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Rheology II

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**Ideal (Newtonian) Viscous Behavior**

Viscosity theory deals with the behavior of a liquid For viscous material, stress, s, is a linear function of strain rate, e.=e/t, i.e., s = he. where h is the viscosity Implications: The s - e. plot is linear, with viscosity as the slope The higher the applied stress, the faster the material will deform A higher rate of flow (e.g., of water) is associated with an increase in the magnitude of shear stress (e.g., on a steep slope)

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**Viscous Deformation Viscous deformation is a function of time**

s = he.= he/t Meaning that strain accumulates over time (next slide) Viscous behavior is essentially dissipative Hence deformation is irreversible, i.e. strain is Non-recoverable and permanent Flow of water is an example of viscous behavior Some parts of Earth behave viscously given the large amount of geologic time available

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**Ideal Viscous Behavior**

Integrate the equation s = he. with respect to time, t: sdt = he. dt st = he or s = he/t or e = st/h For a constant stress, strain will increase linearly with time, e = st/h (with slope: s/h) Thus, stress is a function of strain and time! s = he/t Analog: Dashpot; a leaky piston that moves inside a fluid-filled cylinder. The resistance encountered by the moving perforated piston reflects the viscosity

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Viscosity, h An ideally viscous body is called a Newtonian fluid Newtonian fluid has no shear strength, and its viscosity is independent of stress From s = he/t we derive viscosity (h) h = st/e Dimension of h: [ML-1 T-2][T] or [ML-1 T-1]

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**s = he. (dyne/cm2)/(1/s) poise**

Units of Viscosity, h Units of h : Pa s (kg m -1 s -1 ) s = he. (N/m2)/(1/s) Pa s s = he. (dyne/cm2)/(1/s) poise If a shear stress of 1 dyne/cm2 acts on a liquid, and gives rise to a strain rate of 1/s, then the liquid has a h of 1 poise poise = 0.1 Pa s h of water is 10-3 Pa s Water is about 20 orders of magnitude less viscous than most rocks h of mantle is on the order Pa s

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Nonlinear Behavior Viscosity usually decreases with temperature (effective viscosity) Effective viscosity: not a material property but a description of behavior at specified stress, strain rate, and temperature Most rocks follow nonlinear behavior and people spend lots of time trying to determine flow laws for these various rock types Generally we know that in terms of creep threshold, strength of salt < granite < basalt-gabbro < olivine So strength generally increases as you go from crust into mantle, from granitic-dominated lithologies to ultramafic rocks

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Plastic Deformation Plasticity theory deals with the behavior of a solid Plastic strain is continuous - the material does not rupture, and the strain is irreversible (permanent) Occurs above a certain critical stress (y, yield stress = elastic limit) where strain is no longer linear with stress Plastic strain is shear strain at constant volume, and can only be caused by shear stress Is dissipative and irreversible. If applied stress is removed, only the elastic strain is reversed Time does not appear in the constitutive equation

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Elastic vs. Plastic The terms elastic and plastic describe the nature of the material Brittle and ductile describe how rocks behave Rocks are both elastic and plastic materials, depending on the rate of strain and the environmental conditions (stress, pressure, temp.) Rocks are viscoelastic materials at certain conditions

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Plastic Deformation For perfectly plastic solids, deformation does not occur unless the stress is equal to the threshold strength (at yield stress) Deformation occurs indefinitely under constant stress (i.e., threshold strength cannot be exceeded) For plastic solids with work hardening, stress must be increased above the yield stress to obtain larger strains Neither the strain (e) nor the strain rate (e. ) of a plastic solid is related to stress (s)

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Brittle vs. Ductile Brittle rocks fail by fracture at less than 3-5% strain Ductile rocks are able to sustain, under a given set of conditions, 5-10% strain before deformation by fracturing

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Strain or Distortion A component of deformation dealing with shape and volume change Distance between some particles changes Angle between particle lines may change Extension: e=(l’-lo) / lo = l/ lo [no dimension] Stretch: s = l’/lo =1+e = l½ [no dimension] Quadratic elongation: l = s2 = (1+e)2 Natural strain (logarithmic strain): e =S dl/lo = ln l’/lo= ln s = ln (1+e) and since s = l½ then e = ln s = ln l½ = ½ ln l Volumetric strain: ev = (v’-vo) / vo = v/vo [no dimension] Shear strain (Angular strain) g = tan is the angular shear (small change in angle)

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**Factors Affecting Deformation**

Confining pressure, Pc Effective confining pressure, Pe Pore pressure, Pf is taken into account Temperature, T Strain rate, e.

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Effect of T Increasing T increases ductility by activating crystal-plastic processes Increasing T lowers the yield stress (maximum stress before plastic flow), reducing the elastic range Increasing T lowers the ultimate rock strength Ductility: The % of strain that a rock can take without fracturing in a macroscopic scale

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**e. = de/dt = (dl/lo)/dt [T-1]**

Strain Rate, e. Strain rate: The time interval it takes to accumulate a certain amount of strain Change of strain with time (change in length per length per time). Slow strain rate means that strain changes slowly with time How fast change in length occurs per unit time e. = de/dt = (dl/lo)/dt [T-1] e.g., s-1

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**Shear Strain Rate g . = 2 e. [T-1] Shear strain rate:**

Typical geological strain rates are on the order of s-1 to s-1 Strain rate of meteorite impact is on the order of 102 s-1 to 10-4 s-1

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**Effect of strain rate e. Decreasing strain rate:**

decreases rock strength increases ductility Effect of slow e. is analogous to increasing T Think about pressing vs. hammering a silly putty Rocks are weaker at lower strain rates Slow deformation allows diffusional crystal-plastic processes to more closely keep up with applied stress

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**Strain Rate (e.) – Example**

30% extension (i.e., de = 0.3) in one hour (i.e., dt =3600 s) translates into: e. = de/dt = 0.3/3600 s e. = s-1 = 8.3 x 10-5 s -1

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**Strain Rate (e.) – More Examples**

30% extension (i.e., de = 0.3) in 1 my (i.e., dt = 1000,000 yr ) translates into: e. = de/dt e. = 0.3/1000,000 yr e. = 0.3/( )(365 x 24 x 3600 s)= 9.5 x s-1 If the rate of growth of your finger nail is about 1 cm/yr, the strain rate, e., of your finger nail is: e = (l-lo) / lo = (1-0)/0 = 1 (no units) e. = de/dt = 1/yr = 1/(365 x 24 x 3600 s) e. = 3.1 x 10-8 s-1

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**Effect of Pc Increasing confining pressure:**

inreases amount of strain before failure i.e., increases ductility increases the viscous component and enhances flow resists opening of fractures i.e., decreases elastic strain

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**Effect of Fluid Pressure Pf**

Increasing pore fluid pressure reduces rock strength reduces ductility The combined reduced ductility and strength promotes flow under high pore fluid pressure Under ‘wet’ conditions, rocks deform more readily by flow Increasing pore fluid pressure is analogous to decreasing confining pressure

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**Strength Rupture Strength (breaking strength) Fundamental Strength**

Stress necessary to cause rupture at room temperature and pressure in short time experiments Fundamental Strength Stress at which a material is able to withstand, regardless of time, under given conditions of T, P, and presence of fluids, without fracturing or deforming continuously

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**Factors Affecting Strength**

Increasing temperature decreases strength Increasing confining pressure causes significant increase in the amount of flow before rupture increase in rupture strength (i.e., rock strength increases with confining pressure This effect is much more pronounced at low T (< 100o) where frictional processes dominate, and diminishes at higher T (> 350o) where ductile deformation processes, that are temperature dominated, are less influenced by pressure

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**Factors Affecting Strength**

Increasing time decreases strength Solutions (e.g., water) decrease strength, particularly in silicates by weakening bonds (hydrolytic weakening) (OH- substituting for O-) High fluid pressure weakens rocks because it reduces effective stress

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Flow of Solids Flow of solids is not the same as flow of liquids, and is not necessarily constant at a given temperature and pressure A fluid will flow with being stressed by a surface stress – it does response to gravity (a body stress) A solid will flow only when the threshold stress exceeds some level (yield stress on the Mohr diagram)

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Rheid A name given to a substance (below its melting point) that deforms by viscous flow (during the time of applied stress) at 3 orders of magnitude (1000 times) that of elastic deformation at similar conditions. Rheidity is defined as when the viscous term in a deformation is 1000 times greater than the elastic term (so that the elastic term is negligible) A Rheid fold, therefore, is a flow fold - a fold, the layers of which, have deformed as if they were fluid

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LECTURER 2 Engineering and True Stress-Strain Diagrams

LECTURER 2 Engineering and True Stress-Strain Diagrams

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