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Failure modes and criteria of plastic structures under intense dynamic

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METALS AND MATERIALS, Vol. 4, No. 3 (1998), pp. 219-226

Failure Modes and Criteria of Plastic Structures under Intense

Dynamic Loading: A Review

T. X. Yu and F. L. Chen

Dept. of Mech. Eng., Hong Kong Univ. of Sci. and Tech.

Clear Water Bay, Kowloon, Hong Kong

This paper reviews the failure modes and criteria of plastic structural members (beams, rings, plates and
thin shells, etc.) subjected to intense dynamic loads, such as impulsive velocities or impact. Besides the
three basic failure modes identified by Menkes and Opat [7] for impulsively loaded beams, namely, large
inelastic deformation (Mode I), tensile tearing (Mode II) and transverse shear failure at the supports (Mode
III), more complicated failure behaviours observed in the cases of complex structural members and/or dy-
namic loading conditions are summarized and discussed. All the failure criteria used in the relevant theoret-
ical analyses including elementary failure criterion, energy density criterion and quasi-static equivalent en-
ergy criterion, etc, are commented. Emphasis is particularly put on structural members with macro im-
perfections (for example, containing cracks or notches), where failures which usually originate from the im-
perfections are fracture-dominant so that local fracture criteria should be incorporated into the global struc-
tural failure description.
Key words : intense dynamic loading, plastic structure, failure mode, failure criterion, imperfection

I. I N T R O D U C T I O N arch on structural impact has been gradually switched

from dynamic deformation to dynamic failure [3].
The field of Impact Dynamics covers an extremely Generally speaking, structural failure can be un-
wide range of spatial, temporal and thermal scales, and derstood as the dramatic reduction of its load-car-
also a wide range of materials. It also covers a large var- rying capacity or the loss of its originally specified
iety of loading situations such. as hypervelocity impact, functions due to severe deformation and/or fracture.
blast loading, jet impact, projectile penetration, drop-ob- Typical examples of structural dynamic failure include
ject loading, structural crashing, etc. A non-dimensional catastrophic collapse of buildings, bridges and other
parameter, called Johnson's damage number [1] civil engineering structures, damage suffered by air-
craft, ships and land-based vehicles due to collision,
d n - pV2 (1) and accidents to industrial plants.
crd In order to improve structural crashworthiness and saf-
is commonly used to classify the severity of projectile ety of structures, or to assess the consequence of hazard,
impact, where p is the target density, Vo the impact velo- significant research efforts have been devoted over the
city and ~d the dynamic yield stress of the target ma- years to studying the plastic deformation and failure of
terial. Recently, Corbett, Reid and Johnson [2] published structures under impact or intense pulse loading [4-6].
a comprehensive survey on impact loading of plates and One- and two- dimensional structural members, such as
shells by free-flying projectiles, which concentrates pri- beams, rings, plates and thin shells were examined
marily on the impact of finite thickness metallic target theoretically or experimentally. This paper is aimed to re-
from the viewpoint of investigating the local penetration view the dynamic failure modes of these structural
and perforation processes involved. The present review, members observed in experiments and the failure criteria
however, mainly concerns with somewhat lower velocity used in the relevant theoretical studies. Particular at-
cases, in which both local and global effects need to be tention will be paid to the dynamic behaviours of struc-
taken into account, but the latter predominates. tures with macro imperfections, such as cracks or
In the last two decades the focal point of the rese- notches.
220 T. X. Yu and F. L. Chen

