Viscoelastic Energy Dissipation: Some Constraints from Materials Physics

Reid F. CooperDepartment of Geological Sciences

Brown UniversityAcknowledgements: Marshall Sundberg, Joe Bunton, Kate McMillan, Yun-Che

Wang, Donald Stone; Rod Lakes

Foundation:

K.M. McMillan et al., The viscoelastic behavior of β-In3Sn and the nature of the high-temperature background, J. Mater. Sci. 38:2747-54 (2003).

P.M. Buechner et al., Viscoelastic behavior of superplastic Pb-63wt% Sn over a wide range of frequency and time, Scripta Mater. 41:561-567 (1999).

D.S. Stone, Scaling laws in dislocation creep, Acta Metall. Mater. 39:599–608 (1991).

D.S. Stone et al., Similarity and scaling in creep and load relaxation of single crystal halite (NaCl), J. Geophys. Res., 109:B12201 (2004).

MARGINS Workshop on Interpreting Upper Mantle Images; WHOI; Woods Hole, MA; 17 May 2006

An Energy-Dissipation Perspective…The dynamic geological setting consists of rock being actively deformed—rock experiencing a thermodynamic state that includes a relentless deviatoric stress in the range 1–100 MPa. The accumulation of plastic strain promotes texture—“self-assembly”—on a variety of scales, from nanometer to kilometer:

Seismic waves, at10 ≤ σ (kPa) ≤ 100 and 10–3 ≤ ƒ(Hz) ≤ 1, sample this material. How does the system respond to such “tickling?”

Intrinsic Attenuation: The Absorption Band…How?

Anderson and Given (1982) Liu et al. (1976)

Conventional perspective: superposition of standard (exponential-decay) solid models; a mix of mechanisms and/or microstructural scales…

But: Can You Cite 1010 in Length Scale?QG

–1 v. ƒ; QG–1 ≡ tan δ

McMillan et al. (2003)

Isolating the Physics: Uniform d; No Dislocations‡

Balsam Gap (NC) Dunite: Fo92(this specimen:⎯d = 4.8 ± 0.4 μm)

• ρ ≈ 0.98+ ρth; chemically stable• 4 ≤⎯d (μm) ≤ 17 in five distributions• uniform: no grain growth during testing

• fluid-energy pulverized to ~1-μm particles

• cold-pressed (100 MPa)

• vacuum dried/reacted (0.1 torr; 1000oC, 1 h)

• vacuum sintered (0.1 torr but at Ni:NiO; 1350–1380oC, 1 h; ± Tspike to 1465oC, ¼ h)

‡σxy ≈ 2Gb/λ where λ = d: need olivine d ≤ 20 μm for σxy = 4 MPa

Attenuation Response of Polycrystalline Olivine(uniform grain size; no lattice dislocations)

Fo88; ⎯d = 2.8 ± 0.4 μm

Gribb & Cooper, JGR (1998)

Andrade Fit: single-mechanism, non-exponential-decay relaxation; consistent with εa ∝ t1/2.

Physical Interpretation of Andrade Behavior:Intrinsic transient in diffusional creep (after Raj, 1975)

• Fully Newtonian material; Diffusional Creep (threshold?).

• Grain boundaries shear-inviscid—gives rise to (a)

• Apparent broaddistribution of compliances, but no distribution in a microstructural variable. Andrade behavior intrinsic to the creep mechanism: chemical diffusion in a diminishing potential.

Universal Q–1 Spectrum from Universal Creep Curve

( )2 3

3b

3 3 1 d kT2 ED

− ντ =

π δΩ

xy b xyss 3

ss

132 DkTdσ Ωδ σ

γ ≅ =η

&

Transient—chemical diffusion in the context of a diminishing potential:

Steady-State—Coble (1963) creep:

xy ss ss

ss

10 10 4E E Gσ η η

τ ≅ = ≅γ&

→ Entirety of dynamic behavior can be described by two variables, ηss and E.

