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Stress waves produced by impact on the surface of a plastic medium
Stress Waves Produced by Impact on the Surface of a Plastic Medium b ~ Y. 2yr. TSAI * Brown University, Providence, Rhode Island ABSTX~Cr: This paper discusses certain observations of the effect that local plastic yielding has on the radial surface strain waves produced by the impact of a hard sphere on the surface of a steel block. At impact velocities slightly greater than that required first to initiate local plastic yielding, the effect is almost imperceptible close to the area of impact, but becomes observable at distant points. At higher velocities of impact, however, the experimental results clearly show ~he presence of plastic yielding as a sudden change in the slope of the strain waves for all distances of travel. In the development of the theory the surface displacements of an elastic half-space are written as integrals over the first derivative of an arbitrary vertical loading f(l) applied at a point on tl~efree surface. It is shoym that at large distances r away from the point of impact the amplitude of the radial surface strain generated by f(t) is inversely proportional to rvL However, the radial surface strains excited by finite jumps of f(O) and f'(O) are shown to decay as r- : and r-1, respectively, for large values oft. On the basis of the theory developed here, the slope of the applied forcing function is shown to vary rapidly when local plastic yielding occurs. Introduction When a steel ball impinges on the surface of a block of material, axially symmetrical surface waves are propagated over %he free surface. I n addition, extremely large local stresses of short duration are set tlp in the region of contact and close to it. I f the block is m a d e of glass, then at sufficiently high velocities of impact, localized fracture occurs around the point of impact. I n an earlier work by Tsai and Kolsky (1) experiments were carried out to observe the effect t h a t the occurrence of such fractures had on the stress pulses propagating out from the center of impact. When these fractures occurred, it was observed experimentally t h a t these were sharp jumps in the outgoing stress waves, and by observing the timing of these jumps, the instant of fracture was recorded for different velocities of impact. I f a steel block is subjected to a Hertzian impact of sufficiently high velocity, it will, instead of fracturing, start to yield, not in the surface but ~t .a point inside the matelial. Davies (2) has shown t h a t for an elastic material under Hertzian indentation the m a x i m u m shear stress occurs inside the material Mong the ~xis of symmetry, azld slightly below the area of contact. I f the yielding criterion of m a x i m u m shear stress is adapted, this is the point at which the materials will first deviate fi'om the elastic solution. The transition from one * Now at Iowa State University, Ames, Iowa.
204
Stress WavesProducedby Impact state to another in the material would be expected to affect the stress waves generated b y the impact. I n the present work an experimental set-up similar to that used in the earlier paper (1) was employed and observations recording the effects t h a t local plastic yielding has on the stress waves propagated are given for a large range of impact velocities. These are then discussed in the light of the theoretical predictions. Simultaneously, a set of observations is m a d e to measure the time of contact as a function of the velocity of impact. T h e range of impact velocity covered here is much larger t h a n in (3). B y plotting r against v for steel balls impinging on blocks of mild steel, Lifshitz and Kolsky (3) have shown that, although the velocities of impact v have reached as much as eight times t h a t required first to initiate plastic yield, the time of contact r continued to have the values predicted by the Hertz impact theory for elastic bodies. Since the value of r depends upon Young's modulus and Poisson's ratio, it would be expected, as pointed out in (3), to deviate from elastic values when local plastic deformation occurred. I f instead of plotting v1/~ against r we plot v against rv m, which should be a constant, according to Hertz's treatment, we find a systematic variation in this latter quantity. I t has been shown that the propagation of stress waves in a a elastic halfspace can be v e r y complicated, and in the same way the interpretation of the surface waves obtained experimentMly in this paper is expected to be involved. The response of a half-space to a periodic surface force either localized at a point or uniformly distributed over a finite circular area is given in (1, 4, 5, 6). The exact solutions for an elastic half space subjected to a unit step vertical loading at a point on the surface or inside the material were obtained b y Pekeris (7, 8). I n principle, all of the above solutions can be properly superposed to account for appropriate boundary conditions with a known forcing function. In view of the complexity involved in the calculation of pulse shapes from the above solutions, it wouid appear difficult to examine the effect of local dynamic plastic deformation on the outgoing stress waves. To interpret the experimental results of this paper, a theoretical investigation is carried out for an elastic half space subjected to an arbitrary forcing function at a point either on the surface or inside the material. The dependence of the nature of the pulse shapes on the forcing hmction is then discussed.
Times of Contact The specimen we use is a 1.0 X 10 )< 5 inches mild steel block. One of its largest surfaces is first milled, and then polished with v e r y fine emery paper, and finally finished with silver polishing powder. The surface which has a highly reflecting mirror finish is subjected to the impact of steel balls -~ in. in diameter at velocities rang~ng from 0.44 to 66.6 ft/see. The time of contact is measured on a microsecond counter. For low velocities of impact, the experimental arrangement used is similar to t h a t in (3). The hard steel ball is suspended by a long v e r y fine wire, and both the ball and specimen serve as part of an electrical circuit. While they axe in contact the electrical circuit is completed but is broken
Vol.285,No, 3,~Iarell1968
205
Y. M . T s a i v a n d rv¢=(Io-*} ~--~" orv ----Tv
from Experiment from Hertz's fheor F 't'~ from Experiment II~ from Hertz's theory
zz~
mo
Fn~. 1. Time of contact (~ ~sec) for various velocitiesof impact (v it/see).
eo
eo
4o o.a
0.e
0.8
=,0
1.2
L4
I/V~
when the impact is over. The signal produced is fed to tlle counter in order to measure the time of contact. For higher velocities of impact, when the above an'angement is no longer convenient, an air gun system is used, as in (1). In this arrangement, a piece of aluminum, 1.5 X 1£ × 0.25 in., with a circular hole of I in. in diameter is cemented onto the surface of the specimen, but is electrically insulated from it. The steel ball is fired through this hole. The hole is covered by a piece of thin aluminum foil which has been cut into a number of small strips. Thus, the mechanical resistance to the flight of the ball is minimized and good electrical contact maintained throughout the impact. As before, the steel block and the aluminum foii are part of an electrical circuit and making and breaking this circuit gives the signals for actuating the electronic counter. The measurements of the times of contact for a large range of impact velocities are given in Fig. 1, and for comparison, the theoretical results calculated from Hertz's impact theory are also shown [e.g. see (9)]. For a steel bail of radius R e m impinging on the steel block, the relation between the time of contact in seconds and the velocity of approach v in em/sec is ~" = 2 . 9 4 3 { 2 . 5 ~ ' p [ " ( 1 -
F2)/E3}215/~)-115
whore p is the density of the sphere in g/cm3, v is Poisson's ratio and E Young's modulus in dyn/em2. In Fig. 1 the plot of the theoretical values of r against v-1/~ does not show much deviation from that of the corresponding experimental results. However, if we calculate the value of Tv115which elastic theory predicts to be constant, we can see that this quantity in fact increases with increasing velocity of impact. For the range of impact velocities covered here, the value rvm increases about 10 per cent. The reasons for this increase wouid appear to be twofold: First, the effective value of "Young's modulus" for plastic deformation is smaller than that for the elastic state; secondly, the volume of local plastic defo~znation increases with increasing velocity of impact.
206
Journal of T h e F r a n k l i n I n s t i t u t e
Stress Waves Produced by Impact Stress Wave Observations
As mentioned earlier, the waves of most interest in this investigation are the cylindrical Rayleigh surface waves which travel over the surface of a large steel block when il, is subjected to the impact of a steel bail. These waves propagate at a velocity of about 3,000 m/sec [-see Kolsky (10)]. Since the duration of the impact which depcnds upon the impact velocity is found to be about 70 psec, the length of the surfacc pulse is about 7 in. In addition, dilatational waves in steel travel at about 5,900 m/sec, and it takes about 70 gsec to cover a distance of 10 in. To avoid reflections from the bottom and the sides of the specimen, therefore, the strain gages used for detecting the stress waves are placed at 1 in. off the center of the 10 X 10 in. sm'face of the specimen, and the points of impact are within 3 in. from the gage. The s~rain gages employed, of the semiconductor type manufactured by Baldwin-Lima Hamilton [No. SPB-12100C6], are cemented onto the surface of the specimen with an epoxy~esin cement. The experimental arrangement for very low velocities of impact is similar to that used for the measurements of the time of contact. The steel ball is suspended by a long thin wire, and the electrical pulse produced by the completion of the circuit is used to trigger the cathode-ray oscilloscope. The stress waves detected by the gages were then photographed on the screen of the oscilloscope. When the height of fail is more than 6 in., an electromagnet is used to hold the ball at a known height, and is then released to drop normally on the surface of the specimen. When the impact velocities needed are higher than t h a t conveniently producible by the above method, an air gun system is used to fire steel balls. The experimental arrangements employed in the last two methods are similar to that described in (1). A block diagram of this arrangement is shown in Fig. 2. The steel ball is arranged to traverse a thin beam of light just before SINGLE SWEEP START PULSE
STEEL BALL
9 Fro. 2. Block diagram of apparatus used for detecting surface waves. TRiC 3ER
STRAI GAB[
[
USHZ S O ~ C E
PHOTOC ' ELLt ~ - J
__c::~,
- STRAINGAGE
O~STANCECONTROLLtN6 ]HE'rIil~G OF TRIGG~
~ o , ~ oF , ~ A i
V o L 285, N o . 3; M a r c h 1968
207
Y. M. Tsai
/
/
~ \, ,,, ((1)
i
(b)
oL zo ~o,,.sec.
