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Rheology II: Constitutive Relations
Ch. 6: p.97-‐109; 112-‐
1. Controls on Rheology: The important effect of temperature and pressure on rheology explains why rocks tend to flow plastically or viscously in the middle and lower crust.
Ductile deformation is where rocks deform by solid-‐state flow, causing them to warp and bend (at the scale of observation), perhaps with concurrent recrystallization (mineralogical and chemical changes).
In the shallow crust, rocks tend to be elastic, but may eventually fracture. We are then in the realm of rock mechanics, not rheology. Brittle deformation is when rocks are physically broken by the forces imparted upon them. [Fig. 6.1. Ice in glaciers can flow viscously at Earth’s surface temperatures, but can also behave elastically and eventually fracture in a brittle manner]
2. Continuum Mechanics: We use the assumptions of continuum mechanics, whereby we treat rocks as if they are homogeneous and free of defects (microfractures, mineral grain boundaries, pore spaces) and thus have identical physical properties throughout.
The rock is also considered to be isotropic (deformation properties are independent of direction). [ Figure: Example of progressive viscous deformation of a glacier in a laboratory experiment]
3. Constitutive Relations: Rheology implies that deformation follows explicit laws and is thus quantifiable (i.e., predictable) in response to inherent physical properties and imparted forces.
There are explicit equations that describe the manner in which deformation will proceed under certain conditions. These are the constitutive equations or constitutive laws.
4. Stress and Strain: Although we will return in detail to the concepts of stress and strain later in the course, simple definitions are needed at this stage to understand rheology.
Stress (σ) is the ratio of force divided by area. In other words, it is the cause of deformation. The unit of stress is the Pascal, Pa (i.e., Newtons / m 2 ).
Strain (e or ε) is the measurable change in shape or volume of a deformed material. It is the end result of an applied stress. Strain has no units. Strain may be fast or slow, so we can talk about the strain rate (ė), which has units of per seconds (s -‐1^ ).
5. Elasticity: An elastic material is one that deforms by stretching the bonds between atoms as forces are applied, but it returns to its original shape when the force is removed. In other words, the strain is recoverable.
In a linear sense (1D), strain can be thought of as change in length divided by the original length:
e = (L – L (^) o ) / L (^) o or ΔL/L (^) o
where e stands for extension (which is considered a positive value of strain). Multiply e by 100 to get the percentage of strain. We can also consider the volumetric strain: ΔV/V (^) o.
6. Linear Elasticity: A linear elastic material is one that deforms in direct relation to the force applied (like a spring). Double the force equates to double the extension.
In terms of stress, linear elastic materials show a linear relationship between the stress applied and the amount of strain (whether extension or contraction). The constant of proportionality is E, the Young’s modulus (also called the stiffness). σ = E e
[Fig. 6.2. Elasticity demonstrated as (a) a mechanical analog (a spring); (b) on a stress vs. strain graph; and (c) on a strain history curve]
7. Linear Elasticity: Linear elastic behavior can also be illustrated in terms of a linear relationship between shear stress (σs or τ) and shear strain (γ):
σs = G γ
where G (sometimes called μ) is a linear proportionality constant called the shear modulus (also called the rigidity).
[Fig. 6.7. Shearing of a medium by a principal stress, σ 1 , produces a shear stress, σ s , and a shear strain, γ = tan Θ ]
8. Linear Elasticity: This linear relationship is called Hooke’s Law. Hence, σ = E e is a type of constitutive relationship. As strain is unitless, E must have units of stress and is usually measured in GPa ( gigapascals ). Most geologic materials (rocks) have values of E of the order of 10s of GPa and undergo elastic strains of a few %.
[Fig. 6.3. Linear elastic behavior in 1D. The slope of the σ vs. e graph defines the Young’s modulus, E. Right: values of E for natural materials (Table 6.1)]
9. Nonlinear Elasticity: Although rocks at shallow depths tend to behave in an elastic manner up until the point of brittle failure, not all rocks are linear elastic (although many are).
Some rocks have variable values of E depending on σ and e, but deform and recover the same way, and are called perfect elastic. If the recovery process is different to the deformation process, the rock is elastic with hysteresis.
[Fig. 6.4. Linear elastic behavior of some geologic materials] [Fig. 6.5. Three styles of elastic behavior]
10. The Poisson Effect: We test the elasticity of rocks by compressing them between hydraulic pistons. If rocks were incompressible, any shortening along one direction would need to be balanced out by extension in the orthogonal directions.
But rocks are compressible. Some of the shortening is taken up by a volume change in the rock, and the rest by an equal amount of extension in the two directions perpendicular to the shortening (this is the Poisson effect).
[Fig. 6.6. Experiments showing that contraction of a rock in the z-‐direction results in extension in the x-‐ and y-‐ directions. (a) Unconfined in the lateral directions. (b) Confined in the lateral directions]
16. Plastic Deformation: When the stress is removed, only the elastic portion is recoverable. This elastic-‐plastic behavior defines a Prandtl material.
Although constitutive laws exist for plastic materials, the equations vary depending on the microdeformation mechanisms. The general flow law is:
ė = Aσn^ exp (-‐Q/RT)
where A and n are constants, R is the gas constant, T is temperature, and Q is the activation energy.
[Fig. 6.9. Elastic-‐plastic deformation histories. (a) Recoverable elastic strain. (b) Permanent strain after brittle fracture]
17. Strain Hardening/Softening: Most geologic materials are not perfectly plastic. In some, as deformation proceeds, a greater amount of stress is needed to cause more plastic deformation (i.e., stronger rocks). This phenomenon is called strain hardening.
If rocks get weaker as strain accrues (e.g., by grain size reduction or recrystallization), plastic deformation requires less stress as time proceeds. This phenomenon is called strain softening.
Perfectly plastic materials undergo creep (de/dt = σy = constant), or steady-‐state flow.
[Fig. 6.10. Different types of plastic behavior where the strain depends on the stress]
18. Mixed Rheologies: Most rocks undergo a combination of different rheologies that depend on the time scale and the amount of strain. Examples include: - Elastic-‐plastic (Prandtl) - Viscoplastic (Bingham) e.g. lava with crystals - Viscoelastic (Kelvin) - Viscoelastic (Maxwell) e.g. mantle; ice
[Fig. 6.11. Geologic materials may have different rheological behaviors during the deformation process]
19. Brittle vs. Ductile Deformation: In structural geology, we commonly use the terms brittle and ductile to refer to rock behavior, but they do not explicitly imply specific rheologies. Ductile deformation preserves the continuity of originally continuous structures, but may involve a range of mechanisms (viscous, plastic, microfracture) that vary with conditions (scale, temperature, stress state, strain rate).
Brittle deformation involves actual breakage of the rocks, but this can actually happen in a number of ways…
[Fig. 6.16. Brittle and ductile deformation mechanisms. How these terms are used depends on the scale of observation]