2. D Y N A M I C FAILURE MODES O F STRUC-

TURAL MEMBERS

Different failure modes may develop in a structure

under various dynamic loading conditions. Menkes and
Opat [7] conducted an experimental investigation into
the dynamic plastic response and failure of fully clamp-
ed metal beams subjected to uniformly distributed initial Fig. 2. A clamped beam struck transversely by a projectile.
velocity over the entire span, as shown in Fig. l(a). The
beams were observed to respond in a ductile manner and another critical velocity was reached that was associated
acquire permanently deformed profiles when subjected with shear failure at the supports, see Fig. l(d). Based
to a velocity less than a certain value, see Fig. l(b). on these experiments, Menkes and Opat identified three
However, when the impulsive velocity exceeded this crit- basic failure modes for impulsively-loaded fully clamped
ical value, the beams failed due to tearing of the beam beams, i.e. Large inelastic deformation (Mode I), Tensile
material at the supports, see Fig. l(c). As the impulsive tearing (Mode II) and Transverse shear failure at the sup-
velocity was further increased, the plastic deformation of ports (Mode III). It should be noted that Mode II or
the beams became more localized near the supports until Mode III failure occurred essentially over two cross-sec-
tions of the beam rather than throughout the beam.
The failure of fully clamped beams with flat ends or
with enlarged ends when subjected to impact loads, as il-
lustrated in Fig. 2, was examined by Liu and Jones [8] in
1987. The beam specimens were made either from alu-
minum alloy, which is essentially strain-rate-insensitive,
or from mild steel, a highly strain-rate-sensitive material.
For the aluminum alloy beams, the Mode I response was
obtained for the lower range of impact velocity. The im-
pact velocity was then increased further until a beam was
cracked or broken at the supports or at the impact point.
It transpired that the fiat aluminum beams always suffer a
tensile-tearing (Mode II) under sufficiently large impact
loads except when struck very close to a support (L1/I_~<0.
067, approximately) which then produced a transverse
shear failure (Mode III). The cracking was initiated by
the maximum tensile strain at the supports on the upper
surface (impacted surface) of a beam, or underneath the
impact point on the lower surface. The location of a ten-
sile-tearing failure changed from the impact point to a
support with a reduction in the value of L1.
Large concentrated stresses developed at the interface
between the section of a beam and an enlarged end,
which could cause plastic flow and lead eventually to
the failure of a beam in this region. It was observed that
all of the aluminum beams with enlarged ends were
cracked or broken at a support due to tensile-tearing
(Mode II).
Fig. 1. (a) A clamped beam subjected to an uniformly It was also observed that for sufficiently large impact
distributed impulsive velocity Vo. (b) Failure Mode I, energies, the flat steel beams were broken or cracked at
large inelastic deformation. (c) Failure Mode II, tensile the impact point with a shear mode (Mode IV), possibly
tearing at supports. (d) Failure Mode III, transverse because the dynamic rupture strain er of steel is much
shear failure at supports. larger than that of the aluminum alloy used in the ex-
 Failure Modes and Criteria of Plastic Structures under Intense Dynamicw 221

periments. However, the angle of the broken sections

varied with the position of the impact point. This failure
behaviour is similar to the Mode III response. For steel
beams with enlarged ends, it turned out that they were
cracked or broken at a support due to tensile tearing
(Mode II) with small value of L1, whereas a shear failure
occurred at the impact point (Mode IV) when struck
near the midspan. The difference between the aluminum
and the steel beams indicates the influence of material
strain-rate-sensitivity on the structural failure per-
formance.
Further experimental evidence was reported by Yu
and Jones [9] for the deformation and failure charac-
teristics of clamped beams under local impact loads. The
strain rate-sensitive characteristics of plastic flow and Fig. 3. Velocity fields for an impulsively loaded fully
the uniaxial rupture strain were recorded up to 140s -1 for clamped beam (a) with no shear sliding at supports and
the aluminum alloy and mild steel materials used in the (b) with shear sliding at supports.
beams. The flow stress of mild steel increased with an in-
crease in strain rate while the true strain at fracture was In 1976, Jones [13] used Elementary Failure Criteria
about 0.83 and was almost strain rate-insensitive. In con- (critical tensile strain criterion and critical accumulative
trast, the flow stress of the aluminum alloy was strain-ra- shear sliding criterion) to estimate the threshold velo-
te-insensitive but the true uniaxial rupture strain in- cities corresponding to failure Mode II and Mode III for
creased with an increase in strain rate. Therefore, for the dynamically loaded beams tested by Menkes and
both materials the influence of strain rate was important Opat [7]. The velocity fields are assumed as shown in
in failure analyses. Fig. 3, and the elementary failure criteria can be ex-
Similarly, three basic failure modes were also ob- pressed as
served in impulsively loaded clamped circular [10] and
square plates [11]. Nevertheless, another failure mode, lo-
Emax -----E Mode II (2a)
calized necking, was identified for dynamically loaded A,~ax= H Mode III (2b)
circular plates [12]. In the case of square plates, the
Mode III failure is merely an approximation to shear where ~max is the maximum strain within the structure
failure due to the effect of the four angular points. It and ac the critical tensile strain of the material; 3~ax is
should be noted that for a plate impacted by a projectile the maximum shear sliding at cross-sections and H the
with a higher velocity, a more complex failure mode thickness of the beam. Jones obtained an approximate
with combination of global structural failure and local theoretical prediction of the threshold velocity
penetration/perforation occurs.