Fro. 3. Radial surface waves observed at v = L86 ft/sec. (a) r = (12/32) iu.; (b) r = 1(3/32) in. impinging on the specimen. The light beam is directed onto a small photocell so t h a t an electrical impulse is produced. When amplified, this could be used to trigger the oscilloscope trace. The observations of stress waves are carried out for impact velocities ranging from 0.5 ft/sec to 66.6 ft/sec, and the nature of the pulse shapes is examined as a function of the impact velocity and the distance of travel. I t can be deduced from the results given in (2) and (3) that the impact velocity required first to initiate plastic yielding just below the point of i m ) a c t is about 1 ft/sec. At velocities of impact equal to 0.5 ft/sec and 0.97 ft/sec, which are in the range of elastic defommtion, the pulse shapes detected by the strain gages are recorded as smooth curves. When the impact velocity is increased to 1.86 ft/scc, which falls in the range of local plastic deformation, the pulse shapes detected at distances close to impact point show no noticeable effect of this local yielding ['see Fig. 3(a)7. However, the effect that even this small anmunt of yielding has on the stress pulse observed at distances further away from the point of impact can be clearly seen in Fig. 3 (b). The reasons for this phenomenon will be discussed later in this section. A t an impact velocity of 4 ft/sec, the stress pulse is still only slightly affected by the yielding for small distances of travel. At large distances from the impact point, however, the effect of yielding on the pulse shape is much
(o)
(b)
o 2o ?/.,.sec
Fie. 4. Radial surface waves observed at v = 7.5 ft/sec. (a) r = 1(4/32) in. ; (b) r = 2(8/32) in.
208
Journal of The FranklinInstitttte
Stress Waves Produced by Impact
more noticeable than at the lower velocities of impact and the effect that local plastic yielding has on the pulse shape appears to increase as the impact velocity increases. Thus, at an impact velocity of 7.5 ft/see, the occurrence of plastic yielding around the are~ of contact can be clearly seen for all distances of travel. This is shown in Figs. 4(a) and 4(b). The trace of the stress pulse, which is a smooth curve for purely elastic deformations, has a sharp angle near its peak followed by a curve with almost constant negative slope. We note that the distances of travel of the stress waves in Figs. 4(a) and 4(b) are about the same as in Fig. 5(a) and 5(b), respectively. In the latter figures, the stress pulses are generated by an impact velocity of 66.6 ft/sec and the effect that plastic yielding has on the stress pulses at this velocity is much more serious than in Figs. 4(a) and 4(b). The shapes of the stress pulses, shown in Figs. 3 to 5, are quite different from t h e experimental results obtained by Kolsky and Douch ( 11) in one-dimensional plastic wave propagation. The height of the stress plateaux detected there by
o
(a)
(b)
~o 4,o ~ s e c
I
Fro. 5. Radial surface waves observed at v = 66.6 f t / s e c , (a) r = 1 (4/32) in.; (b) r = 2 ( 1 5 / 3 2 ) in.
strain gages mounted on a long pressure bar is fixed by the dynamic value of the yield point of the metal. Since no plastic deformation is taking place at the gages, the appearance of stress plateaux would be expected from the simple theory of one-dimensional elastic wave propagation. In the half-space problem, however, the nature of the wave propagation is very different from the above. The amplitudes of stress waves propagating spherically in a half space decay rapidly with increasing distance of travel, and this rapid decay restricts the plastic yielding, when it occurs, to a small region around the point of impact. The surface strain is related to the applied surface stress in a complicated manner, and the shapes are very different from each other. In the next section the solution is obtained for an arbitrary point load applied at the surface of an elastic halfspace. Since the region of local plastic yielding is small, this solution might be expected to be a good approximation for stress waves at a large distance from the center of impact. The surface strain wave given in the next section as an integral over the second derivative of the forcing function ff'(t) in Eq. (54). The contribution of
%1.2ss,No.3,Marchi~6S
209
Y. M. Tsai the second term in (54) is small, and from our experience in earlier work, (1) and (a), it is less than about 8 per cent of the first term. Thus, the characteristic features of the first term might be expected to give a good approximation to the shape of the pulse. If in (54) we let t. = "rr/c=, which is the time for the arrival of the Rayleigh wave front at r, we see that the value of eosh 0 is close to unity for t sligh~Iy greater than t,. We m a y thus expect the shape of err to be similar t o f ' ( t ) for t close to t,. For t much larger than t~, however, the shape of er~ would be very different from f'(t). With the above considerations in mind, we see from Figs. 4 and 5 that the total applied forces have abrupt changes in their slopes when local plastic yielding first occurs at ~ time closely after t,. The characteristic features of stress pulses in Fig. 3 are somewhat different. At this velocity of impact, a small nucleus of plastic yielding would be expected to occur inside the material just below the center of impact. According to the results given in (12) the surface waves produced by an interlml disturbance are relatively more marked at distant points than close to the disturbance. This effect is shown by a comparison of Figs. 3(a) and 3(b). Response of an Elastic l l a l f Space to an Arbitrary Loading To make a rigorous analysis of the experimental results described above, we should, first of all, determine the moving elastic-plastic boundary. This would appear almost impossible because of the complex nature of the Hertzian stress distribution. As pointed out before, however, the region of local plastic deformation is extremely small eompared to the size of the specimen. In addition, most of stress wave observations are made at points distant from the center of impact. To make the problem mathematically tractable, therefore, we assume that the source of disturbances is localized at a point. The problem for an elastic halfspace subjected to an arbitrary vertical loading on the surface is first considered, and the similar problem for an arbitrary vertical loading applied at a point inside the materiM is also solved. In the first place, operational techniques are used to tackle the problem, and the inversion integrals are then evaluated by contour integrations. 1. Transform Solution. The equation of motion of an isotropie solid m a y be written (x + 2 ~ ) v v • n - ~V X v × u =
o(oh~/Ot ~)
(1)
where u is the displacemeut vector, o is the density of the medium and X and are the Lama constants. For aal axial symmet~\y problem, the eylindricM polar coordinates (r, e, z) are the most convenient to use, with the z-coordinate as the taxis of symmetry. Let u be equal to (u,, u~, uo). Since the problem considered is an axially symmetrical one, u~ mad all its derivatives with respect to ~, must vaIfish everywhere. Under these conditions, the equation of motion (1) can be written [e,g. see Kolsky (10)
r-0r\
210
~/+0z
7
= p 0t~
Journalof TheFranklinInstitute
Stress Waves Produced by Impact
and r-~
(r~)
4- 7z~ = p ot~
(3)
where the dilatation A is (4)
A = r~l(O/Or) ( r u , ) 4- (Ou~/az)
and the rotation it has only one non-vanishing cireumferentiM component ~ = (ouJoz)
-
(5)
(on,lot).
Now let us consider a senti-infinite elastic medium. Assume that at z = 0 an internal verticM load f ( t ) is suddenly applied and this is uniformly distributed over a disc area of radius a. The body is free from traction on its surface z = - H . Let ~u~ be the radiM horizontal displacement in the medimn between z = 0 and z = - H and ~u,. for z > 0. Similar notation is applied to other quantities. The boundary conditions are as follows: on z = - H the surface is free from traction, i.e. 2ao~ = ~a~,. = 2a~ = 0; whereas on z = 0 we have 2u,. = ~u,, 2uz = luo, 2o-o,. - lO-zrl 20"~0 ~ lO-z~ ~ 0 and 20"zz = lza, 2~r~ -- lo-o~ = f ( t )
for for
r > a r < a
(6)
The formM solution of the problem is quite straight forward. Laplace transforms are first operated over the variable t as in (7), and Hankel transforms are then applied over r as in (1,~6, 5). Now we define
f*(p)
= f~f(t)e Jo
(7)
-~t dt
and u(~)"
= rjo~u(r)rJ,~(sr)
dr.
(8)
For conve~fience, hewever, the asterisk * for Laplace transfl)rm is omitted in this section, but it will be used in subseque~lt sections where the Laplace inverse integral is evainated. If we now operate Laplace transforms followed by Hankel transfol~ms over Eqs. (2) and (3), we have, respectively (d2P~°/dz 2) -- (s 2 - - k l ~ ) ~ ° - 0
(9)
and (d2~tl/dz 2) -- (s ~" 4- k~2)~ 1 - 0
(10)
2t~) = p~/cl ~
(11)
where Ici~
vow. 2as, xo. ~, ~ h
~96s
-
pp2/ (X
4-
21 1
Y. M . Tsai
and (12)
/elz = pp~/# = p2/c2~.
We take the solutions of Eqs. (9) and (10) as 2~o = Ce-~° + De+~,
(13)
2~l = E e - ~ -~ Fe + ~
(14)
and but ~o = A ~ - *
(15)
1~1 = Be - ~
(16)
which remain finite for largo values of z. F r o m Eqs. (13) to (16), we introduce a = (s ~ + k12) ~
(17)
and m (S2 ~- k22) 112,
(18)
Applying successively similar transforms to Eq. (6), we have ~#~o _ i ~ o o
(19)
= (a/s)gl(sa)f(p).
The transformed stresses and displacements ure found to be (pp2/~2) e~o = k4(d~O/dz~) _ ]c~(k~ _ 2)s27~ _ 2s(dO~/dz) (pp~/~-) a~) = (d251/dz ~) + s2~ ~ - 2k~s(d~°/dz) (pp~/~)~ o = ]c2(d~O/dz) _ s~l
(20) (21) (22)
and (23)
(pp2/g)a~l = ( d ~ V d z ) - k2s ?'°
where k = k # k l = E2(1 -
,)/(1
- 2r)] ~.
If we now use Eqs. (13) to (16) and (19) to (23) to satisfy the boundary conditions, we have a system of six linear equations, the soiutions of which give A = C-D,
B =E+F,
D = (~/k2s)F,
(24)
F = pp2aJ~(sa)f(p)/2,u2~k~
ce "H = - - [ F / k ~ G (s)~{[-(2s ~ + k~) ~ + 4 a ~ s ~ ( ~ / s ) e H, _ 4S[~(2S ~ -{- k~2)e- ~ }
(25) aad E e ~H = ~ F / G ( s ) ]{ (2s ~ ~- k~)4o~fle -H" -
['(2s ~ 47 k~) ~ ~- 4s~al~Je -H~}
(26)
where G ( s ) = (2s ~ q-- k~'~)~ - 4s~afl.
