8/14/2019 The Effects of Locked-In Pressure on the Mechanics of Faulting by Larry Barrows

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TheEffects fLocked-inPressuren heMechanics fFaultingLarry arrowsAndrews Engineer ingStatic friction and dynarnic friction between two surfacesaregenerally proportional to the normal force between the sur-faces.Pressuren the Earth is approximately equal to the litho-static overburden. It fbllows that the failure strength of a faultand its resistance o sliding should increasewith depth belowthe surface.Or should it?

Stressdescribes nternal forces that act to deform a mate-rial. It can be separated nto shear stress,which acts to changethe shape; and pressure,which acts to change the volurne. It c:rnalso be separated nto environmenterl tressar-rdocked-in stress.The environmental stress esults from fixed displacements, ur-lace tractions, poir-rt oacls,body forces, and thermal expansionor contraction. The locked-in stress emainsafter all environrnen-ta l factors havebeen removed; it is locked into the shapeof thematerial. A simple example would be the stress hat resu lts fiomcementing an ellipsoid volume of material nto :rsphericalc:rvity.Another example s the pressure hat is ocked into a solid rnate-rial that comes nt o existenccn a high-pressurc nvironnrent. 7eneed o consider he nature and effectso[this locked-in Drcssurc.

S i t t i ngonmy desk sa Wharn -O SuperBa l l . t ' s rwo - inch -diameter, solid-latex,high-bounce ball developed n 1965 byWham-O Manuf;rcturing (the same cornpany th:rt brought usthe Hula Hoop and the Frisbee).The Wharn-O Super Ball "vasmade by first formulating a recipe for an extrernelystrong latex.Melt the latex and inject it under 3,500 pour-rds er squ;rre nch(psi) of gagepressurento a lnold. l Then maintain th e pressurewhile the latex cools and hardens. Tithout the pressure n themanufacturing process,you only get :r normrrl soft rubber ballwith little resiliency or "bounce." More ir-rformation s :rv:rilrrbleat http: / www. uperba Ls. om.The 3,500 ps i of pressurewas ocked into the l:rtex vhen itcooled and hardened. n a normal environtner-rtal ressure foneatmosphere I4.5 ps i - I ba r = l0(' dynes/crnr), he locked-inpressure ends to push the ball :rpart.The stressn the larex s:

6, i = 3,5005where6 -t/ is the Kronecker delta function.r

Gagc pressurc s rcl:rtive o ambient iltmospheric pressure.Absclluteprcssurc s rclative o a pe fcct vilcuul.n.Locked-in prcssllre encls o pr-rshhc mirtcrirrl rpart; l-ris s ;r positivcstrain.Thc corrcsponclir-rgockec-l-ir"rtrcss rlso 'ras rpositive sigtr.

We might do a little thought experiment. Put a Super Ball intoan irnaginary pressure chamber. Increase the environmentalgagepressure n the chamber to 3,500 psi. The pressure n thelatex s the sum of the locked-in pressureand the pressure n thechamber-or zeropsi. i

The locked-in pressureof 3,500 ps i is the pressure n whichthe liquid latex originally cooled and hardened into solid latex.If the environmental pressure n the imaginary chamber is lessthan the locked-in pressure,each ittle piece of latex is slightlylarger than the volume it occupieswhen it is constrained by themateri:rl of the larger ball. The solid latexexists n a stateof elas-tic compression. f the pressure n the chamber s higher than thelocked-ir-r ressure, achpiece s slightly smaller han the volumeit occupies; he latex s n a stateof elasticdilatation. The locked-in pressure s the environrnental pressure n which the materialfits passivelyvithin itself rvithout any associated tress.

Rocks are similarly made of elasticallycompressible mate-rial. It seems easonable o expect rocks that crystallize or meta-morphose at depth rvill possessa locked-in pressureapproxi-mately equal to the lithost:itic overburden. Conceptually, ageometrically perFect, lat-sided cube of rock at depth wouldbecome a highly srressed, lightly larger,slightly distended cubeif it were r:riscd to the surface.One observation that supportsthis contention is the ter-rder-rcyilarge slabsof granite to spalloff the exposed surfaces of granitic batholiths. Exfoliation isattributecl to the pressure relief that accompanies uplift pluserosion-removal f the overburden.

