PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40PTYS 554Evolution of Planetary SurfacesTectonics ITectonics IPYTS 554 – Tectonics I2Tectonics IVocabulary of stress and strainElastic, ductile and viscous deformationMohr’s circle and yield stresses Failure, friction and faultsBrittle to ductile transitionAnderson theory and fault types around the solar systemTectonics IIGenerating tectonic stresses on planetsSlope failure and landslidesViscoelastic behavior and the Maxwell timeNon-brittle deformation, folds and boudinage etc…PYTS 554 – Tectonics I3Compositional vs. mechanical termsCrust, mantle, core are compositionally differentEarth has two types of crustLithosphere, Asthenosphere, Mesosphere, Outer Core and Inner Core are mechanically different Earth’s lithosphere is divided into plates…PYTS 554 – Tectonics I4How is the lithosphere defined?Behaves elastically over geologic timeWarm rocks flow viscouslyMost of the mantle flows over geologic timeCold rocks behave elasticallyCrust and upper mantleMelosh, 2011Rocks start to flow at half their melting temperatureThermal conductivity of rock is ~3.3 W/m/KAt what depth is T=Tm/2PYTS 554 – Tectonics I5Relative movement of blocks of crustal materialMoon & Mercury – Wrinkle RidgesEuropa – Extension and strike-slip Enceladus - ExtensionMars – Extension and compressionEarth – Pretty much everythingPYTS 554 – Tectonics I6The same thing that supports topography allows tectonics to occurMaterials have strengthConsider a cylindrical mountain, width w and height hHow long would strength-less topography last? F =r ghpw2æ è ç ö ø ÷ 2Weight of the mountainh·=2v or v·=12h··Conserve volumewhvF=ma for material in the hemisphereF =1243pw2æèçöø÷3éëêêùûúúr v·Solution for h:  h =h0e-ttwhere t =w6gi.e. mountains 10km across would collapse in ~13s v·=3gwæèçöø÷h pw2æ è ç ö ø ÷ 2h·=124pw2æ è ç ö ø ÷ 2vPYTS 554 – Tectonics I7Response of materials to stress (σ) – elastic deformationLΔLΔLLinear (normal) strain (ε) = ΔL/L Shear Strain (ε) = ΔL/Lsl=Eelss=GesE is Young’s modulusG is shear modulus (rigidity)dP =KdVVVolumetric strain = ΔV/VK is the bulk modulusLPYTS 554 – Tectonics I8Stress is a 2nd order tensorCombining this quantity with a vector describing the orientation of a plane gives the traction (a vector) acting on that planesi ji describes the orientation of a plane of interestj describes the component of the traction on that planeThese components are arranged in a 3x3 matrixT =si jeisi j=s11s12s13s21s2 2s2 3s31s32s33æèççççöø÷÷÷÷s11s22s33Are normal stresses, causing normal strain(Pressure is )s13s23s12Are shear stresses, causing shear strainWe’re only interested in deformation, not rigid body rotation so: sji=sij-13s11+s22+s33( )PYTS 554 – Tectonics I9The components of the tensor depend on the coordinate system used…There is at least one special coordinate system where the components of the stress tensor are only non-zero on the diagonal i.e. there are NO shear stresses on planes perpendicular to these coordinate axessssN1sN1sN2sN2=Shear stresses in one coordinate system can appear as normal stresses in anothersi j=s10 00 s200 0 s3æèççççöø÷÷÷÷s1³ s2³ s3Where:These are principle stresses that act parallel to the principle axesThe tractions on these planes have only one component – the normal componentPressure again: -13s1+s2+s3( )PYTS 554 – Tectonics I10Principle stresses produce strains in those directionsPrinciple strains – all longitudinalStretching a material in one direction usually means it wants to contract in orthogonal directionsQuantified with Poisson’s ratioThis property of real materials means shear stain is always presentExtensional strain of σ1/E in one direction implies orthogonal compression of –ν σ1/EWhere ν is Poisson’s ratioRange 0.0-0.5Where λ is the Lamé parameterG is the shear modulusorLΔLLinear strain (ε) = ΔL/Lsl=EelE is Young’s moduluss1s2s3( )e1e2e3( )PYTS 554 – Tectonics I11Groups of two of the previous parameters describe the elastic response of a homogenous isotropic solidConversions between parameters are straightforwardPYTS 554 – Tectonics I12Typical numbers (Turcotte & Schubert)PYTS 554 – Tectonics I13Materials fail under too much stressElastic response up to the yield stressBrittle or ductile failure after thatMaterial usually fails because of shear stresses firstWait! I thought there were no shear stresses when using principle axis…How big is the shear stress?Brittle failureDuctile (distributed) failureStrain hardeningStrain SofteningSpecial case of plastic flowPYTS 554 – Tectonics I14How much shear stress is there?Depends on orientation relative to the principle stressesIn two dimensions…Normal and shear stresses form a Mohr circleqs1s3ss=s1- s32æèçöø÷sin 2qsN=s1+s32æèçöø÷+s1- s32æèçöø÷cos 2qs1+s32æèçöø÷sssNradius=s1- s32æèçöø÷2qMaximum shear stress:On a plane orientated at 45° to the principle axisDepends on difference in max/min principle stressesUnaffected (mostly) by the intermediate principle stresss2PYTS 554 – Tectonics I15Consider differential stress Failure when:Failure when: Increase confining pressureIncreases yield stressPromotes ductile failure12s1- s3( )³ sy12s1- s3( )2+ s1- s2( )2+ s2- s3( )2éëùû³ syIncrease temperatureDecrease yield stressPromotes ductile failure(Tresca criterion)(Von Mises criterion)PYTS 554 – Tectonics I16Low confining pressure Weaker rock with brittle faultingHigh confining pressure (+ high temperatures)Stronger rock with ductile deformationPYTS 554 – Tectonics I17Crack are long and thinApproximated as ellipsesa >> bEffective stress concentratorsLarger cracks are easier to growabσ σ sTip=so1+2abæèçöø÷What sets this yield strength?Mineral crystals are strong, but rocks are packed with microfracturesPYTS 554 – Tectonics I18Failure envelopesWhen shear