Proceedings of the 9th UK Conference on Boundary Integral Methods, University of Aberdeen, UK, 8-9th July 2013

Transcription

1 Proeedgs of he 9h U Coferee o Bodar Iegra Mehods Uvers of Aberdee U 8-9h J 03 -D ELASTODYNAMIC PROBLEM FOR AN INTERFACE CRAC UNDER AN OBLIQUE HARMONIC LOADING V. MIUCA ad O. MENSHYOV Cere for Mro- ad Naoehas CEMINACS Shoo of Egeerg Uvers of Aberdee Aberdee AB4 3UE U e-as: r0v9@abd.a.; o.eshov@abd.a. Absra. The rre sd s devoed o soo of he wo desoa easoda probe of a raed heerogeeos aera oaded b haro wave. Ths vesgao aes o ao he refeo ad refrao of obe waves a a erfae bewee wo dssar sod haf-spaes whh oas a ra. The probe s soved b he ehod of bodar egra eaos sg a erave agorh. The da sress es faors are gve he wde rage of wave bers for dffere properes of he baera wh he fos o he effe of he ras osre.. INTRODUCTION Srra appaos of opose aeras have reased de o her eee sffess ad ow wegh ogeher wh reded eerg ospo. However oe of he a weaesses of oposes s he presee of varos defes ha or he rego of fber or ar bodg. The growh of ras reases sress he v of he ra p as a oseee gves he rse o sffess degradaos ad overa oad bearg apa of he srre s reded. Cosderabe deberao s pad o he fare aass as a eree pora oo o prove aeras reab ad rede he os whh aso heps preve dsasers asg b preded frare ad o bd p ofdee safe sses. A good owedge of he da respose of he oposes s essea o ahevg deph dersadg of he fare ehas of he baeras. A ber of sdes have orbed o he da aass of heerogeeos aeras vovg erfae ras. Babae ad Laseewsz [] vesgaed he da respose of a erfae ra bewee wo dssar haf-spaes der da oadg b sg da egra eaos whh a epoea fo for he varao of aera properes was sed. Io [] sded he da sress es faors arod a ra a erfaa aer bewee wo dssar eas haf-paes ad he aera properes of he erfaa aer were assed o be o hoogeeos. Ma e a [3] dsssed he da behavor of wo oear ras opose aera der a-pae de haro sress waves. The effes of ra erao wave veo of aeras ad free of de waves o da sress es faors were vesgaed. Jag ad Wag osdered he da ra propagao a erfaa aer wh spaa varg eas properes [4]. The probe of a wodesoa erfae ra was soved b Q [5] for he ase of haro oadg. The effe of he age of he pae wave dee was sded for a rage of freees of he oad. Meshova e a osdered he ase of he shear wave propagag ora o he srfae of a ear erfae ra egeg he fro bewee oppose ra faes [6]. The soo o he wo-desoa probe wh erfae ra der ora eso-opresso oadg aog for he effe of he ra osre was obaed b Meshova e a [7]. The

2 deaed desrpo of he erave agorh for he probe soo was gve ad he sd of he agorh overgee was perfored [8]. I hs paper he da ra aass of wo desoa baera wh ra o he erfae s preseed. For hs prpose he bodar eee ehod was sed ad he haro wave propagao soos for baeras are derved.. METHODOLOGY Le s osder a erfae ra oaed wh wo-desoa baera sbjeed o eera haro oadg. I hs ase der vesgao s sed a boded wo dssar * ear eas hoogeeos sorop sods. The erfae bewee he haf-spaes as as he bodar for he pper haf-spae ad he bodar for he ower haf-spae. The paes ad dffer b he oppose oreao of her oer ora veors. I s * r assed ha srfae = osss of fe par ad fe par. The bodg erfae ad he srfae of he ra are respeve: * r r r. I order o referee he geoera posos of he aera pos a Caresa oordae sse s sed wh org oaed he ere of he ra. If here s o fore apped o he bod he sress-sra sae of boh doas w be defed b he da eaos of he ear eas for he dspaee veor sg Lae eaos: λ + μ grad dv + μ Δ = ρ Ω = [0 where ad ad are Lae eas osas s he spef aera des ad s he Lapae operaor =. The ope apdes of he dspaee veor a be represeed ers of saar ad veor poeas sasfg he hoogeeos Hehoz eao: = gradφ + roψ dvψ = 0. 3 The de eso-opresso wave of e ω e depede propagag he obe dreo a be defed b he saar poea fo as gve beow: Φ 0 = A 0 e p ω 4 = osθ 0 sθ 0 ad represes he poso of veor. Φ 0 = A 0 e p sθ 0 3 osθ 0 e ω Ψ 0 = 0. 5 The grop of refeed ad rased waves assoaed wh a gve haro wave s show Fg.. As a res of her erferee he opoes of sresses a be epressed as he s of for waves wo propagag obe dowwards ad he oher wo pwards.

