Rheology: Chapter 6 (H. Fossen)

• Rheology is the science of flowing matter.
• The study of Rheology helps in gaining insight into the fundamentals of elastic, viscous, and plastic behavior.
• Rheology helps distinguish between brittle and ductile deformation.
• Rheology provides knowledge on the parameters influencing the rheology of the lithosphere and the concept of strength profiles.
• The study material for this topic is H. Fossen, chapter 6.

Elastic Behavior

• Strain is instantaneous, not permanent, and reversible in elastic behavior.
• Strain is small (a few %) and time-independent.
• Hooke's Law for linear elastic material is σ = E.e, where E is Young's modulus (material constant), measured in Pascals [Pa].

Viscous Behavior

• Strain is non-recoverable, it results in permanent deformation.
• Stress is related to strain rate.
• The relationship between stress and strain rate is given by: σₙ = ηέ.
• Viscosity (η) is a measure of a fluid's resistance to flow.
• Strain rate is defined as έ = de/dt = (l-lo/tlo), representing the rate of elongation per unit time, and is measured in [s⁻¹].
• A Newtonian fluid is a linear viscous fluid.

Viscosity Examples

• Water has a viscosity of 10⁻³ Pa·s.
• Olive oil has a viscosity of 10⁻¹ Pa·s.
• Honey has a viscosity of 4 Pa·s.
• Lava has a viscosity ranging from 10 to 10⁴ Pa·s.
• Asphalt has a viscosity of 10⁵ Pa·s.
• Ice has a viscosity of 10¹² Pa·s.
• Steenzout has a viscosity of 10¹⁷ Pa·s.
• Zandsteen has a viscosity of 10¹⁸ Pa·s.
• The asthenosphere has a viscosity of 10²⁰ Pa·s.
• The Onder mantel has a viscosity of 10²¹ Pa·s.

Plastic Behavior

• Plastic behavior is considered the flow of solid rock.
• It results in non-recoverable, permanent deformation without fracturing at constant stress.
• Strain rate is proportional to stress to the power n.
• It has a temperature dependence of rock behavior.
• The strain rate equation is: έ = A · σⁿ exp (-Q/RT), where A is a constant, n is the power-law exponent, Q is the activation energy, R is the gas constant, and T is the absolute temperature.
• This equation is also known as the power law equation and is a non-linear relation.
• The flow of rocks occurs through processes on an atomic scale, where crystal defects like dislocations move through the crystal lattice.

Material Response to Stress

• Elastic materials exhibit immediate response to stress and can be either elastic, then plastic with instant response and permanent plus non-permanent strain.
• Visco-plastic materials have a linear viscous response above a yield point (YP), no response to stress until YP, then time-dependent permanent strain.
• Viscoelastic (Kelvin) show both elastic and viscous behavior at the same time, with an immediate response to stress and strain rate controlled by viscosity and non-permanent strain.
• Viscoelastic (Maxwell) materials are elastic and viscous above a yield stress, with an instant elastic response to stress, then a time-dependent response where the strain rate is controlled by viscosity and exhibit permanent + non-permanent strain.
• General linear materials behave elastically viscous above yield stress, with an instant elastic response to stress and the strain rate is controlled by viscosity and exhibit permanent + non-permanent strain.

Stress-Strain Curves

• Strain hardening occurs when stress needs to increase to strain the rocks; hammering iron makes it stronger.
• Creep is when a material deforms at constant stress.
• Strain softening occurs when less stress is needed for continuous deformation.
• Grain size reduction and introduction of fluids can cause strain softening.

Brittle vs. Ductile Deformation

• Brittle (brose) deformation is discontinuous deformation by fracturing.
• Ductile deformation is continuous deformation by brittle or plastic mechanisms and not by fracturing.
• Term usage is generally for crystal plastic flow.
• The distinction between brittle and ductile deformation is scale-dependent.

Rheology of the Continental Lithosphere

• Strength profiles in the lithosphere are defined by rheological models.
• The Coulomb Failure Criterion is σ = C + μσₙ, representing brittle behavior for intact material.
• Byerlee's law for pre-fractured rocks states:
  • σ = 0.85σₙ for σ < 200 MPa
  • σ = 0.5 + 0.6σₙ for σ > 200 MPa
  • σ is normal stress.
• Dorn's Law, which describes viscous (ductile) behavior, is given by: ¿ = σⁿAe^(Q/RT).

Factors Influencing Rock Behavior

• Temperature and confining pressure affect rock behavior.
• High confining pressure leads to high strength of rocks.
• High temperature leads to low strength of rocks.
• Increasing confining pressure causes rocks to deform in a ductile manner.
• Increasing confining pressure and temperature means greater depth in the crust.
• Strain rate and mode of deformation, fluids and pore-fluid pressure, and composition influence the behavior of rocks.
• Higher pore fluid pressure (Pf) reduces the strength of rocks, where Pe = Pc - Pf.
• At very high temperatures, the influence of the rock fabric at the onset of deformation diminishes and this dependence is also scale dependent.