Rheology What is rrhheeoollooggyy ? From the root work “rheo-” Current: flow Greek: rhein, to flow (river) Like rheostat – flow of current Rheology What physical properties control deformation ? - Rock type - Temperature - Pressure - Deviatoric (differential) Stress - Others ? What are the different types of strain ? - Brittle Low T,P, Shallow - Elastic > - Plastic - Viscous > High T,P, Deep Rheology Strain rate measured by GPS in Southern California What do the GPS measurements indicate ? At what depth do these movements occur ? How can we test this ? Rheology Guest Lecture from Dr. Miranda! - Brittle Deformation - Plastic Deformation - Brittle/Plastic Transition ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion PARPARTT II:: RRHHEOEOLLOOGYGY ANDAND MMAACRCROOSCOSCOPPIICC DDEFEFOORMRMAATITIOONN EarEartthh mmatateerriaialsls rreespspondond ttoo ststrreessseses,s, causcausiningg ssttrrainain.. SSttrreessss andand ststrraainin araree rreelatlateedd tthhrroughough rrhheeoloologygy.. WeWe ddeescrscribeibe tthhee ststrrainain pprroceocesssseess anandd ccatateegorgorizizee tthehemm iinnttoo difdifffeerreenntt bbeehhavaviorior;; tthheessee rreelalattionionsshhipipss araree alsoalso dedescscrribeibedd mmaatthheemmataticallyically.. ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion ThThee nnamamee rrhheeoologylogy dederriviveess ffrromom tthhee GGrreeeekk worwordd “r“rhheeo”,o”, whwhichich mmeeananss ““ffllowow”.”. RRhheeoologilogiccalal ssttudiudieess hhanandldlee tthehe ffllowow ccomomponponeentnt ofof dedeffoorrmmatation,ion, witwithh eemmpphasishasis onon tthhee iinntteerrppllayay bbeettweweeenn ststrreessss,, ststrraainin,, aanndd tthehe rratatee ofof fflow.low. IInn ggeeososccieiennccees,s, rrhheeologologyy rreeprpreesesennttss aa brbrananchch ooff tthhee sciesciencncee ofof rrockock mmeecchhaannicicss.. RReemmeemmbeberr…… eevveerrytythhiningg fflows,lows, eevevenn ssolidolids,s, undeunderr tthehe rrightight conconditditiiononss ofof ttimimee anandd ststrreess!ss! ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion RRhheeoologilogiccalal bbeehhavavioriorss rraanngege ffrromom ppeerrffeectctlyly eelaslastticic solsolidsids atat oneone ExtExtrreemmee ttoo viviscscoousus NeNewtwtonianonian ffluidsluids atat tthhee otothheerr.. HowHoweeveverr,, tthhee rrhheeolologyogy ofof nnaattururalal mmatateerriaialsls ssucuchh asas rroocckkss,, ffallsalls betbetwweeeenn tthheesese eextxtrreemmees.s. QQueuessttionion:: tthhee rreespspoonnsese ofof mmatateerriaialsls ttoo ststrreessss dedepepenndsds onon wwhhaatt pprropopeerrttieies?s? ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion WeWe wwiillll nnooww eenngagegage inin aa mmororee quaquannttititatativeive ststuudydy ooff hhowow mmaatteerrialialss bebehhavavee inin rreesponsponsese ttoo ssttrreesss.s. AA mmatateerrialial ccanan eexhibitxhibit:: • BBrrititttllee bebehhavaviorior • ElasElastticic bebehhaaviorvior • PlasPlastticic bebehhaaviorvior • ViscousViscous bebehaviohaviorr • PowePowerr-law-law bebehhaviavioror ThTheerree araree otothheerr ttypypeess ooff bebehhaviavioror,, butbut wwee willwill ffococusus onon tthheessee mmaiainn ttyyppeess inin tthhisis ccouourrssee.. ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 11.. BBRRIITTLTTLEE MMAATERTERIIALALSS BBrrititttllee bebehhavaviorior ccaann bebe gegenenerralizalizeedd byby MMohohrr CouloCoulommbb ffaiailurluree:: σσ µµ σσ s == ** n ++ CC σ µµ σ where s iis the shear stress, isis the coefficicieient of friictioion, n iis the llininear straiin, and C isis cohesiion. ThThee ComComppososititee FFailurailuree EnEnveveloplopee:: 33 parparttss I: Tensile failure II: Coulomb failure III: von Mises criterion (deep crust) σσ s CC φφ 22θθ τ σσ σσ 3 1 σσ n II IIII IIIIII ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 22.. EELALASSTITICC MMATERATERIIALALSS ElasElastticic mmatateerrialsials dedeffoorrmm byby anan amamounountt pprropopororttioionnalal ttoo tthhee apappliepliedd ststrreess,ss, butbut wwhheenn tthehe ssttrreessss isis rreeleleaseased,d, tthehe mmaatteerrialial rreettuurrnsns ttoo ititss ororiginaliginal undeundeffoorrmmeedd ssttatatee.. ThThee dedefforormmatatiionon isis sasaidid ttoo bebe rreeccooveverrablablee.. ThThiiss rreelatlatioionnshshiipp dedeffinineess eellastasticic bebehhavaviorior:: σσ == YY εε where σ iis the applliied stress, ε isis the lliinear strainin, and Y iis Young’s’s modullus (materiiall specififiic). ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 22.. EELALASSTITICC MMATERATERIIALALSS ThThisis rreelalattionionsshhipip dedeffiinneess eelalaststicic bebehhaviavioror:: σσ == YY εε where σ iis the applliied stress, ε isis the lliinear strainin, and Y iis Young’s’s modullus (materiiall specififiic). σ ε ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 22.. EELALASSTITICC MMATERATERIIALALSS σ == YY εε Ellasticic behaviior isis modelled very wellll by a sprining that iis compressed and then relleased. The sprining recovers the deformatiion. Thisis equatioion isis essentiaialllly the same as Hooke’s’s Law. σ ε ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 33.. PPLALASSTITICC MMATERATERIIALALSS PlasPlastticic mmatateerrialsials dedeffoorrmm byby anan amamounountt pprropopororttioionnalal ttoo tthhee apappliepliedd ststrreessss (e(elaslasttic)ic) atat ffirirstst,, butbut whwheenn aa ccrrititicalical yyieieldld ststrreessss isis rreeachacheedd,, tthheeyy fflowlow rreeadadilyily anandd undeunderrgogo ppeerrmmaanneenntt dedeffoorrmmatation:ion: σ ε ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 33.. PPLALASSTITICC MMATERATERIIALALSS PePerrffeeccttlyly pplalaststicic mmatateerrialialss eexxhhibiibitt nnoo dedeffoorrmmatationion atat aallll bebelowlow tthehe tthhee yiyieeldld ststrreesss:s: σσ s == KK σσ Where s iis the shear stress and K isis the yiielld stress. σ ε ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 33.. PPLALASSTITICC MMATERATERIIALALSS AA mmeecchhaannicicaall ananalalogog ffoorr pplaslastticic dedeffoorrmmatationion isis tthhee ideideaalizelizedd ffrrictictionionaall rreessististananccee ttoo tthhee slidislidinngg ooff aa blblocockk onon aa sursurffaaccee.. Just the plastic deformation: yield stress σ σ ε ε• ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 4.4. VIVISSCCOOUUSS MMATEATERRIIALALSS ViscousViscous mmatateerrialsials dedefforormm byby fflolowwiningg inin rreespsponseonse ttoo aa ststrreesss,s, butbut whwheenn tthhee ststrreessss iiss rreemmoveoved,d, tthhee mmaatteerrialial doedoess nnoott rreettuurrnn ttoo tthhee ununddeefforormmeedd conconffigigururatation.ion. IItt cancan bbee ssaidaid tthhatat ststrreessss isis pprropopoorrttionionalal ttoo sstraintrain raterate durduriningg visvisccououss dedefforormmatationion.. VisVisccououss bebehhavaviorior isis dedessccrribeibedd byby:: σ = 2 η ε Where σ iis stress, η isis viiscosiity, and ε isis shear straiin rate. ViscousViscous bebehaviohaviorr isis comcommmoonn inin NNeewwttononianian ffluiluidsds.. ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 4.4. VIVISSCCOOUUSS MMATEATERRIIALALSS IItt cancan bebe ssaidaid tthhatat ssttrreessss isis pprropopoorrttionionalal ttoo sstraintrain raterate durduriningg viviscscoousus ddeefforormmaattionion.. VViiscscoousus bbeehhavaviorior iiss dedescscrribibeedd bby:y: σ = 2 η ε• Where σ iis stress, η isis viiscosiity, and ε• isis shear straiin rate. σ ε• ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 4.4. VIVISSCCOOUUSS MMATEATERRIIALALSS AA googoodd mmeecchhanicanicalal ananalalogog ffoorr visvisccououss dedefforormmatatiionon isis aa ddashashpotpot.. WhWheenn aa ffororccee isis aapppplieliedd acracrossoss tthhee sysyststeemm,, tthehe mmoottionion ofof tthehe ppisisttonon isis govegoverrnenedd byby tthehe rratatee atat wwhhiicchh tthhee ffluidluid fflowslows tthhrroughough tthhee ppororeess inin tthhee ppisisttonon.. slslopopee == 22ηη σ σ ε• ε ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 55.. PPOOWERWER-L-LAWAW MMATERATERIIALALSS NonNon-Ne-Newtwtononiianan ffluluidsids dodo nnoott hhavavee aa cconstonstantant slslopopee onon tthehe ststrreessss-st-strrainain rratatee grgrapaphh.. IInnststeeadad ofof viviscscoousus bbeehhavaviorior,, tthheeyy eexxhhibitibit ppoweowerr-la-laww rrhheeoloology.gy. • NeNewtwtononiianan σ = 2 η ε slslopopee == 22ηη σ nnon-Non-Neewtwtoonnianian ••εε σσ n == AA(( diff)) ε• ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk DeDefforormmatationion 55.. PPOOWERWER-L-LAWAW MMATERATERIIALALSS NonNon-Ne-Newtwtononiianan ffluluidsids dodo nnoott hhavavee aa cconstonstantant slslopopee onon tthehe ststrreessss-st-strrainain rratatee grgrapaphh.. IInnststeeadad ofof viviscscoousus bbeehhavaviorior,, tthheeyy eexxhhibitibit ppoweowerr-la-laww rrhheeoloology.gy. • •ε NeNewtwtononiianan σ = 2 η ε = σ η slslopopee == 22ηη 2 σ nnon-Non-Neewtwtoonnianian ••εε σσ n == AA(( diff)) ε• ChChapaptteerr 116:6: MMaaccrroscoposcopicic AspeAspeccttss ofof RoRocckk