Journá/ ï[ Sound ánd Vibrátion (1984) 92(4), 517-528

EMISSION OF STRESS W Á VES DURING FRACTURE

Ñ. S. THEOCARIS AND Ç. G. GEORGIADIS

Depártment ÏÉ Theoreticál ánd Applied Mechánics, The Nátionál Technicál University ÏÉ Athens,5 Heroes ÏÉ Polytechnion Avenue, Athens 624, Greece

(Received 10 November 1982, ánd ßð revised form 14 ÁñðÉ 1983)

Emission of both longitudinal and surface (Rayleigh) waves duÞçg fracture of platesunder conditions of plane stress and plane strain were studied experimentally. Thenon-equilibrated tensile stress ßç the fractured section of the plate creates an elastic wave,which travels radially along the plate at the sound speed. Moreover, the high surfacedeformation around the crack tip, due to the high stress concentration there, propagatesas a surface wave following fracture of this zone, at the respective Rayleigh wave speedwith a circular wavefront. The influence of the thickness of the plate and the type offracture (brittle or ductile) was examined and interesting results were derived, by utilizinga high speed photography technique.

1. INTRODUCTION

Previous extensive research ïç fracture produced by stress waves has been expertlydescribed in the book by Kolsky [1]. Kolsky, and also his colleagues Rader, Tsai andPhillips [2-4] gave the first thorough treatment dealing with stress wave emission duringfracture. Another interesting study in this domain has be~n made by Ïß [5], whoinvestigated the formation of longitudinal tensile pulses ßç bars by observing the transientresponse of bonded strain gages.

The topic of pulse emission in plates during crack growth has received ïçÉÕ a fewinvestigations õñ to now. Guz" Zotov and Finkel' [6,7], using photoelasticity, a highspeed camera and piezoelectric transducers, have indicated a correlation between theparameters of the waves emitted duÞçg fracture and the amount of elastic energy storedat the respective instant ßç specimens which have undergone diflerent heat treatments.

More recent experimental work includes a study by Theocaris [8] of after-failurefracture phenomena which are produced by surface waves ßç a thin plate. Rossmanithand Fourney [9] have descÞbed a sketchy study of fracture waves carried out by exploitingthe possibility of gathering information about the crack speed history and the crack tipposition from the stress wave patterns. The latter method may be applied ïçÉÕ ßç opaquematerials, where the longitudinal fracture waves, which intermingle with Rayleigh wavesand therefore confuse the wave patterns, are excluded.

Éç all previous studies of elastic wave emission during fracture ïß plates çï influenceof the stress state (plane stress or plane strain) and the bÞttÉe or ductile character of thefracture has been considered. Finally, Dally [10], ßç a recent publication, has disputedthat stress wave emission during fracture can be observed ßç polymeric materials, althoughhe accepted the existence ïß fracture waves at high energy crack initiations or rapidarrests ïß cracks.

The present paper deals with a series ïß experimental studies of the characteristics ïßtwo types of generated waves duÞçg fracture: that is, longitudinal and Rayleigh waves.

5170022-460×/84/040517 + 12 $03.00/0 @ 1984 Academic Press Inc. (London) Limited

518 Ñ. s. THEOCARIS AND Ç. G. GEORGIADIS

The simultaneous appearance ïß several types ïß waves duÞçg the experiments necessi-tated systematization ïß the study, ßç respect to the thickness ïß the specimens and thedegree ïß brittleness or ductility ïß the materials which were used. Éç order to isolate asingle step ïß fracture propagation and to intensify the emission ïß fracture stress waves,we devised special forms ïß specimens with narrow ligaments between transversal collinearartificial cracks. These forms ïß specimens made possible the separation ïß the varioustypes ïß waves and therefore the separate observation ïß the phenomena studied ßç thecase ïß continuous crack propagation. lnteresting phenomena, which appeared duringthe propagation ïß stress waves ßç the experiments, were compared with theoreticalpredictions.

Since evaluation ïß acoustic wave emission at the beginning of fracture can be usedfor failure indication, it is evident that the results of the present tests may be related toflaw detection ßç ultrasonic reflection and seismic investigations.

