Static Deformation Due To Long Tensile Fault Embedded in an Isotropic Half-Space in Welded Contact with an Orthotropic Half-Space

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1 International Journal of cientific and Research Publications Volume Issue November IN 5-5 tatic Deformation Due To Long Tensile Fault Embedded in an Isotropic Half-pace in Welded Contact with an Orthotropic Half-pace Meenal Malik * Mahabir ingh ** and Jagdish ingh *** * Department of Applied ciences Matu Ram Institute of Engineering and Management Rohtak- ** CoE Deenbandhu Chhotu Ram University of cience and Technology Murthal onepat-9 *** Department of Mathematics Maharshi Dayanand University Rohtak-. Abstract- The Airy stress function for a long tensile fault of arbitrary dip and finite width buried in a homogeneous isotropic perfectly elastic half-space in welded contact with a homogeneous orthotropic perfectly elastic half-space are obtained. This Airy stress function is used to derive closed-form analytical expressions for the stresses at an arbitrary point of the half-space caused by along vertical tensile fault of finite width. The variation of the displacement and stress fields with distance from the fault is studied numerically. I. INTRODUCTION teketee (958 introduced the theoretical formulations describing the deformation of an isotropic homogeneous semi-infinite medium (okada 99. Maruyama (96 calculated the Green s functions for two-dimensional elastic dislocations in a semi - infinite medium and obtained surface displacements due to vertical and horizontal rectangular tensile faults in a semi-infinite Poisson solid. Freund and Barnett (976 developed a -D model of dip-slip faulting in a uniform halfspace using the theory of analytic functions of a complex variable and obtained the relationship between fault slip and surface deformation. Davis (98 modeled the crustal deformation associated with hydro fracture by a dipping rectangular tensile fault beneath the surface of an elastic halfspace. A double-couple source mechanism modeling a shear stress cannot represent all earthquake sources. ipkin (986 developed the theory that the non-double couple mechanism might be due to tensile failure under high fluid pressure. Yang and Davis (986 obtained closed analytical expressions for the displacements strains and stresses due to a rectangular inclined tensile fault in an elastic half-space. Maruyama (96 calculated the Green s functions for two-dimensional elastic dislocations in a semi - infinite medium and obtained surface displacements due to vertical and horizontal rectangular tensile faults in a semiinfinite Poisson solid. ingh and Garg (986 obtained the integral expressions for the Airy stress function in an unbounded medium due to various two- dimensional seismic sources. Beginning with these expressions Rani et al. (99 obtained the integral expressions for the Airy stress function displacements and stresses in a homogeneous isotropic perfectly elastic half-space due to various two-dimensional sources by applying the traction-free boundary conditions at the surface of the half-space. The integrals were then evaluated analytically obtaining closed-form expressions for the Airy stress function the displacements and the stresses at any point of the half-space caused by twodimensional buried sources. Wu and Chou (98 applied the generalized method of images to obtain the elastic field of an inplane line force acting in a two phase orthotropic medium. ingh (986 Garg and ingh (987 and Pan (989a studied the static deformation of a transversely isotropic multilayered half-space by surface loads. The problem of the static deformation of a transversely isotropic multilayered half-space by buried sources has been discussed by Pan (989b. tatic deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads has been investigated by Garg et al. (99. ingh et al. (99 followed a similar approach to obtain closed-form analytic expressions for the displacements and the stresses at any point of either of two homogeneous isotropic perfectly elastic half-spaces in welded contact due to twodimensional sources. Kumar et al. (5 obtained closed-form analytical expressions for the Airy tress function for a tensile line source in two-welded half-spaces which are integrated analytically to derive the Airy stress function In this paper we study the static deformation caused by various two-dimensional seismic sources located in a homogeneous isotropic perfectly elastic half-spaces lying over a homogeneous anisotropic perfectly elastic half-space with which it is in welded contact. Most anisotropic media of interest in seismology have at least approximately a horizontal plane of elastic symmetry. The most general system with one plane of elastic symmetry is the monoclinic system. A material having three perpendicular planes of elastic symmetry at a point is said to possess orthotropic or orthorhombic symmetry. This symmetry is exhibited by olivine and orthopyroxenes the principal rock-forming minerals of the deep crust and upper mantle. Therefore we assume that the lower half-space is orthotropic. In an orthotropic material there are nine elastic constants. The results for a tetragonal material with six constants. The results for a tetragonal material with six elastic constants for a transversely isotropic material with five elastic constants and for a cubic material with three elastic constants can be derived as particular cases. We have verified that the results of ingh et al.

