Lec.21.pptx - 11/6/11 21. Stresses Around a Hole (I)...

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Unformatted text preview: 11/6/11 21. Stresses Around a Hole (I) I Main Topics A Introduc<on to stress fields and stress concentra<ons B Stresses in a polar (cylindrical) reference frame C Equa<ons of equilibrium D Solu<on of boundary value problem for a pressurized hole 11/6/11 GG303 1 21. Stresses Around a Hole (I) hMp://www.pacificautoglass.com 11/6/11 hMp://hvo.wr.usgs.gov/kilauea/Kilauea_map.html GG303 2 1 11/6/11 21. Stresses Around a Hole (I) II Introduc<on to stress fields and stress concentra<ons A Importance 1  Stress (and strain and displacement) vary in space 2  Fields extend deforma<on concepts at a point 3 Stress concentra<ons can be huge and have a large effect 11/6/11 hMp://pangea.stanford.edu/research/geomech/Faculty/crack.html GG303 3 21. Stresses Around a Hole (I) II Introduc<on to stress fields and stress concentra<ons (cont.) B Common causes of stress concentra<ons 1 A force acts on a small area (e.g., beneath a nail being hammered) 11/6/11 hMp://0.tqn.com/d/homerepair/1/0/l/7/ ­/ ­/nail_set.jpg GG303 4 2 11/6/11 21. Stresses Around a Hole (I) II Introduc<on to stress fields and stress concentra<ons (cont.) B Common causes of stress concentra<ons 2 Geometric effects (e.g., corners on doors and windows) hMp://www.basementsystems.com/founda<on ­repair/images/Door ­Crack.jpg 11/6/11 GG303 5 21. Stresses Around a Hole (I) II Introduc<on to stress fields and stress concentra<ons (cont.) B Common causes of stress concentra<ons 3 Material heterogenei<es (e.g., mineral heterogenei<es and voids) 11/6/11 Cracks near a fluid inclusion in halite hMp://www.minsocam.org/msa/collectors_corner/arc/img/halite41.jpg GG303 6 3 11/6/11 21. Stresses Around a Hole (I) III Stresses in a polar (cylindrical) reference frame (on ­in conven<on) A Coordinate transforma<ons 1  x = r cosθ = r cosθxr 2  y = r sinθ = r cosθyr 3  r= (x2 + y2)1/2 4  θ = tan ­1(y/x) 11/6/11 GG303 7 21. Stresses Around a Hole (I) B Stress transforma<ons σi’j’ = ai’k aj’l σkl 1  σrr = arx arx σxx + arx ary σxy +ary arx σyx + ary ary σyy 2  σrθ = arx aθx σxx + arx aθy σxy +ary aθx σyx + ary aθy σyy 3  σθr = aθx arx σxx + aθx ary σxy +aθy arx σyx + aθy ary σyy 4  σθθ = aθx aθx σxx + aθx aθy σxy +aθy aθx σyx + aθy aθy σyy 11/6/11 GG303 8 4 11/6/11 21. Stresses Around a Hole (I) IVEqua<ons of equilibrium (force balance) A In Cartesian (x,y) reference frame 11/6/11 GG303 9 21. Stresses Around a Hole (I) IVEqua<ons of equilibrium (force balance) (cont.) 1 ∑ Fx = 0 = [−σ xx ](dydz ) + [−σ yx ](dxdz ) +[σ xx + 2 ∑ Fx = 0 = [ 3 ∑ F = 0=[ 4 ∂σ xx + x ∂x 11/6/11 ∂σ yx ∂σ xx dx ](dydz ) + [ dy](dxdz ) ∂x ∂y ∂σ xx ∂σ yx +(dxdydz ) ∂x ∂y ∂σ yx ∂y ∂σ yx ∂σ xx dx ](dydz ) + [σ yx + dy](dxdz ) ∂x ∂y =0 The rate of σxx increase is balanced by the rate of σyx decrease GG303 10 5 11/6/11 21. Stresses Around a Hole (I) IVEqua<ons of equilibrium (force balance) (cont.) 1 ∑ Fy = 0 = [−σ yy ](dxdz ) + [−σ xy ](dydz ) +[σ yy + 2 ∑ Fy = 0 = [ 3 ∑ F y 4 ∂σ yy ∂y = 0=[ + ∂σ yy ∂σ xy ∂x ∂y ∂y dy](dxdz ) + [σ xy + dy](dxdz ) + [ ∂y ∂σ yy ∂σ yy + =0 11/6/11 ∂σ xy ∂x ∂σ xy ∂x ∂σ xy ∂y dx ](dydz ) dx ](dydz ) (dxdydz ) The rate of σyy increase is balanced by the rate of σxy decrease GG303 11 21. Stresses Around a Hole (I) IVEqua<ons of equilibrium (force balance) B In polar (r,θ) reference frame (apply chain rule; see supplement) 1 ∑ Fr = 0 = ∂σ rr + 1 ∂σ θ r + σ rr − σ θθ ∂r r ∂θ r 1 ∂σ θθ ∂σ rθ 2σ rθ + + ∂θ ∂r r 2 ∑ Fθ = 0 = r 11/6/11 GG303 12 6 11/6/11 21. Stresses Around a Hole (I) IVEqua<ons of equilibrium (force balance) C Comparison of Cartesian and polar equa<ons ∂σ yx ∂σ yy ∂y ∂y 1a ∂σ + = 0 1b xx ∂x σ ∂σ σ 2a ∂ r + 1 ∂∂θ + σ − σ = 0 2b 1 ∂∂θ r r r rr 11/6/11 θr rr θθ θθ + ∂σ xy + ∂x =0 ∂σ rθ 2σ rθ + =0 ∂r r GG303 13 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole A B C D Homogeneous isotropic material Uniform posi<ve trac<on on wall of hole Uniform radial displacement Radial shear stresses = 0 because radial shear strains = 0 11/6/11 GG303 14 7 11/6/11 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole E Governing equa<on for an axisymmetric problem 1 For constant values of r, stresses and displacements in a polar reference frame do not change with θ, so ∂(ur, qrr, etc.)/∂θ = 0 2 By symmetry the shear stress σrθ = 0 1 ∂σ θθ ∂σ rθ 2σ rθ + + =0 r ∂θ ∂r r 0+0+ 2 (0) =0=0 r ∂σ rr 1 ∂σ θ r σ rr − σ θθ + + =0 ∂r r ∂θ r ∂σ rr σ − σ θθ + 0 + rr =0 ∂r r dσ rr σ rr − σ θθ + =0 dr r Governing equa<on 11/6/11 GG303 15 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole F One General Solu<on Method 1 Replace the stresses by strains using Hooke’s law 2  Replace the strains by displacement deriva<ves to yield a governing equa<on in terms of displacements. 3  Solve differen<al governing equa<on for displacements 4  Take the deriva<ves of the displacements to find the strains 5  Solve for the stresses in terms of the strains using Hooke’s law (not hard, but somewhat lengthy) dσ rr σ rr − σ θθ + =0 dr r 11/6/11 GG303 16 8 11/6/11 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole F One General Solu<on Method 6 The general solu<on will contain constants. Their values are found in terms of the stresses or displacements on the boundaries of our body (i.e., the wall of the hole and any external boundary), that is in terms of the boundary condi,ons for our problem. 