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

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1 I Main Topics A Intoducon to stess fields and stess concentaons B An axisymmetic poblem B Stesses in a pola (cylindical) efeence fame C quaons of equilibium D Soluon of bounday value poblem fo a pessuized hole 11/4/15 GG303 1 hop:// hop://hvo.w.usgs.gov/kilauea/kilauea_map.html 11/4/15 GG303 1

2 II Intoducon to stess fields and stess concentaons A Impotance 1 Stess (and stain and displacement) vay in space Fields extend defomaon concepts at a point 3 Stess concentaons can be huge and have a lage effect hop://pangea.stanfod.edu/eseach/geomech/faculty/cack.html 11/4/15 GG303 3 II Intoducon to stess fields and stess concentaons (cont.) B Common causes of stess concentaons 1 A foce acts on a small aea (e.g., beneath a nail being hammeed) hop://0.tqn.com/d/homeepai/1/0/l/7/- /- /nail_set.jpg 11/4/15 GG303 4

3 II Intoducon to stess fields and stess concentaons (cont.) B Common causes of stess concentaons Geometic effects (e.g., cones on doos and windows) hop:// epai/images/doo- Cack.jpg 11/4/15 GG303 5 II Intoducon to stess fields and stess concentaons (cont.) B Common causes of stess concentaons 3 Mateial heteogeneies (e.g., mineal heteogeneies and voids) Cacks nea a fluid inclusion in halite hop:// 11/4/15 GG

4 III An axisymmetic poblem The geomety dictates use of a pola (cylindical) efeence fame. Displacements ae puely adial by symmety 11/4/15 GG303 7 IV Stesses in a pola (cylindical) efeence fame (on- in convenon) A Coodinate tansfomaons 1 x cosθ cosθ x y sinθ cosθ y 3 (x + y ) 1/ 4 θ tan - 1 (y/x) 5 a x cosθ x cosθ 6 a y cosθ y sinθ 7 a θx cosθ θx - sinθ 8 a θy cosθ θy cosθ 11/4/15 GG

5 B Stess tansfomaons 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 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 11/4/15 GG303 9 V quaons of equilibium (foce balance) A In pola (,θ) efeence fame fo the axisymmetic poblem hee, σ θ σ θ 0 11/4/15 GG

6 V quaons of equilibium (foce balance) B Foce balance in adial diecon 1 0 F The dz tem can be divided out, yielding: ( dθ )( dz) ( + d) ( dθ )( dz) σ + σ + dσ ( σ θθ ) ( d ) sin dθ dz ( dθ ) ( + d) ( dθ ) 0 σ + σ + dσ ( σ θθ ) d sin dθ 11/4/15 GG V quaons of equilibium (foce balance) B Foce balance in adial diecon F ( σ )( dθ ) + ( σ + dσ )( + d) ( dθ ) ( σ θθ ) d Simplifying eq. () by cancelling the undelined tems yields 3 sin dθ ( σ )( d) ( dθ ) + ( dσ )( + d) ( dθ ) ( σ θθ ) ( d ) sin dθ Now as dθ 0, sin( dθ ) dθ, so sin( dθ ) dθ ( σ )( d) ( dθ ) + ( dσ )( + d) ( dθ ) ( σ θθ )( d) ( dθ ) 0 11/4/15 GG

7 C Foce balance in adial diecon ( σ )( d) ( dθ ) + ( dσ )( + d) ( dθ ) ( σ θθ )( d) ( dθ ) 0 Now as d 0and dθ 0,so( dσ ddθ ) 0, hence ( σ )( d) ( dθ ) + ( dσ ) ( dθ ) ( σ θθ )( d) ( dθ ) 0 Now divide though by ( d) ( dθ ) dσ d + σ σ θθ 0 This equaon equilibium in the adial diecon is the govening equaon of fo an axisymmetic poblem. 11/4/15 GG VI Soluon of bounday value poblem fo a pessuized hole ε u A Homogeneous isotopic mateial B Unifom posive tacon on wall of hole C Unifom adial displacement D Radial shea stesses 0 because adial shea stains 0 11/4/15 GG

8 V Soluon of bounday value poblem fo a pessuized hole F One Geneal Soluon Method 1 Replace the stesses by stains using Hooke s law Replace the stains by displacement deivaves to yield a govening equaon in tems of displacements. 3 Solve diffeenal govening equaon fo displacements 4 Take the deivaves 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 0 11/4/15 GG V Soluon of bounday value poblem fo a pessuized hole F One Geneal Soluon Method 6 The geneal soluon 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. 11/4/15 GG

9 V Soluon of bounday value poblem fo a pessuized hole G Stain- displacement elaonships Catesian coodinates ε xx u x ε yy v y ε xy 1 u y + v x Pola coodinates ε u ε θθ u ε θ 0 11/4/15 GG V Soluon of bounday value poblem fo a pessuized hole H Stain- stess elaonships: Plane Stess (σ zz 0) Catesian coodinates Pola coodinates ε xx 1 σ xx νσ yy ε yy 1 σ yy νσ xx ε xy 1 G σ xy ε 1 [ σ νσ θθ ] ε θθ 1 [ σ θθ νσ ] ε θ 1 G σ θ Fom specializing the 3D elaonships 11/4/15 GG

