A Slipping and Buried Strike-Slip Fault in a Multi-Layered Elastic Model

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1 Geosciences 7, 7(): DOI:.59/j.geo.77. A Sipping nd Buried Strike-Sip Fut in Muti-Lyered Estic Mode Asish Krmkr,*, Snjy Sen Udirmpur Pisree Sikshytn (H.S.), Udirmpur, P.O. Knyngr, Pin, Indi Deprtment of Appied Mthemtics, University of Ccutt, Ccutt, Indi Abstrct Modeing of erthquke processes is one of the min concern in the theoretic seismoogy. In most of the theoretic modes, which incorporte the min fetures of ithosphere-sthenosphere system in seismicy ctive regions, the medium is tken to be either singe or two yered hf-spce, estic or viscoestic. But the ithosphere-sthenosphere system hs mny inhomogeneties with respect to their estic properties. In view of this we consider medium which consist of two homogeneous estic yers overying n estic hf-spce. A buried, vertic, ong, strike-sip fut is considered in the second yer. The yers nd the hf-spce re ssumed to be in weded contct. The soutions for strins nd stresses, re obtined for the first yer nd second yer using suitbe mthemtic techniques such s Green s functions, Correspondence principe. Numeric ccutions hs been done by MATLAB. Keywords Strike-sip fut, Green s functions, Estic yers, Lithosphere-sthenosphere system. Introduction Erthquke occurs in cycic order. From regur observtions it is known tht two mjor seismic events re usuy seprted by comprtivey ong seismic period. In this seismic period observtions show sow surfce movements, indicting sow seismic chnge of stress nd strin in the vicinity of the fut. In this pper we deveoped theoretic mode of ithosphere-sthenospohere system represented by three yered estic hf-spce. Such theoretic modes ws considered by Sto [], Rybicki [, ], Mukhopdhyy [4, 5], Mukherjee [6], Sen nd Debnth, [7], Debnth nd Sen [8-], Debnth nd Sen [-], Mond nd Sen [4].. Formution We consider theoretic mode of ithosphere-sthenosphere system consisting of two estic yers nd estic hf-spce. The yers nd hf-spce re ssumed to be in weded contct. The depth of the boundries of two yers from free surfce is tken s nd second yer nd hf-spce s. We consider buried vertic strike-sip fut situted in the second yer nd the ength of the fut is very rge compre to its width. The depth of the upper edge of the fut beow the boundry of * Corresponding uthor: krmkr_sish@yhoo.in (Asish Krmkr) Pubished onine t Copyright 7 Scientific & Acdemic Pubishing. A Rights Reserved two yers is r nd upper nd ower edges of the fut re horizont. We introduce rectngur Crtesin co-ordinte system (y, y, y ) with the pne free surfce s the pne y =, y -xis is tken verticy downwrds in the medium, y -xis is tken ong the strike of the fut on the free surfce. The boundries between two yers nd second yers nd hf-spce re given by y =, y = respectivey. For convenience of nysis we introduce nother set of Crtesin co-ordinte system (y, y, y ) with the upper edge of the fut is tken s y -xis nd the pne of the fut is tken s the pne y =, so tht the fut is given by F: (y =, y ). The retions between two co-ordinte system re given by y = y, y = y, y = y r () The first estic yer, the second estic yer nd estic hf-spce re represented by y, y nd y respectivey. The Figure shows the section of the theoretic mode by the pne y =. It is ssumed tht the ength of the futs re rge compred to their depths. So the dispcements, stresses nd strins re independent of y nd depend ony on y, y. Then the components of dispcement, stress nd strin cn be divided into two groups, one ssocited with strike-sip movement nd nother ssocited with dip-sip movement of the fut. Since in this mode the strike-sip movement of the fut is considered, then the dispcement, stress nd strin components ssocited with ong strike-sip fut for two yers nd hf-spce re u, (τ, τ ), (e, e ); u, (τ, τ ), (e, e ) nd u, (τ, τ ), (e, e ) respectivey.

