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Fracture Interface Waves JOURNAL OF GEOPHYSICAL RESEARCH, VOL. tot, NO. Bt, PAGES 827-835, JANUARY t0, 1996 Fracture interface waves BoliangGu • Departmentof Materials Scienceand Mineral Engineering,University of California, Berkeley Kurt T. Nihei and Larry R. Myer Earth SciencesDivision, Lawrence Berkeley Laboratory, Berkeley, California Laura J. Pyrak-Nolte Departmentof Civil Engineeringand GeologicalSciences, University of Notre Dame, Notre Dame, Indiana Abstract. Interface waves on a single fracture in an elastic solid are investigated theoreticallyand numericallyusing plane wave analysisand a boundaryelement method. The finite mechanicalstiffness of a fracture is modeled as a displacementdiscontinuity. Analysis for inhomogeneousplane wave propagationalong a fracture yields two dispersive equationsfor symmetricand antisymmetricinterface waves. The basic form of these equationsare similar to the classicRayleigh equationfor a surfacewave on a half-space, except that the displacementsand velocities of the symmetricand antisymmetricfracture interface waves are each controlledby a normalizedfracture stiffness. For low values of the normalized fracture stiffness,the symmetric and antisymmetricinterface waves degenerate to the classicRayleigh wave on a traction-freesurface. For large values of the normalized fracture stiffness,the antisymmetricand symmetricinterface waves becomea body S wave and a body P wave, respectively,which propagateparallel to the fracture. For intermediate values of the normalizedfracture stiffness,both interface waves are dispersive. Numerical modelingperformed using a boundaryelement methoddemonstrates that a line source generatesa P-type interfacewave, in addition to the two Rayleigh-typeinterface waves. The magnitudeof the normalizedfracture stiffnessis observedto control the velocities of the interface waves and the partitioning of seismicenergy among the various waves near the fracture. Introduction fracture were observedto be sensitiveto the magnitude of the At the microscale, a fracture in rock appears as two mechanical stiffness of the fracture or, equivalently, to the surfacesof irregular topographywhich contactto form void static stressimposed on the fracture. spacesand asperitiesof contact. A fracture with a sparse Murty [1975] examined the condition for the existence of an interface wave on a nonwelded interface with a populationof asperitiesis morecompliant than a fracturewith closely spacedasperities [Greenwood and Williamson,1966; discontinuityin shear displacementacross the thicknessof the Gangi, 1978; Brown and Scholz, 1985; Hopkins et al., 1987; interface. Pyrak-Nolte and Cook [1987] extended Murty's Cook, 1992]. It has been observed that the additional analysis to the case where both normal and shear compliance of a fracture results in a variety of interesting displacementsare discontinuousacross a nonweldedinterface. elastic wave phenomena. For example, laboratory They found that a single nonwelded interface, such as transmission measurementsof normally incident P and S fracture, can support a fast (symmetric) wave and a slow waves acrossa fractureexhibit low-passfiltering of the source (antisymmetric)wave which propagatewith phase velocities 'waveletaccompanied by a small travel time delay [ Pyrak- between the Rayleigh and shear wave velocities and are Nolte et al., 1990]. In addition, laboratory measurementsof dispersive. Recent laboratory measurementsby Pyrak-Nolte elastic wave propagationparallel to a fracture have revealed et al. [1992] have confirmed the existenceof the fast and slow that a fracture can supportdispersive interface waves [ Pyrak- interface waves in a synthetic fracture in aluminum. Very Nolte et al., 1992]. In both setsof experiments,the amplitude good agreement was found between the measured and and velocity of the waves propagating across or along a predicted group velocities of the fast and slow interface waves. In the field of nondestructive evaluation, the l Now at Earth SciencesDivision, Lawrence Berkeley Laboratory, displacement discontinuity model has been used to Berkeley,California. approximatethin elastic adhesivebonds [e.g., Rokhlin, 1984; Xuand Datta, 1990]. In these models, shear displacements. Copyright1996 by theAmerican Geophysical Union. acrossthe thin bond are discontinuousby an amount that is inverselyproportional to the shearstiffness of the bond while Papernumber 95JB02846. normal displacements are continuous, as in the case of 0148-0227/96/95JB-02846505.00 Murty's [1975] model. 827 828 GU ET AL.' FRACTURE INTERFACE WAVES medium 1 1.02 •( kz)a fracture X 0.98 medium 2 0.96 Figure 1. Problem geometry used in the derivation of OO•"0.94 2.45 interface wave equations. 