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

Stress

Goals

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.

Responses to Stresses

1) Folding

2) Brittle faults

3) Ductile shear zones

4) Joints

ForceForce: 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 = mass x acceleration, F = ma,

mass = density x volume, m = V,

therefore, = m/V,

Weight is the magnitude of the force of gravity (g) acting upon a mass.

The newton (N) is the basic (SI) unit of force.1 newton = 1 kg meter/sec2

1 dyne = 1g cm/sec2 so 1 N = 105 dyne

1 pascal = newton/m2

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.

Load

Force

Units of Stress

1 newton = 1 kg meter/sec2 = this is a unit of force

1 pascal = 1 newton/m2 = unit of stress

• 1 newton is about 0.224 809 pounds of force

•1 pascal is about 0.020 885 lb/ft2, thus pressure is measured in kPa

• 1 kPa = 0.145 lb/in2

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

Stress on a 2-D plane:

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

Normal and shear stresses are perpendicular to one another

Stress ()Stress is force per unit area:

= F/A

Relations between F and

(a) Fn and Fs and angle with top and bottom surface. EF is trace of plane, ABCD is cube with ribs of length AG.

Magnitudes of vectors Fs and Fn is function of angle

Fn = F cos , Fs = F sin

(b) The magnitude of normal and shear stresses is function of angle and the area,

n= cos2

s = sin2

A point represents the intersection of an infinite number of planes and stresses on these planes describe an ellipse.

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

These three axes are normal to the principal

Stress ellipsoid

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

Components of stress

Three normal stresses

Components parallel shear stresses

Reference system x, y, z

Geology sign conventions

Tensional stress is – (negative)

Clockwise shear stress is – (negative)

Counter clockwise shear stress is + (positive)

Stress State

If the 3 principal stresses are equal in magnitude = isotropic stress

Here the state of stress is represented by a sphere, not an ellipsoid.

If the principal stress are unequal in magnitude = anisotropic stress

Here the greatest stress is called 1

The intermediate stress, 2 and minimum stress is called 3

1 > 2 > 3

As a geologist, what is it called if all three principal stresses are equal?

1 = 2 = 3

Hydrostatic Stress

If we calculate stress vectors within a point of a hydrostatic stress field, we find that the stress vectors have the same value. Each stress vector is oriented perpendicular to the plane.

All stress vectors are normal vectors, they have no shear stress components.

Hydrostatic stress = all principal stresses in a plane are equal in all directions. No shear stresses!

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

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

a. Normal stressb. Shear stress3. Mean stress4. Differential stress5. Deviatoric stress6. Hydrostatic state of stress7. Stress and the Mohr circle

Problem set outline

1.Apparent dip2.Angle between lines3.Angle between planes

Stress on a dipping plane in the Earth’s crust

2 componentsNormal stress & Shear stress

n = cos2

s = sin2

Review sign conventions for normal and shear stresses

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) Normal tensional stresses tend to separate rocks along the plane and values are considered negative.

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

Squeeze a block of clay 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

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

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 Diagram Mohr circle: n on x-axiss on y-axis.

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,

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?

Problem set #1.

Handouts in class and go online for additional graph paper in Mohr space

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).

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

For rocks at depth, we use lithostatic pressure.

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

Pl = x g x h if (density) = 2700 km/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

2 ]

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 depth

Stress and strength at depth