Lec.23.pptx - 23. Disloca/ons I Main Topics A...

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Unformatted text preview: 11/14/11 23. Disloca/ons I Main Topics A Disloca/ons and other defects in solids B Significance of disloca/ons C Planar disloca/ons D Displacement and stress fields for a screw disloca/on (mode III) 11/14/11 GG303 1 23. Disloca/ons hHp://volcanoes.usgs.gov/Imgs/Jpg/Photoglossary/fissure4_large.JPG 11/14/11 GG303 2 1 11/14/11 23. Disloca/ons hHp://upload.wikimedia.org/wikipedia/commons/4/43/Mackenzie_dike_swarm.png 11/14/11 GG303 3 23. Disloca/ons II Disloca/ons and other defects in solids A Disloca/ons 1 Originally, extra (or missing) planes or par/al planes of material (e.g., atoms) hHp://www.T.uni ­kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html 11/14/11 GG303 4 2 11/14/11 23. Disloca/ons Transmission Electron Micrograph Of Disloca/ons II Disloca/ons and other defects in solids A Disloca/ons (cont.) 2 Evidence for disloca/ons from electron microscopy hHp://en.wikipedia.org /wiki/Disloca/on 11/14/11 GG303 5 23. Disloca/ons II Disloca/ons and other defects in solids A Disloca/ons (cont.) 3 Con/nuum mechanics usage: surfaces across which displacements are discon/nuous 11/14/11 hHp://volcanoes.usgs.gov/Imgs/Jpg/Photoglossary/fault2_large.jpg GG303 6 3 11/14/11 23. Disloca/ons II Disloca/ons and other defects in solids (cont.) B Point defects 1 Originally, extra (or missing) volumes (e.g., atoms) 2 Displacements are discon/nuous across point defects hHp://www.ndt ­ed.org/Educa/onResources/CommunityCollege/Materials/Graphics/Chrystal ­Defects.jpg 11/14/11 GG303 7 23. Disloca/ons III Significance of disloca/ons A Account for permanent plas/c deforma/on in crystals B Account for the low observed strength of crystals rela/ve to theore/cal predic/ons hHp://www.ndt ­ed.org/Educa/onResources/CommunityCollege/Materials/Graphics/Chrystal ­Defects.jpg 11/14/11 GG303 8 4 11/14/11 23. Disloca/ons Transmission Electron Micrograph III Significance of disloca/ons C They provide useful quan/ta/ve descrip/on of rela/ve mo/ons across surfaces across a broad range of scale (crystals [10 ­6 m] to plate boundaries [106 m]) – ~12 orders of magnitude! 11/14/11 hHp://en.wikipedia.org /wiki/Disloca/on Disloca/on model of subduc/on zone GG303 From Bevis and Martel, 2001 9 23. Disloca/ons III Significance of disloca/ons D Disloca/ons induce tremendous stress concentra/ons and account for “large” deforma/ons under small “average” stresses hHp://pangea.stanford.edu/research/geomech/Faculty/crack.html 11/14/11 GG303 10 5 11/14/11 23. Disloca/ons IV Planar disloca/ons A Represented mathema/cally as infinitely long cut with a straight edge B Rela/ve displacement (of one side of the disloca/on rela/ve to the other) across a disloca/on is called the Burger’s vector b. 11/14/11 GG303 11 23. Disloca/ons D Edge disloca/on 1 Accommodate opening or sliding mo/ons 2 Displacement is exclusively perpendicular to the disloca/on edge 3 Displacement can be parallel or perpendicular to the disloca/on plane View along edge of edge disloca/on hHp://www.T.uni ­kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html 11/14/11 GG303 12 6 11/14/11 23. Disloca/ons D Edge disloca/on (cont.) 4 Analogy: an extra row of corn kernels on a cob of corn Extra row of kernels in an ear of corn Extra half ­plane of atoms in lajce hHp://www.T.uni ­kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html 11/14/11 GG303 blog.faTreevegan.com 13 23. Disloca/ons D Edge disloca/on (cont.) 5 Macroscopic geologic use: modeling dikes or faults hHp://hvo.wr.usgs.gov/gallery/kilauea/erupt/24ds182_cap/on.html 