Fracture criterion developed for elastic-plastic Russian materials

Transcription

1 ndian Jounal of Engineeing & Mateial Science Vol. 1, Octobe 005, pp Factue citeion developed fo elatic-platic Ruian mateial A Ziliuka Depatment of Mechanic of Solid, Kauna Univeity of Technology, Ketucio t. 7, Kauna LT-05, Lithuania Received 16 July 00; accepted 7 June 005 One of the way to maximize the afety of metal tuctue and component i to ue elatic-platic mateial intead of bittle one. The ue of uch mateial educe the pobability of catatophic failue occuing fom cack like flaw that may have eulted fom the manufactuing poce o duing the opeational life. Howeve, within the field of factue mechanic, the citeion of bittle factue i moe developed than bittle-platic factue citeia. Thi pape deal with the analyi of univeal factue citeia, which ae uitable not only fo elatic mateial with the bittle factue, but alo fo elatic-platic mateial with the elatic-platic factue. An analytical two-paametical factue citeion ha been popoed fo tatic loading, which i validated expeimentally and compaed with the behaviou of pefectly platic and elatic pecific Ruian mateial. t allow the factue eitance of tuctual component made of elatic-platic mateial to be evaluated moe peciely. PC Code: G01N Duing the lat two decade, the cience of factue mechanic ha attacted geat attention within the mateial cience and tuctual mechanic communitie. Factue citeia wee developed fo the cae of tatic, a well a cyclic and dynamic loading, in ode to ae and pedict factue behaviou. nitially, the citeia of bittle-platic factue wee ceated, and the citical te intenity facto, known a factue toughne (K C, K C, K C ) wa developed to chaacteize the factue popety of a mateial. n extending factue mechanic to mateial exhibiting high elatic-platic behaviou, the J integal wa uggeted a appopiate equivalent paamete to that of the te intenity facto. Thee citeia ae alo applicable fo evaluating the inceae in the cack tip opening unde vaiable load. Howeve, in the cae of low elatic-platic behaviou the two-paametical citeia ae applicable 1-. The citeion uggeted in ome tudie 1 evaluate both the bittle-platic factue and the platic collape. Accoding to the Dugdale model, the bittle and platic factue ae decibed by the te intenity facto and the load, epectively. Nevethele, the two-paametical citeia decibed have no intene theoetical gounding and do not evidence the peculiaitie and mechanical popetie of the mateial. * antana.ziliuka@ktu.lt Two-Paametical Factue Citeion Let the length of defomation zone,, be diectly popotional to the vaiation of te intenity facto fo evey change of pe-faction zone vaiation (the tep of the cack) fo the igidity of defomed tate D = ε i0 / ε i = σ i / σ 0 ( ε i0 i the intenity of defomation tiaxial tate, ε i i the intenity of defomation linea tate, σ i i te intenity and σ 0 i intenal tee) 5,6. Thi lead to the following elationhip: dk d = D (1) K ntegating Eq. (1) give: 1/ D K = C () C i the contant of the mateial and 1/ D K = 0, KC () Futhemoe, the defomation at the tip of the cack can be analyzed by efeence to the theoy of hea line. Coodinate x of a cetain point ant the top of the cack and lope angle of hea line tangent ae elated by the dependence popoed in the eeach 7 : x= δ F( ) ()

2 5 NDAN J. ENG. MATER. SC., OCTOBER 005 F( ) i a dimenionle function, δ i the opening of the cack at it tip. Let u aume that the defomation ate at the tip of the cack i (, ) ( ) Vx x y = y= 0 V (5) x and y ae the coodinate of the tip of the cack and y i the angle between the hea line. The function V() when = i equal to zeo. Chooing the coodinate x in the un-defomed tate we can wite = F V( ) ( ) V( ) = + δ ' V F F and V ' F' (1) (, δ ) (, δ ) δf = + The following equation thu follow: (1) (1) ( ) ( ) x, δ, δ = F( ) + F' = V (6) Afte the integation of Eq. (6) and taking into account that (, δ ) = when δ =, the F following i obtained: x δ H( ) = (7) H ( ) Thu F' = F exp 0 F( ) V( ) (8) (, δ) F' δf' = + F V F V ' V F d = ln F δ F and (, ) δ δf ' F δ ln =. F V F( ) V( ) Afte integating, we obtain (15) (15a) (15b) ( ) ( ) ( ) '( ) x, δ, δ F' = δf ' = H The defomation at the tip of the cack i ( ) '( ) x F = ln = ln H Refeing to the Eq. (8) ' V F d = ln F δ F t follow that: (9) (10) (11) F δ = ln ln = ln F V F V F (15c) o δ x x = ln (16) x δ Thu 0, δ = δc 1exp (17)

