Inelastic collision and the Hertz theory of impactAmerican Journal of Physics, Vol. 68, No. 10, pp. 920-924, October 2000
©2000 American Association of Physics Teachers. All rights reserved.
H. H. Wills Physics Laboratory, Royal Fort, Bristol BS8 1TL, United KingdomReceived: 19 May 1999; accepted: 2 December 1999
The area, A, and the duration of contact, T, have been measured as a function of impact speed, U, for balls striking a flat surface. The balls lost about 40% of their kinetic energy over the range of speeds studied, but, surprisingly, the results for A(U) and T(U) appear to be consistent with Hertz's elastic theory of impact. Possible reasons are discussed for this unexpected behavior. © 2000 American Association of Physics Teachers. [S0002-9505(00)01405-7]
The physics of collision has been studied at least since Newton's time. Much of the interest has been in the longitudinal collision of cylinders to see whether elastic wave propagation could explain the experimental results, but balls have also been studied, and Hertz devised a quasistatic, elastic theory to describe their behavior. Bell1 gives a survey of much of the early work, while Cross2 has recently performed experiments on balls which are relevant here.
Hertz calculated the deformation at contact between isotropic, homogeneous bodies with spherical surfaces in the static, linear, elastic approximation (see Love3, and Landau and Lifshitz4, for accessible references, while a simpler, approximate treatment has been given by Leroy5 - who also includes the full Hertz results for comparison). Hertz's theory relates the compressions, z1 and z2, at the two surfaces to the force between them, and to the radii and the elastic moduli of the two bodies. Simplifying his general expressions to the case of interest here, where a ball of radius R is pressed onto a flat surface by a normal force, F, one finds that the total compression of the surfaces, z(z1 + z2), is given by
where X1 and X2 are elastic coefficients for the two bodies, with
where and E are Poisson's ratio and Young's modulus, respectively. Integrating (1) gives a static, elastic potential energy function which is proportional to z5/2. The assumption that Eq. (1) can be applied to the dynamic impact of a ball striking the surface, and that the sum of kinetic and elastic stored energy during collision is constant, then yields the maximum compression, zmaxh. h depends on the initial impact speed, U, and on the reduced mass of the colliding bodies - which is just the mass of the ball here, M, since it falls on a body of effectively infinite mass. The Hertz theory thus predicts that h is given by
The time taken to reach maximum compression is given by an elliptic integral over the compression, and for an elastic collision the total duration of contact, T, is twice this calculated time,
In practice, a more convenient parameter than h for experimental measurement is the area of the common circle of contact, A, which Hertz's theory shows is given by A = Rh. The experimental parameters A and T thus have a characteristic dependence on U,
where the power of 0.8 in the first of these expressions arises because the potential energy of compression is proportional to h5/2, and the two exponents in (5) necessarily differ by unity.
The Hertz equations can be rearranged to give ratios of measured quantities which should remain independent of the impact speed, U,
These are used later, as is also a ratio which is, in addition, independent of the elastic moduli involved in the collision,
Hertz's theory can be tested by measurements of the functional form of A(U) and T(U) in Eq. (5), but as is discussed below, the best test is probably obtained by calculating the ratio of Eq. (8). There has been no detailed experimental confirmation of Hertz's theory which fully considers the elastic behavior under impact conditions - both its linearity, and the conservation of energy6 - but it appears to be commonly accepted as essentially correct, at least for impact speeds that are small compared with the speed of sound in the bodies involved3. The assumptions of the theory clearly limit its range, depending on the elastic behavior of the materials involved, and inelastic collision must introduce loss terms into the energy balance equation. These loss terms cannot be specified in general; however, it is clear that the predictions of the theory should be affected systematically as the energy loss increases. A convenient measure of energy loss is the coefficient of restitution, e, the inverse of the ratio of the relative speeds of approach and separation, before and after collision. When, as here, all the kinetic energy is carried by the ball (since the body it collides with is so heavy it is always at rest), the relative loss of kinetic energy is given by K(U)(1-e2(U)).
The present results came from a student project on the coefficient of restitution
between croquet balls and mallets, made in order to establish the variations
which exist for typical equipment. The project included some measurements of
the area and duration of contact, but it was only much later realized that
these appear to agree with Hertz's theory of impact,despite the loss of about
40% of the kinetic energy of the collision. This unexpected agreement was surprising,
and it is discussed in what follows.