11/14/11 GG303 hHp://www.experimentalstress.com/topic/reflect.jpg 14 7 11/14/11 23. Disloca/ons Disloca/on edge C Screw disloca/on 1 Accommodate a tearing mo/on 2 Displacement is exclusively parallel to the disloca/on edge 3 Analogy: a lock washer or a 360° spiral staircase hHp://www.T.uni ­kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html hHp:///www.boltsplus.com 11/14/11 GG303 15 23. Disloca/ons C Screw disloca/on 4 Macroscopic geologic use: modeling faults hHp://volcanoes.usgs.gov/Imgs/Jpg/Photoglossary/fault2_large.jpg hHp://www.T.uni ­kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html 11/14/11 GG303 16 8 11/14/11 23. Disloca/ons V Displacement and stress fields for a screw disloca/on (mode III) A Displacement parallel to the disloca/on edge (w) increases uniformly along a spiral ­like circuit from one side of the disloca/on to the other (for a right ­handed screw disloca/on, point your right thumb along the disloca/on edge; displacement parallel to the edge increases in the direc/on your fingers curl. B Angular posi/on: θ = tan ­1(y/x) C w = bθ/2π D An/ ­plane strain (u,v = 0) 11/14/11 GG303 17 23. Disloca/ons Cylindrical Reference Frame ur = 0, uθ = 0, w = bθ/2π Normal Strains Shear Strains 1 ⎛ ∂u ∂u ⎞ ε rr = ⎜ r + r ⎟ 2 ⎝ ∂r ∂r ⎠ ε rr = 0 1 ⎛ 1 ∂ur ∂uθ uθ ⎞ + −⎟ ⎜ 2 ⎝ r ∂θ ∂r r⎠ ε rθ = εθ r = 0 ε rθ = ur 1 ∂uθ + r r ∂θ εθθ = 0 εθθ = 1 ⎛ ∂w ∂w ⎞ + ⎜ ⎟ 2 ⎝ ∂z ∂z ⎠ ε zz = 0 ε zz = 1 ⎛ ∂w ∂ur ⎞ 1 ⎛ ∂uθ 1 ∂w ⎞ + + ⎜ ⎟ ε zr = ⎜ ⎟ 2 ⎝ ∂z r ∂θ ⎠ 2 ⎝ ∂r ∂z ⎠ b ε zr = ε rz = 0 εθ z = ε zθ = 4π r εθ z = * The deriva/ves are very easy to do. Check them to verify the solu/ons. 11/14/11 GG303 18 9 11/14/11 23. Disloca/ons Cartesian Reference Frame u = 0, v = 0, w = b[tan ­1(y/x)]/2π Normal Strains Shear Strains 1 ⎛ ∂u ∂u ⎞ +⎟ ⎜ 2 ⎝ ∂x ∂x ⎠ ε xx = 0 1 ⎛ ∂v ∂y ⎞ ε yy = ⎜ + ⎟ 2 ⎝ ∂y ∂y ⎠ ε xx = ε xy = ε yy = 0 1 ⎛ ∂u ∂v ⎞ + ⎝ ⎠ 2 ⎜ ∂y ∂x ⎟ ε xy = 0 ε yz = 1 ⎛ ∂v ∂w ⎞ + ⎝ ⎠ 2 ⎜ ∂z ∂y ⎟ b x 4π x 2 + y 2 bx ε yz = 4π r 2 ε yz = 1 ⎛ ∂w ∂w ⎞ + ⎜ ⎟ 2 ⎝ ∂z ∂z ⎠ ε zz = 0 ε zz = 1 ⎛ ∂u ∂w ⎞ + ⎜ ⎟ 2 ⎝ ∂z ∂x ⎠ −b y ε xz = 2 4π x + y 2 −b y ε xz = 4π r 2 ε xz = * The deriva/ves are s/ll rather easy to do. Check them to verify the solu/ons. 11/14/11 GG303 19 23. Disloca/ons Strains Stresses ε rθ = εθ r = 0 εθ z = ε zθ = Gb 4π r ε rθ = εθ r = 0 ε rr = 0 εθθ = 0 ε zz = 0 11/14/11 σ rθ = σ θ r = 2Gε rθ = 0 σ θ z = σ zθ = 2Gεθ z = Gb 2π r σ rθ = σ θ r = 2Gε rθ = 0 σ rr = 0 σ θθ = 0 σ zz = 0 GG303 20 10 11/14/11 23. Disloca/ons Strains Stresses σ xy = σ yx = 2Gε xy = 0 ε xy = ε yx = 0 ε yz = ε xz = b x 2 4π x + y 2 −b y 2 4π x + y 2 σ yz = σ zy = 2Gε yz = σ xz = σ zx = 2Gε xz = Gb x Gb x = 2 2 2π x + y 2π r 2 −Gb y −Gb y = 2 2 2π x + y 2π r 2 ε xx = 0 σ xx = 0 ε yy = 0 σ yy = 0 ε zz = 0 11/14/11 σ zz = 0 GG303 21 23. Disloca/ons Key points a Only the shear stresses ac/ng on a plane normal to the z ­direc/on or in the z ­ direc/on are non ­zero b The stresses are singular (i.e., go to infinity) near the disloca/on edge: a powerful stress concentra/on exists there. c This theore/cal singular stress concentra/on exists no maHer how small G or the rela/ve displacement b is, provided G > 0 and b > 0. 11/14/11 GG303 σ θ z = σ zθ = Gb 2π r 22 11 11/14/11 23. Disloca/ons SUPERPOSTION OF TWO (INFINITE) SCREW DISLOCATIONS (A,B) TO FORM A FINITE DISPLACEMENT DISCONTINTUITY (C) 11/14/11 GG303 23 12 ...
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This note was uploaded on 12/05/2011 for the course GEOLOGY 300 taught by Professor Stephenmartel during the Fall '11 term at University of Hawaii, Manoa.

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