GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD

Transcription

1 Earth Material Lecture 13 Earth Material

2 Hooke law of elaticity Force = E Area Hooke law n = E n Extenion Length Robert Hooke ( ) wa a virtuoo cientit contributing to geology, palaeontology, biology a well a mechanic Contitutive equation where E i material contant, the Young Modulu Unit are force/area N/m 2 or Pa = ij Cijkl kl Thee are relationhip between force and deformation in a continuum, which define the material behaviour.

3 Shear modulu and bulk modulu Young or tiffne modulu: Shear or rigidity modulu: = S = G S µ Bulk modulu (1/compreibility): P = K Can write the bulk modulu in term of the Lamé parameter λ, µ: and write Hooke law a: = n E n v K = λ + 2µ/3 = (λ +2µ) Mt Shata andeite

4 Young Modulu or tiffne modulu Young Modulu or tiffne modulu: n = E n Interatomic force Interatomic ditance

5 Shear Modulu or rigidity modulu Shear modulu or tiffne modulu: = G Interatomic force Interatomic ditance

6 Hooke Law ij and kl are econd-rank tenor o C ijkl i a fourth-rank tenor. For a general, aniotropic material there are 21 independent elatic moduli. In the iotropic cae thi tenor reduce to jut two independent elatic contant, λ and µ. So the general form of Hooke Law reduce to: 2 ij = λδ ij kk + µ ij Thi can be deduced from ubtituting into the Taylor expanion for tre and differentiating. For example: 12 = λ( = 2µ ) + 2µ Normal tre Shear tre

7 Hooke Law: Hooke Law 2 ij = λδ ij kk + µ ij Conider normal tree and normal train: = λ( = λ( In term of principal tree and principal train: = λ( ) + 2µ ) + ) + 2µ 2µ = ( λ µ ) + λ 2 1 = λ + ( λ + 2 2µ ) = λ + λ + ( λ λ 2µ ) 3 + λ 3 3

8 Hooke Law where E i the Young Modulu and υ i the Poion ratio. Poion ratio varie between 0.2 and 0.3 for rock. A principal tre component i produce a train I /E in the ame direction and train (-υ. i / E) in orthogonal direction. Elatic behaviour of an iotropic material can be characterized either by pecifying either λ and µ, or E and υ. Can write in invere form: υ υ υ υ υ υ E E E E E E E E E + = + = =

9 Contitutive equation: uniaxial elatic deformation All component of tre zero except : = λ( + = 0 = λ( = 0 = λ( ) µ ) + ) + 2µ 2µ d /d = E The olution to thi et of imultaneou equation i: 22 µ (3λ + 2µ ) = = E λ + µ = 33 λ = 2( λ + µ ) = ν 33 = 0 22 = 0 where E i Young Modulu and ν i Poion ratio.

10 Contitutive equation: iotropic compreion No hear or train; all normal tree equal to p; all normal train equal to v /3. 33 = -p 22 = -p P = 2 λ + µ V = 3 K V = -p 22 = -p = -p V = = + V v -p 33 = -p P = - 1/3 ( where K i the bulk modulu; hence K = λ + 2/3µ ) = - 1/3 ii -dp/d v = K v

11 Young Modulu (initial tangent) of Material Rubber Normally conolidated clay Boulder clay (overolidated) Concrete Sandtone Granite Baalt Steel Diamond Typical E 7 MPa 0.2 ~ 4 GPa 10 ~20 GPa 20 GPa 20 GPa 50 GPa 60 GPa 205 GPa 1,200 GPa

12 Strength of Material Soil Sandtone Concrete Baalt Granite Rubber Spruce along/acro grain Steel piano wire Uniaxial tenile trength 300 kpa 1 MPa 4 MPa 4 MPa 5 MPa 30 MPa 100 / 3 MPa 3,000 MPa Compreive trength - unconfined 1 MPa 10 MPa 40 MPa 40 MPa 50 MPa 2,000 MPa 100 / 3 MPa 3,000 MPa

13 Fracture Calculate the tre which will jut eparate two adjacent layer of atom x layer apart train energy / m 2 = ½ tre x train x vol U e = ½ n n x Hooke law: n = n / E U e = n2 x / 2E If U i the urface energy of the olid per quare metre, then the total urface energy of the olid per quare metre would be 2U per quare metre Suppoe that at the theoretical trength the whole of the train energy between two layer of atom i potentially convertible to urface energy: 2 n x U E U E 2U or n 2 2E x x For teel: U = 1 J/m; E = 200 GPa; max = 30 GPa E / 10 x = 2 x m x

14 Griffith energy balance Microcrack in lava The reaon why rock don t reach their theoretical trength i becaue they contain crack Crack model are alo ued in modelling earthquake faulting

15 Dilocation (line defect) in hear The reaon why rock don t reach their theoretical hear trength i becaue they contain dilocation Dilocation model are alo ued in modelling earthquake faulting

