Finite Element Simulation on Frictional and Brittle Preseismic fault slip

Transcription

1 Finite Element Simultion on Fritionl nd Brittle Preseismi fult slip Zhishen Wu (1) Yun Go (1) Yutk Murkmi (2) (1) Deprtment of Urn & Civil Engineering. Irki University, Jpn (e-mil: phone exe , Fx: ). (2) Geophysis Deprtment, Geologil Survey, Jpn (e-mil: Astrt In this pper, finite element model for simulting rustl deformtion inluding disontinuous slipping displement long fult is developed, where slip wekening ehvior with simple sher stress-reltive displement reltionship on fult surfe sed on the onept of nonliner frture mehnis is tken into onsidertion. hroughout numeril simultionson fult-end folding with rmp, the ontours nd developments of the sher stress, the seond stress invrint nd the slip veloity vrying with the vlues of frture energy re investigted. Moreover, the rekdown proess is disussed. 1 Introdution It is known tht mny shllow erthqukes our on plte oundries nd tive geologil fults, whih re evidently pre-existing wekness in the shllow rittle prt of the Erth. So fr, numerous experimentl studies on fritionl sliding of pre-existing fults in roks hve een rried out for understnding mehnis of erthquke fulting. From the initil rtionl knowledge, there re two si lws of sliding frition: the fritionl resistne is proportionl to the norml lod nd it is independent of the pprent re of the sliding surfes. Lter, extensive nd quntittive experimentl investigtions of frition hve een performed y Amontons nd Coulom who hve formulted the dry frition lws in the form whih is still widely used nd tught nowdys. In the fritionl lws, the stress drop etween stti frition nd dynmi frition is ssumed to e onstnt. So the stress nd slip veloity exhiit the singulrity t tht time. Although the fritionl lw is prtilly useful for estimting verge soure prmeters of erthqukes, the singulrity is physilly unresonle. o eliminte the stress singulrity t the rk tip, Id, Plmer nd Rie developed slip-wekening model. hen Andrews theoretilly disussed the ritil rk length for unstle rupture for 2D sher rks. Sine the slip veloity nd the slip elertion due to the ohesive fore hve finite vlues ner the rk tip s theoretilly shown y Id, Ohnk nd Ymshit investigted the ehvior of slip veloity nd slip elertion more in detil. Also, the slip-wekening ehvior hs een experimentlly exmined y mny reserhers from whih it is found tht qusi-stle sliding ours on lolized region of fult prior to unstle slip, the length of the lolized region of qusi-stle sliding orresponds to the ritil rk length for unstle rupture. But in generl, present FEM method,

2 this effetive model isn t onsidered. So in this pper, the finite slip-wekening model is introdued to finite element method using Lgrnge desription to void stress singulrity. 2 Bsi theory In this present study, struturl system of fult-end folding in whih the referene onfigurtion of ody exhiiting slipping long fult surfe so tht the whole struturl system is hrterized y two onstitutive reltions. One is volumetri onstitutive lw tht reltes stress nd strin for ody, while the other is ohesive-softening nd fritionl surfe onstitute reltion etween the sher stress nd reltive displement jumps for the fult. Using Lgrnge desription, n ttention is onfined to qusi-stti deformtions nd, with ody fores, oundry fores nd trtion on internl fult surfe onsidered, the inrementl formultion of priniple of virtul work is written s { δε } { s + s} dv = { δ u } { f + f } ds V + + S V S { δ u } { f + f } { δ u } { r + r } he ody is sujeted to ody fore field {r } in V, presried externl trtion {f } on fore oundry S nd internl trtion {f } used y reltive displement jump on disontinuous surfe S. Here {s} is Lgrnge stress vetor; { ε } is Lgrnge strin vetor. τ dv ds (1) τp G frture energy rked zone rekdown zone loked zone G d ritil slip displement rk tip τr d RELAIVE DISPLACEMEN Fig.1 hree zones of different ontt sttes etween sliding surfes on the fult nd reltion etween sher stress nd slip ross fult o get the lne of the internl trtions on the two sides of the fult surfe, mster-slve method is used. For the onstitute reltion on the fult surfe, the slip-wekening model shown in Fig.1 is dopted. he totl fult surfe of roks n e onsist of three zones: loked zone where sliding surfes re strongly interloked, rekdown zone where ll the interloked sperities re wekening until frtured nd reked zone whih is ehind the rekdown zone. he ohesive fore etween inner frture surfes is ssumed to e ontinuously deresing funtion of reltive displement ross the rk in order to eliminte the stress singulrity t the rk tip. he reltive displement ross the fult during the rekdown proess is lled the ritil slip displement d. he shded re for the stress-slip reltion is regrded s the energy required for reting new frture surfes of unit re or the work done y the ohesive fore. his energy hs een often lled the frture energy G, representing the rupture growth resistne. Generlly, in the loked zone, the inrementl formultion of the fult trtion n e otined