2 Geosciences 7, 7(): O O F Estic yer Estic yer y r y τ τ y y = (6) for the second estic yer y, y < τ τ y y = (7) for the estic hf-spce (y, y < ). From eqution ()-(7) we get u = for y, y < (8) u = for y, y < (9) u = for (y, y < ) () y, y Figure. Section of the mode by the pne y =.. Constitutive Equtions (Stress-Strin Retions) For the first estic yer, the stress-strin retion cn be written in the foowing form: u τ = μ y for y u τ = μ, y < () y where μ is the rigidity of the first estic yer. For the second estic yer, the stress-strin retion cn be written in the foowing form: u τ = μ y u for y, y < () τ = μ y where μ is the rigidity of the second estic yer. For hf-spce, the stress-strin retion cn be written in the foowing form: u τ = μ y u for y, y < (4) τ = μ y where μ re rigidity of the hf-spce. The rigidities μ, μ, μ of the estic yers nd hf-spce re ssumed to be constnt... Stress Eqution of Motion Estic f spce For sow, seismic, qusi-sttic deformtion the mgnitude of the inerti terms re very sm compred to the other terms in the stress eqution of motion nd they cn be negected. Hence reevnt stress stisfy the retions: τ y τ y = (5) for the first estic yer y, y <.. Boundry Conditions We ssume tht the upper surfce of the first estic yer is stress-free nd the two yers nd the second yer nd hf-spce re ssumed to be in weded contct. Then the boundry conditions re given beow τ = t y = τ = τ t y = u = u t y = τ = τ t y = () u = u t y = τ s y for y <.4. Initi Conditions We ssume tht the time t is mesured from suitbe instnt when the mode is in seismic stte nd there is no seismic disturbnce in it. (u ), (u ),.,(e ) re the vues of u, u,, e t time t = nd they stisfy the retions stted bove..5. Conditions t Infinity At rge distnce from the fut pne there is sher strin which my chnges with time mintined by the tectonic forces. Then e e g t for ( < y < ) () e e g t for ( < y < ) () e e g t for < y < (4) im where e = y (e ), (e ) = im y (e ), (e ) = im y (e ), where (e ), (e ), (e ) re the vues of e, e, e t time t =. In the medium of ithosphere-sthenosphere the yers nd hf-spce re in weded contct, sme g(t) hs been tken for ech yer nd hf-spce, since strins re continuous t the boundries of yers. From the mjor erthqukes it hs been observed tht the stresses reese my be of the order of 4 brs. Keeping this in view, if we tke g(t) to be inery incresing function of time t with g() =. If we tke g(t) = kt, then the vue of k shoud be of the order of 4.

3 7 Asish Krmkr et.: A Sipping nd Buried Strike-Sip Fut in Muti-Lyered Estic Mode. Dispcements, Stresses nd Strins in the Absence of Fut Movement To obtin the soution for dispcements, stresses nd strins in the bsence of fut movement we sove the boundry vue probem ()-(4) nd get the soution in the foowing form: for the first estic yer u = u y g t τ = τ μ g t τ = τ e = (e ) g(t) (5) u = u y g t = τ μ g t (6) τ = (τ ) τ for the second estic yer for the hf-spce. u = u y g t τ = τ μ g t (7) τ = (τ ) 4. Dispcements, Stresses nd Strins fter the Restortion of Aseismic Stte Foowing Sudden Strike-Sip Movement cross the Fut It is to be noted tht due to sudden fut movement cross the fut F, the ccumuted stress wi be reesed to some extent nd the fut becomes ocked gin when the sher stress ner the fut hs sufficienty been reesed. The disturbnce generted due to this sudden sip cross the fut F wi grduy die out within short spn of time. During this short period, the inerti terms cn not be negected, so tht our bsic equtions re no onger vid. We eve out this short spn of time from our considertion nd consider the mode fresh from suitbe instnt when the seismic stte re-estbished in the mode. We determine the dispcements, stresses nd strins fter the fut movement with respect to new time origin t =. So tht the equtions ()-(4) re so vid. The sudden movement cross F is chrcterized by the discontinuity of u cross F is defined s u = Uf y (8) cross F (y =, y, t ) im where u = y u im y u nd f(y ) is continuous function of y nd U is constnt, independent of y nd y. A the other components u, u, τ,, e re continuous everywhere in the medium. We try to obtin the dispcements stresses for t (with respect to new time origin) due to the movement cross F in the foowing form: u = u u τ = τ τ τ = τ τ e = e e u = u u = τ τ τ τ = τ τ u = u u = τ τ τ τ (9) = τ τ where (u ), (τ ),, (τ ) stisfy the equtions ()-(4) nd continuous throughout the medium. Whie (u ), (τ ),, (τ ) stisfy the retions ()-() nd so the disoction condition (8) together with e e e s y, t () Then the soutions for (u ), (τ ),, (τ ) re given by u = u p y g t τ = τ p μ g t τ = τ p e = e p g t () u = u p y g t = τ p μ g t () τ = τ p τ u = u p y g t τ = τ p μ g t () τ = τ p where (u ) p, (τ ) p,, (τ ) p re the vues of (u ), (τ ),, (τ ) respectivey t t = (i.e. new time origin). Now to sove the boundry vue probem contining (u ), (τ ),, (τ ) we use Green s function technique deveoped by Mruym [5] nd Rybicki [, ] s expined in Appendix. The required soutions re obtined s γ U u = u p y g t ψ π(γ ) (y, y ) τ = τ p μ g t γ μ U π(γ ) ψ (y, y ) τ = τ p γ μ U π(γ ) ψ (y, y ) γ U e = (e ) p g t ψ π(γ ) (y, y ) for the first estic yer (4)