Applicability to Material Deforming by Dislocation Creep?

Polycrystalline data deviate from the Andrade-based model when grain boundary processes start to dissipate mechanical energy.

Data “map” into deformed-single-crystal response, suggestive of a primary role of subgrain boundaries in effecting absorption.

Single Crystal Attenuation : The Role of “Hardness”

Gueguen et al., PEPI (1989)

Synthetic Forsterite

“Pre-deformation:”1600oC; 20 MPa; [111]C

Steady-state creep to 1% strain

Anticipated mean subgrain size: ~20 μm

Attenuation:Sub-resonant torsion

Broadband Viscoelastic Spectrometry: In-Sn System

Broadband Viscoelastic Spectrometer: Lakes (1999)

In-Snβ–“In3Sn”: bct, I4/mmm;

endothermic solutionγ–“InSn4”: hex, P6/mmm→ neither phase meets

vonMises criterion→ both phases have high

Peierls barrier and large b (for metals)

Broadband Spectra for In-Sn Phases (1)

• Spectra show power-law response over broad frequency range• Ultrasonic response clearly related to that at low frequencies• Single-phase aggregates demonstrate “broad-hump” or “plateau” behavior• Eutectic response indicates the primacy of phase boundary effects: rule of

mixtures does not apply

QG–1 v. ƒ;

QG–1 ≡ tan δ

McMillan et al.(2003)

Broadband Spectra for β–In3Sn: Impact of Dislocation Creep

QG–1 v. ƒ;

QG–1 ≡ tan δ

McMillan et al. (2003)

• Dislocation creep to high ε sees QG–1 diminish, despite addition of

dislocations; crept specimens have limited distribution of subgrain boundary structure and orientation.

Fractal Analysis of Erosion Topography of the Grand

CanyonGe 16(5); Autumn 2005

—C.E. Dutton, Tertiary History of the Grand Cañon District, USGS, Washington, 1882.

Spatial Scales of Energy Dissipation: Self-Similarity

Microstructural Self-Similarity

l ≡ SGB dislocation spacing; L ≡ dislocation link length;A½ ≡ subgrain size

Self-similarity implies, too, microstructural aspects at a larger scale, e.g., the statistical angular relationship amongst groups of subgrains.

l = b/θ

A1/2

L10-4 10-2 100 102 104

A1/2 (μm)

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

Freq

uenc

y

This studyLin, Lee and Ardell [1989]Eggeler and Blum [1981]

A1/2Ll = b/θ

σ1= 0.6MPaT = 700oC

Self-Similarity: A State-Variable Approach to

Plasticity?

PathsCreep: constant stress

(decelerating transient)

Constant strain rate(work hardening)

Constant Microstructure?

Hart (1976): Each point in (σ, ε) space (T constant) corresponds to a singular internal structure (“hardness”) of the material—e.g., a specific distribution of lattice defects. Can microstructure bedescribed as a state variable? Can we discover path independence?

A State-Variable Approach to ViscoelasticityHart (1976): Each point in (σ, ε) space

corresponds to a singular internal structure (“hardness”) of the material—e.g., a specific distribution of lattice defects. If the internal structure can be described by a single “state” variable, then curves of constant “hardness” should superpose…

Stone et al., JGR (2004)

Correlation of Subgrain Structure with Hardness—Regardless of Deformation Path…

State-variable model requires microstructural self-similarity; thus, choose subgrain size as proxy for “hardness.” Creep and load-relaxation experiments settling on a specific (σ, ε) point show identical subgrain structures: path independence!

The Physics: Serial Kinetic Processes of Subgrain Diffusional Creep and Dislocation Glide (Stone, 1991)

Diffusional creep of subgrains: linear; n = 1; q = 2

Dislocation glide: non-linear; r ~ 1

Di ≡ subgrain size—shown here as monodispersed

“Hardness” scaling slope

Distribution of subgrain sizes allows broadening of hardness curve

Attenuation “Tickling:” modest excursions along a single hardness curve

Strain (~1%) must be accumulated in order to change the hardness state. Thus, the small excursions in stress potential wrought by seismic waves are confined to the hardness curve characteristic of the large-scale deformation occurring in the medium.