2. MErnOD OF INVESllGAllON

During fracture of a plate different types of waves may be generated. These are thelongitudinál wáves, derived from reduction of the tensile force at the tip of the crack,Ráyleigh wáves due to the spreading of the surface dimples, which are caused by thehigh stress concentration at the tips of the cracks, and ftexurál wáves of the unbalancedmoment of stresses about the neutral axis of the fractured section of the plate. The latterwaves are dispersive by their nature and we are not concerned with them ßn this paper.

The generation ÏÉ secondary waves is also expected according to the theory. Reflectionof the primary longitudinal stress pulses at the boundaries of the plate causes secondaryshear or surface waves. At this point, it must be emphasized that there is a differencebetween the primáry Ráyleigh wáves, due to a point source, which are a consequence ofthe fracture mechanisms, as has been already pointed out, and the secondáry Ráyleighwáves, due to a line source, trailed from theprimary longitudinal waves as they interactwith the lateral faces of the plate.

As to the velocities of the primary fracture waves, it is well known [1] that the waveequation for 10ng waves ßn a thin plate is

pa2u/ar =[4ì(ë +ì)/(ë +2ì)] a2u/ax2 (1)

ßn terms of Lame's constants ë and ì and the density Ñ of the material of the plate.Hence the speed of propagation of the longitudinál wáves,cl' underplane stress conditionsis given by

Cl =[ 4ì(ë + ì)1 ñ(ë + ì)]É/2 = [ÅÉ ñ(1- éé2)]1/2, (2)

where Å and éé denote the modulus of elasticity and Poisson's ratio of the plate,respectively. The influence of the width of the sheet ïç the velocity of long waves hasbeen defined by Gazis and Mindlin [11].

Ïð the other hand, the surfáce wáves created by the lateral deformation of the platearound the tip ïß the propagating crack travel with a speed CR, which is expressed ßçterms of the shear-wave speed C2 (c2=[EI2(1+v)p]l/2) and Poisson's ratio éé of thematerial of the plate, and can be found from a diagram after ÊçïñïßÉ [12]. It is wellestablished that a considerable amount of energy is transmitted by these Rayleigh waves,which are not dissipative.

The detection of the above-mentioned waves has been effected ßç the present studyby using a suitable optical arrangement. When a light beam passes through a transparentspecimen the elastic waves cause a change ïß the optical pathlength and therefore angular

STRESS W Á VES ÉÍ FRAcruRE 519

deflections of light. These angular deflections are imprinted, ßç an optical image of thespecimen, asa variation of light intensity. The optical part of the investigation correspondsto the schlieren phenomenon [13].

Figure 1 shows a photograph of the wave pattern that was produced around a propagat-ing crack ßç a polymethylmethacrylate specimen of thickness d = 0.003 m. Én this photo-graph primary waves emitted at different steps of the fracture process, as well as secondarywaves produced by them, give a complicated pattern, as a consequence of their mutualinterference and the Doppler effect. One may observe intersections of wavefronts,non-constant intervals between fringes and strongly deformed wavefronts near the cracktip, according to the phenomenon of creation of caustics and pseudocaustics [14,15].Also, the crests of waves are situated at curves which resemble more ellipses than circles.

Figure 1. Wave pattern due to crack propagation ßç a ÑÌÌÁ plate ïÉ thickness d = 0.003 m.

Éç order to oíercome these difficulties ßç obseríing the waíe patterns aÑÑeaÞng aroundpropagating cracks ßç typical single-edge notched specimens, a special form of specimenwas deíised.

The types of specimens used ßç the experiments are illustrated ßç Figure 2. Each typeis a thin and long plate with only a narrow ligament, or ligaments, connecting the twoÑÞncßÑaÉ parts of the specimen, which are fixed ßç the grips of the test apparatus. Thelength of the single ligament of the first type (shown ßç Figure 2(a» is less than 0.001 m.With this type of specimen, fracture of the narrow connecting link by the applied forceÑ creates only a short duration pulse, which engenders a short waíe train, coníenientfor studying phenomena of step-propagation of a moíing crack, without any Dopplershift effects, waíe interference, or deformed waíefronts.