(27)
Thus the problem is, in principle, solved, but in order to interpret the experi-
212
Journal of The FranklinInstitute
Stress Waves Produced by Impact mental results a more detailed and indicative result of the behavior on the surface is required. Now, if we insert the above constants into Eqs. (13) and (14) and then substitute the results into Eqs. (22) and (23), we obtain transformed displacement components. If we let z = - - H and perform the inverse Hankel transforms, we have the sin:face displacements
u,(r, p) = af(p) f ~ ~ Ju (~ts)
{(2s ~ "+- ic~)s_, ~ _ 2~f~e-"%/a(sr) as
(28)
u~(r, p) = af(p) f ~ A ( s a ~ .Io G(s)
l(2s ~ I - k~)s -"~ - 2s2e-~'}otJo(sr) ds.
(29)
and
The inverse Laplace transforms of Eqs. (28) and (29) will give meaningful physical quantities, but the inverse integrals would appear intractable unless J~(sa) is replaced by s/2~-a, giving the solution for an internal vertical point load. By letting " H " equal zero in Eqs. (28) and (29), the transform solutions for a sL~rface load can be obtained--integrauds of which i n v o N e singularities in the contour of integration. These singularities m a k e it difficult to reduce to the case of a surface load from the physical solutions for an internal load. Therefore, the inverse integrals for two different locations of ]oading will be evaluated separately, but the main ideas of integrations for the two cases are essentially identical. 2. S u r f a c e Loading. I n Eqs. (28) and (29) if we let s = k:~ and reduce the solutions to the case of a point load as mentioned before, we have
f u~*(r, p) = l~2f* (p) I ~ ~M(~)J~(k~r) d~ ~o
(30)
and
u~*(r, p) = ]co i*(,n~ r]~ ~N (~ )Jo(k~r) d~ ~rlz a o
(31)
where M(~') = {(2.~~ -F 1) - 2a'B'}/g(~)
N(~) = a'/g(~) g(~') = (2{-~ -? 1) 2 - 4~'~d{~' ~' = ( ~ + 1/k~), ~
(32) (33) (34) (35)
and ~' = (~': + 1) u2
(36)
u,*(r, p) = (1/2~r/~) (O/Or)]i*(r, p)
(37)
Ii*(r, p) = f*(p) f ~ fM(f)Jo(k2fr) df. ao
(38)
Eq. (30) can be written
where
VoL 285, No. 3, March ~968
213
Y. M. Tsai If we use the identity for the zeroth order Bessel Function of the first kind, i.e.,
Jo(x) = --
[exp (ixcoshO) -- exp (-ixeoshO)]dO
(x > 0),
(39)
we can write
Ii*(r, p) = -- ~Trf*(p)[lll - 1~2]
(40)
where
In = ~ ~1ll(~) d~ f ~ exp (ik2~r cosh O) dO
(41)
and I1~ = f 0 ~ M ( ~ ) d ~ f o ~ e x p
(--ik2~rcoshO)dO.
(42)
The next step toward integration is to deform the path of integration over ~" in Eqs. (41) and (42) so that both I n and I12 can be recognized as Laplace Transforms. To do this, we consider ~" = ~ + i~ and cut the complex plane, as shown in Fig. 6 in which ~ = :i=iv are roots of g(~'). Under these condition,s, the integrands of both I n and I12 are anMytic on the right half of the complex plane. If we choose the contours of integration, as shown in Fig. 6, and use the Cauehy integral theorem, we have f
Iu = -- !
~M(z~)K*(k2nr) d~ + ~ri Res. ( + V )
(43)
and =
_
r - ~ yM(+in)K*(-k2nr) dy
~iRes. (-7)
ao
(44)
where
K*(k2vr) = rioexp ( -k2~r cosh 0) dO.
(45)
I t m a y be seen that the Cauehy principal values of I11 and 112 are identical for ]~1 < 1/k and ]~] > 1. Thus In-112
=0
for
!~1 < 1/k,
1~] > 1.
(46)
The values of Ill a n d / 1 : are different for 1/]~ < ] ~ ] < 1 and at poles n = :i:% I f we calculate these values and substitute together with Eq. (46) into Eq. (40), we have 4
r 1
Ii*(p, r) = - ~r-if(P) J~]/kQ(n)K*(p,r/c2) d~l ~- 2P(~)f*(p)K*(pTr/c,)
214
(47)
Journal of The Franklin Institute
Stress Waves Produced by Impact
'
t/Contour of integration for
Iu
{:t" B t
I/k ~1
/ FI(~. 6. Contour for integration.
Branch line; _11/k
-~',-B~~ R R~OO
~'Contour of integration for It2
where Q(~) -
~(n ~ - 1/k2)m(1 - ~ 2 ) m ( 1 - 2 ~ ~)
q(n)(q(g))
(48)
q(~) = (1 - 2n~) ~ + i4n2(n 2 - 1/k2)~/2(1 - ~2),/~
(49)
~{(1 --2V) q - 2 ( ~ 2 -- 1/k2)1/2(~ '~ -- 1.) ~/2 P(v) g'(v)
(50)
and g'(,)
= og(i.)/o.l
....
(51)
The second t e r m in Eq. (47) results from poles at ~ = ± % I n the course of calculation of the above results, we use appropriate vahms of a, as shown in Fig. 6, and we also use the property that g ' ( - ~ / ) = - - g ' ( ~ ) . I n Eq. (47), if we consider K* (p'yr/c~) as a function of Laplace Transforms and use the convolution theorem, we have
~-l[f*(P)K*(p~r/c2) ] = foo~a-',°~/,,) fit -- (vr/c2) cosh O] dO. (52)
Using Eqs. (52), (47) and (40), the inverse transform of Eq. (37) gives the radial displacement
ur(r, t) -- P(7) ~'~ 0Or L ~°~h-l°*21"r) fit -- (~r/c2) cosh O] dO ~r~--~o f~ t~ Q(,) dn L °°~h-l(*°~/'~) frt - (nr/c2) coshO]dO.
Vol. 285, N'o. 3, March 1968
(53)
215
Y. M. Tsai
T h e first t e r m in Eq. (53) is a group of waves which travels at the Rayleigh wave velocity, while the second is another group of waves propagating at velocities between t h a t of shear and dilatation waves. F r o m the condition f ( t ) = 0 for t < 0, it m a y be seen from Eq. (53) t h a t at a certain distance r for t < r/c~, u, = 0. For r/c] < t < r/c2, u, has only the contribution of the second t e r m of Eq. (53) in which the upper limit of integration " 1 " m u s t be replaced by tc2/r. T h e first t e r m becomes effective only when t > r~/c2 which corresponds to the time of the arrival of R~yleigh waves. I f we let f ( t ) = Re ~t and also let t --+ ~ , Eq. (53) gives the steady solution for a periodic surface point load. Using a different method of integration in (1), the radial surface displacement is obtained for a periodic force which is uniformly distributed over a circular area. I f we reduce the solution for ~ uniform periodic force over ~ finite are~ (1) to that for a point load Re ~t, the result is identical w i t h the corresponding steady solution deduced from Eq. (53). I f we let f(t) be a unit step function, Eq. (53) gives the same result as t h a t in (7). We now m a k e a comparison with some other propagation problems. For the one-dimensionM wave propagation along a bar, the solution (10) is
u ( x , t) = E fo [t-(~/°)] f(T) dr.
F o r two-dimensional wave propagation, the response of the elastic half space to a surface line source gives the horizoutM displacement, (13) ]
t" t
u = ~ J,Ilof ( t
-- T) Er2 -
(r2/c 2) 3 m dr.
I n the case of an infinite body subjected to a point source, the radial displacem e n t is, u = f ( t -- r/c)/4~rr (10, 13). I n the above three examples we can see t h a t the displacements are written either in terms of f ( t ) or as an integrM over f ( t ) . The response of an elastic h~lf space to a surface point load is, however, different from the above three cases. Equation (53) shows t h a t the displacement
f[t~ )
--
o~ • ~
~
f'(t)/*r i
fit) = Sin=~ - ?
--- flt)w~th ~.iformly
/~///
decreasing slope
0.4
¢(t )hr
o.~i04
"
- --- consto~ slope
o.z 0.4 0.60.B
I.
1.2 L¢
1.6 I.B
.0
O2 oo
'0,0 0,2 0.4 0,6 o,B
,
t ~t
LO 1.2 d.4 1.6 12 2,0
(a)
-o.
-O.S~-
(b}
Fie. 7. Typical shape of the applied force. (a)f(t); (b) its derivativeif(t).
216
J'ournalof The Fr,lnklinInstitute
Stress Waves Produced by Impact is an integral over the derivative of f(t) mu]tlplied b y a fimetion resulting from surface dispersion. I n other words, the displacements, especially the strain components, are sensitive to the gradient of the excitation function f(t). I t Msshown in (7) and in Eq. (53) t h a t a finite j u m p o f f ( t ) will excite displacement waves, and from these waves infinite local strain w~ves are ganerated. To avoid encountering an infinite strain, we m u s t assume that beth f(t) and if(t) are continuous with their values equM to zero at t = 0. Since the radial strain is e~r = Our~Or, from Eq. (53) we can see that a finite value of i f ( 0 ) excites physically unrealistic strain waves which hlcrease with time and eventually become infinite. Under the above conditions, we have from Eq. (53)
~,, (r, t)
~~'~c~ 1
Jo
+ _7...2 f Q ('q) d, ~r ,uco. J1/~
f " [ t -- ('yr/c2) cosh 07 cosh 2 0 dO foeo~h-1(te~/vr)
f"Et - ('qr/c2) cosh 0~, ~ cosh ~0 dO.
