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20.Rheology&LinearElas8city

I MainTopics

ARheology:Macroscopicdeforma8onbehaviorBLinearelas8cityforhomogeneousisotropicmaterials

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hIp://manoa.hawaii.edu/graduate/content/slide-lava

Viscous(fluid)Behavior

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hIp://hvo.wr.usgs.gov/kilauea/update/images.html

hIp://www.hilo.hawaii.edu/~csav/gallery/scien8sts/LavaHammerL.jpg

hIp://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg

Duc8le(plas8c)Behavior

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hIp://www.earth.ox.ac.uk/__data/assets/image/0006/3021/seismic_hammer.jpg

hIps://thegeosphere.pbworks.com/w/page/24663884/Sumatra

Elas8cBehavior

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hIp://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg

BriIleBehavior(fracture)

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II Rheology:Macroscopicdeforma8onbehaviorAElas8city

1 Deforma8onisreversiblewhenloadisremoved

2 Stress(σ)isrelatedtostrain(ε)

3 Deforma8onisnot'medependentifloadisconstant

4 Examples:Seismic(acous'c)waves,rubberball

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hIp://www.fordogtrainers.com

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20.Rheology&LinearElas8city

II Rheology:Macroscopicdeforma8onbehaviorAElas8city

1 Deforma8onisreversiblewhenloadisremoved

2 Stress(σ)isrelatedtostrain(ε)

3 Deforma8onisnot'medependentifloadisconstant

4 Examples:Seismic(acous'c)waves,rubberball

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II Rheology:Macroscopicdeforma8onbehaviorBViscosity

1 Deforma8onisirreversiblewhenloadisremoved

2 Stress(σ)isrelatedtostrainrate()

3 Deforma8onis8medependentifloadisconstant

4 Examples:Lavaflows,cornsyrup

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hIp://wholefoodrecipes.net

!ε

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20.Rheology&LinearElas8city

II Rheology:Macroscopicdeforma8onbehaviorBViscosity

1 Deforma8onisirreversiblewhenloadisremoved

2 Stress(σ)isrelatedtostrainrate()

3 Deforma8onis8medependentifloadisconstant

4 Examples:Lavaflows,cornsyrup

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!ε

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II Rheology:Macroscopicdeforma8onbehaviorC Plas8city

1 Nodeforma8onun8lyieldstrengthislocallyexceeded;thenirreversibledeforma8onoccursunderaconstantload

2 Deforma8oncanincreasewith8meunderaconstantload

3 Examples:plas8cs,soils

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hIp://www.therapypuIy.com/images/stretch6.jpg

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II Rheology:Macroscopicdeforma8onbehaviorCBriIleDeforma8on

1  Discon8nuousdeforma8on

2  Failuresurfacesseparate

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hIp://www.thefeeherytheory.com

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II Rheology:Macroscopicdeforma8onbehaviorDElasto-plas8crheology

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II Rheology:Macroscopicdeforma8onbehaviorEVisco-plas8crheology

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II Rheology:Macroscopicdeforma8onbehaviorFPower-lawcreep

1 =(σ1−σ3)ne(−Q/RT)

2 Example:rocksalt

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!ε

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II Rheology:Macroscopicdeforma8onbehaviorGLinearvs.nonlinearbehavior

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II Rheology:Macroscopicdeforma8onbehaviorH Rheology=f(σij,fluidpressure,strainrate,chemistry,temperature)

I Rheologicequa8onofrealrocks=?

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II Rheology(cont.)JExperimentalresults

CompressiontestdataonTennesseemarbleIIfromWawersikandFairhurst,1970

Axialstress

Pc=confiningpressure

Elas8c

Plas8c

Increa

singconfiningpress

ure

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IIILinearelas8cityAForceanddisplacement

ofaspring(fromHooke,1676):F=kx1 F=force2 k=springconstant

Dimensions:F/L3 x=displacement

Dimensions:lengthL)

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x

F

F

x

k

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IIILinearelas8city(cont.)