2 International Journal of cientific and Research Publications Volume Issue November IN 5-5 (99 for two isotropic half-spaces in welded contact follow from the results of the present paper when the lower orthotropic half-space is replaced by an isotropic half-space II. THEORY Let the Cartesian co-ordinates be denoted by ( x y z ( x x x with z-axis vertically upwards. Consider two homogeneous perfectly elastic half-spaces which are welded along the plane z. The upper half-space ( z is called medium I and the lower half-space ( z is called medium II. Medium I is assumed to be isotropic with stress-strain relation pij ijekk eij ( Medium II is assumed to be orthotropic with stress-strain relation p c c c e p c c c e p c c c e. p c e p c55 e p c66 e ( We consider a two-dimensional approximation in which the ( u displacement components u u are independent of x so that x. Under this assumption the plane-strain problem ( u ( u and the antiplane-strain problem u are decoupled and therefore can be solved separately. The planestrain problem for an isotropic medium can be solved in terms of the Airy stress function U such that U U U p p p = z yz y U. ( ( Figure : A two-phase medium consisting of an isotropic half space lying over an orthotropic half-space with a line source in the isotropic half-space at ( h The plane-strain problem for an orthotropic medium can be solved in terms of the Airy stress function U * such that (Garg et al. 99 * * * U U U p p p = z yz y (5 * a b U y z y z (6 where / a b c c c c c c c a b c / c. (7 For an isotropic medium c c c c c c c c55 c66. (8 This yields a b and eq. (6 reduces to eq. (. Let there be a line source parallel to the x ( -axis passing through the point h of the upper half-space ( z. As shown by U ingh and Garg (986 the Airy stress function for a line source parallel to x ( h -axis passing through the point in an unbounded isotropic medium with Lame constants can be expressed in the form k z h U [ L Mk z h sin ky P Q k z h cos ky] k e d k (9

3 International Journal of cientific and Research Publications Volume Issue November IN 5-5 L where the source coefficients M P Q are independent of k. ingh and Garg (986 have obtained source coefficients for various seismic sources. For a line source parallel to the x -axis acting at the point ( h of medium I ( z which is in welded contact with medium II ( z the Airy stress function in medium I is a solution of Eq. ( and may be taken to be of the form ( kz U U L M kz sin ky P Q kz cos ky k e d k. ( The Airy stress function in medium II is a solution of Eq. (6 and is of the form (assuming a b ( akz bkz akz bkz U L e M e sin kyp e Q e cos ky k d k. ( The superscript ( denotes quantities related to medium I and the superscript ( denotes the quantities related to medium II. L The constants M L M etc. are to be determined from the boundary conditions. ince the half-spaces are assumed to be in welded contact along the plane z the boundary conditions are ( ( ( ( p p p p ( ( ( ( u u u u ( ( at z.the stresses and the displacements for the isotropic medium I in terms of the Airy stress function U are given by (Rani et al. 99 where ( ( ( ( U ( U ( U p p p z yz y ( ( U ( ( y u ( p p d y ( ( U ( ( u p p z ( ( dz ( ( / (. The stresses and the displacements for the orthotropic medium II are given by (Garg et al. 99 where ( ( ( ( U U U p p p = z yz y (6 ( ( ( u ( c p c p d y ( ( ( u ( c p c p dz (7 c c c (5. (8 Using expressions of stresses given by (ingh and Rani 99 and Using source coefficients from the appendix for a vertical tensile fault we obtain the following expressions of Airy stress function and stresses in both medium the expressions of stresses at any point of a two-phase medium consisting of a homogeneous isotropic perfectly elastic half-space in welded contact with a homogeneous orthotropic perfectly elastic half-space having long tensile line source located in the isotropic half-space are given below:

4 International Journal of cientific and Research Publications Volume Issue November IN 5-5 bds X z z h z h U R X D C ( R ( ( ( log log ( log X hz( z h [ X hx z ( zh X hz] (9 ( X X ( X D C ( X X ( zh ( R R ( bds ( zh 8 h( X X h( z h h( X X X ( z h 8 z( X X ( z h ( z h (5 X D C 96 X hz( z h 96 X hz( z h ( bds 6 y( zh 8 y( z h 6 ( ( y[( X X ( z h ( C D( z h X z] R R 8 y[( X X z ( X X h]( z h 8 X hyz( z h 96 X hyz( z h ( ( bds [( D C X] y [ X ( D C] hz(x X + ( R [ Xh ( X X z]( z h 8 y ( z h[( X X z Xh] 96 X hy z( z h ( U bds A C T B D H h A( haz B( hbz ( ( ln ( ln ( T H ( ( ( ( a y b y bds a A C b B D ( A C ( D B ( T H T H a A( h az b B( h bz a A( h az b B( h bz hy 6 6 h 8 T H T H ( bds ( A C( h az ( D B( h bz aa bb ( ay by hy ( +8hy T H T H bds ( A C ( B D ( A C ( B D ( T H T H ( y where and aa( h az bb( h bz 6 6 T H (5 A( h az B( h bz A( h az B( h bz h 8 hy 6 6 T H T H (6 R y ( z h T y ( h az y ( z h H y ( h bz (7

5 International Journal of cientific and Research Publications Volume Issue November 5 IN 5-5 X ( A B X A( a B( b X D( b c( a A ( b s r / W B ( a r s / W C ( b b s / W D ( a a s / W W ( a r ( b b s ( a a s ( b r. (8 y z Y Z We define the following dimensionless quantities h h where L is the finite width of the tensile fault. Let the ( ( Pij P ij dimensionless stresses be denoted by. Then ( h ( ( h ( ij ij Pij ij bds bds P (9 From equations ( - ( and ( - (6 we get the following expressions for the dimensionless stresses for the two mediums for a vertical tensile line source: ( ( Z ( X X ( X D C ( X X( Z P ( E E F F F 8( X X( Z 6 6 6( X X X ( Z 8 Z( X X ( Z F F F DC ( Z (5 X 96 X Z( Z 96 X Z ( Z ( X X + F F F ( Y Z DC 6 Z Z ( XX (5XD C ( X Z ( X X ( Y Z ( Y Z ( Y Z ( 6 Y( Z 8 Y( Z Y X X Z C D Z X P 6 ( ( Z Z YZ YZ [( ( ( ( Z] E E F 8 Y[( X ( X X ]( Z 8 X ( Z 96 X ( Z + + F F F C D X YZ[ ( X X ( ] 8YZ ( X X ( Y Z ( Y Z ( ( [( D C X ] X Y ( Z Y ( C D Z( Z ( X X ( E E F F F ( + P Z 6 8 Z(X X 8 Y ( Z [( X X Z X ] 96 X Y ( Z [ ( D C X ] + + F F F ( Y Z ( D C Z X X 8Y Z X X Y [ X ] ( ( ( Y Z ( Y Z ( Y Z (

6 International Journal of cientific and Research Publications Volume Issue November 6 IN 5-5 ( ( ( a A C b B D ( A C ( D B P a Y b Y ( G J G J P ( ( ( ( a ( A C Y 6 6 a A az b B bz a A az b B bz 8 G J G J ( Y a Z b ( B D a Y ( A C b Y ( B D + ( Y b Z ( Y a Z ( Y b Z ( ay ( az( A C Yb( bz( D B aay bby ( G J G J ( 8 aay ( az 8 bby ( bz a YZ ( A C b YZ ( D B G J ( Y a Z ( Y b Z ( ( A C ( B D ( A C ( B D ( Y P ( G J G J A( az B( bz A( az B( bz ( A C Y G J G J ( Y a Z ( B D Y ( A C Y ( B D ( Y b Z ( Y a Z ( Y a Z (5 III. DICUION AND CONCLUION We have derived the results when an isotropic half-space (medium I lies over an orthotropic half-space (medium II. The results when medium II is tetragonal can be obtained on putting c c c c c55 c (6 The results when medium II is transversely isotropic follow by taking c c c c c55 c c66 ( c c /. (7 imilarly the results when medium II is cubic are obtained on taking c c c c c c c c55 c66 (8 When medium II is isotropic c c c c c c = c c55 c66. (9 This is a degenerate case for which a b (see Eq. (7. However we have verified that when medium II is replaced by an isotropic medium the results of the present paper in the limit coincide with the results of ingh et. al.(99 for two isotropic halfspaces in welded contact. For numerical calculations we assume that medium II is transversely isotropic and use the values of the elastic constants given by Anderson (96. For beryl c / c. c / c.6 c / c.7 c / c.. This yield a.78 b.676. For ice c / c.7 c / c.96 c / c.7 c / c.6. ( (

