Week 5

Dynamic Analysis II Read pages 122-149 in Chapter 3: Dynamic Analysis.

• readings
• terms
• questions
• classnotes

You are expected to read all the sections listed below. Information from the sections in italics will be discussed in class. You are expected to read the other sections and you may be called on in class to answer questions based on that material.

• Experimentally Observed Relationships Between Stress and Strain p.122-142
  Objectives and Hurdles
  The Value of Laboratory Deformational Experiments
  Sample Preparation
  Types of Tests
  Pressures, Stresses, and Loads
  Measuring Shortening
  Measuring Strain Rate
  A Standard Axial Compression Test
  Compression Test at Higher Confining Pressure
  Still Higher Confining Pressure
  Strength and Ductility
• Elastic, Plastic and Viscous Models p.142-149
  The Need for Models of Behavior
  Elastic Behavior
  An Example of the Significance of Young's Modulus
  Plastic Behavior
  Viscous Behavior
  Summary

  return to top of page

You should become familiar with the following terms during this weeks lectures and readings:

return to top of page

You should be able to answer the questions below following this week:

1. Describe three ways by which you could tell that an experimental sample had undergone ductile deformation.
2. How much lateral expansion would occur if a 4 cm cube of limestone (fine grained) was shortened by 1 cm? (use data in Table 3.5)
3. Two rock deformatin experiments were conducted. In experiment A, a specimen underwent brittle deformation and failed at a rupture strength of 200 MPa. In experiment B, a rock failed at 100 MPa under ductile conditions. Describe three differences between experiments A and B that would explain how these results.
4. Consisely define what is meant by: i) rheology; ii) differential stress; iii) effective stress; iv) hysteresis loop; v) Poissons's ratio; vi) anisotropy; vii) hydrostatic stress; viii) Newton.
5. Describe how rock strength varies with changing physical properties and/or deformation conditions. Illlustratre your answer using stress-strain graphs.

return to top of page

Dynamic Analysis II

rheology - response of rocks to stress

Experimental rock deformation

• study interaction of stress and strain under physical conditions typical for the earth's crust

Experimental procedure

1. core rock, grind ends of specimen smooth and parallel.
2. measure width and length of core.
3. surround core with jacket of copper or plastic.
4. place core between two pistons in a sealed pressure vessel.
5. model burial (confining) pressure by pumping fluid (e.g. kerosene) into vessel = hydrostatic stress. Temperature, pore pressure, and strain rate can be varied in some rigs also.
6. apply axial load by driving one piston towards the other. Force measured on a calibrated gauge. Axial compression is most common type of test, assumes triaxial condition with minimum and intermediate stresses equal.
7. Stress measured by dividing force by area of specimen in contact with piston.
8. Strain measured by comparing before and after lengths of specimen outside of rig.
9. Experiments can be conducted in same apparatus that model fault-fold relationships or how fault strength changes with different fault rock composition and/or fabric.

Stress-strain diagrams

stress-strain diagrams plot differential stress vs. strain

• differential stress = axial stress - confining pressure = [maximum principal stress - minimum principal stress]
• use diagrams to understand how a specimen deforms
• stress is applied and only elastic deformation occurs
• elastic strain = strain recovered when applied stress is removed (no permanent deformation)
• see stress-strain diagram: linear relationship between stress and strain
  additional stresses would result in rupture
  yield point = where curve diverges from straight line
  maximum strength = ultimate strength
• under relatively low pressures the specimen breaks and a fault(s) form
• continued stress results in movement on fault surface
• as confining pressure increases the style of deformation changes from fracture on a single fault plane to pervasive microscopic deformation, stress-strain curve "plateaus"

Rock strength (& Ductility)

Lithology

• for any lithology, strength varies with physical conditions
• but relative order of rock strength is maintained

• stratigraphic column may be divided into structural lithic units indicating formations with similar mechanical properties

Confining Pressure (p_c)

• for any given rock type, if all other conditions remain constant, increasing pc results in an increase in yield strength
• pc is equivalent to the minimum principal stress; increasing minimum principal stress in the stress equations results in an increase in normal stress acting across potential failure plane

Pore (Fluid) Pressure (p_p)

• if all pore spaces are filled with fluid, the pore water can partially support the load of the overlying column of rock
• effect of pp counteracts the effect of p_c
• confining pressure - pore pressure = effective stress
• overhead example of Berea sandstone, 1. constant effective stress, 2. constant p_c, p_p changes
• presence of fluids can also decrease strength of rock because it promotes chemical reactions necessary for deformation

Temperature

• higher temperatures result in lower strengths
• at sufficient temperatures, rock deforms as a viscous fluid, flow under any stress
• viscous (no fundamental strength)
• plastic (fundamental strength)
• overhead example: Repetto siltstone, compare ultimate strength vs. yield strength

Strain Rate

• can't deform rocks at geologic deformation rates (10-14/sec)
• lower strain rates = lower strength as increased time allows slow deformation mechanisms to act
• creep - time-dependent strain produced under low differential stress
• primary creep - elastic deformation
• secondary creep - steady-state plastic deformation as strain increases slowly w/time
• tertiary creep - accelerating strain prior to rupture approaches viscous deformation

Anisotropy

• rock has different mechanical properties in different directions
• strength will be greatest when measured parallel or perpendicular to the anisotropy
• strength is least when anisotropy is oriented ~30^o to load direction

Elastic vs. Plastic vs. Viscous Behavior

Elastic behavior

• linear relationship of stress vs. strain during elastic deformation, stress = Ee (Hooke's Law)
• E = Young's (elastic) modulus, measure of rock stiffness
• greater stiffness = steeper slope in stress vs. strain graph, less stiffness = lower slope

• Poisson's ratio (h) = compares lateral to longitudinal strain (also an elastic modulus)
• h = e_lat/ e_long
• perfectly incompressible material would have h = 0.5
• expansion laterally compensates for vertical contraction
• e.g. cube of rock, e_long = 0.2, e_lat = 0.1
  • (examples, fine grained limestone h = 0.25; granite, h = 0.11)
• Poisson effect = the generation of lateral stresses by vertical loading
  (minimum principal stress = [h/1-h] s₁)
  e.g. for fine grained limestone, minimum principal stress = 1/3 maximum principal stress

Plastic behavior

• nonrecoverable (permanent) deformation
• in an ideal plastic body, deformation would not begin until yield stress was overcome, deformation would then continue until stress was removed
• strain hardening or strain softening occur if the rock becomes stronger or weaker as a result of deformation

Viscous behavior

• material deforms under any stress
• no yield strength to overcome
• strain rate is dependent upon stress and viscosity of the deforming medium, stress = viscosity x strain
• viscosity = resistance to flow, measured in poises
• examples of viscosity: mantle, rhyolite lava, basalt lava

return to top of page

return to structure syllabus