3 Fg. Refrao ad rassso a a erfae whe wave s de a a obe age Φ = A 0 e p sθ 0 3 osθ 0 e ω + A e p sθ + 3 osθ e ω Ψ = B e s s γ + 3 os γ e ω Φ = A e p sθ 3 osθ e ω 6 Ψ = B e s s γ 3 os γ e ω. where A ad B are apdes of ogda ad shear waves respeve; ω = π/t s he free T s he perod of vbrao; p = ω/p ad s = ω/s are he geerazed wave ber; p = λ + μ /ρ s = μ /ρ are he veoes of he ogda ad shear wave; λ μ are he eas Lae osas; ρ ad ρ are he deses of aeras. The erao of he de wave ress wo refeed waves a ogda ad a shear wave. Sar here are wo rased waves ogda ad shear. The age of refeo of he ogda wave s he sae as he age of dee b a oher ages deped o he aeras properes aordg o Se s Law whh a be fod a ber of boos [9 0] s θ 0 p = s θ p = s γ s = s θ p = s γ s 7 where eah θ θ γ γ are he refeed ad rased wave ages. There are wo dspaees whh s be oos: he dspaee ora o he erfae ad he dspaee parae o he erfae. Sar here are wo sress opoes whh s be baaed: he sress ora o he erfae ad he shear sress. The he bodar odos of o a he erfae bewee wo sod haf-spaes are [9]: = = 3 = 3 ; σ 3 = σ 3 σ 3 = σ 3 σ 33 = σ 33 8

4 The opoes of he dspaees are defed as: = Φ Ψ 3 = 0 3 = Φ + Ψ 3 9 Copoes of he sress a be epressed b he ers of poeas: = μ Φ 3 σ 3 + μ [ Ψ Ψ ] 3 σ 3 = 0 0 = λ + μ Φ σ λ Φ + μ Ψ ; 3 The approprae dfferea eaos are sbsed o sses 9 ad 0. The he se of bodar odos 8 eads o se of for eaos for he apdes A A B ad A as show -4: A 0 p osθ = A p osθ + B s s γ + A p osθ B s s γ A 0 p sθ = A p sθ + B s os γ + A p sθ + B s os γ A 0 p sθ osθ μ = A p sθ osθ μ B s os γ s γ μ + A p sθ osθ μ + B s os γ s γ μ 3 A 0 p λ + μ os θ = A p λ + μ os θ B s os γ s γ μ + A p λ + μ os θ B s os γ s γ μ 4 As a res he obaed sse of eaos a defe he sresses a he erfae geeraed fro haro eso - opresso wave. To de he oa erao of he oppose ra s faes o osderao he Sgor aera osras s be posed for he ora opoes of he oa fore ad he dspaee veors [ ] 0 0 [ ] 0 [0; T] 5

5 where ] [ s he dspaee dso veor; ad s he oa fore ha arses he oa rego whh s ow beforehad hages e der deforao of he aera ad s be deered as a par of soo. The oa rego aso depeds o he free agde ad dreo of he eera oadg opag he probe eve ore ad ag hgh o-ear. The osras 5 esre ha here s o erpeerao of he oppose ra faes; he ora opoe of he oa fore s aera ad s abse for a o-zero opeg of he ra. Noe ha de o he oa erao he rao veor a he ra faes ~ p s he sperposo of he a rao ased b he de wave g ad he oa fore. I order o ae he oa erao of he oppose ra faes o osderao we asse ha he agea opoes of he dspaee dsoes veor ad veor of oa fores sasf he Coob fro aw: 0 ] [ 6 ]. [0; ] [ ] [ T 7 The oppose ra faes rea ovabe wh respe o eah oher agea dreo whe he are hed b he fro fore. However as soo as he agde of he agea oa fores reahes a era depedg o he ra fro oeffe ad he ora oa fores he ra faes beg o ove ad he sppg effe ors. 3. BOUNDARY INTEGRAL EQUATIONS Wh he prpose of sove he a bodar vae probe wh aera osras he Sogaa da de s sed. The opoes of he dspaee fed he pper ad ower haf-spaes for he raed bod a be wre he foowg for: d W d U p j j 8 where p ad are opoes of ope-vaed agdes of raos ad dspaees a he erfae s he po of observao s he po of oadg ad he egra ere U s he Gree fdaea esor whh has he for r r U j j 9 Here s he roeer dea; r s he dsae bewee he observao po ad he oad po. I he wo-desoa ase fos ad he free-doa are: 0. 0