16 Engineering behaviour of oil Soil are granular material their behaviour i quite different to crytalline rock Deformation i trongly non-linear The curvature of the tre-train i larget near the origin Propertie are highly dependent on water content The contitutive relation for hear deformation, found from hundred of experiment i: = G 0 r + r Uniaxial deformation r i the reference train Shear deformation

17 Contitutive equation for oil Soil are fractal material There i a lognormal ditribution of grain ize (c.f. crack length in rock) Suppoe we ubject a oil to a imple hear train. The hear force applied to each grain mut be lognormally ditributed ince they are proportional to the grain urface. So the hear modulu and rigidity mut be related by a power law: G = c µ d where d i the fractal dimenion of the grain ize ditribution replacing G and µ by their definition in term of hear tre and hear train : d d = c contitutive equation for oil d

18 Contitutive equation for oil From fractal: d d Integrating and etting d = 2: = c d = G 0 r + r Thi i the ame a the empirical contitutive equation! Thi i a hyperbolic tre-train relation (i.e., like a deformation tre-train curve) It may be interpreted a aying that the hear modulu G = d/d of a oil decay inverely a (1 + τ) where τ = / r i the normalied train Note that the tre-train behaviour of oil cannot be linearized at mall train

19 Liquefaction of oil: phae tranition Thi apect of oil behaviour i completely different from crytalline rock Soil liquefaction: Kobe port area Motion on oft ground to trong earthquake i fundamentally different to mall earthquake becaue ediment go through a phae tranition and liquefy Stre-train curve of a oil a compared with that of a crytalline rock note different definition of rigidity

20 Contitutive equation: vicou flow Incompreible vicou fluid For vicou fluid the deviatoric tre i proportional to train-rate: ' ij = 2η ' ij where η i the hear vicoity 1/2η Vicoity i an internal property of a fluid that offer reitance to flow. Vicoity i meaured in unit of Pa (Pacal econd), which i a unit of preure time a unit of time. Thi i a force applied to the fluid, acting for ome length of time. A marble (denity 2800 kg/m 3 ) and a teel ball bearing (7800 kg/m 3 ) will both meaure the vicoity of a liquid with different velocitie. Water ha a vicoity of Pa, a Pahoehoe lava flow 100 Pa, an a'a flow ha a vicoity of 1000 Pa. We can mentally imagine a phere dropping through them and how long it might take.

21 Experimental technique to tudy friction Shear box Direct hear Triaxial tet Rotary hear

22 Experimental reult At low normal tree ( N < 200 MPa) Linear friction law oberved: S = µ N A ignificant amount of variation between rock type: µ can vary between 0.2 and 2.0 but mot commonly between Average for all data given by: S = 0.85 N At higher normal tree ( N > 200 MPa) Very little variation between wide range of rock type (with ome notable exception ep. clay mineral which can have unuually low µ But friction doe not obey Amonton Law (i.e. traight line through origin) but Coulomb Law Bet fit to all data given by: S = N

23 (a) Friction Amonton Law Simple failure criteria 1 t : Friction i proportional normal load (N) Hence: F = µ N - µ i the coefficient of friction 2 nd : Friction force (F) i independent of the area in contact So in term of tree: S = µ N = N tanφ May be imply repreented on a Mohr diagram: S lope µ φ µ= tan φ φ i the angle of friction N

24 Field obervation We are concerned with friction related to earthquake, i.e., friction on fault Fault are interface that have already fractured in previouly intact material and have ubequently been diplaced in hear (i.e., have lipped) Hence they are not mated urface (unlike joint) Joint Fault

25 Summary: Byerlee Friction Law All data may be fitted by two traight line: N < 200 MPa S = 0.85 N N > 200 MPa S = N Thee are largely independent of rock type Independent of roughne of contacting urface Independent of rock trength or hardne Independent of liding velocity Independent of temperature (up to 400 o C)

26 Experimental reult of triaxial deformation tet P C Differential Stre ( 1-3 ) Total Axial Confining Preure P C Stre 1 Hydrotatic P C applied in all direction prior to the differential loading. Mode of brittle fracture in a triaxial ytem P C P C = 2 =

27 Actuator applying axial load To AE tranducer Fluid outlet fitting Thermocouple feedthrough Top wave-guide Preure Veel Load Cell Inulating filler Top teel Fv520 piton Top pyrophillite encloing dic Alumina coil upport Alumina Dic Rock Specimen Pore fluid inlet Fibrou alumina inulation Bottom teel Fv520 piton Bottom encloing pyrophillite block Bottom wave guide Preure fitting Bottom plug

28 Experimental reult Schematic tre-train curve for rock deformation over a range of confining preure Dependence of differential tre at hear failure in compreion on confining preure for a wide range of igneou rock Strength of Weterly granite a a function of confining preure. Alo hown i frictional trength.

29 (b) Faulting Coulomb Law Simple failure criteria S = C + µ i N = N tanφ i C i a contant the coheion Tenile fracture µ i i the coefficient of internal friction Shear fracture S lope µ i φ i ( 2 = - T ) C T tenile trength µ i = tan φ i φ i i the angle of internal friction N