The fractured area of the ligament recoíers its unstrained form within a few micro-seconds, especially for brittle materials. Hence the obseríation of waíes emitted duringfracture takes place ßç an unstrained plate.

As a consequence of the independence of the waíe trains created by narrow fracturedligaments, the shapes of the waíefronts are now circular, eíen near the tips of the ligament.Moreoíer, with this type of specimen one obtains a more symmetrical distribution of thestresses around the fractured area than ßç the case of a crack propagating ßç a wide plate,where the component of the transíerse principal ux-stress along the crack axis is alwayshigher than the longitudinal uy-stress [16,17], which distorts t~e circular paUern of theemitted waíe.

YH;)d

YWWd

ÏÉ.1

61.1

.ò"éå.ï

9Æå.ï

19.Æ

6-ò.-ò

8Å8

Lv'll

688

ÉÅÅÉ

Ô6òô

Å6ÆÆ

(S/w) (EW:>/g)1;) d

(sjw)~;)

4-w Í 601÷)3'

1~!l",~~W (s/w)':1

If

:>!ùåõËá~õ!påoé Jo adA.L

VH::>d

VWWd

Å98

896

816

ïåïéÆÆ91

0081

ÏÆ.É

61'1

09å.ï

-òñå.ï

òÉ"Æ

ñ.å

(åù:)/~)d

(Z_w Í 601÷)g

IR!l;}~RW (sjw)Ç;)

(sjw)Æ;)

(sjù)É:;

é!

:>!~g~s~o!pBOI }Ï ;)dAl,

é1'U!Ñvï/ :>!wvuËÑ ñuû -j!IVJS iapun 'vfI:Jdpuv íÍÍÉÉ .loj spaads aaVM :>psv/a é1'U!Ñuodsa.l.lï?_ñ~.'?séuvésuï:> /v:>!uvf!:>aw

Ig,avj,

p~ç ç:)!çË\ 'u~w!:)~ds }Ï ~dh~ ~lq~~!ns ~ U~!S~P~l o~ Ëé~ss~:)~u p~~pnf S~M ~! s~s~:) ~s~q~ U!'snIU .sn~~l~dd~ ~s~~ ~ç~ }Ï Sd!l~ ~ç~ U! P~XY u~qM '~U!dl~M o~ ~Ë!~!SU~S ~l~M 'ù 100'0q~~u~I }Ï ~U~W~~!1 ~ P~!ll~:) ç;)!çË\ 'ù 100-0 = Ñ ss~u1{:)!q~ }Ï S~~~Id ~~~:)!I~P U!q~ ~éð

.~~~Id p~1{:)~.1:) ~ç~ U! U!~l~S ~U~Id .10 SS~l~S ~u~ld }Ï SUO!~!PUO;)~u!msu~ snq~ 'ù 010.0 pu~ w 100'0 U~~M~~q Ñ~!l~Ë S~Idw~s Ð~}O ss~u1{:)!q~ ~~ 's~m~:)~l}~I!~:)çp .10} (VHJd) V Iou~qds!H }Ï ~~~uïqé~:)ËI0d pu~ s~m~:)~l} ~I~~!lq 10} (VWWd)~~~éË.1:)~q~~wéËq~~wË(ïd }Ï S~~~Id ~U~.1~dSU~l~ ~l~M su~w!;)~ds ~ç~ S~S~~ ~ç~ ð~ uI

'[81 'é 1] d!~ ~U!ËÏW ~ç~ }Ï p~~ç~ S1{:)~D01:)!W pu~ SÑ!ÏË }Ï~:)u~:)s~I~o:) ~ç~ Ëq UO!~~~~dOld 1{;)~1:) O~ P~~~I~l ~u~wou~qd }Ï ËÑn~s ~ç~ lO} ln}~sn ~q u~:)ç:)!çË\ '1{:)~1:) ~ }Ï UO!SU~~X~ ~s!Md~~s ~ ~U!~~lnW!s 10} l~poù ~Iq~~!ns ~ s~q ~uo sn~

d+ d+(:»

0'+

WSOOOI W ~OO'O=P

alDId \1B:)d

.WOOI.O.