7 International Journal of cientific and Research Publications Volume Issue November 7 IN 5-5 and a.89 b.5. For the isotropic medium I we assume that c. We further assume that /. When medium II is also isotropic we take for numerical work. Figure shows that in the model considered here medium I is isotropic and medium II is orthotropic. Numerical calculations of stress components P P and P is done and their figures are drawn by considering the medium II as beryl ice or isotropic. Figures (a - (c (a - (c and (a - (c show the variation of stress components P P and P with distance from the fault. Figures (a (b and (c show the variation of the dimensionless stress component P with distance from the fault when source is at depth x = h/ x = h and x = 5h respectively. These figures depict the comparison of stress component when medium II is beryl or ice and comparison is also made in case medium II is isotropic. The figures show that when x is negative then near the fault P is negative(compressive stress in case of beryl and positive (tensile stress in case of ice and when x is positive then the value of P is more in case of beryl. As we move away from the fault i.e. when x tends to infinity P tends to zero in all the cases. Figures (a (b and (c depict the behavior of stress component P depending on the distance from the fault when source is located at the depth x = h/ x = h and x = 5h respectively. All these figures show that the value of P is negative near the fault but as we move away it becomes positive. Again when x is negative then the value of P is less for beryl than ice but when x is positive the trend get reversed. imilar behavior is observed for the stress component P. Figures (a (b and (c depict the behavior of stress component P depending on the distance from the fault when same is located at the depth x = h/ x = h and x = 5h respectively. Here also the situation is similar to that for P and P in case medium II is beryl ice or isotropicthe comparison is also made when medium II is beryl ice or isotropic. These figures show that the value of stress components do not differ so much for beryl and ice but the difference in values is significant when both media are isotropic. 5 P (beryl P (ice P (iso Figure (a: Variation of dimensionless normal stress P with distance from the fault at x = - h/.

8 International Journal of cientific and Research Publications Volume Issue November 8 IN P (beryl P (ice P (iso Figure (b: Variation of dimensionless normal stress P with distance from the fault at x =h P (beryl P (ice P (iso Figure (c: Variation of dimensionless normal stress P with distance from the fault at x = 5h.

9 International Journal of cientific and Research Publications Volume Issue November 9 IN P (beryl P (ice P (iso Figure (a: Variation of dimensionless shear stress P with distance from the fault at x = -h/ P (beryl P (ice P (iso Figure (b: Variation of dimensionless shear stress P with distance from the fault at x = h.

10 International Journal of cientific and Research Publications Volume Issue November IN P (beryl P (ice P (iso Figure (c: Variation of dimensionless shear stress P with distance from the fault at x = 5h P (beryl P (ice P (iso Figure (a: Variation of dimensionless shear stress P with distance from the fault at x = -h/.

11 International Journal of cientific and Research Publications Volume Issue November IN P (beryl P (ice P (iso Figure (b: Variation of dimensionless shear stress P with distance from the fault at x = h P (beryl P (ice P (iso Figure (c: Variation of dimensionless shear stress P with distance from the fault at x = 5h. APPENDIX. Vertical dip-slip fault d = ( bs L P Q M. Vertical tensile fault

