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1 Inernaional Journal of Geosciences hp://dx.doi.org/.46/ijg..45 Published Online May (hp:// Sress and Srain Accuulaion Due o a Long Dip-Slip Faul Moveen in an Elasic-Layer over a Viscoelasic Half Space Model of he Lihosphere-Ashenosphere Syse Sanjay Sen Subraa Kr. Debnah Deparen of Applied Maheaics Universiy of Calcua Kolkaa India Deparen of Basic Science and Huaniies Meghnad Saha Insiue of Technology (A Uni of Techno India Group) Kolkaa India Eail: dr.sanjaysen@rediff.co skd.sxccal@gail.co Received January ; revised February 4 ; acceped March Copyrigh Sanjay Sen Subraa Kr. Debnah. This is an open access aricle disribued under he Creaive Coons Aribuion License which peris unresriced use disribuion and reproducion in any ediu provided he original work is properly cied. ABSTRACT Mos of he earhquake fauls in Norh-Eas India China id Alanic-ridge he Pacific seisic bel and Japan are found o be predoinanly dip-slip in naure. In he presen paper a dip-slip faul is aken siuaed in an elasic layer over a viscoelasic half space represening he lihosphere-ashenosphere syse. A oveen of he dip-slip naure across he faul occurs when he accuulaed sress due o various econic reasons e.g. anle convecion ec. exceeds he local fricion and cohesive forces across he faul. The oveen is assued o be slipping in naure expressions for displaceens sresses and srains are obained by solving associaed boundary value proble wih he help of inegral ransforaion and Green s funcion ehod and a suiable nuerical ehods is used for copuaion. A deailed sudy of hese expressions ay give soe ideas abou he naure of sress accuulaion in he syse which in urn will be helpful in forulaing an earhquake predicion prograe. Keywords: Aseisic Period; Dip-Slip Faul; Earhquake Predicion; Manle Convecion; Plae Moveens; Sress Accuulaion; Teconic Process; Viscoelasic-Layered Model. Inroducion I is he observaional fac ha while soe fauls are srike slip (finie or infinie in lengh) in naure here are fauls (e.g. Sierra Nevada/Owens valley: Basin and Range fauls Rocky Mounains Hialayas Alanic faul of cenral Greece a seeply dipping faul wih dip 6 8 (deg)) where he surface level changes during he oion i.e. he fauls are dip-slip in naure. A pioneering work involving saic ground deforaion in elasic edia was iniiaed by []. Ref. [] did a wonderful work in analyzing he displaceen sress and srain for dip-slip oveen. Laer soe heoreical odels in his direcion have been forulaed by a nuber of auhors like [4-]. Ref. [] has discussed various aspecs of faul oveen in his book. Ref. [] has discussed sress accuulaion near buried faul in lihosphere-ashenosphere syse. The work of [] can also be enioned in hese connecions. In os of hese works he ediu were aken o be elasic and/or viscoelasic bu a layered odel wih elasic layer(s) over elasic or viscoelasic half space will be a ore realisic one for lihosphere-ashenosphere syse. In he presen case we consider a long dip-slip faul siuaed in an elasic layer over a viscoelasic half space which reaches up o he free surface. The ediu is under he influence of econic forces due o anle convecion or soe relaed phenoena. The faul is assued o undergo a slipping oveen when he sresses in he region exceed cerain hreshold values. In our paper we consider an elasic layer over a viscoelasic half space o represen he lihosphere-ashenosphere syse wih consan rigidiy (. 5 Mpa) and viscosiy ( - pa s) following he observaional daa enioned by [45]. Analyical expressions for Copyrigh SciRes.

