Transcript No Slide Title

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 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
 = 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
Mohr Circle Diagram
Measuring Present-Day Stress
Stress in the United States