affects the pressure and velocity at the interface during impact. It is interesting to derive the energy flow rate into the half-space as a fufiction of time. Figure 2 shows total energy as the integrated pressure over penetration. The energy flow rate, IV, through the interface per unit area is
W=pv.
(12)
It can be made dimensionless by expressing it as a fraction of kinetic energy flow of the bar: i.e., the kinetic energy of material reached by the first impact signal propagating through the bar. Hence the dimensionless energy flow rate is W W*
=
89 V2 c~
and with equations (2), (9), (10) and (12) can be expressed as
Equation (13) is represented in Figure 6 for different impedance ratios, PI cI[p2 c2. The time
wT 0'5-
0.25
Figure 6. Dimensionlessenergy flow rate through interface. is dimensionless relative to the time constant, T, which in turn depends on the impedance ratio and Poisson's ratio in the half-space. The integral of IV* up to a given time represents the energy transmitted to the half-space as a fraction of kinetic energy of that part of the bar which has been reached by the wave front. With this analysis one can optimize the length of a finite bar for maximum energy transfer.
4. EXPERIMENTS An experimental set-up can only duplicate the situation of a semi-infinite bar impacting on a half-space for a limited time span, as the finite dimensions will give rise to reflected waves. The dimensions of a finite block and a bar were chosen so as to allow for at least 3 to 4 times the time constant, 7". before reflections from the free side surfaces would reach the zone of
ELASTICIMPACTOF A BAR ON A HALF-SPACE
341
contact." In order to keep the impact pressures low and avoid plastic deformation at ihe periphery of the interface, the impact velocity was limited to 1 to 2 m/s. Such a velocity is easily reached by gravitational acceleration of the bar from a height of 0.05 to 0.2 m. As material~,plexiglas was chosen for the half-space and aluminium and plexiglas for the bar. Plexiglas has the adx;antage of a low wave propagation velocity and a low elastic modulus, giving rise to large strains at low stresses. Its viscoelastic behaviour was considered to be negligible at the very low stress levels encountered in the experiments. The dimensions of the block representing the half-space were 0.25 x 0.25 x 0.125 m, the aluminium bar had a diameter of 12 m m and was 0.5 m long, while the plexiglas measured 30 m m in diameter and was 1 m long. The elastic constants and wave velocities of these materials were calculated from measurements of the resonance frequency of the bars oscillating in various modes. Bending waves were used for the elastic modulus and the bar wave velocity, torsion pendulum tests for the shear modulus and the shear wave velocity. The longitudinal wave velocity and Poisson's ratio can then be calculated. These measurements gave the material properties shown in Table 1. TABLE 1
Properties of bar attd block materials
Density, p Elastic modulus, E Bar wave velocity, cb~r Shear modulus,/.t Shear wave velocity, cs Longitudinal wave velocity, c Poisson's ratio, v Rayleigh wave velocity
Plexiglas
Aluminium
1186 44"5 1935 16.86 1190 2315 0-32 1110
2700 723 5175 270 3165 6375 0-336
kg/m 3 kb m/s kb m/s m/s -m/s
The impact pressure was measured with two strain gauges on opposite sides of the bar at a distance of 3 diameters from the impacting end. The agreement of the two signals was a check on the flatness of the impact. The strain gauges were connected with 4 loosely suspended wires of 0.05 m m diameter. The measured stress wave at a location of 3 diameters up in the bar is delayed relative to the impact and represents an average for the pressure at the interface. At the same time an electromagnetic gauge was installed on the block to measure either the horizontal or vertical component of the surface velocity at some distance from the impacting bar. This was done by gluing a thin wire to the surface and suspending a small permanent magnet above it. As the stress waves pass the gauge, the wire moves in the stationary magnetic field and induces a voltage. The signals from these gauges were recorded on oscilloscopes, triggered by a closing contact at impact. This contact trigger consisted of a thin aluminium foil glued to the half-space and a second contact on the front surface of the bar. The aluminium bar had to be grounded to avoid triggering noise. The degree of flatness of the impact establishes the rise time of the initial pressure jump. Keeping the rise time below the travel time of an elastic wave over one bar diameter requires the two impacting surfaces to be parallelto within 10 -3 rad at an impact velocity of 2 m/s with plexiglas. The surface roughness is not to exceed a comparable value. An estimation of the effect of the air cushion formed between the impacting surfaces showed that it is not negligible but within the required tolerance for the rise time. The top surface of the block was aligned to the horizontalwith the help o f a water-level with a precision of 2 x 10-5 rad. Then
W. JANACH
342 Impact
I
11"35 b/di' I0/~s/div
(a)
Impact
Arrival of stress wave
I 11.55 b/div
lOFsldiv
I
I
ill I
(b)
Figure 7, Impact pressure forplexiglas bar onpleMglas block, ~ = 30 ram, V = 1.98 m/s. (a) Oscilloscope record from strain gauges, 90 mm up on the bar; (b) calculation. Impact
i 36,5 b/div IO/zs/div
[a) Impact Arrivalof stresswave
I /~,~ .~
/ t
lO/zs/div
//
,
(b)
Figure 8. Impact pressure for alumiMum bar on plexiglas block, ~ = 12 mm, V = 1 m/s. (a) Oscilloscope record from strain ~auaes. 36 mm uo on the bar: (b/ calculation.
