Chapter 3 Rock Mechanics Stress

Basic Physics Force Stress 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 3n 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 m3 x 2,750 kg m-3 x 9.8 ms-2 Plane is 1 x 1 m, A = 1 m2 What is the Stress?

Stress on a plane σ=F/A F = (2.7 x 108 kg ms-2)/1m2 2.7 x 108 kg m-1s-2 or 2.7 x 108 Pa or 269.5MPa

Stress on a plane Inclined Plane at 45º Through the same 1m x 1m space, actually has a larger surface area, now 1.41 m2 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 m2 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

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

Mohr Circle Diagram

Measuring Present-Day Stress

Stress in the United States