The existenceof locked-ir-r ressure n solid material seemsreasonable-the $Zham-O Super Ball is a simple example. Theproblem is rve do r-rotusually include locked-in pressure n theconstitutive equations that describe he behavior of earth mate-ria.ls. -r he abscnccof locked-in shear stress, his relation is:Tot:rl stress environmental stress+ locked-in pressure

The environmental stress s the sum of the effects of appliedforces, surface tractions, lithostatic overburden, body forces,therrnal expansion, tc. n a tensor format the relation can bervritten as

o)l- ' o::" ' F6,,3. Thc prcssurcon a hon-rogeneor-rspheredirectl,v ransfers nt o prcssure

u'ithin t hc nratcriirl. his is not generallv ru e fo r clbjects vith rnorecorrr lutc. l h,r trcs.

DSII

( .l l . t = |{l 0 , i + jt .

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544 Seismologica lesearchet tersVolume9,Number Ju ly /August008 doi: 1.0.1.75 gssrl .T.4.544

8/14/2019 The Effects of Locked-In Pressure on the Mechanics of Faulting by Larry Barrows

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where o1:"') s the total srressn the material, o1i'"' is rhe envi-.on. .nr l l sr ress,nd F is hc locked- inpr. r rui" . '

IFcheabove reasoning s correct, the locked-in pressureu'illgenerallybalance he effectsof the weight oIthe overlying rocks.An active fault zone can be regarded as a very thin planar vol-ume of mechanicallv weak n-raterial. he lithostatic overburdenacting on the fault is then balanccd by rl-re ocked-in pressure nthe fault zone. The total normal force acting acrossa fault zoneat depth can then be zero or even slightly negarive.The f-ailurestrength and resistance o slidine of a faulr at depth are hen notexpected o b e larger than those of a fault near the surface.

Fur rher mpl icar ions re:o \7e don't need high pore-water pressure to explain the

mecharnicsf low-angledecollement hrust faults.Extensivelor,r-angle hrust faults are characteristicof the thrust/foldbeits that fank some rnountain ranges,such as alor-rg hewestern side of the Appalachians. A depth-dependentfrictional resistance mplies rhe scrarra houlcl be crusheclimrnediately adjacent to the rnount:rin range rather thanslide along the fault. Hubbert and Ruby (1959) proposedthese faults were weakened by pore-water pressllres hatapproached or exceeded he Iithost:rtic overburc{en.

o The stress drops associar.ted itl-r earthquakes ure notexpected to be dependent on fbcal depth. Stressclropscanbe determir-red ronl the spectralanalvsisof seismic waves;thesedo not shorva first-order depcnclcncyon clepth (Steinan d \Tysession 003, 269 27 3) .

o High pore-water pressures,partial n'relting, or mineralphase change are not necded to expl:r in the mechanicsof deep-focus earthquakes. The rnechanicsof clccp-focusearthquakeshavebeen problernatic since heir depths weref i rs t detennined r.g, Kasaharm98l, 151-155).

. Thc San Andreas heart-flowparacloxgoes a\ver.y.he par;i-dox comes rorn converting the frictional rvork clone r-r hefault zone into an irnplied thern-ralanomaly. If the resis-tance to siiding is proportior-ral o the lithost;rtic overbur-den, an observable heat flow anorn,rly shoulcl exisr. Fielcirreasurementsshow the predicted anornaly does not exisr(Brune et al. 7969).

. The Poisson ffect sasubtleproblem inherent ro consrrucr-ing cornputer models of gravitarional tectonics. Ir predictsunrealistic shear stresscs n gr;rvity-loeided,compressible,e rrst ic nodels. hesc hcar rrcsscsrc nor prcscnr rrgraviry-loaded, incompre.s.sible,iscousrnodels rhar accornnroclarepressure sa separate ari:rble.The pressure erm is biilancedby the locked-in pressure Barrows ar-rd aul 1998).

