UDK :519.61/.B4:539.3 Analiza napetostnega stanja v tritočkovnem upoglbncm preizkušancu z enostransko razpoko Stress Analysis of a Three Point B

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1 UDK :519.61/.B4:539.3 Analiza napetostnega stanja v tritočkovnem upoglbncm preizkušancu z enostransko razpoko Stress Analysis of a Three Point Bend Specimen With Single Ended Crack ZORAN ŽUNIČ - JANKO LEGAT Analizirali smo napetostno stanje v okolici vrha razpoke in primerjali rezultate z eksperimentalnimi. Kot primerjalne vrednosti smo izbrali faktor intenzivnosti napetosti, odprtje ustja razpoke, odprtje vrha razpoke in pomik obremenitvene črte. Uporabili smo program BEASY, ki, med drugim, omogoča tudi elastično analizo ravninskih teles z razpokami. Analysis of stress distribution near the crack tip was performed. Results obtained are compared with experimental data. The following factors have been compared: stress intensity factor, crack mouth opening displacement, crack tip opening displacement and load line displacement. Elastic analysis of cracked bodies was performed with the program BEASY. 0 UVOD Zanimalo nas je, kako se obnaša tritočkovni upogibni preizkušanec z razpoko, če povečujemo silo do trenutka, ko se začne razpoka širiti. Naš namen je bil ugotoviti kriterij večanja razpoke. Dobljene vrednosti smo primerjali z eksperimentalnimi, in sicer za preizkušanec z dvema različno dolgima razpokama. Za preračun smo uporabili program BEASY, ki temelji na metodi robnih elementov, za modeliranje razpoke pa uporablja metodo dvojnih robnih elementov. 0 INTRODUCTION A three point bend specimen with single end crack behaviour under a load which is increased until crack grow occurs was considered. The goal of the research was to discover the crack growth criterion. The results were compared with experimental data for examples with two different crack lengths. A commercial program, BEASY, based on the boundary element method and using the dual boundary element method for crack modelling was applied. 1 TEORETIČNE OSNOVE 1.1 Osnove mehanike loma Obravnavali bomo primer neskončne stene z debelino B, obremenjene z natezno napetostjo o, v kateri imamo razpoko dolžine 2a (sl. 1). Napetostno stanje v delčku stene, oddaljenem od vrha razpoke za r in pod kotom 0, je določeno z izrazoma: in a i j 1 THEORETICAL BASIS 1.1 Fracture mechanics Consider a crack length 2a in an infinite plate of thickness B. The plate is subjected to a tensile stress ö (Fig. 1). The elastic stress field in an element of the plate at a distance r from the crack tip and at an angle 0 with respect to the crack plane is: K i y/2irr and K j = cry/g a ' t it t t! t t f t /«( ) (l) Sl. 1. Razpoka v neskončni steni Fig. 1. Crack in an infinite plate

2 /L, imenujemo faktor intenzivnosti napetosti. Napetostno stanje okoli razpoke je natančno določeno takoj, ko poznamo faktor intenzivnosti napetosti. Okoli konice razpoke dobimo zaradi singularnosti napetosti zmeraj določeno plastično cono (sl. 2). Oblika in velikost plastične cone sta odvisni od napetostnega stanja ob konici razpoke, napetostno stanje pa je odvisno od razmerja med velikostjo plastične cone in debelino stene. S preizkusi je bilo ugotovljeno, da se pojavi ravninsko deformacijsko stanje, če je to razmerje reda 1/40. The factor /L, is known as the stress intensity factor. The whole stress field at the crack tip is known when the stress intensity factor is determined. Because of the stress singularity at the crack tip, there is always a certain region in which plastic deformations occur (Fig. 2). The state of stress influences the size of the plastic zone. On the other hand, the ratio of the plastic zone size to thickness is an important factor for the state of stress. Experimentally, it has been determined that cracking behaviour is typical of plane strain if this ratio is the the order of 1/40. Si. 2. Oblika plastične cone Fig. 2. The shape of the plastic zone Plastična cona pri ravninskem deformacijskem stanju je manjša od plastične cone pri ravninskem napetostnem stanju zaradi tega, ker lahko material pri ravninskem deformacijskem stanju prenese večjo napetost od meje tečenja (c?ys), določene z nateznim preizkusom (sl. 