Chapter 3

Force and Stress

In geology, the force and stress have very specific meaning.• Force (F): the mass times acceleration (ma) (Newton’s

second law); F=ma.

• Stress (σ): Force per unit area (F/A); σ = F/A

Stress can be considered as the intensity of force.

σ = (ma/A) = (kg.m-1.s-2), which is called Pascal (Pa).

Stress

Megapascal (MPa) = 106 Pa and Gigapascal (GPa) = 109 Pa

In geology we used (bar) = 105 Pa

1kbar = 1000 bar = 108 Pa = 100 MPa = 0.1GPa

• Tensors: The forces within the earth act over surfaces or through volumes of material, and they generate stresses which are three-dimensional entities.

• Mechanics: is the branch of science which concerned with the action of forces on bodies and their effect.

• In nature we can recognize four basic forces:

1. Gravitational force

2. Electromagnetic force

3. Nuclear or strong force

4. Weak force

• Body forces: are forces that result from action of a field at every point within the body.

• Surface forces: are forces that act on a specific surface area in a body.

Normal Stress (σn) and Shear (σs) or τ

Stress in 3 Dimension can be expressed by Stress ellipsoid which describes the stress in 3 D and enable us to determine the principal stresses

σ1 , σ2 and σ3 acting on any plane.

σ

F

Stress in 3 Dimension

Principal Stresses

Stress ellipse: stress in 2 D

σ1

σ3

Normal stress and shear stress

Stress acting on a plane is a vector quantity, which meaning that it has both magnitude and direction.

Fn = F * cosθ = σ EF cos2θ  

Fs = = F * sinθ =½σ EF (sin 2θ)

 

Thus the stresses are:

 σn = Fn/EF = σ cos2θ

 

σs = Fs/EF = ½σ (sin 2θ)

Relation between Normal Stress σn, Shear Stress σs and Principal

Stresses σ1 , σ2 and σ3

σ2

Applications to Geological structures

σn=1/2(σ1+σ3)+1/2(σ1-σ3)cos2θσs =1/2(σ1-σ3)sin2θ

Application on Mohr Circle

• Problem 1

Given the principle stresses of σ1 =100 MPa (vertical) and σ3 = 20 MPa (horizontal), determine the normal stress σn and shear stress σs on a fault plane that strikes parallels to σ2 and dips 32° with σ3 .

Problem 2 Determine the σn and σs on planes 2 through 5 and plot them on Figure below, and plot them on the Mohr diagram. (Recall that trigonometric functions of angles in the second and fourth quadrants are negative, e.g., cos180°=-1).

PROBLEM 3

• If σ1 is vertical and equal to 50MPa and σ3 is horizontal, E-W, and equal to 22MPa, using a Mohr circle construction to determine the normal and shear stresses on a fault striking N-S and dipping 60o E.

Stress statesσ1 σ2 σ3

The following relationships between the principal stresses define common stress states:

General triaxial stress: σ1 > σ2 > σ3 0

Biaxial (plane) stress: σ1 > 0 > σ3, one axis=0

Uniaxial tension :σ1 = σ2 = 0; σ3 < 0

Uniaxial compression: σ2 = σ3 = 0; σ1> 0

Hydrostatic stress (pressure): σ1 = σ2 = σ3

4 cases represented on Mohr circles

Isotropic and Anisotropic stresses

• Isotropic stress:

a. hydrostatic pressure

b. lithostatic pressure

They form change in volume

• σ= ρ*g*h (Pa)

=2700*9.8*1500=39690000=396.9 Bar

• Anisotropic (Deviatoric) stress :

It forms changes in the shape of a body (strain). It is due to the tectonic stressestectonic stresses

Homogenous and Heterogeneous stress field:

•Homogenous: the stress field is the same in magnitude and orientation at every point throughout the body.

•It is heterogeneous if it is not the same.

Here, you must differentiate between homogenous and isotropic.

Source of inhomogenity in the crust are:1. Fracture in rocks 2. Contrast in viscosity3. The complex interplay of more than one stress field.

Stress Trajectories

Methods of stress measurement

• Bore-hole breakouts

• Hydrofracturing

• Strain release (In-situ stress measurements)

• Analysis of faults and fractures

• Fault-plane solutions (Earthquake focal mechanisms)

2. Paleostress: through determine stresses on fault plane by measuring attitudes of fault and slickenlines pitches.

3. Stress in the earth: We can divided the global stress field into "stress provinces", which generally correspond to geologic provinces (Fig. 3.11).

1. Present-day stress: by analysis of fault planes and fractures

• The main present-day driving forces (stress field) of plate tectonic include:

1. Pull of the down-going slab in subduction zones (Slab pull)

2. Push at ocean ridges (Ridge Push)

3. Continent-continent collision