Dr. Andrea Bizzarri, Ph.D.

Transcription

1 Università degi Studi di Boogna Dottorato di Ricerca in Geofisica XX Cico Dr. Andrea Bizzarri, Ph.D. Istituto Nazionae di Geofisica e Vucanoogia Sede di Boogna Sezione di Sismoogia e Tettonofisica September 2005

2 1. EARTHQUAKE SOURCE DYNAMICS - Easto dynamic probem - Rupture description - Disocation vs. crack modes - Forward modeing scheme - Rupture stages 2. FAULT GOVERNING LAWS ( CONSTITUTIVE EQUATIONS ) - Faut modes - Physica phenomena in fauting - Fracture criteria and constitutive aws - Strength and constitutve aws - Sip dependend friction aws - Rate and state dependent friction aws

3 3. EARTHQUAKE NUCLEATION 4. 1 D SPRING SLIDER MODEL - Numerica method - Faut seismic cyce modeing - Anaytica stress perturbations - Rheaistic stress perturbations 5. RUPTURE PROPAGATION IN 2 D FAULT MODELS - Numerica methods - BIE vs. FD - Sip weakening vs. Dieterich Ruina aw - The cohesive zone and the breakdown processes - Theoretica interpretations and correspondency formua - The estimate of d 0 and reated probems - The importance of the evoution equation

4 6. RUPTURE PROPAGATION IN A TRULY 3 D FAULT MODEL - The numerica method - The reference case. Comparison between 2 D and 3 D modes - Couping of two modes of propagation. The rake variation - Dependence on the absoute stress eve. Simmetry eak - Heterogeneous configurations 7. RHEOLOGICAL HETEROGENEITIES, CRACK ARREST AND HEALING PHENOMENA - The crack and arrest modes - The barrier heaing - The pseudo sef heaing - The puse - Different materias - Anaytica modifications to friction aws - I modeo aa Niesen

5 8. CONVERGENCE

6 1. Beardinei M. E., Bizzarri A.,, Cocco M. ( 2003 ), JGR, 108, No. B3, 2135 BBC Bizzarri A.,, ( 2003 ), Ph.D. Thesis B Bizzarri A.,, Cocco M. ( 2003 ), JGR, 108, No. B8, 2373 BC Bizzarri A.,, Cocco M. ( 2005 ), Ann. Geophys., 48, No. 2 BC Bizzarri A.,, Cocco M., Andrews D. J., Boschi E. ( 2003 ), GJI, 144, 1 30 B Cocco M., Bizzarri A. ( 2002 ), GRL, 29, No. 11, CB Cocco M., Bizzarri A.,, Tinti E. ( 2003 ), Tectonophys., 378, CBT Tinti E., Bizzarri A.,, Cocco ( 2005 ), Ann. Geophys., in press TBC2005

7 Earthquake source dynamics

8 * Soution of the fundamenta easto dynamic equation ( i. e. the II aw of dynamic for continuum media ): ρ(d 2 /dt 2 )U i = σ ij,j + f i ; i = 1, 2, 3 where: ρ is the mass cubic density, U is the dispacement vector ( U = x x ), {σ ij } is the stress tensor; σ ij = C ijk e k ; i,j,k, = 1, 2, 3, where C ijk is the eastic constant tensor, accounting for the rheoogy of the medium and e k is the strain tensor ( e k = ½ (U k, + U,k ) ), f is the body force vector.

9 * Choice of the dimensionaity d of the probem ( 1 D, 2 D, 3 D ). ( d = rank of the u array, i. e. number of equations ) 1. Wave propagation probem: Hyperboic PDE D Aembert wave equation: 2 U (1/c 0 ) ( 2 / t 2 )U = 0 where c 0 is the wave speed. 2. Rupture propagation probem

10 Foowing Schoz ( 1990 ) the rupture can be described by using: * CRACK MODELS: The energy dissipation at crack edge ( or crack tip ) is paramount. Describe expicitey the crack propagation. * FRICTION MODELS FRICTION MODELS: The effects at the edges are not expicitey considered. Expicitey aow for the cacuation of the evoution of stress tensor components in terms of materia properties of the faut.

11 DISLOCATION MODELS * Study of dispacement discontinuity * Sip is assumed to be constant on the faut; The faut evoution is represented by uniatera or biatera motion ( rectanguar disocations: Haske s mode ) * Kinematic description: it accounts for time evoution of rupture front and it negects dynamics of fauting Long period seismic waves modeing ( λ L faut ) constant disocation is inadmissibe; strain energy at crack tip is unbounded; stress drop is infinite

12 CRACK MODELS * Impose finite energy fow into the rupture * Sip is not prescribed, but it is cacuated from the stres drop and from the faut strength S faut * Dynamic description: the shear stress drops inside the crack ( after nuceation processes ), increases the stress outiside the crack near the crack tip ) and tends to faciitate further grow of the rupture The motion is determined by fracture criterion ( and eventuay by the assumed constitutive aw on the faut ) The probem is characterized by assuming the boundary conditions on the faut pane. It has mixed b. c.: sip assigned outside the crack tip and stress tensor components inside the crack tip

13 1. Faut mode: - Faut geometry ( orientation, panar or non panar, ) - Faut system ( mutipe segments, mutipe fauts, ) 2. Medium surrounding the faut surface( s ) - Properties of the medium surrounding the faut(s): cubic mass density structure, veocity structure, anysotropy, attenuation 3. Choice of the dimensionaity d of the probem ( 1 D, 2 D, 3 D, 4 D ). ( d = number of the independent variabes in the soutions ) 4. Choice of the representation