A test anvil was made from a cylinder of steel weighing 70 kg (about 150 times the mass of the ball) equipped so that mallet heads could be firmly clamped to it. The coefficient of restitution, e, was found by dropping a ball from varying heights either on to the mallet face, or onto the smooth surface of the anvil, and measuring its speed before and after impact. The apparatus is shown in Fig. 1. Vertical drops gave impact speeds of 2-6 m s-1, while for speeds of 0.5-2 m s-1 the anvil assembly was turned on its side and the ball swung by a long, light, bifilar suspension so as to make impact at the bottom of its swing; in both cases the impact was normal to the surface, with negligible rotation. The ball interrupted a laser ray as it fell, which triggered a digital timer and thus allowed its mean speed at the ray height to be found. The ratio of the times measured before and after impact, rt1/t2, gives an approximate value of e; however, correction is needed for the extra acceleration between the ray position and the point of impact. Impact speeds calculated from corrected values of t1 agreed with the values expected for fall under gravity from the known initial heights. Values of e(U) were usually reproducible within ±0.002. For example, ten vertical drops at 2 m s-1 gave a speed ratio r = 0.733(1) (i.e., 0.733±0.001), giving e = 0.827(1) after correction, while at the same speed in the pendulum tests, r was 0.823(1), which corrected to e = 0.824(1).
The area of contact, A, was obtained from impressions made on a piece of carbon paper or thin foil; the impressions were usually close to circular, though somewhat irregular, and the values of A were not significantly affected by the thickness of the layer. The duration of contact, T, was measured by adapting the timing system so that the start and stop pulses were produced by the make and break of electrical contact of the impact surfaces, each of which was covered by a very thin layer of conductor. The standard deviations of the measured contact areas were ±5 mm2, and of the times, ±0.02 ms.
The ball used was a Jaques "Eclipse" ball (commonly used in the game of croquet),
which fell on a wooden mallet. The ball is rather similar to a golf ball (but
bigger, with M = 454 g and R = 46 mm), and has a core of
a proprietary composite material contained within a plastic shell 2
mm thick. The ball had a coefficient of restitution close to the value of the
golf ball reported by Cross2 (cf. Sec. III C 2).
The mallet was a typical, well-used old mallet of unrecorded wood and grain,
but with good flat faces.
III. RESULTS AND DISCUSSION
A. Contact areas, A, and times, T
Results averaged over ten measurements are shown in Table I. The data were compared with Hertz's theory using weighted, least-squares, log-log fits. This gave exponents of 0.77±0.03 and -0.23±0.03 forA(U) and T(U), respectively, compared with values of 0.8 and -0.2 given by Eqs. (5).
The last three columns of Table I show that the terms predicted to be constant by Eqs. (6,7,8) are indeed constant at the level of experimental accuracy, provided that one uses for U the initial impact speeds. The mean value of A/UT from Eq. (8) is then 51±2 mm, close to the theoretical value of 49.1mm, whereas using the recoil speed gives values of 65mm, which increase systematically with U.
Hertz's equations arise because the elastic strain field is localized to about the depth of the radius of the circle of contact, cf. the diagram in Love.3 However, this must be the case for any solid which is reasonably large compared with the area of impact. For example, for cylinders, Prowse7 found that T(U) follows a power law with an exponent which varies from -0.14 for long cylinders, to -0.23 as the length tends to zero, i.e., straddling the Hertz value. Some of his cylinders had ball-ended impact surfaces for which one can calculate the ratio (A/UT) from the data in his paper; it was constant for each cylinder, but decreased with length, falling to 7.1 mm, significantly smaller than the value of 7.7 mm expected from Hertz's theory. The exponents measured in the present work agree with Hertz's theory, but the results for cylinders show that this is not a critical test. Equation (8) is a stronger test because (i) it is independent of the value or behavior of the elastic coefficients X, (ii) the value of the numerical term is a direct consequence of Hertz's theory for a sphere, and (iii) it involves the magnitude of the impact speed explicitly. The ratio (A/UT) discriminates better between the shapes of the colliding bodies than does the exponent of T(U); moreover, it gives a strong indication that it is the initial speed which should be used in the comparison with theory (cf. Sec. III C 1).
The mean values of A1/2T2 and A5/2/U2 from Table I lead to values of (X1 + X2) equal to 14×10-10 N-1 m2 and 15×10-10 N-1 m2, respectively, and a plot of A vs U0.8 gives a good fit through the origin, despite the restricted range of the data, with a value of 15.2±0.4×10-10 N-1 m2. The value of (X1 + X2) is the sum of terms from the ball and the wood of the mallet, (XJ + XW) say, neither of which is known independently; however, the results discussed in Sec. III B lead to a value of the ratio XW/XJ, and thus to their separate values.