3) The plane strain plastic zone is significantly smaller then the plane stress plastic zone. This is a result of the fact that the effective yield stress in plane strain is larger than yield stress (<?ys), obtained with tension test (Fig. 3). \ RNS r SI. 3. Porazdelitev napetosti okoli konice razpoke Fig. 3. Stress distribution in plane stress and plane strain Razmerje med največjo napetostjo v plastični coni in mejo tečenja imenujemo faktor vklenitve materiala: The ratio of the maximum stress to the yield stress is called the plastic constraint factor: p c f = ~!L cr, ys Teoretično, pri čistem ravninskem deformacijskem stanju in z v = 1/3, je vrednost faktorja vklenitve materiala pcf = 3, v praksi pa je zaradi ravninskega napetostnega stanja na površini ta vrednost manjša in znaša po Irwinu, pcf = V2 72 = = 1, Metoda dvojnih robnih elementov Modeliranje razpoke je pri metodi robnih elementov problem, saj dobimo zaradi ujemajočih se površin razpoke singularen sistem linearnih (3). Theoretically, in the plane strain and by taking v = 1/3 the plastic constraint factor is pcf = 3. In a practical case, plane strain does not exist at the specimen surface and the average plastic constraint factor is much lower. Irwin uses a pcf= -/2 72 = Dual boundary elem ent method The solution of general crack problems cannot be achieved with the direct application of the boundary element method (BEM) because the coincidence of the crack surfaces gives rise to a singular system of linear equations. The equations

3 enačb. Enačba za točko, ki leži na eni strani razpoke, je enaka enačbi za točko, ki leži na nasprotni strani razpoke, saj uporabljamo isto integralsko enačbo, koordinate obeh točk pa so enake. Metoda dvojnih robnih elementov vključuje dve neodvisni robni integralski enačbi. Prvo za en rob razpoke in drugo za drugi rob ter preostanek območja. Čeprav je integracijska pot še vedno ista, z ujemajočimi se izvornimi točkami, matrika ni singularna, saj uporabljamo različni enačbi. Enačbi, na katerih temelji metoda dvojnih robnih elementov, sta robni integralski enačbi za pomike in napetosti. Če zanemarimo prostorninske sile in predpostavimo zveznost pomikov v robni točki x \ lahko zapišemo robno integralsko enačbo v naslednji obliki: for a point located at one of the surfaces of the crack are identical to those equations for the point on the opposite surface, with the same coordinates, because the same integral equation is applied with the same integration path, at both coincident points. The dual boundary element method (DBEM) incorporates two independent boundary integral equations, firston one of the crack surfaces and the second on the other and the remainder of the boundary. Although the integration path is still the same for coincident points on the crack surfaces, the respective boundary integral equations are now distinct. The dual equations on which the D BEM is based are the displacement and the traction boundary integral equations. In the absence of body forces and assuming continuity of the displacements at a boundary point x\ the boundary integral equation is given by: Cij(y:')uj(x) + j- Tij (x*, x)uj (x)dr(x) = J f7y(x', x)i,(x)dt (x) (4), kjer i in J označujeta kartezične koordinate, 7Jj (x \ x) in (./[j (x \ x) pomenita Kelvinov! osnovni rešitvi za pomike in sile v robni točki x. Razdaljo med točkama x in x označujemo z r. Integrala v enačbi sta regularna za r / 0. Ko se r približuje 0, postaneta osnovni rešitvi singularni; močno singularna reda 1/r v 7^ in šibko singularna reda ln(l/r) v Uiy Simbol / v enačbi označuje integral v smislu Cauchyjeve osnovne vrednosti, koeficient c^tx ) pa ima vrednost 5Xi/2 za gladek rob v točki x. Če zanemarimo prostorninske sile in predpostavimo zveznost tako pomikov kakor sil v točki x gladkega roba, se enačba za določitev napetosti glasi: where / and j denote Cartesian components; 7^ (.v, x) and Uxi (x \ x) represent the Kelvin traction and displacement fundamental solutions, respectively, at a boundary point x. The distance between the points X' and x is denoted by r. The integrals in the equation are regular, provided r 0. As the distance r tends to 0, the fundamental solutions exhibit singularities; a