1. Chapter 3 Force and Stress

2. Stress can be considered as the intensity of force.
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).

3. 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

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

5. In nature we can recognize four basic forces: Gravitational force
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:
Gravitational force
Electromagnetic force
Nuclear or strong force
Weak force

6. 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.

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

9. Stress in 3 Dimension 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. F σ

10. Principal Stresses

11. Stress ellipse: stress in 2 Dσ3 σ1
Stress ellipse: stress in 2 D

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

13. 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)

15. Relation between Normal Stress σn, Shear Stress σs and Principal Stresses σ1 , σ2 and σ3

16. Applications to Geological structures

17. σn=1/2(σ1+σ3)+1/2(σ1-σ3)cos2θ σs =1/2(σ1-σ3)sin2θ

18. 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 .

19. 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).

20. 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.

21. Stress states
σ1  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

22. 4 cases represented on Mohr circles

23. Isotropic and Anisotropic stresses
a. hydrostatic pressure
b. lithostatic pressure
They form change in volume
σ= ρ*g*h (Pa)
=2700*9.8*1500= =396.9 Bar
Anisotropic (Deviatoric) stress :
It forms changes in the shape of a body (strain). It is due to the tectonic stresses

25. 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 viscosity
3. The complex interplay of more than one stress field.

26. Stress Trajectories

27. Methods of stress measurement
1. Present-day stress: by analysis of fault planes and fractures
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).

30. The main present-day driving forces
(stress field) of plate tectonic include:
Pull of the down-going slab in subduction zones (Slab pull)
Push at ocean ridges (Ridge Push)
Continent-continent collision