Izvod. Abstract NAUKA ISTRAŽIVANJE RAZVOJ SCIENCE RESEARCH DEVELOPMENT. B. Međo, M. Rakin, O. Kolednik, N.K. Simha, F. D. Fischer
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1 B. Međo, M. Rakin, O. Kolednik, N.K. Simha, F. D. Fischer UTICAJ ZAOSTALIH NAPONA NA PONAŠANJE ZAVARENIH SPOJEVA I DRUGIH NEHOMOGENIH MATERIJALA SA PRSLINAMA THE INFLUENCE OF RESIDUAL STRESSES ON WELDED JOINTS AND OTHER INHOMOGENEOUS MATERIALS WITH CRACKS Originalni naučni rad / Original scientific paper UDK / UDC: : Rad primljen / Paper received: April 28. Ključne reči: Nehomogena struktura, zaostali naponi, sila rasta prsline. Izvod Zaostali naponi u mnogim konstrukcijama nastaju tokom hlađenja u procesu proizvodnje ili spajanja materijala. Primeri su: zavareni i zalemljeni spojevi, kompoziti, materijali sa izraženom anizotropijom, prevlake i dr. Zaostali naponi u slučaju ovakvih materijala utiču na polje napona i deformacija u okolini vrha prsline i dovode do promene vrednosti sile rasta prsline u odnosu na materijal bez zaostalih napona. U ovom radu, uticaj zaostalih napona je analiziran metodom konačnih elemenata, primenom programskog paketa ABAQUS, uz korišćenje postprocesorskog programa razvijenog primenom metode konfiguracionih sila. Ovaj pristup je ranije primenjen na ispitivanje uticaja nehomogenosti mehaničkih osobina (modula elastičnosti, granice tečenja, koeficijenta deformacionog ojačavanja) na spojeve niskolegiranih čelika povišene čvrstoće, ali bez uzimanja u obzir nehomogenosti koeficijenta termičkog širenja materijala. Cilj ovog rada je analiza uticaja termički unetih zaostalih napona na silu rasta prsline u bimaterijalnim spojevima i korišćenje dobijenih rezultata za obezbeđenje integriteta i sigurnog rada zavarenih konstrukcija i struktura sastavljenih od više materijala. * Rad je izlagan na Savetovanju sa međunarodnim učešćem ZAVARIVANJE 28 - Subotica UVOD Zaostali naponi su prisutni u mnogim mašinskim komponentama i delovima, zbog procesa proizvodnje, uključujući hlađenje sa povišenih temperatura. Sa stanovišta inženjerske prakse, izuzetno je važna analiza heterogenih materijala kod kojih se uticaj zaostalih napona ne može zanemariti. Uticaj nehomogenosti je važan za razumevanje ponašanja i ZAVARIVANJE I ZAVARENE KONSTRUKCIJE (2/29), str Adresa autora / Author's address: B. Međo, M. Rakin, Tehnološko metalurški fakultet Univerziteta u Beogradu, 11 Beograd, Srbija. bojanmedjo@gmail.com O. Kolednik, Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, A-87 Leoben, Austria; Materials Center Leoben, A-87 Leoben, Austria. N.K. Simha, University of Minnesota, MMC 289, Minneapolis, MN 55455, USA. F. D. Fischer, Materials Center Leoben, A-87 Leoben, Austria; Institute of Mechanics, Montanuniversität Leoben, A-87 Leoben, Austria. Keywords: Inhomogeneous structure, residual stresses, crack driving force. Abstract Residual stresses in many structural components are often produced during the fabrication or bonding processes, due to the cooling from elevated temperatures. Examples for such behaviour are: welded and soldered joints, composites, functionally graded materials, coatings, etc. The residual stresses in these materials affect the stress and strain fields in the vicinity of the crack tip, thus making the crack driving force to become different compared to a material without residual stresses. In this paper, the influence of residual stresses is analyzed using FEM software ABAQUS and post processing routine based on the method of configurational forces. A previous application of this method has included investigations on the influence of mechanical inhomogeneities (Young s modulus, yield stress, hardening coefficient) in