Isostasy, Rheology, Cross Sections, Fault Mechanics

2/16/12

GG612 Structural Geology Secon Steve Martel POST 805 [email protected] Lecture 4 Isostasy Rheology Strike-view Cross Secons Fault Mechanics

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Isostasy

• Refers to gravitaonal equilibrium • Provides a physical raonale for the existence of mountains • Based on force balance and buoyancy concepts h P = ρ(h)g(h)dh ∫0 P = pressure (convenon: compression is posive) ρ = density g = gravitaonal acceleraon

For constant ρ and constant g, P = ρgh

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Isostasy

• Assumes a “compensaon depth” at which pressures beneath two prisms are equal and the material beneath behaves like a stac ﬂuid, where P1 = P2 • Flexural strength of crust not considered • Gravity measurements yield crustal thickness and density variaons • Complemented by seismic techniques

hp://en.wikipedia.org/wiki/File:Iceberg.jpg 2/16/12 GG612 3

Isostasy History • Roots go back to da Vinci • Term coined by Clarence Edward Duon (USGS) • Post-1800 interest triggered by surveying errors in India • Two main models: Pra, Airy

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John Henry Pra (6/4/1809-12/28/1871) • Pra, J.H., 1855, On the aracon of the Himalaya Mountains, and of the elevated regions beyond them, upon the Plumb- line in India. Philosophical Transacons of the Royal Society of London, v. 145, p. 53-100. • Brish clergyman and mathemacian • Archdeacon of India

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Sir George Biddell Airy (7/27/1801 - 1/2/1892 • Airy, G.B., 1855, On the computaon of the Eﬀect of the Aracon of Mountain-masses, as disturbing the Apparent Astronomical Latude of Staons in Geodec Surveys. Philosophical Transacons of the Royal Society of London. v. 145, p.101-104. • Brish Royal Astronomer from 1835-1881 • Determined the mean density of the Earth from pendulum experiments in mines • Contributor to elascity theory (telescope deformaon) • Opponent of Charles Babbage from 1842 to ?? hp://www.computerhistory.org/babbage/georgeairy/img/5-2-1.jpg

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Isostasy and Gravity Measurements

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Comparison of Isostac Models

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Thermal Isostasy (e.g., Turcoe and Schubert, 2002) • Oceanic crust thickens and increases in density as it cools with me • Oceanic crust thickens and increases in density with distance from ridge

hp://openlearn.open.ac.uk/ﬁle.php/2717/!via/oucontent/course/414/s279_1_014i.jpg

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Isostac Rebound: Lake Bonneville

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Shorelines of Lake Bonneville Tilt Away from Lake

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20. Rheology & Linear Elascity

I Main Topics A Rheology: Macroscopic deformaon behavior B Linear elascity for homogeneous isotropic materials

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20. Rheology & Linear Elascity Viscous (ﬂuid) Behavior

hp://manoa.hawaii.edu/graduate/content/slide-lava 2/16/12 GG303 13

20. Rheology & Linear Elascity

Ducle (plasc) Behavior

hp://www.hilo.hawaii.edu/~csav/gallery/sciensts/LavaHammerL.jpg hp://hvo.wr.usgs.gov/kilauea/update/images.html

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20. Rheology & Linear Elascity

Elasc Behavior

hps://thegeosphere.pbworks.com/w/page/24663884/Sumatra hp://www.earth.ox.ac.uk/__data/assets/image/0006/3021/seismic_hammer.jpg

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20. Rheology & Linear Elascity

Brile Behavior (fracture)

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior A Elascity 1 Deformaon is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformaon is not me dependent if load is constant 4 Examples: Seismic (acousc) waves, hp://www.fordogtrainers.com rubber ball

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20. Rheology & Linear Elascity

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior B Viscosity 1 Deformaon is irreversible when load is removed 2 Stress (σ) is related to strain rate (ε) 3 Deformaon is me dependent if load is constant 4 Examples: Lava ﬂows, corn syrup hp://wholefoodrecipes.net