(54)
Thus, the radiM strMn wave is t h e integral of the second derivative of the excitation function F ( t ) , and consequently, it is quite sensitive to the "smoothness" off(t) as shown schematically in Fig. 7. This is consistent with the experimental results described above, where surface waves produced by the impact of steel balls on steel block are detected b y semi-conductor strain gages. When the velocity of impact is sufficiently small, the total force applied to the surface of metal smoothly followed the Hertz solution and hence purely elastic waves are observed which do not show any drastic fluctuations. However, when the velocity of impact is sufficiently large, local plastic deformation occurs ~romld the point of impact and the total applied force would presumably deviate from the smooth elastic solution. The effect of yielding therefore distorts the shape of the applied force and hence the obsel~ed strain waves which depend upon the second derivative of the applied forcing function as shown in (54) are drastically disturbed. We now examine the asymptotic behavior of Eqs. (53) and (54) for large values of r. This can be detelTMned if we know the behavior of the following integral
t" eosh-l(t/t,) Io = Jo f ( t -- t~ cosh O) cosh 20 dO
(55)
where t~ = ~,r/c~ and t > t,. W e first examine the magnitude of the iaterval of integration. Let the upper limit of integration be Oo = cosh -~ (tit,,) this can be written cosh 00 = 1 -k (r/t,)
(56)
where • = t -- t,. When r tends to infinity i.e., t, --~ ~ , the second t e r m in Eq. (56) becomes vanishingly small as does 00. W e can, therefore, introduce in the
Vol.285,No. 3, March1968
217
Y. M. Tsai integrand eosh 0 = 1 + (r'/t~)
(57)
a~ld drop higher order terms in the series expansion of eosh 0. If we insert the above substitutions into Eq. (55), we immediately have
° -
(ha)
(,~')'~"
\2~r)
If we apply Eq. (58) to Eqs. (53) and (54), we can readily see that they both are iuversely proportional to (r) m as r tends to infinity. We now come to the integration of the vertical displacement Eq. (31), and the procedures for iutegration are similar to that for Eq. (38). We let
I2*(r, p) = f*(p) fo ~ ~N(~)Jo(k2~r) d~
(59)
and choose the same contour as Fig. 6 for integration. The contributions of the poles at =t=i~"cancel out and the values of Is* also vanishes fOl I ~ I < l / k . The value of/2* for I ~ ] > 1/k can be readily calculated. Since k2 = p/c~, the inverse of integral of Eq. (31) can be obtained by differentiating the inverse of Eq. (59) with respect to time. In view of Eqs. (39), (59) and (52), the Cauchy principal value of the vertical displacement is
u,(r,t) =0
for
t
1 -0 ft°~/~ vl(~) d~ fo¢°~1~'(t~2/'') f['t -- (nr/c2) cosh 0] dO ~'~gc: Ot J11k
for ~r2.c~ at J ,/,~
(rA~) < t < (r/c,~)
[-vl ( ~ ) + v~ ('q ) J dn
f ~osh-1(tc~/~r) X [ f i t - (vr/c~) cosh0~d0 ~o
for
t > r/c~
(60)
where vl(~) -
~(v~
--
1/k'2),/~(1
q0?) (q0?))
--
2~)
~
(61)
and v2(,) =
1 / [ d ) ( ~ -- 1) 1/2 q(~) (q(~))
4.~8(~7 ~ - -
(62)
If we carry out the differentiation, we can show that the vertical displacement Mso behaves as the integral of if(t). If we let f(t) be the Heaviside function, Eq. (60) immediate]y reduces to the same results as obtained in (7).
218
Journalof TheFranklinInstitute
Stress Waves Produced by Impact 3. I n t e r n a l Loading. If we let s = ~'k2 and replace J~(sa) by s/27ra in Eqs. (28) aald (29), we have the displacements for an internal point load u,.*(r, p)
if(P)2~r~~r0 fo~ ~ELI(~.) exp (--Hlc~') -- L2(£) exp ( - I I k 2 a ' ) j X Jo(k2~r) d~
(63)
and
f*(p)k2 [ ~ u,*(r, p) = ~ Jo rELy(r) exp ( - H k 2 d ) -- L,(~) exp (--Hlc~') 3 Jo(k2~r) d~
(64)
where LI(~) = (2£e + 1)/e(~) L2(~) = 2d13'/g(~) L.~(£) = a'(2f 2 + 1)/g(r)
(65) (66) (67)
L~(~) = 2 ~ d / g ( ~ ) .
(6S)
and The inversions of Eqs. (63) and (64) are achieved if we can carry out the inverse integral of the following typical expression in which Jo(ls~D') is written as an integral similar to Eq. (39) :
Ia* (r, p) = - (i/~r)ff (p) EI~* (r, p) - I~2"(r, p) l
(69)
where
I0~*(r, p) = f ~ d0 f ~ ~L(~) exp ~-HZ~(;~ + ~)~'~ + ik~r eosh 0~ d~
(70)
and
z~* (r, p) = f0~ d0 fo° ~L (~) exp [-Hk~(~ + ~ ' " -- il~r eosh 0] ~
(7~)
L(~') is an even function and e is equal to either "1" or 1/k. We can easily see that I,~* is the complex conjugate function of Ia*. In view of Eq. (69), therefore, we are interested in only the imaginary part of Eq. (70). For integration, we consider ~" = $ A- in and deform the contour of integration of Eq. (70) fi'om the positive real axis to a path such that along it
g,,EIt (~~ + e~) l~ - i~r cosh OJ = 0.
(72)
For Eqs. (70) and (72), we again cut the ~-plane, as shown in Fig. 6. Equation (72) gives ~ = 0 and = ('Oo~ + p~ cosh ~ 0~) al~ (73) where p = r/tt, ~ = ~p~ eosh ~ 0/(1 + p~ eosh ~ 0). (74) The purpose of this deformation is to make (70) recognizable as a Laplace
voL2~5.No.3. M~rch19~S
219
Y. M. Tsai Transform. Hence, for ,1 < T0 we choose } = 0, but from then on we take the p a t h (73), along which = [ ( ~ _ no~)l/2/p cosh 0] + iT
(75)
and the exponent in (70) is --p7 where
r = (H/c) ['p cosh 0 + (p cosh 0)-11~.
(76)
This deformatinn of contour is justified, because the integral (70) vanishes at infinity along the arc connecting the real axis and the path (73). Along the line = 0, the exponent in (70) is - - p r ' where r r = (l/c) [-H(e2 -- ~2)1/2 + ~r cosh 01.
(77)
Therefore, the integral (70) can now be written
181*(r,p)=
f0® d0 {fore ~'L(~ ') exp ( - p r ' ) ( O ~ ' / O r ' )
where r0' and ro are, respectively, the values of r ~ and ~ at ~ = 70. ~' is the quailtity ~"writtan in telTnS of r ' from (77), and ~ in the second term of (78) is from Eqs. (75) and (76). Taking the inverse transform of (78), we have
I3~(r, z) = fo ~ d~I~tL(~')(O~'/Or)H(ro ' - .r) + ~L(~)(O~/Or)H(r
-- to) }.
(79)
If we recall that I82" is the complex conjugate of In1*, we have from Eqs. (69) and (79), P t
Is(r, t) = (2/~r) Jo f ( t -- r)¢,,[Is~(r, r) ] dr.
(80)
Applying (69) and its inversion Eq. (80) to Eqs. (63) and (64), we have immediately the solutions for u,(r, t) and u,(r, t). The expressions for u~ and u, are, respectively, similar to (53) and (60). If we let f(t) be a Heaviside fullctinn, we have reduced the solution to the same displacement fields as obtained in (8).
Discussion
and Conclusions
The surface displacements of an elastic half-space have been written as integrals (53) and (60) over the first derivative of an arbitrary vertieM loading f(t) which is applied at a point on the surface of material. At large distances r away from the point of impact, it was shown in (58) that the amplitude of the radial surface strain is inversely proportional to r 1/2. However, the radial strain waves excited by the finite jumps of f(0) and f ' ( 0 ) , which involve infinite
220
Journal of The F r a n k l i n I n s t i t u t e
Stress Waves Produced by Impact
strains, can be shown from Eq. (53) to decay as r -2 and r -l, respectively, as r --+ ~o. Thus, the stress waves observed at large distances of travel would be the disturbances produced, not b y the j u m p f(0) or f'(O), b u t b y the continuous vaa'iation o f / ( t ) . The integral representations for surface displacements generated b y internM vertical loading h a v e also been obtained, and the nature of these integrals are similar to those for surface loading. The effect t h a t local p]astie yielding has on the stress pulses has been observed by v a r y i n g the velocity of impact and the distance of travel. A t impact velocities slightly higher t h a n t h a t required first to initiate local plastic yielding, this effect was almost imperceptible close to the area of impact, b u t became noticeable at distant points. This was discussed and shown to be consistent with the result in (12). A t higher velocities of impact, the experimental ~esults clearly show the presence of plastic yielding, b u t instead of producing a stress plateau as it does for plane waves, it appears as a sudden change of slope of the cylindrical stress pulses propagating out from the area of impact. On the basis of the theory developed here and the simplified assumptions made, the shape of the initial part of the strain pulses is similar to t h a t of the first derivative of the applied forcing function. Thus, the slope of the applied forcing function varies rapidly when locnl plastic yielding has occun'ed. The effect of plastic yielding on the stress pulses increases with increasing velocity of impact. The time of contact was measured for a l~rge range of impact velocity. During plastic deformation the amount of contact time in excess of that calculated from ttertz's elastic impact theory increases with increasing velocity of impact. F o r the range of velocities covered, the value (1 -- ~2)/E has increased b y about 10 per cent. Acknowledgment
The author wishes to thank Professor H. Koisky for many useful discussions in the course of the investigation. Thanks are due also to Messrs. L. Daubney and R. Stanton for help with the experimental work. The present research was sponsored partly by the U. S. Army Research Office (Durham) under contcact DA-ARO(D)-358, partly by the National Science Foundation under grant GP-2010 and partly by the Advanced Research Projects Agency under contract SD-86, all at Brown University. References
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
Y. M. Tsai and H. Kolsky, J. Mech. Phys. Solids, Vol. 15, pp. 263-278, 1967. R. M. Davies, Prec. Roy. Soc. A, Vol. 197, p. 416, 1949. J. M. Lifshitz and N. Kolsky, J. Mech. Phys. Solids, Vol. 12, p. 35, 1964. H. Lamb, Phil. Trans. A, VoI. 203, p. 1, 1904. G. F. Miller and H. Pursey, Prec. Roy. Soc. A, Vol. 223, p. 521, 1954. Y. M. Tsai, to appear, J. Mech. Phys. Solids, 1968. C. L. Pekeris, Prec. Nat. Aca. Sci., Vol. 41, p. 469, 1955. C. L. Pekeris, Prec. Nat. Aca. Sci., Vol. 41, p. 629, 1955. A. E. H. Love, "A Treatise on the Mathematical Theory of Elasticity," p. 195, Camridge Univ. Press, 1927. H. Kolsky, "Stress Waves in Solids," Oxford, Clarendon Press, 1953. H. Kolsky and L. S. Douch, J. Mech. Phys. Solids, Vol. 10, p. 195, 1962. C. L. Pekeris and H. Lifson, J. Accoust. Soc. Am., Vol. 29, p. 1233, 1957. A. T. de Hoop, Appl. Sd. Res. B, Vol. 8, p. 349, 1960.