BHooke’sLawforuniaxialstress:σ=Eε

1 σ=uniaxialstress2 E=Young’smodulus

Dimensions:stress3 ε=strainDimensionless

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σ

ε

E

σ

ε=ΔL/L0L0L0+ΔL

20.Rheology&LinearElas8cityTypicalrockmoduliandstrengths

Young’s Poisson’s Modulus Ra0o

(GPa)

Rocktype Emin Emax νmin νmax

Quartzite 70 105 0.11 0.25

Gneiss 16 103 0.10 0.40

Basalt 16 101 0.13 0.38

Granite 10 74 0.10 0.39

Limestone 1 92 0.08 0.39

Sandstone 10 46 0.10 0.40

Shale 10 44 0.10 0.19

Coal 1.5 3.7 0.33 0.37

UniaxialStrengths(MPa)

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Rocktype Tensile(low)

Tensile(high)

Comp.(low)

Comp.(high)

Quartzite 17 28 200 304

Gneiss 3 21 73 340

Basalt 2 28 42 355

Granite 3 39 30 324

Limestone 2 40 48 210

Sandstone 3 7 40 179

Shale 2 5 36 172

Coal 1.9 3.2 14 30

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20.Rheology&LinearElas8city

IIILinearelas8city(cont.)BHooke’sLawforuniaxial

stress(cont.):ε1=σ1/E1  σ2=σ3=02 ε2=ε3=-νε1

a ν=Poisson’sra8ob νisdimensionlessc Straininone

direc8ontendstoinducestraininanotherdirec8on

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σ1

ε=ΔL/L0L0L0+ΔL

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IIILinearelas8city(cont.)

CLinearelas8cityin3DforhomogeneousisotropicmaterialsBysuperposi8on:1  εxx=σxx/E–(σyy+σzz)(ν/E)2  εyy=σyy/E–(σzz+σxx)(ν/E)3  εzz=σzz/E–(σxx+σyy)(ν/E)

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σ1

ε=ΔL/L0L0L0+ΔL

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IIILinearelas8city(cont.)CLinearelas8cityin3Dfor

homogeneousisotropicmaterials(cont.)4 Direc8onsofprincipal

stressesandprincipalstrainscoincide

5 Extensioninonedirec8oncanoccurwithouttension

6 Compressioninonedirec8oncanoccurwithoutshortening

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σ1

ε=ΔL/L0L0L0+ΔL

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IIILinearelas8city

E Specialcases1  Isotropic(hydrosta8c)stress

a σ1=σ2=σ3b Noshearstress

2  Uniaxialstraina εxx=ε1≠0b εyy=εzz=0

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x

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IIILinearelas8cityE Specialcases

3  Planestress(2D)σz=0“Thinplate”case

4  Planestrain(2D)εz=0a Displacementinz-direc8onisconstant(e.g.,zero)

b Plateisconfinedbetweenrigidwalls

c “Thickplate”case10/30/17 GG303 25

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IIILinearelas8city

E Specialcases5  Pureshearstress(2D)

σxx=-σyy;σzz=0

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IIILinearelas8city

F Strainenergy(W0)foruniaxialstress

1

2 W=(1/2)(σxxdydz)(εxxdx)

3 W=(1/2)(σxxεxx)(dxdydz)

4 W0=W/(dxdydz)

5 W0=(1/2)(σxxεxx)10/30/17 GG303 27

εxxdx

W = Fdu0

u

∫ = σ xxdydz( ) dudx

⎛⎝⎜

⎞⎠⎟dx

0

du

∫

W0=strainenergydensity

σxxdydz

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IIILinearelas8cityGStrainenergy(W0)in3D

W0= (1/2)(σ1ε1+σ2ε2+σ3ε3)

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εxxdx

σxxdydz

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20.Rheology&LinearElas8city

IIILinearelas8cityD Rela8onshipsamong

differentelas8cmoduli1 G=μ=shearmodulusG=E/(2[1+ν])

εxy=σxy/2G2 λ=Lame'constant

λ=Ev/([1+ν][1-2ν])3 K=bulkmodulusK=E/(3[1-2ν])

4 β=compressibilityβ=1/K

∆=εxx+εyy+εzz=-p/Kp=pressure

5  P-wavespeed:Vp

6 S-wavespeed:Vs

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Vp = K + 43µ⎛

⎝⎜⎞⎠⎟ ρ

Vs = µ ρ