GY403 Structural GY403 Structural GeologyGeology
Lecture 7: Dynamic AnalysisLecture 7: Dynamic Analysis
Dynamic Analysis GoalsDynamic Analysis Goals
Determine the magnitude and orientation of Determine the magnitude and orientation of forces that produce rigid and nonforces that produce rigid and non--rigid body rigid body strain.strain.Determine the mechanical factors in earth Determine the mechanical factors in earth materials that favor/disfavor deformationmaterials that favor/disfavor deformationRelate forces to tectonic evolution of deformed Relate forces to tectonic evolution of deformed terranesterranes
StressStress
Stress: force applied to an area (i.e. Stress: force applied to an area (i.e. p.s.ip.s.i. in tire . in tire inflation specifications).inflation specifications).Stress Ellipsoid: the magnitude of stress in any Stress Ellipsoid: the magnitude of stress in any direction relative to a point in a rock mass can direction relative to a point in a rock mass can be conceptualized as a stress ellipsoid. The be conceptualized as a stress ellipsoid. The larger the size of the ellipsoid, the higher the larger the size of the ellipsoid, the higher the stress on the rock.stress on the rock.Lithostatic stress: the stress on a rock mass Lithostatic stress: the stress on a rock mass due to the overlying column of rock (stress due to the overlying column of rock (stress ellipsoid is a sphere).ellipsoid is a sphere).Directed stress (Differential stress): produced Directed stress (Differential stress): produced when plate motion produces a maximum (when plate motion produces a maximum (σσ11) ) and minimum (and minimum (σσ33) compressive stress direction ) compressive stress direction (stress ellipsoid: (stress ellipsoid: σσ11> > σσ2 2 > > σσ33). ).
Stress ConventionsStress Conventions
Stress is not a vector, instead it is considered a 2Stress is not a vector, instead it is considered a 2ndnd order order tensor. This means that you cannot add 2 stress tensors tensor. This means that you cannot add 2 stress tensors ““headhead--toto--tailtail”” as you can with 2 force vectors (1as you can with 2 force vectors (1stst order order tensors)tensors)Compressive (Normal) Stress: The stress tensor acting Compressive (Normal) Stress: The stress tensor acting perpendicular to an imaginary plane passing through a perpendicular to an imaginary plane passing through a rock mass. It is considered positive if it would cause rock mass. It is considered positive if it would cause shortening of material along axis of stress tensor. If shortening of material along axis of stress tensor. If normal stress is negative it would potentially cause a normal stress is negative it would potentially cause a stretching along this direction. The symbol stretching along this direction. The symbol σσ is used to is used to represent normal stress.represent normal stress.
Normal StressNormal Stress
Normal Stress: stress tensor acts perpendicular to a Normal Stress: stress tensor acts perpendicular to a reference plane passing through the rock mass.reference plane passing through the rock mass.
+15 Mpa-15 Mpa
Compressive Stress Tensile Stress
Reference plane
Stress Conventions cont.Stress Conventions cont.
Shear stress (Shear stress (ττ): produced when differential ): produced when differential stress field exists (i.e. stress ellipsoid)stress field exists (i.e. stress ellipsoid)
If shear would cause rightIf shear would cause right--lateral offset in a rock it is lateral offset in a rock it is positive. The shear plane thus produced has a positive. The shear plane thus produced has a positive angle positive angle θθ to to σσ11..If shear would cause a leftIf shear would cause a left--lateral offset in a rock it is lateral offset in a rock it is negative, and the shear plane would have a negative negative, and the shear plane would have a negative θθ angle relative to angle relative to σσ11..
Shear Stress (Shear Stress (ττ))
Shear Stress (Shear Stress (ττ): Positive shear stress is by ): Positive shear stress is by convention rightconvention right--lateral (dextral), negative shear lateral (dextral), negative shear stress is leftstress is left--lateral (sinistral).lateral (sinistral).
σN
+τRight-lateralSlip offset
Stress tensor θ= +50
σ1
Resolution of Stress via Vector Resolution of Stress via Vector AdditionAddition
Stress is a 2Stress is a 2ndnd order tensor therefore it cannot be order tensor therefore it cannot be directly resolved by simple vector additiondirectly resolved by simple vector additionIf the stress tensors can be converted to forces If the stress tensors can be converted to forces vectors (1vectors (1stst order tensors) then the overall stress order tensors) then the overall stress can be evaluated by vector additioncan be evaluated by vector additionExample: Example: http://www.usouthal.edu/geography/allison/Ghttp://www.usouthal.edu/geography/allison/GY403/stress.pdf Y403/stress.pdf
Mohr Circle & Fracture EnvelopeMohr Circle & Fracture Envelope
Fracture envelope displays the stress state requirements Fracture envelope displays the stress state requirements for fracture formationfor fracture formation
Fracture Envelope: parabolic shape
Lithostatic Load for #3Tensile strength
Cohesivestrength
A
B
2θ=60
A & B represent conjugate fractures
at 30° to σσ11
Brittle/Ductile transition
Fracture Envelope PropertiesFracture Envelope Properties
Increasing lithostatic Increasing lithostatic σσ requires greater requires greater differential differential σσ to produce fracturesto produce fracturesBecause of the parabolic shape of the fracture Because of the parabolic shape of the fracture envelope conjugate fractures will tend to form at envelope conjugate fractures will tend to form at 3030°° to to σσ1.1.