Vol.285,No. 3, March1958
22 1
204
Stress WavesProducedby Impact state to another in the material would be expected to affect the stress waves generated b y the impact. I n the present work an experimental set-up similar to that used in the earlier paper (1) was employed and observations recording the effects t h a t local plastic yielding has on the stress waves propagated are given for a large range of impact velocities. These are then discussed in the light of the theoretical predictions. Simultaneously, a set of observations is m a d e to measure the time of contact as a function of the velocity of impact. T h e range of impact velocity covered here is much larger t h a n in (3). B y plotting r against v for steel balls impinging on blocks of mild steel, Lifshitz and Kolsky (3) have shown that, although the velocities of impact v have reached as much as eight times t h a t required first to initiate plastic yield, the time of contact r continued to have the values predicted by the Hertz impact theory for elastic bodies. Since the value of r depends upon Young's modulus and Poisson's ratio, it would be expected, as pointed out in (3), to deviate from elastic values when local plastic deformation occurred. I f instead of plotting v1/~ against r we plot v against rv m, which should be a constant, according to Hertz's treatment, we find a systematic variation in this latter quantity. I t has been shown that the propagation of stress waves in a a elastic halfspace can be v e r y complicated, and in the same way the interpretation of the surface waves obtained experimentMly in this paper is expected to be involved. The response of a half-space to a periodic surface force either localized at a point or uniformly distributed over a finite circular area is given in (1, 4, 5, 6). The exact solutions for an elastic half space subjected to a unit step vertical loading at a point on the surface or inside the material were obtained b y Pekeris (7, 8). I n principle, all of the above solutions can be properly superposed to account for appropriate boundary conditions with a known forcing function. In view of the complexity involved in the calculation of pulse shapes from the above solutions, it wouid appear difficult to examine the effect of local dynamic plastic deformation on the outgoing stress waves. To interpret the experimental results of this paper, a theoretical investigation is carried out for an elastic half space subjected to an arbitrary forcing function at a point either on the surface or inside the material. The dependence of the nature of the pulse shapes on the forcing hmction is then discussed.
Times of Contact The specimen we use is a 1.0 X 10 )< 5 inches mild steel block. One of its largest surfaces is first milled, and then polished with v e r y fine emery paper, and finally finished with silver polishing powder. The surface which has a highly reflecting mirror finish is subjected to the impact of steel balls -~ in. in diameter at velocities rang~ng from 0.44 to 66.6 ft/see. The time of contact is measured on a microsecond counter. For low velocities of impact, the experimental arrangement used is similar to t h a t in (3). The hard steel ball is suspended by a long v e r y fine wire, and both the ball and specimen serve as part of an electrical circuit. While they axe in contact the electrical circuit is completed but is broken
Vol.285,No, 3,~Iarell1968
205
Y. M . T s a i v a n d rv¢=(Io-*} ~--~" orv ----Tv
from Experiment from Hertz's fheor F 't'~ from Experiment II~ from Hertz's theory
zz~
mo
Fn~. 1. Time of contact (~ ~sec) for various velocitiesof impact (v it/see).
eo
eo
4o o.a
0.e
0.8
=,0
1.2
L4
I/V~
when the impact is over. The signal produced is fed to tlle counter in order to measure the time of contact. For higher velocities of impact, when the above an'angement is no longer convenient, an air gun system is used, as in (1). In this arrangement, a piece of aluminum, 1.5 X 1£ × 0.25 in., with a circular hole of I in. in diameter is cemented onto the surface of the specimen, but is electrically insulated from it. The steel ball is fired through this hole. The hole is covered by a piece of thin aluminum foil which has been cut into a number of small strips. Thus, the mechanical resistance to the flight of the ball is minimized and good electrical contact maintained throughout the impact. As before, the steel block and the aluminum foii are part of an electrical circuit and making and breaking this circuit gives the signals for actuating the electronic counter. The measurements of the times of contact for a large range of impact velocities are given in Fig. 1, and for comparison, the theoretical results calculated from Hertz's impact theory are also shown [e.g. see (9)]. For a steel bail of radius R e m impinging on the steel block, the relation between the time of contact in seconds and the velocity of approach v in em/sec is ~" = 2 . 9 4 3 { 2 . 5 ~ ' p [ " ( 1 -
F2)/E3}215/~)-115
whore p is the density of the sphere in g/cm3, v is Poisson's ratio and E Young's modulus in dyn/em2. In Fig. 1 the plot of the theoretical values of r against v-1/~ does not show much deviation from that of the corresponding experimental results. However, if we calculate the value of Tv115which elastic theory predicts to be constant, we can see that this quantity in fact increases with increasing velocity of impact. For the range of impact velocities covered here, the value rvm increases about 10 per cent. The reasons for this increase wouid appear to be twofold: First, the effective value of "Young's modulus" for plastic deformation is smaller than that for the elastic state; secondly, the volume of local plastic defo~znation increases with increasing velocity of impact.
206
Journal of T h e F r a n k l i n I n s t i t u t e
Stress Waves Produced by Impact Stress Wave Observations
As mentioned earlier, the waves of most interest in this investigation are the cylindrical Rayleigh surface waves which travel over the surface of a large steel block when il, is subjected to the impact of a steel bail. These waves propagate at a velocity of about 3,000 m/sec [-see Kolsky (10)]. Since the duration of the impact which depcnds upon the impact velocity is found to be about 70 psec, the length of the surfacc pulse is about 7 in. In addition, dilatational waves in steel travel at about 5,900 m/sec, and it takes about 70 gsec to cover a distance of 10 in. To avoid reflections from the bottom and the sides of the specimen, therefore, the strain gages used for detecting the stress waves are placed at 1 in. off the center of the 10 X 10 in. sm'face of the specimen, and the points of impact are within 3 in. from the gage. The s~rain gages employed, of the semiconductor type manufactured by Baldwin-Lima Hamilton [No. SPB-12100C6], are cemented onto the surface of the specimen with an epoxy~esin cement. The experimental arrangement for very low velocities of impact is similar to that used for the measurements of the time of contact. The steel ball is suspended by a long thin wire, and the electrical pulse produced by the completion of the circuit is used to trigger the cathode-ray oscilloscope. The stress waves detected by the gages were then photographed on the screen of the oscilloscope. When the height of fail is more than 6 in., an electromagnet is used to hold the ball at a known height, and is then released to drop normally on the surface of the specimen. When the impact velocities needed are higher than t h a t conveniently producible by the above method, an air gun system is used to fire steel balls. The experimental arrangements employed in the last two methods are similar to that described in (1). A block diagram of this arrangement is shown in Fig. 2. The steel ball is arranged to traverse a thin beam of light just before SINGLE SWEEP START PULSE
STEEL BALL
9 Fro. 2. Block diagram of apparatus used for detecting surface waves. TRiC 3ER
STRAI GAB[
[
USHZ S O ~ C E
PHOTOC ' ELLt ~ - J
__c::~,
- STRAINGAGE
O~STANCECONTROLLtN6 ]HE'rIil~G OF TRIGG~
~ o , ~ oF , ~ A i
V o L 285, N o . 3; M a r c h 1968
207
Y. M. Tsai
/
/
~ \, ,,, ((1)
i
(b)
oL zo ~o,,.sec.
Fro. 3. Radial surface waves observed at v = L86 ft/sec. (a) r = (12/32) iu.; (b) r = 1(3/32) in. impinging on the specimen. The light beam is directed onto a small photocell so t h a t an electrical impulse is produced. When amplified, this could be used to trigger the oscilloscope trace. The observations of stress waves are carried out for impact velocities ranging from 0.5 ft/sec to 66.6 ft/sec, and the nature of the pulse shapes is examined as a function of the impact velocity and the distance of travel. I t can be deduced from the results given in (2) and (3) that the impact velocity required first to initiate plastic yielding just below the point of i m ) a c t is about 1 ft/sec. At velocities of impact equal to 0.5 ft/sec and 0.97 ft/sec, which are in the range of elastic defommtion, the pulse shapes detected by the strain gages are recorded as smooth curves. When the impact velocity is increased to 1.86 ft/scc, which falls in the range of local plastic deformation, the pulse shapes detected at distances close to impact point show no noticeable effect of this local yielding ['see Fig. 3(a)7. However, the effect that even this small anmunt of yielding has on the stress pulse observed at distances further away from the point of impact can be clearly seen in Fig. 3 (b). The reasons for this phenomenon will be discussed later in this section. A t an impact velocity of 4 ft/sec, the stress pulse is still only slightly affected by the yielding for small distances of travel. At large distances from the impact point, however, the effect of yielding on the pulse shape is much
(o)
(b)
o 2o ?/.,.sec
Fie. 4. Radial surface waves observed at v = 7.5 ft/sec. (a) r = 1(4/32) in. ; (b) r = 2(8/32) in.
