2.3 Rheology & Viscous Flow

How Rocks Deform

Rheology is the study of how materials deform and flow under applied stress. At the timescales of seismic waves (seconds), the mantle behaves as an elastic solid. At geological timescales (millions of years), the same mantle flows like an extremely viscous fluid. This dual behavior is central to understanding plate tectonics: the lithosphere is rigid on human timescales but the underlying asthenosphere creeps slowly, enabling plate motion and mantle convection.

The transition from elastic to viscous behavior is governed by the Maxwell relaxation time, which compares the elastic modulus to the viscosity. Understanding mantle rheology requires combining laboratory experiments on rock deformation with geophysical observations such as postglacial rebound and the geoid.

Elastic (Hookean) Behavior

For short-timescale loading (seismic waves, tidal forces), rocks behave as linear elastic solids obeying Hooke's law. The stress is proportional to strain, and deformation is fully recoverable:

\(\sigma = E\,\varepsilon \quad \text{(1D, Young's modulus)}\)

\(\sigma_{ij} = \lambda\,\delta_{ij}\,\varepsilon_{kk} + 2\mu\,\varepsilon_{ij} \quad \text{(3D, generalized Hooke's law)}\)

where E is Young's modulus, λ and μ are the Lamé parameters, and δ_ij is the Kronecker delta. Typical values for mantle rocks:

μ ≈ 80 GPa

Shear modulus (upper mantle)

K ≈ 130 GPa

Bulk modulus (upper mantle)

E ≈ 170 GPa

Young's modulus (olivine)

Viscous (Newtonian) Behavior

Over long timescales, the mantle deforms as a viscous fluid. In the simplest (Newtonian) case, the deviatoric stress is proportional to the strain rate:

\(\sigma_{ij}^{\prime} = 2\eta\,\dot{\varepsilon}_{ij}\)

where η is the dynamic (shear) viscosity in Pa·s and σ′_ij is the deviatoric stress tensor. The mantle viscosity is extraordinarily high compared to everyday fluids:

Material	Viscosity (Pa·s)	Relative Scale
Water	10³	1
Honey	~10	10&sup4;
Pitch (at 20°C)	~10&sup8;	10¹¹
Glacier ice	~10¹³	10¹&sup6;
Asthenosphere	~10¹&sup9; - 10²°	10²² - 10²³
Upper mantle	~10²° - 10²¹	10²³ - 10²&sup4;
Lower mantle	~10²¹ - 10²³	10²&sup4; - 10²&sup6;

Viscoelastic (Maxwell) Behavior

The Maxwell model combines elastic and viscous elements in series, capturing the transition from short-timescale elastic to long-timescale viscous behavior. The constitutive equation relates stress and strain rate:

\(\frac{1}{\mu}\frac{d\sigma}{dt} + \frac{\sigma}{\eta} = \frac{d\varepsilon}{dt}\)

The characteristic Maxwell relaxation time τ_M defines the timescale at which behavior transitions from elastic to viscous:

\(\tau_M = \frac{\eta}{\mu}\)

For loading times t << τ_M, the material responds elastically. For t >> τ_M, the material flows viscously. Under a step stress σ₀applied at t = 0, the stress relaxes exponentially:

\(\sigma(t) = \sigma_0 \, e^{-t/\tau_M}\)

Mantle Maxwell Time Estimate

Using typical upper mantle values:

• Viscosity: η ≈ 10²¹ Pa·s
• Shear modulus: μ ≈ 10¹¹ Pa (100 GPa)
• τ_M = 10²¹ / 10¹¹ = 10¹⁰ s ≈ 300 years

This means that for processes shorter than ~300 years (seismic waves, tidal loading), the mantle is effectively elastic. For processes longer than ~300 years (postglacial rebound, mantle convection), the mantle flows as a viscous fluid.

Creep Mechanisms

At mantle temperatures (1000-2000 K) and pressures (1-140 GPa), rocks deform by solid-state creep. Two dominant mechanisms operate depending on stress level, temperature, and grain size:

Dislocation Creep (Power-Law Creep)

Deformation occurs by the glide and climb of crystal lattice dislocations. The strain rate depends nonlinearly on stress (power-law) and is independent of grain size:

\(\dot{\varepsilon} = A\,\sigma^n\,\exp\!\left(-\frac{E^* + PV^*}{RT}\right)\)

• A: pre-exponential constant
• n: stress exponent (n ≈ 3.5 for olivine)
• E*: activation energy (~500 kJ/mol for olivine)

• V*: activation volume (~10-20 cm³/mol)
• P: pressure, T: absolute temperature
• R: gas constant (8.314 J/mol·K)

Dominates at high stress (> 1-10 MPa) and large grain size. Produces a lattice-preferred orientation (LPO) that causes seismic anisotropy. The effective viscosity decreases with increasing stress (shear-thinning or pseudoplastic behavior).

