RHEOLOGY #2: Anelascity (aenuaon and modulus dispersion) of rocks, an organic, and maybe some ice

Chrisne McCarthy Lamont-Doherty Earth Observatory …but first, cheese

Team Havar 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 Viscoelascity: Deformaon at a range of me scales Viscoelascity: Deformaon at a range of me scales Viscoelascity Elasc behavior is Viscous behavior; strain rate is instantaneous elascity and proporonal to stress: instantaneous recovery. σ = ηε Follows Hooke’s Law: σ = E ε

Steady-state viscosity Elasc Modulus k or E ηSS

Simplest form of viscoelascity is the Maxwell model: t 1 J(t) = + ηSS kE

SS kE Viscoelascity

How do we measure viscosity and elascity in the lab? Steady-state viscosity Elasc Modulus k or EU ηSS

σ σ η = η = effective [Fujisawa & Takei, 2009] ε ε1 Viscoelascity: in between the two extremes? Viscoelascity: in between the two extremes?

Icy satellites

velocity (at grounding line)

tidal signal glaciers velocity (m per day) per velocity(m

Vertical position (m) Vertical

Day of year 2000 Anelasc 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 Anelasc 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 des velocity (at grounding line)

tidal signal velocity (m per day) per velocity(m

Vertical position (m) Vertical

Day of year 2000 [Anandakrishanan et al., 2003] Anelasc behavior in Earth and Planetary science

Aenuaon: 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 Viscoelascity: 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 Viscoelascity 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]

Viscoelascity How do we measure anelascity in the lab?

[custom apparatus used in many Jackson, Faul, Farla papers; described in Jackson and Paterson, 1993] Viscoelascity How do we measure anelascity in the lab?

Crosshead δ Stress Strain Transducer EXTENSOMETER DETAIL Housing and Cage LVDT core σ Cryostat andstress 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 Viscoelascity 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 mulple frequencies

All grain sizes and temperatures

We also measured viscosity at each condion

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 mulple studies

high temperature background Raj and Ashby (1971) Model of anelascity 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(τn), 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, on 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) V 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 (km/s) V −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. • They use geophysics and petrology (garnet peridote) to come up with independent esmates of T,P, which they use for Vs(T,P), Q-1(T,P) and η(T,P), the laer is used to calculate Maxwell frequency fM. • They use our polynomial descripon of the Master curve, and scale it by that

independent fM and compare that to the frequency dependent aenuaon determined from seismology [Dalton et al, 2009] The master curve approach

Method used by Priestley and McKenzie, 2012 Empirical Observaons: Normalizaon by Maxwell frequency comparison of mulple studies

11 fM=10 Pa/ 1020Pa s =10-9Hz Normalized seismic frequency range − p ⎛ ⎞ ⎡ ⎛ ⎞ ⎤ ⎡ ⎛ ⎞ ⎤ d U 1 1 V P Pr τ M (d,T,P) = JU (T,P)⋅η0 ⎜ ⎟ exp⎢ ⎜ − ⎟ ⎥exp⎢ ⎜ − ⎟ ⎥ ⎝ dr ⎠ ⎣⎢ R ⎝ T Tr ⎠ ⎦⎥ ⎣⎢ R ⎝ T Tr ⎠ ⎦⎥

[Jackson, Faul, Skelton, 2013] Problems with master curve approach

Maxwell EU frequency scaling fM = has limits: η Problems with master curve approach

Maxwell frequency scaling has limits:

[Sundberg and Cooper, 2010] High frequency peak

G − G Δ = U R GR

[Sundberg and Cooper, 2010] High frequency peak

[Takei, Karasawa, and Yamauchi, 2014] High frequency peak

Faul et al., 2004 How does melt affect diff-GBS?

Poroelastic effect of effect of melt on GB- melt on sliding at ultrasonic seismic properties frequencies

Poroelastic effect vs. Anelastic effect Well known! Not well known! Empirical observaons: melt fracon Anelasticity Viscosity

Significant d≈23μm decrease in viscosity even with small % melt

[McCarthy and Takei, GRL38, 2011] Viscous conguity model: melt becomes fast path for diffusion

[Takei and Holtzman, 2009] Empirical Observaons: Normalizaon by Maxwell frequency Anelasticity melt fracon

Variation in modulus due to chemical effect suggests a high frequency anomaly

Melt and chemical effects captured by viscosity. Similitude persists. No shape change of spectra, no Debye peaks but at high frequency something else is happening… [McCarthy and Takei, GRL38, 2011] Dissipaon peak due to melt “squirt”