10 V Soluon of bounday value poblem fo a pessuized hole J Stess- stain elaonships: Plane Stess (σ zz 0) Catesian coodinates σ xx σ yy ( 1 ν ) ε + νε xx yy 1 ν ε yy + νε xx σ xy Gε xy Pola coodinates σ ( 1 ν ) ε + νε θθ σ θθ 1 ν [ ] [ ε + νε θθ ] σ θ Gε θ Fom specializing the 3D elaonships 11/4/15 GG V Soluon of bounday value poblem fo a pessuized hole L Axisymmetic Govening quaon Plane Stess (σ zz 0) In tems of stess In tems of displacement See Appendix fo the algeba to convet the govening equaon fom a funcon of stesses to adial displacement dσ d + σ σ θθ 0 d u + 1 du d d u 0 11/4/15 GG

11 Appendix quaons of equilibium in Catesian fom Plane stain elaonships (e zz 0) Convesion of axisymmetic govening equaon to a funcon of adial displacement 11/4/15 GG303 1 IVquaons of equilibium (foce balance) A In Catesian (x,y) efeence fame 11/4/15 GG303 11

12 IVquaons of equilibium (foce balance) A In Catesian (x,y) efeence fame 11/4/15 GG303 3 IVquaons of equilibium (foce balance) (cont.) 1 F x F x 0 [ σ xx ](dydz) + [ σ yx ](dxdz) +[σ xx + σ xx x dx](dydz) + [σ yx + σ yx y dy](dxdz) 0 [ σ xx x dx](dydz) + [ σ yx y dy](dxdz) 3 F x 0 [ σ xx x + σ yx y ](dxdydz) 4 σ xx x + σ yx y 0 The ate of σxx incease is balanced by the ate of σ yx decease 11/4/15 GG

13 IVquaons of equilibium (foce balance) (cont.) 1 F y 0 [ σ yy ](dxdz) + [ σ xy ](dydz) +[σ yy + σ yy y dy](dxdz) + [σ xy + σ xy y dx](dydz) 3 F y F y 0 [ σ yy y dy](dxdz) + [ σ xy x dx](dydz) 0 [ σ yy y + σ xy x ](dxdydz) 4 σ yy y + σ xy x 0 The ate of σyy incease is balanced by the ate of σ xy decease 11/4/15 GG303 5 V Soluon of bounday value poblem fo a pessuized hole K Stess- stain elaonships: Plane Stain (ε zz 0) σ xx σ yy Catesian coodinates ( 1+ ν) ε + ν xx 1+ ν ε yy + 1 ν ε xx + ε yy ν 1 ν ε yy + ε xx σ σ θθ Pola coodinates ( 1 + ν) ε + ν 1+ ν ε θθ 1 ν ( ε + ε θθ ) ν 1 ν ( ε θθ + ε ) σ xy Gε xy ε θ Gε θ Fom specializing the 3D elaonships 11/4/15 GG

14 V Soluon of bounday value poblem fo a pessuized hole I Stain- stess elaonships: Plane Stain (ε zz 0) Catesian coodinates ε xx 1 ν ν σ xx 1 ν σ yy ε yy 1 ν ν σ yy 1 ν σ xx ε xy 1 G σ xy Pola coodinates ε 1 ν ν σ 1 ν σ θθ ε θθ 1 ν ν σ θθ 1 ν σ ε θ 1 G σ θ Fom specializing the 3D elaonships 11/4/15 GG303 7 Convesion of axisymmetic govening equaon to a funcon of adial displacement dσ d + σ σ θθ 0 σ 1 ν [ ε +νε θθ ] ( 1 ν ) σ θθ 1 ν dσ [ ε +νε θθ ] ( 1 ν ) u du d +ν u +ν du d d d u 1 ν d +ν u du d σ σ θθ 1 ν 1 du d +ν u u ν du d dσ d + σ σ θθ d u 1 ν d +ν u du d + 1 du d +ν u u ν du d dσ d + σ σ θθ d u 1 ν d dσ d + σ σ θθ d u 1 ν d d u d + 1 du d u du d 1 u + 1 du d 1 +ν u + 1 du d + 1 ν u ν du d 11/4/15 GG303 8 u 0 14

15 V Soluon of bounday value poblem fo a pessuized hole L Axisymmetic Govening quaons Plane Stess (σ zz 0) Plane Stain (ε zz 0) In tems of stess In tems of displacement In tems of stess In tems of displacement σ + σ σ θθ 0 d u + 1 du d d u 0 σ + σ σ θθ 0 d u + 1 du d d u 0 Govening equaons ae the same fo plane stess and plane stain 11/4/15 GG