CHAPTER 5

1. CHAPTER 5 RHEOLOGY Dr. Masdouq Al-Taj

2. Introduction • Rheology is the study of the flow of material. • In Other words: The relation between stress and strain with time. • Do rocks flow? • Geologically speaking and due to the availability of time, rocks are able to flow. Not in the same physical meaning, but with the final result, which can be achieved by rocks.

3. Strain Rate Strain rate (ė)is the time interval it takes to accumulate a certain amount of strain; and it is defined as the elongation per time. • ė = e/t = l/l0t t in second Example: 30% finite longitudinal strain (|e|= 0.3) is achieved in an experiment that lasts one hour (3600 s). The correspond strain rate is ė = 0.3/3600 = 8.3 x 10–5/s Now let’s see what happens to the strain rate when we change the time interval, but maintain the same amount of finite strain of 30%. t=1 day (86.4 x 103 s) ė =3.5 x 10–6/s t=1 year (3.15 x 107 s) ė =9.5 x 10–9/s t=1 m.y. (3.15 x 1013 s) ė =9.5 x 10–15/s

4. Q:What is the strain rate of a 200 km long thrust sheet moving 50 km in 1 m.y.? Answer: Now we can calculate the strain rate ė = e/t = l/l0t = 50/200 x 3.15 x 1013 ė = 0.8 x 10-14/s In some cases, shear strain rate preferred to be used rather than strain rate.The relation between shear strain rate (γׂ) and strain rate is: γׂ= 2 ė

5. A widely used estimate is based on the Quaternary displacement along the San Andreas Fault in California, which gives a strain rate on the order of 10–14/s(moderate geological strain). • Which agrees well with present-day observation of plate velocities. • For geological processes, the typical range of strain rate is between 10-12/s to 10-15/s. There are some exceptions.

6. General behavior: the creep curve • Three creep regimes are obtained: I. Primary or transient creep (decreasing strain rate) II. Secondary or steady-state creep (constant strain rate) III. Tertiary or accelerated creep (increase strain rate)

7. Rheologic Relationships • The behavior of materials can be divided into three types: 1. Elastic behavior: recoverable and instaneous response to stress (time independent) 2. Viscous behavior (Plastic behavior in rock) 3. Combination of elastic and viscous behaviors (visco-elastic and elastico-viscous)

8. The stress-strain relation is commonly expressed in graphs known as stress-strain diagrams. 1. Curve A is a brittle substance. 2. Curve B is an ideal plastic substance. 3. Curve C more normal type of plastic behavior 4. Curve D another common type of plastic deformation.

9. Stress-strain curve showing three types of rheological behavior: 1. Elastic region: Stress is proportional to strain (Hook’s law) 2. Plastic region: Behave according to ductile flow. 3. Brittle Failure region : fracture. * Elastic limit is the maximum stress under which the material exhibits elastic strain. Beyond this value, the material undergoes permanent deformation by ductile flow or by brittle fracture.

10. Elastic Behavior- Modulus of Elasticity • We can express elastic behavior by the equation: • σ = E • e, • where E is the modulus of elasticity or Young’s Modulus

11. 1. Young’s Modulus Description • Another description of Young’s modulus is the slope of the line in a σ - e diagram • The slope is the tangent of angle  • Figure 5.3a in text

12. Dimensions of Young’s Modulus • Since strain I dimensionless, Young’s modulus has the same units as stress, Pascals • A typical value of Young’s modulus for rocks is -1011 Pa • The sign is negative because we apply a positive stress (compression) to produce a negative elongation (shortening)

13. 2. Rigidity • Another expression for elastic behavior is given by • σs = G • γ • where G is the rigidity and γ is shear strain

14. 3. Bulk Modulus and Compressibility • We can also write equations for dilation: • σ = K • ((V - V0)/V0) • K is the bulk modulus, or incompressibility • Its reciprocal is the compressibility, β

15. 4. Poisson’s Ratio • This ratio may be expressed as: ν = e ┴/e // • where e┴ is the elongation perpendicular to the compressive stress • e// is the elongation parallel to compressive stress

16. Plastic Behavior • After stress is removed, there is a rapid drop in strain, followed by a period of relaxation, during which the strain decreases a bit more before

17. Viscous Behavior • The symbol to the left is a “dashpot”, a type of leaky piston-cylinder apparatus • Resistance encountered by the piston depends on the viscosity of the fluid

18. Definition of Viscosity • Viscosity is defined by the term η in the equation relating stress and strain rate: • σ = η ė • Viscosity dimensions may be deduced from the equation • Stress is pressure per unit area, or Pa • Strain rate has units of reciprocal time, or s-1 • So, viscosity must have units of Pa. s

19. Low Viscosity Position after short period of flow Initial position

20. Moderate Viscosity • Videos of pahoehoe flows, Hawaii

21. Relative Viscosities

22. Strain Rate of the Mantle • We can make an estimate of the strain rate for the mantle using a viscosity of 1021 Pa.s and a differential stress of 50 MPa • 50 x 106 Pa ÷ 1021 Pa • s = 5 x 10-14 sec-1

23. CombinationsViscoelastic Behavior • The equation for visco-elastic behavior is: • σ = η ė +E • e • An example is a sponge filled with water

24. Viscoelastic Diagram • When a weight is placed on the sponge, the sponge loses water (viscous behavior) until the water is gone. • Then the sponge supports the load elastically

25. Example of Viscoelastic Behavior • Ropy pahoehoe surface - Upper crust behaves in a viscoelastic fashion before solidifying into a brittle crust • The soft crust can deform into folds and ropy structures as the result of flow beneath the viscoelastic layer

26. Elasticoviscous • The spring responds instantly to stress, whereas the dashpot moves and keeps moving as long as stress is present • When stress is removed, the dashpot stops moving and stays put.

27. Elasticoviscous Equation • Elasticoviscous behavior is given mathematically as: • ė = σ/E + σ/η Initial sigma should have a dot symbol above it

28. Relaxation Time of Mantle • Thus, the elasticoviscous model seems to fit the mantle. • What is the relaxation time of the mantle? • If the viscosity is 1021 Pa • s and G = 1011 Pa, we get a relaxation time of 1010 seconds, or about 300 years

29. General Linear Behavior • General linear behavior is a combination of elasticoviscous and viscoelastic behavior in series.

30. Comparison of Rock Strengths • General linear behavior can be rewritten as a function of stress: • σ = (ė• A)1/nexp(E*/RT) • Substituting known values of A, n, and E* at constant T allows us to compare the relative strength of different rock types.

31. Creep Strength versus Depth

32. Natural Rocks • It is very important to examine the relationships between stress, strain and strain rate by using natural rocks to have better understanding of the flow of rock (Rheology). • What are the main reasons of doing experiments on natural rocks? 1. We observe the actual behavior of natural rocks. 2. We can vary several parameters in our experiments such as pressure, temperature, time and fluid pressure, and to examine their roles in rock deformation.

33. The effect of the following parameters in the triaxial tests are:

34. Effective Pressure • pc - pf is equal to the effective pressure • Pistons at both ends of the cylinder allow the experimenter to impose pressure along the vertical axis, called the axial stress

35. 1. Confining (Lithostatic) Pressure (Pc) • Acts equally in all direction (1 = 2 = 3) • By changing the confining pressure during the experiments, we observe a very important characteristic. 1. With increasing confining pressure greater amounts of strain accumulate before failure occurs. 2. Increasing confining pressure, increases the viscous component and the rock’s ability to flow. 3. Higher confining pressures increasingly resist the opening of fractures.

36. Clapping Underwater • Clapping one’s hand in the air is easy, but doing it under water is much harder • The water resists movement

38. By increasing the experimental temperatures the effects of confining pressure become clear. 4. Larger strain can be achieved before failure with increasing depths in the Earth (lithostatic).

39. The amount of strain before failure(ductility) differs significantly among the various rock types

40. 2. Temperature • A change in temperature conditions also produces a marked change in response. 1. Most rocks are ruptured at low T. 2. At these conditions most of the strain prior to rupture is recoverable (elastic). 3. When T increases, the elastic portion of the strain decreases while the plasticity increases. 4. Rocks have lower strength and become more ductile with depth in the Earth, where we find higher T.

42. Differential Stress vs. Temperature • Effect of increasing temperature on the ability of various rocks to withstand a given differential stress • In all cases, the rocks grow weaker as temperature increases, but the effect is much higher on pure calcite than other materials.

43. 3. Strain Rate 1. Decreasing the strain rate results in decreased rock strength and increased ductility (viscosity) 2. T. changes produce similar effects as strain rate variations in rocks experiments (h. T  s. ė)

44. Young Experimentalist • Human factors also play a role • Old experimenters may die or become incapacitated • Younger experimenters, who face lower odds of either event happening, have to obtain results quickly to justify continued funding

45. 4. Pore-fluid Pressure (Pf) • Some rock types (sandstone, shale) contain a significant fluid component that affects their behavior under stress. • Experiments show that increasing the pore-fluid pressure produces a drop in the strength and reduces the ductility of the sample.

46. Quartz, Wet and Dry • Effect of water on the mineral quartz • Strength decreases as temperature increases, while the quartz is dry, but when it is wet ……….

47. Rocks are weaker when the pore-fluid pressure is high. • Increasing the pore-fluid pressure will have the same effect as decreasing the confining pressure of the experiment. Effect pressure = Confining pressure – fluid pressure (Pe = Pc – Pf)

48. Significance of Experiments to Natural Conditions • Increasing the confining pressure (Pc) and fluid pressure (Pf) have an opposite effects. • Increasing temperature (T) and lowering strain rate (ė) have the same effects. • Confining pressure and temperature, which both increase with depth in the Earth, result in rocks that increasingly resist failure, while at the same time they allow larger strain accumulation, and increase the ability for rocks to flow. • High fluid content is more complex and may promote fracturing if Pf is high.

50. Strain Softening and Hardening • D no elastic component, with strain softening • E is elastic-plastic behavior, with permanent strain at constant stress above the yield stress • F is elastic strain followed by permanent strain that requires increasingly higher stresses to accumulate (strain hardening)