Fracture envelope only predicts fracture failure Fracture envelope only predicts fracture failure via brittle behaviorvia brittle behavior-- ductile deformation is not ductile deformation is not addressed by the envelopeaddressed by the envelope
Mohr Circle & Fluid OverMohr Circle & Fluid Over--pressurepressure
If Fluid pressure approaches that of the If Fluid pressure approaches that of the σσ1 the 1 the Mohr circle is shifted left toward the origin Mohr circle is shifted left toward the origin making fracture much more likely:making fracture much more likely:PPEE = P= PLL –– PPF F (Effective Pressure = Lithostatic (Effective Pressure = Lithostatic Pressure Pressure –– Fluid Pressure)Fluid Pressure)Petroleum companies exploit this property by Petroleum companies exploit this property by ““FracingFracing”” the reservoir after traditional the reservoir after traditional production declines (fracturing dramatically production declines (fracturing dramatically increases porosity & permeability)increases porosity & permeability)
Mohr Fracture Envelope & Fluid Mohr Fracture Envelope & Fluid OverOver--PressurePressure
Fluid overFluid over--pressure shifts the Mohr circle toward the pressure shifts the Mohr circle toward the origin greatly increasing chances for fracture formationorigin greatly increasing chances for fracture formation
+σ
+τ
σ3 σ1
PE = PL - PF
Rock Mechanical PropertiesRock Mechanical Properties
Triaxial stress apparatus is used to test the mechanical Triaxial stress apparatus is used to test the mechanical strength of various rocks under a variety of different strength of various rocks under a variety of different conditions (Lithostatic load, Temp, Fluid pressure, conditions (Lithostatic load, Temp, Fluid pressure, Strain Rate, etc.)Strain Rate, etc.)Several Several ““IdealIdeal”” mechanical behaviors are useful in mechanical behaviors are useful in understanding rock mechanics:understanding rock mechanics:
Elastic: stress produces strain up to the yield strength at Elastic: stress produces strain up to the yield strength at which point the rock fractures, however, releasing stress which point the rock fractures, however, releasing stress before the yield strength allows the rock to recover all strain before the yield strength allows the rock to recover all strain ((““Rubber BandRubber Band””))Plastic: any amount of applied stress will produce permanent Plastic: any amount of applied stress will produce permanent strain (strain (““Silly PuttySilly Putty””))
Stress v. Strain GraphsStress v. Strain Graphs
Graphical plots of rock Graphical plots of rock mechanical behavior with mechanical behavior with strain (strain (εε) on the X axis ) on the X axis and differential stress (and differential stress (σσ= = σσ1 1 –– σσ3) on the Y axis3) on the Y axisRock mechanics Rock mechanics dominated by elastic dominated by elastic component is component is ““BrittleBrittle””A significant plastic flow A significant plastic flow component is termed component is termed ““DuctileDuctile””
Brittle
Ductile
Examples of Brittle v. Ductile TestsExamples of Brittle v. Ductile Tests
Same rock (limestone) deformed to 15% Same rock (limestone) deformed to 15% εε under under various Lithostatic and Temperature conditionsvarious Lithostatic and Temperature conditions
Original High confining Low confining High confining Low confining
Lithostatic Load (Confining)Lithostatic Load (Confining)
Mechanical Effects of Increasing Lithostatic Mechanical Effects of Increasing Lithostatic Load (i.e. Depth of Burial):Load (i.e. Depth of Burial):
> Lithostatic = > Rock Strength> Lithostatic = > Rock Strength> Lithostatic = > Ductility> Lithostatic = > Ductility
Strain (ε)
Diff.Stress
(σ)
Elastic Limit
Low Lithostatic
Release σ
Strain (ε)
Elastic Limit
High Lithostatic
Release σ
0.5Kb 5.0KbElastic Limit
Diff.Stress
(σ)
Lithostatic Load cont.Lithostatic Load cont.
