DETERMINATION OF STRESS INTENISITY FACTORS USING FINITE ELEMENT METHOD

Transcription

1 Fist Pape The oiginal pape (in Sebian) appeas in monogaph Intoduction to Factue Mechanics and Factue-Safe Design, edited by S. Sedmak, jointly published by GOŠA Institute (Smedeevska Palanka) and the Faculty of Technology and Metallugy (Belgade), pp , The monogaph contains lectues fom the Fist Intenational Factue Mechanics Summe School (IFMASS 1) held in Smedeevska Palanaka, Oiginalna vezija ovog ada objavljena je na spskom jeziku u monogafiji Uvod u mehaniku loma i konstuisanje sa sigunošću od loma, uednik S. Sedmak, zajedničko izdanje Instituta GOŠA (Smedeevska Palanka) i Tehnološko-metaluškog fakulteta (Beogad), st , Monogafija sadži pedavanja sa Pve međunaodne letnje škole mehanike loma (IFMASS 1) u Smedeevskoj Palanci, Mladen Beković DETERMINATION OF STRESS INTENISITY FACTORS USING FINITE ELEMENT METHOD Oiginal scientific pape, UDC: : INTRODUCTION Abstact The stess intensity facto is one of Factue Mechanics basic paametes. In this pape its detemination is consideed by using the finite element method (FEM). Futhemoe, effective detemination of stess intensity factos by a pack of geneal pupose pogammes in stuctual mechanics SMS, developed in the Aeonautical Institute (VTI) Žakovo, will be epesented. The most impotant poblem in factue mechanics is how to assess stess distibution nea a shap tip of a cack. Assuming linea elasticity, the stess on the tip is singula, theefoe the fundamental paametes taken into account ae stess intensity factos I, II and III, which epesent a measue of intensity of stess singulaity of thee basic foms of cack development (opening, in-plane sheaing and out-of-plane sheaing), as shown in Fig. 1. Bittle factue occus when loading and cack length coespond to citical value of the stess intensity facto. In addition, cack gowth unde effect of cyclic loading is often expessed using a simple Pais equation /3/: da C( I ) m (1) dn whee N is the numbe of loading cycles, a cack length, C and m expeimentally obtained mateial constants, and I epesents the ange within which the stess intensity facto changes unde cyclic loading, in othe wods () I Imax Imin Figue 1. Thee foms of cack development. DISPLACEMENTS AND STRESSES NEAR THE CRAC TIP Displacements and stesses aound the cack tip can be epesented using fomulae /1,5/: INTEGRITET I VE ONSTRUCIJA Vol. 4, b. (004), st Vol. 4, No (004), pp. 57 6

2 Detemination of Stess Intensity Factos Using Finite Element Method u1 1 3 θ θ I ( k 1) cos cos θ 3θ + II ( k + 3) sin + sin µ u 1 3 θ θ I ( k 1) sin sin 3 θ θ + + II ( k 3) cos + cos µ (3) u3 1 θ III sin µ π 11 1 θ 3 I cos θ θ 1 sin sin θ II sin θ 3θ + cos cos π 1 1 θ θ 3θ 3θ I cos sin cos II cos θ 3θ + 1 sin sin 13 1 θ III sin 1 θ 3 I cos θ θ 1 + sin sin θ θ 3θ + II sin cos cos 3 1 θ III cos 33 3 ν k ( 1 + ν ) 1 θ θ I cos II sin ν In the expessions above, u 1, u and u 3 ae the displacement components in diections x 1, x, x 3 paallel to the nomal, binomal and tangent of the cack tip, in x 1, x plane with pola coodinates, θ elative to the penetation point of the cack tip though the plane; µ epesents the shea modulus, ν is Poisson s atio, k 3 4ν, except in case of plane stess, whee k (3 ν)/(1 + ν). Linea elastic factue mechanics can be applied only when the plastic stain, which always occus since the stess singulaity 0, is limited to the egion nea the cack tip which is much smalle than othe elevant dimensions /1, /. One of these applications, typical fo small plastic zone, is the bittle factue of high-stength mateials duing cack gowth in small amplitude egime and at a lage numbe of loading cycles. Stess intensity factos ae also elated to stain enegy U by expessions [], [4], [6] and [7]: du k G ( I + II ) + III (5) da 8µ µ and in case of plane stess: du k + 1 G ( I + II + III ) (6) da 8µ whee G is the measue of eleased stain enegy duing cack gowth; the element of the cack aea is da hda (7) with h thickness, and a cack width. In case of plane stain, h 1. Anothe possibility fo calculating the measue of eleased stain enegy, in case of a plane poblem, is based on a path-independent J integal /6,1/, ij ui J Wdx n j ds 1 Γ x Stain enegy pe unit volume is epesented as (8) 1 ij W e ij (9) Taking into account the symmety of stess tensos ij, one can wite: 1 ij W u ij (10) whee stain tenso e ij is eplaced by displacement gadient ui uij j (11) x In the plane poblem, nomal vecto coodinates ae: 1 dx dx n1, n (1) ds ds In a special case of genealized plane stess, stess tenso coodinates ae: ij µ νδ ij δ kl + ( 1 ν ) δ ik δ jl u ( kl, ) (13) 1 ν By placing [10], [1] and [13] in [8] the following expession is finally obtained: dx + 1 µ J (14) 1 ν Γ ( 1 ν) u11, u1, + ( 1+ ν) u11, u1, dx + u1, u, ( 1 ν )( u, u, )( u, u, ) + ( u, + u11, )( u, u11, ) USING FINITE ELEMENT METHOD The finite element method, as well as factue mechanics epesent an aea that has developed apidly. Theefoe, a lage numbe of possibilities fo calculating factue mechanics paametes by finite element method ae available. Patial eview of these possibilities can be found in /4,6,8 and 9/. (4) INTEGRITET I VE ONSTRUCIJA Vol. 4, b. (004), st Vol. 4, No (004), pp. 57 6

