TRANSIENT PLANE WAVES IN MULTILAYERED HALF-SPACE

Transcription

1 acta echaca et autoatca vol.7 o. (3) DOI.478/aa-3- TRANSIENT PLANE WAVES IN MULTILAYERED HALF-SPACE Ihor TURCHYN * Olga TURCHYN ** * Iva Frako Natoal Uversty of L vv Uverstetska L vv Ukrae ** Ukraa Natoal Forestry Uversty Ge. Chupryky 3 L vv Ukrae horturchy@gal.co olgaturchy@.ua Abstract: Cosdered the dyac proble of the theory of elastcty for ultlayered half-space. Boudary surface of hoogeeous half-space loaded wth oral load ad the boudares of separato layers are codtos of deal echacal cotact. The forulato volves o-classcal separato of equatos of oto usg two fuctos wth a partcular echacal eag voluetrc expaso ad fucto of accelerato of the shft. I ters of these fuctos obtaed two wave equato wrtte boudary codtos ad the codtos of deal echacal cotact of layers. Usg the Laguerre ad Fourer tegral trasforatos was obtaed the soluto of the forulated proble. The results of the calculato of the stress-stra state the half-space wth a coatg for a local pact loadg are preseted. Key words: Dyacal Proble of Elastcty Multlayered Half-Space Aalytcal Soluto Fourer Itegral Trasforato. INTRODUCTION Questos cocerg the propagato of a elastc waves layered bodes uder local loadg of ther surfaces are urget due to great uber of up to date probles of sol echacs acoustc flaw detecto probles volvg stregth of coposte aterals etc. I geeral the aalyss of ths pheoeo s coected wth solvg a spatal dyacal elastcty proble wth correspodg tal ad boudary codtos ad og codtos o flat or curvlear surfaces. At preset there have bee fored dfferet approaches to solvg dyacal elastcty probles. Aog the aalytcal ethods the frst place there should be etoed ethods coected wth the applyg of Fourer ad Laplace tegral trasforatos (Slyep ya ad Yakovlyev 98). However the exact verso of the receved trasfors ca be perfored oly few splest cases ad that's why here varous uercal ad asyptotcal ethods ust be appled. Ad t's obvous t flueces the accuracy ad relablty of the receved results. Aog other approaches to solvg dyacal elastcty probles the ethod of characterstcs (Chou ad Gref 968 Yag ad Achebach 97) the ethod of usg fte tegral trasforatos (Slyep ya ad Yakovlyev 98 Wakhede ad Bhose 98) the ethod of sug up of eleetary waves (Bedford ad Druheller 994) ad others (Pao ad Mow 973 Poruchkov 986) ca be poted out. But a spatal case ad the case of great uber of separato boudares the etoed above ethods are ot always effectve. I ths paper Laguerre tegral trasfor wth respect to te varable (Galayuk 98) s offered for solvg ths kd of probles. The advatage of ths approach s the splcty of perforg a verso whch cossts sug up a orthogoal seres as well as the possblty of costructo of sple algorth of seres coeffcets exact fdg. The applcato of ths ethod as well as the covergece of all procedures s ustfed by kow theores fro a orthogoal polyoals theory (Sego 959). More over the scalg factor troduced Laguerre trasfor perts us to "stretch" or "cotract" te terval to deped o uercal aalyss eeds ad ake use of correspodg theores about ltg value.. BASIC EQUATIONS Let us a edu whch cossts of flat layers wth dfferet thckesses ad dfferet physcal-echacal propertes. Each -th layer has thckess (uber deotes the half-space) ad s charactersed by Lae elastc costats ad desty. Whe body forces appled to the edu are detty equal to ero the deforatoal feld for every -th layer ca be foud fro equato of oto: c grad ( dvu ) c rot( rotu ) U (.) ( ) ( ) ( ) t wth correspodg tal ad boudary codtos. The sae as Galayuk ad Chuak (99) vector equatos (.) xed cyldrcal coordates referred to soe lear deso h wth the help voluetrc expaso: h( ) [ ( Du ) ( Cv )] w (.) ( ) ( ) ( ) ( ) twst fuctos: h( ) [ ( Dv ) ( Cu )] (.3) ( ) ( ) ( ) ad fuctos of accelerato of a shft: (.4) ( ) ( ) ( ) ca be dvded to 3M scalar wave equatos: h D C ( ) ( ) ( ) ( ) 53

