Chapter 4Chapter 4Rock MechanicsRock Mechanics

StrainStrain

StrainStrain

►By comparing rocks in the deformed By comparing rocks in the deformed state to the original undeformed state, state to the original undeformed state, we get a better understanding of we get a better understanding of tectonic structures.tectonic structures.

DefinitionsDefinitions

►Deformation - The displacement field Deformation - The displacement field for tectonically driven particle motionsfor tectonically driven particle motions

►Distortion - Involves a change in Distortion - Involves a change in shape, rotation, and translation.shape, rotation, and translation.

►Strain - Aspects of shape change Strain - Aspects of shape change measured in line length, angular measured in line length, angular relationship between lines, or volume.relationship between lines, or volume.

DeformationDeformation

Strain

DefinitionsDefinitions

►Homogenous Strain - Lines that are Homogenous Strain - Lines that are straight and parallel before straight and parallel before deformation remain straight and deformation remain straight and parallel after deformation. parallel after deformation.

► Inhomogenous Strain - The landscape Inhomogenous Strain - The landscape is distorted and lines may be broken.is distorted and lines may be broken.

Homogenous and Homogenous and Inhomogenous StrainInhomogenous Strain

Homogenous Strain

Inhomogenous Strain

DefinitionsDefinitions

► Incremental Strain - Describes the Incremental Strain - Describes the deformation history of the rock body.deformation history of the rock body. Usually cannot differentiate the Usually cannot differentiate the

deformation history.deformation history.

►Finite Strain - A comparison can be Finite Strain - A comparison can be made between the present shape and made between the present shape and some previous shape. some previous shape. Finite strain is path independent.Finite strain is path independent.

Measures of StrainMeasures of Strain

►Strain may be recognized as a change Strain may be recognized as a change in line length, angles between lines, or in line length, angles between lines, or volume.volume.

Linear StrainLinear Strain

►Elongation - The ratio of the length of Elongation - The ratio of the length of the line in the deformed mass (lthe line in the deformed mass (l11) ) minus the length of the original line minus the length of the original line (l(l00).).

ε = (lε = (l11 - l - l00)/l)/l00 = Δl/l = Δl/l

Linear StrainLinear Strain

►Stretch - also called engineer’s stretch Stretch - also called engineer’s stretch - is simply the length of the deformed - is simply the length of the deformed line divided by the original length.line divided by the original length.

S = lS = l11/l = l + ε/l = l + ε

Linear StrainLinear Strain

►Quadratic Elongation - The square of Quadratic Elongation - The square of the stretch.the stretch.

λ = (lλ = (l11/1/100))22 = (1 + ε) = (1 + ε)22 = S = S22

Linear StrainLinear Strain

Elongation?

Stretch?

Quadratic elongation?

ε = (l1 - l0)/l0 = (5-3)/3 = 0.67

S = l1/l = 5/3 = 1.67

λ = (l1/10)2 = (5/3)2 = 2.78

Shear StrainShear Strain

►Shear StrainShear Strain (γ) - Strain that (γ) - Strain that results when parts of a rock body results when parts of a rock body are deformed so that angles are deformed so that angles between originally orthogonal between originally orthogonal reference lines are rotated. reference lines are rotated.

γ = tan ψ γ = tan ψ

Dilation StrainDilation Strain

►Dilation StrainDilation Strain (Δ) - Changes in volume (Δ) - Changes in volume Three possible mechanismsThree possible mechanisms

►Closing voids - Negative volume changeClosing voids - Negative volume change►Dissolving by pressure solution - Negative Dissolving by pressure solution - Negative

volume changevolume change►Fracturing the mass - Positive volume changeFracturing the mass - Positive volume change

Δ = (VΔ = (V11 - V - V00)/V)/V00 = δV/V = δV/V00

Dilation StrainDilation Strain

Strain EllipsoidStrain Ellipsoid► Strain EllipsoidStrain Ellipsoid - Graphical tool that provides - Graphical tool that provides

a reference object for estimating shape a reference object for estimating shape change from an assumed initial sphere.change from an assumed initial sphere.

► Elliptical sections through these are sometimes printed on Elliptical sections through these are sometimes printed on geologic maps to indicate geologic strain.geologic maps to indicate geologic strain.

► Made of three mutually perpendicular axes x, y, and z, Made of three mutually perpendicular axes x, y, and z, where Xwhere X Y Y Z. Z.

The Strain Ellipsoid usually has an inverse relationship with the Stress Ellipsoid. X corresponds to σ3.

ShearShear►Simple ShearSimple Shear - Rotational homogenous - Rotational homogenous

shear with motion between layers.shear with motion between layers.

►Pure ShearPure Shear - Distortion by homogenous - Distortion by homogenous deformation without rotation.deformation without rotation.

Strain MeasurementsStrain Measurements►Strain MarkersStrain Markers - A deformed - A deformed

feature in the rock that can be feature in the rock that can be measured to determine strain.measured to determine strain. Have to know the original shape for Have to know the original shape for

comparison.comparison. Should have the same mechanical Should have the same mechanical

characteristics as the original rock.characteristics as the original rock.

Strain MarkersStrain Markers

►Good strain markers include:Good strain markers include: Reduction SpotsReduction Spots PebblesPebbles OoidsOoids FossilsFossils VesiclesVesicles Pillow BasaltsPillow Basalts BurrowsBurrows

Flinn DiagramFlinn Diagram

►Flinn Diagram - Most useful means of Flinn Diagram - Most useful means of displaying constant-volume finite strain.displaying constant-volume finite strain.

► Invented by the British structural Invented by the British structural geologist Derek Flinn in 1962.geologist Derek Flinn in 1962.

k = (Rk = (Rxyxy-1)/(R-1)/(Ryzyz-1)-1)

RRxyxy= (1+ε= (1+ε11)/(1+ ε)/(1+ ε22))

RRyzyz= (1+ε= (1+ε22)/(1+ ε)/(1+ ε33))

Wellman’s MethodWellman’s Method► Wellman’s MethodWellman’s Method - A simple geometric - A simple geometric

technique for determining the technique for determining the orientation and shape of the strain orientation and shape of the strain ellipse.ellipse. Requires at least ten strain markers.Requires at least ten strain markers. All must be on the same plane.All must be on the same plane. Most commonly use brachiopods or trilobites.Most commonly use brachiopods or trilobites.

R MethodR Method