Geodynamics Lecture 11 Brittle deformation and faulting

Transcription

1 Geodamic Lecture 11 Brittle deformatio ad faultig Lecturer: David Whipp Geodamic 1

2 Goal of thi lecture Preet mai brittle deformatio mechaim()! Dicu the role of frictio! Relate both item above to faultig 2

3 Termiolog, clarified There are five word ued to decribe rock deformatio that are frequetl miued, cofued ad poorl udertood: Brittle Ductile Elatic Platic Vicou 3

4 Termiolog, clarified There are five word ued to decribe rock deformatio that are frequetl miued, cofued ad poorl udertood: Brittle Ductile Elatic Platic Vicou Deformatio tpe Deformatio mechaim 4

5 Termiolog, clarified Ductile deformatio, Cap de Creu, Spai!!!! Tpe of deformatio Brittle: Fracture of rock with poible lip alog the fracture urface (fault); Relativel low T ad P, large force or rapid impoed deformatio Ductile: Flow or coheret chage i the rock i the olid crtallie tate; Relativel high T ad P, mall force, low impoed deformatio 5

6 Termiolog, clarified Deformatio mechaim (or law) Elatic: Liear relatiohip betwee tre ad trai; recoverable (Lecture 5-6) Platic: Ifiite trai poible above ield tre; orecoverable (Thi lecture) Vicou: Stre i proportioal to trai rate; orecoverable (Lecture 12) 6

7 Elaticit What wa the tre-trai relatiohip for elatic material? 7

8 Elaticit or or Twi ad Moore, 2007! / " Stre i proportioal to trai For 1-D ormal tre xx = E" xx E : Youg modulu (1D) G : Shear modulu (1D) If tre 0, trai 0 (recoverable) xx = E" xx E or 2G " or " " or " 8

9 Perfectl platic behavior Twi ad Moore, 2007 Cotat tre required for deformatio No deformatio prior to exceedig ield tre " Ifiite deformatio if applied tre equal (or exceed) ield tre < o deformatio! = failure; ifiite deformatio Norecoverable " 9

10 Perfectl platic behavior Twi ad Moore, 2007 Cotat tre required for deformatio No deformatio prior to exceedig ield tre " Ifiite deformatio if applied tre equal (or exceed) ield tre < o deformatio! = failure; ifiite deformatio Norecoverable " 10

11 Perfectl platic behavior Twi ad Moore, 2007 Cotat tre required for deformatio No deformatio prior to exceedig ield tre " Ifiite deformatio if applied tre equal (or exceed) ield tre < o deformatio! = failure; ifiite deformatio Norecoverable " 11

12 Perfectl platic behavior Twi ad Moore, 2007 Cotat tre required for deformatio No deformatio prior to exceedig ield tre " Ifiite deformatio if applied tre equal (or exceed) ield tre < o deformatio! = failure; ifiite deformatio Norecoverable " 12

13 Elatic-Perfectl platic behavior What might the cotitutive relatiohip look like for a elatic-perfectl platic material? 13

14 Elatic-Perfectl platic behavior Twi ad Moore, 2007 Combiatio of behavior of elatic ad perfectl platic behavior Iitial behavior i elatic, the platic after reachig ield tre " " Elatic portio of total trai i recoverable, platic portio i ot " 14

15 Elatic-Perfectl platic behavior Twi ad Moore, 2007 Combiatio of behavior of elatic ad perfectl platic behavior Iitial behavior i elatic, the platic after reachig ield tre " " Elatic portio of total trai i recoverable, platic portio i ot " 15

16 Elatic-Perfectl platic behavior Twi ad Moore, 2007 Combiatio of behavior of elatic ad perfectl platic behavior Iitial behavior i elatic, the platic after reachig ield tre " " Elatic portio of total trai i recoverable, platic portio i ot " 16

17 Elatic-Perfectl platic behavior Twi ad Moore, 2007 Combiatio of behavior of elatic ad perfectl platic behavior Iitial behavior i elatic, the platic after reachig ield tre Elatic portio of total trai i recoverable, platic portio i ot Thi combied behavior i imilar to that of a fault durig the earthquake ccle Fault " " Rock urroudig " fault 17

