1. Chapter 3 Rock Mechanics Stress

2. 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

3. 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

4. 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

5. Basic Physics Stress can be Tensional - Pulling apart
Compressional - Pushing together

6. 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

7. 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

8. 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

9. 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?

10. 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

11. 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?

12. 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

13. 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.

14. Triaxial Test Apparatus

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

16. Mohr Circle Diagram τ
τ, P

17. Mohr Circle Diagram

18. Mohr Circle Diagram

19. Measuring Present-Day Stress

20. Stress in the United States