3. T H E O R E T I C A L STUDIES AND FAILURE (3a)

CRITERIA
for the onset of Mode II failure of the beams with rec-
Compared with the limited experimental investigations tangular cross-sections, and
summarized above, much more theoretical works have
been published concerning structural dynamic failure dur- 2x/~ A cry
ing the last decades. Several structural dynamic failure cri- Vc3- "~/ (3b)
3 p
teria have been proposed and used in the relevant theoret-
ical analyses. In fact, one- or two- dimensional structural for the Mode III failure, where 2L is the length of the
members such as beams, plates or thin shells usually fail beam and (~v the material yield stress. Thus for a given
(Mode II or Mode III) over a cross-section, so the failure material the threshold velocity for Mode II failure of a
criterion should be established over a cross-section of beam does not depend on L and H but only on the ratio
these structural members. of 2L/H while for Mode III failure it is a constant, as ob-
222 T. X. Yu and F. L. Chen

served by Menkes and Opat [7]. However, although the observations [19,20] on the rupture cross-section, it was
theoretical predictions give surprisingly good agreement noted that in most cases the realistic failure mode is a
with the corresponding experimental values, these sim- combination of shear sliding and crack propagation, so
ple theoretical formulations are on a far less firm foun- that the elementary failure criterion for Mode III failure
dation; for example, the interaction between bending mo- should be modified as A~,• The factor k (0<k<l)
ment M, membrane force N and shear force Q was en- slightly depends on the material properties, geometric
tirely neglected. constraint of the structure as well as loading conditions.
In rigid-plastic analyses deformation is ideally lo- One may easily conjecture that the value of k for a
calized at plastic hinges only, so that information about tough material is larger than that for a brittle material
the strain distribution in a structure can not be obtained [21]. However, for a wide range of situations it can be
directly. To calculate the maximum strain within the approximately taken as a constant, say 0.3. With the vari-
structure involved in the failure criterion (2a), an ef- ation of shear force Q during the failure process taken
fective length of the plastic hinge, Le, has to be defined. into account, the analysis [18] results in a lower bound
At beginning of deformation, membrane force N=0 and of the threshold velocity for Mode III failure, which
Le may be taken as H, see Fig. 4(a). With the influence may be smaller than that for Mode II failure.
of membrane force considered, Nonaka [14] and thereaft- By employing an interaction yield function of M, N
er other authors estimated and Q
Le=(Z-5)H (4) [M* ]~/(1-Q*Z)+N*2+Q*2 = 1, (5)
See Fig. 4(b). In 1988, Liu and Jones [15,16] used the a- where M*=M/Mo, N*=N/N0, Q*=Q/Q0, Shen and Jones
bove elementary failure criterion to evaluate the failure [22,23] proposed an Energy Density Criterion which
conditions of Mode II and Mode III for a clamped beam states that rupture occurs in a rigid-plastic structure
struck transversely by a mass. Reasonable agreement when the plastic energy absorption per unit volume, 0,
was found between the theoretical predictions and most reaches a critical value
of the experimental results observed by themselves [8].
By means of the slip-line theory of plasticity, Yu and 0 = 0c, (6)
Hua [17] took account of the influence of shear force on where 0 contains the plastic work contributions related
the effective length of a plastic hinge and found it to all the stress components. For a rigid-plastic beam it
reduces as the shear force increases (Fig. 4(c)). can be further expressed as follows
Recently we [18] further investigated the Mode III
s <s /3 < tic. Mode 1 (7a)
failure of dynamically loaded beams. From experimental
s =.o_~, /3 </~ Mode II (7b)
s /3</~ Mode III (7a)
where f2 is the total amount of plastic energy dissipated
at a plastic hinge, ~ the plastic energy dissipated in the
same plastic hinge through shearing deformation and 13=
f~/f2; f2c=0cBHLe, with B being the width and H the
thickness of the beam. According to the experimental
measurement reported by Menkes and Opat [7], Shen
and Jones also suggested an approximate formula re-
garding the length of the plastic hinge and the dissipated
energy ratio
a + 1.2/3 = 1.3, (8)
where ~=Le/H. Since 0<_13~1, it follows that 0 . 1 < o r <
1.3. Obviously this formula is empirical. The energy den-
Fig. 4. Effective length of a plastic hinge: (a) bending sity criterion is universally applicable for all the three
only, (b) with the influence of axial force, and (c) with failure modes of beams [24] and circular plates [25].
the influence of shear force. However, there are some problems associated with this
 Failure Modes and Criteria of Plastic Structures under Intense Dynamicw 223