EL;kSTICIMPACTOF A BAR ON A HALF-SPACE Impact
I
343
Reflection
,R
I
Out
l
~O.I m/s/div I0 ~s/div
Horizontal
Impact
Reflection
Up
l
~0.065 m/s/div I0 bts/div
Vertical 6o
Figure 9. Oscilloscoperecords of electromagnetic velocity measurements on block surface at 45 mm from axis for plexiglas impacting plexiglas. the bar, vertically suspended with a thin nylon thread, was lowered on to the block. The attachment point of the thread at the top ofthe bar was laterally adjusted until the bar could be lowered on to the blocl~ without any angular motion taking place at the moment of contact. For the experiment, the bar was suspended at a given level and the thread was melted by heating a thin electric wire wound around it. Figure 7 shows the measured and calculated impact pressure for the plexiglas bar impacting at 1.98 m/s and generating a final pressure of 45.4 bars. The signals from the two identical strain gauges on the bar at a distance of 90 mm from the impacting end are almost parallel and indicate that the impact was fiat. The time lag between impact and arrival of the wave corresponds to the propagation along the bar. As the bar is dispersive for wavelengths shorter
Impact
i
Reflection
IP
IR
f
Up
-0.025 rn/s/div lO#s/div
.Figure 10. Oscilloscope record of electromagnetic velocity measurement (vertical) on block surface at 45 mm from axis for alumlntum impactingplexiglas.
344
9
w. JANACH
than its diameter, the initial pressure jump is deformed. The impact pressure from the smaller aluminium bar at 1 m/s can be seen in Figure 8. It reaches a final value of 139-7 bars. Here, dispersion does not allow one to see the initial jump at all. The oscilloscope triggered about 5/~s.before complete contact, which represents a travel of 5 l t m of the bar. Figures 9 and 10 show the records from the electromagnetic gauges on the surface of the block at a distance of 45 mm from the axis for the same impact velocities. After arrival of the longitudinal wave the velocity is first outwards and upwards and reverses to inwards and downwards with the arrival of the surface wave. Here, the reflection from the nearest block side arrives after 75 l~s. 5. DISCUSSION - When using the results from the simple model, it should be borne in mind that it is based on a pressure-penetration relationship which was not derived but assumed as being the most plausible. As can be seen from Figure 5, the solution for the average pressure at the interface is composed of an initial jump, followed by an asymptotic exponential approach to the final value. The initial jump and the final value are given exactly. The fraction represented by the initial jump depends on the impedance ratio of bar to half-space. This fraction decreases towards zero for increasing impedance ratio. Looking at the assumed pressure-penetration relationship in Figure 2, one can notice that it approaches the static line for increasing impedance ratio. This means that for large ratios the impact transient is controlled almost entirely by the static elastic behaviour of the half-space. As a consequence, the exponential term of the solution becomes more and more accurate as the impedance ratio increases. In the limit of infinit~ ratio there is no inertia contribution from the half-space. The process becomes analogous to the interaction of a drill rod and a failing rock surface with proportionality between force and penetration as treated by Lundberg [5].
(al
ntL L 1
2
'
t/~
(b)
Pr (c)
Figure 11. Reflection of a stress pulse propagating in a bar which rests on a half-spaCewith Pa cz[p~ cz = .(a) Incident pulse; (b) average pressure at interface; (c) reflected wave. I
3.