3 ZLUKAS: FRACTURE CRTERON DEVELOPED FOR ELASTC-PLASTC RUSSAN MATERALS 5 Taking into account Eq. (), the following can be witten 1/ D 1 expε x,0 K δ = KC δc (18) ndicating p = 1 exp ε x,0 fo idealized platic mateial when ε x,0 0, p = 1 o idealized bittle mateial ε x,0 0, p = 0. The tain factue citeion conit of two tain component that ae eponible fo factue 8, ε the local tain at the tip of the cack, and ε the inal tain, which lead to the equation: ε ε ε max ε 0, + = 1, (19) i intepolation paamete and ε 0, i yielding tain. Conideing that the paamete coheent can be witten 8 a =, then Eq. (17) can be witten a: Dp δ ε + = 1 δ ε C 0, (0) and by ubtituting = in Eq. (0), we obtain the equation: Dp δ ε + = 1 δ ε C 0, (1) The popoed factue citeion i not vey atifactoy fom a pactical point of view becaue the meauement of cack tip opening and the defomation ae complicated 6. Thu, afte the ejection of high accuacy it can be implified in the following way: The fit tem δ δc Dp with Eq. (18) can be ubtituted by econd tem ε ε 0, of Eq. (1) in accodance K K C and the can be implified, accoding to n the equation σ = Aε (wee A i the contant of the mateial, n i the hadening index) into the / n σ equation. Heeby, Eq. (1) can be witten a σ 0, C 0, / n K σ + = 1 () gnoing mateial hadening, Eq. () can be expeed a K σ + = 1 C 0, () The enegetic Gifft factue theoy fo elatic mateial eult in a imila elationhip 9 : K σ + = 1 C 0, () Reult and Dicuion The ageement between Eq ()-() and expeimental eult wa veified fo two type of autenitic teel: 1Ch17N and 08Ch10N10T. Heewith, tandad pecimen (ASME E 8M) wee ued fo tandad compact pecimen (ASTM E ) fo failue in tenion (Fig. 1). 1.8 N max=.1 * lmax=7.5 ρ=0.mm Gmin= Fig. 1 The geomety of the compact tet pecimen 5

4 5 NDAN J. ENG. MATER. SC., OCTOBER 005 Baic cack gowth data wee obtained on tandad compact pecimen with a thickne B=5 mm and width of the ample W=55 mm (l max =7.5 mm, N max =.1 mm, G min =7.5 mm). The K C value wa computed uing the following fomula: K Table 1 Veification of failue citeion peciion fo teel 1Ch17N Citeion C FY = BW 1/ σ, МPa Eq. (), A 00 1 Eq. (), A 00 Eq. (), A 00 K, C MPa m 1/ A, i Δ A= (i=1,, ) Ai Y= 9.6( l/ W) 185.5( l/ W) ( l/ W) 1017( lw / ) ( lw / ) 1/ / 5/ 7/ / Table Veification of failue citeion peciion fo teel 08Ch10N10T Citeion Eq. (), Eq. (), Eq. (), σ, МPa A 00 1 A 00 A 00 K, C MPa m 1/ A, i Δ A= (i=1,, ) Ai F-the citical load. nvetigation wee pefomed by tenile teting machine SDMPu-10 (Gemany). Fo teel 1Ch17N it wa obtained that σ 0, =560 Mpa; n=0.7; K C =8 Mpa m 1/, while fo teel 08Ch10N10T: σ 0, =60 Mpa; n=0.65; K C =78 Mpa m 1/, epectively. Fo the eae of application the following calculation will be maked a: / n K σ K C σ 0, K + = A ; σ 1 + = A ; K C σ 0, K σ K C σ 0, + = A The eult of calculation ae peented in Table 1. The eult of calculation have poved that the bet confomity with the expeimental data i obtained uing Eq. (), which wa popoed by the autho (ee Table ). Fo teel 08Ch10N10T Eq. () give the highet dicepancy between calculation and expeimental data, while the bet confomity i obtained uing citeion (), popoed by the autho. Concluion The following concluion may be dawn fom the peent tudy: (i) The dependencie of cack opening veu cack defomation at the cack tip and the ize of pe-factue zone have been obtained on the bai of the egulaitie of hea line theoy. (ii) A factue citeion fo elatic-platic mateial ha been validated expeimentally taking into account the complex effect of te intenity facto and cack opening togethe with the effect of defomation paamete upon elatic-platic factue. (iii) The citeion fo elatic-platic mateial decibe the factue poce moe peciely compaed to the othe citeia, which do not take account of hadening (epecially platic) of the mateial. Nomenclatue a, a c = the cack length and citical cack length; δ, δ c, = the cack opening and citical cack opening; K, K C = te intenity facto and citical te intenity facto; σ = the inal te; σ 0, = the 0.% offet yield te; ε, max ε = maximum tain and inal tain; ε = the yielding tain; 0, ε C n = the citical tain; = the hadening index.

5 ZLUKAS: FRACTURE CRTERON DEVELOPED FOR ELASTC-PLASTC RUSSAN MATERALS 55 Refeence 1 Dowling A R & Townley C H A, nt J Peue Veel Piping, (1975) 77. Andeon T L, Factue Mechanic: Fundamental and Application (CRC Pe, Boca Raton), Schwalbe K H & Zebt U, EFAM ETM 97: the ETM method fo aeing cack- like defect in engineeing tuctue (GKSS Fochungzentum, Geethacht), 1998 Dugdale D S, J Mech Phy Solid, 8 (1960) Mahutov N, Defomation Factue Citeia and Calculation Stength of Stuctual Element (Machinotoenija, Mocow), 1981, (in Ruian). 6 Ziliuka A, Factue of Stuctue Element (Mokla, Vilniu), 1988, (in Ruian). 7 Rice J R & Johnon M A, in nelatic Behavio of Solid, edited by Kanninen M F, et al. (McGaw-Hill, New Yok), 1970, Panayuk V, Phyic Chemical Mechanic of Mateial, 1 (1986) 7, (in Ruian). 9 Moozov E, Stength Calculation in the Cae of Elatic Platic Factue. Poduction of Machine (Machinotoenija, Mocow), 1989, (in Ruian).