criterion, e.g. 1) the magnitude of energy density 0 dis- cussed and the critical loading conditions for these two
plays a mesh-dependence; 2) it is unclear whether the failure modes were obtained.
critical energy density 0c depends on the state of stress
and strain; 3) it is unclear whether a real structure fails if 4. IMPERFECT STRUCTURAL MEMBERS
0=0c is only satisfied at a single "point" or in a small re-
gion; 4) 1]c=0.45 was suggested by Shen and Jones [22], While most of the previous studies were focused
but a further justification seems needed. on "perfect" structures, real structures are usually
In the dynamic plastic shear failure analyses of a can- "imperfect" in their geometry and/or in containing
tilever or an infinitely large plate subjected to a pro- cracks, notches, holes, defects and voids. Engineering
jectile impact, Yu and Zhao [26-28] obtained an Energy practice indicates that these imperfections will trap a sig-
Criterion for the shear failure at the impact interface. nificant part of the work done by the dynamic loads and
The rotary inertia of both the projectile and the struc- thereby alter the dynamic failure features of structures.
tural element was found to be of significant influence on Since mid 1980's, relevant published research works
the shear failure conditions. have most concerned the theoretical or experimental in-
More recently, based on the principle of energy con- vestigations on dynamic loaded beams or rings with a
servation and the assumption of equivalent work done, crack or notch at various locations.
Wen and Yu employed a Quasi-static Equivalent Energy In 1983, by introducing a reduction factor "/<1 on the
Criterion to analyze impulsively loaded clamped beams fully plastic moment at the cracked section, Petroski [36]
[29] and circular plates [30]. In spite of the simplicity of explored the effect of the cracks on beam's plastic
this theoretical procedure, their predictions were in good response. Later, this simple technique was used to in-
agreement with the relevant experimental results in vestigate the dynamic response of cracked cantilevers.
terms of maximum permanent transverse displacements, He [37] first studied the effect of a crack at the root on
failure modes, and the critical input impulses for either the dynamic deformation of a cantilever subjected to an
tensile tearing (Mode II) or transverse shear failure initial velocity imparted on an attached mass at the tip.
(Mode III). Failure maps were given to facilitate prac- The transient phase characterized by travelling plastic
tical applications. hinges was completely neglected, and only the modal-
Other works include a rare elastic-plastic computer phase motion was considered. Thus the final rotation at
simulation carried out by Yu and Jones [31]. For a the root was easily found. Later a crack at an interior
clamped aluminum alloy beam struck transversely by a cross-section of a cantilever was considered [38]. The ro-
mass which produce large inelastic deformation, the evo- tation angle at the cracked section after impact was es-
lution of the equivalent plastic strain contours was ob- timated by employing a mechanism wherein stationary
tained, which is valuable to a better understanding of the plastic hinges were located at the root and the cracked
failure process. cross-section throughout the entire response. The sub-
Besides, Yu and Chen [32] theoretically examined the sequent stability of the crack was considered by cal-
large deflection dynamic response of thin rectangular culating the tearing modulus based on the J-integral as-
plates impulsively loaded by a uniform velocity. By em- sociated with the deflecting beam. Also, experiments
ploying a membrane factor method, both the interaction were conducted to demonstrate the structural response
between bending moment and membrane force and the and failure modes of cantilever beams with a crack sub-
travelling hinge phase were treated. This analysis pro- jected to impact loading [39]. The permanent damage
vides an estimate on the Mode I failure of rectangular suffered by a cracked beam was found to be sig-
plates. The plastic deformation of rectangular plates nificantly different in magnitude and character from that
struck transversely by a wedge was then examined by of a similarly loaded uncracked beam. The failure mode
Zhu [33] and Shen [34]. For the Mode II and Mode III depended strongly upon the size and location of the
failure of rectangular plates, no theoretical work has crack; and in most cases the initial crack propagated and
been published up to now. final rupture failure occurred at the pre-cracked section
Duffey [35] once studied the dynamic behaviours (Mode V), as illustrated in Fig. 5.
of cylindrical shells loaded by a rectangular or an ex- Woodward and Baxter [40] reported their experi-
ponential pressure pulse. Biaxial ductile strain lo- mental study on impact bending of unnotched and notch-
calization and failure (Mode II) and the transverse shear ed free-flee steel beams; they commented that the notch-
failure (Mode III) at supports (hard points) were dis- ed beam problem is an ideal case for application of the
224 T. X. Yu and F. L. Chen

