This week:PS #1 assigned (due next Tuesday)

Lab #1 due tomorrow at 11:00

Lectures: ch. 3- Force and Stress

Lab #2: Field trip to Mt Baldy- Will collect field books at end of field trip

This week:PS #1 assigned (due next Tuesday)

Lab #1 due tomorrow at 11:00

Lectures: ch. 3- Force and Stress

Lab #2: Field trip to Mt Baldy- Will collect field books at end of field trip

The stresses that cause deformationUnderstand "stress calculations”

Spend some time with these calculations to convince yourself that stress on a given plane resolves itself into a single stress tensor.

Stress () = force/unit area

= F/A

StressGoals

1) Interpret the stresses responsible for deformation.

2) Describe the nature of the forces that cause the stresses.

3) Understand the relations between stress, strain and rock strength.

Describing stress and force is a mathematical exercise.

All the deformation we see is due to Responses to Stresses

1) Folding

2) Brittle faults

3) Ductile shear zones

4) Joints

Stress = Force/AreaForce: changes in the state of rest or motion of a body.

Only a force can cause a stationary object to move or change the motion (direction and velocity) of a moving object.

force =

Weight is the magnitude of the force due to gravity (g) acting upon a mass. You weigh less on the moon (gravity is less)

mass x acceleration, F = ma,

ForceUnits of Force:

The newton (N) is the basic (SI) unit of force.

1 newton = 1 kg meter/sec2

Units of Force:

The newton (N) is the basic (SI) unit of force.

1 newton = 1 kg meter/sec2

1 newton is about the weight of one small apple (remember Sir Newton and the apple!)

Forces as Vectors

Force is a vector - it has magnitude and direction. Vectors can be added and subtracted using vector algebra. We can evaluate vectors in order to determine whether the forces on a body are in balance.

If sum of the forces=0,Then there is no change in motion (no acceleration)

Forces as Vectors

Magnitude and direction

Load

Force

If sum of the forces=0,Then there is no change in motion (no acceleration)

Forces in the Geologic World

Typically we think of the Earth as at rest - in static equilibrium, or moving very slowly.

When there are net forces, they cause accelerations that are usually one of 2 kinds:

1) slow ponderous motion of a tectonic plate that increases or decreases velocity over a very long time, or;

2) sudden, short lived, strong accelerations during fault slip accompanying earthquakes.

Two Types of Forces

1) Body forces, that act on the mass of a body and are independent of forces applied by adjacent material.

As geologists, there are two body forces that we really care about. What are they?

2) Contact forces, are pushes and pulls across real or imaginary surface of contact such as faults.

Three different type of loading due to contact forces:

1) gravitational loading - pushing on adjacent rock.

2) thermal loading - expansion or contraction.

3) displacement loading - push due to motion.

Gravity, magnetic field

Stress ()Stress is force per unit area: = F/A

In class demonstration of force, stress, and deformation.

Units of Stress1 pascal = 1 newton/m2 (Force/area)

• 1 newton is about 0.2 pounds of force

•1 pascal is about 0.02 lb/ft2, thus for geologists, we often measure pressure in MPa

• 1 MPa = 145 lb/in2

• 9.81 Pa is the pressure caused by a depth of 1mm of water

•What is the pressure at the bottom of a swimming pool filled with water?•Filled with quartzite?•What is the pressure at the bottom of a swimming pool filled with water?•Filled with quartzite?

Units of Stress

Sometimes you will see stress in the units of bars or kbars1 bar=105 Pa

What is the pressure at the bottom of a swimming pool in kbars?

As geologists, we have to deal with situations when stresses act at an angle to planes (like faults and shear zones).In this case we have to resolve the stresses into components

Stress as Vectors

Stress is a vector - it has magnitude and direction.

As geologists, we’re interested in stresses that compress things, and stresses that shear things.

So, we resolve the stress vector into into it’s components

Stress on a plane:

Normal stress act perpendicular to the plane Shear stress act along the plane.

Normal and shear stresses are perpendicular to one another

Relations between F and

Stress on a 2-D plane:

Force F is applied to a plane with the trace EF.

Magnitudes of vectors Fs and Fn is function of angle

At what angle is the normal force the maximum?

At what angle is the shear force the maximum

Fn = F cos ,

Fs = F sin

What is the equation for the normal force? For the Shear force?

Relations between F and

The magnitude of normal and shear stresses are a more complicated function of angle ,

n= Fn/Area

where

Fn= F cos

Area=AB/(cos

Therefore

ncos2

s = sin2

Stress on a 2-D plane:

Stress=Force/Area

=F/AB

Stress on a 2-D plane:

Stress=Force/Area

=F/AB

At what angle is the normal stress the maximum?At what angle is the shear stress the maximum?