ELASTIC IMPACT OF A BAR ON A HALF-SPACE
345
The present analysis easily can be extended to find the average pressure at the interface of a bar resting on a half-space for the case of an incident stress pulse in the bar (see Figure I l). The stress pulse in the stationary bar is equivalent to an impact with a velocity of two times the particle ve.l,ocity of the. pulse. This is explained by the fact that if the incident pulse would reflect at a free end, the resulting particle velocity would double and the stress vanish. The reflected wave is obtained as tile difference between the pressure at the interface and the incident ~pulse. Figure l I shows the average pressure at the interface and the reflected wave for an impedance ratio of bar to half-space of 3. The end of the square pulse generates an identical but negative signal which is superposed on the first one. The initial inversion of the reflected wave is only present for impedance ratios greater than one. Kolsky [3] relates this phenomenon to what he calls a "cross-over effect" and compares it to the wave reflection at the boundary between an elastic and a linear viscoelastic bar. Th6 experimental results obtained from strain gauge measurements on the bar at 3 diameters from the impacting end agree well with the model predictions. Using the bar to transmit the impact pressure to the gauges puts a limitation on the higher frequency components of the signal. As a result, the initial jump has a rise time of about 20/~s fo r the plexiglas bar (Figure 7) and is not visible at all for the aluminium bar (Figure 8). The record from the plexiglas bar shows weak oscillations of the stress after the initial jump. These are due to two-dimensiorlal effects in the contact region, their wavelength corresponding to the bar radius ifcalculated with the Rayleigh wave velocity. In Figure 8, showing the impact of an aluminium bar on the plexiglas block, one can clearly see the exponential transition towards the final pressure. The impedance ratio of 5.08 is sufficiently high for the half-space to behave almost statically. This is desp!te the fact that its wave propagation velocity is 3 times lower than in the bar. The static behaviour of the half-space is dominant because the pressure increases by a large factor above its initial value whereas the interface velocity decreases. The inertia effects ofthe dynamic process in the half-space are related to the initial interface velocity and their influence decreases as the pressure increases above its initial value. The electromagnetic velocity measurements on the surface of the half-space in Figures 9 and l0 can be interpreted with the help of the analytical solution for the suddenly applied point load given by Pekeris [I]. P and R mark the arrivals of the longitudinal and Rayleigh waves coming from the nearest point of the bar. It is characteristic that both the horizontal and vertical components of velocity change sign with the arrival of the surface wave. The analytic solution has an upward infinity for vertical displacement just before the Rayleigh wave arrival, the effect of which can be clearly seen in Figure 10. The horizontal infinity is not visible in the measurement because it comes after the Rayleigh wave and is inwards. Taking the static solution for the half-space by Sneddon [4] one can calculate the horizontal and vertical displacements at the position of the electromagnetic gauge for the final pressure. This value then can be compared to the integral of the measured velocity curves. '.The integrals were computed up to the moment of the arrival of the reflection. For the impact with the plexiglas bar (Figure 9) the measured horizontal displacement is~1-4/~m and the vertical 4.7 llm, which have to be compared to 1.21/~m and 4.67 llm as calculated from the static solution. For the aluminium bar (Figure I0) the measured vertical displacement was 2.0/~m and the calculated one 2.26 l~m. In most rock breakage methods involving impact, the impedance of the impacting body is 9 higher than for the rock. As a consequence, the initial pressure developed in an elastic flat impact is only a small fraction of the final v'alue gradually built up with time. The rise-time required to attain this final value depends only on the ratio of bar to half-space impedance and not on impact velocity. It was shown that the transition from the initial to the final pressure is exponential with a characteristic time constant. Under normal conditions, the developed pressures are well above the load bearing capacity of the rock and failure takes I
346
w.
JANACH
place. It is well known that rocks can withstand higher loads than their static strength for a short period of time, the reason being that failure requires time to develop. In an impact on rock the pressure at the interface therefore depends on two competing processes: the first is due~o the elastic behaviout and increases the pressure, whereas the second tends to unload the interface through material failure. REFERENCES 1. C. L. PEKERlS 1955 .ProcJeedings of the National Academy of Science USA 41,469--480. The seismic surface PUlSe. 2. G. EASON 1966 Journal of the Institute of Mathematics and its Applications 2, 299-326. The displacements produced in an elastic half-space by a suddenly applied surface force. 3. S. BOOCHERand H. KOLSKY 1972 Journal of the Acoustical Society of America 52, 884-898. Reflection of pulses at the interface between an elastic rod and an elastic half-space. 4. I. N. SNEDDON 1951 Fourier Transforras. New York: McGraw-Hill Book Company, Inc. See pp. 458-462. 5. B. LUNDBERG1971 Dissertation, Chalmers University of Technology, G6teborg. Some basic problems in percussive rock destruction.