Éo~oo O~O.O

É. '\ .1. \ .1 É

wIOO.O é

--1\0- "

==d~

9é,éÏn

1015 IOI:J!!ljJ\1

juawD5!l

WIOOO=Póéïéd l1ííííd

w 0100 =Ñ~IDId l1ííííd

É UJ kOOO =Ñ

ajDId l1ííííd

d+ d+SIGVID~O3D ¼ .Ç áÍí SI~V;:)O3HL .S .d

é:é+

ÏÆò ~

.s~u;lwIJ8!1 ç~!M su;lW!:J;lds )0 !..l~;lwo;lD "Æ ;I.ln8!d

(q) (Ï)

STRESS WAVES ÉÍ FRACTURE 521

an extra bond of length 0.020 m, besides the initial ligament of length 0.001 m (seeFigure 2(b» ßç order to avoid warping. Finally, ßç order to simulate phenomena appearingat the last stage of fracture, close to the opposite longitudinal boundary of the plate,another type of specimen, which had two edge ligaments (see Figure 2(c», was used ßçsome tests.

Table 1 shows the static and dynamic elastic constants of the two materials which wereused ßç the experiments [19] and the corresponding wave speeds Ct, C2 and CR for thinplates made of these materials.

3. EXPERIMENTAL ARRANGEMENT

For the study of the fracture waves a Cranz-Schardin high speed camera was used,disposing 24 sparks with a maximum frequency of 106 frames per second. The light beamfrom each spark was first reflected by a spherical mirror of high reflectivity, having adiameter of 0.50 m and a focal distance of 7.00 m. The light beam, after passing throughthe specimen, was focussed ïç each respective lens of the camera. This optical set-up hasbeen described ßç detail ßç reference [20] and has been extensively used for the study ofcaustics.

ÁÉÉ quasi-static loadings were applied by a Schenck tester with a piston velocity of0.0002 mjs, which created a strain rate equal to Ý = 0.0008 S-l.

The synchronization of the fracture process with the high speed camera was achievedby means of a convenient silver contact circuit, which triggered the first spark of theseries with the initiation of the crack propagation.

4. RESULTS ÁÍÏ DISCUSSION

Ôï give a clear picture ïß the various types ïß stress waves, created during fracture,the following three cases were studied.

4.1. BRITI1..E MATERIALS. PLANE STRAIN CONDITIONS

For plane strain conditions ïß the fractured specimens the surface deformation at thecrack tip is always negligible, as compared with the thickness ïß the plate (d = 0.010 m).

Also, insignificant plastic zones are formed around the crack tips because ïß the modeïß brittle fracture ïß the plate. As a consequence ïß this situation çï Rayleigh waves wereobseríed ßç such plates, as is indicated ßç Figure 3. ÏçÉÕ longitudinal wave pulses appearwith circular wavefronts. The speeds ïß these waves are ßç good agreement with therespective theoretical íalue C} (c} = 1800 m/ s) for the material of the specimen and thecase ïß static loading shown ßç Table 1.

These dilatational waves, due to infinitesimal fractures ïß the short ligament, give theimpression ïß being concentric circularly crested wavefronts. Íï trailing dilatational pulsesalong the faces ïß the plate were formed ßç any ïß the similar experiments carried out,unlike the behaíiour shown ßç cases ïß impact compressional waves propagating ßç thickplates [21]. Áç explanation ïß this lack ïß trailing pulses may be based ïç the fact that,ßç these tests an almost normal incidence ïß the wavefront to the lateral faces of the platewas always achieíed. Therefore, an almost complete absence ïß the waveguide effect waseffected [22].

The traíeling cylindricallongitudinal waves initially meeting at the faces ïß the artificialcrack ïç both sides of the fractured ligament trail a shear wave and a secondary refiectedlongitudinal waíe, depending ïç the angle ïß incidence. Thus, three separate pulses traíelnllt tn the ~irie~ nf the Cl1t. Thi~ nhennmenon corresDonds to the !!lancin!!-an!!le refiection

522 Ñ. S. THEOCARIS AND Ç. G. GEORGIADIS

1ì5 2 ìs

3 ì5 4 ìsFigure 3. Series of photographs showing fracture of the ligament and longitudinal waves propagation, ßn a

ÑÌÌÁ specimen of thickness d = 0.010 m.

of longitudinal waves from a free boundary descÞbed ßç references [23] and [24].Experiments related to this phenomenon have been pedormed õñ to now only withdetonation or impact pressure pulses.