2 Ihor Turchy Olga Turchy Traset Plae Waves Multlayered Half-Space h D C ( ) ( ) ( ) ( ) h D C ( ) ( ) ( ) ( ) Here ad the followg desoless orthogoal coordates o the plae copoets of the dsplaceet vector s the dyacal te are the logtudal ad trasversal waves propagato veloctes the ateral of -th layer velocty of wave propagato soe edu (t s selected to deped fro a uercal aalyss tasks) coeffcets of the frst quadrc for the syste. I the followg we suppose that the traset processes source the tally ovable layered edu s the shock loadg of ts boudary. Ths loadg tur coes to acto of a syetrcal to -axs oral loadg the crcle rego of radus. O the og surfaces codtos of perfect echacal cotact take place ad o fty dsplaceets ad stresses are abset. I the cosdered stateet our proble of deteratoof stress-stra state s plae ad that s why all the characterstcs of the proble wll be fuctos oly of ad fucto troduced by (.3) s detcally equal to ero. I the desoless rectagular coordates coeffcets of the frst quadrc for wll be. The ters of separato fuctos ad proble reduces to tal- boudary proble of atheatcal physcs for the syste of wave equatos: ( ) ( ) ( ) x M (.5) ( ) ( ) ( ) x M (.6) uder ero tal codtos: (.7) ( ) ( ) ( ) ( ) boudary codtos o the surface : () () () p () () () x (.8) (.9) Jog codtos o the surfaces : ( ) ( ) ( ) ( ) ( ) ( ) (.) ( ) ( ) ( ) ( ) ( ) ( ) r x x (.) ( ) ( ) ( ) ( ) [ ] [ ] (.3) ( ) ( ) ( ) ( ) x x ad codto o the fty: (.4) I the equatos (.5)-(.3) kow oral force loadg. Usg the fuctos ad foud fro the proble (.5)-(.4) oral copoets of dsplaceet vector ca be detered as the solutos of Cauchy probles: w w w ( ) ( ) ( ) ( ) ( ) ad radal copoets help of tegrals: x ( ) ( ) ( ) (.5) ca be detered wth the u ( x ) [ ( y ) w ( y )] dy. (.6) 3. SOLUTION OF PROBLEM (.) Laguerre tegral trasforato ca be troduced by the expresso: F ( r ) e F( r ) L ( ) d... (3.) Laguerre polyoals. The or- where thogoal seres: F( x ) F ( x ) L ( ) (3.) serve as the verso forula for the trasforato (3.). Uder soe ltatos o the fucto (Sego 959) the tegral (3.) exsts ad seres (3.) uforly cocdes wht arbtrary terval fro. Applyg Fourer tegral trasforato wth respect to spatal coordate ad trasforato (3.) wth respect to ad usg the ero tal codtos (.6) ad dfferetato forulae: exp( ) L ( ) exp( ) ( ) L ( ) we'll receve a tragular sequece of boudary value probles of the followg kd: 54

3 acta echaca et autoatca vol.7 o. (3) DOI.478/aa-3- ( ) ( ) ( ) d ( ) M (3.3) ( ) ( ) ( ) d ( ) M (3.4) () () () ( ) ( ) d d p (3.5) d ( ) (3.6) () () () ( ) ( ) ( ) ( ) d d ( ) ( ) ( ) ( ) d d M d ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) M ( ) ( ) ( ) ( ) d d M ( ) ( ) ( ) ( ) d d M. (3.) I the expressos (3.3) (3.) are the Fourer Laguerre trasfors: ( ) ( ) F ( ) rj ( r) e F ( r ) L ( ) ddr As t s kow (Galayuk ad Gorekcho 983) the geeral soluto of the tragular sequece of ordary dfferetal equatos (3.3) (3.4) ca be represeted as the algebracally covoluto: ( ) ( ) ( ) ( ) ( ) ( ) A G ( ) B W ( ) (3.) ( ) ( ) ( ) ( ) ( ) ( ) C G ( ) D W ( ) (3.3) where ukow fuctos whch ca be foud fro boudary codtos ad respectvely systes of fudaetal solutos of sequeces (3.3) ad (3.4). Usg the ukow coeffcets ethod they ca be represeted the for: () ( ) k l l G l ( ) exp( l ) a k k! k (3.7) (3.8) (3.9) (3.) () ( ) k l l W l ( ) exp( l ) a k (3.4) k! k where coeffcets expressos: ca be detered fro sple recurret l l l l a k a k ( p ) ap k (3.5) l pk uder arbtrary. I the expressos (3.5) ad all for are detcally equal to ero. Sce fuctos ultedly crease whle t's evdet that accordg to codtos (3.) t should be take: B D... (3.6) We use codtos (3.5)-(3.) for the deterato of the ukow fuctos. For ths we substtute represetato (3.) (3.3) to the codto (3.5). Usg the eag for fro the equato (3.3) ad after regroupg we obta: () () () () () B W () W () () () () () C dg () D dw () ( ) p. () () () A G G (3.7) 55