2 55 S. SEN S. K. DEBNATH displaceens sresses and srains in he syse are obained boh before and afer he faul oveen using appropriae aheaical echnique involving inegral ransforaion and Green s funcion. Nuerical copuaional works have been carried ou wih suiable values of he odel paraeers and he naure of he sress and srain accuulaion in he ediu have been invesigaed.. Forulaion We consider a long dip-slip faul F and widh D siuaed in an elasic layer over a viscoelasic half space of linear Maxwell ype. A Caresian co-ordinae syse is used wih a suiable poin O on he srike of he faul as he origin Y axis along he srike of he faul Y axis is as shown in Figure and y axis poining downwards. We choose anoher co ordinae syse Y Y and Y axes as shown in Figure below so ha he faul is given by F : y y D. Le be he dip of he faul F and v w be he displaceen coponens along y and y axes respecively for he layer and v w be ha for he k k half-space. ij eij are sress and srain k = for he layer and k = for he half-space and i j =. Figure. Secion of he odel by he plane y =... For an Elasic Mediu he Consiuive Equaions Are Taken as For he layer: M v y v w y y w y.. For a Viscoelasic Maxwell Type Mediu he Consiuive Equaions Are Taken as For half-space: M v e y v w ( e) y y w e y (.) (.) (.) (.4) (.5) (.6) where μ μ are he effecive rigidiy of he layer and he half-space respecively and is he effecive viscosiy of he half-space... The Sresses Saisfy he Following Equaions (Assuing No Changes in he Exernal Body Force) For M: y y y y where y y H For M: y y y y where y y H (.7) (.8) (.9) (.) and assuing quasi-saic deforaion for which he ineria er are negleced. The boundary condiions are aken as wih = represening an insan when he ediu is aseisic sae. Copyrigh SciRes.

3 S. SEN S. K. DEBNATH 55 y y as y y H (.) On he free surface y = > y y (.) y y (.) y y y (.4) y y a y H( y ) Also as y y y a y H y y y say w y y w y y (.7) is he shear sress ainained by anle Le v w and e rs r s be he rs value of (v ) (w ) rs and ers a ie = H (.5) (.6) [where convecion and oher econic phenoena hroughou he ediu]..4. The Iniial Condiions Are which are funcions of y y and saisfy he relaions (.)-(.7). a) Soluions in he absence of any faul dislocaion [67]: The boundary value proble given by (.)-(.7) can be solved by aking Laplace ransforaion wih respec o ie of all he consiuive equaions and he boundary condiions. On aking he inverse Laplace ransforaion we ge he soluions for displaceen sresses as: For M: v y y v y d wy y w y H y y y y d (A) Fro he above soluion we find ha reain unchanged fro he iniial one while increases line- arly wih ie if we assue ha o be consan. We assue ha he geological condiions as well as he characerisic of he faul is such ha when reaches soe criical value say c he faul F undergoes a sudden slip along he dip direcion. The agniude of he sudden slip shall saisfy he following condiions as discussed by []. (C) Is value will be axiu on he free surface. (C) The agniude of he slip will decrease wih y as we ove downwards and uliaely ends o zero near he lower edge of he faul. y y D b) Soluions afer he faul oveens [67]: We assue ha afer a ie T he sress coponen (which is he ain driving force for he dip-slip oion of he faul) exceeds he criical value c and he faul F undergoes a sudden slip along he dip direcion characerized by a dislocaion across he faul given in (Appendix). We solve he resuling boundary value proble by odified Green s funcion ehod following [4] and correspondence principle (as shown in Appendix) and ge he soluion for displaceens sresses and srain as: For M: or v y y v y d where and F [( ) g x A B C D A y y d x y H w y y w y + y y say H y y U π g x A B ( C D) A F A y y dx s s e e a r a r r s a Copyrigh SciRes.