ELASTIC IMPACT OF A BAR ON A HALF-SPACE W. JANACH Institut CERA C, Chemin des Larges Pi~ces, CH-1024 Ecublens, Switzerland (Received I0 December 1974, and in revised form 31 January 1975) When a semi-infinite bar impacts on a half-space, the pressure and particle velocity at the contact surface jump to a value which is given by the impedances of the two materials. "The presence of the half-space is gradually felt by the bar as an increase of effective impedance. As a result, the contact pressure starts to increase while the velocity keeps decreasing. Whert the contact surface eventually comes to rest, the pressure reaches a final value. This transition between the initial jump and the final state is investigated. A simple model is derived for small impact velocities and linear elastic material behaviour based on the assumption of a linear pressure-penetration relationship. Impact experiments of an aluminium and a plexiglas bar on a plexiglas block with a velocity of 1 to 2 m/s show good agreement with results from the model. 1. INTRODUCTION When two elastic bodies come into contact with a finite velocity, elastic waves start propagating from the region of contact. The pressures developed will depend on impact velocity, impedance and shape of the two bodies. As soon as the pressure exceeds the material strength, plastic flow or brittle fracture will take place locally. Here only the elastic domain will be considered. The particular geometry investigated is a semi-infinite fiat-ended bar impacting on a half-space (Figure 1). A system of dispersing wave s is set up in the half-space, whereas the waves are not attenuated as they propagate through the bar. The interaction of these two different behaviours results in a characteristic impact process. This process is fundamental to m a n y rock breakage methods such as impact by a hammer, drill rod, impactor or projectile. All ofthese cases are of more complex nature than the process that is investigated here mainly because of more complicated geometry and material failure. When the fiat end of the bar impacts on the half-space, the pressure and particle velocity at the interface j u m p to a value which is given by the impedances of the two materials. The conditions are the same as if the bar were impacting on a second bar. The initial transition from a uniaxial strain to a uniaxial stress situation in the bar is not considered. As the interface moves into the half-space, the increasing effective impedance of the half-space begins to be felt by the bar and as a consequence the contact pressure starts to increase while the velocity
vI Ip,
Figure I. Contact region of semi-infinite bar impacting on half-space. 335
336
w. JANACH
keeI3'S decreasing. In the limit the interface will come to rest, corresponding to an effective impedance of infinity for the half-space. The pressure and penetration will then correspond to the static loading of a half-space by a cylinder. The average final pressure will be the same as if the hat had impacted on a rigid half-space. At the instant of impact, the average pressure a{ the interface between the bar and the half-space jumps to a value that is the same as if the half-space were replaced by a second bar of equal diameter and then increases gradually with time to the final value corresponding to the impact on a rigid half-space. The dynamic response of an elastic half-space to various kinds of loading has been exten9sively investigated by previous workers [l, 2, 3]. Pekeris [1] treated the case of a suddenly applied point load at the surface for an elastic material with equal Lam6 constants: i.e., a Poisson ratio of 0-25. He derived closed form expressions for the horizontal and vertical displacements at the surface. The case of the load being uniformly distributed over a circular area was investigated by Eason [2], who derived and evaluated expressions for the displacement inside the half-space, again for a Poisson ratio of 0.25. Boucher and Kolsky [3] have treated the problem ofthe reflection of elastic stress pulses at the interface between a bar and a half-space already in contact. They computed the reflected pulse from the Fourier components of the incident pulse using the dynamic response of the half-space in the frequency domain and compared it to experimental results. The problem of flat impact of a bar on a half-space could be treated in the same way as done by Boucher and Kolsky. This would require considerable computation in order to cover a wide range of impedance ratios. Here, however, a simplified approach is used that takes into account the main features of the process. The resulting expressions are of closed form and offer tlle advantage of showing immediately the influence of the impedance ratio. The main purpose of the model is to estimate the rise-time of the pressure for the transition from its initial bar on bar to its final bar on rigid half-space value. For the particular case of a steel cylinder impacting on rock the impedance ratio is typically 3. In this case, the average impact pressure will build up to a final value of 4 times the initial value if no failure takes place and reflections from the back end of the cylinder have not reached the interface.