4 Ihor Turchy Olga Turchy Traset Plae Waves Multlayered Half-Space Let. I the left part of the equato (3.7) we leave oly expressos for the rest we wrte the rght part. As a result we obta: () () () () A B C D ( ) p () () () () () () A B ( ) C dg () D dw (). Usg the sae techque as for equatos (3.6) (3.) we receve the recurret sequece of the equato systes of the deso. Wth the help of these systes the ukow fuctos ca be detered. Thus the costructo of the soluto of the proble (.5)-(.4) s copleted. The fal expressos for fuctos ad oero copoets of dsplaceets vector receved fro Fourer ad Laguerre tegral trasforatos verso forulae: ( ) ( x ) ( ) ( )cos( x) d ( ) ( x ) ( ) ( )cos( x) d p( x ) p xs( ) (3.8) ad the area equal to ero. The calculato refers for codto wth the followg paraeters. The dsplaceets ad stresses are the sutable soluto to related of statcally proble. It s based o the Laguerre tegral trasfor ethod wth lt trasto. For exaple we ay wrte where s the dsplaceets the statc proble trasfor whe. w*(..5 ) s the value of the w( ) ( x ) ( ) w ( )cos( x) d ( ) ( ) () ( ) ( ) ( ) w u x s( x) d where { } { ad for calculato of we ca use forulae: } w d d d. ( ) ( ) ( ) ( ) ( ) ( ) ( ) Usg the kow dsplaceets the stress tesor copoets ca be detered accordg to Hook's law: () / ( ) ( w ) () / ( ) ( xx xu ) () / ( ) ( ) x u xw where s the shear odulus of the -th layer. 4. NUMERICAL SOLUTION I ths exaple we show the results of calculato of dsplaceet ad oral stresses o the terfaces of elastc layer ad half-space loaded by shock effect o the surface whe the area the oral pressure s gve by the law: Fg.. Te dstrbuto of oral dsplaceet at the boudary of coatg ad half-space Fgures ad gve the desoless dsplaceets ad stresses o the terface of elastc layer ad half-space at the pot. As see fro the results of the te varato of dsplaceets ad oral stresses the process of trasto s the for of oscllatos aroud the statc equlbru posto (ut values). The frst wave of the dsplaceets ad stresses arrves at ad t s accordace wth the physcs of the pheoeo. The peak value of apltude gets ts axu ad soo after arrval of the copresso wave ad thereafter the apltudes decrease. The extree values of apltudes are attaed the begg of trastoal perod whe the copresso wavereflecto fro the terfaces of elastc layer ad half-space arrves. The ultple reflecto results crease of frequecy-wave. 56

5 acta echaca et autoatca vol.7 o. (3) DOI.478/aa-3-.5 (..5 ) Fg.. Te dstrbuto of oral stresses at the boudary of coatg ad half-space The uercal results have deostrated the advatages of the preset theory ters of effectveess ad effcecy whch see to ustfy ts ore tesve forulato. Future work wll address the exteso of the preset theory for the dyac probles of layered bodes. 5. CONCLUSION The paper proposes a ew atheatcal forulato of the plae dyac proble of elastcty theory for a layered halfspace. The forulato volves o-classcal separato of equatos of oto usg two fuctos wth a partcular echacal eag voluetrc expaso ad fucto of accelerato of the shft. I ters of these fuctos obtaed two wave equato wrtte boudary codtos ad the codtos of deal echacal cotact of layers. Usg the Laguerre ad Fourer tegral trasforatos was obtaed the soluto of the forulated proble. The results of the calculato of the stress-stra state the half-space wth a coatg for a local pact loadg are preseted. REFERENCES. Bedford A. Druheller D. S. (994) Itroducto to elastc wave propagato. Wley New York.. Chou S.-C. Gref R. (968) Nuercal soluto of stress waves layered eda AIAA Joural Vol Galayuk V.A. (98) Chebyshev-Laguerre polyoals ethod xed proble for a lear dfferetal equato of the secod order wth costat coeffcets Dopovydy AN USSR Ser. A No 3-7 ( Russa). 4. Galayuk V.A. Chuak A.C. (99) Nostatoary processes a elastc layer uder hgh-speed shock-wave loadg of the lted rego of ts surface Prkl. Mekhaka Vol ( Russa). 5. Galayuk V.A. Gorechko A.N. (983) The geeral soluto of the fte tragular syste of a ordary dfferetal equatos Ukrae atheatcal oural Vol ( Russa). 6. Pao Y.-H. Mow C.-C. (973) Dfracto of elastc waves ad dyac stress cocetratos New York: Crae Russak. 7. Poruchkov V.B. (986) The dyacal elastcty theory ethods Moskow Nauka ( Russa). 8. Slyep'ya L.I. Yakovlyev Yu.S. (98) Itegral trasforatos ostatoary probles of echacs Legrad Sudostroyeye ( Russa). 9. Sego G. (959) Orthogoal polyoals Aer. Math. Soc. Colloquu Publcatos.. Wakhede P.C. Bhose B.R. (98) Elastc vbrato coposte cylders or spheres Proc. Nat. Acad. Sc. Ida Sec. A. Vol Yag J.C.S. Achebach J.D. (97) Stresses ultlayered structures uder hgh-rate pressure loads Pap. ASME WA/Ut.-4. Ackowledgets: The authors gratefully ackowledge the fudg by Natoal Scece Cetre of Polad uder the grat No. N