Study of the conditions of fracture at explosive compaction of powders

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1 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. Spcial Issu: Russian Fractur Mchanics School Study of th conditions of fractur at plosiv compaction of powdrs Andry E. Buzyurkin, Evgny I. Kraus Khristianovich Institut of Thortical and Applid Mchanics of Sibrian Branch of RAS, 4/ Instituskaya str., Novosibirsk, , Russia buzjura@itam.nsc.ru, kraus@itam.nsc.ru Yaroslav L. Lukyanov Lavrntyv Institut of Hydrodynamics of Sibrian Branch of RAS, 5 Lavrntyv pr., Novosibirsk, , Russia lukyanov@hydro.nsc.ru ABSTRACT. Joint thortical and primntal invstigations hav allowd to raliz an approach with us of mathmatical and physical modling of procsss of a shock wav loading of powdr matrials. In ordr to gain a bttr insight into th ffct of loading conditions and, in particular, to study th ffct of dtonation vlocity, plosiv thicknss, and plosion prssur on th proprtis of th final sampl, w numrically solvd th problm about powdr compaction in th aisymmtric cas. Th prformd analysis shows that an incras in th dcay tim of th prssur applid to th sampl du to an incras of th plosiv thicknss or th trnal loading causs no shrinkag of th dstructd rgion at a fid propagation vlocity of th dtonation wav. Simultanously, a dcras in th propagation vlocity of th dtonation wav rsults in an apprciabl shrinkag of this rgion. KEYWORDS. Shock wavs; Fractur; Powdr. INTRODUCTION M thods of plosiv loading of powdr matrials in consrvation ampouls ar applid in ordr to obtain nw matrials including composit ons with th uniqu physical and mchanical proprtis. In addition, ths mthods can b usd to study phas transitions occurring in matrials at high prssurs and tmpraturs taking plac bhind shock wavs, as wll as for th synthsis of mtastabl phass. In rcnt dcads, significant dvlopmnt has bn achivd in such a scintific and tchnical branch of matrials scinc as powdr mtallurgy. This trm is currntly undrstood a whol compl of problms connctd with th dsign of matrials and products from mtal and nonmtal powdrs. Intrst in ths problms is quit undrstandabl sinc th opportunity to crat nw classs of matrials with uniqu and controllabl proprtis which can not b obtaind by ordinary mtallurgy mthods has arisn. A spcial plac in th powdr mtallurgy is occupid by plosiv compaction of powdr matrials. It is asy to plain th strong intrst in th plosiv compaction. It consists in th fact that virtually all th mthods of composit matrials' production from powdr miturs lad to a chang in initial matrial proprtis du to high tmpraturs and rlativly long duration of th procss. 0

2 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. Sinc th powdrs bing in th form of granuls, fibrs, ndls and ribbons, possssing th ncssary proprtis in th initial stat, can not b usd dirctly to produc smi-finishd products or componnts, th mthods of compaction of ths matrials prform two tasks at onc. On th on hand th compaction changs th shap and siz of th powdrs, and on th othr hand it producs th matrial itslf. From this point of viw, th short posur to high tmpraturs and prssurs during plosiv compaction allows, in gnral, to kp th original structur and proprtis of th componnts. At th sam tim, varying of th intnsity and tim posur to high prssur and tmpratur in shock comprssion allows to modify, if ncssary, th structur and proprtis of th compacts a controlld mannr. Th loading of th powdr matrials in th consrvation ampouls can b carrid out by mans of both plan and obliqu shock wavs. Each of th mthods has its pros and cons. Th plosiv loading by obliqu shock wav is charactrizd by high valus of shar strain, in comparison with th plan impact, which lads to strongr bonds btwn th compactd particls. In addition, this schm allows to obtain th compacts not only in th form of plats, but pips, rods, cons, tc as wll. On can also gt th compacts of larg sizs. Th loading by plan shock wavs allow to vary th