2. SIMPLE MODEL The model is based on one-dimensional wave motion in the bar and a linear functional relationship between average impact pressure, p, and penetration, st (see Figure 1). The following assumptions are made: linear elastic bar and half-space; one-dimensional treatment of the bar; bar impedance based on bar wave velocity, V ~ ; impact velocity small in comparison to wave velocity in bar, no failure; contact surface remains flat. Immediately after impact the pressure, p, and the velocity, v, at the interface will jump to the initial valuesp~ and v~ given by the impedances Pl Cl of the bar and P2 c2 of the half-space. The impedance of the bar is based on the bar velocity for wave propagation, whereas in the half-space it is based on the propagation velocity of longitudinal waves in an infinite medium. For an impact velocity V and under the stated assumptions the initial pressure and interface velocity are Pt = V Pl cl P2 c2
(1)
pt ct + P2 c2' and v~ = V,
Pl Cl
(2)
Pl cl + P2 c2" As the stress waves in the semi-infinite bar travel in only one direction, one has a simple wave
337
ELASTIC IMPACT OF A BAR ON A HALF-SPACE
region w'i'th parallel characteristics. This gives a simple relation between average pressure, p, and velocity, v, at the interface: (3)
P = Pl c l ( V - v).
The interface velocity, v, gradually decreases to zero. When the interface has come to rest, the half-space will have deformed to the same extent as if it had been statically loaded to the same average pressure with a flat ended cylinder. Not considered here is the actual pressure distribution over the contact surface, which is singular at the edge. Sneddon [4] gives the expression for the static displacement, us, at the interface with a rigid cylinder of radius R under the total load P as
P0. + 2t,). us = 8nt~0. + t0'
(4)
2 and p are the Lam6 constants. The displacement for a rigid cylinder is taken to be also representative for the average displacement with an elastic cylinder, where the interface will not remain fiat. It should be noted that the displacement us is proportional to the load P. Using the following relations between the elastic constants 2, p, E, v and the longitudiiial wave propagation velocity, c, in an infinite medium, 2pv
)" =
E
1 -- 2v'
P = 2 ( 1 + v)
~fp E0 c=
v)
(l
one can rewrite equation (4) in terms of e2, P2 and the Poisson ratio, v, of the half-space. After introducing the average final pressure, p:, over the circular surface of radius R, one obtains ~zRps(1 - v) z us = 2c2ZPz (1 - 2v)" (5) The final pressure, p:, is obtained from equation (3) as Pj = Pl cx V,
(6)
and corresponds to the case when the bar would impact on a rigid body of infinite impedance. P
Radiated wave energy
3= Us
U
Figure 2. Assumed pressure-penetration relationship.
338
w. JANACH
It is'possible to establish the evolution of the pressure, p, over time if one states how the pressure changes with penetration, tt. As in the static relation (4) the force is proportional to displacement, the most plausible assumption for the dynamic case is a linear relationship between pressure and penetration. As shown in Figure 2 one assumes that the pressure, p, increases linearly' with penetration, u, from Pt to Ps and obtains u
P =P~ + (Ps - P i ) - . u,
(7)
For the case of low half-space impedance, pi becomes small and e'xpression (7) approaches the static pressure-penetration relationship. If one considers that (8)
v = dxldt
the time functions of pressure and velocity at the interface can be derived. Taking equation (8) with equations (7) and (3) one obtains u~pl cl do
(P~"-Pt)
dt '
which, by using equations (1), (5) and (6), can be expressed as d/3 V=-T--
dt'
with T=-1+ 2c2 P2 c2 ] 1 - - ~ v ' the solution of which is
o o exp(
PI CI
vi
~
pzC2-3
7" p
a,T Figure 3. Velocity and average pressure at the interface between bar and half-space.
(9)
339
ELASTIC I M P A C T OF A BAR ON A H A L F - S P A C E I00
I
I
i
I
I
I
I
50
20
'~
5
~0"
2
I
OI
I
0-2
!
0-5
I
I
I
2
I
5
I
I
I0
I
gO
50
p~c~
Pzcz
Figure 4. Dimensionlesstime constant. The pressure is obtained from equation (3) with equations (9) and (2) as
Pl ci -I--P2 c2 exp
(lO) - -
.
Equations ~9) and (10) represent the velocity and pressure at the interface after impact and are plotted in Figure 3 for an impedance ratio of 3. The remarkable property of this solution is that the time constant, T, does not depend on impact velocity, V. The time constant is made dimensionless with the longitudinal wave velocity c2 in the half-space and the bar radius R: c.r
n=i
1
(ID
p c2}I- "
Figure 4 shows plots of the dimensionless time constant, c2 T/R, for various Poisson ratios of the half-space. It may be noted that it has a finite lower bound for an impedance ratio, Pl c~/p2c2, going towards zero. In this case the initial pressure, p~, approaches the final pressure, p:, and the initial interface velocity, v~, is almost zero. The exponential terms in equations "(9) and (10) become negligible and their time constant meaningless. Figure 5 shows in dimensionless representation the pressure as a function of time for various impedance ratios and a Poisson ratio of 0.3.
p/p, f
5
I0
15
20
czt/R
Figure 5. Pressure at interface for various impedance ratios.
