21. Stresses Around a Hole (I) 21. Stresses Around a Hole (I) I Main Topics

Transcription

1 I Main Topics A Intoduc<on to stess fields and stess concenta<ons B Stesses in a pola (cylindical) efeence fame C qua<ons of equilibium D Solu<on of bounday value poblem fo a pessuized hole GG303 1 hmp:// hmp://hvo.w.usgs.gov/kilauea/kilauea_map.html GG

2 II Intoduc<on to stess fields and stess concenta<ons A Impotance 1 Stess (and stain and displacement) vay in space 2 Fields extend defoma<on concepts at a point 3 Stess concenta<ons can be huge and have a lage effect hmp://pangea.stanfod.edu/eseach/geomech/faculty/cack.html GG303 3 II Intoduc<on to stess fields and stess concenta<ons (cont.) B Common causes of stess concenta<ons 1 A foce acts on a small aea (e.g., beneath a nail being hammeed) hmp://0.tqn.com/d/homeepai/1/0/l/7/- /- /nail_set.jpg GG

3 II Intoduc<on to stess fields and stess concenta<ons (cont.) B Common causes of stess concenta<ons 2 Geometic effects (e.g., cones on doos and windows) hmp:// epai/images/doo- Cack.jpg GG303 5 II Intoduc<on to stess fields and stess concenta<ons (cont.) B Common causes of stess concenta<ons 3 Mateial heteogenei<es (e.g., mineal heteogenei<es and voids) Cacks nea a fluid inclusion in halite hmp:// GG

4 III Stesses in a pola (cylindical) efeence fame (on- in conven<on) A Coodinate tansfoma<ons 1 x cosθ cosθ x 2 y sinθ cosθ y 3 (x 2 + y 2 ) 1/2 4 θ tan - 1 (y/x) GG303 7 B Stess tansfoma<ons σ i j a i k a j l σ kl 1 σ a x a x σ xx + a x a y σ xy +a y a x σ yx + a y a y σ yy 2 σ θ a x a θx σ xx + a x a θy σ xy +a y a θx σ yx + a y a θy σ yy 3 σ θ a θx a x σ xx + a θx a y σ xy +a θy a x σ yx + a θy a y σ yy 4 σ θθ a θx a θx σ xx + a θx a θy σ xy +a θy a θx σ yx + a θy a θy σ yy GG

5 IVqua<ons of equilibium (foce balance) A In Catesian (x,y) efeence fame GG303 9 IVqua<ons of equilibium (foce balance) (cont.) 1 2 F x F x [ σ xx ](dydz) + [ σ yx ](dxdz) +[σ xx + σ xx x dx](dydz) + [σ yx + σ yx y dy](dxdz) [ σ xx x dx](dydz) + [ σ yx y dy](dxdz) 3 F x [ σ xx x + σ yx y ](dxdydz) 4 σ xx x + σ yx y The ate of σxx incease is balanced by the ate of σ yx decease GG

6 IVqua<ons of equilibium (foce balance) (cont.) 1 F y [ σ yy ](dxdz) + [ σ xy ](dydz) +[σ yy + σ yy y dy](dxdz) + [σ xy + σ xy y dx](dydz) 2 3 F y F y [ σ yy y dy](dxdz) + [ σ xy x dx](dydz) [ σ yy y + σ xy x ](dxdydz) 4 σ yy y + σ xy x The ate of σyy incease is balanced by the ate of σ xy decease GG IVqua<ons of equilibium (foce balance) B In pola (,θ) efeence fame (apply chain ule; see supplement) 1 F σ + 1 σ θ θ + σ σ θθ 2 F θ 1 σ θθ θ + σ θ + 2σ θ GG

7 IVqua<ons of equilibium (foce balance) C Compaison of Catesian and pola equa<ons 1a σ xx x + σ yx y 1b σ yy y + σ xy x 2a σ + 1 σ θ θ + σ σ θθ 2b 1 σ θθ θ + σ θ + 2σ θ GG V Solu<on of bounday value poblem fo a pessuized hole A Homogeneous isotopic mateial B Unifom posi<ve tac<on on wall of hole C Unifom adial displacement D Radial shea stesses because adial shea stains GG

