Fracture Mechanics: Linear Elastic Fracture Mechanics 2/2

Transcription

1 Task 6 - Safety Review and Licensing On the Job Taining on Stess Analysis Factue Mechanics: Linea Elastic Factue Mechanics / Davide Mazzini Cio Santus Pisa (taly) June 15 July 14, 015

2 Table of content Class V.b. Content Stess singulaity - Notch degeneating into a cack - Multi-axial stess at notch oot/ cack tip - The Williams poblem Linea Elastic Factue Mechanics (LEFM) - The Westegaad stess function - Definition and calculation of the Stess ntensity Factos (SFs) - LEFM Validity limitations

3 Fom notch to cack n-plane stesses y/a y/a x y x y 0 n mid egion x y y x 0 n a x/a x 0,at the notch oot feesuface x/a 3

4 Fom notch to cack Westegaad stess function 1/ ij 1 1/ 1 0 s Othe tems nonsingula

5 Westegaad/ win stesses Catesian/ Cylindical coodinates xx yy K K K 3 cos 1sin sin 3 cos 1 sin sin 3 xy cos sin cos yy xy xx K cos cos 4 4 K cos cos 4 4 K sin sin 4 4 Cack Cack 5

6 Westegaad stesses Stesses ahead of the cack tip 0: xx yy xy 0 K K Notes: xx xy yy even at the notch oot 0 symmetyof the poblem Cack 0 yy xy xx 6

7 Westegaad stesses Stesses ahead of the cack tip x K y x y Cack Singulaity dominated zone y x 0 Fa field zone n 7

8 Westegaad stesses Stess ntensity Facto (SF) K is the(fist)stess ntensity Facto y K Diffeent K values, weake and stonge singulaities K k (same intensification meaning) What is the eason of? Giffith Enegy eleaseate" G" Cack Now thisequation holds without any tem 8

9 Stess ntensity Facto The Stess ntensity Facto is NOT the Stess Concentation Facto y K y/a K max t n Cack x/a 9

10 Stess ntensity Facto Units, two options MPa mm MPa y K MPa m m MPa y N mm K mm mm N mm 3 Convesion: xmpa m xmpa 1000 mm 1000 xmpa mm 31.6 xmpa mm Example: 8.0 MPa m 5.8MPa mm 10

11 Stess ntensity Facto Logaithm scale y K Let'slog both sides log( y ) log( K) log( ) log( ) y (log) 1 1 Cack Cack 0goesto (log) 11

12 Stess ntensity Facto ANSYS Apdl (classic) MATLAB: Veify the -1/ slope (log-log) and calculate the SF fo a plate with a lateal cack 100MPa Element type to be used a 15mm h 70mm b 100mm 1

13 Stess ntensity Facto - Half model with Symmety - Full model 13

14 Stess ntensity Facto - Half model with Symmety - Full model Kidney bean shape 14

15 Stess ntensity Facto MATLAB elaboation Linea scales 10 4 log-log scales Stess components, MPa , mm x y xy y, MPa 10 3 K = 1047 MPa mm 1/ , mm MATLAB "polyfit": P, P P 0.5(good appox.) 1 1 K 10 P 1 log ( ) 10 15

16 Stess ntensity Facto Homewok: Veify the angle dependency of the stess distibution at the cack tip xx yy K 3 cos 1 sin sin K 3 cos 1 sin sin K 3 cos sin cos xy Path at any diffeent angle 0 16

17 n-plane Stess ntensity Factos Thee ae two distinct ways to apply in-plane loading K 3 K 3 K 3 K 3 K 3 K 3 cos sin cos cos 1sin sin xx cos 1sin sin sin cos cos yy cos 1sin sin sin cos cos xy Cack yy xy xx Catesian coodinates Mode and Mode Stess ntensity Factos Symmetical and Nonsymmetical stess components 17

18 n-plane Stess ntensity Factos Thee ae two distinct ways to apply in-plane loading K K cos cos sin sin K K cos cos sin sin K K sin sin cos cos Cylindical coodinates Mode and Mode Stess ntensity Factos Cack Symmetical and Nonsymmetical stess components 18

19 Plane Stess/ Stain Same solution n-plane K 3 K 3 cos 1 sin sin sin cos cos xx K 3 K 3 yy cos 1sin sin sin cos cos xy K 3 K 3 cos sin cos cos 1 sin sin Similaly to the notch, any point even vey close to thesingulaity: zz 0 plane stess xx yy plane stain 19

20 Plane Stess/ Stain Plane stess, tansvesal stess/ stain zz 0 K K zz ( xx yy ) cos sin E E Plane stain, tansvesal stess/ stain K K zz ( xx yy ) cos sin 0 zz 0

21 Plane Stess/ Stain How plane stess can be possible at the cack tip? Plane stess solution if a>>b Plane stain if a<<b 0 Any thickness, the local adius is infinitely smalle being just zeo. Why talking about plane stess fo a cack? Plastic zone need to intoduced. a B t 0 t:thickness :adius 0 1

22 Fully thee-dimensional poblem The thee cack loading modes and elated Stess ntensity Factos n-plane: Out-of-plane: K, K K

