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

RheidityRheidity

Definition (Carey, 1953):Definition (Carey, 1953):

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