6 s he odfed Besse fo of he seod d ad order ; r / / r. The veoes of he ogda wave ad he rasverse shear wave are / ad The dfferea operaor / respeve. [ ] [ ] [ ] P [ ] s sed o oba he egra ere W. The operaor s apped o he Gree fdaea soo U ad afer he dffereao he egra ere obas he for: W U j U U j The ope of he probe aog for he effe of ra osre s opoded b he fa ha he area of oa vares e; he sze ad he for of he oa area s ow beforehad ad s be deere as a par of he soo. As he res he proess s o oger a haro proess b a sead-sae perod oe. Moreover opoes of he sress-sra sae ao be represeed a fo of oordaes ped b a epoea fo. Hee he opoes of he dspaee ad rao veors shod be epaded o he Forer seres as foowg: f 0os f f os os f s s where 3 T T f os f os d f s s d Se he erpeerao of he oa srfaes s prohbed he aera osras are posed ad he bodar odos beoe oear he probe reres a erave soo proedre. The soo of he easoda probe for he raed bod egeg he effe of he ra osre s obaed drg he frs par of he agorh. I he seod par he orreo of he soo s perfored wh respe o he foowg aera osras wh fro of he ra faes. The agorh s apped o he probe of he obe dee of a haro wave he baera wh a erfae ra. Drg he erave proess he Forer oeffes are hagg he dsrbo of he veors of dspaees ad oa fores sasfg he osras 5 7 s fod. 4. NUMERICAL RESULTS As a era eape a ear erfae ra wh he egh R s osdered he prese sd. The aeras of he pper ad ower haf-spaes have he pa properes of see ad a: E 07 GPa E 70 GPa; ; 7800g/ g/ 3. ad

7 Fg. The sress es faors as fo of he wave ber a for he ase α = 30 a he opeg ode ad b he rasverse shear ode a he eadg ra p Fg. 3 The sress es faors as fo of he wave ber a for he ase α = 30 a he opeg ode ad b he rasverse shear ode a he rag ra p I Fgs. ab-3ab he a e vaes of sress es faors SIF for opeg ad he rasverse shear odes are gve for boh ra ps as fos of he wave ber s a for he age of he wave dee α = 30. I a ases he vaes of he SIFs ed o rease frs ad aheve a a ad he derease. There s a sgfa dfferee bewee he vaes of he sress es faors obaed for he ase of aog for he effe of he ra osre ad egeg hs effe. The sress es faors evaaed for he ase of aog for he ra osre aheve her a vae a he ower freees of oadg ha hose obaed for he ase of egeg he ra osre. The obaed dsrbo of sress es faors he v of he rag ra p dffer fro he vaes he v of he eadg ra p de o o-ser of soo wh respe he spae ad e varabes.

8 5. CONCLUSIONS The soo of he wo-desoa probe of a raed heerogeeos sod sbjeed o obe dee of he eso-opresso wave was soved aog for he effe of he ra osre. The sse of bodar egra eaos whh s sed for era soo of he osdered probe was obaed. The da sress es faors opeg ad rasverse shear odes are oped as fos of wave ber ad opared wh hose obaed egeg he ra s osre. The oparso bewee he ress obaed a he rag ad eadg ps of he ra are preseed. Aowedgees The ahors wod e o aowedge ajor faa sppor reeved fro he Coege of he Phsa Sees of he Uvers of Aberdee ad he Egeerg ad Phsa Sees Researh Co. REFERENCES. R. Babae S.A. Laseewsz 998 Da respose of a ra a foa graded aera bewee wo dssar haf-paes der a-pae shear pa oad Eg. Fra. Meh S. Io 00 Trase da sress es faors arod a ra a o hoogeeos erfaa aer bewee wo dssar eas haf-paes I. J. Sods Sr L. Ma L.Z. W ad L.C. Go 00 Da behavor of wo oeara-pae shear ras a foa graded aer boded o dssar haf-paes Meh. Res. Co L.Y. Jag X.D. Wag 00 O he da ra propagao a er phase wh spaa varg eas properes der pae oadg. I.J.Fra J. Q 994 Ierfae ra oaded b a e-haro pae wave I. J. Sods Sr M.V. Meshova O.V. Meshov ad I.A. Gz 009 Lear erfae ra der pae shear wave. CMES: Cop. Mode. Eg. S M.V. Meshova O.V. Meshov ad I.A. Gz 00 Modeg ra osre for a erfae ra der haro oadg. I. J. Fra M.V. Meshova O.V. Meshov ad I.A. Gz 0 A erave BEM for he da aass of erfae ra oa probes Eg. Aa. Bod. Ee V.T. Grheo V.V. Meesho 98 Haro Waves Eas Bodes Da ev Rssa. 0. J.A. Hdso 980 The Eao ad Propagao of Eas Waves Cabrdge Uvers Press Cabrdge.