208
Journal of The FranklinInstitttte
Stress Waves Produced by Impact
more noticeable than at the lower velocities of impact and the effect that local plastic yielding has on the pulse shape appears to increase as the impact velocity increases. Thus, at an impact velocity of 7.5 ft/see, the occurrence of plastic yielding around the are~ of contact can be clearly seen for all distances of travel. This is shown in Figs. 4(a) and 4(b). The trace of the stress pulse, which is a smooth curve for purely elastic deformations, has a sharp angle near its peak followed by a curve with almost constant negative slope. We note that the distances of travel of the stress waves in Figs. 4(a) and 4(b) are about the same as in Fig. 5(a) and 5(b), respectively. In the latter figures, the stress pulses are generated by an impact velocity of 66.6 ft/sec and the effect that plastic yielding has on the stress pulses at this velocity is much more serious than in Figs. 4(a) and 4(b). The shapes of the stress pulses, shown in Figs. 3 to 5, are quite different from t h e experimental results obtained by Kolsky and Douch ( 11) in one-dimensional plastic wave propagation. The height of the stress plateaux detected there by
o
(a)
(b)
~o 4,o ~ s e c
I
Fro. 5. Radial surface waves observed at v = 66.6 f t / s e c , (a) r = 1 (4/32) in.; (b) r = 2 ( 1 5 / 3 2 ) in.
strain gages mounted on a long pressure bar is fixed by the dynamic value of the yield point of the metal. Since no plastic deformation is taking place at the gages, the appearance of stress plateaux would be expected from the simple theory of one-dimensional elastic wave propagation. In the half-space problem, however, the nature of the wave propagation is very different from the above. The amplitudes of stress waves propagating spherically in a half space decay rapidly with increasing distance of travel, and this rapid decay restricts the plastic yielding, when it occurs, to a small region around the point of impact. The surface strain is related to the applied surface stress in a complicated manner, and the shapes are very different from each other. In the next section the solution is obtained for an arbitrary point load applied at the surface of an elastic halfspace. Since the region of local plastic yielding is small, this solution might be expected to be a good approximation for stress waves at a large distance from the center of impact. The surface strain wave given in the next section as an integral over the second derivative of the forcing function ff'(t) in Eq. (54). The contribution of
%1.2ss,No.3,Marchi~6S
209
Y. M. Tsai the second term in (54) is small, and from our experience in earlier work, (1) and (a), it is less than about 8 per cent of the first term. Thus, the characteristic features of the first term might be expected to give a good approximation to the shape of the pulse. If in (54) we let t. = "rr/c=, which is the time for the arrival of the Rayleigh wave front at r, we see that the value of eosh 0 is close to unity for t sligh~Iy greater than t,. We m a y thus expect the shape of err to be similar t o f ' ( t ) for t close to t,. For t much larger than t~, however, the shape of er~ would be very different from f'(t). With the above considerations in mind, we see from Figs. 4 and 5 that the total applied forces have abrupt changes in their slopes when local plastic yielding first occurs at ~ time closely after t,. The characteristic features of stress pulses in Fig. 3 are somewhat different. At this velocity of impact, a small nucleus of plastic yielding would be expected to occur inside the material just below the center of impact. According to the results given in (12) the surface waves produced by an interlml disturbance are relatively more marked at distant points than close to the disturbance. This effect is shown by a comparison of Figs. 3(a) and 3(b). Response of an Elastic l l a l f Space to an Arbitrary Loading To make a rigorous analysis of the experimental results described above, we should, first of all, determine the moving elastic-plastic boundary. This would appear almost impossible because of the complex nature of the Hertzian stress distribution. As pointed out before, however, the region of local plastic deformation is extremely small eompared to the size of the specimen. In addition, most of stress wave observations are made at points distant from the center of impact. To make the problem mathematically tractable, therefore, we assume that the source of disturbances is localized at a point. The problem for an elastic halfspace subjected to an arbitrary vertical loading on the surface is first considered, and the similar problem for an arbitrary vertical loading applied at a point inside the materiM is also solved. In the first place, operational techniques are used to tackle the problem, and the inversion integrals are then evaluated by contour integrations. 1. Transform Solution. The equation of motion of an isotropie solid m a y be written (x + 2 ~ ) v v • n - ~V X v × u =
o(oh~/Ot ~)
(1)
where u is the displacemeut vector, o is the density of the medium and X and are the Lama constants. For aal axial symmet~\y problem, the eylindricM polar coordinates (r, e, z) are the most convenient to use, with the z-coordinate as the taxis of symmetry. Let u be equal to (u,, u~, uo). Since the problem considered is an axially symmetrical one, u~ mad all its derivatives with respect to ~, must vaIfish everywhere. Under these conditions, the equation of motion (1) can be written [e,g. see Kolsky (10)
r-0r\
210
~/+0z
7
= p 0t~
Journalof TheFranklinInstitute
Stress Waves Produced by Impact
and r-~
(r~)
4- 7z~ = p ot~
(3)
where the dilatation A is (4)
A = r~l(O/Or) ( r u , ) 4- (Ou~/az)
and the rotation it has only one non-vanishing cireumferentiM component ~ = (ouJoz)
-
(5)
(on,lot).
Now let us consider a senti-infinite elastic medium. Assume that at z = 0 an internal verticM load f ( t ) is suddenly applied and this is uniformly distributed over a disc area of radius a. The body is free from traction on its surface z = - H . Let ~u~ be the radiM horizontal displacement in the medimn between z = 0 and z = - H and ~u,. for z > 0. Similar notation is applied to other quantities. The boundary conditions are as follows: on z = - H the surface is free from traction, i.e. 2ao~ = ~a~,. = 2a~ = 0; whereas on z = 0 we have 2u,. = ~u,, 2uz = luo, 2o-o,. - lO-zrl 20"~0 ~ lO-z~ ~ 0 and 20"zz = lza, 2~r~ -- lo-o~ = f ( t )
for for
r > a r < a
(6)
The formM solution of the problem is quite straight forward. Laplace transforms are first operated over the variable t as in (7), and Hankel transforms are then applied over r as in (1,~6, 5). Now we define
f*(p)
= f~f(t)e Jo
(7)
-~t dt
and u(~)"
= rjo~u(r)rJ,~(sr)
dr.
(8)
For conve~fience, hewever, the asterisk * for Laplace transfl)rm is omitted in this section, but it will be used in subseque~lt sections where the Laplace inverse integral is evainated. If we now operate Laplace transforms followed by Hankel transfol~ms over Eqs. (2) and (3), we have, respectively (d2P~°/dz 2) -- (s 2 - - k l ~ ) ~ ° - 0
(9)
and (d2~tl/dz 2) -- (s ~" 4- k~2)~ 1 - 0
(10)
2t~) = p~/cl ~
(11)
where Ici~
vow. 2as, xo. ~, ~ h
~96s
-
pp2/ (X
4-
21 1
Y. M . Tsai
and (12)
/elz = pp~/# = p2/c2~.
We take the solutions of Eqs. (9) and (10) as 2~o = Ce-~° + De+~,
(13)
2~l = E e - ~ -~ Fe + ~
(14)
and but ~o = A ~ - *
(15)
1~1 = Be - ~
(16)
which remain finite for largo values of z. F r o m Eqs. (13) to (16), we introduce a = (s ~ + k12) ~
(17)
and m (S2 ~- k22) 112,
(18)
Applying successively similar transforms to Eq. (6), we have ~#~o _ i ~ o o
(19)
= (a/s)gl(sa)f(p).
The transformed stresses and displacements ure found to be (pp2/~2) e~o = k4(d~O/dz~) _ ]c~(k~ _ 2)s27~ _ 2s(dO~/dz) (pp~/~-) a~) = (d251/dz ~) + s2~ ~ - 2k~s(d~°/dz) (pp~/~)~ o = ]c2(d~O/dz) _ s~l
(20) (21) (22)
and (23)
(pp2/g)a~l = ( d ~ V d z ) - k2s ?'°
where k = k # k l = E2(1 -
,)/(1
- 2r)] ~.
If we now use Eqs. (13) to (16) and (19) to (23) to satisfy the boundary conditions, we have a system of six linear equations, the soiutions of which give A = C-D,
B =E+F,
D = (~/k2s)F,
(24)
F = pp2aJ~(sa)f(p)/2,u2~k~
ce "H = - - [ F / k ~ G (s)~{[-(2s ~ + k~) ~ + 4 a ~ s ~ ( ~ / s ) e H, _ 4S[~(2S ~ -{- k~2)e- ~ }
(25) aad E e ~H = ~ F / G ( s ) ]{ (2s ~ ~- k~)4o~fle -H" -
['(2s ~ 47 k~) ~ ~- 4s~al~Je -H~}
(26)
where G ( s ) = (2s ~ q-- k~'~)~ - 4s~afl.
(27)
Thus the problem is, in principle, solved, but in order to interpret the experi-
212
Journal of The FranklinInstitute
Stress Waves Produced by Impact mental results a more detailed and indicative result of the behavior on the surface is required. Now, if we insert the above constants into Eqs. (13) and (14) and then substitute the results into Eqs. (22) and (23), we obtain transformed displacement components. If we let z = - - H and perform the inverse Hankel transforms, we have the sin:face displacements
u,(r, p) = af(p) f ~ ~ Ju (~ts)
{(2s ~ "+- ic~)s_, ~ _ 2~f~e-"%/a(sr) as
(28)
u~(r, p) = af(p) f ~ A ( s a ~ .Io G(s)
l(2s ~ I - k~)s -"~ - 2s2e-~'}otJo(sr) ds.
