Materials 337. Lecture 7. Topics covered Introduction to fracture mechanics The elastic stress field Superposition principle Fracture toughness

Transcription

1 Mterils 337 Lecture 7 Topics covered Introduction to frcture mechnics The elstic stress field Superposition principle Frcture toughness Deprtment of Mechnicl Engineering Curtin University of Technology CRICOS Provider Code 00301J D:\Dt\Work\Curtin\Lecture courses\mterils 337\2006\Lectures\Lecture 7\Lecture_07_005.doc 7/04/2006

2 Introduction to frcture mechnics In severl of our erlier lectures we hve discussed the ftigue filure of steel nd luminium components under different types of loding. In ll cses we hve ssumed tht eventul filure ws due to the growth of crcks when the stress ws bove prticulr stress, i.e., C L C D C S S n for specimen with no stress concentrtion. Alterntively, we cn stte tht ftigue filure cn be overcome through preventing the growth of crcks. However, in order to do this we need quntittive description of crck growth under the previling conditions. The subject which provides this quntittive description is known s frcture mechnics nd we shll initilly consider the filed of liner elstic frcture mechnics (LEFM). Consider three identicl specimen, ech precrcked to crck length of o nd then subjected to ftigue stresses S 1 > S 2 > S 3 s follows: Ftigue crck length versus pplied cycles. Frcture is indicted by the x [1]. For lrger stress levels the crck propgtion rte is lrger nd thus the ftigue life is shorter. Hence, it cn be shown tht the totl life to filure is dependent on: (i) initil crck length, o, (ii) the stress mgnitude, nd (iii) the finl frcture resistnce. How cn this informtion be used to predict ftigue life nd for the design of components expected to fil under ftigue? A formt more useful for design is plot of the crck growth rte, d/dn, versus quntity known s the stress intensity fctor rnge, ΔK, which will be defined lter

3 Schemtic behviour of ftigue crck growth versus ΔK [2]. The generl shpe of this curve hs been confirmed for mny mterils, including the exmple shown below: Ftigue crck dt for Ti-6222 titnium lloy [3]

4 It cn be shown tht the S-shped curve is independent of initil crck length. The quntity ΔK tkes into ccount the crck length,, nd the stress rnge, ΔS. Hence, if the stress intensity fctor is correctly chosen for given component nd crck geometry then integrtion of the S-shped curve cn provide the ftigue crck growth life for components subjected to different stress levels nd different initil crck sizes. For exmple, the liner portion of the d/dn vs ΔK curve cn be described by the Pris eqution of the form [4]: d c( K) m dn = Δ (1) where c nd m re constnts. Typicl vlues of c nd m for vrious metls re given s follows: Approximte region II ftigue crck growth rte properties for the Pris eqution for vrious metls [5-10]. Mteril Slope, m Intercept, c (m/cycle) Ferritic-perlite steels x Mrtensitic steels x Austenitic stinless steels x T6 wrought luminium x A356-T6 cst luminium x Ti-6-4 mill nneled titnium x Ti mill nneled titnium x AZ91E-T6 cst luminium x Also, s we will find shortly, for simple crck geometries: Δ K =Δ σ π (2) Combining equtions (1) nd (2) results in: d dn = c Δσ π m (3) If we ssume constnt mplitude loding then Δσ is constnt nd hence: d dn = c Δσ π m (4) - 4 -

5 Integrting the bove eqution gives: end 1 m /2 N = d m strt c Δσ π (5) Hence, we cn generte curve of vs N, even if Δσ is some other vlue nd the initil crck size is different. Combined ftigue crck growth dt in the liner portion (i.e., region II) of the d/dn vs ΔK curve for wide vriety of steels with S y vrying from 250 to 2070 MP is found to fll into two min groups for ferritic-perlite nd mrtensitic steels s follows: Summry of ftigue crck growth rte dt for steels: () ferritic-perlite, nd (b) mrtensitic steels [10]. Wheres the crck growth dt vried by fctor of 2 for the ferritic-perlite dt, the mrtensitic dt vried by fctor of 5. The upper boundries of the dt represent conservtive vlues when no other dt is vilble

6 A generl schemtic sigmoidl sctter bnd for the bove dt, in ddition to ustenitic steels, is shown s follows: Superposition of Brsom s sctter bnds on the generl ftigue crck growth sctter bnds for steel [10]. For given crck size there is criticl vlue of the stress intensity fctor, K C, which, when exceeded, results in the crck propgting t very high speed, i.e., ctstrophic (finl) frcture. Alterntively we cn stte tht, for ny given stress loding, there exists criticl crck size bove which the component will fil ctstrophiclly. Note tht the stress required for ctstrophic filure in crcked component will be lower thn tht required to fil n uncrcked component, e.g., S y or S u, due to the stress concentrtion effect of the crck. Note: It will be implied lter thn in mny prcticl cses n ddition geometricl term, Y, needs to be included the eqution (2) relting the flw size to stress intensity fctor. The vlue - 6 -