340
w. JANACtt 3. ENERGY PARTITION
It has been seen how the impedance ratio of bar to'half-space affects the pressure and velocity at the interface during impact. It is interesting to derive the energy flow rate into the half-space as a fufiction of time. Figure 2 shows total energy as the integrated pressure over penetration. The energy flow rate, IV, through the interface per unit area is
W=pv.
(12)
It can be made dimensionless by expressing it as a fraction of kinetic energy flow of the bar: i.e., the kinetic energy of material reached by the first impact signal propagating through the bar. Hence the dimensionless energy flow rate is W W*
=
89 V2 c~
and with equations (2), (9), (10) and (12) can be expressed as
Equation (13) is represented in Figure 6 for different impedance ratios, PI cI[p2 c2. The time
wT 0'5-
0.25
Figure 6. Dimensionlessenergy flow rate through interface. is dimensionless relative to the time constant, T, which in turn depends on the impedance ratio and Poisson's ratio in the half-space. The integral of IV* up to a given time represents the energy transmitted to the half-space as a fraction of kinetic energy of that part of the bar which has been reached by the wave front. With this analysis one can optimize the length of a finite bar for maximum energy transfer.
4. EXPERIMENTS An experimental set-up can only duplicate the situation of a semi-infinite bar impacting on a half-space for a limited time span, as the finite dimensions will give rise to reflected waves. The dimensions of a finite block and a bar were chosen so as to allow for at least 3 to 4 times the time constant, 7". before reflections from the free side surfaces would reach the zone of
ELASTICIMPACTOF A BAR ON A HALF-SPACE
341
contact." In order to keep the impact pressures low and avoid plastic deformation at ihe periphery of the interface, the impact velocity was limited to 1 to 2 m/s. Such a velocity is easily reached by gravitational acceleration of the bar from a height of 0.05 to 0.2 m. As material~,plexiglas was chosen for the half-space and aluminium and plexiglas for the bar. Plexiglas has the adx;antage of a low wave propagation velocity and a low elastic modulus, giving rise to large strains at low stresses. Its viscoelastic behaviour was considered to be negligible at the very low stress levels encountered in the experiments. The dimensions of the block representing the half-space were 0.25 x 0.25 x 0.125 m, the aluminium bar had a diameter of 12 m m and was 0.5 m long, while the plexiglas measured 30 m m in diameter and was 1 m long. The elastic constants and wave velocities of these materials were calculated from measurements of the resonance frequency of the bars oscillating in various modes. Bending waves were used for the elastic modulus and the bar wave velocity, torsion pendulum tests for the shear modulus and the shear wave velocity. The longitudinal wave velocity and Poisson's ratio can then be calculated. These measurements gave the material properties shown in Table 1. TABLE 1
Properties of bar attd block materials
Density, p Elastic modulus, E Bar wave velocity, cb~r Shear modulus,/.t Shear wave velocity, cs Longitudinal wave velocity, c Poisson's ratio, v Rayleigh wave velocity
Plexiglas
Aluminium
1186 44"5 1935 16.86 1190 2315 0-32 1110
2700 723 5175 270 3165 6375 0-336
kg/m 3 kb m/s kb m/s m/s -m/s
The impact pressure was measured with two strain gauges on opposite sides of the bar at a distance of 3 diameters from the impacting end. The agreement of the two signals was a check on the flatness of the impact. The strain gauges were connected with 4 loosely suspended wires of 0.05 m m diameter. The measured stress wave at a location of 3 diameters up in the bar is delayed relative to the impact and represents an average for the pressure at the interface. At the same time an electromagnetic gauge was installed on the block to measure either the horizontal or vertical component of the surface velocity at some distance from the impacting bar. This was done by gluing a thin wire to the surface and suspending a small permanent magnet above it. As the stress waves pass the gauge, the wire moves in the stationary magnetic field and induces a voltage. The signals from these gauges were recorded on oscilloscopes, triggered by a closing contact at impact. This contact trigger consisted of a thin aluminium foil glued to the half-space and a second contact on the front surface of the bar. The aluminium bar had to be grounded to avoid triggering noise. The degree of flatness of the impact establishes the rise time of the initial pressure jump. Keeping the rise time below the travel time of an elastic wave over one bar diameter requires the two impacting surfaces to be parallelto within 10 -3 rad at an impact velocity of 2 m/s with plexiglas. The surface roughness is not to exceed a comparable value. An estimation of the effect of the air cushion formed between the impacting surfaces showed that it is not negligible but within the required tolerance for the rise time. The top surface of the block was aligned to the horizontalwith the help o f a water-level with a precision of 2 x 10-5 rad. Then
W. JANACH
342 Impact
I
11"35 b/di' I0/~s/div
(a)
Impact
Arrival of stress wave
I 11.55 b/div
lOFsldiv
I
I
ill I
(b)
Figure 7, Impact pressure forplexiglas bar onpleMglas block, ~ = 30 ram, V = 1.98 m/s. (a) Oscilloscope record from strain gauges, 90 mm up on the bar; (b) calculation. Impact
i 36,5 b/div IO/zs/div
[a) Impact Arrivalof stresswave
I /~,~ .~
/ t
lO/zs/div
//
,
(b)
Figure 8. Impact pressure for alumiMum bar on plexiglas block, ~ = 12 mm, V = 1 m/s. (a) Oscilloscope record from strain ~auaes. 36 mm uo on the bar: (b/ calculation.