prssur and tmpratur bhind th shock front in a widr rang and to rach much highr valus of ths paramtrs. At th sam tim, th mthod is mor matrial-consuming and has limitations on th siz of th loadd sampls. Invstigation into th intraction btwn obliqu shock wavs in porous matrials and powdrs is a topical problm in optimization of loading conditions for obtaining, from a givn sampl, a compactd matrial with spatially uniform physical and mchanical proprtis. In compacting a powdr in th cylindrical schm, an irrgular intraction btwn shock wavs occurs. Th compactd powdr displays substantial non-uniformity in particl displacmnts, rsulting in inhomognity of powdr charactristics and, in som cass, vn in matrial failur. In compacting porous matrial and powdrs, th strong bonding btwn particls is achivd through th combind prssur-shar loading. During th compacting, a substantial nrgy is rlasd at th intrfacs btwn powdr particls, rsulting in surfac claning and matrial mlting in narrow intrfacial rgions. As a rsult, por collapsing, giving ris to strong bonding btwn particls, occurs. Blow, this phnomnon is trmd compaction. V.F. Nstrnko proposd th following critrion for th formation of a strong compact: P>H V () whr, according to [], H 3Y. Following [], w can writ critrion (), dducd from primntal data, as V s P >6Y s () In turn, R. Prummr [] uss th following condition for obtaining a uniform, in its physical proprtis, cylindrical compact with no Mach rflction inducd singularitis at its cntr: P H V, whr P is th dtonation prssur. Comparing condition () with th condition P H V, Nstrnko [] arrivs at a conclusion that it is impossibl in principl, without a cntral rod, to obtain a spatially uniform compact in th cylindrical loading schm sinc th shock prssur rquird for obtaining a dns compact () will always lad to Mach rflction at th cntr of th sampl. Anothr important problm is prsrvation of th finish compact aftr loading. With th arrival of unloading wavs, thr ariss a tnsil strss that rsults in partial or complt dstruction of th sampl. W assum that th sampl undrgos mchanical failur if th maimum tnsil strss ma rachs a crtain critical valu *. In lin with th adoptd hypothsis, th following condition for th sampl failur should b assumd: > (3) ma * whr ma is th highst strss among th principal strsss for th straind stat undr study and * is th critical strss. In th prsnt work, th critical strss * is stimatd as =(/3) Yln(/ m ) * s whr m is th rsidual porosity. Taking th finish-compact dnsity to qual 99%, w obtain m =0.0 and * 3Y s. For th principal strsss, w hav: yy = ( yy ) 4y yy = ( yy ) 4y 03

3 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. = 3 EXPERIMENTAL STUDY OF THE COMPACT STRUCTURE E primnts on th plosiv compaction hav bn conductd by a cylindrical schm without a cntral rod. Th powdr has consistd of particls of narly sphrical shap and with siz of µm (Fig. ). Bulk dnsity of th powdr has bn in all primnts for coppr qual to -5.0 ± 0.05 g/cm 3, for aluminum -.4 g/cm 3. Th primntal arrangmnt is shown in Fig.. Figur: Initial coopr powdr. Figur : Schm of plosiv compaction: dtonator, plosiv charg, 3 stl plugs, 4 containr (stl tub mm in diamtr), 5 coppr powdr, 6 momntum trap. Eplosiv compaction occurs undr th action of th dtonation products of th contact plosiv chargs. For varying of th dtonation vlocity th chargs wr mad from ammonit, RDX and mitur of ammonit with RDX in diffrnt proportions. Th dtonation vlocity (D) was masurd by lctrical contact tchniqu, and rangd from 3.9 to 5.6 km/s. Containr wall was thin compard to th thicknss of plosiv layr and diamtr of th powdr sampl. Structur of compacts cross-sctions wr studid using an optical microscop NEOPHOT. In Fig. 3 th structurs of th cross sctions nar th ais of th sampls is shown for thr diffrnt valus of dtonation vlocity (3.9, 3.95 and 5.6 km/s) and at th sam thicknss of plosiv charg 5 mm. (a) ( (c) Figur 3: Th structurs of th cross sctions of th sampls for thr diffrnt valus of D and at th