strong singularity of the order 1/r in Ty and a weak singularity of the order In (1/r) in The symbol f stands for the Cauchy principal-value integral, and the coefficient Cytx ) is given by Jy/2 for a smooth boundary at the point x \ In the absence of body forces and asuming continuity of both strains and tractions at x' on a smooth boundary, the stress components are given by: ia y (x ')+ ^ S 'y fc (x ',x )u fc(x)(ir(x) = D ijki'x1 ) x)tfc(^)dr(x) (5), V tej enačbi <S,jk(x \ x) in ZJ,jk (x, x) vsebujeta odvode 7',j(x, x) in f/y tx, x). Integrala v enačbi (5) sta regularna za r / 0. Ko gre r proti 0, postane funkcija ^ijk hipersingularna, reda 1 /r2 in Dljk močno singularna, reda 1/r. Simbol / označuje integral v smislu Hadamardove osnovne vrednosti. Na gladkem robu so komponente sile tj podane z: In this equation, S,jk(x \ x) and D1Jk(x \ x) contain derivatives of TxA x \ x) and C/jj (x \ x), respectively. The integrals in eq. (5) are regular, provided r / 0. As the distance r tends to 0, 5,jk exhibits a hypersingularity of the order 1 /r, while Z)ljk exhibits a strong singularity of the order 1/r. The symbol /stan d s for the Hadamard principal-value integral. On a smooth boundary, the traction components, fj, are given by: - i j ( x ' ) + n j ( x ' ) S i j k ( x ', x.) u k ( x ) đ r ( x ) = n, - ( x ' ) / > y fe( x ', x ) f * ( x ) d r ( x ) (6 ) kjer /7j označuje i-to komponento normale na rob območja v točki x \ Enačbi (4) in (6) sta osnova metode dvojnih robnih elementov Modeliranje razpoke Osnove modeliranja razpoke bi lahko strnili v naslednje: rob razpoke diskretiziramo z nezveznimi kvadratnimi elementi (sl. 4); where n, represents i-th component of the contour outward normal in x'. Equations (4) and (6) are governing equations of the DBEM Crack modelling The general modelling strategy can be summarized as follows: the crack boundaries are modelled with discontinuous quadratic elements (Fig. 4);

4 vozliščne točke element node mrežne točke mesh points enačba pomikov displacement equation enačba sil traction equation ostanek območja diskretiziramo z zveznimi kvadratnimi elementi, z izjemo ustja razpoke, kjer uporabimo nezvezna ali polzvezna elementa, da se izognemo skupnemu vozlišču; na prvem robu razpoke uporabimo integralsko enačbo pomikov (4); na drugem robu razpoke uporabimo integralsko enačbo sil (6); po preostalem robu območja uporabimo integralsko enačbo pomikov. Sl. 4. Modeliranje razpoke Fig. 4. Crack modelling continuous quadratic elements are used along the remaining boundary of the structure, except to the crack mouth, where discontinuous or semi-continuous elements are used in order to avoid a common node; the displacement equation (4) is applied for collocation on one of the crack surfaces; the traction equation (6) is applied for collocation on the other crack surface; the displacement equation (4) is applied for collocation on all non-crack boundaries. 1.3 Izračun faktorja Intenzivnosti napetost! K j Eksperimentalno določanje Kj 1.3 Calculation of stress Intensity factor K j Experimental evaluation of Kj Neposredno merjenje vrednosti faktorja intenzivnosti napetosti ni mogoče, zato uporabljamo različne posredne metode. Najbolj preprosto je posredno določanje prek izmerjenih pomikov. Izkustvena enačba odprtja ustja razpoke tritočkovnega upogibnega preizkušanca 131 se glasi: CM OD = pri čemer je razdalja S med podporama v mm, sila P v N, dolžina razpoke a v mm, elastični modul E v MPa, debelina preizkušanca t v mm, višina preizkušanca W v mm in geometrijski faktor V, ki je funkcija dolžine razpoke in višine preizkušanca: 7 = ^ ^ Izkustvena enačba za faktor intenzivnosti napetosti se glasi: kjer je F geometrijski faktor: 1.99 F Iz enačbe (7) in (8) dobimo enačbo za CMOD v odvisnosti od Kp. Direct measurement of the value of the stress intensity factor is not possible, so we use indirect methods. The simplest way is evalution from measured displacements. Empirical expression of crack mouth opening displacements for a three point bend specimen 131 is given by: 6SPa E tw 2 V (7), 3 SP K j = --prpjvtta F 2 tw 2 where S denotes the distance between bears in mm, P is load in N, a crack length in mm, E Youngs modulus in MPa, t specimen thickness in mm, W specimen height in mm and V geometrical factor which is a function of the crack length and specimen height: a3 a4 a5-5-96^ ^ WF5 Empirical expression for stress intensity factor is given by: where P is a geometrical factor: 2 ) f (1 ~ w )( w + 2-7^ ^ (1 + 2^ )(1" F )3/2 (8), From (7) and (8) an expression for CMOD as a function of Kj is obtained:

5 CM OD = AKj a Ey/irä V F Obrazec se glasi: K i = C M O D -E y ß ä F Aa ' V The formula for Kj is: Integral J J Integral Za numerični izračun faktorja intenzivnosti napetosti smo uporabili integral J. Rice 121 je definiral od integracijskih poti neodvisen integral J kot: kjer je S poljubna krivulja, ki obdaja konico razpoke, W je gostota deformacijske energije, tj je komponenta sile, podana z izrazom tj = d, =7? in n, je komponenta normale na rob krivulje. Gostoto deformacijske energije W izračunamo iz enačbe: J = [ [W n\ tjuj,i)ds JS (9). For a numerical evaluation of the stress intensity factor the J integral has been used. Rice 121 has defined a path independent J integral as: (10), where S is an arbitrary contour surrounding the crack tip, W is the strain energy density, <j are the traction components, given by fj = cr,j /7, and n, are the components of the unit outward normal to the contour path. Strain energy density W is: W r ii i] d ij = L Ui- Zveza med integralom J in faktorjem inten- The relationship between the J integral and zivnosti napetosti je: the stress intensity factor is given by: kjer je E ' = i? pri ravninskem napetostnem stanju in E ~ E /i 1 - v 2) pri ravninskem deformacijskem stanju. Z razdelitvijo napetostnega tenzorja na simetrični in antisimetrični del lahko dobimo ločena izraza za faktorja intenzivnosti napetosti za primer odpiranja M = I in M = II: iz česar izhaja, da je: J M J 1 E Uporaba tega postopka v metodi robnih elementov je preprosta. Zaradi preprostejše integracije izberemo za integracijsko krivuljo krožnico s središčem v konici razpoke. Po krožnici razporedimo notranje točke, ki morajo biti v dvojicah simetrične glede na os x] (sl. 5). Začetna in K f + K% J = E (11), where E = E for a plane stress condition and E'= E/{\ - v 2) for a plane strain condition. With decomposition of the elastic field into symmetric and antisymmetric mode components, it is possible to obtain different expressions of the stress intensity factors for mode M = 1 and mode A/ = II: = [ (w Mm tf< x ) d S (12), Js and the following relationships hold: K l J n = n E' The implementation of this procedure into the boundary element method is straightforward. A circular contour path, around the crack tip, is defined with a set of internal points, located at symmetrical positions relative to the at, direction, as shown in fig. 5. At these points, it is always SI. 5. Integracijska pot za integral J Fig. 5. Contour path for J integral

6 končna točka krožnice ležita na robu razpoke. V teh dveh točkah ima normala na integracijsko krivuljo smer u= (- 1, 0), iz česar Izhaja, da je t2 = 0 za razpoko brez napetosti po površini. Za integracijo po krožnici uporabimo trapezno formulo Irwinova korekcija plastične cone V skladu z linearno elastično mehaniko loma dobimo v materialu ob konici razpoke neskončno napetost oy. Ker pa iz prakse vemo, da se materiali lahko elastično deformiranjo samo do določene meje, nad njo pa se deformirajo plastično, pomeni, da imamo okoli konice razpoke zmeraj določeno področje s plastičnimi deformacijami (sl. 6). Irwin je dokazal, da se razpoka z določeno plastično cono obnaša enako kakor elastična razpoka večje dolžine. Podaljšek razpoke je ocenil z: p verified that n = (- 1, 0) and thus, for a traction free crack t2 = 0. The integration along the contour can be accomplished with the trapezoidal rule The Irwin plastic zone correction According to linear elastic fracture mechanics, a stress singularity exists at the tip of a crack. In practice, materials tend to exhibit a yield stress above which they deform plastically. This means that there is always a region around the tip of a crack, where plastic deformations appear (Fig. 6). Irwin has argued that a crack with a certain plastic zone behaves the same as a longer elastic crack. The