GG612 Structural Geology Sec3on Steve Martel POST 805 smartel@hawaii.edu Lecture 4 Mechanics: Stress and Elas3city Theory 11/6/15 GG611 1 Topics 1. Stress vectors (trac3ons) 2. Stress at a point 3. Cauchy s formula 4. Principal stresses 5. Elas3c stress- strain rela3onships 6. Geologic examples 1. Radia3ng dikes 2. Faults 11/6/15 GG611 2 1
Stresses Control How Rock Fractures hup://hvo.wr.usgs.gov/kilauea/update/images.html 11/6/15 GG611 3 I Stress! vector! (trac3on) A τ = lim F / A A 0 B Trac3on vectors can be added as vectors C A trac3on vector can be resolved into normal (τ n ) and shear (τ s ) components 1 A normal trac3on (τ n ) acts perpendicular to a plane 2 A shear trac3on (τ s ) acts parallel to a plane D Local reference frame 1 The n- axis is normal to the plane 2 The s- axis is parallel to the plane Stress Vector 11/6/15 GG611 4 2
II Stress at a point (cont.) A Stresses refer to balanced internal "forces (per unit area)". They differ from force vectors, which, if unbalanced, cause accelera3ons B "On - in conven3on": The stress component σ ij acts on the plane normal to the i- direc3on and acts in the j- direc3on 1 Normal stresses: i=j 2 Shear stresses: i j Stress at a Point 11/6/15 GG611 5 Stress at a Point II Stress at a point C Dimensions of stress: force/unit area D Conven3on for stresses 1 Tension is posi3ve 2 Compression is nega3ve 3 Follows from on- in conven3on 4 Consistent with most mechanics books 5 Counter to most geology books 11/6/15 GG611 6 3
II Stress at a point Stress at a Point C D σ ij = σ xx σ yx σ xx σ xz σ ij = σ yx σ yz σ zx σ zy σ zz 2- D 4 components 3- D 9 components E For rota3onal equilibrium, = σ yx, σ xz = σ zx, σ yz = σ zy F In nature, the state of stress can (and usually does) vary from point to point 11/6/15 GG611 7 Cauchy s formula II Cauchy s formula A Relates trac3on (stress vector) components to stress tensor components in the same reference frame B 2D and 3D treatments analogous C τ i = n j σ ji Note: all stress components shown are posi3ve 11/6/15 GG303 8 4
Cauchy s formula II Cauchy s formula (cont.) C τ i = n j σ ji 1 Meaning of terms a τ i = trac3on component b n j = direc3on cosine of angle between n- direc3on and j- direc3on c σ ji = stress component d τ i and σ ji act in the same direc3on n j = cosθ nj 11/6/15 GG303 9 Cauchy s formula II Cauchy s formula (cont.) D Expansion (2D) of τ i = n j σ ji 1 τ x = n x σ xx + n y σ yx 2 τ y = n x + n y n j = cosθ nj 11/6/15 GG303 10 5
II Cauchy s formula (cont.) E Deriva3on: Contribu3ons to τ x Cauchy s formula Note that all contribu3ons must act in x- direc3on 1 2 3 τ x = w ( 1) σ xx + w ( 2) σ yx F x A n = A x A n ( 1) F x + A y A x τ x = n x σ xx + n y σ yx A n ( 2) F x A y n x = cosθ nx n y = cosθ ny 11/6/15 GG303 11 II Cauchy s formula (cont.) E Deriva3on: Contribu3ons to τ y Cauchy s formula Note that all contribu3ons must act in y- direc3on 1 2 3 τ y = w ( 3) + w ( 4) F y A n = A x A n ( 3) F y + A y A x τ y = n x + n y A n ( 4) F y A y n x = cosθ nx n y = cosθ ny 11/6/15 GG303 12 6
II Cauchy s formula (cont.) F Alterna3ve forms 1 τ i = n j σ ji 2 τ i = σ ji n j Cauchy s formula 3 τ x τ y τ z σ xx σ yx σ zx = σ xz σ yz σ zz n x n y n z n j = cosθ nj 11/6/15 GG303 13 Principal Stresses III Principal Stresses (these have magnitudes and orienta3ons) A Principal stresses act on planes which feel no shear stress B The principal stresses are normal stresses. C Principal stresses act on perpendicular planes D The maximum, intermediate, and minimum principal stresses are usually designated σ 1, σ 2, and σ 3, respec3vely. E Principal stresses have a single subscript. 11/6/15 GG611 14 7