3 Detemination of Stess Intensity Factos Using Finite Element Method The possibilities of detemining factue mechanics paametes by standad softwae, available to eveyone who uses the finite element method, will be consideed. As it is well known, the application of this method equies foming of a coesponding mesh that divides the obseved body into a numbe of unifom elements. In a case whee standad pogammes of the finite element method ae used, it is necessay to efine significantly the mesh nea the cack tip. If the coesponding pogamme, such as the SMS package has the pentagonal plana element option /10,11/, this kind of efinement is quite easy to pefom. The application of this method to an obseved poblem esults in mesh displacement, which leads to detemination of stesses. Once these quantities ae known, one can detemine the coesponding factue mechanics paametes by using Eqs. [3], [4], [5], [6] and [14]. BASED ON GIVEN DISPLACEMENT VALUES It is obvious consideing Eqs. [3] that they can be solved in egad to I, II, and III fo known coodinates and θ and values u 1, u and u 3. Fo pactical eason, it will be adopted that θ π, meaning that the displacements on the cack suface ae consideed, so fom Eqs. [3] it follows: I µ u k + 1 µ u1 II (15) k + 1 π III µ u3 Fo effective detemination of these values, it is usual to calculate seveal of them along the line θ const, and then detemine the valid one by extapolation to 0. This pocedue, simple as it is, still equies time and effot fo gaphic elaboation of esults. BASED ON GIVEN STRESS VALUES This pocedue is analogous to the pevious one, with the use of Eqs. [4]. Again, we will detemine I, II, III assuming θ 0: I II 1 3 (16) III This pocedue is usually less accuate than the above, because the stess is less accuate compaed to the displacement when using FEM. BY USING THE MEASURE OF RELEASED STRAIN ENERGY DURING CRAC GROWTH When using this appoach, one can explicitly detemine the stess intensity factos only in case of a pue fom of defomation (Fig. 1). Fo example, if II III 0, it follows fom [5] and [6]: I 8µ G k + 1 (17) Released stain enegy, G, can be appoximately calculated using two consecutive FEMs fo two close cack lengths which diffe by an incement a. Fom [5] and [6] follows: U G (18) h a whee U U U 1 (19) epesents the diffeence due to two close cack lengths, exposed to the same extenal loading. As fo stain enegy, while solving the static poblems with the theoy of linea elasticity using FEM, it equals: 1 T U R u (0) whee R is an n dimensional vecto designating the extenal foces affecting the stuctue, and u is the vecto of thei coesponding displacements. The disadvantage of this pocedue is that it equies two analyses, o in pefoming cetain coections in the pogamme that ae not simple /7/. USING RICE S J INTEGRAL It can be shown that J-integal equals G, theefoe, fom Eq. [17], one can wite: 8µ I J (1) k + 1 As fo the effective detemination of the J-integal, the autho evaluates Eq. [14], developed in this pape, as vey convenient. On pats of the integation path that ae paallel to the x 1 axis, dx equals 0, and vice-vesa, which educes the amount of calculations equied in [14], e.g. in case of ectangula path (Fig. ). Figue. Effective calculation of J-integal fo ectangula paths. INTEGRITET I VE ONSTRUCIJA Vol. 4, b. (004), st Vol. 4, No (004), pp. 57 6