Finally, u,e might consider the u.ay iquids :.rccolnrnodare res-sure. Viscosity is :rt least p:rrtially due to fiiction rvithin che4. This relation s dcntical vith thc cflcctivestrcss cparior.rhrrtdcscribes

thc rtrcchanicalbcl-raviorof rvatcr-siltrrrated nconsolicl,itcc'l lrrsticscxccpt tl-rc prcssurc" s thar oI th e solid nurrcrill ir-rstcaclf rhe porc\vatcr. Prior to thc mech,rnicaldevclopnrcr.rt f a concinlrotiswatcr-sttl-rportedriult, l-re lllctive strcss qllation should incluclea polosirvfactor. \{ost civil enuineering cxts do not inclucle he porr>sin.frrctorbu r it is requircc{ [ che pore \\'rtte clTlcr is ro upprorrchzcro rrs hcporclst, vappmaches ero.

material, bu t the viscosity ofu,ater doesno t increase vith depthbelolv the surfaceof a lake or ocean. This is because he pres-sure acting on a small volume of water at depth is exactly bal-anced by the presslrrewithin the volurne. Similarly, the lithos-tatic overburden acring on the rocks of the Earth is balancedbythe locked-in pressure n the rocks. The difference s that liquidpressure ir-rstantly rcljusts o the local pressure environmenr;rock pressure s locked into the solid material until it eithermelts or is metamorphosed.Stress s a unique second-order tensor. Its representationdepends on the tvpe and orientarion of the coordinace sysremused co describe a particular problem, buc al l represenrarionsdescribe he same system of internal fbrces.Strain is also a sec-ond-order tensor,but strain is not unique. Rather it describeinternirl defbrmation relative to some baseline configuration.Hooke's l;rw sayssrress s proportional to strain, so the correctbaseline configuration corresponds with zero stress.There isnothing in the continuum mechanicsof stress nd strain thatrequires that the baseline configuration be the configurarionthat rvould exist if the solid rnaterial forrned in a perfect pres-sure-freevacuum (c.q.,outer space).The \Wham-O Super Ballobscrvations sllggest t is the actual phvsical configuration atthe tinre the solid n'raterialcame into existence.After all, theonly parent material available for the creation of a solid musthave already been compresseclby the local pressure environ-rnent beFore hc solid was creared.The solid materi:r lcanbe rhephysical matrix of crvstals n an igneous or meramorphic rockor the representativeelernental volumes (REV) that composea theoretic-irlel;rstic continuullr. In a developing solid there isno net force betu,een the growing cryst,rlsor betu,eenadjacentREV. Hooket 111y, treSS nd strain, and locked-in pressureare:rl lconsistent oncepts rovided the baseline onfigurat ion usedto defir-rehe strain tensor s correct ly dentif ied.

I scurnbled nto this problem while writ ing computer codethat rvould deal u'ith the arbove-mentionedPoissoneffect. Myir-ritial ,rpproach \\r;rs o vastly overcomplicate the rnechanics:rncl rssociated onstitutivc cquations. The Wham-O SuperBallput things into a rnuch simple perspective.E1REFERENCESBirrrou's, .J.,anclK. M. Paul 1998).A f in itc-c lcrnentmodclingapproach

to qrirvitational tectorric strcss rnd earrhquakes Journn/ of'GeosciencL,ducdriott 6, 1-17.

Brunc,J. N. , T. L. Hcnyer,,anclR. F. Roy (1961)).Hcat flow, srrcss nd rarccrf slip alor-rg he San Anclrels fauh.-fout'rtrt/of'Geoplt.ysica/ esearcl74,3,82r-3,827

Hubbcrr, lvl. K., and \\". W. Ruby ( t q;q). Role of lluid prcssurc n mcchan,ics of verthmst rrtrltins. BuIIcrin of't/ta Gco/ogicalSocitt_y f-Americrt- 0 ( 2 ) , l 5 - 1 6 6 .

Krsalrrira, K. (1981). Edrthqu,zkeMechnttics.Cambridge: CanbridgcUniversitl, Prcss, 4t l pps.

Stcin, S.. ar-rcl vl. \\''r'session (2003). An Introduction to Seistnologl,Lartltquakt.r, dud Enrtlt .Strttt'htrc. Nlaldcn, MA: BlackwellPtrb l ishing, 9tt pps.

Andreus EngineeringSpringfield, illinois 6271I U.S.A.lb a rows@snd ews- ng.com

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