5. Fundamental fracture mechanics

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Georgia Ramsey
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1 5. Fudmetl frcture mechic

3 Stre cocetrtio Frcture New free urfce formtio Force Mteril Eviromet Frcture mechic Ect crck growth drivig force? Reitce to frcture of mteril? Iititio Growth Stre cocetrtio Notch Cro ectio uddel chge P P Stre lie Stre lie bpth Thick tre lie Stre cocetrtio P P

4 Stre cocetrtio fctor Stre cocetrtio fctor S tre cocetrtio fctor K t m ( 5.3) Stre cocetrtio of hole Ifiite plte = Teio of plte with hole At poit(r,θ), tre of θdirectio 4 3 θ co θ 4 r r i mimum t hole edge m co 0 3 (θ=0 =r) Stre cocetrtio K t m 3 K t 3 ( 5.)

5 Stre cocetrtio fctorⅡ Stre cocetrtio fctor of ellipe hole = Ifiit plte with ellipe hole( d b) pplied re Stre cocetrtio fctor (ellipe hole) m K t + b ρ ( 5.) Ifiite plte For circle hole = ρ K t + 3 ρ

6 Stre cocetrtio fctorⅢ Applictio ρ= Coider ellipe hole Stre cocetrtio fctor ( = 0 ρ=) 0 K t 0 ρ 7.3 = 0 Ifiite plte Stre cocetrtio fctor K t A ρ A

7 Stre cocetrtio fctorⅣ Eercie Obti the tre cocetrtio fctor for followed otch Equtio K t A ρ A 4 Ifiite plte 4 A K t ρ= A 3.83 ρ

8 Stre cocetrtiofctorⅤ Stre cocetrtio fctor P=(b-) = 3 (Rdiu ) Ifiite plte (Rdiu ) K t =3 Fiite plte () Kt of circle hole For rdiu d, Sme K t = 3 (b) Stre cocetrtio fctor for fiite plte Stre i omil tre, t Miimum cro ectio

9 Deformtio mode of crck Compoet with crck form uder pplied lod. z z z Crck Crck Crck Mode ModeⅡ ModeⅢ Deformtio mode of crck

10 Stre cocetrtio fctor of ellipe hole 3 3 ( 5.4) m 3 ρ ( 5.) ρ K t m 0 Notch tip of ellipe =0

11 Stre ditributio t crck tip Stre i viciit of crck, Stre / 0 0 ρ/=0 K t = (Crck) ρ/=0.0 K t = ρ/=0. K t =7.3 = b Kt ρ ρ/=0.5 K t =3.83 K t m K t ρ For ce of crck (ρ= 0), ( 5.) Ditce from crck tip / K t (Idepedet of crck legth d crck tpe) Stre ditributio t ellipe otch tip

12 Stre iteit fctor Stre t otch tip OK Differet otch d tre Kt M. tre t otch root Stre cocetrtio fctor Epre tre t otch root NO For crck Mimum tre (Idepedet of crck tpe d legth) Stre iteit fctor K Iteit of tre field i viciit of crck Stre iteit fctor of ifiite plte with crck K π Net

13 Stre iteit fctorⅡ Stre ditributio t crck tip ( 5.4) 3 3 For crck Notch rdiu ρ 0 0 ρ 3 ( 5.5) Proportio to qure root Ivere proportio ~ Stre ditributio i viciit of crck ~ I viciit of crck i er 0

14 Stre iteit fctorⅢ Stre iteit fctor of ifiite plte with crck ( 0) π π K :tre iteit fctor K π Mode I K π K π Oe crck d ifiite plte [ Uit MP m ] For fiite plte d three dimeio Stre ditributio i viciit of crck tip d tre iteit fctor K π F Corrected fctor deped o crck geometr

15 Stre iteit fctor Ⅳ Crck legth d tre ditributio Log crck legth tre K π Log crck Short crck < Short crck legth tre K ' ' π A K equl to K Sme iteit of tre field Stre ditributio of differet crck legth Iteit of tre i viciit of crck i decided b ol tre iteit fctor

16 Stre iteit fctorⅤ Summr Stre t crck tip i Stre i viciit of crck i ivere proportio to qure root ( ditce from crck tip) Iteit of tre t crck tip i decided b ol tre iteit fctor (No reltio betwee eterl force, pecime dimeio d crck legth)

17 Stre iteit fctor of differet tpe crck

18 Smll cle ieldig Eltic bod ~ Uder pplied lod, tre i Itt frcture Prctice No lier ivlid of K Smll cle ieldig Pltic zoe i ver mller th crck legth I eltic zoe roud pltic zoe, it c coider the me o pltic deformtio. Stre c be evluted b tre iteit fctor, K. It clled mll cle ieldig tte. Pltic deformed zoe Vlid K

19 Smll cle ieldigⅡ Pltic zoe t crck tip Yield tre E A r p B Pltic zoe corrected eltic tre ditributio r p F Eltic tre ditributio φ O O D R=r p Crck Imgir eltic crck C Stre ditributio After ieldig Smll cle ieldig i ple tre tte K ( 5.6) π K π r p K π Eltic perfect pltic bod Applied lod i ot chged b pltic deformtio. Pltic zoe eted util two re re the me. After correctio, pltic zoe ize R R r p K K π K π ( 5.3)

20 Smll cle ieldig Ⅲ Crck opeig diplcemet E Stre ditributio corrected b pltic zoe C Stre ditributio fter B Yield tre pltic deformtio A r p r p F Eltic tre ditributio φ O O D R=r p Crck Imgir eltic crck Crck opeig diplcemet b pltic deformtio 4K φ πe K Coditio of mll cle ieldig Pltic zoe ize R d Crck tip opeig diplcemet φ Proportio to K over two

21 Pltic zoe ize = R/ Smll cle ieldig Ⅳ To dipper igulrit, deciio of R/ (Dgdle model ) R π ec S For mll cle ieldig ( 5.5) Rge of mll Scl ieldig 応力比 = / Smll cle ieldig π K 8 Pltic zoe ize t crck tip Applied tre < ield tre S Whe <<, R ec π 8 S S Equl to ( 5.3) Smll cle ield ( 5.6) Reltive pplied lod / S = 0.4 Reltive pltic zoe R/ = 0.