(29)
and
The inverse Laplace transforms of Eqs. (28) and (29) will give meaningful physical quantities, but the inverse integrals would appear intractable unless J~(sa) is replaced by s/2~-a, giving the solution for an internal vertical point load. By letting " H " equal zero in Eqs. (28) and (29), the transform solutions for a sL~rface load can be obtained--integrauds of which i n v o N e singularities in the contour of integration. These singularities m a k e it difficult to reduce to the case of a surface load from the physical solutions for an internal load. Therefore, the inverse integrals for two different locations of ]oading will be evaluated separately, but the main ideas of integrations for the two cases are essentially identical. 2. S u r f a c e Loading. I n Eqs. (28) and (29) if we let s = k:~ and reduce the solutions to the case of a point load as mentioned before, we have
f u~*(r, p) = l~2f* (p) I ~ ~M(~)J~(k~r) d~ ~o
(30)
and
u~*(r, p) = ]co i*(,n~ r]~ ~N (~ )Jo(k~r) d~ ~rlz a o
(31)
where M(~') = {(2.~~ -F 1) - 2a'B'}/g(~)
N(~) = a'/g(~) g(~') = (2{-~ -? 1) 2 - 4~'~d{~' ~' = ( ~ + 1/k~), ~
(32) (33) (34) (35)
and ~' = (~': + 1) u2
(36)
u,*(r, p) = (1/2~r/~) (O/Or)]i*(r, p)
(37)
Ii*(r, p) = f*(p) f ~ fM(f)Jo(k2fr) df. ao
(38)
Eq. (30) can be written
where
VoL 285, No. 3, March ~968
213
Y. M. Tsai If we use the identity for the zeroth order Bessel Function of the first kind, i.e.,
Jo(x) = --
[exp (ixcoshO) -- exp (-ixeoshO)]dO
(x > 0),
(39)
we can write
Ii*(r, p) = -- ~Trf*(p)[lll - 1~2]
(40)
where
In = ~ ~1ll(~) d~ f ~ exp (ik2~r cosh O) dO
(41)
and I1~ = f 0 ~ M ( ~ ) d ~ f o ~ e x p
(--ik2~rcoshO)dO.
(42)
The next step toward integration is to deform the path of integration over ~" in Eqs. (41) and (42) so that both I n and I12 can be recognized as Laplace Transforms. To do this, we consider ~" = ~ + i~ and cut the complex plane, as shown in Fig. 6 in which ~ = :i=iv are roots of g(~'). Under these condition,s, the integrands of both I n and I12 are anMytic on the right half of the complex plane. If we choose the contours of integration, as shown in Fig. 6, and use the Cauehy integral theorem, we have f
Iu = -- !
~M(z~)K*(k2nr) d~ + ~ri Res. ( + V )
(43)
and =
_
r - ~ yM(+in)K*(-k2nr) dy
~iRes. (-7)
ao
(44)
where
K*(k2vr) = rioexp ( -k2~r cosh 0) dO.
(45)
I t m a y be seen that the Cauehy principal values of I11 and 112 are identical for ]~1 < 1/k and ]~] > 1. Thus In-112
=0
for
!~1 < 1/k,
1~] > 1.
(46)
The values of Ill a n d / 1 : are different for 1/]~ < ] ~ ] < 1 and at poles n = :i:% I f we calculate these values and substitute together with Eq. (46) into Eq. (40), we have 4
r 1
Ii*(p, r) = - ~r-if(P) J~]/kQ(n)K*(p,r/c2) d~l ~- 2P(~)f*(p)K*(pTr/c,)
214
(47)
Journal of The Franklin Institute
Stress Waves Produced by Impact
'
t/Contour of integration for
Iu
{:t" B t
I/k ~1
/ FI(~. 6. Contour for integration.
Branch line; _11/k
-~',-B~~ R R~OO
~'Contour of integration for It2
where Q(~) -
~(n ~ - 1/k2)m(1 - ~ 2 ) m ( 1 - 2 ~ ~)
q(n)(q(g))
(48)
q(~) = (1 - 2n~) ~ + i4n2(n 2 - 1/k2)~/2(1 - ~2),/~
(49)
~{(1 --2V) q - 2 ( ~ 2 -- 1/k2)1/2(~ '~ -- 1.) ~/2 P(v) g'(v)
(50)
and g'(,)
= og(i.)/o.l
....
(51)
The second t e r m in Eq. (47) results from poles at ~ = ± % I n the course of calculation of the above results, we use appropriate vahms of a, as shown in Fig. 6, and we also use the property that g ' ( - ~ / ) = - - g ' ( ~ ) . I n Eq. (47), if we consider K* (p'yr/c~) as a function of Laplace Transforms and use the convolution theorem, we have
~-l[f*(P)K*(p~r/c2) ] = foo~a-',°~/,,) fit -- (vr/c2) cosh O] dO. (52)
Using Eqs. (52), (47) and (40), the inverse transform of Eq. (37) gives the radial displacement
ur(r, t) -- P(7) ~'~ 0Or L ~°~h-l°*21"r) fit -- (~r/c2) cosh O] dO ~r~--~o f~ t~ Q(,) dn L °°~h-l(*°~/'~) frt - (nr/c2) coshO]dO.
Vol. 285, N'o. 3, March 1968
(53)
215
Y. M. Tsai
T h e first t e r m in Eq. (53) is a group of waves which travels at the Rayleigh wave velocity, while the second is another group of waves propagating at velocities between t h a t of shear and dilatation waves. F r o m the condition f ( t ) = 0 for t < 0, it m a y be seen from Eq. (53) t h a t at a certain distance r for t < r/c~, u, = 0. For r/c] < t < r/c2, u, has only the contribution of the second t e r m of Eq. (53) in which the upper limit of integration " 1 " m u s t be replaced by tc2/r. T h e first t e r m becomes effective only when t > r~/c2 which corresponds to the time of the arrival of R~yleigh waves. I f we let f ( t ) = Re ~t and also let t --+ ~ , Eq. (53) gives the steady solution for a periodic surface point load. Using a different method of integration in (1), the radial surface displacement is obtained for a periodic force which is uniformly distributed over a circular area. I f we reduce the solution for ~ uniform periodic force over ~ finite are~ (1) to that for a point load Re ~t, the result is identical w i t h the corresponding steady solution deduced from Eq. (53). I f we let f(t) be a unit step function, Eq. (53) gives the same result as t h a t in (7). We now m a k e a comparison with some other propagation problems. For the one-dimensionM wave propagation along a bar, the solution (10) is
u ( x , t) = E fo [t-(~/°)] f(T) dr.
F o r two-dimensional wave propagation, the response of the elastic half space to a surface line source gives the horizoutM displacement, (13) ]
t" t
u = ~ J,Ilof ( t
-- T) Er2 -
(r2/c 2) 3 m dr.
I n the case of an infinite body subjected to a point source, the radial displacem e n t is, u = f ( t -- r/c)/4~rr (10, 13). I n the above three examples we can see t h a t the displacements are written either in terms of f ( t ) or as an integrM over f ( t ) . The response of an elastic h~lf space to a surface point load is, however, different from the above three cases. Equation (53) shows t h a t the displacement
f[t~ )
--
o~ • ~
~
f'(t)/*r i
fit) = Sin=~ - ?
--- flt)w~th ~.iformly
/~///
decreasing slope
0.4
¢(t )hr
o.~i04
"
- --- consto~ slope
o.z 0.4 0.60.B
I.
1.2 L¢
1.6 I.B
.0
O2 oo
'0,0 0,2 0.4 0,6 o,B
,
t ~t
LO 1.2 d.4 1.6 12 2,0
(a)
-o.
-O.S~-
(b}
Fie. 7. Typical shape of the applied force. (a)f(t); (b) its derivativeif(t).
216
J'ournalof The Fr,lnklinInstitute
Stress Waves Produced by Impact is an integral over the derivative of f(t) mu]tlplied b y a fimetion resulting from surface dispersion. I n other words, the displacements, especially the strain components, are sensitive to the gradient of the excitation function f(t). I t Msshown in (7) and in Eq. (53) t h a t a finite j u m p o f f ( t ) will excite displacement waves, and from these waves infinite local strain w~ves are ganerated. To avoid encountering an infinite strain, we m u s t assume that beth f(t) and if(t) are continuous with their values equM to zero at t = 0. Since the radial strain is e~r = Our~Or, from Eq. (53) we can see that a finite value of i f ( 0 ) excites physically unrealistic strain waves which hlcrease with time and eventually become infinite. Under the above conditions, we have from Eq. (53)
~,, (r, t)
~~'~c~ 1
Jo
+ _7...2 f Q ('q) d, ~r ,uco. J1/~
f " [ t -- ('yr/c2) cosh 07 cosh 2 0 dO foeo~h-1(te~/vr)
f"Et - ('qr/c2) cosh 0~, ~ cosh ~0 dO.
