23. Dislocations

I Main Topics A Dislocations and other defects in solids B Significance of dislocations C Planar dislocations D Displacement and stress fields for a screw

dislocation (mode III)

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23. Dislocations

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http://volcanoes.usgs.gov/Imgs/Jpg/Photoglossary/fissure4_large.JPG

23. Dislocations

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23. Dislocations

II Dislocations and other defects in solids A Dislocations

1 Originally, extra (or missing) planes or partial planes of material (e.g., atoms)

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http://www.tf.uni-kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html

23. Dislocations

II Dislocations and other defects in solids A Dislocations (cont.)

2 Evidence for dislocations from electron microscopy

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http://en.wikipedia.org/wiki/Dislocation

Transmission Electron MicrographOf Dislocations

23. Dislocations

II Dislocations and other

defects in solids

A Dislocations (cont.)

3 Continuum

mechanics usage:

surfaces across

which

displacements

are discontinuous

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http://volcanoes.usgs.gov/Imgs/Jpg/Photoglossary/fault2_large.jpg

23. Dislocations

II Dislocations and other defects in solids (cont.) B Point defects

1 Originally, extra (or missing) volumes (e.g., atoms)

2 Displacements are discontinuous across point defects

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http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Graphics/Chrystal-Defects.jpg

23. Dislocations

III Significance of dislocations

A Account for permanent plastic deformation in crystals

B Account for the low observed strength of crystals relative to theoretical predictions

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http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Graphics/Chrystal-Defects.jpg

23. Dislocations

III Significance of dislocations

C They provide useful quantitative description of relative motions across surfaces across a broad range of scale (crystals [10-6 m] to plate boundaries [106 m]) –~12 orders of magnitude!

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http://en.wikipedia.org/wiki/Dislocation

Dislocation model of subduction zone

From Bevis and Martel, 2001

Transmission Electron Micrograph

23. Dislocations

III Significance of dislocationsD Dislocations induce

tremendous stress concentrations and account for “large” deformations under small “average” stresses

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http://pangea.stanford.edu/research/geomech/Faculty/crack.html

23. DislocationsIV Planar dislocations

A Represented mathematically as infinitely long cut with a straight edge

B Relative displacement (of one side of the dislocation relative to the other) across a dislocation is called the Burger’s vector b.

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23. Dislocations

D Edge dislocation 1 Accommodate opening

or sliding motions 2 Displacement is

exclusively perpendicular to the dislocation edge

3 Displacement can be parallel or perpendicular to the dislocation plane

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http://www.tf.uni-kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html

View along edge of edge dislocation

23. Dislocations

10/10/18 GG303 13blog.fatfreevegan.comhttp://www.tf.uni-kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html

Extra half-plane of atoms in lattice

Extra row of kernels in an ear of cornD Edge dislocation (cont.) 4 Analogy: an extra row of corn kernels on

a cob of corn

23. Dislocations

D Edge dislocation (cont.)

5 Macroscopic geologic use: modeling dikes or faults

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http://www.experimentalstress.com/topic/reflect.jpghttp://hvo.wr.usgs.gov/gallery/kilauea/erupt/24ds182_caption.html

23. Dislocations

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C Screw dislocation

1 Accommodate a

tearing motion

2 Displacement is

exclusively parallel to

the dislocation edge

3 Analogy: a lock

washer or a 360°

spiral staircase

Dislocation edge

http://www.tf.uni-kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html

http:///www.boltsplus.com

23. Dislocations

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http://volcanoes.usgs.gov/Imgs/Jpg/Photoglossary/fault2_large.jpg

http://www.tf.uni-kiel.de/matwis/amat/def_ge/kap_5/backbone/r5_2_2.html

C Screw dislocation4 Macroscopic geologic use: modeling faults

23. DislocationsV Displacement and stress fields for

a screw dislocation (mode III) A Displacement parallel to the

dislocation edge (w) increases uniformly along a spiral-like circuit from one side of the dislocation to the other (for a right-handed screw dislocation, point your right thumb along the dislocation edge; displacement parallel to the edge increases in the direction your fingers curl.