14 5. Choice of the numerica method - ( FE, FD, BE, BIE, SE, hybrid ) 6. Specification of the Boundary Conditions - Domain Boundaries Conditions ( DBCs ) - Faut Boundary Condition ( FBCs ) - Auxiiary Conditions ( ACs ) 7. Specification of the Initia Conditions - Initia conditions on the faut: ( initia sip, sip veocity, state variabe, pre stress ); - Initia conditions outside the faut: ( tectonic oad, ( state of neighbouring fauts: the faut is not an isoated system ) ) 8. Evauation of the soutions - Convergence anaysis ( consistency + stabiity )

15 1. Nuceation ( quasi static to dynamic evoution ) - How can we simuate nuceation? - How can we promote faut instabiity? 2. Propagation - What is the faut constitutive equation ( governing aw )? 3. Heaing - What type of heaing occurs? - What contros faut heaing? 4. Rupture arrest - What is responsibe of rupture arrest? - How can we represent it? Earthquake energy baance? 5. Faut re strengthening - How can we mode further instabiities episodes on the faut?

16 This side is empty intentionay.

17 Support Sides: Parameters, Notes, etc. To not be dispayed directy. Referenced above.

18 Dimensionaity d

19 1 D Sping Sider ( mass spring ) mode u u 0 eff Frictiona siding ( rheoogica properties ) Eastic behaviour ( surrounding medium ) Loading veocity ( tectonic oad )

20

21 Representation Representation 3 ) ( 3, 1,2; 1,2,3; ; ), ( ), ( d d ), ( S = = = + ξ ξ ξ ξ x x x α σ α α n t t t G t t u t ' ' p ' n ' n S = + = + eff n p p t x t x t x C t x u µσ τ τ τ ), ( ), ( ), ( ), ( L Source integra rapresentation ( Betti s theorem, Integration in time ( Green Voterra s reation ), imit in faut surface, Lamb s probem ): First neighbours decouping ( in the case of a 2 D, pure in pane rupture ): INTEGRAL REPRESENTATION INTEGRAL REPRESENTATION Friction Traction

22 2. DISCRETIZATION OF EQUATIONS ( FE, FD APPROACHES )

23 Domain Boundaries Conditions * BOUNDARY: - Bottom - Fixed - Absorbing - Top - Free surface - Topography - Coasts - Latera - Cycic - Absorbing

24 Let us consider a boundary on i direction. Indeces i, j and k identify x 1, x 2 and x 3 axes, respectivey. Apex m indicate the actua time eve, whie index stands for component ( = 1, 2, 3 ). Fixed Boundary ( FB ):) U U m 1 jk m i jk end = 0, 1 = 0, m i jk end m jk = 0 = 0

25 Absorbing Boundary ( AB ):) Left boundary: m 1 jk = + + A A m 1 1 jk m 2 1 jk + + A A A m 2 jk m 1 2 jk m 2 2 jk A A A m 3 jk m 1 3 jk m 2 3 jk Right boundary: m i jk end = + + A A m 1 i jk end end m 2 i jk + + A A A m i end m 1 i 1 jk end m 2 i 1 jk end 1 jk A A A m i end m 1 i 2 jk end m 2 i 2 jk end 2 jk

26 In the previous compact representation of ABCs ( that foows Moczo, 1998 ): - the coefficients {A pq } p,q = 1, 2, 3 depend on the choice of ABC scheme ( e. g. Cayton and Engquist, 1977; Reynods, 1978; Emerman and Stephen, 1983; Higdon, 1991; Peng and Toksöz, 1994, 1995; Liu and Archueta, 2000, ); - dispacement components at actua time eve m are derived by numerica integration from partice veocity components, after update; - vaues in edges and in corners are derived from agebraic averaging of vaues of quantities beonging to was.

27 km

28 (m) (km)

29 Number of nodes Number of eements Type of buiding bock Minimum node distance Maximum node distance References Triange 200 m 2000 m Armigiato A., Tinti S., (2005), EGU Genera Asseby; Tinti S., Armigiato A., Bortoucci E. (2001), J. Seismo., 5,

30 1. TYPE Faut Boundary Conditions - Traction at Spit Nodes ( TSN ): in 2 D by Andrews ( 1973 );) ; in 3 D by Day ( 1977 ),) Archueta and Day ( 1980 ),) Day ( 1982a, 1982b ),) Andrews ( 1999 ),) Bizzarri ( 2003 ),) Bizzarri and Cocco ( 2005 ) - Stress Gut ( SG ): Backus and Muchay ( 1976 ),) Andrews ( 1976 ) - Thin zone ( TnZ ): Virieux and Madariaga ( 1982 ) - Thick zone ( TkZ ): Madariaga et a. ( 1998 ) 2. CONSTITUTIVE LAW - Faut rheoogy - Different physica phenomena occurring during the rupture process

31 Auxiiary Conditions * COLLINEARITY BETWEEN FAULT SHEAR TRACTION AND FAULT SLIP VELOCITY: v ( i. e. Τˆ = ). v Τ // v

32 * COLLINEARITY VS. ANTIPARALLELISM 1. Definition of the faut sip u ( i. e. dispacement discontitunity ): x 2 U + Faut surface U - u = U + U - reative motion of the + side with respect to the side

33 2.. Faut surface orientation. Possibiity A x 2 nˆ Τ Faut surface Τ Shear traction that a partice oacted on the + side exercised on partice ocated in the side In this case the traction vector Τ is coinear to the direction of motion ( namey to the faut sip vector u and thererore to the faut sip veocity vector v ).

34 Possibiity B x 2 Faut surface Τ Τ nˆ Shear traction that a partice oacted on the side exercised on partice ocated in the + side In this case the traction vector Τ is antiparae to the direction of motion ( namey to the faut sip vector u and thererore to the faut sip veocity vector v ).