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Rheology and
the Lithosphere
Processes in Structural Geology & Tectonics
Ben van der Pluijm
© WW Norton+Authors, unless noted otherwise
3/18/2016 4:03 PM
We Discuss …
Rheology and the Lithosphere
• What is rheology?
• Insights from rock deformation
experiments
• Characteristics stress-strain
behaviors
• General rock creep
• Elastic, viscous and composite
linear rheologies
• Maxwell relaxation time and Earth
behavior
• Non-linear rheologies
• Plastic flow stresses
• Crust and upper mantle strength
curve(s)
• Defining lithosphere and
asthenosphere
• Lithologic vs rheologic layering
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Rheology is ....
… the study of deformation
(flow) of materials.
Associated concepts:
 Stress
 Strain, Strain rate
 Elasticity
 Viscosity
 Failure and Friction
 Plasticity
Malaspina Glacier, AK (NASA)
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Earth’s Lithosphere
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Insights from Rock Experiments
Triaxial deformation apparatus
Compression and extension tests
Brittle
Plastic
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Confining Pressure
Suppresses fracturing
Promotes ductility (distributed strain)
Increases strength (maximum stress)
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Temperature
Suppresses fracturing
Promotes ductility
Reduces strength
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Fluid Pressure
Pf
Pc
Inverse form Pc: Pf ~ 1/Pc
Peff = Pc - Pf
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Strain rate
ė = 10-6/sec is 30% change in 4 days
ė = 10-14/sec is 30% change in 1 million years
“Fast” ~100 MPa
“Slow” ~10 MPa
Small ė (significantly) reduces rock strength
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Summary of Rock Responses to Pc, Pf, T, ė
Low Pc, high Pf, low T, (high ė): promotes fracturing
High Pc, low Pf, high T, (low ė): promotes viscous flow
The Lithosphere ©PSG&T
= upper crust
= lower crust, mantle
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Characteristic Stress-Strain Behaviors
Representative stress-strain curves.
(A) Elastic behavior followed by
failure.
(B) Small viscous (permanent)
strain before failure.
(C) Significant viscous
(permanent) strain before failure.
Yield stress marks stress at change from
elastic (recoverable) to viscous
(permanent) strain;
Failure stress is stress at fracturing.
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Work hardening/softening.
(D) No elastic component and
requiring lower stresses to
deform: work softening.
(E) Elastic-plastic behavior, with
permanent strain accumulation at
constant stress.
(F) Elastic-plastic behavior,
requiring increasingly higher
stresses to deform: work
hardening.
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Strength and Competency
Rock strength is maximum
stress a material can
support before failure.
Competency is relative term
that compares resistance
of rocks to deformation.
Rock competency scale:
rock salt < shale < limestone
< greywacke < sandstone
< dolomite
schist < marble < quartzite <
gneiss < granite < basalt
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General Creep Curve of Rocks: Elastic, Viscous, Failure
Creep: I. Elastic, II. Viscous, III. Accelerated viscous.
(a) Under continued stress a material will fail.
(b) If we remove stress before failure, material relaxes (elastic component)
while permanent (viscous) strain remains.
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Rheologic Models: Elastic and Viscous Behavior
Elastic behavior
rubber band
s = E . e (ss = G. g); Pa
Viscous behavior
water, syringe
s =  . ė ; t-dependent; Pa.s
∫s dt = s.t
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Elastic+Viscous Rheologies: Visco-elastic
General creep curve (strain –time)
Visco-elastic behavior
water-soaked sponge, memory foam
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Elastic+Viscous Rheologies: Elastico-viscous
General creep curve (strain –time)
Visco-elastic behavior
water-soaked sponge, memory foam
Elastico-viscous behavior
mayonnaise, toothpaste
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Elastic+Viscous Rheologies: General Linear Behavior
General creep curve (strain –time)
Visco-elastic behavior
water-soaked sponge, memory foam
Elastico-viscous behavior
mayonnaise, silly putty
General Linear behavior
~natural rock
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Summary: Linear Rheologic Models
Elastic behavior
rubber band
s = E . e (ss = G. g)
Viscous behavior
water
s =  . ė ; ∫s dt = s.t
Visco-elastic behavior
water-soaked sponge, memory foam
Elastico-viscous behavior
mayonnaise, silly putty
General Linear behavior
~rock
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Earth Behavior
Earth is like silly putty®: elastic
or break (fast) or viscous
(slow) as fion of time (strain
rate). For example, both
earthquakes and mantle
plumes.
Maxwell relaxation time:
tM = /G
is dominance of viscosity ()
over elasticity (G)
Crust
Mantle
Unit: tM = (s/(e/t)) / s/e = time
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Maxwell relaxation time, tM
viscosity is temperaturedependent, so tM (= /G) is
T-dependent.
tM range plotted in t-T space.
Elasticity dominates on seismic
timescales (failure).
Viscosity dominates on
tectonic timescales (flow).
Crust
Mantle
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Elastic and Viscous Earth
Elastic Earth:
Mantle viscosity of 1021 Pa⋅s,
rigidity of 1011 Pa (olivinedominated mantle)
tM = 1010 s
that is, on order of 1000 years
(e.g., Earth’s “fast” glacial
rebound).
Viscous Earth:
Mantle viscous flow stress:
s=.ė
= 1021 . 10-14 = 107 Pa
= 10MPa
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Linear vs. Non-linear Rheologies
General linear behavior
Non-linear (or elastic-plastic) behavior
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Linear and Non-linear Viscous Behavior
Effective viscosity:
Linear rheology
Non-linear rheology
ė = 1/ . s
A = proportionality constant
E = activation energy (100-500 kJ/mol)
R = gas constant (8.3 J/K.mol)
T = temperature (K)
n = stress exponent (2-5)
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Rock Experiments
Brittle
Plastic
Triaxial deformation apparatus
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Plastic Flow Stresses
Salt
Quartzite
Anorthosite
Dunite
solve for σ
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Friction vs. Plasticity (strength curves)
Salt
Quartzite
Anorthosite
Friction: s = 2sn . m [s = 2ss]
Flow:
Dunite
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Stress and Deformation Regimes
Frictional Regime
•
normal stress and
Pf dependent
(“effective stress”)
•
temperature and strain
insensitive
•
shear stress is primarily
function of normal stress
Plastic Regime
•
normal stress and Pf insensitive
•
temperature and strain rate
dependent
•
shear stress is primarily function of
temperature and strain rate
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Earth’s Lithosphere
Lithologic layering
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Rheologic layering
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