extent is calculated from: K Ì 2n(pcf ays)2 kjer pomenijo: rp polmer plastične cone v mm, Kj faktor intenzivnosti napetosti v MPa -/mm, pcf faktor vklenitve materiala in c?ys napetost, pri kateri začne material teči, v MPa. Če razpoko podaljšamo, se spremeni tudi velikost faktorja intenzivnosti napetosti: kjer so K * popravljeni faktor intenzivnosti napetosti v MPa T mm, o referenčna napetost na robu v neskončnosti v MPa, rp polmer popravljene plastične cone v mm in a začetna dolžina razpoke v mm. Vrednost za K* dobimo z iterativnim postopkom iz enačbe: Prava velikost plastične cone je: Fig. 6. Irwin plastic zone correction K i - \ 1 + Tp Odprtje vrha razpoke lahko definiramo: where rp is the radius of the plastic zone in mm, Kj is the stress intensity factor in MPa Vmm. pcf is the plastic constraint factor and öys is the yield stress in MPa. If the plastic zone correction is applied consistently, a correction to stress intensity factor is also necessary: r ~ T a + 2tt(pc/ ays)2 where K * is the corrected stress intensity factor in MPa Vmm, a is the reference stress at infinity in MPa, rp is the corrected radius of the plastic zone in mm and a the initial crack length in mm. The value of K* is detemined by following an iteration procedure from equation: K*2 2ira(pcf -ays)2 - K ] = 0 (13). The corrected plastic zone size is now: K*2 2 tv(pcf ays)2 Crack tip opening displacement is given by: CTOD = yj(a + r;)2 - a 2

7 in če upoštevamo, da je za ravninsko deformacijsko stanje E = 7(1- v *2), se enačba odprtja vrha razpoke glasi: 4iT/(l - CTOD v2) Eyfixä and E for a plane strain state is E ' = E /( 1 - v 2), which leads to: i 2 ar* + r*2 (14). Podrobnejše Izpeljave teh formul lahko najdemo v literaturi Dugdalova korekcija plastične cone Dugdalov model tudi temelji na navideznem podaljšanju razpoke. Singularnost napetosti v konici razpoke izničimo z zaporno napetostjo, ki deluje na podaljšku razpoke d in je enaka meji tečenja materiala (sl. 7). Velikost podaljška izberemo tako, da dobimo v konici navidezne razpoke vrednost faktorja intenzivnosti napetosti nič. To pomeni, da faktor intenzivnosti Ka, ki ga povzroči napetost d, izničimo s faktorjem intenzivnosti napetosti Kd ki ga povzroči zaporna napetost dys 141. Detailed information about deriving the above expressions can be found in [ The Dugdale plastic zone correction Dugdale also considers an effective crack which is longer than the pysical crack. Stress singularity at the tip of the crack is compensated by the compressive stress equal to yield stress, acting along the distance cf (Fig. 7). The size of d is chosen such that the stress intensity factor in the crack tip vanishes. This means that the stress intensity Ka, due to the uniform stress d has to be compensated by the stress intensity Kd due to the wedge forces dys [41. Velikost podaljška razpoke d je enaka: d = a SI. 7. Dugdalov postopek Fig. 7. Dugdale approach The length d is found as: a (p cf crys) Če v tej enačbi upoštevamo še (2) in (3), do- Including (2) and (3) in the above equation bimo: it follows: 7r K / d = a sec (,p c f -dys)y/nä/ Vrednost faktorja intenzivnosti napetosti The stress intensity factor is obtained from dobimo iz enačbe: the equation: Po preoblikovanju dobimo: Odprtje vrha razpoke je: K} = (PC/ dys) \ 8 7r a m sec ( 7r L V 2 d (p cf dys) _ K j2 ~ ~ W After transforming it follows: K, Jys) Crack opening displacement is: (15). CTOD = I. 7T E a In K i N 2 (pcf dys)y/tm Upoštevamo še, da za ravninsko deformacijsko stanje velja E '= E/( 1- v 2) in dobimo izraz: to: For plane strain E ' = E/(\ - v 2), which leads