The only comparable work on inelastic collision appears to be due to Tait8,
who made experiments on materials with e between 0.3 and 0.8. Tait's
ingenious experiments preceded Hertz's theory and do not allow a systematic
comparison with it; however, they provide some support for the agreement found
here. For example, for a sample of vulcanized India rubber (e0.78),
he found a compressive force proportional to z3/2, in agreement
with (1), and he used it to calculate an impact time which
is in good agreement with Hertz's theory.
B. Coefficient of restitution, e
The results for the Jaques ball striking both steel and wood are shown in Fig. 2. Since the ratio of maximum linear compressions normal to the area of contact of bodies 1 and 2 is given by the Hertz theory as
and since (see below) XJ for the Jaques ball is 4×10-10 N-1 m2 while X for steel9 = 0.043×10-10 N-1 m2, then the ball accounts for 99% of the total compression for the impact on steel, and e(U) on steel is almost entirely due to the intrinsic energy loss of the ball. The values of e(U) shown for impact on wood in Fig. 2 are thus surprising at first sight since they indicate a loss lower than the intrinsic loss of the ball; however, the important variable is not U, but the compression of the colliding bodies, which is related to their elastic coefficients and to U by Eq. (6). The shapes of the curves in Fig. 2 are similar, and in fact by transforming the data to give the relative loss of kinetic energy using K(U)(1-e2(U)), both sets of data can be expressed by K(U) = K0 + K1U0.4, as shown in Fig. 3, with K0 and K1 equal to 0.198(2) and 0.112(2), respectively, for the intrinsic loss on steel, and to 0.213(5) and 0.077(4) on wood.
The expressions for K(U) show that by making a small shift in the value of K0, and by scaling U in the ratio UW/US = 2.5(1), the loss curves on steel and on wood can be superposed exactly. Evidence from the present experiments (not given in detail here) shows that the energy loss in the wooden mallet can account for the small increase in the value of K0, but makes negligible contribution to the value of K1, from which it follows that the ratio UW/US is just the speed ratio which produces the same compression of the ball, h1, on impact with the two surfaces. This is a ratio which can also be calculated from Eq. (6) in terms of the elastic coefficients of the bodies (recalling that A = Rh and that h = h1 + h2), giving
where XW and XJ refer to wood and to the ball, and X for steel has been assumed to be negligible. Using (10) and the measured value of UW/US gives XW/XJ = 2.5, and combining this with (XW + XJ) as found before, gives XJ = 4.3(2)×10-10 N-1 m2 and XW = 10.7(2)×10-10 N-1 m2.
The original aim of this work did not require the equipment used to be well
characterized, and no independent value exists for XJ.
For the wood, neither the sort used nor its grain alignment is known, but since
all woods have the same value of Young's modulus within a factor of 2 (Ref. 9)
(about the same factor as arises from accidents of growth), a value for X along
the grain of 1×10-10 N-1 m2 is typical,
i.e., only one-tenth of the value of XW found here. Wood,
however, is extremely anisotropic, with values of X along, across,
and tangential to the grain typically in ratios 1:10:209,10.
The effect of elastic anisotropy on the Hertz theory has never been calculated,
but the present results show that the higher value of X associated
with transverse stresses must be important (cf. the diagram of the stress field
given in Love3 and the considerations of elastic
stored energy in Leroy5).
C. The Hertz theory and inelastic collision
The results of Sec. III A show good agreement with Hertz's theory of impact, which is unexpected since about 40% of the initial kinetic energy is lost. The theory assumes that mechanical energy is conserved, which is clearly not true, so why do Hertz's equations appear to hold?
1. Quasi-elastic initial compression
A considerable part of the initial kinetic energy does not contribute to the kinetic energy of recoil, but is almost all ultimately dissipated by thermal diffusion; this dissipated energy is not initially distinct from that which is recovered, however, and one may expect that both fractions are identically stored throughout the strain field. If this is correct, then the elastic potential energy function used by Hertz is valid for the initial compression, and the calculated values of A and of T/2 are given correctly using the initial impact speed U, even for inelastic collisions. Of course, the recoil speed is reduced after impact and Hertz's theory shows that the subsequent separation must take longer than the initial compression by an amount which can be calculated from the known values of e(U). However, this systematic increase of T in the present experiments is only 0.018 ms for each of the speeds in Table I, an effect at the limit of uncertainty (but see also Sec. III C 3)
2. Asymmetric mechanical response
The combination of measurements of A(U) and T(U) suggests that the energy loss occurs after the time of maximum compression of the ball, and this is confirmed by the recent experiments of Cross2 on the mechanical hysteresis which gives rise to energy loss. Cross used a piezo-transducer to obtain force-time data for a variety of bouncing balls, and obtained the dynamic force-distance response by numerical integration. His results relate to Hertz's theory in a number of ways, but the important point here is that all his hysteresis curves are asymmetric, as shown schematically in Fig. 4 (cf. his Fig. 2): The compression is at least approximately Hertzian, but the force during recoil is notably depressed, and it is also clear that the ball has not fully recovered its shape at separation. Cross also shows quasistatic force-distance curves for the softer balls, using a 3-min strain cycle, and even at this much slower strain rate, 10-4 of that on impact, most balls show hysteresis and stress relaxation during unloading, e.g., from the areas of Cross's hysteresis loops one finds that the relative energy loss of the golf ball is17% under quasistatic test (corresponding to e = 0.91), and 27% under impact (i.e., e = 0.86; cf. the value of e = 0.844 at an impact speed of 1.47 m s-1 from Cross's Table I).