III Principal Stresses (cont.) F Principal stresses represent the stress state most simply Principal Stresses G σ ij = σ 1 0 0 σ 2 2- D 2 components H σ 1 0 0 σ ij = 0 σ 2 0 0 0 σ 3 3- D 3 components * If σ 1 = σ 2 = σ 3, the state of stress is called isotropic. This occurs beneath a s3ll body of water. 11/6/15 GG611 15 Principal Stresses III Principal stresses (eigenvectors and eigenvalues) A B C τ x τ y τ x τ y σ xx = τ n x n y = σ xx σ yx σ yy σ yx n x n y Let λ = τ = λ n x n y n x n y The form of (C ) is [A][X]=λ[X], and [σ] is symmetric 11/6/15 GG303 16 8
Principal Stresses 1 [σ][x]=τ[x] 2 This is an eigenvalue problem (e.g., [A][X]=λ[X]) A [σ] is a stress tensor (represented as a square matrix) B τ is a scalar C [X] is a vector D [X], [σ][x], and τ[x] all point in the same direc3on 3 Solving for [X] yields vectors that do not rotate Principal stresses are normal stresses and are mutually perpendicular 11/6/15 GG611 17 Principal Values (eigenvalues) and Principal Direc3ons (eigenvectors) Mathema'cs 1 [σ][x] = τ[x] (Eigenvector equa3on) 4 Since [σ] = [σ] T, X i X j = 0 Meaning For solu3ons of (1), [X] (eigenvector) is not rotated by If [F] is symmetric, then its eigenvectors are perpendicular 11/6/15 GG611 18 9
Principal Values (eigenvalues) and Principal Direc3ons (eigenvectors) Mathema'cs 1 [σ][x] = τ[x] (Eigenvector equa3on) 5 d(x X) = 0!X dx = 0 Meaning Max/min lengths of λx are where X and its tangent(s) dx are perpendicular, so eigenvectors are also in direc3ons of maxima and minima 11/6/15 GG611 19 Principal Values (eigenvalues) and Principal Direc3ons (eigenvectors) Mathema'cs 1 [σ][x] = τ[x] (Eigenvector equa3on) 6 Solving for τ yields the principal stress magnitudes (Most tensile σ 1, Intermediate σ 2, least tensile σ 3 ) σ ij = σ xx σ yx σ 1 0 0 σ 2 σ xx σ xz σ 1 0 0 σ ij = σ yx σ yz 0 σ 2 0 σ zx σ zy σ zz 0 0 σ 3 11/6/15 GG611 20 10
Elas3c Stress- Strain Rela3onships Stress components are linearly related to the (small) strain components 11/6/15 GG611 21 Geologic Problem: Radia3ng Dikes Shiprock, New Mexico Aerial view showing radial dikes Both images from hup://en.wikipedia.org/wiki/shiprock#images 11/6/15 GG611 22 11
Displacement Field Around a Pressurized Hole in an Elas3c Plate Geometry and boundary condi'ons Displacement field ( u r u r,hole ) = a r ( ) 11/6/15 GG611 23 Displacement and Stress Fields Around a Pressurized Hole Displacement field Stress field ( u ( σ rr σ rr(at hole) ) = ( σ 1 P) = ( a r) 2 r u r,hole ) = ( a r) ( σ θθ σ rr(at hole) ) = ( σ 2 P) = a r 11/6/15 GG611 24 ( ) 2 12
Dikes Open in Direc3on of Most Tensile Stress, Propagate in Plane Normal to Most Tensile Stress Most tensile stress hup://hvo.wr.usgs.gov/kilauea/update/images.html 11/6/15 GG611 25 Fault Mechanics: Ver3cal Strike- slip Faults 11/6/15 GG611 26 13
Fault Mechanics: Dip- slip Faults 11/6/15 GG611 27 Modeled vs. Measured Displacement Fields Around a Fault Model for Fric3onless, 2D Fault in an Elas3c Plate Model Displacement field GPS Displacements, Landers 1992 11/6/15 GG611 28 14
Modeled vs. Measured Co- seismic Displacement, Landers Earthquake Co-seismic displacement field due to the 1992, Landers EQ Co- seismic deforma3on can be modeled using elas3c disloca3on theory (based on Massonnet et al., 1993) 11/6/15 GG611 29 Stress Fields Around a Fric3onless, 2D Model Fault in an Elas3c Plate vs. Observa3ons Model stress field: Most tensile stress & stress trajectories Tail cracks at end of lel- lateral strike- slip fault Note loca3on and orienta3on of tail cracks 11/6/15 GG611 30 15