3 through the inrementl formultion of reltive displement { ( u)} nd stiffness mtrix [K ] on the fult surfe f = K u (2) { } [ ]{ ( )} τ = f / lt σ = f lt (3) t n / where τ nd σ re the sher nd norml stress on the fult surfe; l is the length of interfe element nd t is the thikness. When the sher stress τ rehes to the vlue of pek stress τ p whih is equl to µ s σ, the wekening of the sher stress is eginning: µ s is the stti frition oeffiient. hen the frture energy egins to relese. hus, the inrementl trtion used y reltive displement is dded to the totl inrementl trtion w f t = K t ( ( u t ) u ) (4) where u w is the reltive inrementl wekening displement whih is the funtion of G, d, the tngent stiffness K t nd the reltive displement d from the eginning of initil wekening G lt w u 2 = 1 + d (5) 2 K t d As the frture energy releses ompletely, the residul stress eomes to dynmi fritionl stress whih is ssumed to oey the rte- nd stte-dependent frition lw proposed y Ruin τ r = µ d σ µ = µ + θ + ln V / V (6) d dθ V = dt L ( ) [ θ + ln( V / V )] where µ d is the dynmi frition oeffiient, V is sliding veloity of n element nd V * is referene veloity given ritrrily. he onstnts µ,, nd L hrterize the fritionl property. Generlly, dθ / dt = mens stedy-stte. * * 3 Numeril simultion Y p p U x Ux= Ux=Uy= node 1 node 2 node 3 Uy= Fig.2 Initil, undeformed grid for the finite-element model, showing oundry onditions X Fig.2 shows finite element mesh of struturl model with fult-end folds, whose responses in long time hve een investigted very well. In this pper, we fous on the effet of the response in short time nd wekening mteril ehviors on the fult surfe. Eh struturl model ontins 48 6-node isoprmetri, qudrti tringle elements. Plne strin is ssumed. he initil fult geometry onsists of 2m long nd 5m high rmp onneting lower nd upper flts. A surfe pressure of 75 MP is pplied to the top of the hnging wll, whih simultes 3km overurden. here is zero sher stress long this top surfe of the model. A zero displement oundry

4 ondition, U x =U y =, is used long the left (hinterlnd) side of the hnging wll nd the footwll, U y = long the se of the model nd U x = long the right (forelnd) side of the footwll. A displement of 5m per 5 yer time step is imposed on the left side of the hnging wll, the veloity (1 m y -1 ) tht is onsistent with estimtes of nturl thrust sheet motion ut the time step n utomti vry with different periods nd different onditions. he shded elements from left to right re nmed s element 1,2 nd 3. Moreover, the elow-right orner-node of element 1 is nmed s node 1 nd y the sme method, node 2,3 re nmed. d Fig.3 Contours of (J 2 ) 1/2 ()in initil time () in wekening of node 1 () in wekening of node 2 (d) in wekening of node 3 Fig.3 shows the ontours of (J 2 ) 1/2 in different period s µ s =.25, µ d =.2, K t = P nd G =1 1 5 Jm -2, where it is 1MP per lyer of right r. It is found tht t every time, the higher vlues umulte in the right side of the forelnds of the hnging wll nd footwll. Collting the initil ontour, t the time when the node 1 is wekening, in the hnging wll, the re of low vlue enlrges from the hinterlnd to the forelnd; nd lso, in the footwll, the re with higher J 2 in the hinterlnd enlrges. When the node 2 is wekening, ollting Fig.3, in the hnging wll, the re of low vlue enlrges to the forelnd ut redues from the hinterlnd; nd the re with higher J 2 in the hinterlnd enlrges ontinuously. Contrsting with Fig.3, when the node 3 is wekening, in the hnging wll, the re of low vlue lso enlrges to the forelnd nd lso redues from the hinterlnd; nd the re with higher J 2 in the hinterlnd lso enlrges ontinuously. Furthermore, y the totl trend, the nodes on the fult surfe re yielding from left to right, ut in some lol res speilly, some right nodes re yielding even erlier thn their ner left nodes. By following two individul prtiles s shown in Fig.2, the stress pths in J spe, sher stress nd reltive veloity n e trked. From Fig.4, node 1 egins to e wekening in the 27th yer nd node 3 egins in the 67th yer. In the totl period, though sher stresses hs een wekened, the vlue of the (J 2 ) 1/2 tremles very smll. Moreover, they eome lower until some time nd from this time, they egin to rise. For the reltive veloity, the unstedy phenomenon n e oserved. From Fig.4, the vlue of the veloity is smller of m/s, nd osionlly, the mgnitudes exeed m/s t some time. For most of the totl period, it is stedy slip. But t some time when there re some jumps or drops produed y the wekening of the sher stress, the unstedy slip ours. From Fig.4d nd Fig.4e, the iggest mgnitude s G =1 1 3 Jm -2 exeeds m/s nd the iggest mgnitude s G = Jm -2 exeeds m/s. Bse on the sme reson, the unstedy slips emerge, nd with the derese of the G, the unstedy slips inrese. his unstedy phenomenon is s sme s the results from rok experiments y whih the slow erthquke nd silent erthquke n e