4 Geosciences 7, 7(): u = u p y g t U π φ (y, y ) τ = τ p μ g t μ U φ π (y, y ) τ = (τ ) p μ U φ π (y, y ) for the second estic yer u = u p y g t U χ π(γ ) (y, y ) τ = τ p μ g t μ U χ π(γ ) (y, y ) τ = (τ ) p μ U χ π((γ ) (y, y ) (5) (6) for the hf-spce. where γ = μ, γ μ = μ μ nd nytic form of ψ, ψ, ψ ; φ, φ, φ ; χ, χ, χ re given in Appendix. It is found tht the dispcements, stresses nd strins wi be finite nd singe vued ny where in the mode if the foowing conditions re stisfied (i) f(y ) nd f (y ) re both continuous functions of y for y. (ii) f() =, f() = nd f = f = (iii) Either f (y ) is continuous in y or f (y ) is continuous in y except for finite number of points of finite discontinuity in y or f (y ) is continuous in < y < except possiby for finite number of points of finite discontinuity nd for the end points of (, ), there exist re constnt m, n both < such tht y m f (y ) or finite imit s y nd ( y ) n f (y ) or to finite imit s y. the observed rte of strin ccumution in seismicy ctive regions during the seismic period is of the order of 6 to 8 per yer. The vues of mode prmeters re tken from the book of Aki [6], Cthes [7], Buen nd Bot [8] nd reserch ppers by Cift [9], Krto []. We now compute the foowing quntities (i) E = The residu/ ddition surfce sher strin due to fut sip ner the fut fter restortion of seismic stte = [e (e ) p g(t)] y = = γ U π(γ ) ψ (y, y ) y = (ii) T = Chnge in sher stress in the first yer due to fut movement. = τ (τ ) p μ g(t) = γ μ U π γ ψ y, y y =4 km. (ii) T = Chnge in sher stress in the second yer due to fut movement. = τ (τ ) p μ g(t) = μ U π φ (y, y ) y =46 km. The chnge in surfce sher strin with y ner fut fter restortion of seismic stte is shown in Figure. The figure shows tht the fut movement eds to reese of the surfce sher strin nd the effect is symmetric bout the fut trce. The mgnitude of this sher strin reese is mximum ner fut trce nd then fs off rpidy s we move wy from the fut nd become very sm for rge vue of y. 5. Numeric Computtions To study the surfce dispcements, stresses nd strin ccumution/ reese nd the sher stress ner fut tending to cuse strike-sip movement s we choose f(y ) = y (y ) 4. = km. is the width of the fut F. = 4 km., = km. from free surfce, representing the upper prt of the ithosphere nd upper prt of the srhenosphere. r = 5 km. We tke μ =.6 dynes/cm, μ =.75 dynes/cm, μ =.4 dynes/cm. U = 4 cm, is the sip cross F nd for the function. It is ssumed tht due to some tectonic reson there is sow but stedy ccumution of sher strin t distnce fr wy from the fut. Keeping this in view we tke g(t) to be inery incresing with time nd g() =. With this ssumption, we tke g(t) = kt. From mjor erthqukes it hs been observed tht the stress reese my be of the order of 4 brs. So we ssume k =. 4, noting so tht Figure. Surfce sher strin due to fut movement The Figure nd 4 shows the contour mp in the first nd second yer respectivey due to fut movement cross the fut F fter restortion of seismic stte.