7 of Y depends on the prticulr loding sitution, specimen geometry, nd crck geometry. Therefore, equtions (2) to (5) should more generlly be given by the following: Δ K = YΔ σ π (2) d dn = c YΔσ π m (3) d dn = c YΔσ π m (4) end 1 m /2 N = d m strt c YΔσ π (5) For exmple, the influence of crck width on finite width plte is shown s follows: Finite width correction for centre crcked plte [11]

8 Exmple 1 A trnsverse crck extends 6 mm inwrds from the surfce of 320 mm dimeter steel tie rod of n extrusion press. The press lod is pproximtely 2000 tonnes nd is eqully shred by four such rods. Between press cycles, which occur 1500 times per month, there is no lod on the tie rods. A frcture mechnics evlution yields shpe fctor, Y = 2.2 (ΔK = YΔσ(π) 1/2 MP m 1/2 ) nd is ssumed to be independent of crck length,. The evlution concludes tht when the crck is 75 mm deep the tie rod will be in dnger of brittle frcture. The ftigue behviour of the steel is known from experience to be: d dn ( K) = Δ m per cycle x It is desirble to keep the press operting. How long will it be possible to put off replcing the tie rod? We will use the following expression to solve this problem: d dn = c YΔσ π m The vlue of the stress difference, Δσ, will be: x10 x9.81 Δ σ = = MP 2 4 π 0.16 ( ) d dn = c YΔσ π m end 1 m /2 N = d m strt c YΔσ π Integrting with respect to gives: N 1 m /2 1 = m c Y σ π 1 m /2 Δ end strt 1 3.5/ = / x10 2.2x60.99x π - 8 -

9 Note tht, in the bove eqution, the vlue of Δσ in units of MP must be used = 267.8x = x = cycles Mximum time to filure = = 9.39 months 1500 Note tht in prctice we would obviously wnt to use suitble sfety mrgin nd not let the component be used until the point of ctstrophic filure. The elstic stress field When discussing the growth of crcks, the type of loding is very importnt. All stress systems t the tip of crck cn be derived from the following three loding modes: The three modes of loding [12]. In this discussion we will only concern ourselves with Mode I loding, such s often occurs in components subjected to tensile loding. The derivtion of the stress field t crck tip is rduous nd includes dvnced solution techniques for the Airy stress function which is beyond the scope of this course. We will simply stte the results

10 Consider liner crck in n infinite plte subject to bixil stress field s follows: σ y σ y τ xy x r θ σ x σ Schemtic representtion of liner crck in n infinite plte. The following boundry conditions must pply: (i) At y = 0, σ y = 0 - < x < (ii) (iii) At x = ±, σ y = σ At x = ± (i.e., crck tip), σ y An exmple of complex Airy stress function # which stisfies these conditions is: σ ( z) 2 φ = 1 z (6) # To be discussed in lter lectures

11 Solving for the stresses gives: σ x σ π θ θ 3θ = cos 1 sin sin 2π r (7) σ y σ π θ θ 3θ = cos 1+ sin sin 2π r (7b) τ xy σ π θ θ 3θ = sin cos cos (7c) 2π r Note tht s r 0, σ The stresses re products of the position reltive to the crck tip, i.e., f(θ)/(2πr) 1/2 nd fctor σ π which is function of the remote stress nd the crck length. Thus, σ π describes the mgnitude of the elstic stresses in the vicinity of the crck tip. This fctor is clled the mode I stress intensity fctor: K I = σ π (units: MP m 1/2 ) (8) Note tht: (i) This solution is vlid when r <<. (ii) In the vicinity of the crck tip, the totl stress field due to two or more different mode I loding systems cn be obtined by superposition of the respective stress intensity fctors. (iii) In generl, the term π does not pper nd hence the π 1/2 terms re not cncelled out. The geometry of finite sized specimens hs n effect on the crck tip stress field nd so expressions for stress intensity fctors hve to be modified by the ddition of correction fctors to enble their use in prcticl problems. Severl generl forms exist: KI = σ π f w (9) KI = cσ π f w (9b)

12 K = σy (9c) I In ech cse, the σ π term is modified to ccount for the geometry. Note tht w tkes into ccount the specimen width in, for exmple, finite width plte. The modifiction fctors, f(/w), c, nd Y hve to be determined from stress nlysis. Most re obtined from numericl solutions. An exmple of this is plte of finite width, w, subject to unixil tension with trnsverse crck of width, 2. Mny solutions exist with vrying ccurcies: σ w 2 σ K I w π = σ π tn π w (10) or KI = σ π f w where: 2 3 f = w w w w (10b)

