Schedule Rheology basics Viscous, elastic and plastic Creep processes Flow laws Yielding mechanisms Deformation maps Yield strength envelopes Constraints on the rheology from the laboratory, geology, geophysics and numerical modelling (next time)!2
Rheology: What is it? The branch of science concerned with how material flows More precisely, the response of a material to deformation (e.g. applied strain, or strain-rate, or stress)! Some examples!3
Rheology of Fluids
steel
concrete
visco-elastic fluids
Strain original shape Tensile strain - or - direct strain xx, yy deformation due to! displacement Shear strain xy Volumetric strain (dilatation)!8
Strain-rate Analogous to the strain tensor, but involves gradients of velocity (and not displacement)! x =(x, y)!!!!! v =(v x,v y ) xx = @v x @x position velocity Strain-rate invariant: a scalar measure of the magnitude of a tensorial quantity xx, yy, xy [1/s] xy = 1 yy = @v y @y 2 ij = 1 @vi + @v j 2 @x j @x i II =!9 q 1 2 2 ij @vx @y + @v y @x
World strain-rate map!10
Three basic rheologies Elastic! Viscous! Plastic!11
0D elastic and viscous media Elastic Straintime = E 0 = tan-l E Viscous =2 = tan 1 (2 )!12
Characteristic values Table 5.5 Some Representative Viscosities (in pa.s) Air i 0-5 Water 10r Olive oil 10-l Honey 4 Glycerin 8A Lava 10-104 Asphalt 105 Pitch 10e lce 1012 Rock salt 1017 Sandstone slab 1018 Asthenosphere (upper mantle) 1020 Lower mantle 1021 Soarces: Several sources, including Turcotte and Schubert (1 gg2).
Visco-elasticity Visco-elastic Kelvin-Voigt body = E +2 Elastico-viscous Maxwell body = E + 2!14
Visco-elasticity = E + 2 = E +2 stress strain stress strain strain stress strain stress!15
Maxwell relaxation time Maxwell relaxation time: t m =2 /E Time required for initial stress to reduce by 1/e Parameters values: E ~ 10 10-10 11 Pa, η = 10 17-10 27 Pa s! Implying tm = 11 days 3000 million years
Material behaviour classification Stress and strain (strain-rate) relationships 300 Linear. Yield stress X Failure stress Work hardening Non-linear Ductile : 2OO (L Plastic o o \Permanent strain Brittle E co (I) i5 100 - Elastic strain \Ultimate strength 0 4 8 12 16 200 4 I 12 16 20 Strain (shortening elongation; o/o)
Non-linear (non-newtonian) behaviour / n General stress - strain-rate relation n = 1: Newtonian n > 1: non-newtonian (shear thinning) n infinite: pseudo-brittle Local slope: Effective viscosity Application: Different viscosities within the lithosphere due to different absolute plate motions!18
Rocks are not rheologically simple: Loading experiment Simple compression experiment Constant stress loading Many effects observed (viscous, elastic, brittle)
Rocks are not rheologically simple: Unloading experiment!20
hich should give us e the significance of cales,from mountain edict broad rheologven assumptionson.figure 5.18 usesexn this chapter,to calthe crust and upper ble,first-orderpredicre,which can then be thdynamics. fusing use of theseterms, they do allow a convenientdistinction for the behavior of natural rocks. Figure 5.19 schematically highlights the observational aspect of this Combined effects Brittle Brittle-ductile Brittle-ductile Ductile NNNN (a) (b) (c) (d) (a)to brittle-ductile (b,c)to ductile (d)deformation, Figure 5.19 Brittle reflecting thegeneral thatisused inthesubsequent chapters. subdivision ultimate strength Rheology95 Flgur andtl that
Pressure-temperature effects!22 (Ranalli, Rheology of the Earth, 1995)
Temperature dependence SHIVA test High speed, torsion experiment Gabbro test sample Axial load of ~ 8 MPa Rotational velocity ot 5 m/s!23
Creep The slow, continuous deformation of a material over time Mechanism occurs under applied stress, due to thermally activated motion of atoms and ions associated with crystal defects Thermally activated diffusion process Viscous behaviour (strain-rate) Flow law! = f(, t, T,...)! Solid state creep is a major deformation mechanism in the Earth s crust and mantle!24
Diffusion creep! Creep processes Migration of atoms through the (i) interior of the crystalline lattice (Herring-Nabarro), or (ii) along grain boundaries (Coble)!!!!! = A di Function of: * crystal grain size (d) * pressure (P) * temperature (T) Dominant at low stresses! Linear (Newtonian) flow law
Creep processes Dislocation creep! Migration of defects (dislocations) within the crystalline lattice. Dislocations may assume line or point geometries!!!!!! = A disc n Function of: * crystal grain size (d) * pressure (P) * temperature (T) Dominant at high stresses! Non-linear (non-newtonian) flow law!26
Viscous creep law Experimental data The viscosity of rocks is strongly dependent on pressure, temperature, stress (strain-rate),! grain size, water content, melt and mineralogy, Arrhenius flow law = A n d p C s OH exp( )exp apple (E + PV) RT Effective viscosity e ective = 2!27
Viscous creep laws typically used ij = 1 2 e ij e = B ( II ) 1 n exp apple E + pv RT, II = q 1 2 ij ij B depends on grain size (in the linear domain) n = 1 > diffusion creep n > 1 > dislocation creep Common simplification > Frank-Kamenetskii approx. / exp( T) Satisfactory for a limited p,t range!28
Upper mantle (1)- *'[-"=#2] e: ^ (;)' Thble 5.3. Parameter Values for Dilfusion Creep and Dislocation Creep in a Dry Upper Mantlea Quantity Diffusion Creep Dislocation Creep Pre-exponential lactor A (s-l) Stress exponent n Grain size exponent m Activation energy E* (kj mol Activation volume V* (m3 mol r) l) 8.7 x l0r5 I 3 300 6 x 10-6 3.5 x 1022 3.5 0 540 2x105 " AfterKaratoandWu(1993).Otherrelevantparamctervaluesarepsheil. :80GPa, nodulus h : 0.55nm. and R - 8.3144JK I mol-1.