11/6/11 GG303 17 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole G Strain ­displacement rela<onships Cartesian coordinates ∂u ∂x ∂v ε yy = ∂y ∂ ur ∂r u ε yy = r r ε xx = ε xy = 11/6/11 Polar coordinates ε rr = 1 ⎛ ∂u ∂ v ⎞ + ⎝ ⎠ 2 ⎜ ∂y ∂x⎟ ε rθ = 0 GG303 18 9 11/6/11 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole H Strain ­stress rela<onships: Plane Stress (σzz = 0) Cartesian coordinates 1 ⎡σ xx − νσ yy ⎤ ⎦ E⎣ 1 ε yy = ⎡σ yy − νσ xx ⎤ ⎦ E⎣ 1 [σ rr − νσ θθ ] E 1 εθθ = [σ θθ − νσ rr ] E ε xx = ε xy = Polar coordinates ε rr = 1 σ xy 2G ε rθ = 1 σ rθ 2G From specializing the 3D rela<onships 11/6/11 GG303 19 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole I Strain ­stress rela<onships: Plane Strain (εzz = 0) Cartesian coordinates Polar coordinates ε xx = 1− ν2 E 1− ν2 ⎡ ⎤ ⎛ν⎞ σ xx − ⎜ σ yy ⎥ ε rr = ⎢ ⎝1−ν ⎟ ⎠ E ⎣ ⎦ ε yy = 1− ν2 E 1−ν ⎡ ⎤ ⎡ ⎤ ⎛ν⎞ ⎛ν⎞ ⎢σ yy − ⎜ 1 − ν ⎟ σ xx ⎥ εθθ = ⎢σ θθ − ⎜ 1 − ν ⎟ σ rr ⎥ ⎝ ⎠ ⎝ ⎠ E⎣ ⎣ ⎦ ⎦ ε xy = ⎡ ⎤ ⎛ν⎞ ⎢σ rr − ⎜ 1 − ν ⎟ σ θθ ⎥ ⎝ ⎠ ⎣ ⎦ 2 1 σ xy 2G ε rθ = 1 σ rθ 2G From specializing the 3D rela<onships 11/6/11 GG303 20 10 11/6/11 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole J Stress ­strain rela<onships: Plane Stress (σzz = 0) Cartesian coordinates E 1− ν2 E σ yy = 1− ν2 σ xx = Polar coordinates E [ε rr + νεθθ ] 1− ν2 E = ε + νε (1 − ν 2 ) [ θθ rr ] ⎡ε xx + νε yy ⎤ ⎣ ⎦ σ rr = ⎡ε yy + νε xx ⎤ ⎦ σ θθ ( ) ( )⎣ σ xy = 2Gε xy ( ) σ rθ = 2Gε rθ From specializing the 3D rela<onships 11/6/11 GG303 21 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole K Stress ­strain rela<onships: Plane Strain (εzz = 0) Cartesian coordinates Polar coordinates σ xx = E⎡ ⎤ ⎛ν⎞ ε xx + ⎜ ε + ε yy ⎥ ⎝ 1 − 2ν ⎟ xx ⎠ 1+ ν) ⎢ ( ⎣ ⎦ σ rr = E⎡ ⎛ν⎞ ε rr + ⎜ (ε + εθθ )⎤ ⎥ ⎝ 1 − 2ν ⎟ rr ⎠ 1+ ν) ⎢ ( ⎣ ⎦ σ yy = E⎡ ⎤ ⎛ν⎞ ⎢ε yy + ⎜ 1 − 2ν ⎟ ε yy + ε xx ⎥ ⎝ ⎠ (1 + ν ) ⎣ ⎦ σ θθ = E⎡ ⎤ ⎛ν⎞ ⎢εθθ − ⎜ 1 − 2ν ⎟ ( εθθ + ε rr ) ⎥ ⎝ ⎠ (1 + ν ) ⎣ ⎦ ( ( ) ) σ xy = 2Gε xy ε rθ = 2Gε rθ From specializing the 3D rela<onships 11/6/11 GG303 22 11 11/6/11 21. Stresses Around a Hole (I) V Solu<on of boundary value problem for a pressurized hole L Axisymmetric Governing Equa<ons (Incomplete) Plane Stress (σzz = 0) In terms of stress Plane Strain (εzz = 0) In terms of stress In terms of displacement ∂σ rr σ rr − σ θθ d 2ur 1 dur ur + − =0 + =0 dr 2 r dr r 2 ∂r r 11/6/11 In terms of displacement ∂σ rr σ rr − σ θθ d 2ur 1 dur ur + − =0 + =0 dr 2 r dr r 2 ∂r r 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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