EL;kSTICIMPACTOF A BAR ON A HALF-SPACE Impact
I
343
Reflection
,R
I
Out
l
~O.I m/s/div I0 ~s/div
Horizontal
Impact
Reflection
Up
l
~0.065 m/s/div I0 bts/div
Vertical 6o
Figure 9. Oscilloscoperecords of electromagnetic velocity measurements on block surface at 45 mm from axis for plexiglas impacting plexiglas. the bar, vertically suspended with a thin nylon thread, was lowered on to the block. The attachment point of the thread at the top ofthe bar was laterally adjusted until the bar could be lowered on to the blocl~ without any angular motion taking place at the moment of contact. For the experiment, the bar was suspended at a given level and the thread was melted by heating a thin electric wire wound around it. Figure 7 shows the measured and calculated impact pressure for the plexiglas bar impacting at 1.98 m/s and generating a final pressure of 45.4 bars. The signals from the two identical strain gauges on the bar at a distance of 90 mm from the impacting end are almost parallel and indicate that the impact was fiat. The time lag between impact and arrival of the wave corresponds to the propagation along the bar. As the bar is dispersive for wavelengths shorter
Impact
i
Reflection
IP
IR
f
Up
-0.025 rn/s/div lO#s/div
.Figure 10. Oscilloscope record of electromagnetic velocity measurement (vertical) on block surface at 45 mm from axis for alumlntum impactingplexiglas.
344
9
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than its diameter, the initial pressure jump is deformed. The impact pressure from the smaller aluminium bar at 1 m/s can be seen in Figure 8. It reaches a final value of 139-7 bars. Here, dispersion does not allow one to see the initial jump at all. The oscilloscope triggered about 5/~s.before complete contact, which represents a travel of 5 l t m of the bar. Figures 9 and 10 show the records from the electromagnetic gauges on the surface of the block at a distance of 45 mm from the axis for the same impact velocities. After arrival of the longitudinal wave the velocity is first outwards and upwards and reverses to inwards and downwards with the arrival of the surface wave. Here, the reflection from the nearest block side arrives after 75 l~s. 5. DISCUSSION - When using the results from the simple model, it should be borne in mind that it is based on a pressure-penetration relationship which was not derived but assumed as being the most plausible. As can be seen from Figure 5, the solution for the average pressure at the interface is composed of an initial jump, followed by an asymptotic exponential approach to the final value. The initial jump and the final value are given exactly. The fraction represented by the initial jump depends on the impedance ratio of bar to half-space. This fraction decreases towards zero for increasing impedance ratio. Looking at the assumed pressure-penetration relationship in Figure 2, one can notice that it approaches the static line for increasing impedance ratio. This means that for large ratios the impact transient is controlled almost entirely by the static elastic behaviour of the half-space. As a consequence, the exponential term of the solution becomes more and more accurate as the impedance ratio increases. In the limit of infinit~ ratio there is no inertia contribution from the half-space. The process becomes analogous to the interaction of a drill rod and a failing rock surface with proportionality between force and penetration as treated by Lundberg [5].
(al
ntL L 1
2
'
t/~
(b)
Pr (c)
Figure 11. Reflection of a stress pulse propagating in a bar which rests on a half-spaCewith Pa cz[p~ cz = .(a) Incident pulse; (b) average pressure at interface; (c) reflected wave. I
3.