sam thicknss of plosiv charg 5 mm: a) D=3.9 km/s; D=3.95 km/s; c) D=5.6 km/s. It can b notd that with th incras of th dtonation vlocity, and hnc th shock prssur, a compacts structurs chang. At minimum valu of D compact is homognous. Thn, in th cntr du to th irrgular rflction of th convrging shock wav a zon of mlt appar. And at D=5.6 km/s in th compact th radial cracks aris. A furthr incras in D can lad to th dstruction of th containr. To invstigat th ffct of th thicknss of th plosiv charg on th crack formation, an primnt was conductd in which th thicknss of th charg was incrasd to 0 mm, and th dtonation vlocity was 5.7 km/s. Th structur of this compact is shown in Fig

4 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. Figur 4: Th structurs of th cross sction of th sampl loadd at D=5. km/s and thicknss of th plosiv charg 0 mm. Comparing structurs from Fig. 3а and Fig. 4, it is visibl, that in spit of approimatly idntical dtonation rgims, in a compact in Fig. 4 cracks ar absnt. NUMERICAL SIMULATION OF THE EXPLOSIVE LOADING I n ordr to gain a bttr insight into th ffct of loading conditions and, in particular, to study th ffct of dtonation vlocity, plosiv thicknss, and plosion prssur on th proprtis of th final sampl, w numrically solvd th problm about powdr compaction in th aisymmtric cas using conditions of abov mntiond primnts. Th problm statmnt according to primntal schm is clar from Fig. 5. Y P D plosiv powdr Figur 5: Th problm statmnt. X W solvd th full systm of quations govrning th dformation of a porous lastic-plastic matrial [3]. Th action of th plosion products on th sampl was modld with a prssur applid to th uppr bordr of th sampl. Th prssur was calculatd by th approimation formula for th prssur upon unrstrictd disprsion of dtonation products [4]: 3( ) D ()= H p( ( / )/ ), = 4( ) Pt P t D t t Hr is th plosiv thicknss and is th adiabatic ponnt of th dtonation products. Sinc th problm is symmtric, a half of th primntal assmbly is considrd. Th symmtry ais is th ais of th containr with th powdr. On th symmtry ais rigid wall boundary conditions ar st. Th right boundary is considrd to b fr of strss, and at th lft boundary condition of a rigid wall is put. Computation of th contact boundaris is prformd by using a symmtric algorithm [5]. Th calculations ar carrid out by th M.L. Wilkins schm [6]. Th shock wav propagats from lft to right. Gomtric dimnsions and valus of th physical paramtrs corrspond to th primntal data mntiond abov. In this papr a fw-paramtric quation of stat is applid [7], which has allowd to simulat shock-wav procsss with a minimal numbr of physical paramtrs as th initial data. 05

5 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. /3 V EE cv, lt cv, 0T V 0 3 de lcv, lt cv, 0T V P dv V 3 V V0 or, in trms of fr nrgy /3 ( V) V v, l v, 0 T V0 FVT (, ) E( V) c Tln c T whr P and E - prssur and spcific intrnal nrgy of th zro isothrm, T - tmpratur, cv cv, l cv, - hat capacity at constant volum, ( V ) - th Dby tmpratur. Th quation of stat prsntd hr is basd on th dpndnc of th Grunisn cofficint γ on th volum [] ( V) /3/( av / V) a /( ) P / K s t,0 s 0 whr s KV s 0 / cv, K s - adiabatic bulk modulus, -officint of thrmal pansion, P t,0 - hat prssur undr th normal conditions. V is usd: To find th lastic curvs a gnralizd modl dscribing th Grunisn cofficint t /3 t V d PV / dv V /3 t 3 dpv / dv at t 0 th quation corrsponds to th Landau and Slatr thory [, 9], at t it corrsponds to th Dugdal and MacDonald hypothsis [0], and at t to th thory of fr volum []. In th physics of shock wavs a mthod of calculating th prssur at th Hugoniot adiabat of th porous matrial by prssur on th rfrnc Hugoniot adiabat of monolithic matrial [] is known: P h, p 0 V V V Ph V V V 00 Hr V is th spcific volum of th Hugoniot adiabats, V0 and V 00 ar spcific volums of monolithic and porous matrials, rspctivly, at th normal initial conditions. CALCULATION RESULTS AND DISCUSSION F Ig. 6, a and b shows th prssur isolins for th cass