4 55 S. SEN S. K. DEBNATH and n i e n b n r Brn b Ar b r r a z z e z sin cos A y y B x x ycos ysin C y sin y cos D r y y x x y cos y sin y y where where y y C D A B A B C D A y H y A y H y B x x y H y sin cos [ cos sin B x x y H y y H y A y H y sin cos cos sin sin cos ( B) x x ycos H y sin y H y C ysin H ycos D x x y H y y H y C ysin H ycos D x x y H y + yh y y y y y y cos sin cos sin d y y d y y say where y y y y y (B). Nuerical Copuaions Following [8] and recen sudies on rheological behavior of crus and upper anle by [45] he values of he odel paraeers are aken as: dyne c.5 dyne c poise D = Deph of he faul = k [noing ha he deph of all ajor earhquake fauls are in beween - 5 k]. H = Thickness of he layer = 4 k. say (hough he hickness varies fro region o region of he Earh). T. 8 dyne c ( bars) [pos seisic observaions reveal ha sress released in ajor earhquake are of he order of bars in exree cases i ay be 4 bars] 7 5 dyne c 5 bars and 4 We ake he funcion g x W[ x D D wih W = c/year saisfying he condiions saed in C C. We now copue he following quaniies: For layer M: W y y w y y w y H y y (.) y y (.) d 4. Resuls and Discussions The paraeers involved in he expressions for displaceens and sresses have been assigned appropriae values available fro observed daa hrough repeaed geodeic surveys prior o and afer he seisic evens in seisically acive regions in Souh and Norh Aerica and China. Nuerical copuaions are carried ou using his observed values of he paraeers as discussed in. 4.. Variaion of Verical Coponen of Displaceen Due o Sudden Slip across he Faul afer = Year Equaion (.) gives us he verical coponen of displaceen a (y y ) due o he oveen across he faul Copyrigh SciRes.

5 S. SEN S. K. DEBNATH 55 for differen dip angle and a differen ie afer he faul oveen. We ake = year. In Figure he graph shows he naure of surface displaceen W agains y he disance fro he srike of he faul wih = 9 (in deg). I is observed ha he displaceens are in opposie direcions across he srike of he faul. Their agniudes gradually decrease and end o zero as we ove away fro he faul. This is quie expeced as he effec of he faul oveen gradually dies ou wih disance. The sudden changes of W near y = is due o he dip-slip oion along he faul. This is in good conforiy wih he resul shown in Paul Segal (). Figure shows he variaion of W wih deph y along he verical hrough a poin y = 5 k. for a verical faul wih = 9 (in deg). I shows ha W decreases sharply up o a deph of abou 8 k. and hereafer diinishes o zero a a slower rae and coninuously decreases o zero for y > k. 4.. Variaion wih Deph of he Main Driving Sress in he Dip-Slip Direcion Due o he Moveen across F Figures 4-7 show he variaion of wih deph y for various and soe specific values of y. In Figure 4 i is found ha for a dip-slip faul wih dip angle = and along he line y = 8 k undergoes a change (in one year) due o he slip oveen across F. Iniially here is a very sall region of sressrelease jus below he free surface ( < y < 4.5 k). Thereafer he rae of sress-release decreases up o a deph of k and hen sress begin o accuulae in he region fro y = k o y = k afer ha he rae of sress accuulaion gradually decreases coninuously and becoes negligibly sall a a deph of 5 k. Figure 5 shows he variaion of ain driving sress coponen wih he deph of y = 8 k = year Figure. Variaion of verical coponen of displaceen W wih y for y = k = 9 (deg) = year due o he faul oveen. Figure 4. Variaion of sress wih deph y for y = 8 k = (deg) = year due o he faul oveen. Figure. Variaion of verical coponen of displaceen W wih deph y for y = k = 9 (deg) = year due o he faul oveen. Figure 5. Variaion of sress wih deph y for y = 8 k = 45 (deg) = year due o he faul oveen. Copyrigh SciRes.

