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RHEOLOGY #2: Anelasicity RHEOLOGY #2: Anelas2city (aenuaon and modulus dispersion) of rocks, an organic, and maybe some ice Chris2ne McCarthy Lamont-Doherty Earth Observatory …but first, cheese Team Havar2 Team Gouda Team Jack Stress and Strain Stress σ(MPa)=F(N)/A(m2) 1 kg = 9.8N 1 Pa= N/m2 or kg/(m s2) Strain ε = Δl/l0 = (l0-l)/l0 l0 Cheese results vs. idealized curve. Not that far off! Cheese results σ n ⎛ −E + PV ⎞ ε = A exp A d p ⎝⎜ RT ⎠⎟ n=1 Newtonian! σ Pa viscosity η = ε s-1 Havarti,Jack η=3*107 Pa s Gouda η=2*108 Pa s Muenster η=6*108 Pa s How do we compare with previous studies? Havarti,Jack η=3*107 Pa s Gouda η=2*108 Pa s Muenster η=6*108 Pa s Despite significant error, not far off published results Viscoelas2city: Deformaon at a range of 2me scales Viscoelas2city: Deformaon at a range of 2me scales Viscoelas2city Elas2c behavior is Viscous behavior; strain rate is instantaneous elas2city and propor2onal to stress: instantaneous recovery. σ = ηε Follows Hooke’s Law: σ = E ε Steady-state viscosity Elas1c Modulus k or E ηSS Simplest form of viscoelas2city is the Maxwell model: t 1 J(t) = + ηSS kE SS kE Viscoelas2city How do we measure viscosity and elascity in the lab? Steady-state viscosity Elas1c Modulus k or EU ηSS σ σ η = η = effective [Fujisawa & Takei, 2009] ε ε1 Viscoelas2city: in between the two extremes? Viscoelas2city: in between the two extremes? Icy satellites velocity (at grounding line) tidal signal glaciers velocity (m per day) (m per velocity Vertical position (m) Vertical Day of year 2000 Anelas2c behavior in Earth and Planetary science Ocean on Europa How can there be liquid water way out there? Tidal dissipation as significant energy source Geysers on Enceladus Anelas2c behavior in Earth and Planetary science forces on an ice stream gravity Tidally modulated ice streams h(t) friction from weight till Velocity fluctuaons of Bindschadler Ice bedrock stream compared to 2des velocity (at grounding line) tidal signal velocity (m per day) (m per velocity Vertical position (m) Vertical Day of year 2000 [Anandakrishanan et al., 2003] Anelas2c behavior in Earth and Planetary science A8enuaon: seismic wave amplitude decreases with each cycle Magnitude of damping should depend on the character of the material the wave is passing through. Large global variaons in Q-1 Even larger variaons in regional studies [Rychert et al., 2008] Low Q < 77 High Q > 1000 Why we measure Q-1 in the lab • So we need to describe the intrinsic aenuaon of materials and characterize damping for different variables (T, P, d, Φ, H2O…) • We can use mechanical models (spring/dashpot) • We can also use experiment Viscoelas2city: in between the two extremes? 