ELASTIC IMPACT OF A BAR ON A HALF-SPACE
345
The present analysis easily can be extended to find the average pressure at the interface of a bar resting on a half-space for the case of an incident stress pulse in the bar (see Figure I l). The stress pulse in the stationary bar is equivalent to an impact with a velocity of two times the particle ve.l,ocity of the. pulse. This is explained by the fact that if the incident pulse would reflect at a free end, the resulting particle velocity would double and the stress vanish. The reflected wave is obtained as tile difference between the pressure at the interface and the incident ~pulse. Figure l I shows the average pressure at the interface and the reflected wave for an impedance ratio of bar to half-space of 3. The end of the square pulse generates an identical but negative signal which is superposed on the first one. The initial inversion of the reflected wave is only present for impedance ratios greater than one. Kolsky [3] relates this phenomenon to what he calls a "cross-over effect" and compares it to the wave reflection at the boundary between an elastic and a linear viscoelastic bar. Th6 experimental results obtained from strain gauge measurements on the bar at 3 diameters from the impacting end agree well with the model predictions. Using the bar to transmit the impact pressure to the gauges puts a limitation on the higher frequency components of the signal. As a result, the initial jump has a rise time of about 20/~s fo r the plexiglas bar (Figure 7) and is not visible at all for the aluminium bar (Figure 8). The record from the plexiglas bar shows weak oscillations of the stress after the initial jump. These are due to two-dimensiorlal effects in the contact region, their wavelength corresponding to the bar radius ifcalculated with the Rayleigh wave velocity. In Figure 8, showing the impact of an aluminium bar on the plexiglas block, one can clearly see the exponential transition towards the final pressure. The impedance ratio of 5.08 is sufficiently high for the half-space to behave almost statically. This is desp!te the fact that its wave propagation velocity is 3 times lower than in the bar. The static behaviour of the half-space is dominant because the pressure increases by a large factor above its initial value whereas the interface velocity decreases. The inertia effects ofthe dynamic process in the half-space are related to the initial interface velocity and their influence decreases as the pressure increases above its initial value. The electromagnetic velocity measurements on the surface of the half-space in Figures 9 and l0 can be interpreted with the help of the analytical solution for the suddenly applied point load given by Pekeris [I]. P and R mark the arrivals of the longitudinal and Rayleigh waves coming from the nearest point of the bar. It is characteristic that both the horizontal and vertical components of velocity change sign with the arrival of the surface wave. The analytic solution has an upward infinity for vertical displacement just before the Rayleigh wave arrival, the effect of which can be clearly seen in Figure 10. The horizontal infinity is not visible in the measurement because it comes after the Rayleigh wave and is inwards. Taking the static solution for the half-space by Sneddon [4] one can calculate the horizontal and vertical displacements at the position of the electromagnetic gauge for the final pressure. This value then can be compared to the integral of the measured velocity curves. '.The integrals were computed up to the moment of the arrival of the reflection. For the impact with the plexiglas bar (Figure 9) the measured horizontal displacement is~1-4/~m and the vertical 4.7 llm, which have to be compared to 1.21/~m and 4.67 llm as calculated from the static solution. For the aluminium bar (Figure I0) the measured vertical displacement was 2.0/~m and the calculated one 2.26 l~m. In most rock breakage methods involving impact, the impedance of the impacting body is 9 higher than for the rock. As a consequence, the initial pressure developed in an elastic flat impact is only a small fraction of the final v'alue gradually built up with time. The rise-time required to attain this final value depends only on the ratio of bar to half-space impedance and not on impact velocity. It was shown that the transition from the initial to the final pressure is exponential with a characteristic time constant. Under normal conditions, the developed pressures are well above the load bearing capacity of the rock and failure takes I
346
w.
JANACH
place. It is well known that rocks can withstand higher loads than their static strength for a short period of time, the reason being that failure requires time to develop. In an impact on rock the pressure at the interface therefore depends on two competing processes: the first is due~o the elastic behaviout and increases the pressure, whereas the second tends to unload the interface through material failure. REFERENCES 1. C. L. PEKERlS 1955 .ProcJeedings of the National Academy of Science USA 41,469--480. The seismic surface PUlSe. 2. G. EASON 1966 Journal of the Institute of Mathematics and its Applications 2, 299-326. The displacements produced in an elastic half-space by a suddenly applied surface force. 3. S. BOOCHERand H. KOLSKY 1972 Journal of the Acoustical Society of America 52, 884-898. Reflection of pulses at the interface between an elastic rod and an elastic half-space. 4. I. N. SNEDDON 1951 Fourier Transforras. New York: McGraw-Hill Book Company, Inc. See pp. 458-462. 5. B. LUNDBERG1971 Dissertation, Chalmers University of Technology, G6teborg. Some basic problems in percussive rock destruction.