Actual test with limestone at various levels of lithostatic Actual test with limestone at various levels of lithostatic stressstress
LithologyLithology
The type of Lithology The type of Lithology exerts a strong influence exerts a strong influence on rock strengthon rock strength
Strain (ε)
Diff.Stress
(σ)
2.5Kb
Shale
Quartzite
SaltSalt22 22 MPaMPa
AnhydriteAnhydrite45 45 MPaMPa
ShaleShale120 120 MPaMPa
MarbleMarble140 140 MPaMPa
SchistSchist167 167 MPaMPa
LimestoneLimestone213 213 MPaMPa
BasaltBasalt250 250 MPaMPa
GraniteGranite280 280 MPaMPa
QuartziteQuartzite300 300 MPaMPa (strongest)(strongest)
LithologyLithologyRock StrengthRock Strength
NOTE: a strong anisotropy such as
foliation may cause a dramatic weakening of the rock parallel to the
fabric
Pore Fluid PressurePore Fluid PressureIncreasing pore fluid pressure counteracts increasing Increasing pore fluid pressure counteracts increasing lithostatic:lithostatic:
Rock is weakenedRock is weakenedRock is likely to deform by brittle failureRock is likely to deform by brittle failure
TemperatureTemperature
Increasing T will decrease Increasing T will decrease the rock strength and the rock strength and favor ductile behavior (test favor ductile behavior (test is on basalt at various is on basalt at various TT°°C)C)
Strain RateStrain Rate
Increasing the Increasing the strain rate will strain rate will increase the rock increase the rock strength and favor strength and favor brittle behaviorbrittle behavior
Time FactorTime Factor
Given enough time any solid material will Given enough time any solid material will ““flowflow”” below below the elastic limitthe elastic limit-- this is termed this is termed ““Mechanical CreepMechanical Creep””
Differential Stress <Elastic Limit Fundamental
Strength
Rheidity ExamplesRheidity Examples
Marble Bench: Marble Bench: (see photo)(see photo)Continental Crust: Continental Crust: has suffered no has suffered no appreciable creep appreciable creep since Archeansince Archean
YoungYoung’’s Moduluss Modulus
YoungYoung’’s Modulus (E): relationship between s Modulus (E): relationship between σσ and and εε (i.e. (i.e. the slope on a stress v. strain graphthe slope on a stress v. strain graph
Bulk and Shear ModulusBulk and Shear Modulus
Bulk Modulus (K): measures resistance to Bulk Modulus (K): measures resistance to ∆∆ volume volume (dilation)(dilation)Shear Modulus (G): measures resistance to shear (Shear Modulus (G): measures resistance to shear (ττ))
PoissonPoisson’’s Ratios Ratio
PoissonPoisson’’s Ratio (s Ratio (νν ““nunu””): ration of lateral E to ): ration of lateral E to longitudinal Elongitudinal E
ViscosityViscosity
Viscosity (Viscosity (ηη ““etaeta””): resistance of a fluid to flow): resistance of a fluid to flow
Exam 2: Dynamic Analysis SummaryExam 2: Dynamic Analysis Summary
Be able to solve strain equations for S, Be able to solve strain equations for S, λλ, , γγ, , ΨΨ, , ααBe able to discuss the difference between homogenous and inhomogBe able to discuss the difference between homogenous and inhomogeneous straineneous strain-- give geological examplesgive geological examplesKnow how to calculate lithostatic stress given depth and densityKnow how to calculate lithostatic stress given depth and densityKnow how to solve a resolution of stress by vector addition probKnow how to solve a resolution of stress by vector addition problemlemKnow the general equations for Know the general equations for σσ and and ττ for the Mohr Circle, and know the relationship between terms infor the Mohr Circle, and know the relationship between terms in the the equation and circle geometryequation and circle geometryBe able to interpret the Mohr Circle/Mohr Fracture envelope diagBe able to interpret the Mohr Circle/Mohr Fracture envelope diagramramKnow the definitions and positions on a Know the definitions and positions on a σσ versus versus εε graph of Yield strength, Ultimate Strength, and Rupture graph of Yield strength, Ultimate Strength, and Rupture StrengthStrengthBe able to discuss the effects of the following on a Be able to discuss the effects of the following on a σσ versus versus εε graph:graph:
Lithostatic loadLithostatic loadTemperatureTemperatureStrain rateStrain ratePore Fluid pressurePore Fluid pressureLithologyLithology
Be able to discuss the Rheid concept and describe examples; knowBe able to discuss the Rheid concept and describe examples; know where Fundamental Strength is locatedwhere Fundamental Strength is locatedBe able to discuss YoungBe able to discuss Young’’s Modulus, Bulk Modulus, Shear Modulus, Poissons Modulus, Bulk Modulus, Shear Modulus, Poisson’’s Ratio, and Viscositys Ratio, and Viscosity