Interpreting seismic attenuation in terms of e.g., water content, temperature, strain-rate, etc., requires understanding the physics of extrapolation of the hardness curves.

Kinetic Factors Affecting Hardness

Curve

Translation StepsI: T effect on unrelaxed modulusII: Thermokinetic effects: ∝ fH2O

m × fO2p × fox

q × exp(–A/T): ± ϕ effectIII: Stress effect to match strain rateWhile attenuation samples hardness state, note the convolution

of a number of thermodynamic factors!Note, too, changing (work-hardening “rate”)log / logν = ∂ σ ∂ ε&

Matching Data to the Model

The more compliant nature of the data compared to the model suggests that microstructural features at a scale larger than the subgrain size—and yet related by similarity—are an important contributor to the effective viscosity of the specimens.

Δ10 ≡ standard deviation of lognormal distribution of subgrain size

Measured: Δ10 = 0.27

Reasonable maximum:Δ10 = 0.50

Best fit: Δ10 = 0.95

Solid-State Phase Boundaries: Potent Absorbers

The trend of the data is logical: materials that are very stiff are also low-loss. The diagonal line represents E′′ = E′ tan δ = 0.6 GPa—an apparent physical limit to stiffness-damping behavior…

Stiffness-Loss Map(Lakes, 1999)

T ~ 300Kƒ ~ audio/subaudio

E ≡ Young’s modulustan δ ≡ QE

–1

Solid-State Phase Boundaries: Potent Absorbers

Finely-divided phases—here composites with eutectic microstructures—produce high damping while retaining high stiffness. Non-polymorphic phase transformations can create materials in this behavioral regime.

Stiffness-Loss Map(Lakes, 1999)

T ~ 300Kƒ ~ audio/subaudio

E ≡ Young’s modulustan δ ≡ QE

–1

Example: RUS of Dunite & Harzburgite

Resonant Ultrasound

Spectroscopy(Lan et al., 2003)

synthetic dunite and harzburgite

d ~ 5 μmT = 300K

first torsional mode(~700 kHz for

specimen size)

The play-off between phase boundary absorption effects and those associated with deformation-induced sub-microstructure (“hardness”) at seismic frequencies remains to be studied…

Fabric: Dissipation-Structure Effect on Q? on ηeff?E.g., Strain- and Strain-Rate Effected Solid-State Phase Separation

—Marshall Sundberg

Synthetic harzburgite: opx/ol (vol): 65/35; 1200oC; 1.6GPa; γ ~1.5→ Fully Newtonian aggregates: grain/phase boundary diffusion creep→ No static grain growth: coarsening only possible across grain boundaries→ Periodicity related to strain rate?A three-phase rheology problem — viscosities: ηol-ol , ηopx-opx & ηol-opx

τ ≤ 20MPa (resolution of load cell)

& 410 / s−ε = & 510 / s−ε =

SEM/BEI: opx ≡ dark gray

Summary/ConclusionsHierarchical dissipation structures characterize large-

strain deformation. These structures have significant implications for phenomena like melt segregation and rock homogeneity, with ramifications for anisotropy and attenuation.

The rock physics community is in the early stages of characterizing such structures—with an emphasis on scaling.

Attenuation consists of small-strain mechanical dissipation within the hierarchical structure. In materials where that structure is created by dislocation creep, subgrain boundaries may be the primary source of absorption. Effects of T, chemical activities on Q–1 are manifest in these structures. Stress-relaxation protocol may prove an experimentally efficient approach to discriminating, e.g., T and fH2O effects on relaxation/attenuation.

Phase boundaries seem particularly potent absorbers; in their presence one achieves high absorption while retaining high stiffness.