1. Introduction to Rheology and Rock Mechanics
Lecture 10, Geology for Engineers

2. Rheology and rock mechanics
Goal: Relating stress to strain
Rock mechanics: the study of rock under stress
Near the earth’s surface, rocks are elastic solids characterized by elastic and brittle behavior
Rheology: Study of flow properties in fluids and solids, Typically ductile behavior, occurs in rocks at depth
Terms
Plastic/elastic: permanent/springs back, nonpermanent
Ductile deformation: Marked by a flow-like behavior (e.g. silly putty)
Brittle deformation: marked by a failure – breaking of the rock (fracturing)

3. Rheology and rock mechanics
Stress and strain are related
Physical properties modify simple relationships between the two
Pressure state, temperature and time relationship are important
Fast, low P,T  elastic-brittle
Slow, high P,T  ductile
Common: combined mechanisms
Does the glacier flow, or break?

4. Models of deformation
Common plots
We use idealized models to describe mechanical behavior on small scales. Assumptions:
Isotropic material (same physical properties throughout rock vol.)
Primary stretching axes = Primary stress axes
Good for small degrees of strain
PLOTS: Either e vs. σ, or time vs. e σ e e t

5. Models of deformation The creep curve
real rock behaviors are complex!
Plotting strain vs. time
Behavior under compression
Three strain regimes over time observed:
dec. strain rate
Constant strain rate
Inc. strain rate (until failure)
One goal: Understanding the elements!
Generalized strain-time curve
Primary
Secondary
Tertiary
Assumption (or experimental condition):
Constant applied stress

6. Deformation behaviors
Examples
Stress vs. strain
Strain over time

7. Strain rate
Interval over which a strain accumulates, or elongation over time (ė)
ė = e/t, measured in 1/sec or sec-1
(recall: strain is unitless!) e
time

8. Elastic materials Resist deformation
Strains as stress is applied; recovers shape when stress ceases
Internal bonds stretched, not broken
Consider a spring…

9. Elastic materials Three behaviors Linear elastic
Linear relationship between stress and strain  strain (elongation) is proportional to stress applied
Double the weight on the spring, double the length!
Deformation isn’t permanent

10. Elastic materials
Hooke's Law defines the relationship in terms of mechanical properties
σ = E*e
E is Young’s modulus
Ratio of stress to strain
Stiffness, or resistance to shape change
Related term – Shear modulus (μ)
Higher E generally = higher strength
σ inc.
(loading)
σ dec.
(loading)
Strain recovers proportionally, too

11. Elastic materials
Common geologic materials behave this way (to a point…)
Keep in mind, strains are very small (just a few %)!
From experimental data

12. Elastic materials Non-linear elasticity: more common
Due to variable stress to strain relationship
Perfect elastic: Non linear σ/e values, loading and unloading follow the same path
Elastic with hysteresis: Different loading and unloading paths

13. Elasticity and Poisson’s ratio
Poisson effect: The balance between elongation in one direction and flattening in another, relative to compression
Balanced in an incompressible material (no vol. change during def.)
Poisson’s ratio
ν (nu) = eperpendicular/eparallel
(recall: (-) e is shortening, (+) e is lengthening)
Rubber: almost totally incompressible

14. Elasticity and Poisson’s ratio
ν = e perpendicular/e parallel
Punch line:
confining stress
limits e in
both v and h!
Unconfined state: no restriction
To elongation
Horiz. stress restricts
Elongation in both directions!
νrock ≈ 0.25

15. Plasticity Elastic behavior explains strains in rock near surface
Plastic strain is permanent – not recovered like elastic strain
At depth, rock flows!
Dependent on time (strain rate)

16. Viscous materials Viscosity: The ease of flow in a material
Dependence of strain rate on applied stress:
στ = η*γ.
(η = viscosity constant, γ. = shear strain rate)
Newtonian fluid (ideal) or linear viscosity
Viscous material
More applied stress = faster shear strain rate

17. Non-linear viscous behavior
9/19/2014
Linear viscosity – only in true fluids
Geologic e.g’s: Magma, salt, overpressured mud
Non-linear viscosity: viscosity (η) changes w/ strain rate (ė)
E.g. hot rock in the crust
Folding
Boudinage (the sausage-like layers) – also contrasts in viscosity!
η for water 10-3 pa s
η for glacial ice 1011 pa s
η for glass 1014 pa s
η for rock salt 1017 pa s
η for asthenosphere 1021 pa s

18. Plastic behavior Strain with no loss of continuity
Both models of flow…
Strain with no loss of continuity
Permanent strains, without fracturing, accumulated over time
Essentially, flow in solid material!
Crystal plasticity: Bonds break but coherency is maintained
Different responses to stress!

19. Plastic behavior
Due to microscale deformation mechanisms  harder to define parameters (e.g. migration of crystal lattice defects)
Flow laws: quantify these relationships
ė = Aσn exp(-Q/RT)
 [power law]
A = material constant, Q = activation energy, R = gas constant, T = Absolute temp.

20. Plastic behavior Perfectly plastic material Elastic plastic material
Stress cannon exceed yield stress
100% incompressible
Strength not strain rate sensitive
Elastic plastic material
Elastic behavior until yield stress  “instant” strain
Plastic behavior beyond yield stress

21. Elastic-Plastic behavior
Steady state or creep: Const. stress produces const. strain (Blue)
Strain hardening (Red)
Stress inc. to keep strain inc.
Due to defect migration and accumulation in strain zones
Strain softening (Green)
Less stress needed to maintain strain
Physical property changes
Strain hardening e.g.
Restoring a bent wire

22. Elastic-Plastic behavior
Strain hardening outcomes:
Recoverable elastic portion (if stress is removed)
permanent plastic portion
If the rupture strength/ultimate strength is reached… failure results!

23. Viscoplastic behavior
9/19/2014
No deformation below yield stress
Perfect viscous behavior above yielded stress (time dependent)
Example: Silicic magma
Crystals and liquid  cohesive, must be overcome
Example: Thin coat of pain on a wall

24. Viscoelastic behavior
Recoverable strain elements
Rate for loading and unloading is different
General Eqn: σ = E*e + η*ė
Two flavors:

25. General linear behavior
9/19/2014
Closest to natural rock
Strain begins at yield stress
Combines Kelvin and Maxwell viscoelastic properties
Recoverable elastic and permanent strains

26. Deviations from models
Temperature: Inc. T lowers yield stress
Rate of strain: slow strain favors plastic def., fast strain favors elastic to brittle
Fluids: Inc. fluid content lowers yield stress
Confining pressure: Inc. confining pressure strengthens rock
Grain size: Small grains favor plastic deformation
Fabric orientations
Experimental data: Yule marble
Normal to fabric
Parallel to fabric
Diff. rates