welded joints of high-strength low-alloyed steels (HSLA). However, these analyses did not include the effect of inhomogeneity in thermal properties (coefficient of thermal expansion). The main goal of this paper is to find out the effect of thermally induced residual stresses on the crack driving force in bimaterials and to use the obtained data to improve the integrity and work safety of welded and other inhomogeneous structures. loma kompozitnih materijala, materijala sa izraženom anizotropijom, zavarenih i zalemljenih spojeva, itd. Materijali kod kojih je urađena termička obrada površinskog sloja, ili materijali prevučeni tankim ili debljim prevlakama, su takođe tipični primeri kod kojih postoji oštra granica ili postepeni prelaz između dva materijala. Eksperimentalna istraživanja su pokazala da čak i male razlike u koeficijentu termičkog širenja (CTE 43
2 coefficient of thermal expansion) mogu imati značajan uticaj na ponašanje struktura sa početnom prslinom [1-4]. Štaviše, u [1] je pokazano da pritisni zaostali naponi u materijalu sa manjom vrednošću CTE mogu pod određenim uslovima dovesti i do zaustavljanja rasta zamorne prsline u ispitivanoj bimaterijalnoj strukturi. Kvantitativni opis uticaja zaostalih napona je važan za razumevanje ponašanja nehomogenih materijala ili komponenti pod dejstvom statičkog i cikličnog opterećenja. Poznato je da različite mehaničke osobine materijala mogu uticati na efektivnu silu rasta prsline [5, 6]. U novijim radovima [6, 7], koncept materijalnih (konfiguracionih) sila se koristi za ispitivanje uticaja nehomogenosti. Pokazano je [8-11] da nehomogenost materijala u pravcu rasta prsline, izaziva pojavu dodatne sile rasta prsline faktora nehomogenosti materijala,. Efektivna sila rasta prsline, izražena preko J - integrala izračunatog na oblasti u blizini vrha prsline, J tip, se dobija kao suma J - integrala izračunatog na celoj oblasti,, i faktora nehomogenosti materijala: J + tip = Cinh 1) Vrednost faktora nehomogenosti materijala može se izračunati koristeći post procesorsku proceduru, posle analize metodom konačnih elemenata. U dosadašnjim istraživanjima, uticaj različitih nehomogenosti materijala je analiziran u slučaju linearno elastičnih i elastoplastičnih bimaterijalnih uzoraka sa diskontinuitetom i sa postepenim prelazom. Dosadašnji rezultati pokazuju da je faktor nehomogenoosti materijala,, pozitivan i J tip je veće od, ukoliko prslina raste ka materijalu sa manjom krutošću ili nižom granicom tečenja. Važi i obrnuto, tj. ukoliko prslina raste ka materijalu sa većom krutošću ili višom granicom tečenja, faktor nehomogenoosti materijala je negativan, a je veće od J tip. Negativna vrednost odgovara pojavi shieldinga u okolini vrha prsline (pojave da kombinacija dva materijala otežava ili čak sprečava rast prsline [1]), dok pozitivna vrednost odgovara pojavi antishieldinga (kombinacija dva materijala pogoduje rastu prsline). U navedenim istraživanjima zanemaren je uticaj zaostalih napona. Postavlja se pitanje: kakav je uticaj zaostalih napona nastalih tokom proizvodnje/montaže usled hlađenja sa povišenih temperatura? Za istraživače koji se bave naukom o materijalima, ali i za inženjere, veoma je važno predvideti da li i u kojim situacijama je ovaj uticaj značajan, i kada ga je opravdano zanemariti. NUMERIČKA ANALIZA Za Numerička analiza je sprovedena na modelu CT epruvete koju sačinjavaju dva materijala, izložene prvo hlađenju a zatim mehaničkom opterećenju. Bimaterijal se sastoji od dva homogena materijala, spojena duž oštre dodirne površine (bez prelazne oblasti). Materijal levo od dodirne površine označen je kao materijal 1, a materijal desno od dodirne površine kao