12 International Journal of cientific and Research Publications Volume Issue November IN 5-5 bs d L M P = Q = (. Horizontal tensile fault bs d L M P = Q = ( x The upper sing is for > h x the lower sign is for < h b is the magnitude of the displacement dislocation and ds is the width of the line fault. REFERENCE [] Bonafede M. & Danesi Near-field modifications of stress induced by dyke injection at shallow depth Geophys. J. Int [] Bonafede M. & Rivalta E The tensile dislocation problem in a layered elastic medium Geophys. J. Int [] Davis P. M. 98. urface deformation associated with a dipping hydrofracture Geophys. J. Res [] Dundurs J. & Hetenyi M Transmission of force between two semiinfinite solids AME Journal of Applied Mechanics [5] Garg N. R. ingh. J. & Manchanda. 99. tatic deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads Proc. Ind. Acad. ci. (Earth Planet. ci [6] Heaton T. H. & Heaton R. E tatic deformation from point forces and point force coupls located in welded elastic Poissonian half-spaces: implications for seismic moment tensors Bull. eism. oc. Am [7] Kumar A. ingh. J. & ingh J. 5. Deformation of two welded halfspaces due to a long inclined tensile fault J. Earth yst. ci [8] Kumari G. ingh. J. & ingh K. 99. tatic deformation of two welded half-spaces caused by a point dislocation source Phys. Earth. Planet. Inter [9] Maruyama T. 96. tatical elastic dislocations in an infinite and semiinfinite medium Bull. Earthquake Res. Inst [] Pan E. 989a. tatic response of a transversely isotropic and layered halfspace to general surface loads Phys. Earth Planet. Inter [] Pan E. 989b. tatic response of a transversely isotropic and layered halfspace to general surface loads Phys. Earth Planet. Inter [] Rani. & ingh. J. 99. tatic deformation of two welded half-spaces due to dip-slip faulting Proc. Indian Acad. ci.(earth Planet. ci [] Rani. ingh. J. & Garg N. R. 99 Displacements and stresses at any point of a uniform half-space due to two-dimensional buried sources Phys. Earth Planet. Inter [] Rongved L Force interior to one of the two joined semi-infinite solids in Proc. Of the nd Midwestern Conf on olid Mech.ed Bogdanoff J L Purdue University Indiana Res. er.9 -. [5] ingh M. & ingh. J.. tatic deformation of a uniform half-space due to a very long tensile fault IET J. Earthquake Techn [6] ingh. J. & Garg N. R On two-dimensional elastic dislocations in a multilayered half-space Phy. Earth Planet. Int [7] ingh. J. & Rani. 99. tatic deformation due to two-dimensional seismic sources embedded in an isotropic half-space in welded contact with an orthotropic half space J. Phys. Earth [8] ingh. J. Rani. & Garg N. R. 99. Displacements and stresses in two welded half-spaces caused by two dimensional sources Phys. Earth Planet. Int [9] ingh. J. & Garg N. R. 986 On the representation of two-dimensional seismic sources Acta Geophy. Pol. -. [] ingh. J tatic deformation of a transversely isotropic multilayered half-space by general surface loads Phys. Earth Planet. Inter [] ipkin. A Interpretation of non-double- couple earthquake mechanisms derived from moment tensor inversions J. Geophys. Res [] Wu R.. & Chou Y.T. 98. Line force in a two-phase orthotropic medium AME. Appl. Mech [] Yang X. M. & Davis P. M Deformation due to a rectangular tensile crack in an elastic half-space Bull. eism. oc. Am AUTHOR First Author Meenal Malik Ph.D Department of Applied ciences Matu Ram Institute of Engineering and Management Rohtak- -id-meenaljalaj@gmail.com econd Author Mahabir ingh Ph.D CoE Deenbandhu Chhotu Ram University of cience and Technology Murthal onepat-9 -id-msdhankhar@gmail.com Third Author Jagdish ingh Ph.D Department of Mathematics Maharshi Dayanand University Rohtak-. Correspondence Author Meenal Malik Department of Applied ciences Matu Ram Institute of Engineering and Management Rohtak- -id-meenaljalaj@gmail.com

Shear Stresses and Displacement for Strike-slip Dislocation in an Orthotropic Elastic Half-space with Rigid Surface

International Journal of Applied Science-Research and Review (IJAS) www.ijas.org.uk Original Article Shear Stresses and Displacement for Strike-slip Dislocation in an Orthotropic Elastic Half-space with