1. Aspects of Crustal Rheology
ESCI 302

2. What is rheology? Heraclitus (~535 BC – 475 BC):
“panta rei” (“everything flows”)
E.C. Bingham & M. Reiner (1920s):
“[Rheology is] the study of the deformation and flow of all types of matter (under the influence of an applied stress)”
CD / Ch 9 / Intro

3. What matters? Elastic (Hooke) materials Viscous (Newtonian) materials
Plastic (Von Mises) materials
Elastico-viscous (Maxwell) materials
Elastic-plastic (Prandtl) materials
Visco-plastic (Bingham) materials
Firmoviscous (Kelvin) materials

4. Elastic (Hooke) materials
R. Hooke (1678):
“Ut tensio, sic vis” (“As the extension, so the force”)
Linear elasticity (1D)
Key properties:
Strain is instantaneous, recoverable
(stores energy)
CD/Ch. 9 /Spring Models/Elastic
Rubber band

5. Elastic strain s11 L0 DL s11 w w0 Dw/2 L
= normal force, F1, divided by area of face 1 (wo) = a pressure L0 DL
s11 w w0
Dw/2 L

6. Elastic moduli Young’s modulus 40-90 GPa (lab) BANG! Stress [MPa] s11ds
Young’s modulus
40-90 GPa (lab) de
e11
Strain [%]

7. Elastic moduli Poisson’s ratio= lateral strain / axial strain
-de11
de22
e11
Poisson’s ratio=
lateral strain / axial strain
(lab)
0.5 (incompressible)

8. Elastic moduli Bulk modulus Shear modulus t Phydrostatic:
s11 = s22 = s33

9. Viscous (Newtonian) materials
I. Newton (1687):
“The resistance which arises from the lack of slipperiness originating in a fluid – other things being equal – is proportional to the velocity by which the parts of the fluid are being separated from each other.”
Linear viscosity or
CD/Ch. 9 /Spring Models/Viscous

10. Viscous (Newtonian) materials
shear stress [MPa]
slope = viscosity
Viscosity:
measure of “stickiness”
Mantle Pa s
Molten basalt Pa s
Glacier ice 1.2 x 1013 Pa s
shear strain rate [s-1]
Silly putty

11. Viscous (Newtonian) materials
Deformation is permanent and accumulates as a function of time (not instantaneously)
given large amounts of time, even tiny stresses can cause huge finite strains. No concept of a “threshold” or yield stress (all materials are infinitely weak).
shear stress on
off
off
time
shear strain
time

12. Plastic (Von Mises) materials
R. Von Mises ( ):
- threshold concept
Suction ball
CD/Ch. 9 /Spring Models/Rigid-Plastic-Von Mises

13. Plastic (Von Mises) materials
tyield
tyield
shear stress
shear stress
any strain rate possible
time
shear strain rate
Von Mises yield criterion states fixed minimum threshold on shear stress but no information about strain rate at that yield stress.
what apparent viscosity??
If post-yield material flows viscously, material is said to be a visco-plastic (Bingham body), e.g. paint, yoghurt, rocks.
shear strain
time

14. Rheological models depicted by analogue symbols
Elastic: spring
Viscous: dash-pot or hydraulic piston
Plastic: slider block
Examples of hybrid models:
Visco-plastic (Bingham)
Elastic-plastic
Elastico-viscous (Maxwell)

15. Real rock behaviour
elastic
upper
crustal
rocks
ductile behaviour of rocks best compared at same homologous temperature
Temp strongly affects the viscous component (flow-rate)
elastic-plastic
visco-elastic
metamorphic
rocks
(lower crust)
rock salt
magma
Viscous
(stress/flow-rate)
Plastic
(yield pt)
visco-plastic
silly putty

16. Experimentally deformed rocks
triaxial testing apparatus
Pc from 100’s to 1000’s MPa
T from 0 C to melting temperature of material
slowest strain rate ~ 10-7 s-1

17. Experimentally deformed rocks
triaxial tests:
Keep stress constant, see how strain rate varies (creep tests) or
Impose constant strain rate , see how stress varies with time

18. Real Material Behaviour (experimental)
consider constant strain rate experiment
Elasticity at small strains + Plasticity concept of YIELD:
beyond yield point:
NO recoverable elastic strain, BUT ductile creep or flow (= permanent damage)
Catastrophic failure (may be brittle fracturing) at ultimate strength (= max. sustainable stress)
Brittle (elastic-plastic)
ultimate
strength
yield
point
ductile flow
(elasticoviscous)
stress s
elastic
elastic
limit
accumulated strain e or time
Paper clip, silly putty