13 or π f = sec w w ±0.3 % for 0.35 w (10c) or 1 f = w 2 1 w 2 (10d) In the prcticl ppliction of frcture mechnics solutions, stress intensity fctors re obtined from compendiums. The most commonly encountered stress intensity fctors hve been shown on the following pge. Exmple 2 A centre-crcked plte hs unixil tensile lod, P, of 500 kn. Choose the most pproprite loding condition from Figures 4(-c) nd determine the following (ssuming b = 180 mm, t = 15 mm, h b, nd h t): () Wht is the stress intensity fctor, K I, for crck length of = 15 mm? (b) Wht is the stress intensity fctor, K I, for crck length of = 90 mm? (c) Wht is the criticl crck length, c, for frcture if the mteril is 2014 T651 luminium with K IC = 24 MP m 1/2. Effect of /b rtio, α, on the constnt, F, for different loding conditions [13]

14 Expressions for F s function of /b rtio, α, for the loding conditions shown bove: () (b) α α F = 1 α 4 πα 2 πα F = cos tn 2 πα 2 h/ b 1.5 h/ b α = + (c) F 0.265( 1 α ) ( 1 α ) 3/2 () Wht is the stress intensity fctor, K I, for crck length of = 15 mm? S g P = 2bt 3 500x10 = 2x0.18x0.015 = 92.6 MP α = b = = F 1 (pproximtion from figure for the cse α 0.4) K = FS π I g = (1)(92.6 x 10 6 )(π x 0.015) 1/2 = 20.1 MP m 1/2 (b) Wht is the stress intensity fctor, K I, for crck length of = 90 mm? α = b 0.09 = 0.18 =

15 In this cse, α > 0.4 so we will use the generl eqution: α α F = 1 α h/ b 1.5 ( ) ( ) = = K = FS π I g = (1.176)(92.6 x 10 6 )(π x 0.09) 1/2 = 57.9 MP m 1/2 (c) Wht is the criticl crck length, c, for frcture if the mteril is 2014 T651 luminium with K IC = 24 MP m 1/2. In this cse, is not known, therefore, α c is not known. If we first ssume α 0.4 then we cn use K 1 K = FS π IC g c 24 x 10 6 = (1)(92.6 x 10 6 )(π x c ) 1/2 Solving for c gives: c = m This would result in: α = c b = = The originl ssumption of α 0.4 ws correct nd thus c = m is resonbly ccurte

16 Superposition principle One importnt point resulting from the bove nlysis is tht the stress field equtions (7 c) would be the sme for ll mode I cses. Thus, the stress intensity fctor for components subjected to number of lod systems, p, q, r,..., cn be obtined simply from superposition, i.e., K I = K Ip + K Iq + K Ir +... (11) The principle of superposition cn lso be used for situtions where the loding is entirely mode II or III. However, note tht this superposition principle is not vlid for the cse of loding under combintion of different modes. The superposition principle cn occsionlly be used to derive stress intensity fctors. For exmple, consider the cse of crck with internl pressure s follows: Illustrtion of the superposition principle [14]. Figure () illustrtes plte without crck nd subject to unixil tensile stress, σ. The stress intensity fctor for this sitution, K I, must be zero s there is no crck present (i.e., = 0). Imgine tht crck of length 2 is then introduced into the centre of the plte. This would be llowed if unixil compressive stress, -σ, ws pplied to the crck s shown in Figure (b). Figure (b) is thus superposition of plte with centrl crck under unixil tension, σ, nd plte with crck hving compressive stress, -σ, t its edges (Figures (d) nd (e)). It thus follows tht: K Id + K Ie = K Ib = 0 (12) or K Ie = -K Id = -σ π (12b)

17 The cse of crck with n internl pressure, p, is equivlent to Figure (e) except tht the pressure cts in direction opposite to σ. In this cse the sign of K would thus be reversed, resulting in the following: K = p π (13) I Frcture toughness Using equtions such s K I = σ π we hve been ble to relte the stress, σ, to the stress intensity fctor, K I. We cn imgine tht if σ is incresed to progressively lrger vlues then eventully mximum vlue will be reched, σ mx, fter which the specimen will fil ctstrophiclly. From exmining the previous equtions it is esy to see tht the stress intensity fctor will lso hve mximum vlue, known s the criticl stress intensity fctor, K IC, or frcture toughness, such tht: K IC = σ mx π (14) The following generl eqution cn thus be noted: K σ IC mx (15) The mximum stress, i.e., strength, for component contining crck is thus proportionl to the frcture toughness nd inversely proportionl to the squre root of the crck hlf-length. The units of K IC re MP m 1/2 with typicl vlues for vrious metls being shown below: Mechnicl properties of typicl mterils [15]