8 C T O D = - 7T (Pcf 'Oys){ 1 ~ ^ 2) E a ln (16). 2 IZRAČUN 2 EVALUATION 3.1 G eom etrijska oblika testn ih prim erov Analizirali smo dva preizkušanca, enega s kratko in drugega z dolgo razpoko. Dimenzije so podane na sliki 8. V prvem primeru je dolžina razpoke a = 3,264 mm in v drugem primeru je a = 19,37 mm. 2.1 Specimen geom etry Two specimens were analysed, one with a short and the other with a long crack. Dimensions are depicted on Fig. 8. In the first case, crack length a = mm and the second case a = mm. V prvem koraku smo naredili linearno elastično napetostno analizo obeh primerov z razpoko. Dobili smo vrednosti faktorja intenzivnosti napetosti, odpiranja ustja razpoke in pomika obremenitvene črte. V drugem koraku smo dobljene vrednosti popravili z uvedbo plastične cone. Uporabili smo Irwinov in Dugdalov korekcijski faktor. Elastični preračun smo izvedli s programom BEASY, ki rešuje integralske enačbe po metodi robnih elementov. Preizkušanca sta bila iz mikrolegiranega jekla s povečano trdnostjo Nionicral 70 A, z naslednjimi lastnostmi: E = 198 GPa... modul elastičnosti, Rpo,; = 718 MPa... meja plastičnosti, /?m = 778 MPa... natezna trdnost, e0 2 = 0,563 %... skupni raztezek pri /?p0i2, eu = 7,5 %... raztezek pri največji napetosti. Za zgornji material uporabimo bilinearno poenostavitev in dobimo naslednje materialne podatke: E = 198 GPa... modul elastičnosti, " S p i = 1848 MPa... tangentni modul plastičnosti, óys = 705 MPa... meja tečenja, ou = 831 MPa... porušna natezna trdnost. In the first step, linear elastic stress analysis of both specimens was performed. The obtained values were: stress intensity factor, crack mouth opening displacement, crack tip opening displacement and load line displacement. In the second step, the obtained values were corrected by the introduction of the plastic zone applying the Irwin and Dugdale correction factors. Elastic analysis was done with the BEASY program for solving integral equations by the boundary element method. Speciments material was high strength low alloy steel Nionicral 70 A, with the following properties: E = 198 GPa... elastic modulus, Rp0,2 = 718 MPa... yeald stress, Rm = 778 MPa... ultimate strength, e0 2 = %... total strain by Rp0 2> eu = 7.5 %... ultimate strain. Bilinear simplification gives for the above material: E = 198 GPa... elastic modulus, Epi = 1848 MPa... tangent plastic modulus, öys = 705 MPa... yield stress, ou = 831 MPa... ultimate tensile stress.

9 2.2 Diskretni model 2.2 D iscrete model Na slikah 9 in 10 imamo diskretizirana modela za preračun. Uporabili smo parabolične zvezne elemente, razen na robu razpoke, kjer smo uporabili parabolične nezvezne elemente. Za boljšo predstavitev rezultatov v območju smo uporabili notranje točke. Figures 9 and 10 show discretized models analysis. Parabolic continuous elements were used, except on the crack surface, where parabolic discontinuous elements were used. For better representation of the results in the domain, internal points were applied. SI. 9. Diskretiziran model, a = 3,264 mm Fig. 9. Discretized model, a =3.264 mm SI. 10. Diskretiziran model, a = 19,37 mm Fig. 10. Disceti zed model, a = mm Točkovno silo smo postavili v sredinsko vozlišče elementa, kar da razporeditev napetosti po funkciji cry = (1 - <f2) efmax (sl. 11), pri čemer je dmax = 3/2 PBle\ f je lokalna spremenljivka in pomeni pot po elementu od -1 do +1, P je sila v N, B je debelina v mm in le dolžina elementa v mm. The point load was set in the middle node of the element, wich gives stress distribution oy = = (1 - f 2) 7max (Fig. 11), where is <7max = 3/2 PBle- I is an intrinsic variable representing path over element from -1 to +1, P is load in N, B is thickness in mm and le is element length in mm. SI. 11. Obremenitev modela Fig. 11. Model load Obe podpori smo enako kakor točkovno obremenitev postavili v sredinski vozlišči ustreznih elementov. Za analizo smo uporabili največjo silo, preden se začne razpoka večati. Vrednosti smo dobili iz eksperimentalnih podatkov. V prvem primeru, s kratko razpoko, je sila P = 100 kn in v drugem primeru P = 27 kn. Model s kratko razpoko je imel 115 zveznih in 4 nezvezne elemente, model z dolgo razpoko pa 115 zveznih in 24 nezveznih elementov. Vsak model je imel 144 notranjih točk. As with the point load, the supports were set in the middle node of the appropriate elements. For analysis, maximum force before crack growth occurred, was used. The values were obtained from experimental data. In the first case, with the short crack, the force is P = 100 kn and in the second case, with the long crack, P= 27 kn. The model with the short crack had 115 continuous and 4 discontinuous elements, and the model with the long crack 115 continuous and 24 discontinuous elements. Both models had 144 internal points each.