At what angle is the normal stress the maximum?At what angle is the shear stress the maximum?

Relations between F and

ncos2

s = sin2

Maximum normal stress when force is perpendicular (90°) to the plane

Maximum shear stress when force 45° to plane

ncos2

s = sin2

Maximum normal stress when force is perpendicular (90°) to the plane

Maximum shear stress when force 45° to plane

Stress ellipsoidDescribing the state of stress at a point:

A point represents the intersection of an infinite number of planes. The stress at this point can be represented by the stresses on all of these planes.

Graphically, all the stresses on all these planes describe an ellipse.

(Draw stress ellipse)

Stress ellipsoid Describing the state of stress at a point:

Graphically, all the stresses on all these planes describe an ellipse, with one long axis (the direction of maximum stress), and one short axis (the direction of minimum stress). These two axis are perpendicular

Stress in 3-D Stress ellipsoid

In 3-D, all the stresses on all these planes describe an ellipsoid, with one long axis (the direction of maximum stress), and one short axis (the direction of minimum stress), and one intermediate axis.

In 3-dimensions, the ellipsoid is defined by three mutually perpendicular principal stresses (> 2 > 3).

What is important about the principal stresses (> 2 > 3)?

The axes are perpendicular to each other.

They do not contain shear stresses

The state of stress of any body is described by the orientation and magnitude of the principal stresses.

Stress ellipsoid

Geology sign conventions

Tensional stress is – (negative)

Clockwise shear stress is – (negative)

Counter clockwise shear stress is + (positive)

Mt Baldy Mapping Field Trip

Mapping Guidelines

General Geology of area-stratigraphy-general structures

KML of foldHand out Map

-figure out general structure, -strike and dip of north limb

Axial Planar Cleavage

Mt Baldy Regional Stratigraphy:

Colulmbia River Basalts, Basalt, basalt…

Mt BaldyRegional Structures:

Yakima Fold Belt

Mt. Baldy Lidar on Google Earth

Mt Baldy Map Contoured Shaded Relief from lidar

Axial Planar Cleavage

Map symbol for attitude of beddingMap symbol for attitude of cleavage

Lecture outline1.Overview of stress2.Minimum and maximum stress3.Types of stress on a plane

a. Normal stressb. Shear stress4. Hydrostatic state of stress5. Stress and the Mohr circle6. Mean stress7. Differential stress8. Deviatoric stress

Relations between F and

ncos2

s = sin2

Maximum normal stress when force is perpendicular (90°) to the plane

Maximum shear stress when force 45° to plane

ncos2

s = sin2

Maximum normal stress when force is perpendicular (90°) to the plane

Maximum shear stress when force 45° to plane

Reference system x, y, zImagine a cube of rock:Every face experiences normal and shear stress =>

Each face has one component of normal stress

Each face has two components of shear stresses

Stress in 3 D:We can also think about stresses on the sides of imaginary cubic rocks:

Stress StateIf the principal stress are unequal in magnitude = anisotropic stress –The greatest stress is called 1

-The intermediate stress, 2 and minimum stress is called 3

If the 3 principal stresses are equal in magnitude = isotropic stress and the state of stress is represented by a sphere, not an ellipsoid.Referred to as:

Pressure,Hydrostatic stressLithostatic stressIsotropic stress

1 > 2 > 3

1 = 2 = 3

Hydrostatic Stress a.k.a.PressureLithostatic stressIsotropic stressIf we calculate stress vectors within a point of a hydrostatic stress field, we find that all of the stress vectors in all directions have the same value. Stress ellipse is a stress sphere.

Hydrostatic stress = all stresses are equal in all directions. Therefore, No shear stresses!

Equal stress magnitudes in all directions. Dive into a pool. All stresses have the same values.

Stress on a dipping plane in the Earth’s crust

n = cos2

s = *1/2(sin2

Review sign conventions for normal and shear stresses

2 componentsNormal stress & Shear stress

Stress on a plane examples…

We resolve stress into two components

Normal stress, n and the component that is parallel to the plane, shear stress, s

1) Normal compressive stresses tend to inhibit sliding along the plane and are considered positive if they are compressive.

2) Shear stresses tend to promote sliding along the plane, labeled negative if its right-lateral shear and possitive if its left-lateral shear.

Squeeze a block of cement between two planks of wood

AB, trace of fracture plane that makes an angle with

The 2-D case is simple, since

(atmospheric pressure)

Important: What is angle

Maximum principal stress (1) and minimum stress (3) act on plane P that makes an angle with the 3 direction.

In Mohr space, we plot 1 and 3 on n-axis

These principal stress values are plotted on the n-axes because they are the normal stresses acting on plane P.