B Angular position: θ = tan-1(y/x)C w = bθ/2πD Anti-plane strain (u,v = 0)

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23. Dislocations

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NormalStrains

Shear Strains

Cylindrical Reference Frameur = 0, uθ = 0, w = bθ/2π

ε rr =12

∂ur∂r

+ ∂ur∂r

⎛⎝⎜

⎞⎠⎟

ε rr = 0

ε rθ =121r∂ur∂θ

+ ∂uθ∂r

− uθr

⎛⎝⎜

⎞⎠⎟

ε rθ = εθr = 0

εθθ =urr+ 1r∂uθ∂θ

εθθ = 0

εθz =12

∂uθ∂z

+ 1r∂w∂θ

⎛⎝⎜

⎞⎠⎟

εθz = ε zθ =b4πr

ε zz =12

∂w∂z

+ ∂w∂z

⎛⎝⎜

⎞⎠⎟

ε zz = 0

ε zr =12

∂w∂r

+ ∂ur∂z

⎛⎝⎜

⎞⎠⎟

ε zr = ε rz = 0

* The derivatives are very easy to do. Check them to verify the solutions.

23. Dislocations

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NormalStrains

Shear Strains

Cartesian Reference Frameu = 0, v = 0, w = b[tan-1(y/x)]/2π

ε xx =12

∂u∂x

+ ∂u∂x

⎛⎝⎜

⎞⎠⎟

ε xx = 0

ε xy =12

∂u∂y

+ ∂v∂x

⎛⎝⎜

⎞⎠⎟

ε xy = 0

ε yy =12

∂v∂y

+ ∂y∂y

⎛⎝⎜

⎞⎠⎟

ε yy = 0

ε yz =12

∂v∂z

+ ∂w∂y

⎛⎝⎜

⎞⎠⎟

ε yz =b4π

xx2 + y2

ε yz =b4π

xr2

ε zz =12

∂w∂z

+ ∂w∂z

⎛⎝⎜

⎞⎠⎟

ε zz = 0

ε xz =12

∂u∂z

+ ∂w∂x

⎛⎝⎜

⎞⎠⎟

ε xz =−b4π

yx2 + y2

ε xz =−b4π

yr2

* The derivatives are still rather easy to do. Check them to verify the solutions.

23. Dislocations

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σ rθ =σθr = 2Gε rθ = 0

σθz =σ zθ = 2Gεθz =Gb2πr

σ rθ =σθr = 2Gε rθ = 0ε rr = 0εθθ = 0ε zz = 0

Strains Stresses

σ rr = 0σθθ = 0σ zz = 0

ε rθ = εθr = 0

εθz = ε zθ =b4πr

ε rθ = εθr = 0

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ε xy = ε yx = 0 σ xy =σ yx = 2Gε xy = 0

ε yz =b4π

xx2 + y2

σ yz =σ zy = 2Gε yz =Gb2π

xx2 + y2

= Gb2π

xr2

ε xz =−b4π

yx2 + y2

σ xz =σ zx = 2Gε xz =−Gb2π

yx2 + y2

= −Gb2π

yr2

ε xx = 0 σ xx = 0

ε yy = 0 σ yy = 0

ε zz = 0 σ zz = 0

Strains Stresses

23. DislocationsKey pointsa Only the shear stresses

acting on a plane normal to the z-direction or in the z-direction are non-zero

b The stresses are singular (i.e., go to infinity) near the dislocation edge: a powerful stress concentration exists there.

c This theoretical singular stress concentration exists no matter how small G or the relative displacement bis, provided G > 0 and b > 0.

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σθz =σ zθ =Gb2πr

23. DislocationsSUPERPOSTION OF TWO (INFINITE) SCREW DISLOCATIONS (A,B)

TO FORM A FINITE DISPLACEMENT DISCONTINTUITY (C)

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