10 2.3 Rezultati elastične analize 2.3 R esults of elastic analysis Na slikah 12 in 13 imamo izobare Misesovih Figures 12 and 13 show Mises isobars for primerjalnih napetosti za obe dolžini razpoke. both crack lengths. The equation for Mises stres- Enačba Misesovih primerjalnih napetosti se glasi: ses is given by: Sl. 12. Primerjalne Misesove napetosti, o < 70,5 i MPa, a = 3,264 mm Fig. 12. Mises stresses, ö < 70.5 i MPa, a = mm Sl. 13. Primerjalne Misesove napetosti ó < 70,5 i, a = 19,37 mm Fig. 13. Mlses stresses, ö < MPa, a = mm Iz rezultatov vidimo, da dobimo v prvem primeru precej večje območje elastičnih napetosti, večjih od porušne natezne trdnosti, kakor v drugem primeru. 2.4 Primerjava z eksperimentalnimi rezultati Slika 14 prikazuje vrednosti faktorja intenzivnosti napetosti Kj za primer kratke razpoke. Slika 15 prikazuje vrednosti faktorja intenzivnosti napetosti K\ za primer dolge razpoke. It can be seen from the results that in the first case a much larger area with stresses higher than the yield stress than in the second case is obtained. 2.4 Comparison w ith experimental results Figure 14 shows a comparison of the stress intensity factors Kj for the specimen with the short crack. Figure 15 shows a comparison of the stress intensity factors K\ for the specimen with the long crack. SI. 14. Faktor intenzivnosti napetosti Kj, a = 3,264 mm Fig. 14. Stress intensity factor Kj, a = mm Eksperimentalne vrednosti faktorja intenzivnosti napetosti smo dobili iz enačbe (9) z uporabo izmerjenih vrednosti CMOD. Rezultate elastične analize, dobljene z BEASY smo korigirali z Irwinovim (13) in Dugdalovim (15) faktorjem kn 30 F Sl. 15. Faktor intenzivnosti napetosti Kj, a = 19,37 mm Fig. 15. Stress intensity factor Kj, a = mm The experimental values of stress intensity factors are obtained from equation (9) over the measured values of CMOD. Results of elastic analysis obtained by BEASY are corrected with the Irwin (13) and Dugdale (15) factors.

11 Slika 16 prikazuje vrednosti odprtja vrha razpoke CTOD za primer kratke razpoke. Slika 17 prikazuje vrednosti odprtja vrha razpoke CTOD za primer dolge razpoke. Figure 16 shows crack tip opening displacement CTOD for the specimen with the short crack. Figure 17 shows crack tip opening displacement CTOD for the specimen with the long crack. Sl. 16. Odprtje vrha razpoke CTOD, a = 3,264 mm Fig. 16. Crack tip opening displacement CTOD, a = mm Za izračun odprtja vrha razpoke smo uporabili Irwinovo (14) in Dugdalovo (16) enačbo. Odstopki izračunanih od izmerjenih vrednosti so precej veliki, kot zelo vpliven faktor pa se izkaže faktor vklenitve materiala. V enačbah smo uporabili p cf = 1,68. Z zmanjševanjem te vrednosti se približujemo eksperimentalnim vrednostim. Slika 18 prikazuje vrednosti odprtja ustja razpoke CMOD za primer kratke razpoke. Slika 19 prikazuje vrednosti odprtja ustja razpoke CMOD za primer dolge razpoke. SI. 17. Odprtje vrha razpoke CTOD, a = 19,37 mm Fig. 17. Crack tip opening displacement CTOD, a = mm For the calculation of the crack tip opening displacement, the Irwin (14) and Dugdale (16) equations were used. The difference between the calculated and measured values is significant. It has been concluded that the plastic constraint factor has a great influence. In the equations pcf = 1.68 was used. With a decrease in this value, a better coincidence can be obtained. Figure 18 shows crack mouth opening displacement CMOD for the specimen with the short crack. Figure 19 crack mouth opening displacement CMOD for the specimen with the long crack. Sl. 18. Odprtje ustja razpoke CMOD, a = 3,264 mm Fig. 18. Crack mouth opening displacement CTOD, a = mm Odprtje ustja razpoke smo popravili tako, da smo elastični rešitvi odprtja ustja razpoke prišteli velikost odprtja vrha razpoke (14), (16). Slika 20 prikazuje vrednosti pomika obremenitvene črte Vjj za primer kratke razpoke. Slika 21 prikazuje vrednosti pomika obremenitvene črte vn za primer dolge razpoke. Kakor vidimo, se rezultati zadovoljivo ujemajo z eksperimentalnimi. Na sliki 21 vidimo, da smo vse do največje obremenitve v linearnem elastičnem območju, na sliki 20 pa je opazen močan vpliv plastifikacije pri večji obremenitvi. SI. 19. Odprtje ustja razpoke CMOD, a = 19,37 mm Fig. 19. Crack mouth opening displacement CTOD, a = The crack mouth opening displacement was corrected by adding the value of the crack tip opening displacement to the elastic solution of the crack mouth opening displacement (14), (16). Fig. 20 shows a comparison of load line displacement vn for the specimen with the short crack. Fig. 21 shows comparison of the load displacement vn for the specimen with the long crack. It can be seen that the agreement of results with the experimental values is good enough. From fig. 21 it can be seen that only elastic deformations are present in contrast to fig. 20, where strong nonlinear behaviour can be seen.