of planar and cylindrical symmtris with idntical loading conditions. It is sn from Fig. 6 that, in th planar statmnt of th problm, a rgular rflction of th incidnt shock wav taks plac. In th cas of th cylindrical loading schm, th incidnt shock wavs bnds as it approachs th cylindr ais, and, undr th sam loading conditions, an irrgular rflction occurs. Fig. 7a shows th prssur profil nar th symmtry ais for th cass of planar and cylindrical statmnts (solid and dashd curvs, rspctivly). An apprciabl prssur ris nar th symmtry ais is obsrvd in th cas of cylindrical configuration compard to th planar problm du to th divrgnc of th shock wav to th ais. Fig. 7b shows th profil of th longitudinal vlocity u across th sampl undr loading bhind th shock front. Th solid and dashd lins show th data for th planar and aisymmtric problm statmnts, rspctivly. It is clarly sn that th vlocity in th cylindrical cas is much gratr than in th planar variant. As statd abov, an important problm is prsrvation of finish compact, i.., prvnting its mchanical failur and obtaining a sampl with uniform proprtis. Using critrion (3), w can find th intrfac btwn th solid and distructd matrials. Th rgions of th compactd and porous matrials for various plosiv thicknsss for th trnal prssurs 06

6 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. P =0.05 Mbar and P =0.075 Mbar ar shown in Fig., a and b, rspctivly. In ths calculations, th dtonation vlocity was D =5 km/s. Th solid, dashd, and dot-and-dash lins outlin th dstruction rgions for th plosiv thicknsss = cm, =3 cm, and =5 cm, rspctivly. Rgion is th compactd rgion, and Rgion, th dstruction rgion. An analysis of ths graphs shows that an incras in th plosiv thicknss and, hnc, an incras of th loading dcay tim dos not caus any substantial shrinkag of th dstruction rgion...5 P, Mbar 0.35 a) Figur 6: Prssur isolins: a) planar gomtry; cylindrical configuration U, cm/msc Figur 7: Prssur profil (a) and longitudinal-vlocity profil u ( y ) ( for th planar and cylindrical gomtris (solid and dashd lins, rspctivly) Figur : Compactd and dstruction rgions for various plosiv thicknsss undr trnal prssurs P =0.05 Mbar (a) and P =0.075 Mbar (. Th dtonation vlocity is D =5 km/s. Th solid, dashd, and dot-and-dash lins rfr to th plosiv thicknsss = cm, =3cm, and =5cm, rspctivly. Th compactd and dstruction rgions ar indicatd by and. It should b mphasizd that this conclusion is valid for critrion (3). In drivation of (3), it was implicitly assumd that th intrfacial mltd zons ar narrow, and th matrial in ths zon rapidly solidifis as th particls in th bulk of th matrial undrgo cooling. If this condition dos not hold, thn thr can b a situation in which, by th momnt of arrival of th unloading wav, th matrial in th intrfacial zons still rmains mltd, which will prvnt compaction. In this cas, th dimnsions of th dstruction rgion will b dpndnt on th loading dcay tim and on th plosiv thicknss. Th plosiv thicknss should b larg nough to prvnt shock wav damping in th powdr and to nabl complt por collapsing in th sampl. Fig. 9, a and b shows th dnsity isolins for th plosiv thicknss = cm and th trnal prssur P =0.05 Mbar. Parts a and b of Fig. 9 dpict th data for th aisymmtric and planar problm statmnts. Damping of th incidnt shock wav is vidnt from th figur. This rsults in incomplt powdr compaction; th lattr is clar from Fig. 0, which shows th distribution of porosity m across th sampl. Th solid and dashd lins in this figur corrspond to th planar cas and to th cylindrical configuration, rspctivly. An analysis of 07