Fig. 5. Permanent deformation of cracked cantilever beams

subjected to impact at the tip (refer to [39]).

rigid-plastic approach because a notch localizes de- Fig. 6. Failure modes of circular rings with notches at
formation, making it more like a hinge. varied locations, subjected to impact from the top.
Inspired by the success of employing a double-hinge
mechanism proposed by Hua, Yu and Reid [41] in stepp- ducted a series of impact tests on unnotched and notch-
ed cantilever problem, Yang and Yu [42] made a com- ed circular rings made of mild steel. Their experiments
plete analysis to an impact-loaded cantilever with an in- revealed that notches in some regions affected the dy-
itial crack at an arbitrary interior cross-section. They ob- namic behaviour of the ring seriously and even led to a
tained the partition of the dissipated energy and a cri- local breakage, whilst notches in other regions had negli-
terion for stability of the crack. A typical stress cor- gible influence on the global performance of the ring.
rosion circumferential crack in a tubular cantilever beam Therefore, the effect of the notches on the dynamic beha-
was analyzed as an example; and the J-integral and tear- viour of rings is position-sensitive. Three failure modes
ing modulus were calculated to predict the stability of were observed for impact-loaded notched circular rings;
the crack after impact. It was concluded that in this case namely, 1) the whole ring suffered a large inelastic de-
most of the input energy is dissipated by crack extension formation with negligible influence of the notches lo-
in the radial direction, leading to a leak-before-break pro- cated outside the sensitive region, see Figs. 6(a), (d) ; 2)
cess that is desirable from a safety point of view. This only a part of the ring suffered a large inelastic de-
analytical procedure has also been extended to the cases formation but partial tearing occurred at the notched sec-
of a clamped beam [43] and a simply-supported beam tion, Fig. 6(b) ; 3) the whole ring suffered a large ine-
[44], both containing initial cracks and subjected to dy- lastic deformation with complete rupture at the notched
namic loads. Experimental study for clamped beams section, Fig. 6(c). It appeared that the failure occurred at
with initial cracks was performed by the same group and the notched section are fracture-dominant [49]. A 2-D
reported in Ref. [45]. In the experiments, the projectiles numerical simulation of this problem has also been con-
were launched by an air gun, and the opening of pre- ducted [50]. The determination of the sensitive region(s)
made notches was measured to estimate the relevant en- could be of significance in practice.
ergy dissipation.
Recently, we [46] have examined the impact on a 5. C O N C L U D I N G REMARKS
clamped beam with cracks at supporting ends. By em-
ploying a three-hinge mechanism, a complete solution in- Different failure modes may develop in a structure
corporating the interaction of bending moment M and ax- under various dynamic loading conditions. Based on the
ial force N is obtained, and then the criterion for failure experiments of clamped metal beams subjected to un-
occurred at the pre-cracked cross-section is discussed. iformly distributed impulsive loading, Menkes and Opat
Zhao, Fang and Yu [47,48] in Peking University con- identified three basic failure modes, i.e. Large inelastic
 Failure Modes and Criteria of Plastic Structures under Intense Dynamicw 225