1 V S = µ ρJ1 µ −1 J2 QS = µ J1 Mechanical models for dissipaon Burgers Model Steady state instantaneous Burgers ε(t) t 1 ⎡ ⎛ −t ⎞ ⎤ 1 1 J(t) = = + 1− exp + transient ⎢ ⎜ ⎟ ⎥ σ ηSS k2 ⎣ ⎝ τ ⎠ ⎦ kE SS kE k1 Extended Burgers Model Steady state τ M transient instantaneous ε(t) t 1 − t 1 1 1 J(t) = = + Δ ⎡1− e τ ⎤D(τ )dτ + σ η k ∫ ⎣⎢ ⎦⎥ k SS E τ m E SS k1 k1 Andrade Model And ε(t) t m 1 Steady state instantaneous J(t) = = + βt + i σ ηSS kE transient SS kE k i Mechanical models for dissipaon Viscoelas2city Time domain ε(t) t 1 J(t) = = + F(t) + σ ηSS kE Creep compliance Laplace transform frequency domain Complex compliance * σ(t) = σ 0 exp(iωt);ε(t) = σ 0 J exp(iωt) J *(ω) = J1 (ω) + iJ2 (ω) ω = 2π f 1 J -Loss compliance Q− = 2 J1 -Storage compliance 1 E = 2 2 J1 + J2 Mechanical models for dissipaon Burgers Model 1 k instantaneous B 2 Steady state J1 (ω) = + 2 2 2 kE k2 + η2ω 1 transient η ω 1 J B (ω) = 2 + 2 k 2 + η2ω 2 η ω SS kE 2 2 SS k1 ∞ Extended Burgers Model EB 1 ⎡ 2 2 ⎤ J1 (ω) = 1+ Δ D(τ )dτ / (1+ ω τ ) ⎢ ∫0 ⎥ kE ⎣ ⎦ Steady state ∞ transient instantaneous EB ω 2 2 1 J2 (ω) = Δ τ D(τ ) / (1+ ω τ ) + 1 1 ∫0 kE ηSSω ατ −(1−α ) H(τ − τ )H(τ − τ ) D(τ ) = m M SS α α τ M − τ m k1 k1 Andrade Model And 1 −m ⎛ mπ ⎞ Steady state instantaneous J1 (ω) = + βΓ(1+ m)ω cos⎜ ⎟ kE ⎝ 2 ⎠ i transient And −m ⎛ mπ ⎞ 1 J2 (ω) = βΓ(1+ m)ω sin ⎜ ⎟ + ⎝ 2 ⎠ η ω SS kE SS k i Mechanical models for dissipaon Burgers Model Steady state instantaneous 1 transient SS kE k1 Extended Burgers Model Steady state transient instantaneous 1 1 SS k1 k1 Andrade Model Steady state instantaneous i transient SS kE ki [Cooper, 2002] Burgers vs. Andrade: work equally well [Tan, Jackson and Fitz Gerald, 2001] Viscoelas2city How do we measure anelascity in the lab? [custom apparatus used in many Jackson, Faul, Farla papers; described in Jackson and Paterson, 1993] Viscoelas2city How do we measure anelascity in the lab? Crosshead δ Stress Strain Transducer EXTENSOMETER DETAIL Housing and Cage LVDT core σ Cryostat stress and strain 0 ε0 Phase lag = δ Macor TM Sample normalized Platens and moving Pistons angle (ωt) [radians] sample Load Cell and Cage periodic stress o!set (tectonic) stress Actuator stress time Viscoelas2city How do we measure anelascity in the lab? Analogue samples: borneol (C10H18O) Piezoelectric crossbar actuator load cells mirrors platen laser displacement motor sample meter T/Tm meter 0.5 0.6 0.7 sample housing assembly Acrylic thermal Forsterite" 900oC 1100oC movable barriers o o o stage Tm=1900 C 1000 C 1200 C Electromagnet (1273K) (1473K) stepping Borneol" o o o 40oC 60oC motor Tm=204 C 0 C 20 C [Takei, Fujisawa, McCarthy, JGR116, 2011]" (273K) (293K) (313K) (333K) Empirical observaons: Grain size, temperature dependences Anelasticity Raw data – we take it at mul2ple frequencies All grain sizes and temperatures" We also measured viscosity at each condi2on Viscosity [McCarthy, Takei, Hiraga, JGR116, 2011] Empirical observaons: Grain size, temperature dependences All grain sizes and temperatures" Normalized data" normalize data E Unrelaxed by Maxwell U modulus fM = viscosity frequency:" η [McCarthy, Takei, Hiraga, JGR116, 