7 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. ths graphs shows that, in th aisymmtric cas, du to th wav divrgnc to th ais, th pors undrgo collapsing in a largr volum than in th planar variant Figur 9: Dnsity isolins for = cm: cylindrical configuration; planar statmnt porosity Figur 0: Porosity m for = cm: solid lin is planar statmnt; dashd lin is cylindrical configuration. A twofold incras in th plosiv thicknss maks th dcay of th incidnc shock wav lss intnsiv. Th dnsity isolins for th plosiv thicknss = cm and th trnal prssur P =0.05 Mbar ar shown in Fig., a and b. Parts a and b of this figur shows th calculation data for th aisymmtric and planar statmnts, rspctivly. It is clarly sn that in th cas of cylindrical symmtry th shock wav bnds nar th ais, giving ris to an irrgular rflction; in th planar configuration, a rgular intraction occurs. Fig., which dpicts th distribution of porosity m across th sampl compactd in th cylindrical gomtry (th dashd lin in Fig. ), is indicativ of complt collapsing of pors ovr th ntir thicknss of th sampl. In th planar cas (s th dashd lin in Fig. ), th complt collapsing of pors is obsrvd approimatly ovr half th thicknss of th sampl, and th porosity nar th symmtry ais is clos to th initial on, m Figur : Dnsity isolins for = cm: a) cylindrical configuration; planar configuration porosity Figur : Porosity m for = cm: solid lin - planar statmnt; dashd lin - cylindrical configuration. Th dnsity isolins for th plosiv thicknss = cm ar shown in Fig. 3, a and b. Th trnal prssur was takn to b P =0.05 Mbar. Parts a and b of this figur show th calculation data for th aisymmtric and planar statmnts. In th cylindrical cas (s Fig. 3, a), an irrgular rflction is clarly obsrvd, whras in th planar cas (s Fig. 3, th 0

8 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. incidnt shock wav intracts with th rigid wall in th rgular mannr. In both cass, all pors in th sampl collaps compltly. Furthr calculations wr carrid out for th plosiv thicknsss = cm, =3 cm and =5 cm. Fig. 4, a and b illustrats th ffct of applid prssur on th thicknss of th dstruction rgion. In th calculations, th trnal prssurs wr P = 0.05 Mbar and P = Mbar, rspctivly, and th dtonation vlocity in both cass was D =7 km/s. Th solid and dashd lins show th data for th plosiv thicknsss =3 cm and =5 cm. Rgions and ar th compactd and dstruction rgions. As is sn from th figur, an incras in th trnal load causs no shrinkag of th dstruction zon. Thus, it can b concludd that an incras in th dcay tim of th prssur applid to th sampl rsulting from an incras in th plosiv thicknss or in th valu of th trnal load dos not mak th dstruction zon shrink at a fid propagation vlocity of th dtonation wav Figur 3: Dnsity isolins for = cm: a) cylindrical configuration; planar statmnt Figur 4: Compactd and dstructd rgions for two valus of trnal prssur, P =0.05 Mbar (a) and P = Mbar (. Th dtonation vlocity is D =7 km/s. Th solid and dashd lins show th calculation data for th plosiv thicknsss = 3 cm and = 5 cm. As shown by abov mntiond primnts an incras of th vlocity of th dtonation wav rsults in a considrabl shrinkag of th dstruction rgion. Fig. 5 show th compactd () and dstructd () rgions in th sampl for th dtonation vlocitis D =3, 5, 7 km/s at a fid plosiv thicknss =5 cm and at a fid trnal prssur P =0.05 Mbar. Th solid, dashd, and dot-and-dash lins show th calculation data for th dtonation vlocitis D =3 km/s, D =5 km/s, and D =7 km/s Figur 5: Compactd () and dstructd () rgions for thr valus of th dtonation vlocity. Th solid, dashd, and dot-and-dash lins rfr to D =3 km/s, D =5 km/s, and D =7 km/s. Th isolins of prssur for th indicatd loading paramtrs ar shown in Fig. 6, a-c. It is sn from th graphs that, as th shock-wav propagation vlocity incrass, th angl of incidnc dcrass and th rflctd shock causs matrial dstruction (s Fig. 6, b and c). As th vlocity of th dtonation wav incrass, th angl of incidnc of th incidnt shock wav incrass and, as it is sn from Fig. 6, a, at th vlocity D =3 km/s th incidnt shock wav is clos to th normal shock and th amplitud of th rflction wav is almost zro. Sinc in th cas of cylindrical symmtry no rgular rflction occurs, th final sampl turns out to b inhomognous. Fig. 7 shows th distribution of th longitudinal vlocity u (Fig. 7, a) and tmpratur T (fig. 6, across th sampl 09