23 Fully thee-dimensional poblem Many figues available to show the thee cack modes 3

24 Mode stess distibution Out of plane shea stesses, o altenatively, xz yz xz yz wee zeo fo Mode and Mode eithe plane Stess o plain Stain xz yz K sin K cos othe stess component ae zeo 4

25 How to calculate SFs fo stuctues Diffeent appoaches Dimensional appoach, with gaph and tabula data Weight function Finite elements with diffeent techniques, peviously an example with the stess asymptotic appoach has been shown 5

26 How to calculate SFs fo stuctues Dimensional appoach S K FS a aand Sae equied just fo the dimensionalanalysis F is the "shape function" it is dimensionless and depends on the loading configuation and elative dimensions(similaly to K ) t Fa boundaies a K S a Fo this specific poblem: F 1.0(theoetical esult) 6

27 How to calculate SFs fo stuctues Dimensional appoach K FS a n g Usually S g gossstess is used in thefomula athe than S which is the net stess Finite with, fa stess S g Fo this specific case: F depends on a/ batio a... tabula cases available on textbooks and atlas to find (appoximated) values fo F b 7

28 How to calculate SFs fo stuctues Cack on plate cases F 1.1 fo a single-edge-cacked plate with a smallcack 8

29 How to calculate SFs fo stuctues Pevious example solved with FE simulation 100MPa a 15mm b 100mm h 70mm Hee h bis not satisfied, howeve, appoximately: a/ b0.15 F 1.8(not much lage than1.1) K FS a K (ANSYS) 880 MPa mm MPa mm 1 ( 16% ) 9

30 Pevious example modified, lage height How to calculate SFs fo stuctues 100MPa a 15mm b 100mm h 300 mm Now h bis satisfied: a/ b0.15 F 1.8(not much lage than1.1) K FS a 880 MPa mm K (ANSYS) 899 MPa mm ( % )

31 How to calculate SFs fo stuctues Thee-dimensional elliptical o cicula (penny-shaped) cacks nfinite space Half space Ellipt. Cic. Suf. Ellipt. 31

32 Many cases available on books and atlas How to calculate SFs fo stuctues T.L. Andeson, Factue Mechanics: Fundamentals and Applications, thid edition. CRC Pess 005. S.A. Laham, R.A. Ainswoth, Stess ntensity Facto and Limit Load Handbook. EPD/GEN/REP/0316/98, SSUE, Y. Muakami, Stess ntensity Factos Handbook. Pegamon, ASTM standads (to be shown next) and othes 3

33 How to calculate SFs fo stuctues S.A. Laham, R.A. Ainswoth, Stess ntensity Facto and Limit Load Handbook. EPD/GEN/REP/0316/98, SSUE,

34 How to calculate SFs fo stuctues Homewok: Calculate thesf fo a suface cack in a half-space at A and B points: a t a mm, c.5mm 100 MPa (unifom) Compae the esults Andeson vs. Laham 34

35 win 1961 Cack tip plastic zone Mateial model: elastic pefectly plastically S Y Plane stess: y y S Y 0 1 K Y SY Afte equlibium coection: 1 K ( ) p SY p Y 35

36 win 1961 Cack tip plastic zone Mateial model: elastic pefectly plastically S Y Plane stain: y y S Y 0 1 K Y 6 SY Afte equlibium coection: 1 K ( ) p 3 SY p Y 36

37 Cack tip plastic zone Plane Stess/ Plain Stain Kidney bean shape Pl. Stain at intemediate thickness, only fo a thick plate Always Pl. Stess at the suface 3 times size 37

38 Cack tip plastic zone Plane Stess/ Plain Stain Within the Linea Elastic assumption, always plain stain at the singulaity, local adius = 0 Afte consideing the plastic zone, the size to be compaed to the thickness is p To distinguish between plane Stess/ Stain p 0 38

39 Cack tip plastic zone Plane Stess/ Plain Stain To have fully developed planestain: B K.5 SY K.5 S Y 1 K (.5 6 ) 6 SY (.56 ) 50 Y Y 39

40 Cack tip plastic zone Small Scale Yielding fo LEFM validity a To have LEFM validity: a p this way the plastic zone is dominated by the K-field. Usuallyit isstated that: a 8 (fo pl.stess) Y a 8 (fo pl.stess) Y fo Pl. Stess: fo Pl. Stain: a 4 K 4 K a S 3 S Y Y 40

41 Cack tip plastic zone Small Scale Yielding fo LEFM validity Y ( ) has to be limited also p with espect to the geomety boundaies fo Pl. Stess: fo Pl. Stain: 4 K 4 K a ba h a ba h,( ),,( ), SY 3 SY 41

42 Cack tip plastic zone ANSYS Wb ASTM CT specimen: At suface (Pl. Stess) nteio section (Pl. Stain) Elastic-Plastic mateial 4

43 Cack tip plastic zone ANSYS Wb ASTM CT specimen: Plane Stess (small thickness with espect to the plastic zone) Same F/thick. 43

44 Cack tip plastic zone ANSYS Wb ASTM CT specimen: Plane Stain, inteio thick specimen Plane Stess, at thick specimen suface Plane Stess, small thickness specimen 44