The patterns of longitudinal and shear (trailing) waves developed in these tests hadcircular and elliptical shapes, respectively. The measured values ïÉ Ct, C2, á, and ás confirm

Figure 4. Photograph of the propagation of ÑÞmarÕ longitudinal wavesand trailing shear waves, ßç a ÑÌÌÁspecimen of thickness d = 0.010 m.

STRESS WAVES ÉÍ FRACTURE 523

the validity of the well-known relation [1]

(sin á,)/ CI = (sin á.)/ C2' (3)

ÂÕ referring to Figure 4 (7 ìs) one may derive the values á, = 900 and á. = 340. Then,since CI = 1800 m/s and C2 = 1030 m/s it is evident that these deÞved values well satisfyrelation (3), thus confirming its validity.

4.2. BRI~E MATERIALS, PLANE STRESS CONDInONS

Éç typical fracture tests with thin plates made of a brittle material a dimple of sufficientmagnitude, as compared with the plate thickness, is always formed ïç either lateral faceof the fractured plate surrounding the crack tip. This dimple is responsible for the creationof the cáustic. Éç the e×ÑeÞments, during loading and before fracture, a sufficient amountof surface displacement was created in the ligament zone. Thus, duÞçg fracture, Rayleighsurface waves were launched from this area with circular wavefronts.

2 ìs1 ìs

É. ìs3 ìs

Figure 5. Series ïÉ photographs showing fracture ïÉ the !igament and both !ongitudina! and Ray!eigh wavepropagation, ßç a ÑÌÌÁ specimen ïÉ thickness d = 0.001 m.

Figure 5 indicates the creation ïß both longitudinal and surface waves during fractureïß a ÑÌÌÁ specimen ïß thickness d = 0.001 m. The Rayleigh waves caused a cleardeformation ïß the lips ïß the artificial slot as the sketch ïß Figure 6 indicates. Theexistence ïß this deformation provides one with a way ïß distinguishing the Rayleighwaves from the longitudinal ones ßç these experimental results.

It is well known [8] that, at the instant ïß arrival ïß the crack tip at the oppositelongitudinal boundary ïß the plate, a large amount ïß elastic energy is released ßç theform ïß a travelling wave. Figure 7 presents a series ïß photographs showing the propaga-tion ïß both longitudinal and Rayleigh waves generated during fracture ïß an edge ligament

524 Ñ. S. nfEOCARIS AND Ç. G. GEORGIADIS

()()()()~ m

d: 0001 m

Figure 6. Sketch of wavefro~ts of longitudinal and Rayleigh pulses caused by fracture ßç a thin plate.

2ì51 ìs

3ì5 Sìs

Figure 7. Series ïß photographs showing ..Jongitudina! apdRay!eigh waíe propagation produced by lai!ureïß a ÑÌÌÁ specimen ïß thickness d = 0.003 m.

ßç a ÑÌÌÁ specimen of thickness d= 0.003 m, with the geometry of Figure 2(c). Én thisexperiment, the deformation, which is induced by Rayleigh waves at the dihedral angleof the longitudinal boundary of the plate, deflects the light outside the image ïß thespecimen. This is due to the formation of a groove, which connects the moving disturbanceof the Rayleigh wave and the longitudinal boundary of the plate. Therefore, one maydeduce that the "ridges" ïß light rays ïç the respective photographs correspond to troughs(valleys) ïç the disturbed surface of theplate.

4.3. DUCTILE MATERIALS

The difference already observed previously between thin and thick plates ßç brittlematerials regarding the launching of the Rayleigh waves has not been detected ßç these

STRESS WAVES ÉÍ FRACTURE 525

tests. Both longitudinal and surface waves were emitted ßç our experiments with ductilethin and thick plates. The longitudinal waves were less clear than the surface waves ßçthe tests with PCBA specimens, because they are formed by the schlieren effect (seeFigure 8). The speeds of the waves appearing ßç these tests were very close to therespective values indicated ßç Table 1 under static loading.