4 Detemination of Stess Intensity Factos Using Finite Element Method Deivatives in Eq. [14] can be detemined fom known displacement fields in an element o, if these ae not known, by using the finite element method. The deivative of displacement in the tangential diection can be calculated fo the assumed linea displacement field, fom: u u u J () t t t J J and in the nomal diection they ae an aveage value of: u 1 u um uj ul + n n nm nj nl (3) In these expessions, t and n ae coodinates of point in Catesian coodinate system NT. Also, in Eq. [14] dx 1 is eplaced by x 1 x 1 J and dx by x x J. Finally, the integal in [14] is eplaced with a sum of segments of the obseved contou: j< k N 1, N µ 1, 1, 1, 1,, 11,, 11, J 1 ν J, 1, 1 ν 1 ν ( 1 ν )( u + u )( u u ) + ( u + u ) ( u u ) ( x x ) 1 1 ( ) u11, u, + ( + ) u11, u1, + u1, u, ( x xj ) + (4) EXAMPLE A squae plate with a cental symmetical cack has been consideed as an example. The used mesh is shown in Fig. 3. It is automatically geneated by a special pe-pocesso fo pogamme package SMS. The same figue shows vey good ageement of stess esults with some known solutions. Futhe, Fig. 4 shows the lines of local stess values nea the cack, calculated as Mises stesses: whee 1 and epesent pinciple stesses. Figue 3. Squae plate with a cental symmetical cack. Figue 5 shows a defomed configuation of the consideed plate, in an exaggeated scale. Gaphs in Figs., 4 and 6 ae dawn using a standad postpocesso fo pogamme package SMS, on a GERBER plotte. INTEGRITET I VE ONSTRUCIJA Vol. 4, b. (004), st Vol. 4, No (004), pp. 57 6

5 Detemination of Stess Intensity Factos Using Finite Element Method Figue 4. Local iso-stess lines. Refeence /7/ Figue 5. Defomed stuctue. Table 1. Stess intensity factos I (πa) 1/ F(a/b). Stess intensity facto Figue 6. Gaphic detemination of stess intensity factos based on displacement I (u ) and stess I ( y ). Figue 6 shows gaphic detemination of stess intensity factos, based on displacements, shown in Fig. 5, and based on stesses shown in Fig. 3, calculated using standad SMS. Some advantage should be given to the J-integal method, because it equies a simple post-pocesso, based on Eq. [4] and on the diffeent scheme peviously descibed. Hence, the esults ae obtained automatically, and ae a bit geate, theefoe safe, compaed to othe pocedues. In Table 1 the esults of stess intensity facto detemination using vaious pocedues ae pesented. Appaently, with developed mesh, all pocedues ae sufficiently accuate. Coection facto Coection facto Cack length to plate width atio Plate length to width atio I F(a/b) F(a/b,L) a/b L/b Infinite plate Infinite band (ISIDA) Infinite band (PARIS+SIH) Finite plate (HELLEN) This pape Displacements Stesses G measue J (I) J (II) J (III) J (IV) J (V) J (VI) J (VII) J (VIII) INTEGRITET I VE ONSTRUCIJA Vol. 4, b. (004), st Vol. 4, No (004), pp. 57 6

6 Detemination of Stess Intensity Factos Using Finite Element Method REFERENCES 1. S. Sedmak: The effect of notches and cacks on factue occuence at elastic and plastic defomation, PhD Thesis (Uticaj zaeza i pslina na pojavu loma pi elastičnoj i plastičnoj defomaciji, Doktoska disetacija), Faculty of Mechanical Engineeing, Univesity of Belgade, M.P. aplan, J.A. Reiman: Use of factue mechanics in estimating stuctual life and inspection intevals, J. of Aicaft, Vol. 13, No., B. Aamodt, F. lem: Application of numeical techniques in pactical factue mechanics in Engineeing Pactice, Applied Science Publishes, Baking, Essex, England. 4. T.H.H. Pain: Cack elements, Poceedings Wold Con. FEM Stuct. Mech., Robinson and Ass., Woodlands, Wimbone, Doset, England, W.S. Blackbun: Calculation of stess intensity factos at cack tips using special finite elements, The Mathematics of Finite Elements and Applications, Academic Pess, S.E. Bengley, D.M. Paks: Factue Mechanics, Stuctual Mechanics Compute Pogams, Univesity Pess of Viginia, T.. Hellen: On the method of vitual cack extension, Int. J. of Num. Methods in Engng, Vol. 9, , R.H. Gallaghe: A eview of finite element techniques in factue mechanics, MARC Euope Semina, Potoož, G. Batelds, A.U. de oning: Finite element analysis of cack gowth, II Wold Congess of Finite Element Methods in Stuctual Mechanics, Robinson and Ass., Woodlands, Wimbone, Doset, England, M. Beković: Hybid finite elements in plane poblem of theoy of elasticity, (Hibidni konačni elementi u avnom poblemu teoije elastičnosti, XII Jugoslovenski konges acionalne i pimenjene mehanike, Ohid J.R. Whiteman, J.E. Akin: Finite elements, singulaities and factue, The Mathematics of Finite Elements and Applications III, Academic Pess, J. Jaić: Consevation laws of the J-Integal type in micopola elastostatics, Int. J. Engng Sci. Vol. 16, pp , INTEGRITET I VE ONSTRUCIJA Vol. 4, b. (004), st Vol. 4, No (004), pp. 57 6