22 Smll cle ieldig Ⅵ Stre tte Ple tre (thi plte) dimeio Ple tri (thick plte) 3dimeio Surfce, ple tre δ crck Pltic zoe Thicke B r p K π Crck opeig diplcemet Ple tre Ple tri Pltic zoe i thick plte Iide mteril, ple tri r K 6 π p

23 Frcture toughe Frcture toughe? For crcked bod to pltic deform, Whe tre iteit fctor i over the criticl vlue, Crck uddel propgte d frcture occur Frcture toughe me the reitce to crck propgtio of mteril uder ttic lod

24 Frcture tougheⅡ Frcture toughe K C Ple tre regio Regio() Stble growth Tritio regio Slt Regio(Ⅱ) Ple tri frcture toughe Thicke B K C Ple tri regio Regio(Ⅲ) Verticl Utble growth Sher lip Ftigue crck otck

25 Frcture toughe Ⅲ Frcture toughe K C Ple tri tte + Smll cle ieldig B, Ple tri Frcture toughe Thicke B K.5 S K C C Thi plte Thick plte Ple tre i pltic zoe Stble growth, Slt tpe frcture urfce High frcture toughe Ple tri i pltic zoe Utble crck growth Verticl frcture urfce A cott frcture toughe

26 Frcture toughe Ⅳ K C t room temperture 材料アルミニウム合金 04-T T65 降伏応力 (MP) 平面ひずみ破壊靭性 K C チタン合金 Ti-6Al-4V 鋼 AISI 4340 A B

27 Stre d diplcemet Stre i the viciit of crck tip(r,θ) Plte r E,ι θ τ K θ θ 3θ co i i π r K θ θ 3θ co i i π r K π r θ θ 3θ co i i τ v u τ, directio diplcemet u K G Ple tre Ple tri r θ co κ i θ π 3ν ν 3 4ν v K G r θ i κ co θ π

28 Smll ieldigⅤ (Pltic zoe i viciit of crck tip) Stre / m R Smll ellipe =ρ/4 Circle =ρ Due to m Regio of me Pltic zoe Lrge ellipe =4ρ Pltic zoe R/ρ m (Eltic mimum tre) ρ (Notch rdii) Sme R/ 0.4 Applied tre i the me

29 Cocept of lier frcture mechic K = K ρ=0 Pltic zoe ()Sme eltic tre field (b)eltic-pltic tre filed Cocept of frcture mechic For differet crck legth, if tre iteit fctor i the me, Eltic tre i the me, d the eltic d pltic tre i lo the me At crck tip, the me frcture pheomeo occur

30 Cocept of lier otch mechic m = m ρ =ρ ρ = ρ ρ 一定 t t t t Pltic zoe ()Sme eltic tre field Cocept of lier otch mechic (b)sme eltic pltic tre field For two otche otch rdii ρd eltic m. tre re the me Additio to eltic tre, eltic-pltic tre i lo the me. At ech otch tip, the me frcture pheomeo occur.

31 Stri eerg relee rte I(Griffith theor) Eerg relee rte clculted tri eerg Crck grow Stri eerg whe crck grow uit legth π E Free urfce Stri eerg relee rte ρ G π E π E K E Griffith equtio Eγ π K G E γ

32 Stri eerg relee rte II (Irwi tud) Eerg relee rte clculted from tre t crck tip 0 K π 0 v K 4 E Δ Δ π () (b) Chge of tri eerg with icreig crck growth Crck grow Eltic tri eerg relee ΔU Releed eltic Stri eerg ΔU = After crckig, lod pplie to Δ Ad revered diplcemet, v, occur Before crckig, workig,δw

33 Stre d Stri eerg relee rte III Eerg relee rte clculted from tre t crck tip After growth Coider crck upper ide Before growth 0 (c) Dplcemet d Workig t =Stri eerg (d) Stri eerg chge with icreig crck growth U 0 Δ K π 4K E K Δ Δ d πe 0 K U Δ GΔ E Δ π G ; Stri eerg relee rte (Drivig force of crck) d

34 Stre cocetrtio 0 Ulimited plte = (Notch) Teile tre bed o I the me compreive tre bed o Pi = + P P () (b) (c) (b)for o hole plte ppled + Stre cocetrtio (Teile tre) (c)circumferece of mll hole m cocetrted force, Pi ditribute (Compreive tre) Stre ditributio b cocetrted force

35 Stre ditributio i the viciit of crck tip Smll ellipe =ρ/4 Circle =ρ Agreemet of tre Lrge ellipe =4ρ 0.3 ρ Ditce from crck tip /ρ If ρd m re the me For differet otch, Stre ditributio i equl Whe Notch rdii ρd m re the me, Stre ditributio Stre ditributio log i of ellipe pplied 3 3

S(x)along the bar (shear force diagram) by cutting the bar at x and imposing force_y equilibrium.

S(x)along the bar (shear force diagram) by cutting the bar at x and imposing force_y equilibrium. mmetric leder Bems i Bedig Lodig Coditios o ech ectio () pplied -Forces & z-omets The resultts t sectio re: the bedig momet () d z re sectio smmetr es the sher force () [ for sleder bems stresses d deformtio