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20. Rheology & Linear Elascity

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior C Plascity 1 No deformaon unl yield strength is locally exceeded; then irreversible deformaon occurs under a constant load 2 Deformaon can increase with me under a constant load 3 Examples: plascs, soils hp://www.therapypuy.com/images/stretch6.jpg

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior C Brile Deformaon 1 Disconnuous deformaon 2 Failure surfaces separate

hp://www.thefeeherytheory.com

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior D Elasto-plasc rheology

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior E Visco-plasc rheology

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior F Power-law creep n (−Q/RT) 1 ė = (σ1 − σ3) e 2 Example: rock salt

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior G Linear vs. nonlinear behavior

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20. Rheology & Linear Elascity

II Rheology: Macroscopic deformaon behavior

H Rheology=f (σij ,ﬂuid pressure, strain rate, chemistry, temperature) I Rheologic equaon of real rocks = ?

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20. Rheology & Linear Elascity

III Linear elascity x A Force and displacement F of a spring (from Hooke, 1676): F= kx 1 F = force 2 k = spring constant k Dimensions:F/L F 3 x = displacement Dimensions: length L) x

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20. Rheology & Linear Elascity σ III Linear elascity (cont.) B Hooke’s Law for L0+ΔL L0 ε=ΔL/L uniaxial stress: σ = Eε 0 1 σ = uniaxial stress 2 E = Young’s modulus Dimensions: stress σ E 3 ε = strain Dimensionless ε

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20. Rheology & Linear Elascity σ1 III Linear elascity (cont.) B Hooke’s Law for uniaxial L0+ΔL L0 stress (cont.): ε1 = σ1/E ε=ΔL/L0 1 σ2 = σ3 = 0

2 ε2 = ε3 = -νε1 a ν = Poisson’s rao b ν is dimensionless c Strain in one direcon tends to induce strain in another direcon ε

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20. Rheology & Linear Elascity

III Linear elascity (cont.) C Linear elascity in 3D for homogeneous isotropic materials By superposion:

1 εxx = σxx/E – (σyy+σzz)(ν/E)

2 εyy = σyy/E – (σzz+σxx)(ν/E)

3 εzz = σzz/E – (σxx+σyy)(ν/E)

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20. Rheology & Linear Elascity

III Linear elascity (cont.) C Linear elascity in 3D for homogeneous isotropic materials (cont.) 4 Direcons of principal stresses and principal strains coincide 5 Extension in one direcon can occur without tension 6 Compression in one direcon can occur without shortening ε

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20. Rheology & Linear Elascity

III Linear elascity E Special cases 1 Isotropic (hydrostac) stress

a σ1 = σ2 = σ3 b No shear stress 2 Uniaxial strain x a εxx= ε1≠ 0

b εyy = εzz = 0

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20. Rheology & Linear Elascity

III Linear elascity 4 β = compressibility D Relaonships among β = 1/K diﬀerent elasc moduli ∆ = εxx+εyy+ εzz= -p/K 1 G = μ = shear modulus p = pressure G = E/(2[1+ν]) 5 P-wave speed: Vp εxy = σxy/2G ⎛ 4 ⎞ V = K + µ ρ 2 λ = Lame' constant p ⎝⎜ 3 ⎠⎟ λ = Ev/([1 + ν][1 - 2ν]) 6 S-wave speed: Vs 3 K = bulk modulus V = µ ρ K = E/(3[1 - 2ν]) s

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Strike-view Cross Secons

• Prepared by projecng features along strike onto a cross secon plane, where the cross secon plane is perpendicular to strike • Shows the true inclinaon and thickness of features • Lines of strike lie in geologic planes and connect points of equal elevaon

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Strike-View Cross Secons

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Fault Mechanics: Vercal Strike-slip Faults

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Fault Mechanics: Dip-slip Faults

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