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Chapter 3 Rock Mechanics Stress

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

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Chapter 3 Rock Mechanics Stress

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  1. Chapter 3Rock MechanicsStress

  2. Basic Physics • Force • that which changes the state of rest or the state of motion of a bodyF=ma • Stress • force applied to an areaσ=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 • = 191 MPa x 0.707 • = 135 MPa

  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

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