GY403 StructuralGeologyLecture 7: Dynamic Analysis
Dynamic Analysis Goals Determine the magnitude and orientation offorces that produce rigid and non-rigid bodystrain.Determine the mechanical factors in earthmaterials that favor/disfavor deformationRelate forces to tectonic evolution of deformedterranes
Stress Stress: force applied to an area (i.e. p.s.i. in tireinflation specifications).Stress Ellipsoid: the magnitude of stress in anydirection relative to a point in a rock mass canbe conceptualized as a stress ellipsoid. Thelarger the size of the ellipsoid, the higher thestress on the rock.Lithostatic stress: the stress on a rock massdue to the overlying column of rock (stressellipsoid is a sphere).Directed stress (Differential stress): producedwhen plate motion produces a maximum (σ1)and minimum (σ3) compressive stress direction(stress ellipsoid: σ1 σ2 σ3).
Stress Conventions Stress is not a vector, instead it is considered a 2nd ordertensor. This means that you cannot add 2 stress tensors“head-to-tail” as you can with 2 force vectors (1st ordertensors)Compressive (Normal) Stress: The stress tensor actingperpendicular to an imaginary plane passing through arock mass. It is considered positive if it would causeshortening of material along axis of stress tensor. Ifnormal stress is negative it would potentially cause astretching along this direction. The symbol σ is used torepresent normal stress.
Normal Stress Normal Stress: stress tensor acts perpendicular to areference plane passing through the rock mass.Reference planeCompressive Stress 15 Mpa-15 MpaTensile Stress
Stress Conventions cont. Shear stress (τ): produced when differentialstress field exists (i.e. stress ellipsoid)If shear would cause right-lateral offset in a rock it ispositive. The shear plane thus produced has apositive angle θ to σ1. If shear would cause a left-lateral offset in a rock it isnegative, and the shear plane would have a negativeθ angle relative to σ1.
Shear Stress (τ) Shear Stress (τ): Positive shear stress is byconvention right-lateral (dextral), negative shearstress is left-lateral (sinistral).σNσ1Stress tensorθ 50 τRight-lateralSlip offset
Resolution of Stress via VectorAddition Stress is a 2nd order tensor therefore it cannot bedirectly resolved by simple vector additionIf the stress tensors can be converted to forcesvectors (1st order tensors) then the overall stresscan be evaluated by vector allison/GY403/stress.pdf
Mohr Circle & Fracture Envelope Fracture envelope displays the stress state requirementsfor fracture formationBrittle/Ductile transitionACohesivestrength2θ 60TensilestrengthBFracture Envelope:parabolic shapeA & B representconjugate fracturesat 30 to σ1Lithostatic Load for #3
Fracture Envelope Properties Increasing lithostatic σ requires greaterdifferential σ to produce fracturesBecause of the parabolic shape of the fractureenvelope conjugate fractures will tend to form at30 to σ1.Fracture envelope only predicts fracture failurevia brittle behavior- ductile deformation is notaddressed by the envelope
Mohr Circle & Fluid Over-pressure If Fluid pressure approaches that of the σ1 theMohr circle is shifted left toward the originmaking fracture much more likely:PE PL – PF (Effective Pressure LithostaticPressure – Fluid Pressure)Petroleum companies exploit this property by“Fracing” the reservoir after traditionalproduction declines (fracturing dramaticallyincreases porosity & permeability)
Mohr Fracture Envelope & FluidOver-Pressure Fluid over-pressure shifts the Mohr circle toward theorigin greatly increasing chances for fracture formationPE PL - PFτ σ3σ1σ
Rock Mechanical Properties Triaxial stress apparatus is used to test the mechanicalstrength of various rocks under a variety of differentconditions (Lithostatic load, Temp, Fluid pressure,Strain Rate, etc.)Several “Ideal” mechanical behaviors are useful inunderstanding rock mechanics: Elastic: stress produces strain up to the yield strength atwhich point the rock fractures, however, releasing stressbefore the yield strength allows the rock to recover all strain(“Rubber Band”)Plastic: any amount of applied stress will produce permanentstrain (“Silly Putty”)