deformation (Mode I), Tensile tearing (Mode II) and 25, 549 (1995), in Chinese.
Transverse shear failure at the supports (Mode III). Be- 4. N. Jones and T. Wierzbicki (eds), Structural
sides these basic modes, more complex failure beha- Crashworthiness and Failure, Elsevier Applied Science,
viours were also observed for more complicated struc- London and New York (1993).
tural members and/or dynamic loading conditions. 5. W.J. Stronge and T.X. Yu, Dynamic Models for Struc-
tural Plasticity, Springer-Verlag (1993).
Several failure criteria have been proposed and used
6. T. Wierzbicki and N. Jones (eds), Structural Failure,
in the relevant theoretical analyses. They could be clas-
John Wiley & Sons (1989).
sified as (1) the elementary failure criterion (critical ten- 7. S.B. Menkes and H.J. Opat, Exp. Mech. 13, 480 (1973).
sile strain/critical accumulative shear sliding criterion), 8. J.H. Liu and N. Jones, Int. J. Impact Engng. 6, 303
(2) the energy density criterion and (3) the quasi-static e- (1987).
quivalent energy criterion, etc. Comments have been 9. J.L. Yu and N. Jones, Int. J. Solids Struct. 27, 1113
made on all these failure criteria. It should be pointed (1991).
out that there exists no universal criterion for broad class 10. R.G. Teeling-Smith and G.N. Nurick, Int. J. Impact
of structures and loading conditions. Engng. 11, 77 (1991).
While most of the previous studies were focused on 11. M.D. Olson, G.N. Nurick and J.R. Fagnan, Int. J. Im-
"perfect" structures, real structures are usually "im- pact Engng. 13, 279 (1993).
perfect" in their geometry or containing cracks, notches, 12. G.N. Nurick and R.G. Teeling-Smith, Proc. of the
Second Int. Conf. on Strut. Impact, p. 431, Portsmouth
holes, defects and voids. These imperfections sig-
(1992).
nificantly affect the dynamic failure features of struc-
13. N. Jones, Trans ASME J. Eng. Indust. 98, 131 (1976).
tures. For example, the strength failure modes are usu- 14. T. Nonaka, ASME J. AppL Mech. 34, 623 (1967).
ally yield-dominant for perfect structures, however, im- 15. J.H. Liu and N. Jones, Int. J. Solids Strut. 24, 251
pact tests on cracked/notched beams and rings have re- (1988).
vealed that the failure modes for the imperfect structures 16. J.H. Liu and N. Jones, in Inelastic Solids and Structures
usually become fracture-dominant. Moreover, the failure (eds., M. Kleiber and J.A. Konig), p. 361, Pineridge
features of the imperfect structures were found to de- Press, Swansea, U.K. (1990).
pend strongly upon the shape, size and location of the 17. T.X. Yu and Y.L. Hua, Proc. AEPA'96, p. 845 (1996).
imperfections. The "position-sensitivity" of the effect of 18. T.X. Yu and F.L. Chen, A further investigation into the
imperfections would be of significance in practice. The plastic failure of dynamically loaded beams, in pre-
presence of a crack or a notch in one-dimensional beams/ paration.
19. W.S. Jouri and N. Jones, Int. J. Mech. Sci. 30, 153
rings not only reduces the fully plastic bending moment
(1988).
at the damaged cross-section but also creates a 2-D ef-
20. Q. Zhou and T. Wierzbicki, Int. Y. Mech. Sci. 38, 303
fect so as to complicate the dynamic behaviour. For (1996).
structures with crack-typed imperfections, local fracture 21. Y.P. Zhao, J. Fang and T. X. Yu, Int. Y. Solids Struct.
failure criteria should be incorporated into the global 31, 1585 (1994).
structural failure description. New reliable criteria need 22. W.Q. Shen and N. Jones, Int. J. Impact Engng. 12, 101
to be investigated. (1992).
Finally, it must be noted that structural failure always 23. N. Jones and W. Q. Shen, in Structural Crashworthiness
originates from a material failure, so the dynamic failure and Failure (eds., N. Jones and T. Wierzbicki), p. 95, El-
analysis has to combine structural behaviour and ma- sevier Applied Science, London and New York (1993).
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need to be studied in meso- and micro- scales, e.g. using (1993).
damage mechanics. 25. W.Q. Shen and N. Jones, Int. J. Impact Engng. 13, 259
(1993).
26. T.X. Yu, Explosion and Shock Waves 13, 97 (1993), in
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