19. Real Material Behaviour (experimental) for creeping rock
Ultimate strength
constant strain rate
strain hardening:
material becomes progressively stronger w/ dfm.
(favors dfm. widening)
ultimate
strength
strain hardening
stress s
steady-state flow
strain softening:
material becomes weaker as a result of continuing dfm.
(favors dfm. localization)
strain softening
elastic
strain e or time

20. Real Rocks: Now consider Constant Stress experiment (slope of graph is strain-rate)
Th < 0.5 (cold working)
Th > 0.5 (hot working)
work hardening at constant stress
(logarithmic creep)
secondary creep =
steady state flow
tertiary creep
primary creep
work
slow
strain rate
strain e
strain e
softening
work
hardening
fast
strain rate
slope = de/dt
constant strain rate
yield
yield
time
time
low temperature (high stress)
high temperature (low stress)

21. Effect of temperature on viscosity
Consider constant strain rate (10-14 s-1)
relationship temperature-strength is non-linear (decreasing flow-stress w/ temperature increase)
= effective viscosity reduction
yield points and eff. viscosities in
natural materials decrease
exponentially with increasing
temperature

22. Flow-stress vs Temp for minerals
Strain-rate per sec (steady state creep)
at fixed strain-rate
depends on the mineral
Flow Stress (differential, in MPa)
Temperature (deg C)

23. Effect of strain-rate on flow-stress (recall the concept of viscosity: reducing strain-rate is another way to reduce flow-stress. Slow geological strain-rates mean geological flow stresses are low relative to lab)
Trick to Do Fast Experiments:
To mimic the low flow stress that accompanies slow (geological) strain rates, conduct experiments at unnaturally high temperatures (reduces viscosity)
constant
Fast
strain-rate (lab)
yield
Flow Stress (differential, in MPa)
Slow (nature)
Twiss & Moores, 2007
Time or deformation progress

24. Steady-state creep of mid-lower crustal rocks
concept of viscosity is valid beyond yield stress and hardening phase ( )
but… the relationship is not linear (non-Newtonian) any more! (power law instead )
at > 15 km in the crust, Th > 0.5, so that steady-state creep is typical
steady-state plastic creep can be described by a power-law equation

25. Steady-state creep Non-Newtonian Newtonian power law slope = 1/
no strength
treat rocks as if they were liquids

26. Power-law creep activation pre-exponential energy factor stress
Assumes that yield stress has been exceeded and that there is neither hardening or softening (= steady-state creep) during flow. In this case, the stress level uniquely determines a fixed strain rate.
Power-law creep
pre-exponential
factor
activation
energy
stress
exponent
strain rate
differential
stress
universal
gas constant
= J K-1 mol-1
temperature

27. Steady-state creep
activation energy ‘Q’ [J/mol], relates to temperature sensitivity of viscosity
high s
slope = -Q/R
low s

28. Steady-state creep
stress exponent ‘n’, relates to stress sensitivity of viscosity
high T
slope = n
n = 1 Newtonian flow (e.g., melts)
n > 1 power law creep
most silicates:
3 < n < 5
low T

29. What else affects steady-state creep?
pressure
grain size
presence of water
presence of other minerals in rock aggregates
time and length scale of deformation (extrapolation to geol. time scales…)

30. Flow-stress vs Temp for minerals
Faster strain-rate
Strain-rate per sec (steady state creep)
at fixed strain-rate
depends on the mineral
Flow Stress (differential, in MPa)
Temperature (deg C)

31. Temperature (deg C)
Flow Stress (differential, in MPa)

32. Strength vs Depth Profile in the Crust
Strength (Diff. Stress, MPa)
Peak strength
Shallower BDT at higher geothermal gradient
Effect of increased geothermal gradient
BDT: km (depends on geothermal gradient)
Depth (~geothermal gradient)
Temperature
Assumes fixed, steady strain-rate

33. Strength vs Depth Profile in the Crust
Strength (Diff. Stress, MPa)
Peak strength
Depth (~geothermal gradient)
Temperature
Effect of increased strain-rate
Deeper BDT at faster strain-rate
Assumes fixed, steady strain-rate