23. Disloca0ons. 23. Disloca0ons. I Main Topics

Transcription

1 I Main Topics A Disloca0ons and other defects in solids B Significance of disloca0ons C Planar disloca0ons D Displacement and stress fields for a screw disloca0on (mode III) 11/10/16 GG303 1 hhp://volcanoes.usgs.gov/imgs/jpg/photoglossary/fissure4_large.jpg 11/10/16 GG

2 hhp://upload.wikimedia.org/wikipedia/commons/4/43/mackenzie_dike_swarm.png 11/10/16 GG303 3 II Disloca0ons and other defects in solids A Disloca0ons 1 Originally, extra (or missing) planes or par0al planes of material (e.g., atoms) hhp:// kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html 11/10/16 GG

3 II Disloca0ons and other defects in solids A Disloca0ons (cont.) 2 Evidence for disloca0ons from electron microscopy Transmission Electron Micrograph Of Disloca0ons hhp://en.wikipedia.org/wiki/disloca0on 11/10/16 GG303 5 II Disloca0ons and other defects in solids A Disloca0ons (cont.) 3 Con0nuum mechanics usage: surfaces across which displacements are discon0nuous hhp://volcanoes.usgs.gov/imgs/jpg/photoglossary/fault2_large.jpg 11/10/16 GG

4 II Disloca0ons and other defects in solids (cont.) B Point defects 1 Originally, extra (or missing) volumes (e.g., atoms) 2 Displacements are discon0nuous across point defects hhp:// ed.org/educa0onresources/communitycollege/materials/graphics/chrystal- Defects.jpg 11/10/16 GG303 7 III Significance of disloca0ons A Account for permanent plas0c deforma0on in crystals B Account for the low observed strength of crystals rela0ve to theore0cal predic0ons hhp:// ed.org/educa0onresources/communitycollege/materials/graphics/chrystal- Defects.jpg 11/10/16 GG

5 III Significance of disloca0ons C They provide useful quan0ta0ve descrip0on of rela0ve mo0ons across surfaces across a broad range of scale (crystals [10-6 m] to plate boundaries [10 6 m]) ~12 orders of magnitude! Transmission Electron Micrograph hhp://en.wikipedia.org/wiki/disloca0on Disloca0on model of subduc0on zone From Bevis and Martel, /10/16 GG303 9 III Significance of disloca0ons D Disloca0ons induce tremendous stress concentra0ons and account for large deforma0ons under small average stresses hhp://pangea.stanford.edu/research/geomech/faculty/crack.html 11/10/16 GG

6 IV Planar disloca0ons A Represented mathema0cally as infinitely long cut with a straight edge B Rela0ve displacement (of one side of the disloca0on rela0ve to the other) across a disloca0on is called the Burger s vector b. 11/10/16 GG D Edge disloca0on 1 Accommodate opening or sliding mo0ons 2 Displacement is exclusively perpendicular to the disloca0on edge 3 Displacement can be parallel or perpendicular to the disloca0on plane View along edge of edge disloca0on hhp:// kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html 11/10/16 GG

7 D Edge disloca0on (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:// kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html blog.faureevegan.com 11/10/16 GG D Edge disloca0on (cont.) 5 Macroscopic geologic use: modeling dikes or faults hhp://hvo.wr.usgs.gov/gallery/kilauea/erupt/24ds182_cap0on.html hhp:// 11/10/16 GG

8 C Screw disloca0on 1 Accommodate a tearing mo0on 2 Displacement is exclusively parallel to the disloca0on edge 3 Analogy: a lock washer or a 360 spiral staircase Disloca0on edge hhp:// kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html hhp:/// 11/10/16 GG C Screw disloca0on 4 Macroscopic geologic use: modeling faults hhp://volcanoes.usgs.gov/imgs/jpg/photoglossary/fault2_large.jpg hhp:// kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html 11/10/16 GG

9 V Displacement and stress fields for a screw disloca0on (mode III) A Displacement parallel to the disloca0on edge (w) increases uniformly along a spiral- like circuit from one side of the disloca0on to the other (for a right- handed screw disloca0on, point your right thumb along the disloca0on edge; displacement parallel to the edge increases in the direc0on your fingers curl. B Angular posi0on: θ = tan - 1 (y/x) C w = bθ/2π D An0- plane strain (u,v ) 11/10/16 GG Cylindrical Reference Frame u r, u θ, w = bθ/2π Normal Strains ε rr = 1 u r 2 r + u r r ε rr ε θθ = u r r + 1 r ε θθ u θ θ ε zz = 1 w 2 z + w z ε zz Shear Strains ε rθ = 1 1 u r 2 r θ + u θ r u θ r ε rθ = ε θr ε θz = 1 2 ε θz = ε zθ = u θ z + 1 r b 4πr w θ ε zr = 1 w 2 r + u r z ε zr = ε rz * The deriva0ves are very easy to do. Check them to verify the solu0ons. 11/10/16 GG

10 Cartesian Reference Frame u, v, w = b[tan - 1 (y/x)]/2π Normal Strains ε xx = 1 u 2 x + u x ε xx ε yy = 1 v 2 y + y y ε yy ε zz = 1 w 2 z + w z ε zz Shear Strains ε xy = 1 u 2 y + v x ε xy ε yz = 1 v 2 z + w y ε yz = b 4π ε yz = b x 4π r 2 11/10/16 GG x x 2 + y 2 ε xz = 1 u 2 z + w x ε xz = b y 4π x 2 + y 2 ε xz = b y 4π r 2 * The deriva0ves are s0ll rather easy to do. Check them to verify the solu0ons. Strains ε rθ = ε θr Stresses σ rθ = σ θr = 2Gε rθ ε θz = ε zθ = ε rθ = ε θr ε rr ε θθ ε zz b 4πr σ θz = σ zθ = 2Gε θz = Gb 2πr σ rθ = σ θr = 2Gε rθ σ rr σ θθ σ zz 11/10/16 GG

11 Strains Stresses ε xy = ε yx σ xy = σ yx = 2Gε xy ε yz = b 4π ε xz = b 4π x x 2 + y 2 y x 2 + y 2 σ yz = σ zy = 2Gε yz = Gb 2π σ xz = σ zx = 2Gε xz = Gb 2π x x 2 + y = Gb x 2 2π r 2 y x 2 + y = Gb 2 2π y r 2 ε xx σ xx ε yy σ yy ε zz σ zz 11/10/16 GG Key points a Only the shear stresses ac0ng on a plane normal to the z- direc0on or in the z- direc0on are non- zero b The stresses are singular (i.e., go to infinity) near the disloca0on edge: a powerful stress concentra0on exists there. c This theore0cal singular stress concentra0on exists no maher how small G or the rela0ve displacement b is, provided G > 0 and b > 0. σ θz = σ zθ = Gb 2πr 11/10/16 GG

12 SUPERPOSTION OF TWO (INFINITE) SCREW DISLOCATIONS (A,B) TO FORM A FINITE DISPLACEMENT DISCONTINTUITY (C) 11/10/16 GG