Cross did not study a croquet ball, but the ball studied here appears to be similar in its behavior to his golf ball, with a compression stage which is essentially elastic and Hertzian, and an energy loss which occurs mostly during the hysteresis on recoil, and which includes energy carried off by the still partly compressed ball. The energy loss of the ball measured here, K(U) = K0 + K1U0.4, can now be understood as an irreducible part, K0, arising from the area under the quasistatic hysteresis loop, and a speed-dependent part which increases as the impact time decreases and the hysteresis loop expands. Using the measured value of K(U) for the intrinsic loss of the Jaques ball, its value of e falls from 0.90 for very low impact speeds to 0.82 for a speed of 1.47 m s-1, values which are very close to those given above for the golf ball studied by Cross.
3. Residual deformation at separation
The systematic increase of T(U) by 0.018 ms calculated
in Sec. III C follows from assuming that the ball recoils at reduced speed,
but in accord with Hertz's theory. This implies that the ball has regained
a strain-free, spherical shape, and carries off no elastically stored energy:
This is one possible model of collision; however, it is not compatible with
the mechanical hysteresis curves measured by Cross. The observed residual compression
at separation can be modelled by a different assumption, i.e., that the collision
is elastic and reversible up to the recoil speed eU, when separation
occurs (when the flux of elastic stored energy is too small to produce further
acceleration), while the ball carries off the rest of the energy, which is
ultimately dissipated. The Hertz equation for conservation of kinetic and elastic
stored energy can be solved to obtain the fraction of the maximum compression
which is retained at separation, f (= (1-e2)0.4),
and also the resulting decrease of the collision time. For the speeds in Table
I, f varies from 63.3% to 66.8% when the decrease of
the collision times varies from 0.193 to 0.168 ms. The two models above represent
extremes of behavior, and the true correction for inelastic effects must lie
between their limits. In fact, the experimental value of f for Cross's
golf ball is about 16%, much lower than the 61% which can be calculated from
its value of e, and if this value of f is also appropriate
for the croquet ball, then the contact times of Table I will
all be smaller by 0.04
ms than predicted by Hertz's theory. A systematic deviation of 0.04 ms is at
only twice the level of the experimental uncertainty, and cannot be ruled out;
indeed there is some evidence for it, since if the values of T observed
here are all increased by 0.04 ms to correct for this effect, then the value
of the ratio (A/UT) becomes 49±2 mm, in exact agreement
with the prediction from Hertz's theory.
IV. SUMMARY AND CONCLUSIONS
The object of this work was not to make a test of Hertz's theory of collision, which can be regarded as proven for elastic impact6, but to show that measurements on a croquet ball of contact area, A(U), and contact time, T(U), are well described by his theory, even though about 40% of the kinetic energy is lost on collision.
This result can be understood since the values of A, and T/2,
are largely determined bythe compression stage of impact, before appreciable
energy loss has occurred, and are thus given correctly by the Hertz theory.
The contribution to the total contact time from recoil needs correction, since
it is increased by a lower recoil speed, but is decreased by
a residual compression of the ball at separation. However, since TU-0.2 [cf.
Eq. (5)], both effects give rise to rather small corrections,
and Hertz's equations therefore continue to be a good approximation for inelastic
collision even when the energy loss is large. The two effects lie outside Hertz's
theory, so that the observed agreement with it is partly fortuitous, but many
inelastic materials will have mechanical hysteresis loops similar to that for
Cross's golf ball and the croquet ball studied here, so that the present initially
unexpected agreement with Hertz's elastic theory should in fact not be uncommon.
This remains to be seen.
These experiments began as undergraduate projects. I am grateful to several
students for their enthusiastic interest; in particular, to D. G. McDowall
for developing the timer, to him and J. M. Marshall for the contact time results,
and to A. Jennings and S. Tanner for the contact area results. I am also grateful
to Professor R. G. Chambers for helpful comments on an earlier draft of this
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