Figure 8. Photographof the propagation of longitudinal and Rayleigh waves, in a PCBA specimen of thicknessd =0'008 m.

4-ìs1 ìs

16 ìs10ìs

Figure 9. Series of photographs showing fracture of the ligament and both longitudinal and Rayleigh wavepropagation, ßç a PCBA specimen of. thickness d = 0.003 m.

526 Ñ. S. 1¹ÅÏCÁRÉS AND Ç. G. GEORGIADIS

Fracture of PCBA plates is always associated with the creation ïÉ plastic zones due tothe ductility of the material, which causes a significant surface deformation around thecrack tip. Accordingly, it was expected that the ligament region ßï the tests would developplastic zones. Éç fact, this phenomenon was always observed. The plastic zones thusdeveloped after fracture of the ligament created Rayleigh waves propagating along theplate.

'ii]~

ï ìs 3 ìs

7 ìs 21 ìs

Figure 10. Series of photographs showing fracture of the ligament and both longitudinal and Rayleigh wavepropagation, ßç a PCBA specimen ïÉ thickness d = 0.008 m.

Indeed, ßç the first photographs ïß Figures 9 and 10 the Rayleigh wave fronts werenot circularly crested. This phenomenon may be attributed to the fact that the forms ïßthese fronts are int1uenced by the shapes ïß the plastic zones. Therefore, the shapes ïßthe spreading elastic surface disturbances yield information about the geometry ïß theplastic zone formed around crack tips.

5. CONCLUSIONS

With the e×ÑeÞmeÞts descÞbed above, and especially with the introduction ïß newtypes ïß specimens, the difficulties ßç observing waves emitted duÞçg fracture have beenovercome. The specimens were aÑÑrÏÑÞateÉÕ machined, so that ïçlÕ a single fracturestep was observed and an intensification ïß the emission ïß the fracture waves was achieved.The latter effect occurred because ïß the abrupt disappearance ïß the singular stress fieldafter failure ïß the ligament. .

From the seÞes ïß e×ÑeÞmeçts executed one may conclude that the generation ïß thelongitudinal wave pulses is independent ïß the state ïß the stresses and the bÞttÉ~çess orductility ïß the mateÞal. óç the contrary, the emission ïß Rayleigh waves is suppressed

STRESS WAVES ÉÍ FRACTURE 527

ßï the cases ïß brittle materials and plane strain conditions, indicating a dependence ïßthese waíes ïç the amount ïß ductility ïß the specimens.

Emission ïß the Rayleigh waíes during fracture is also obseríed during ductilelracture,a fact which is related to the formation ïß plastic zones ßç the neighbourhood ïß fracturefronts.

Shear waíes appear as trailing waíes from incidence at glancing angles ïß the prim~,rylongitudinal waíes ïç the lips ïß the propagating crack.

Complicated waíe patterns, generally obseríed during crack propagation ßç wide plates,are considerably clarified by the procedure and the new forms ïß specimeos used ßç thesetests. Éç further iníestigations one can proceed to analyze waíe interference and Dopplerphenomena, to obtain insight ïç the mode ïß fracture.

ACKNOWLEDGMENT

The authors wish to acknowledge valuable discussions with Dr D. Pazis, Lecturer ïÉthis Department.. One of the authors (H.G.G.) was partly supported by a researchprogramme subsidised by the Hellenic Aluminium Co. The authors express their gratitudefor this financial support.

REFERENCES

1. Ç. KOLSKY 1963 Stress Wáves in Solids. New York: Dover Publications.2. Ç. KOLSKY and D. RADER 1968 Treátise onFrácture 1,553-569 (editor Ç. Liebowitz). New

Õ ork: Academic Press.3. Õ. Ì. TSAI andH. KOLSKY 1967 Journál ÏÉ Mechánics ánd Physics ÏÉ Solids 15, 263-278.