18 Vlues of K for prcticl geometries [16]

19 Vlues of K IC for specific metl lloys hve been shown in the following tble: Frcture toughness vlues for typicl mterils [17]. Note tht K IC is essentilly n indictor of how much energy is required to propgte crck within the component. Following from this, it is cler tht K IC is mteril property which mesures toughness with tough mterils generlly hving K IC vlues greter thn 20 MP m 1/2 whilst the vlue of K IC for brittle mterils my be s low s 1 MP m 1/2 for certin cermics such s mgnesium oxide (MgO). Incidentlly, rgubly the min reson behind the development of composite mterils ws to increse the frcture toughness of brittle mterils. Exmple 3 The following long rectngulr br of thickness, b = 20 mm, nd width, w = 100 mm, is mde from 4340 steel with frcture toughness, K IC, of 60 MP m 1/2, nd subjected to lod, P, of 250 kn. During mintennce, 20 mm deep edge crck is found. Assuming LEFM, is it sfe to return the br to service without repir? P w 20 mm 60 mm P The lod is not centrl nd we don t hve Y fctor for this geometry nd loding. However, both tension nd bending lods re present nd for ech of these we do hve Y fctor. Recll tht K I σ m ; this linerity llows superposition s follows: 3 σ = P 250x MP A = 0.1x0.02 = M = 250 x 10 3 x ( ) = 2.5 kn m

20 + = w σ From the list of stndrd stress intensity fctors we get for this geometry: K = Yσ (16) I where Y α α α α = (16b) nd α = (16c) w For our sitution, α = 0.2, which gives: Y = K = Yσ = x 125 x 10 6 x (0.02) 1/2 I = 43.0 MP m 1/2 w Μ The stndrd stress intensity fctors for this geometry would be: 6M KI = Y π (17) 2 bw

21 where Y ( α ) α = 1 0.7α (17b) nd α = (17c) w For our sitution, α = 0.2, which gives: Y = M K Y π bw I = 2 3 ( x ) = π x ( ) = 19.8 MP m 1/2 K I = = 62.5 MP m 1/2 repir component becuse 62.5 MP m 1/2 > K IC

22 References 1. R. I. Stephens, A. Ftemi, R. R. Stephens, nd H. O. Fuchs, p. 143 in Metl ftigue in engineering, 2nd edition (John Wiley & Sons, Inc., New York) (2001). 2. Unknown reference 3. R. I. Stephens, A. Ftemi, R. R. Stephens, nd H. O. Fuchs, p. 144 in Metl ftigue in engineering, 2nd edition (John Wiley & Sons, Inc., New York) (2001). 4. P. C. Pris, M. P. Gomez, nd W. E. Anderson, A rtionl nlyticl theory of ftigue, Trend Eng., 13(9) p. 9 (1961). 5. Dmge tolernt design hndbook, A compiltion of frcture nd crck growth dt for high strength lloys, CINDAS/Purdue University, Lfeyette (1994). 6. R. I. Stephens, in Ftigue nd frcture toughness of A356-T6 cst luminium lloy, SP-760 (Society of Automotive Engineers, Wrrendle) (1988). 7. J. Feiger, in Smll ftigue crck growth from notches in Ti-6Al-4V under constnt nd vrible mplitude loding (M. S. thesis, University of Idho) (1999). 8. H. O. Liknes nd R. R. Stephens, Effect of geometry nd lod history on ftigue crck growth in Ti , p. 175 in Ftigue crck growth thresholds, endurnce limits, nd design (ASTM STP 1372, J. C. Newmn nd R. S. Piscik, eds., ASTM) (1999). 9. D. L. Goodenberger nd R. I. Stephens, Ftigue of AZ91E-T6 cst mgnesium lloy, J. Eng. Mter. Tech., 115, p. 391 (1993). 10. J. M. Brsom, Ftigue-crck propgtion in steels of vrious yield strengths, Trns. ASME, J. Eng. Ind., Series B, Number 4, p (1971). 11. D. Broek, p. 83 in Elementry engineering frcture mechnics, 4th edition (Kluwer Acdemic Publishers, Dordrecht) (1986). 12. D. Broek, p. 8 in Elementry engineering frcture mechnics, 4th edition (Kluwer Acdemic Publishers, Dordrecht) (1986). 13. R. J. Juvinll, p.234 in Engineering considertions of stress, strin, nd strength (McGrw-Hill Book Compny, New York) (1967). 14. D. Broek, p. 86 in Elementry engineering frcture mechnics, 4th edition (Kluwer Acdemic Publishers, Dordrecht) (1986). 15. M. F. Ashby nd D. R. H. Ashby, in Engineering mterils II (Pergmon Press, Oxford, UK) (1992). 16. D. Broek, p. 85 in Elementry engineering frcture mechnics, 4th edition (Kluwer Acdemic Publishers, Dordrecht) (1986). 17. R. W. Hertzberg, in Frcture mechnics of engineering mterils (John Wiley nd Sons, New York, USA) (1995)