Rheology two basic relationships between stress and strain (geologic rheology) Elastic (Hookes law) Viscous Combinations of elastic and viscous Strain rate Shear stress and deformation (shear modulus) Stress and dilation (Poissons ratio) Adventures with rocks Rheology two basic relationships between stress and strain (geologic rheology) Elastic (Hookes law) Viscous Combinations of elastic and viscous Strain rate Shear stress and deformation (shear modulus) Stress and dilation (Poissons ratio) Adventures with rocks

Rheology Relations between stress and strain. Rheology: describes the how stressed materials deform. strain rate creep regimes elastic behavior viscous types of behavior

Rheologic Behavior Two types of behavior 1) Elastic behavior 2) Viscous behavior

Experiments to understand the relaionship between stress and strain Springs vs Silly Puddy

Rheologic Behavior 1)Elastic behavior: Strain is directly related to stress strain varies linearly with stress The equation is known as Hookes Law E = Young’s modulus (slope of stress/strain diagram)  Seismic waves travel thru elastic medium Stress vs strain Strain vs time

Rheologic Behavior Hooke’s Law  = Ee Stress is linearly related to strain by the constant E, known as Young’s modulus

Hookes Law 1)This straight line relation between stress and strain is called Hookes law.  = Ee Strain (e) is linearly proportional to stress (  ) where E = Young’s modulus E =  / e = stress/strain The value of E, or Young’s modulus describes the slope of a straight line, stress-strain curve. Stress and strain are directly and linearly related = the slope of the line. Rheologic Behavior Young’s modulus, How much stress is required to achieve a given amount of length-parallel elastic shortening of a rock.

Rheologic Behavior 1)Elastic behavior: Stress and strain are linear Reversible. Once stress is removed, the material returns to its original shape – strain is recoverable Instantaneous response to strain

Rheologic Behavior 2) Viscous behavior: Strain is not directly related to stress But! Strain is directly related to strain rate!!!! where  is viscosity Non-recoverable strain and permanent. Shock absorbers Geologic Examples: Upper mantle, lower mantle, magmas, ice, salt domes

Strain rate The time it takes material to accumulate a certain amount of strain. Elongation (e) per unit time. Dimensions of strain rate are ???? Example: What is the strain rate of a box that experiences 30% finite strain one hour? Known: e = 0.3 Time =(3600 sec)  Strain rate= 0.3/(3600 sec) = 8.3 x /sec 1.5 cm long fingernail grows 1 cm/yr What is strain rate? (check this math) 0.67/yr or 2 x /s Typical geologic rates are /s – /s Geologic time often measured in My, Strain rate typically in s -1 How many seconds in a million years? Geologic time often measured in My, Strain rate typically in s -1 How many seconds in a million years?

Rheologic Behavior 3) Viscoelastic behavior: [elastic plus viscous) Reversible deformation (the spring will pull the dash-pot back) Strain accumulation and recovery is delayed. Water soaked sponge that is loaded on top

Rheologic Behavior 4) Elastico-viscous behavior: Elastic deformation with initial stress (soft spring extends), then viscous behavior (spring continues to pull on dash pot) Total strain accumulation and recovery is delayed. Some deformation is recoverable (the spring returns to original length) Maxwell relaxation time – stress relaxation decays exponentially

Creep curve Behavior of rocks to compression is not simple. Three creep regimes: 1) Primary or transient creep: strain rate decreases with time following rapid initial accumulation 2) Secondary or steady state creep: strain accumulation is linear with time 3) Tertiary or accelerated creep: strain rate increases with time. Instant deformation => Deforms over time => Elastic: Non-linear viscous Linear viscous Non linear viscous

Creep curve Behavior of rocks to compression is not simple. Removing stress in steady state creep. 1) Immeidate drop in strain “springing back” 2) Permanent strain remains

Elastic behaviour and shear stress Shear modulus (G): resistance of elastic solids to shearing. Divide shear stress (  s ) by shear strain (  ) G = shear modulus =  s /  Elastic behaviour and shear stress Shear modulus (G): resistance of elastic solids to shearing. Divide shear stress (  s ) by shear strain (  ) G = shear modulus =  s /  ss

Elastic behaviour and dilation (important in seismology) Bulk Modulus (K): resistance of elastic solids to dilation. Elastic behaviour and dilation (important in seismology) Bulk Modulus (K): resistance of elastic solids to dilation. Another relationship between stress and volume change Poisson’s Ratio =e perpendicular /e parallel (perpendicular and parallel to compression direction) Common values 0 to 0.5 (fully compressible, to fully incompressible) Another relationship between stress and volume change Poisson’s Ratio =e perpendicular /e parallel (perpendicular and parallel to compression direction) Common values 0 to 0.5 (fully compressible, to fully incompressible)

Poisson’s ratio, Greek letter nu ( ). This describes the amount that a rock bulges as it shortens. The ratio describes the ratio of lateral strain to longitudinal strain: = e lat /e long Poisson’s ratio is unit-less, since it is a ratio of extension. What does a low ratio mean? What does a high ratio mean? Typical values for are: Fine-grained limestone: 0.25 Apilite: 0.2 Oolitic limestone: 0.18 Granite: 0.11 Calcareous shale: 0.02 Biotite schist: 0.01

Poisson’s ratio If we shorten a granite and measure how much it bulges, we see that we can shorten a granite, but it may not be compensated by an increase in rock diameter. So stress did not produce the expected lateral bulging. Somehow volume decreases and stress was stored until the rock exploded! Thus low values of Poisson’s ratio are significant.