materijal 2. Širina uzorka je W=5 mm, debljina je B=25 mm, a dužina početne prsline je a =29 mm (slika 1). Slika 1: Mreža sa konturama za izračunavanje J-integrala: J-integral izračunat na oblasti u blizini vrha prsline, J tip, J-integral izračunat na celoj oblasti,, J-integral izračunat na dodirnoj površini, J int Figure 1: Mash with contours for calculation of the J- integral: J - integral calculated for the area near the crack tip, J tip, J - integral calculated for the whole area,, J-integral calculated for the contact surface, J int Oštra granica između materijala (dodirna površina) je normalna na ravan prsline, a rastojanje vrha prsline od dodirne površine, L, je varirano tokom numeričke analize (četiri vrednosti su uzete u obzir: 2.5 mm, 1.25 mm,.3 mm i.15 mm). Korišćene su sledeće vrednosti modula elastičnosti: E 1 GPa (koja odgovara čeliku), E = 7 GPa (koja odgovara legurama 44 aluminijuma), E = 14 GPa i E = 15 GPa. Za Poasonov koeficijent oba materijala je uzeta vrednost ν =.3. Dva materijala imaju različite vrednosti CTE, korišćene su takođe vrednosti koje odgovaraju čeliku i legurama aluminijuma, 12 x 1-6 K -1, odnosno 24 x 1-6 K -1 ; ZAVARIVANJE I ZAVARENE KONSTRUKCIJE (2/29), str
3 prema tome, odnos CT može imati vrednost.5 ili 2. Analizirane su dve vrste nehomogenosti: a) nehomogenost koeficijenta termičkog širenja, CTE, oba materijala imaju iste mehaničke osobine; b) spregnuta nehomogenost CTE i modula elastičnosti E. Mreža se sastoji od izoparametarskih osmočvornih elemenata. Pošto se u ovom radu analizira stacionarna prslina, a materijali su elastični, singularni elementi su korišćeni u okolini vrha prsline. Da bi se postigla zadovoljavajuća tačnost rezultata, mreža mora biti dovoljno fina u blizini vrha prsline i duž cele dodirne površine. J-integrali (J tip i ) su dobijeni korišćenjem metode virtuelnog rasta prsline, a detalji ovog postupka su prikazani u literaturi [12]. C φ ( ε, x ) Analiza metodom konačnih elemenata je sprovedena korišćenjem programskog paketa ABAQUS ( Zaostali naponi su uvedeni tako što je simulirano hlađenje uzorka sa povišene temperature. Nakon hlađenja, simulirano je statičko opterećenje, i to tako što je zadato pomeranje napadne tačke sile (gornji levi ugao na slici 1). Da bi se skratilo vreme potrebno za proračun i izbegli problemi sa zaostalim naponima u pravcu normalnom na ravan CT epruvete, korišćen je dvodimenzionalni model, u ravnom stanju napona. Primer analize elastično idealno plastičnog bimaterijala je prikazan u literaturi [13]. Faktor nehomogenosti materijala,, je izračunat korišćenjem postprocesorske procedure, koje je nastala na osnovu rezultata iz literature [13]: ([[] φ ] σ [ ε ]) k pq p i inh = e j da D n e Σ i pq pq ( j j ) x j i = 1 gde je φ gustina energije deformacije, koja zavisi od linearne deformacije, ε, koja opet zavisi od referentne koordinate x. Oblast D označava površinu koja se nalazi između kontura koje se koriste za izračunavanje J tip i. Pretpostavlja se da postoji i=1,2,..,k oštrih dodirnih površina, Σ i, u okviru oblasti D. [ φ ] označava skok vrednosti gustine energije deformacije, a [ ε ] ik je skok vrednosti komponenata deformacije na dodirnoj površini. σ označava srednju vrednost komponenata napona ik na dve strane dodirne površine; n j je jedinični vektor normale na dodirnu površinu, a e j je jedinični vektor u pravcu rasta prsline. Ukoliko su oba materijala homogena, površinski integral u (2) ne postoji, pa je za bimaterijal sa oštrom dodirnom površinom (kakav se razmatra u ovom radu) u post procesorskoj proceduri za izračunavanje [14] korišćen samo drugi deo ove jednačine. U radu [9] je pokazano da je faktor nehomogenosti materijala