Introduction to Rheology and Rock Mechanics

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

2. Rheology and rock mechanics 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) 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

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 Generalized strain-time curve • 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! 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 σinc. (loading) σdec. (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 • 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 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 = Aσnexp(-Q/RT)  [power law] A = material constant, Q = activation energy, R = gas constant, T = Absolute temp.

20. Plastic behavior • Perfectly 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 • 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 • 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