18 Frictio i rock Turcotte ad Schubert, 2002 Fault lip accout for a large portio of deformatio of the upper crut What mut be overcome for lip to occur? Frictio After exceedig the frictioal reitace, rock will deform platicall withi the hear zoe Kow a frictioal platicit Normal tre Shear tre = = mg co A mg i A The baic relatiohip for tatic frictio i c = µ where μ i the iteral coefficiet of frictio, ad σ c i the critical hear tre required for lip 18

19 Frictio i rock Turcotte ad Schubert, 2002 Fault lip accout for a large portio of deformatio of the upper crut What mut be overcome for lip to occur? Frictio After exceedig the frictioal reitace, lip will occur o the fault or hear zoe Kow a frictioal platicit Normal tre Shear tre = = mg co A mg i A The baic relatiohip for tatic frictio i c = µ where μ i the iteral coefficiet of frictio, ad σ c i the critical hear tre required for lip 19

20 Frictio i rock Fig. 8.5, Turcotte ad Schubert, 2014 Fault lip accout for a large portio of deformatio of the upper crut What mut be overcome for lip to occur? Frictio After exceedig the frictioal reitace, lip will occur o the fault or hear zoe Kow a frictioal platicit Normal tre Shear tre = = mg co A mg i A The baic relatiohip for tatic frictio i f = f (Amoto law) where f i the coefficiet of tatic frictio, ad τf i the tatic frictioal tre required for lip 20

21 Frictio i rock Fig. 8.5, Turcotte ad Schubert, 2014 A tpical value for the coefficiet of tatic frictio i rock i f = 0.85 Aumig thi value, at what agle θ would the block at the left begi to lip? Normal tre Shear tre = = mg co A mg i A You ca aume that τf = σ, ad recall: f = f 21

22 Frictio i rock Upper crutal rock geerall behave a elatic-perfect platic For frictioal lip, rock propert meauremet ugget the hear tre required for fault lip icreae with ormal force i two domai Elatic Platic c =0.85 [MPa] for 5 MPa < apple 200 MPa c = [MPa] for 200 MPa Thee are kow a Berlee law Twi ad Moore,

23 Frictio i rock Upper crutal rock geerall behave a elatic-perfect platic For frictioal lip, rock propert meauremet ugget the hear tre required for fault lip icreae with ormal force i two domai Elatic Platic c =0.85 [MPa] for 5 MPa < apple 200 MPa c = [MPa] for 200 MPa Thee are kow a Berlee law Twi ad Moore,

24 Mohr-Coulomb criterio Amoto law, a we aw it, doe ot accout for rock coheio! f = f Icludig coheio c we ca modif Amoto law to! f = c + f Thi i kow a the Coulomb criterio 24

25 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 Mohr foud a elegat graphical repreetatio of the Coulomb criterio that illutrate umerou item of iteret, icludig The failure evelope, coheio ad iteral agle of frictio 25

26 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 Mohr foud a elegat graphical repreetatio of the Coulomb criterio that illutrate umerou item of iteret, icludig The failure evelope, coheio ad iteral agle of frictio 26

27 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 Mohr foud a elegat graphical repreetatio of the Coulomb criterio that illutrate umerou item of iteret, icludig The failure evelope, coheio ad iteral agle of frictio 27

28 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 Mohr foud a elegat graphical repreetatio of the Coulomb criterio that illutrate umerou item of iteret, icludig The failure evelope, coheio ad iteral agle of frictio 28

29 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 Plottig the tate of tre of a rock a a circle with a diameter of (σ1 - σ3), failure will occur if/whe the circle iterect the failure evelope I thi cae, failure occur at critical hear tre τf 29

30 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 The Mohr circle alo portra everal other importat relatiohip The agle Θ i the agle betwee a coidered plae i a rock ad the pricipal tre directio, give b 2 i(2 ) = ( 1 3 ) 30

31 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 At what agle θ i the hear tre larget? 31

32 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 The Mohr circle alo portra everal other importat relatiohip You ca alo eail ee the maximum allowable hear tre i the rock ad the maximum ad miimum ormal tree,max =

