Chapter 3 Force and Stress

1. Chapter 3Force and Stress

2. 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 andGigapascal (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. 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 ellipsoidwhich describes the stress in 3 D and enable us to determine the principal stresses σ1 , σ2 andσ3 acting on any plane. F σ

10. Principal Stresses

11. σ3 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 andσ3 σ2

16. Applications to Geological structures

17. σn=1/2(σ1+σ3)+1/2(σ1-σ3)cos2θσs =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 σnand 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 • 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 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