5 7 Asish Krmkr et.: A Sipping nd Buried Strike-Sip Fut in Muti-Lyered Estic Mode Appendix Soutions of dispcement, stress nd strin in seismic stte fter sudden movement cross the fut: The dispcements, stresses nd strins for t with new time origin fter restortion of seismic stte foowed by sudden movement hve been found in the form given by (9) where u, τ,, τ re given by ()-() nd u, τ,, τ stisfy ()-(), (8) nd ().This boundry vue probem invoving u, τ,, τ cn be soved by using modified Green s function technique deveoped by Mruym [5] nd Rybicki [] nd correspondence principe. According to them we get, Figure. Contour mp of sher stress in the first yer u Q = u Q = F G u P Q, P dx } F G u P Q, P dx } {G Q, P dx {G Q, P dx (A) (A) u Q = u P {G Q, P dx F G Q, P dx } (A) where Q (y, y, y ), Q (y, y, y ), Q (y, y, y ) re the fied points in the first yer, second yer nd hf-spce respectivey nd P(x, x, x ) is ny point on the fut F nd [(u ) (P)] is the mgnitude of discontinuity of u cross the fut F. According to Rybicki [], the vues of G () (Q, P), G () (Q, P), G () (Q, P), G () (Q, P), G () (Q, P), G () (Q, P) re given beow Figure 4. Contour mp of sher stress in the second yer 6. Concusions It is observed tht the movement cross the fut system significnty effect the nture of stress ccumution in the region. The rte of ccumution of stress in the system fter the fut movement my give us some ide bout the time to the next mjor event. Such resuts my be used for the purpose of prediction of erthqukes. ACKNOWLEDGEMENTS One of the uthors Asish Krmkr thnks the Hed Mster of Udirmpur Pisree Sikshytn (H.S.) for owing me to pursue the reserch, nd so thnks the Geoogic Survey of Indi; Deprtment of Appied Mthemtics, University of Ccutt for providing the ibrry fciities. Q, P = A λ e λy B λ e λy G G sin λ x y dλ Q, P = C λ e λy D λ e λy G cos λ x y dλ Q, P = A λ e λy B λ e λy G sin λ x y dλ x y π (x y ) (x y ) Q, P = C λ e λy D λ e λy G cos λ x y dλ π x y x y x y Q, P = A λ e λy B λ e λy sin λ x y dλ Q, P = [C (λ)e λy D (λ)e λy ] G where cos λ x y dλ (A4) (A5) (A6) (A7) (A8) (A9)

6 Geosciences 7, 7(): A = B = γ γ πδ e λ x γ e λ x ] A = πδ γ γ e λ x γ γ e λ 4 x γ γ e λ 4 x γ γ e λ x B = γ πδ γ e λ 4x e λx γ e λ x e λ x A = πδ γ e λ 4 x e λ x γ eλ x e λ x C = D = γ πδ γ e λ x C = πδ γ e λ x ] γ γ e λ x γ γ e λ 4 x γ γ e λ 4 x γ γ e λ x D = (γ ) πδ γ e λ 4x e λx γ e λ x e λ x nd C = πδ γ e λ 4 x e λ x γ eλ x e λ x Δ = γ e λ γ γ e λ γ e λ γ γ e λ nd γ = μ μ, γ = μ μ. Now A λ e λy B λ e λy = γ πδ γ e λ x γ γ e λ 4 x γ γ e λ 4 x γ γ e λ x e λy γ γ e λ 4x e λx πδ γ e λ x e λ e λy x First prt of (A) (A) (A) (A) = π γ γ Δ e λ x y γ γ e λ 4 x y Δ γ γ e λ 4 x y Δ Now second prt of (A) π Now Where γ γ Δ e λ x y γ (γ ) Δ e λ 4 x y e λ x y γ (γ ) Δ e λ x y e λ x y γ γ = e λ Δ M γ γ = c e λ Δ M γ γ = c e λ Δ M γ γ = e λ ( ) Δ M M = e λ c e λ( ) c e λ where = γ γ nd c = γ γ. Therefore using the resut of (A), we get πm A λ e λy B λ e λy = (A) eλ x y c e λ x y e λ x y c e λ x y e λ x y e λ x y c e λ x y c e λ x y (A6) Now the term e λ c e λ( ) c e λ < (Mond nd Sen [4]) nd we cn express M s n infinite geometric series nd negecting the higher order term nd we get from (A6) A λ e λy B λ e λy = π e λ x y c e λ x y e λ x y c e λ x y e λ x y e λ x y c e λ x y c e λ x y e λ c e λ( ) c e λ