9 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS.4. in th compactd rgion for th dtonation vlocity D =5 km/s. An apprciabl non-uniformity in th distribution of paramtrs is vidnt from th graphs. Nar th ais, both th vlocity and tmpratur ar gratr than in th rgion som distanc away from it c) Figur 6: Prssur isolins: a) dtonation vlocity D =7 km/s; dtonation vlocity D =5 km/s; c) dtonation vlocity D =3 km/s U, cm/msc T, K Figur 7: Prdictd distributions of th longitudinal vlocity u (a) and tmpratur T ( across th compactd rgion of th sampl for th dtonation vlocity D =5 km/s. Parts a and b of Fig. show th distributions of th longitudinal vlocity u and tmpratur T across th compactd rgion of th sampl prdictd for th dtonation vlocity D =3 km/s. Hr, undr idntical loading paramtrs, th final sampl is quir homognous. As a rsult, it bcoms possibl to obtain spatially uniform compactd sampls. Th ncssary condition for this is sufficintly low dtonation vlocity, qual, for th aluminum powdr, to 0.3 cm/ m sc. Hr, on th on hand, compaction condition () should b fulfilld and, on th othr, th uniformity of loading paramtrs across th sampl should b nsurd. Thus, th compaction of powdrs with low dtonation vlocitis rsults in a considrabl shrinkag of dstruction zons in finish sampls and in spatial uniformity of matrial paramtrs in thir compactd parts. Th prformd analysis shows that an incras in th dcay tim of th prssur applid to th sampl du to an incras of th plosiv thicknss or th trnal loading causs no shrinkag of th dstructd rgion at a fid propagation vlocity of th dtonation wav. Simultanously, a dcras in th propagation vlocity of th dtonation wav rsults in an apprciabl shrinkag of this rgion. 0

10 Andry E. Buzyurkin t alii, Frattura d Intgrità Struttural, 4 (03) 0-; DOI: 0.3/IGF-ESIS U, cm/msc T, K Figur : Distributions of th longitudinal vlocity u (a) and tmpratur T ( across th compactd rgion of th sampl for th dtonation vlocity D =3 km/s. CONCLUSIONS J oint thortical and primntal studis hav allowd to implmnt an approach that uss mathmatical and physical simulation of shock-wav loading of powdrd matrials. A numrical simulation of shock wav propagation and dformation of th primntal assmbly has bn prformd. Th tmpratur distributions ovr th sampl thicknss in th compactd zon for svral valus of dtonation vlocity show that at highr spds thr is considrabl htrognity in th tmpratur distribution ovr th sampl thicknss. Nar th ais of th sampl th tmpratur has a highr valu than in distanc from it. An incras in th prssur dcay tim du to incrasing ithr th plosiv thicknss or th trnal loading intnsity causs no shrinkag of th dstruction zon at a fid propagation vlocity of th dtonation wav. Compaction of powdrs with low dtonation vlocitis rsults in a considrabl shrinkag of dstruction zons in finish sampls and in a uniform distribution of matrial paramtrs in th compactd rgion. REFERENCES [] V.F. Nstrnko, High-rat dformation of htrognous matrials, Nauka, Novosibirsk (99) (in Russian). [] R. Prümmr, Powdr compaction. Eplosiv wlding, forming and compaction, London; Nw York: Appl. Sci. Publ., (93). [3] S.P. Kislv, V.M. Fomin, J. of Applid Mchanics and Tchnical Physics. 34(6) (993) 6. [4] V.V. Pai, G.E. Kuz'min, I.V. Yakovlv, Combustion, Eplosion, and Shock Wavs, 3(3) (995) 4. [5] A.I. Gulidov, I.I. Shabalin, Numrical Ralization of th Boundary Conditions in Dynamic Contact Problms, Prprint ITAM SB RAS, Novosibirsk (97) (in Russian). [6] M. L. Wilkins, Computr simulation of dynamic phnomna, Springr, Brlin, Hidlbrg (999). [7] E. I. Kraus, Vstnik NGU. Fizika, () (007) 65 (in Russian). [] C. Slatr, Introduction in th chmical physics. Nw-York-London: McGraw Book company, Inc., (935) 39. [9] L. D. Landau, K.P. Stanyukovich, Dokl. Akad. Nauk SSSR, 46 (945) 399 (in Russian). [0] J. S. Dugdal, D. McDonald, Phys. Rv., 9 (953) 3. [] V.Ya. Vaschnko, V.N. Zubarv, Sov. Phys. Solid Stat, 5 (963) 653. [] R. G. McQun, S. P. Marsh, J. W. Taylor, J. N. Fritz, High-Vlocity Impact Phnomna, d. by R. Kinslow, Nw York: Acadmic Prss (970) 93.