Chapter 3 Rock Mechanics Stress Basic Physics Force

Chapter 3 Rock Mechanics Stress

Basic Physics ► Force § that which changes the state of rest or the state of motion of a body F=ma ► Stress § force applied to an area σ=F/A

Basic Physics Scalar § Possesses only a magnitude at some point in time or space ► Vector § Possesses both magnitude and direction ► Tensor § A field of data with magnitudes and directions ►

Basic Physics ► Tensors § Zero-order tensor is a scalar like temperature and has only 1 component § First-order tensor is a vector like wind direction and is described by 3 components (time, magnitude, direction) § Second-order tensor relates sets of tensors to each other and has 9 components The number of components may be determined from 3 n where n in the order of the tensor

Basic Physics ► Stress can be § Tensional - Pulling apart § Compressional - Pushing together

Basic Physics ► Stress on a surface can be broken into two vector components § Normal Stress (σn) - acts perpendicular to the reference surface § Shear Stress (τ)- acts parallel to the surface

Basic Physics ► Principal normal stress components (σ1, σ2, and σ3) § These are oriented perpendicular to each other and σ1 σ2 σ3 § Differential stress is the difference between the maximum (σ1) and the minimum (σ3) § Mean stress is (σ1 + σ2 + σ3)/3 § If the differential stress exceeds the strength of the rock, permanent deformation occurs

Basic Physics Lithostatic state of stress § Occurs where the normal stress is the same in all directions ► Hydrostatic Pressure § Confining stress acting on a body submerged in water ► Lithostatic Pressure § Confining stress acting on a body under ground ►

Stress on a plane ► Horizontal plane ► F = ma = volume x density x acceleration ► F = 104 m 3 x 2, 750 kg m-3 x 9. 8 ms-2 ► Plane is 1 x 1 m, A = 1 m 2 ► What is the Stress?

Stress on a plane ► σ=F/A ►F = (2. 7 x 108 kg ms-2)/1 m 2 ► 2. 7 x 108 kg m-1 s-2 or 2. 7 x 108 Pa or 269. 5 MPa

Stress on a plane ► Inclined Plane at 45º ► Through the same 1 m x 1 m space, actually has a larger surface area, now 1. 41 m 2 ► Still F = 2. 7 x 108 kg m s-2 ► So σ=F/A § σ= (2. 7 x 108 kg m s-2)/1. 41 m 2 § or 191 MPa § How does that compare to the stress on the horizontal plane?

Stress on a plane ► Stress can be broken down into components of normal and shear stress. § σn = σ cos 45º § = 191 MPa x 0. 707 § = 135 MPa § τ = σ sin 45º § = 191 MPa x 0. 707 § = 135 MPa

Stress Ellipsoid ►A Shear Ellipsoid is a graphical means of showing the relationship between the principal stresses § The axes represent the principle normal stress components σ1, σ2, and σ3 § The planes of maximum shear stress are always parallel to σ2 and at 45º to σ1 and σ3.

Triaxial Test Apparatus

Mohr Circle Diagram ► Created by Otto Mohr, a german engineer, in 1882 ► Enables us to determine the normal and shear stress across a plane

Mohr Circle Diagram τ τ, P

Mohr Circle Diagram

Measuring Present-Day Stress

Stress in the United States