1.87 antisymmetric 0.92 1.63 interface wave 1.50 This paperinvestigates interface waves on a planar fracture symmetric in an elastic solid using plane wave and numerical analysis. interface wave The fracture is modeled as a displacementdiscontinuity 0.88 10-3 10-2 lfi 1 10ø 101 102 boundarycondition in both shear and normal displacements. ix and [z Closed-form equations for fracture interface waves are developed. The characteristicsof the dispersionand particle Figure 3. Normalized phasevelocities of the symmetricand motions of trapped and leaky interface waves are examined. antisymmetricinterface waves as a function of the normalized The partitioning of seismic energy among waves near the fracture stiffnessfor a rangeof C•,/Cs ratios. The numbersof fracture is also evaluated. 1.50, 1.63, 1.87, and 2.45 labeledon the curvesare the C•,/Cs ratios, which are equivalent to Poisson'sratios of 0.1, 0.2, 0.3 Plane Wave Analysis and0.4, correspondingly. Symbol(•z)cf is thecut-off normalized fracture stiffness above which the symmetric The seismic behavior of a thin fracture compared to the interface wave ceases to exist in the normal mode. wavelength can be well described by the displacement discontinuity model. Across such a discontinuity, seismic stresses are continuous and particle displacements are discontinuousby an amount which is determinedby the ratio on the fracturesurfaces, kx and kz arethe x andz components• of the stresson the fracture surface to the fracture specific of the fracturespecific stiffness, and subscripts 1 and 2 referto stiffness[ Kendall and Tabor, 1971; Schoenberg, 1980; Rokhlin the mediaabove and below the fracture,respectively. and Wang, 1991; Cook, 1992]. This boundary condition The generalized potentials for an inhomogeneousplane degenerates to that for a welded fracture as the fracture wave propagatingin the x direction with exponentially specific stiffnessapproaches infinity and, for two traction-free decayingamplitude in the z directioncan be expressedas surfaces,as the fracture specific stiffnessreaches zero. The displacement discontinuity model was found to accurately •l=Ale-PøøZeiøø(x/C-t)' z->O (2) predictthe frequency-dependenttransmission of a plane ,wave •2= A2ePøøZeiøø(x/C-t),z_<O normally incidentupon a fracture [ Schoenberg,1980; Myer e t for P wave motion and a/., 1985] and the existence of interface waves along a I//1= Ble-qøøZeiøø(x/C-t), z -> 0 fracture [Pyrak-Nolte and Cook, 1987]. (3) Consider a fracture located in the x-y plane, as shown in ¾t2= B2eqøø zeiOO(x/C-t), z -<0 Figure 1. The displacementdiscontinuity boundary conditions for in-plane motion (P and SV waves) are for SVwave motion, where i=-•/•-•, w is the angular frequency, t is time, A1, A2, B1, and B2 are undetermined Ul- u2= •'xzl/kx, constants,C is the phase velocity of the inhomogeneous Wl- w2 = •'zzl/kz, (1) wave, and p and q are defined as •'zx l = •'zx 2 , •'zzl = •'zz2, where u and w are the x and z componentsof the particle P='c2 c2' (4) displacement,•:• and •:zz are the shearand normalstresses q=I C12 C}' ' where C.• and Cs are the P and S wave velocities, respectively. Using the potentials in (2) and (3) and the Helmholtz symmetricinterfacewave theorem, u- &- &, (5) w + the particle displacementsare derived with the form Ul=toi-•Ale-PøøZ+qBle-q øøzeiOO(x/C-t), [' (6) Figure 2. Schematic of symmetric and antisymmetric interface waves. Wl= tO I-PAle-PtøZ+i•Ble-q 1 tøz]eitO(x/C_t) ' GU ET AL.: FRACTURE INTERFACE WAVES 829 1.6 respectto the fracture;and when B2= B1 and A2 =-A1, the wave field is antisymmetricwith respectto the fracture. The C•,/Cs leakysymmetric ' 1.4 interfacewave •' symmetricand antisymmetricwave fields are schematically shownin Figure 2. SubstitutingB2 = -B1 and A2 = A1 into 1.2 normalsymmetric / (9) resultsin two linearly independentequations, 1.0 interfacewv• CAl+ +q2Bl=O ' \ antisymmetric 0.8 (10) interface wave i i i i 10-a 10-2 10-1 10ø 10• 102 C2sC2 ]go)AI +'•'(•'•+q BI=0. •z and • a) a symmetricinterface wave Figure 4. Normalized phasevelocities of the symmetricand antisymmetricinterface waves, including the leaky mode of the symmetricinterface wave, as a function of the normalized fracture stiffnesswhen C•,/Cs=1.53. lowersurface propagation ofa fracture for the uppermedium and u2= O) '•A2ePeøZ-q B2 O. 11)•sm Ii 1 eqøazleiøa(x/C-t), (7) w2=o)PA2e•'ø•z +i• B2eqø•z eiOa(x/C-t), for the lower medium. Seismic stressesin the upper and lower media obtained 0.24)•sm from (6) and (7) and Hooke's Law are Tzxl=(,02 _2ipkt c Ale_vO•z_k t +p2 Ble-qO•z eiOa(x/C-t), O. 47)•sm - A1 B1e-qm z eim(x/C-t) 0.96)•sm0 •,=•2• • •) C 1 ' (s) z 1 ß•2=•'L•A2eP• + B2e• ei•(x/C-t) b) an antisymmetricinterface wave w •2=•2F(2L••-• 1• A2e• z+2i•c B2e•Zei•(x/C-•)' lower surface of a fracture where X •d • are the Lame'sconstants. By substituting(6) through(8) into the boundaryconditions propagation•
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