(54)
Thus, the radiM strMn wave is t h e integral of the second derivative of the excitation function F ( t ) , and consequently, it is quite sensitive to the "smoothness" off(t) as shown schematically in Fig. 7. This is consistent with the experimental results described above, where surface waves produced by the impact of steel balls on steel block are detected b y semi-conductor strain gages. When the velocity of impact is sufficiently small, the total force applied to the surface of metal smoothly followed the Hertz solution and hence purely elastic waves are observed which do not show any drastic fluctuations. However, when the velocity of impact is sufficiently large, local plastic deformation occurs ~romld the point of impact and the total applied force would presumably deviate from the smooth elastic solution. The effect of yielding therefore distorts the shape of the applied force and hence the obsel~ed strain waves which depend upon the second derivative of the applied forcing function as shown in (54) are drastically disturbed. We now examine the asymptotic behavior of Eqs. (53) and (54) for large values of r. This can be detelTMned if we know the behavior of the following integral
t" eosh-l(t/t,) Io = Jo f ( t -- t~ cosh O) cosh 20 dO
(55)
where t~ = ~,r/c~ and t > t,. W e first examine the magnitude of the iaterval of integration. Let the upper limit of integration be Oo = cosh -~ (tit,,) this can be written cosh 00 = 1 -k (r/t,)
(56)
where • = t -- t,. When r tends to infinity i.e., t, --~ ~ , the second t e r m in Eq. (56) becomes vanishingly small as does 00. W e can, therefore, introduce in the
Vol.285,No. 3, March1968
217
Y. M. Tsai integrand eosh 0 = 1 + (r'/t~)
(57)
a~ld drop higher order terms in the series expansion of eosh 0. If we insert the above substitutions into Eq. (55), we immediately have
° -
(ha)
(,~')'~"
\2~r)
If we apply Eq. (58) to Eqs. (53) and (54), we can readily see that they both are iuversely proportional to (r) m as r tends to infinity. We now come to the integration of the vertical displacement Eq. (31), and the procedures for iutegration are similar to that for Eq. (38). We let
I2*(r, p) = f*(p) fo ~ ~N(~)Jo(k2~r) d~
(59)
and choose the same contour as Fig. 6 for integration. The contributions of the poles at =t=i~"cancel out and the values of Is* also vanishes fOl I ~ I < l / k . The value of/2* for I ~ ] > 1/k can be readily calculated. Since k2 = p/c~, the inverse of integral of Eq. (31) can be obtained by differentiating the inverse of Eq. (59) with respect to time. In view of Eqs. (39), (59) and (52), the Cauchy principal value of the vertical displacement is
u,(r,t) =0
for
t
1 -0 ft°~/~ vl(~) d~ fo¢°~1~'(t~2/'') f['t -- (nr/c2) cosh 0] dO ~'~gc: Ot J11k
for ~r2.c~ at J ,/,~
(rA~) < t < (r/c,~)
[-vl ( ~ ) + v~ ('q ) J dn
f ~osh-1(tc~/~r) X [ f i t - (vr/c~) cosh0~d0 ~o
for
t > r/c~
(60)
where vl(~) -
~(v~
--
1/k'2),/~(1
q0?) (q0?))
--
2~)
~
(61)
and v2(,) =
1 / [ d ) ( ~ -- 1) 1/2 q(~) (q(~))
4.~8(~7 ~ - -
(62)
If we carry out the differentiation, we can show that the vertical displacement Mso behaves as the integral of if(t). If we let f(t) be the Heaviside function, Eq. (60) immediate]y reduces to the same results as obtained in (7).
218
Journalof TheFranklinInstitute
Stress Waves Produced by Impact 3. I n t e r n a l Loading. If we let s = ~'k2 and replace J~(sa) by s/27ra in Eqs. (28) aald (29), we have the displacements for an internal point load u,.*(r, p)
if(P)2~r~~r0 fo~ ~ELI(~.) exp (--Hlc~') -- L2(£) exp ( - I I k 2 a ' ) j X Jo(k2~r) d~
(63)
and
f*(p)k2 [ ~ u,*(r, p) = ~ Jo rELy(r) exp ( - H k 2 d ) -- L,(~) exp (--Hlc~') 3 Jo(k2~r) d~
(64)
where LI(~) = (2£e + 1)/e(~) L2(~) = 2d13'/g(~) L.~(£) = a'(2f 2 + 1)/g(r)
(65) (66) (67)
L~(~) = 2 ~ d / g ( ~ ) .
(6S)
and The inversions of Eqs. (63) and (64) are achieved if we can carry out the inverse integral of the following typical expression in which Jo(ls~D') is written as an integral similar to Eq. (39) :
Ia* (r, p) = - (i/~r)ff (p) EI~* (r, p) - I~2"(r, p) l
(69)
where
I0~*(r, p) = f ~ d0 f ~ ~L(~) exp ~-HZ~(;~ + ~)~'~ + ik~r eosh 0~ d~
(70)
and
z~* (r, p) = f0~ d0 fo° ~L (~) exp [-Hk~(~ + ~ ' " -- il~r eosh 0] ~
(7~)
L(~') is an even function and e is equal to either "1" or 1/k. We can easily see that I,~* is the complex conjugate function of Ia*. In view of Eq. (69), therefore, we are interested in only the imaginary part of Eq. (70). For integration, we consider ~" = $ A- in and deform the contour of integration of Eq. (70) fi'om the positive real axis to a path such that along it
g,,EIt (~~ + e~) l~ - i~r cosh OJ = 0.
(72)
For Eqs. (70) and (72), we again cut the ~-plane, as shown in Fig. 6. Equation (72) gives ~ = 0 and = ('Oo~ + p~ cosh ~ 0~) al~ (73) where p = r/tt, ~ = ~p~ eosh ~ 0/(1 + p~ eosh ~ 0). (74) The purpose of this deformation is to make (70) recognizable as a Laplace
voL2~5.No.3. M~rch19~S
219
Y. M. Tsai Transform. Hence, for ,1 < T0 we choose } = 0, but from then on we take the p a t h (73), along which = [ ( ~ _ no~)l/2/p cosh 0] + iT
(75)
and the exponent in (70) is --p7 where
r = (H/c) ['p cosh 0 + (p cosh 0)-11~.
(76)
This deformatinn of contour is justified, because the integral (70) vanishes at infinity along the arc connecting the real axis and the path (73). Along the line = 0, the exponent in (70) is - - p r ' where r r = (l/c) [-H(e2 -- ~2)1/2 + ~r cosh 01.
(77)
Therefore, the integral (70) can now be written
181*(r,p)=
f0® d0 {fore ~'L(~ ') exp ( - p r ' ) ( O ~ ' / O r ' )
where r0' and ro are, respectively, the values of r ~ and ~ at ~ = 70. ~' is the quailtity ~"writtan in telTnS of r ' from (77), and ~ in the second term of (78) is from Eqs. (75) and (76). Taking the inverse transform of (78), we have
I3~(r, z) = fo ~ d~I~tL(~')(O~'/Or)H(ro ' - .r) + ~L(~)(O~/Or)H(r
-- to) }.
(79)
If we recall that I82" is the complex conjugate of In1*, we have from Eqs. (69) and (79), P t
Is(r, t) = (2/~r) Jo f ( t -- r)¢,,[Is~(r, r) ] dr.
(80)
Applying (69) and its inversion Eq. (80) to Eqs. (63) and (64), we have immediately the solutions for u,(r, t) and u,(r, t). The expressions for u~ and u, are, respectively, similar to (53) and (60). If we let f(t) be a Heaviside fullctinn, we have reduced the solution to the same displacement fields as obtained in (8).
Discussion
and Conclusions
The surface displacements of an elastic half-space have been written as integrals (53) and (60) over the first derivative of an arbitrary vertieM loading f(t) which is applied at a point on the surface of material. At large distances r away from the point of impact, it was shown in (58) that the amplitude of the radial surface strain is inversely proportional to r 1/2. However, the radial strain waves excited by the finite jumps of f(0) and f ' ( 0 ) , which involve infinite
220
Journal of The F r a n k l i n I n s t i t u t e
Stress Waves Produced by Impact
strains, can be shown from Eq. (53) to decay as r -2 and r -l, respectively, as r --+ ~o. Thus, the stress waves observed at large distances of travel would be the disturbances produced, not b y the j u m p f(0) or f'(O), b u t b y the continuous vaa'iation o f / ( t ) . The integral representations for surface displacements generated b y internM vertical loading h a v e also been obtained, and the nature of these integrals are similar to those for surface loading. The effect t h a t local p]astie yielding has on the stress pulses has been observed by v a r y i n g the velocity of impact and the distance of travel. A t impact velocities slightly higher t h a n t h a t required first to initiate local plastic yielding, this effect was almost imperceptible close to the area of impact, b u t became noticeable at distant points. This was discussed and shown to be consistent with the result in (12). A t higher velocities of impact, the experimental ~esults clearly show the presence of plastic yielding, b u t instead of producing a stress plateau as it does for plane waves, it appears as a sudden change of slope of the cylindrical stress pulses propagating out from the area of impact. On the basis of the theory developed here and the simplified assumptions made, the shape of the initial part of the strain pulses is similar to t h a t of the first derivative of the applied forcing function. Thus, the slope of the applied forcing function varies rapidly when locnl plastic yielding has occun'ed. The effect of plastic yielding on the stress pulses increases with increasing velocity of impact. The time of contact was measured for a l~rge range of impact velocity. During plastic deformation the amount of contact time in excess of that calculated from ttertz's elastic impact theory increases with increasing velocity of impact. F o r the range of velocities covered, the value (1 -- ~2)/E has increased b y about 10 per cent. Acknowledgment
The author wishes to thank Professor H. Koisky for many useful discussions in the course of the investigation. Thanks are due also to Messrs. L. Daubney and R. Stanton for help with the experimental work. The present research was sponsored partly by the U. S. Army Research Office (Durham) under contcact DA-ARO(D)-358, partly by the National Science Foundation under grant GP-2010 and partly by the Advanced Research Projects Agency under contract SD-86, all at Brown University. References
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
Y. M. Tsai and H. Kolsky, J. Mech. Phys. Solids, Vol. 15, pp. 263-278, 1967. R. M. Davies, Prec. Roy. Soc. A, Vol. 197, p. 416, 1949. J. M. Lifshitz and N. Kolsky, J. Mech. Phys. Solids, Vol. 12, p. 35, 1964. H. Lamb, Phil. Trans. A, VoI. 203, p. 1, 1904. G. F. Miller and H. Pursey, Prec. Roy. Soc. A, Vol. 223, p. 521, 1954. Y. M. Tsai, to appear, J. Mech. Phys. Solids, 1968. C. L. Pekeris, Prec. Nat. Aca. Sci., Vol. 41, p. 469, 1955. C. L. Pekeris, Prec. Nat. Aca. Sci., Vol. 41, p. 629, 1955. A. E. H. Love, "A Treatise on the Mathematical Theory of Elasticity," p. 195, Camridge Univ. Press, 1927. H. Kolsky, "Stress Waves in Solids," Oxford, Clarendon Press, 1953. H. Kolsky and L. S. Douch, J. Mech. Phys. Solids, Vol. 10, p. 195, 1962. C. L. Pekeris and H. Lifson, J. Accoust. Soc. Am., Vol. 29, p. 1233, 1957. A. T. de Hoop, Appl. Sd. Res. B, Vol. 8, p. 349, 1960.
Vol.285,No. 3, March1958
22 1