Garapic et al. [G3, 2013] show weed grain boundaries in Olivine + melt

[Jackson et al., 204; Faul et al., 2004] Summary of Empirical Observaons

• Although “apparent” relaonships of Q-1 on d, T, P, Φ etc. exist, these are related through viscosity (scaling by Maxwell freq. negates the effects). • At these laboratory condions* there is similitude in the “high temperature background” response of polycrystalline materials (*low stress, low dislocaon density, small grain size). The similitude means that a single mechanism is acng in all of the studies. • Unfortunately, if you scale the master curve to condions in the mantle (very high viscosity), our lab data is not at the right frequencies for seismic waves. Extrapolaon is needed, which requires knowledge of the physics (at higher freq) and/or use of a “good” mechanical model. Many observaons of seismic anisotropy

Shear wave spling method using SKS and SKKS waves Many regional studies [this is Fouch and Rondenay, 2006] Anisotropy comes from alignment of grains, aka CPO or LPO. LPOs form from deformaon in dislocaon creep

periodic stress

o!set (tectonic) stress stress

time

[Lee et al., 2002]

periodic stress

o!set (tectonic) stress stress

Dislocaon creep: time relave displacement of two parts of a crystal (b) dislocation creep n=3;p=0

m3 diffusion (Newtonian) Diffusion creep: creep 2 m mass diffusion at 1 m n=1;p=3 the grain boundary How do dislocaons affect Q-1?

This study used samples pre-deformed in the dislocaon creep regime (longitudinal and torsional)

In both cases, the Q-1 was greater than the predicon (of diffusion- GBS)

[Farla et al., 2012] (b) GBS-accommodated dislocation basal slip creep m3 Strain effects in ice bs-accom. GBS

m2

FLOW LAW m1 diffusion (Newtonian) FOR ICE-I creep

T/TM = 0.7-0.9

5µm All 25µm normalized 150µm 350µm by fM How do dislocaons affect Q-1? Granato Lücke Theory [1956]

Some combinaon of subgrain formaon and moon of lace dislocaons LN Lc

INCREASING STRESS

From Uli Faul Conclusions about experimental measurement of aenuaon • Master curve shows geometry accounted for, so either tesng method fine (torsion vs. long.); T,P,Φ etc. determined from fM • Some difference in interpretaon about d-scaling • McCarthy et al. gave a parameterized curve of experimental data, but… • High frequency peak is sll poorly constrained • Peak due to melt poorly constrained • Effect of strain/dislocaon density poorly constrained Observaons can help constrain things THANK YOU! Studies on organic analogue material samples (longitudinal) Side note: longitudinal studies measure young’s modulus-type attenuation, which contains both bulk and shear attenuation. The fraction of each can be determined from the Poisson's ratio of your material and the standard relationships between elastic moduli. µ 3k Q−1 = Q−1 + Q−1 E 3k + µ k 3k + µ µ −1 −1 −1 QE ≅ 0.12Qk + 0.88Qµ Ice Poisson ratio = 0.325 −1 −1 −1 QE ≅ 0.086Qk + 0.914Qµ Borneol Poisson ratio = 0.37 iωt σ(t) = σ 0e Defining J1 and J2 * iωt ε(t) = σ 0 J e

Complex compliance

J*(ω) =J1- iJ2 Relation between Q and E and J1 and J2 J Q−1 = 2 J1 1 E = 2 2 J1 + J2

From Q and E to J1 and J2 1 J1 (ω) = E(ω) 1+ (Q−1 (ω))2 −1 J2 (ω) = Q (ω)J1 (ω) Defining J1 and J2

τ =∞ 1 dτ J1 (ω) = JU + X(τ ) ∫τ =0 1+ (ωτ )2 τ τ =∞ ωτ dτ 1 J2 (ω) = X(τ ) + ∫τ =0 1+ (ωτ )2 τ ωη

elascity anelascity viscosity

Well approximated by: P 2 ln P J1 (P) + 2 = J1 (PR ) + J2 dln P π η π ∫ln PR π ⎛ P ⎞ P J2 (ω) = X ⎜τ − ⎟ + 2 ⎝ 2π ⎠ 2πη

From Nowick and Berry How do we measure anelascity in the lab? Gribb & Cooper’s “Universal curve” based on Raj, 1975

[Gribb and Cooper, 1998]

2 3 3 3(1− ν )d kT 10ηss 4ηss τ = 3 ≅ ≅ 2π EDgbδΩ E G -1 Q (fn): physical model of GBS

GBS

5nm Hiraga et al. (2002)

model III Grain with  8 edges Takei & McCarthy, A granular model for 56 GB steps anelascity due to grain boundary sliding, AGU2010. High frequency peak

• Raj and Ashby [1971] Fourier series approx: G R = (0.57(1− ν) + 1)−1;Δ = 0.43 GU Background/offset stress determines dislocaon density

(b) dislocation creep

3 m tides diffusion

(Newtonian) creep stress m2

m1 convection time