Stress v. Strain Graphs Graphical plots of rockmechanical behavior withstrain (ε) on the X axisand differential stress (σ σ1 – σ3) on the Y axisRock mechanicsdominated by elasticcomponent is “Brittle”A significant plastic flowcomponent is termed“Ductile”DuctileBrittle
Examples of Brittle v. Ductile Tests Same rock (limestone) deformed to 15% ε undervarious Lithostatic and Temperature conditionsOriginalHigh confiningLow confiningHigh confiningLow confining
Lithostatic Load (Confining) Mechanical Effects of Increasing LithostaticLoad (i.e. Depth of Burial): Lithostatic Rock Strength Lithostatic Ductility High LithostaticLow Lithostatic0.5KbElastic LimitDiff.Stress(σ)Release σStrain (ε)Elastic Limit5.0KbDiff.Stress(σ)Elastic LimitRelease σStrain (ε)
Lithostatic Load cont. Actual test with limestone at various levels of lithostaticstress
Lithology NOTE: a stronganisotropy such asfoliation may cause adramatic weakening ofthe rock parallel to thefabricThe type of Lithologyexerts a strong influenceon rock strengthRock StrengthLithology300 MPa (strongest)Quartzite280 MPaGranite250 MPaBasalt213 MPaLimestone167 MPaSchist140 MPaMarble120 MPaShale45 MPaAnhydrite22 MPaSalt2.5KbQuartziteDiff.Stress(σ)ShaleStrain (ε)
Pore Fluid Pressure Increasing pore fluid pressure counteracts increasinglithostatic: Rock is weakenedRock is likely to deform by brittle failure
Temperature Increasing T will decreasethe rock strength andfavor ductile behavior (testis on basalt at variousT C)
Strain Rate Increasing thestrain rate willincrease the rockstrength and favorbrittle behavior
Time Factor Given enough time any solid material will “flow” belowthe elastic limit- this is termed “Mechanical Creep”Differential Stress Elastic LimitFundamentalStrength
Rheidity Definition (Carey, 1953):
Rheidity Examples Marble Bench:(see photo)Continental Crust:has suffered noappreciable creepsince Archean
Young’s Modulus Young’s Modulus (E): relationship between σ and ε (i.e.the slope on a stress v. strain graph
Bulk and Shear Modulus Bulk Modulus (K): measures resistance to volume(dilation)Shear Modulus (G): measures resistance to shear (τ)
Poisson’s Ratio Poisson’s Ratio (ν “nu”): ration of lateral E tolongitudinal E
Viscosity Viscosity (η “eta”): resistance of a fluid to flow
Exam 2: Dynamic Analysis Summary Be able to solve strain equations for S, λ, γ, Ψ, αBe able to discuss the difference between homogenous and inhomogeneous strain- give geological examplesKnow how to calculate lithostatic stress given depth and densityKnow how to solve a resolution of stress by vector addition problemKnow the general equations for σ and τ for the Mohr Circle, and know the relationship between terms in theequation and circle geometryBe able to interpret the Mohr Circle/Mohr Fracture envelope diagramKnow the definitions and positions on a σ versus ε graph of Yield strength, Ultimate Strength, and RuptureStrengthBe able to discuss the effects of the following on a σ versus ε graph: Lithostatic loadTemperatureStrain ratePore Fluid pressureLithologyBe able to discuss the Rheid concept and describe examples; know where Fundamental Strength is locatedBe able to discuss Young’s Modulus, Bulk Modulus, Shear Modulus, Poisson’s Ratio, and Viscosity
Triaxial stress apparatus is used to test the mechanical strength of various rocks under a variety of different conditions (Lithostatic load, Temp, Fluid pressure, Strain Rate, etc.) Several “Ideal” mechanical behaviors are useful in understanding rock mechanics: Elastic: stress produces strain up to the yield strength at
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