Á study of the fractures produced ßç glass blocks by impact.4. J. W. PHILLIPS 1970 Internátionál Journál ÏÉ Solids ánd Structures 6,1403-,1412. Stress pulses

produced during the fracture of brittle tensile specimens.5. Ê. Ïé 1966 Experimentál Mechanics 23, 463-469. Transient response of bonded strain gages.6. É. S. Guz' and Á. D. Æïôïí 1974 Problemy Prochnosti 4, 63-65. Emission of elastic waves

during crack growth ßç steel that has passed through different heat treatments.7. É. S. Guz' and V. Ì. FINKEL' 1973 SovietPhysics-Solid Státe 14, 1619-1622. Relationship

between the spectrum of waves emitted by a growing crack and the energy reserve at its tip.8. Ñ. S. THEOCARIS 1981 Engineering Frácture Mechánics 15, 283-290. Secondary afterfailure

fractures due to transversely reflected waves.9. Ç. Ñ. ROSSMANITH and W. L. FOURNEY 1982 Experimentál Mechánics 22, 111-116.

Determination of crack-speed history and tip locations for cracks moving with nonuniformvelocity.

10. J. W. DALL Õ 1980 Proceedings ÏÉ the XVth Internátionál Congress ÏÉ Theoreticál ánd AppliedMechánics 79-89 (editors F. R. J. Rimrott and Â. Tabarrok). Amsterdam: North-Holland.Experimental studies of dynamicfracture.

11. D. C. GAZIS andR. D. MINDLIN 1957 Journál ÏÉ Applied Mechánics 24, 541-546. Influenceof width ïç velocities of long waves ßç plates.

12. É. TOLSTOY 1973 Wáve Propágátion. New York: McGraw-Hill.13. Ç. SCHARDIN1943 Glástechnische Berichte 21, 73. Die Schlierenverfahren und ihre Anwen-

dungen.14. Ñ. S. THEOCARIS 1980 Engineering Frácture Mechánics 12,235-242. The caustic as a means

to define the core region ßç brittle fracture.15. Ñ. S. THEOCARIS and D. PAZIS 1982 InternátionálJournál ÏÉ Mechánicál Science 25, 121-136.

The topography of the core-region around cracks under modes É, ÉÉ and ÉÉÉ of fracture.16. Â. R. BAKER 1962 Joumál ÏÉ Applied Mechánics 29,449-458. Dynamic stresses created by

a moving crack.17. Â. COTTERELL 1964 Journál ÏÉ Applied Mechánics 31, 12-16. Ïð the nature of moving cracks.18. J. R. RICE and Ì. Á. JOHNSON 1970 ßç Inelástic Behávior ÏÉ Solids, ññ. 641-672 (editors

Ì. F. Kanninen, W. Adler, Á. R, Rosenfield and R, Jaffee). New York: McGraw-Hill. Therole of large crack tip geometry changes ßç plane strain fracture.

528 Ñ. s. THEOCARIS AND Ç. G. GEORGIADIS

19. Ñ. S. THEOCARIS andN. Ñ. ANDRIANOPOULOS 1982 SESA Seventhlntemátionál ConferenciIsráel, 473-482. Strain rate effects ïç the mechanical properties of polymers.

20. Ñ. S. THEOCARIS 1981 ßç Mechánics of Frácture 7, 189-252 (editor G. C. Sih). The HagufThe Netherlands: Nijhoff Publishers. Elastic stress intensity factors evaluated by caustics.

21. J. Ì. BEINERT 1975 Joumál of Applied Mechánics 42, 5-8. Schlierenoptical stress analysisof short duration pulses ßç elastic plates.

22. Ô. R. ÊÁÍÅ 1957 Joumálof Applied Mechánics 24, 219-227. Reflection of dilatational wavesat the edge of a plate.

23. F. C. ROESLER 1955 Philosophicál Mágázine 46, 517-526. Glancing angle reflection of elasticwaves from a free boundary.

24. D. G. CHRISTlE 1955 Philosophicál Mágázine 46, 527-541. Reflection of elastic waves froma free boundary.

é

.É

~

ß.,

~r~"i:

É.~".,,~'.'=,

j,..'"jf

.:

É~

;ciic.~