Diffusion Creep

Deformation occurs by the diffusion of atoms or vacancies through the crystal lattice (Nabarro-Herring creep) or along grain boundaries (Coble creep). The strain rate is linear in stress and depends strongly on grain size:

\(\dot{\varepsilon} = A\,\sigma\,d^{-m}\,\exp\!\left(-\frac{E^* + PV^*}{RT}\right)\)

• n = 1 (linear stress dependence, Newtonian)
• d: grain size
• m = 2 (Nabarro-Herring, lattice diffusion)

• m = 3 (Coble, grain boundary diffusion)
• Dominant at low stress (< 1 MPa) and small grain size
• Does not produce LPO (no seismic anisotropy)

Effective Viscosity & Deformation Maps

For non-Newtonian rheology, we define an effective viscosity that depends on the applied stress (or strain rate):

\(\eta_{\text{eff}} = \frac{\sigma}{2\dot{\varepsilon}} = \frac{1}{2A}\,\sigma^{1-n}\,\exp\!\left(\frac{E^* + PV^*}{RT}\right)\)

For dislocation creep (n ≈ 3.5), increasing stress by a factor of 10 decreases effective viscosity by a factor of ~10^2.5 ≈ 300. This strain-rate weakening is crucial for localizing deformation in shear zones and plate boundaries.

Deformation Mechanism Maps

Deformation mechanism maps plot stress vs. grain size (or temperature) and show fields where each mechanism dominates:

High stress, large grains

→ Dislocation creep dominates

→ Non-Newtonian (n ≈ 3.5)

→ Produces seismic anisotropy

Low stress, small grains

→ Diffusion creep dominates

→ Newtonian (n = 1)

→ No seismic anisotropy

The boundary between these fields shifts with temperature. At higher temperatures, the transition to dislocation creep occurs at lower stress. The upper mantle likely deforms mainly by dislocation creep (explaining the observed seismic anisotropy), while the lower mantle may be dominantly in the diffusion creep regime.

Strength Envelope (Yield Strength Profile)

The strength of the lithosphere varies dramatically with depth. At shallow depths, rocks fail by brittle fracture; at greater depths, they flow by ductile creep. The strength envelope combines these two regimes:

Brittle Regime (Shallow: 0 to ~15-30 km)

Governed by Byerlee's law -- frictional sliding on pre-existing faults. The shear stress for failure increases linearly with depth (lithostatic pressure):

\(\tau = \mu_f \, \sigma_n = \mu_f(\rho g z - P_f)\)

where μ_f ≈ 0.6-0.85 (friction coefficient from Byerlee's law), σ_n is the effective normal stress, and P_f is pore fluid pressure. Strength increases linearly with depth in this regime.

Ductile Regime (Deep: > 15-30 km)

Controlled by power-law creep. Strength decreases exponentially with depth as temperature increases. At a fixed strain rate, the flow stress is:

\(\sigma = \left(\frac{\dot{\varepsilon}}{A}\right)^{1/n} \exp\!\left(\frac{E^* + PV^*}{nRT}\right)\)

Strength decreases rapidly with increasing temperature. The brittle-ductile transition (BDT) occurs where the brittle and ductile strength curves intersect, typically at 300-400°C for quartz-rich continental crust and 600-700°C for olivine-rich mantle.

Jelly Sandwich vs Crème Brûlée Models

The continental lithosphere strength envelope has been debated. The jelly sandwichmodel predicts a strong upper crust, weak lower crust, and a second strong layer in the uppermost mantle (quartz BDT at ~15 km, olivine BDT at ~40 km). The alternative crème brûlée model argues that all lithospheric strength resides in the crust, with the mantle being uniformly weak. Earthquake depth distributions and flexural rigidity studies provide constraints but have not fully resolved this debate.

Key Concepts Summary

τ_M ≈ 300 yr

Maxwell time for the mantle; elastic below, viscous above

n ≈ 3.5

Dislocation creep stress exponent for olivine

μ_f ≈ 0.6-0.85

Byerlee friction coefficient for brittle failure

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