6 554 S. SEN S. K. DEBNATH due o he slip oveen across he faul a a deep angle = 45 (degree). The rae of sress release increases up o a deph of 4 k. Then he rae decreases up o a deph of abou 4 o 8 k. And hen begin o accuulae fro 8 o k. wih a axiu of bar per year a a deph of abou k. And coninuously decreases o zero a abou 75 k fro he free surface. Figure 6 shows he variaion of sress coponen for y = 9 k = year and dip angle = 6 (degree) wih he deph due o he slip across he faul. We see sress releases in he region y 4 k. wih increasing rae up o y = 6.5 k and hen wih decreasing rae and becoe a a deph abou k. For verical dip-slip faul he naure is he sae wih less nuerical values which is explained by Figure 7. For verical dip-slip faul he naure is he sae wih less nuerical values which is explained by Figure. 5. Conclusions Thus in he above discussion we see ha due o slip oveen in he dip-slip faul here are regions where sress ge released and here are cerain oher regions where sress accuulaes. The rae of sress release/accuulaion depends essenially on he dip-angle and he disance y fro he faul. If here were a second faul siuaed in he region where sress ge released due o he oveen across he firs faul he possibiliy of a oveen across he second faul would likes o be deferred furher. On he oher hand if he second faul be siuaed in he region where sress accuulaes he possibiliy of a oveen across he second faul will be enhanced. The sudy of such ineracions aong neighboring fauls is very iporan in seisically acive regions where here are a nuber of neighboring fauls exis. 6. Acknowledgeens One of he auhors (Subraa Kr. Debnah) hanks he Principal and Head of he Deparen of Basic Science and Huaniies Meghnad Saha Insiue of Technology a uni of Techno India Group (INDIA) for allowing e o pursue he research and also hanks he Geological Survey of India ISI Kolkaa for providing e he library faciliies. Deparen of Applied Maheaics Universiy of Calcua for providing he library faciliies. Figure 6. Variaion of sress wih deph y for y = 9 k = year = 6 (deg) due o faul oveen. Figure 7. Variaion of sress wih deph y for y = k = year = 9 (deg) due o faul oveen. REFERENCES [] T. Maruyaa Saical Elasic Dislocaions in an Infinie and Sei-Infinie Mediu Bull Earhquake Research Insiue Tokyo Universiy Tokyo 964. [] T. Maruyaa On Two Diensional Dislocaions in an Infinie and Sei-Infinie Mediu Bull Earhquake Research Insiue Tokyo Universiy Tokyo 966. [] J. C. Savage and L. M. Hasie Surface Deforaion Associaed wih Dip Slip Fauling 966. [4] K. Rybicki The Elasic Residual Field of a Very Long Srike Slip Faul in he Presence of a Disconinuiy Bullein of he Seisological Sociey of Aerica Vol pp [5] L. Mansinha and D. E. Syllie The Displaceen Fields of Inclined Fauls Bullein of he Seisological Sociey of Aerica Vol. 6 No pp [6] R. Sao Sress Drop of Finie Faul Journal of Physics of he Earh Vol. 97 pp [7] S. J. Singh and M. Rosenan Quasi Saic Deforaion of a Viscoelasic Half-Space by a Displaceen Dislocaion Physics of he Earh and Planeary Ineriors Vol pp [8] A. Nur and G. Mavko Pos-Seisic Viscoelasic Rebound Science Vol pp [9] R. Sao and T. Yaashia Saic Deforaions in an Ob- Copyrigh SciRes.