12 Sl. 20. Pomik obremenitvene črte vn, a = 3,264 mm Fig. 21. Load line displacement vn, a = 3.264mm 3 SKLEPI Iz rezultatov za kratko razpoko (sl. 14, 18) lahko vidimo, da so razlike precejšnje. Največje razhajanje je pri odprtju ustja rarzpoke C M OD in zaradi tega pri faktorju intenzivnosti napetosti /Lj. Kakor je razvidno s slike 12 je velikost plastične cone precejšnja v primerjavi s širino preizkušanca, kar močno vpliva na natančnost izračuna faktorja intenzivnosti napetosti. Rezultati za dolgo razpoko (sl. 15, 19) se precej bolje ujemajo z izmerjenimi vrednostmi. S slike 13 je razvidno, da je tudi plastična cona precej manjša kakor v prvem primeru. Če primerjamo rezultate za odpiranje vrha razpoke CTOD (sl. 16, 17) vidimo, da je ujemanje obeh numeričnih primerov z eksperimentalnimi približno enako. Močan je vpliv faktorja vklenitve materiala, ki pri dvodimenzionalnih analizah zajame vpliv debeline materiala. ZAHVALA P r v i a vto r članka se zah valju je prof. dr. Carlosu B re b b ii (W e sse x In stitu te o f Technology. Southam pton. England) za njegovo pom oč in podporo m ed b ivanjem na in štitu tu. O benem se za h va lju je še p ro jek tu TEM PUS, k i m u je s š tip e n d ijo omogoči1 izpopolnjevanje. 4 Sl. 21. Pomik obremenitvene črte vn. a = 19,37 mm Fig. 21. Load line displacement vfi, a = mm 3 CONCLUSIONS From the results (Figures 14, 18) it can be seen that the differences are significant. The largest deviation is at the crack mouth opening displacement CMOD and, as a consequence, by the stress intensity factor K^. It can be also noticed (Fig. 12) that the plastic zone size is not negligable compared with the specimen thickness, which strongly affects the accuracy of the stress intensity factor calculation. Results for the long crack (Figures 15, 19) are in better agreement with the experimental values. It can also be seen from fig. 13 that the size of the plastic zone is much smaller than in the first case. Comparing the results for crack tip opening displacements CTOD (Figures 16, 17) shows that the agreement of the two numerical calculations with experimental values is more or less the same. The plastic constraint factor has a great influence, which takes into account the specimen thickness in two-dimensional analysis. A K N O W LEDCEM ENT F irs t author o f th is paper w ish e s to e x p re ss h is thanks to P r o f C arlos B reb b ia for W IT. Southam pton help and su p p o rt during h is s ta y at the In stitu te. E order he w ish es to thank TEM PU S p ro je c t for grantin g h is r e search. 4 LITERATURA 4 REFERENCES (li Portela, A.: Dual Boundary Element Incremental Analysis of Crack Growth, Wessex Institute of Technology, Southampton, Rice, J.R.: A Path Independent Integral and the Approximate Analysis of Strain Concentracion by Notches and Cracks. Trans. ASME, J. Appi. Mech Naslov avtorjev: Zoran Žunič, dipl. inž., doc. dr. Janko Legat, dipl. inž. Fakulteta za strojništvo Univerza v Mariboru Smetanova Maribor Prejeto: Received: Murakami, Y.: Stress Intensity Factors Handbook. Pergamon Press. Oxford, Volume 1, [41 Broek, D.: Elementary Engineering Fracture Mechanics. Martinus Nijhoff Publishers, Dortrecht, A u th o r's Address: Zoran Žunič. Dipl. Ing., Doc. Dr. Janko Legat, Dipl. Ing. Faculty of Mechanical Engineering University of Maribor Smetanova Maribor. Slovenia Sprejeto: Accepted: