lecture 5 rheology - impacttectonics.org
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Rheology
Earth Structure (2nd Edition), 2004
W.W. Norton & Co, New York
Slide show by Ben van der Pluijm
© WW Norton; unless noted otherwise
Lecture 5
© EarthStructure (2nd ed) 29/23/2014
Rheology is .... the study of deformation and flow of materials
Rheology is
Associated concepts:
� Stress
� Strain, strain rate
� Elasticity
� Viscosity
� Failure and friction
� Plasticity
� Brittle behavior
� Ductile behavior
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Ideal material behaviours
Elastic behaviour – Hooke’s law
Viscous behaviour – Newtonian fluids
Viscoelasticity - Kelvin and Maxwell bodies
Other ideal rheologies
Definitions
• A material’s response to an imposed stress is called the rheology.
• A mathematical representation of the rheology is called a constitutive law.
Simple, 1D Hooke body (linear elasticity)
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General Behavior: The Creep Curve
Generalized strain vs. time curve shows
primary (I) or “transient”, |
secondary (II) or “steady-state”,
and tertiary (III) or “accelerated” creep.
Under continued stress the
material will fail
When stress is removed, the material relaxes,
but permanent strain remains
Shear strain rate:
Strain rate: time interval it takes to
accumulate a certain
amount of strain:
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Rheologic models
• Physical models consisting of strings and dash pots, and associated
strain–time, stress–strain, or stress–strain rate curves for (a) elastic, (b) viscous, (c) viscoelastic, (d) elasticoviscous, and (e) general linear behavior.
• A useful way to examine these models is to draw your own strain–time
curves by considering the behavior of the spring and the dash pot
individually, and their interaction.
• An ideal dashpot or damper creates a force proportional to its velocity; it
cannot be displaced instantaneously by a finite force.
Symbols used:
e = elongation, e˙ = strain rate, σ=stress, E = elasticity, η (eta)=viscosity, t = time, el denotes elastic component, vi denotes viscous component.
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Rheologic models
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a) Elastic or Hookean behavior: rubber band σ = E . e (σs = G. γ)
b) Viscous or Newtonian behavior: water σ = η . eo ; ∫σ dt = σ.t = η . e
FIGURE 5.3 Models of linear rheologies
Symbols used: e = elongation, e˙ = strain rate, σ=stress, E = elasticity, η (eta)=viscosity, t = time, el denotes elastic
component, vi denotes viscous component.
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Rheologic models
c) Viscoelastic orKelvin behavior:
water-soaked sponge, memory foam
D) Elastoviscous or
Maxwell behavior: paint
E) General Linear
behavior or
Burgers behavior
handout
Symbols used: e = elongation, e˙ = strain rate, σ=stress, E = elasticity, η (eta)=viscosity, t = time, el denotes elastic
component, vi denotes viscous component.
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Rheologic models (interactive)
DePaor, 2002
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Elastic Constants
DePaor, 2002
Elastic constants equate two 2nd order tensors, so 4th order
tensor (81 components;
9 independent)
Interdependent moduli:
E = 2G (1+ υ) = 3K (1-2υ)
Symbols used: e = elongation, e˙ = strain rate, σ=stress, E = elasticity, η (eta)=viscosity, t = time, el denotes elastic
component, vi denotes viscous component.
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Representative moduli and viscosities for rocks
K = σ/∆V; G = σs/γ; η = σ/eo
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Maxwell relaxation time, tM
Like silly putty, the mantle is elastic (earthquakes) or viscous (plumes); it can
behave elastically (earthquakes) or viscously (plumes) as function of time
Elastic behavior < tM < viscous behavior
Maxwell relaxation time: tM = η/G
or, dominance of viscosity over rigidity
Stress relaxes to 1/e (=0.37) of original
stress (exponential decay)
Mantle viscosity of 1021 Pa⋅s, rigidity of 1011 Pa (olivine-dominated mantle)
tM = 1010 s] or on order of 1000 years (e.g., glacial rebound).
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Maxwell relaxation time, tM
Mantle flow stress: σ = η . Eo = 1021 . 10-14 = 107 = 10MPa
= η/G is also temperature dependent
FIGURE 5.4 The Maxwell
relaxation time, tM, is plotted as a
function of time and temperature.
• The curve is based on
experimentally derived
properties for rocks and their
variability.
• The diagram illustrates that hot
mantle rocks deform as a
viscous medium (fluid),
whereas cooler crustal rocks
tend to deform by failure.
• This first-order relationship fits
many observations reasonably
well, but is incomplete for
crustal deformation.
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Linear vs. non-linear models
FIGURE 5.6 Nonlinear rheologies: elastic-plastic behavior. (a) A physical
model consisting of a block and a spring; the associated strain–time curve (b) and stress–strain rate curve (c).
General Linear is Non-linear (elastic-plastic behavior)
(linear for comparison)Linear
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General linear behavior is modeled by placing the elastico-viscous and viscoelastic models in series.
Elastic strain accumulates at the first application of stress (the elastic
segment of the elasticoviscous model).
Subsequent behavior displays the interaction between the elastico-viscous
and viscoelastic models.
When the stress is removed, the elastic strain is first recovered, followed by
the viscoelastic component. However, some amount of strain (permanentstrain) will remain, even after long time intervals (the viscous component of
the elastico-viscous model).
The creep curve for
general linear closely
mimics the creep curve that is observed in
experiments on natural rocks
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Flow stresses (strength curves)
FIGURE 5.7 Variation of creep strength with
depth for several rock types using a geothermal
gradient of 10 K/km and a strain rate of 10–14/s.
Rs = rock salt, Gr = granite, Gr(w) = wet granite, Qz = quartzite, Qz(w) = wet quartzite,
Pg = plagioclase-rich rock, QzD = quartz diorite, Db = diabase, Ol = Olivine-rich rock.
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Rock experiments
FIGURE 5.8 Schematic diagram of a triaxial compression apparatus and states of stress in
cylindrical specimens in compression and extension tests.
• The values of Pc, Pf, and σ can be varied during the experiments.
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P, T, depth relationship
Moho
The dashed line is the adiabatic gradient, which is the increase of temperature with
depth resulting from increasing pressure and the compressibility of silicates.
FIGURE 5.9 Change of temperature (T) and pressure (Pc) with depth.
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Pressure Suppresses fracturing , promotes ductility, Increases strength
FIGURE 5.10 Compression stress–strain curves of Solnhofen limestone at
various confining pressures (indicated in MPa) at (a) 25°C and (b) 400°C.
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Temperature Suppresses fracturing, promotes ductility, reduces strength
FIGURE 5.12 Compression stress–strain curves of Solnhofen limestone at various
temperatures (indicated in °C) and pressures
0.1 MPa confining pressure
40 MPa confining pressure
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Temperature
FIGURE 5.13 The effect
of temperature changes
on the compressive strength of some rocks
and minerals.
The maximum stress that a rock can support
until it flows (called the
yield strength of a material) decreases with
increasing temperature.
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Strain rate
FIGURE 5.14 Stress versus
strain curves for extension
experiments in weakly foliated Yule marble for
various constant strain rates
at 500°C.
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Strain rate
eo = 10-6/sec is 30% change in 4 days
eo = 10-14/sec is 30% change in 1 million years
FIGURE 5.15 Log stress
versus log strain rate
curves for various temperatures based on
extension experiments in Yule marble.
The heavy lines mark the
range of experimental data; the thinner part of the
curves is extrapolation to
slower strain rates.
A representative geologic
strain rate is indicated by the dashed line.
Small eo reduces strength
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Fluid pressure
Pf ~ 1/Pc Peff = Pc - Pf
FIGURE 5.16 Comparing the effect of fluid pressure on the behavior of limestone
(b) varying confining pressure.(a) varying pore-fluid pressure
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Pore-Fluid Pressure
• Natural rocks commonly contain a fluid phase that may originate from the depositional history or may be secondary in origin (for example, fluids released from prograde metamorphic reactions).
• In particular, low-grade sedimentary rocks such as sandstone and shale, contain a significant fluid component that will affect their behavior under stress.
• Experiments show that increasing the pore-fluid pressure produces a drop in the sample’s strength and reduces the ductility.
• In other words, rocks are weaker when the pore-fluid pressure is high.
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Fluid pressure
FIGURE 5.17 The effect of water content on the
behavior of natural quartz.
Dry and wet refer to low
and high water content, respectively.
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SIGNIFICANCE FOR GEOLOGIC CONDITIONS
Low Pc, high Pf, low T, high eo: promotes fracturing
High Pc, low Pf, high T, low eo: promotes flow
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Representative stress-strain curves
A elastic behavior followed immediately by failure, representing brittle behavior.
B, small viscous component (permanent strain) present before brittle failure.
C, permanent strain accumulates before material fails, representing transitional behavior between brittle and ductile.
D no elastic component and work softening.
E ideal elastic-plastic behavior, in which permanent strain accumulates at constant stress above yield stress.
F typical behavior of many experiments, displaying component of elastic strain followed by permanent strain that requires increasingly higher stresses to accumulate (work hardening).
Yield stress marks stress at change from elastic (recoverable or nonpermanent strain) to viscous (nonrecoverable or permanent strain) behavior
Failure stress is stress at fracturing.
of brittle and ductile behavior
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Linear and non-linear viscous behavior
Linear rheology Non-linear rheology
A = proportionality constantE = activation energy (100-500 kJ/mol)R = gas constant (8.3 J/K.mol)T = temperature (K)n = stress exponent (2-5)
eo = 1/η . σ
The viscosity is
defined by the slope of the
linear viscous line in (a) and the effective viscosity
by the slope of
the tangent to the curve in (b).
Effective viscosity:
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Brittle and Ductile Behavior
Brittle behavior: response of a solid material to stress during which rock loses continuity (cohesion). Brittle behavior reflects the occurrence of fracture mechanisms. It occurs only when stresses exceed a critical value, after body has already undergone some elastic and/or plastic behavior.
Ductile behavior: response of a solid material to stress such that the rock appears to flow mesoscopically like a viscous fluid. In a material that has deformed ductilely, strain is distributed, i.e., strain develops without formation of mesoscopicdiscontinuities. Ductile behavior can involve brittle (cataclastic flow) or plastic deformation mechanisms.
FIGURE 5.21 Brittle (a) to brittle-ductile (b, c) to ductile (d) deformation,
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Ductile strain: behavior vs. mechanism
FIGURE 5.19 Deformation experiment with two cubes containing marbles (a),
and balls of clay (b)
frictional sliding of undeformed marbles
Ductile strain accumulates by
different deformation mechanisms.
The finite strain is equal in
both cases and the mode of
deformation is distributed (ductile
behavior) on the scale of the block.
However, the mechanism
by which the deformation
occurs is quite different
Plastic distortion of clay balls into ellipsoids.
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Strength and Competency
Strength is stress that material can
support before failure.
Competency is relative term that
compares resistance of rocks to
flow.
FIGURE 5.20 Rheological stratification of
the lithosphere based on the mechanical
properties of characteristic minerals.
Computed lithospheric strength (i.e., the
differential stress) changes not only as a
function of composition, but also as a
function of depth (i.e., temperature).
Rock competency scale: rock salt < shale < limestone
< greywacke < sandstone < dolomite < schist < marble < quartzite < gneiss < granite < basalt
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Strength and Rheologic Stratification
Combining friction and flow laws (dry and wet)
(Fossen 2010)