7 S. SEN S. K. DEBNATH 555 liquely Layered Mediu Par-II Dip-Slip Faul Journal of Physics of he Earh Vol. 975 pp. -5. [] L. B. Freund and D. M. Barne A Two-Diensional Analysis of Surface Deforaion Due o Dip-Slip Fauling Bullein of Seisological Sociey of Aerica Vol. 66 No. pp [] J. B. Rundle Viscoelasic Crusal Deforaion by Finie Quasi-Saic Sources Journal of he Geophysical Research Vol. 8 No. B 978 pp [] A. Mukhopadhyay e al. On Sress Accuulaion near Finie Recangular Faul Indian Journal of Meeorology Hydrology and Geophysics (Mausa) Vol. 979 pp [] A. Mukhopadhyay e al. On Sress Accuulaion and Faul Slip in Lihosphere Indian Journal of Meeorology Hydrology and Geophysics (Mausa) Vol. 979 pp [4] T. Iwasaki and R. Sao Srain Field in a Sei-Infinie Mediu Due o an Inclined Recangular Faul Journal of Physics of he Earh Vol pp [5] Cohen Pos Seisic Viscoelasic Surface Deforaions and Sress Theoreical Consideraions Displaceens and Srains Calculaions Journal of Geophysical Research Vol. 85 No. B6 98 pp. -5. [6] S. Rani and S. J. Singh Saic Deforaion of a Unifor Half Space Due o a Long Dip-Slip Faul Geophysical Journal Inernaional Vol pp [7] U. Ghosh A. Mukhopadhyay and S. Sen On Two Ineracing Creeping Verical Surface-Breaking Srike-Slip Fauls in a Two-Layered Model of Lihosphere Physics of he Earh and Planeary Inerior Vol pp [8] J. W. Rudnicki and M. Wu Mechanics of Dip-Slip Fauling in an Elasic Half-Space Journal of he Geophysical Research Vol. No. B 995 pp [9] Y. Ting-To J. B. Rundle and J. Fernandez Deforaion Produced by a Recangular Dipping Faul in a Viscoelasic Graviaional Layered Earh Model Par-II: Srike-Slip Faul-Sraegy and Srengh Forran Progras Copuers and Geosciences Vol. No pp [] S. J. Singh M. Punia and G. Kuari Deforaion of a Layered Half-Space Due o a Very Long Dip-Slip Faul Proceedings of Indian Naional Science Acadey Vol. 6A No. 997 pp [] J. C. Savage Displaceen Field for an Edge Dislocaion in Layered Half Space Journal of Geophysical Research Vol. No. B 998 pp [] S. K. Toar and Dhian D-Deforaion Analysis of a Half-Space Due o a Very Long Dip-Slip Faul a Finie Deph Indian Acadey Science (Earh Plane Science) Vol. No. 4 pp [] D. D. Oglesby The Dynaics of Srike-Slip Sep Overs wih Linking Dip-Slip Fauls Bullein of Seisological Sociey of Aerica Vol. 95 No. 5 5 pp [4] C. Zhang D. D. Oglesby and G. Xu Earhquake Nucleaion on Dip-Slip Fauls wih Deph-Dependen Fricional Properies Journal of Geophysical Research Vol. No. 6 Aricle ID: B7. [5] M. Masu ura and R. Sao Saic Deforaion Due o Faul Spreading over Several Layers in Muli-Layered Mediu Par-II-Srain and Til Journal of he Physics of he Earh Vol. No. 975 pp. -. [6] A. Mukhopadhyay S. Sen and B. P. Paul On Sress Accuulaion in a Viscoelasic Lihosphere Conaining a Coninuously Slipping Faul Bullein Sociey of Earhquake Technology Vol. 7 No. 98 pp. -. [7] A. Mukhopadhyay S. Sen and B. P. Paul On Sress Accuulaion near a Coninuously Slipping Faul in a Two Layered Model of Lihosphere Bullein Sociey of Earhquake Technology Vol. 7 No pp [8] M. Masu ura and R. Sao Saic Deforaion Due o Faul Spreading over Several Layers in Muli-Layered Mediu Par-II-Srain and Til Journal of he Physics of he Earh Vol. No. 975 pp. -. [9] D. A. Spence and D. L. Turcoe An Elasosaic Deforaion of a Unifor Half Space Due o a Long Dip-Slip Faul Geophysical Journal Inernaional Vol pp [] S. Sen S. Sarker and A. Mukhopadhyay A Creeping and Surface