8 V Solu<on of bounday value poblem fo a pessuized hole Govening equa<on fo an axisymmetic poblem 1 Fo constant values of, stesses and displacements in a pola efeence fame do not change with θ, so (u, q, etc.)/ θ 2 By symmety the shea stess σ θ 1 σ θθ θ + σ θ + 2σ θ ( 0) σ + 1 σ θ θ + σ σ θθ σ σ σ θθ dσ d + σ σ θθ Govening equa<on GG V Solu<on of bounday value poblem fo a pessuized hole F One Geneal Solu<on Method 1 Replace the stesses by stains using Hooke s law 2 Replace the stains by displacement deiva<ves to yield a govening equa<on in tems of displacements. 3 Solve diffeen<al govening equa<on fo displacements 4 Take the deiva<ves of the displacements to find the stains 5 Solve fo the stesses in tems of the stains using Hooke s law (not had, but somewhat lengthy) dσ d + σ σ θθ GG

9 V Solu<on of bounday value poblem fo a pessuized hole F One Geneal Solu<on Method 6 The geneal solu<on will contain constants. Thei values ae found in tems of the stesses o displacements on the boundaies of ou body (i.e., the wall of the hole and any extenal bounday), that is in tems of the bounday condi,ons fo ou poblem. GG V Solu<on of bounday value poblem fo a pessuized hole G Stain- displacement ela<onships Catesian coodinates ε xx u x ε yy v y ε xy 1 u 2 y + v x Pola coodinates ε u ε yy u ε θ GG

10 V Solu<on of bounday value poblem fo a pessuized hole H Stain- stess ela<onships: Plane Stess (σ zz ) Catesian coodinates Pola coodinates ε xx 1 σ xx νσ yy ε yy 1 σ νσ yy xx ε xy 1 2G σ xy ε 1 [ σ νσ θθ ] ε θθ 1 [ σ νσ θθ ] ε θ 1 2G σ θ Fom specializing the 3D ela<onships GG V Solu<on of bounday value poblem fo a pessuized hole I Stain- stess ela<onships: Plane Stain (ε zz ) Catesian coodinates ε xx 1 ν 2 ν σ xx 1 ν σ yy ε yy 1 ν 2 ν σ yy 1 ν σ xx ε xy 1 2G σ xy Pola coodinates ε 1 ν 2 ν σ 1 ν σ θθ ε θθ 1 ν 2 ν σ θθ 1 ν σ ε θ 1 2G σ θ Fom specializing the 3D ela<onships GG

11 V Solu<on of bounday value poblem fo a pessuized hole J Stess- stain ela<onships: Plane Stess (σ zz ) Catesian coodinates σ xx σ yy ( 1 ν 2 ) ε + νε xx yy 1 ν 2 ( ) ε yy + νε xx σ xy 2Gε xy Pola coodinates σ ( 1 ν 2 ) ε + νε θθ σ θθ 1 ν 2 [ ] [ ( ) ε + νε θθ ] σ θ 2Gε θ Fom specializing the 3D ela<onships GG V Solu<on of bounday value poblem fo a pessuized hole K Stess- stain ela<onships: Plane Stain (ε zz ) σ xx σ yy Catesian coodinates ( 1+ ν) ε + ν xx 1+ ν ( ) ε yy + ( ) 1 2ν ε xx + ε yy ( ) ν 1 2ν ε yy + ε xx σ σ θθ Pola coodinates ( 1 + ν) ε + ν 1+ ν ( ) ε θθ 1 2ν ( ε + ε θθ ) ν 1 2ν ( ε θθ + ε ) σ xy 2Gε xy ε θ 2Gε θ Fom specializing the 3D ela<onships GG

12 V Solu<on of bounday value poblem fo a pessuized hole L Axisymmetic Govening qua<ons (Incomplete) Plane Stess (σ zz ) Plane Stain (ε zz ) In tems of stess In tems of displacement In tems of stess In tems of displacement σ + σ σ θθ d 2 u + 1 du d 2 d u 2 σ + σ σ θθ d 2 u + 1 du d 2 d u 2 GG