More experimental data Table 5.6 Experimentally Derived Creep Parameters for Some Common Rock Types Rock type tolog A (MPa-ns-r) E* (kj.mol-l) Values valid for the following form of the flow law = A n exp apple E RT Albite rock Anorthosite Clinopyroxenite Clinopyroxenite (wet) Diabase Granite Granite (wet) Marble Olivine rock Olivine rock (wet) Quartz diorite 0uartzite 0uartzite (wet) Rock salt 18 16 17 5.17 17 6.4 7.7 33.2 4.5 4.0 11.5 10.4 10.8-1.59 3.9 3.2 2.6 3.3 3.4 3.4 1.9 4.2 3.6 3.4 2.4 2.8 2.6 5.0 234 238 335 490 260 139 137 427 535 444 219 184 134 82 Source: Kuby and Kronenberg (l 987).
Rock and mineral aggregates Diffusion creep and dislocation creep mechanisms are not independent - Both simultaneously occur at a given stress state Composite rheology Effective composite viscosity 1 = 1 + 1 e di disc under strain rate decomposition assumption (Maxwell like) = di + disc!31
Rocks have a finite strength Differential stress plastic elastic strain The differential stress ( ) is limited in nature 1 3 Caused by micro-defects, breaking bonds between atoms, growth of micro cracks!32
Differential stresses in the crust p / ( 1 3 ) (Townend & Zoback Geology 2000)!33
Plastic vs. viscous deformation brittle/plastic!34 viscous
o*l h O1 ill il l* 01 U $ T I /l os-l Tensile fracture 'tr"n.itona-tensle" / i- 01 fracture \ VJ A co'ro'bshear Fracture style as a function of confining Brittl -plastic transition pressure \ r,*8 o1 Plastic (b) Figure 6.25 (a) A representative composite failurenvelope on a Mohr diagram. The different parts of the envelope are labeled, and are discussed in the text. (b) Sketches of the fracture geometry that forms during failure. Note thathe geometry depends on the part of the failurenvelope that represents lailure conditions, because the slope ol the envelope is not c0nstant.!35
Coulomb failure criteria = C + tan( ) friction angle normal stress cohesion max shear stress = m m sin( ) = m sin( ) m = 1 2 ( 1 3) m = 1 2 ( 1 + 3 ) m = C cos( )+ m sin( )! 1 3 = C 0 + pµ 0
Byerlee s law Coulomb plasticity is empirical theory, however seems to work reasonably well for upper crustal rocks ~ 50 km
Byerlee s law http://geophysics.eas.gatech.edu/people/anewman/classes/geodynamics/misc/5_7_10.jpg 200 MPa apple n apple 2000 MPa n < 200 MPa
Numerical models of (brittle) localisation!39
Numerical models of (brittle) localisation Brittle failure in the upper crust may result in localised zones of deformation due to Mohr-Coulomb plasticity (which mimics Byerlee s law)
Peierls creep and strength of rocks Problem: (Kameyama et al, 2001)! Byerlee is valid for upper crustal rocks. At deeper levels, the mechanism limiting stress is not very clear Low temperature plasticity (Peierls creep) has been suggested based on; experiments, numerical calculations and theoretical considerations Consequences This plasticity form does not produce localised faults as easy, requires shear-heating feedbacks to break the lithosphere Possible explanation of the small number of earth quakes in the lithospheric mantle compared to upper crustal rocks?!41
Deformation maps Question: Is diffusion creep or dislocation creep the dominant deformation mechanism in the upper mantle?! Considerations: 1. For a given stress - the mechanism with the largest strain-rate is dominant! 2. For a given strain-rate - the mechanism with the lowest stress is dominant!42
Mantle deformation maps (Schubert et al, 2001) Mantle:! T ~ 1600 [K]! e ~ 1e-15 [1/s] Independent! of stress Dry upper mantle, p = 0 e: (1)- *'[-"=#2] ^ (;)'
Olivine deformation map 10-2 Exponential-law creep_ 0 E 1 0-3 1 o-4-5 -6..8' 9-102 ^(g L a 100 1 o-5 1 1 0-6 o2 Depth (10'km) 1 0-1 Figure 9.35 Deformation mechanism map lor olivine with a grain size of '100 pm. Variables are the same as in Figure 9.32, except that depth is substituted lor temperature given an exponentially decreasing geotherma gradient with 300"C at the surface and 1850'C at 500 km depth.!44
Viscous deformation map (Kameyama et al, 2001)!
Strength envelopes Low strain-rate (or high temp.) High strain-rate (or low temp.) Byerlee's law Arrhenius flow law!46
Strength of the mantle-lithosphere Brittle ductile! transition (Kohlstedt et al., Strength curves for different materials: lithosphere, 1995)!
Compression versus extension Difference come from the dependence of Byerlee s law on the normal stress Compression results in large normal stress (tectonic loading)!48 (Burov E., Treatise on Geophysics V. 6, 2007)
Compression versus extension Difference come from the dependence of Byerlee s law on the normal stress Compression results in large normal stress (tectonic loading)!49 (Burov E., Treatise on Geophysics V. 6, 2007)
Summary Learned the vocabulary of rheology Examined three basic classes elastic viscous plastic / brittle Deformation maps and strength envelops!50