Breaking Long Srike-Slip Faul Inclined o he Verical in a Viscoelasic Half-Space Mausa Vol pp [] P. Segal Earhquake and Volcano Deforaion Princeon Universiy Press Princeon. [] U. Ghosh and S. Sen Sress Accuulaion near Buried Faul in Lihosphere-Ashenosphere Syse Inernaional Journal of Copuing Vol. No. 4 pp [] S. G. Fuis D. Scheirers E. V. Langenhei D. M. Koh- Ler A New Perspecive on he Geoery of he San Andreas Faul of Souh California and Relaionship o Lihospheric Srucure Bullein of Seisological Sociey of Aerica Vol. pp [4] P. Chif J. Lin and U. Barckiausen Marine and Peroleu Geology Vol. 9 pp [5] S.-I. Karao Rheology of he Earh s Manle: A Hisorical Review Gondwana Research Vol. 8 No. pp [6] S. Sen and S. K. Debnah A Creeping Verical Srike- Slip Faul of Finie Lengh in a Viscoelasic Half-Space Model of he Lihosphere Inernaional Journal of Copuing Vol. No. pp [7] S. Sen and S. K. Debnah Long Dip-Slip Faul in a Viscoelasic Half-Space Model of he Lihosphere Aerican Journal of Copuaional and Applied Maheaics Vol. No. 6 pp [8] K. Aki and P. G. Richards Quaniaive Seisology: Theory and Mehods W. H. Freean San Francisco 98. Copyrigh SciRes.

8 556 S. SEN S. K. DEBNATH Appendix Soluions afer he Faul Moveen We assue ha afer a ie T he sress coponen (which is he ain driving force for he dip-slip oion of he faul) exceeds he criical value c he faul F undergoes a sudden slip. Then we have an addiional condiion characerizing he dislocaion in w due o he slipping oveen as: w Ug x H () where U is he axiu dislocaion along he dip of he faul occurred a he free surface i s value gradually diinishes wih increasing direcion of x governed by he funcion g x and H is he Heaviside funcion and w = The disconinuiy of w i across F given by w = li w () y' i y ' y' D li w y ' i Taking Laplace ransforaion in () we ge U p w gy ' () where P is Laplace ransforaion variable and p w w e dp The faul-slip coences across F afer ie T. clearly w for where T F is locaed in he region (y = y D). We ry o find he soluion as: w w v v v w i i i i i i i i i i i i i i i v w rs (4) where i = and are coninuous everywhere in he odel and are given by (A). While he i v are obained by solving second par v i rs v creep so ha while w saisfies odified boundary value proble as saed below. We noe ha is coninuous even afer he faul he dislocaion condiion given by (). where i = and r s =. The resuling boundary value proble can now be saed as: v w Copyrigh SciRes. v w saisfies D Laplace equaions as w v v w w where is he Laplace ransforaion of w wih he odified boundary condiions For M: y y as y y (6) y y as y y ( y y ) as y y and he oher boundary condiions are as usual. For M: y y p as y y (9) y y p as y y y y p as y y (5) (7) (8) () () and he oher boundary condiions are as usual. We solve he above boundary value proble by odified Green s funcion ehod following [4] and he correspondence principle. Le Q(y y ) be any poin in he layer Q (y y ) be any poin in he half-space and P(x x ) be any poin in he faul hen we have For M: w Q Ug x G y y x x dx F Gy y x x G y y x x x where y y dx and Gy y x x [ A B C D where < y < H and A y sin y cos cos cos B x x y y y y C y sin y cos cos sin D x x y y y y where y y A B A B C D C D

9 S. SEN S. K. DEBNATH 557 where A y H y B x x y A y H y B x x y sin cos cos H y y H y sin cos cos H y y H y sin sin C y sin H y cos D x x y H y y H y C ysin H ycos D x x y H y y H y cos sin ( cos sin Copyrigh SciRes.