7 74 Asish Krmkr et.: A Sipping nd Buried Strike-Sip Fut in Muti-Lyered Estic Mode Now we ssume d = x y, d = x y, d = y x in the bove expressions nd putting this vue in (A6) nd fter integrtion we get G () = d π d d d d d d d d c d d d c d 4 d d c d d d c d d d d d d c d d d c d 4 d d c d 4 d d d d 4 d d 4 d d c d d d d d d c d d 4 d c d 4 d d d 4 d d c d 4 4 d d c d d 4 d c d d d c d d d c d 4 d d c d 4 4 d d c d 4 d d c d 4 d d c d d d c d d d c d d d c d d d d d d Using simir process we obtin tht G c nd c = γ π γ d d d d d d d d d d d d d d d d d d d d 4 d d 4 d d d d d d d d d 4 d d c d d d d d d d d d d 4 d d d d d (A7) (A8) G () = c c π γ d d d d d d d d d d d d d d d d d d c d 4 d d d d d c d d d d d d d d d d d d d d d d d 4 d (A9) Let P(x, x, x ) is point on the fut F with respect to the origin O nd (ξ, ξ, ξ ) is ny point on F with respect to the origin O nd chnge of co-ordinte system from P(x, x, x ) to (ξ, ξ, ξ ) is connected by the foowing retions x = ξ, x = ξ, x = ξ r. Then on the fut F, ξ = nd ξ, dx = dξ =, dx = dξ. The discontinuity in u is [(u ) (P)] = Uf(ξ ). Then from (A), (A), (A) we get (u ) Q = U f ξ G u Q = U f ξ G u Q = U f ξ G (Q, P)dξ Q, P dξ Q, P dξ (A) (A) (A) In chnge co-ordinte system d = ξ r y,d = y, d = y ξ r. Putting it in (A8), (A7), (A9) nd we get the new form of G (), G (), G (). Using these new form of G (), G (), G () in (A), (A), (A), we get the foowing resuts nd u Q = τ Q = μ γ U π(γ ) ψ (y, y ) u y = γ μ U π(γ ) ψ y, y τ Q = μ u y = γ μ U π(γ ) ψ y, y u Q = U π φ (y, y ) τ Q = μ u y Q = μ U π φ (y, y ) τ Q = μ u y Q = μ U φ π (y, y ) (A) (A4) (A5) (A6) (A7) (A8)

8 Geosciences 7, 7(): where ψ y, y = f(ξ ) nd φ y, y = f(ξ ) nd χ y, y = u Q = U π(γ ) χ (y, y ) τ Q = μ u y Q = μ U χ π(γ ) (y, y ) τ Q = μ u y Q = μ U π(γ ) χ (y, y ) y B 4 y B 7 y B y B 9 y y y y B B 5 B 9 B 8 c y c y c y c y B B B B 5 c y B c y B 6 c y B c y B dξ ψ y, y = ψ y y, y ψ y, y = ψ y y, y y c y c y y B B B B 4 c y B 5 y B 7 y B 8 y B 9 c y c y y c y B B B B c y y y c y B 4 B 5 B 6 B c y c y c y c y B 7 B 8 B B 9 c y c y c y c y B 5 B B B c y c y c y c y B B B 4 B 5 f ξ c y B 6 c y B 8 y B 9 φ y, y = φ y y, y φ y, y = φ y y, y c y B y B 9 y B c y B c y B y B 4 y c y B 9 B c y B 4 c y B 4 c y B c y c y c y B B 5 B χ y, y = χ y y, y (A9) (A) (A) (A) (A) (A4) dξ (A5) (A6) (A7) dξ (A8) (A9) χ y, y = χ y y, y Where B = ξ r y y, B = y ξ r y, B = ξ r y y, B 4 = y ξ r y, B 5 = y ξ r y, B 7 = ξ r y y, B 8 = ξ r y y, B 9 = ξ r y y, B = y ξ r 4 y, B = ξ r y y, B = y ξ r 4 y, B = 4 y ξ r y, B 4 = 4 ξ r y y, B 5 = 4 ξ r y y, B 6 = 4 ξ r y y, B 7 = y ξ r 4 5 y, B 8 = ξ r y y, B 9 = 4 y ξ r y, B = 4 ξ r y y, B = ξ r y y, B = ξ r y y, B = ξ r y y, B 4 = ξ r y y, B 5 = y ξ r y, B 6 = ξ r y y, B 8 = ξ r y y, B 9 = ξ r y y, B = y ξ r y, B = y ξ r y, B = y ξ r y, B = y ξ r y, B 4 = y ξ r y, B 5 = y ξ r y, (A4) (A4)

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