The principal stresses always have zero shear stress values (s = 0).

Mohr Stress DiagramMohr circle: n on x-axiss on y-axis.

Mohr Stress Diagram

a)This give us a useful picture or diagram of the stress equations.

b) They solve stress equations on page 49 (Eqs 3.7 and eq. 3.10)

c) Plot N versus S

d) Rearrange Eqs. 3.7 and 3.10 and square them yields

[n – ½(1 + 2]2 + s2 = [½(1 – 3

2 )]

form (x –a)2 + y2 = r2

Important: What is angle

In Mohr space, we use 2

a) Mohr circle radius = ½(1 – 3] that is centered on ½(1 + 3] from the origin.

b) The Mohr circle radius, ½(1 - 2] is the maximum shear stress s max.

c) The stress difference (1 – 3), called differential stress is indicated by d.

Mohr Stress Diagram

Construct a circle thought points 1 and 3 with 0, the midpoint, at ½(1 + 3) as the center with radius, ½(1 - 3].

Now draw a line OP, so that angle PO1 is equal to 2 – confusing step, plot twice the angle which is the angle between the plane and 3.

Remember we measure 2 from the 1 side on the n-axis.

We can read the values of n,p along the n-axis, and s,p along s-axis for our plane P.

Mohr Stress Diagram

n,p = ½(1 + 3] + ½(1 - 3] cos 2

s,p = ½(1 - 3] sin 2

Remember,

Because a body’s response to stress, we subdivide the stress into two components, mean and deviatoric stress.

Mean stress = [1 + 2 + 3]/3 or m

In 2-D, [1 + 3]/2

Deviatoric stress is the difference between the mean stress and total stress. total mean + dev

mean is often called the hydrostatic component (1 = 2 = 3)

Mean stress and deviatoric stress

When the principal stress magnitudes change w/o differential stress, the Mohr circle moves along the n-axis without changing s

Mohr Stress Diagram

How is this achieved?

Suggest geologic examples?

When the principal stress magnitudes change w/o differential stress, the Mohr circle moves along the n-axis without changing s

Mohr Stress Diagram

1) Change confining pressure (Pc). Increase air pressure on our clay experiment, or carry the experiment underwater.

2) Burial of rocks changes confining pressure. Which way along the n-axis?

3) Exhumation of rock changes confining pressure. Again, in what direction along the n-axis?

Various states of stress

Uniaxial compression, two of the three principal stresses are zero.

Hydrostatic stress, a single point on the Mohr circle that lies on the x-axis. All normal stresses are the same, and no shear stresses.

Various states of stress

Triaxial stress, all three principal stresses are different.

Biaxial stress, all three principal stresses are non-zero, but two of the principal stresses have the same value. Typical stress ellipse (plane stress).

For rocks at depth, the weight from the overlying rocks creates lithostatic pressure.

Consider a rock at 3 km depth. Lithostatic pressure (weight of rock of overlying column).

Pl = * g * h if (density) = 2700 kg/m3, g (gravity) = 9.8 m/s2 and h (depth) is 3000 m, we get:

Pl = 2700 x 9.8 x 3000 = 79.4 x 206 Pa ~ 80 Mpa

For every km in the Earth’s crust, the lithostatic pressure increases 27 MPa.

The lithostatic pressure is equal in all directions (isotropic stress), [1 = 2 = 3]

Lithostatic pressure (Pl).

So we divide the rocks state of stress into an isotropic (lithostatic/hydrostatic) and an anisotropic (deviatoric).

Isotropic stresses act equally on all directions, resulting in a volume change of the rock – increase water pressure on a human, or air pressure on take-off or landing.

Deviatoric stress, changes the shape of the body. The difference between isotropic stress and additional stress from tectonic forcing.

Lithostatic pressure (Pl).

Present day stressDifficult to measure

EQ focal mechanisms Bore-hole breakouts in situ measurements

Measuring Stress

in situ borehole measurements of d (1 –3) with depth.

World stress map and topography showing maximum horizontal stress.

Stress in the Earth

Generalized pattern based on stress trajectories for individual plates.

Stress in the Earth

Strength – the ability of a material to support different stress

Maximum stress before a rock fails

Strength curves: differential stress magnitude versus depth.

Stress and strength at depth

A. Regional with low geothermal gradient

B. Regional high geothermal gradients

Give some geologic examples?

This is important and will be on exam 1!

Stress and strength at depthVery sensitive to rock type, And presence of water (hydrous minerals)

Stress and strength at depthIntegrate the strength of the column, to get strength of lithosphere. Can predict where rifts will occur (where lithosphere is weakest)Helps to understand Wilson Cycle