5 simulted. From these experiments, when the stiffness of test mhine is K, the dependene of fritionl trtion f nd slip displement u is desried with df/du, if df/du <K, the slip is stedy. On the ontrry, the slip is unstedy. For the se of Fig.4, the vlue of the df/du is lwys smller thn K in most of the period, so its unstedy slips re very few. On the other hnd, for the se of Fig.4d nd Fig.4e, the order tht the vlue of the df/du is lrger thn K in most of the period inreses, so the unstedy slip inreses. 1 Fig.4 ()sher stress () (J 2 ) 1/2 ()reltive veloity s G =1 1 5 Jm -2 (d)reltive veloity s G =1 1 3 Jm -2 (e)reltive veloity s G= with time for element 1 & 3 or node 1 & 3 d1 e1 2 d2 e2 Fig.5 () sher stress of node 3 ()reltive veloity of node 3 s G =1 1 4 Jm -2 nd G =1 1 5 Jm -2

6 Fig.5 show the wekening proesses of node 3 with different vlues of G. If the G is lrger, the wekening time is longer nd the mgnitude of the reltive veloity is smller s shown in Fig.5. Furthermore, the time eginning to weken is sme nerly. When the simultions re ompred with different µ d while the other mteril prmeters re the sme, the se with smller µ d will e weken erlier with shorter wekening period s shown in Fig.6. Even though its mgnitude of reltive veloity is lrger, the mgnitude order is sme. When the simultions re ompred with different K t, it is found tht the se with lrger K t will e weken erlier with shorter wekening period ut its mgnitude order of the mgnitude of reltive veloity is lrger s shown in Fig.7. 4 Conlusions Fig.6 () sher stress of node 3 ()reltive veloity of node 3 s µ d =.1,.2 nd µ s =.25 Fig.7 () sher stress of node 3 ()reltive veloity of node 3 s K t = MP nd K t = MP From the ove numeril simultions, it n e onluded tht: (1) here re unstedy slips s G =, nd Jm -2, ut with the derese of the G, the unstedy slips inrese. (2) When the wekening of the sher stress ppers, the seond invrint vries hrdly, however, the vriety of the reltive veloity is huge. (3) When the K t inreses, or the µ d dereses, or the G dereses, remrkly, the mgnitude of the reltive veloity inreses oviously. Moreover, the strt time of the wekening proess is lso effeted y the mteril prmeters K t nd µ d. Referene 1) Erikson & Jmison: Visous-plsti finite-element models of fult-end folds, Jour. Stru. Geol, Vol 17, No.4, , ) Zhishen Wu, Yun Go & Yutk Murkmi: A finite element model for rustl deformtion with lrge slipping on fult surfe, Interntionl workshop on solid erth simultion nd ACES WG meeting, 2 3) Yun Go, Zhishen Wu & Yutk Murkmi: Visous-plsti nlysis of rustl deformtion of fult-end folds, Jour. Appl. Meh., Vol.3, ,2