33 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 Latl, if pore preure pw i coidered, the Mohr-Coulomb criterio i = c + f ( p w ) which ca be formulated a = c + f ( w g) aumig the pore preure i hdrotatic (pw = ρwg) 33

34 Mohr-Coulomb criterio Critical hear tre (failure) τf f = ta c f = c + f Fig. 5.6, Stüwe, 2007 So, what i the effect of pore fluid? How might the affect fault lip? = c + f ( w g) 34

35 Faultig Beartooth Plateau, Womig, USA 35

36 Faultig Beartooth Plateau, Womig, USA 36

37 Adero theor a 2 Adero formulated the Mohr-Coulomb criterio i term of pricipal tree ad lithotatic preure (σ = ρg)! Adero aumed σ = σ3 for a revere fault, σ = σ1 for a ormal fault ad σ = σ2 = (σ1 + σ3)/2 for a trike-lip fault 1 Normal fault 3 b 2 3 Revere fault 1 c 2 1 Strike lip fault 3 Fig. 5.8, Stüwe,

38 Adero theor c = 0; f = 0.85 Fig. 5.7, Stüwe, 2007 I term of differetial tre, Adero theor i Revere: Normal: Strike-lip: d = 1 3 = 2(c + f ( w g) p f 2 +1 f d = 1 3 = 2(c f ( w g) p f 2 +1+f d = 1 3 = 2(c + f ( w g) p f

39 Predictig fault orietatio Now we ll look at how to appl Adero theor to predict the dip agle β of ormal ad revere fault Aume pricipal tree = g xx = g + xx where Δσxx i the tectoic deviatoric tre, which i poitive for a revere fault ad egative for a ormal fault 39

40 Predictig fault orietatio Fig. 8.8, Turcotte ad Schubert, 2014 We firt eed to relate σxx ad σ to σ ad σ i order to appl Amoto law b uig the equatio for the ormal ad hear tree i a coordiate tem rotated b agle θ with repect to the pricipal tree (ee Lecture 3) = 1 2 ( xx + )+ 1 2 ( xx ) co 2 = 1 2 ( xx )i2 Note that here, θ i with repect to vertical, θ = π/2 - β 40

41 Predictig fault orietatio Fig. 8.8, Turcotte ad Schubert, 2014 If we plug i the value for σxx ad σ, we fid xx = g + (1 + co 2 ) 2 Iertig the value above ito the form of Amoto law that iclude pore fluid preure, τ = f(σ - pw), ield ± xx 2 = xx 2 i 2 = f apple g p w + i 2 xx 2 (1 + co 2 ) Note that the upper ig i for revere fault (Δσxx > 0) ad the lower for ormal fault (Δσxx < 0) 41

42 Predictig fault orietatio Fig. 8.8, Turcotte ad Schubert, 2014 The previou expreio ca be rearraged to olve for Δσxx 2f ( g p w ) xx = ± i 2 f (1 + co 2 ) If we aume that faultig will occur with the miimum tectoic tre, the Δσxx hould be miimized B ettig dδσxx/dθ = 0, we fid ta 2 = 1 or ta 2 = ± 1 f f where the upper ig agai applie to revere fault ad the 42 lower to ormal fault

43 Predictig fault orietatio!!!!! Dip agle Fig. 8.9, 8.10, Turcotte ad Schubert, 2014 Tectoic tre = 2700 kg m 3 p w = w g w = 1000 kg m 3 =5km Fiall, the two equatio from the previou lide ca be combied to ield the tectoic tree correpodig to agle θ or β xx = ±f ( g p w ) (1 + f 2 ) 1/2 f where the upper ig agai correpod to revere fault ad the lower to ormal fault 43

44 Recap Rock i the brittle domai deform elaticall util their ield tre i reached! Oce ield i reached, rock exhibit platic behavior, where ifiite deformatio i poible at a cotat tre! Uig baic geometr, fault orietatio ca be predicted from pricipal tree uig Adero theor 44

45 Referece Twi, R. J., & Moore, E. M. (2006). Structural Geolog, 2d Editio. W.H. Freema Co. 45