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) = 10 6 Pa and Gigapascal (GPa) = 10 9 Pa In geology we used (bar) = 10 5 Pa 1kbar = 1000 bar = 10 8 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 (σ n ) and Shear (σ s ) 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 σ 1, σ 2 and σ 3 acting on any plane. σ F Stress in 3 Dimension

Principal Stresses

Stress ellipse: stress in 2 D σ1σ1 σ3σ3

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

F n = F * cosθ = σ EF cos 2 θ F s = = F * sinθ =½σ EF (sin 2θ) Thus the stresses are: σn = F n /EF = σ cos 2 θ σs = F s /EF = ½σ (sin 2θ)

Relation between Normal Stress σ n, Shear Stress σ s and Principal Stresses σ 1, σ 2 and σ 3 σ2

Applications to Geological structures

σ n =1/2(σ 1 +σ 3 )+1/2(σ 1 -σ 3 )cos2θ σ s =1/2(σ 1 -σ 3 )sin2θ

Application on Mohr Circle Problem 1 Given the principle stresses of σ 1 =100 MPa (vertical) and σ 3 = 20 MPa (horizontal), determine the normal stress σ n and shear stress σ s on a fault plane that strikes parallels to σ 2 and dips 32° with σ 3.

Problem 2 Determine the σ n and σ s 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 σ 1 is vertical and equal to 50MPa and σ 3 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 60 o E.

Stress states σ 1  σ 2  σ 3 The following relationships between the principal stresses define common stress states: General triaxial stress: σ 1 > σ 2 > σ 3  0 Biaxial (plane) stress: σ 1 > 0 > σ 3, one axis=0 Uniaxial tension :σ 1 = σ 2 = 0; σ 3 < 0 Uniaxial compression: σ 2 = σ 3 = 0; σ 1 > 0 Hydrostatic stress (pressure): σ 1 = σ 2 = σ 3

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= =396.9 Bar Anisotropic (Deviatoric) stress : tectonic stresses It forms changes in the shape of a body (strain). It is due to the tectonic 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 viscosity 3. 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