Aspects of Crustal Rheology ESCI 302

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

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

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

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

Elastic moduli Young’s modulus 40-90 GPa (lab) BANG! Stress [MPa] s11 ds Young’s modulus 40-90 GPa (lab) de e11 Strain [%]

Elastic moduli Poisson’s ratio= lateral strain / axial strain -de11 de22 e11 Poisson’s ratio= lateral strain / axial strain 0.1-0.3 (lab) 0.5 (incompressible)

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

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

Viscous (Newtonian) materials shear stress [MPa] slope = viscosity Viscosity: measure of “stickiness” Mantle 1020-1022 Pa s Molten basalt 102-104 Pa s Glacier ice 1.2 x 1013 Pa s shear strain rate [s-1] Silly putty

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

Plastic (Von Mises) materials R. Von Mises (1883-1953): - threshold concept Suction ball CD/Ch. 9 /Spring Models/Rigid-Plastic-Von Mises

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

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)

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

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

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

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

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

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)

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

Flow-stress vs Temp for minerals Strain-rate 10-14 per sec (steady state creep) at fixed strain-rate depends on the mineral Flow Stress (differential, in MPa) Temperature (deg C)

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

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

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

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 = 8.314 J K-1 mol-1 temperature

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

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

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…)

Flow-stress vs Temp for minerals Faster strain-rate Strain-rate 10-14 per sec (steady state creep) at fixed strain-rate depends on the mineral Flow Stress (differential, in MPa) Temperature (deg C)

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

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: 10-15 km (depends on geothermal gradient) Depth (~geothermal gradient) Temperature Assumes fixed, steady strain-rate

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