2011] Empirical Observaons: Normalizaon by Maxwell frequency comparison of mul2ple studies high temperature background Raj and Ashby (1971) Model of anelas2city Morris and Jackson (2009) Grain boundary sliding Local slip at Local stress No slip planer segments concentraon Global slip (creep) and relaxaon Grain boundary elastic larger time scale viscous The master curve approach Q is a unique function of normalized frequency fn = f/fM! −1 −1 ⎛ f ⎞ Q ( f ,d,T ) = Q ⎜ ⎟ ⎝ fM ⎠ ⎛ U ⎞ f (d,T ) = M (T )⋅η−1d− p exp − M U 0 ⎝⎜ RT ⎠⎟ α " Frequency dependence of Q-1:" ⎛ ∂lnQ−1 ⎞ = −α ⎜ ∂ln f ⎟ ⎝ ⎠ d,T Grain size dependence of Q-1:" ⎛ ∂lnQ−1 ⎞ All depend on α α is not a constant!" = −α ⋅ p ⎜ ∂lnd ⎟ ⎝ ⎠ f ,T Temperature dependence of Q-1:" ⎛ ∂lnQ−1 ⎞ U = α ⋅ ⎜ ∂lnT ⎟ RT ⎝ ⎠ f ,d [McCarthy, Takei, Hiraga, JGR116, 2011] The master curve approach So how do we extrapolate into a desired range?! Just need these two things:" −1 −1 Q ( f ,d,T ) = Q ( fn ) ≈ Xn (τ n ) − p ⎛ d ⎞ ⎡U ⎛ 1 1 ⎞ ⎤ ⎡V ⎛ P P ⎞ ⎤ τ (d,T,P) = J (T,P)⋅η exp ⎢ − ⎥ exp ⎢ − r ⎥ M U 0 ⎜ d ⎟ R ⎜ T T ⎟ R ⎜ T T ⎟ ⎝ r ⎠ ⎣⎢ ⎝ r ⎠ ⎦⎥ ⎣⎢ ⎝ r ⎠ ⎦⎥ -1 Note: Relaxation Xn("n), Q (fn), storage J1, and loss J2 are all related through the Kramers-Kronig Relations.! The master curve approach n fitting curve (this study) Faul & Jackson [2005] X 0 10 data (d=3-8μ m) seismic range for d=1mm Using a polynomial fit -1 d=1mm 10 to the data" d=10μ m PREM on spectrum, i 10-2 axat l re d 10-3 fit2 fit1 ze li 10-4 norma 100 10-2 10-4 10-6 10-8 10-10 10-12 10-14 normalized time scale, τ n = τ/τ M [McCarthy, Takei, Hiraga, JGR116, 2011] 4.6 observation 4.4 depth=50km S Priestley & McKenzie [2006] V (km/s) V 4.2 0.24 4.6 The master curve approach 0.2 anharmonic 4.4 anelastic 0.16 VS 4.2 Q S 0.12 -1 S 4 V (km/s) Shear wave velocity V and shear 4.6 This study 0.08 S wave attenuation Q-1 as functions 3.8 f = 0.01 Hz observation 4.4 d = 1 mm Q -1 0.04 of Temperature can be then 3.6 depth=50kmP = 1.5 GPa S S Priestley & McKenzie [2006] calculated directly from J and J :" (km/s) V 4.2 0 1 2 0.24 4.6 0.2 anharmonicanharmonic 1 4.4 extra-anelastic anelastic V = 0.16 S µ VS anelastic 4.2 Q V Q ρJ S S 1 -1 0.12 S 0.12 -1 S 4 S µ V (km/s) −1 J2 (km/s) V ThisFaul study& Jackson [2005] 0.08 QS = µ 3.8 f = 0.01 Hz d = 1 mm -1-1 0.04 J1 d = 1 mm QQ 0.04 3.6 P = 1.5 GPa SS 0 600 700 800 900 1000 1100 1200 13000.24 R 4.6 Temperature T( C) anharmonic 0.2 Note: Relaxation X (τ ), Q-1(f ), storage J , 4.4 extra-anelastic n n n 1 0.16 and loss J are all related through the anelastic 2 4.2 V Q Kramers-Kronig Relations.! S S 0.12 -1 4 S V (km/s) V Faul & Jackson [2005] 0.08 3.8 f = 0.01 Hz d = 1 mm Q -1 0.04 3.6 P = 1.5 GPa S 0 600 700 800 900 1000 1100 1200 1300 R Temperature T( C) The master curve approach Method used by Priestley and McKenzie, 2012 • They take a velocity model that is based on a large number of seismograms.
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