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CHAPTER 5 RHEOLOGY Dr. Masdouq Al-Taj

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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.

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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/l 0 t 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 10 3 s) ė =3.5 x 10 –6 /s t=1 year (3.15 x 10 7 s) ė =9.5 x 10 –9 /s t=1 m.y. (3.15 x s) ė =9.5 x 10 –15 /s

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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/l 0 t = 50/200 x 3.15 x ė = 0.8 x /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

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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 /s to /s. There are some exceptions.

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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)

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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)

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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.

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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.

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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

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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

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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 Pa The sign is negative because we apply a positive stress (compression) to produce a negative elongation (shortening)

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

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3. Bulk Modulus and Compressibility We can also write equations for dilation: σ = K ((V - V 0 )/V 0 ) K is the bulk modulus, or incompressibility Its reciprocal is the compressibility, β

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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

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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

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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

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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

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Low Viscosity Initial position Position after short period of flow

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Moderate Viscosity Videos of pahoehoe flows, Hawaii

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SubstanceViscosity (Pa s) Air (at 18 o C)1.9 x ( ) Water (at 20 o C)1 x (0.001) Canola Oil at room temp.0.1 Motor Oil at room temp.1 Corn syrup at room temp.8 Pahoehoe lava100 to 1,000 A'a lava1000 to 10,000 Andesite lava10 6 to 10 7 Rhyolite lava10 11 to Relative Viscosities

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Strain Rate of the Mantle We can make an estimate of the strain rate for the mantle using a viscosity of Pa.s and a differential stress of 50 MPa 50 x 10 6 Pa ÷ Pa s = 5 x sec -1

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

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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

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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

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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.

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

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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 Pa s and G = Pa, we get a relaxation time of seconds, or about 300 years

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

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

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Creep Strength versus Depth

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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.

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

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Effective Pressure p c - p f 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

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1. Confining (Lithostatic) Pressure (P c ) 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.

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

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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).

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

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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.

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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.

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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)

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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

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4. Pore-fluid Pressure (P f ) 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.

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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 ……….

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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 (P e = P c – P f )

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Significance of Experiments to Natural Conditions Increasing the confining pressure (P c ) and fluid pressure (P f ) 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 P f is high.

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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)

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Terminology We also need to introduce some other terms –Strength is the stress a rock can withstand without failure –Competency compares the resistance of various rock to flow Qualitative competency guides have been developed from field observations and from experimental data

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Competency Guides, Sedimentary Rocks In increasing order of competency: Salt (Low) Shale Limestone Graywacke Sandstone Dolomite (High)

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Competency Guides, Crystalline Rocks In increasing order of competency: Schist (Low) Marble Quartzite Gneiss Granite Basalt (High)

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Rheological Properties of Minerals Experimental data can also be used to categorize rheological properties on the basis of properties of the major minerals which comprise the rocks.

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From the previous observation we would predict that: 1. Brittle behavior (fracturing) is largely restricted to the upper crust. (faulting and earthquakes < 15km depth). 2. Ductile behavior (flow) dominates at greater depth.

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One of the geological structure’s classification based on the cohesiveness during deformation: Brittle Fracture system Ductile Fold system Brittle-ductile Shear zones

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Brittle-Ductile Classification Scheme (a) brittle (Faults) (b) brittle-ductile (faults and folds) (c ) brittle-ductile (small crack faulting and folding) (d) ductile (folding)

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