jednak negativnoj vrednosti J-integala izračunatog na dodirnoj površini; = J int. REZULTATI I DISKUSIJA U prvom delu rada prikazan je uticaj temperaturnog opsega na primeru bimaterijalnih uzoraka napravljenih od dva materijala koja se razlikuju samo po CTE, a imaju iste mehaničke osobine. Na Slici 2 je prikazana varijacija u zavisnosti od rastojanja L, za bimaterijal kod koga je E 1 GPa i CT, za tri temperaturna opsega ΔT. Sa ove slike vidi se da se vrednost povećava sa smanjenjem L i povećavanjem temperaturnog opsega. Ove vrednosti, dobijene tokom hlađenja modela, su korišćene kao polazni podaci za mehaničku analizu. U svakom slučaju, ove vrednosti su male, osim u slučaju kada je vrh prsline veoma blizu dodirne površine. ds 2) E 1 GPa, CT ΔT = -15 K ΔT = -3 K ΔT = -45 K L (mm) Slika 2: Bimaterijali sa nehomogenošću CTE; Uticaj temperaturnog opsega, ΔT, i rastojanja između vrha prsline i dodirne površine, L, na nakon hlađenja Figure 2: Bimaterials with inhomogeneity CTE, Effects of the temperature range, ΔT, and distance between crack tip and contact surface, L, for after cooling ZAVARIVANJE I ZAVARENE KONSTRUKCIJE (2/29), str
4 Tokom mehaničkog opterećivanja uzorka analiziran je uticaj modula elastičnosti, odnosa CT i rastojanja između vrha prsline i dodirne površine. Rast spoljnog opterećenja povećava vrednosti sila rasta prsline (J tip i ). Zbog neuniformne raspodele zaostalih napona, spomenute sile rasta prsline ne rastu istim intenzitetom, iako oba materijala imaju isti modul elastičnosti. Na slici 3. prikazana je promena sila rasta prsline tokom mehaničkog opterećivanja (u zavisnosti od pomeranja napadne tačke sile, v LL ): J tip,, i vrednost J - integrala dobijena pomoću krive sila - pomeranje napadne tačke, na osnovu procedure [15], J : J ηu = 3) bb gde je U površina ispod krive sila pomeranje napadne tačke, b = W a je dužina ligamenta, a η je faktor uticaja oblika i za datu geometriju ima vrednost približno 2.2. Zaključak je da J - integral izračunat na celoj oblasti ne daje pravu sliku otpornosti materijala prema početku rasta prsline u slučaju linearno elastičnog bimaterijala u kome se javljaju zaostali naponi, uprkos homogenosti modula elastičnosti i Poasonovog koeficijenta. Za posmatrani slučaj sa odnosom CT, J tip >, što znači da je veće od nule (videti j-nu 1) i javlja se antishielding efekat. Uticaj E na je prikazan na Slici 3b, na primeru sa L =.3 mm. Prikazane su vrednosti za različite vrednosti E, koje odgovaraju različitim materijalima (čelik, legure bakra, legure aluminijuma, legure titanijuma)., J tip, J, 8 ΔT = -15 C, E 1 GPa, 7 CT L.5 mm J tip J V LL (mm) a) Δ T = -15 C, CT L =.3 mm 8 6 GPa) 4 (E = 14 GPa) 2 (E = 15 GPa) (E = 7 GPa) b) Slika 3: Bimaterijali sa nehomogenošću CTE; (a) Sile rasta prsline J tip,, J i u zavisnosti od pomeranja napadne tačke sile, v LL, tokom mehaničkog opterećivanja; (b) Uticaj modula elastičnosti E na Figure 3: Bimaterials with inhomogeneity CTE, (a) Crack growth forces J tip,, J and in dependence of the displacing of impact force point, v LL, during the mechanical loading, (b) Effect of the elasticy modulus E on 46 ZAVARIVANJE I ZAVARENE KONSTRUKCIJE (2/29), str
5 U nastavku je razmatrano mehaničko opterećenje uzorka u slučajevima gde postoji spregnuta nehomogenost po CTE i E. Kao i u prethodnoj analizi, simulirano je hlađenje uzorka, zbog uvođenja zaostalih napona, a zatim mehaničko opterećenje. Uticaj modula elastičnosti je prikazan na Slici 4a, za konstantan odnos CT. Može se videti da smanjivanje odnosa smanjuje, poboljšavajući otpornost bimaterijala prema rastu prsline. Na slici 4b je prikazana promena u odnosu na promenu vrednosti, za dva slučaja nehomogenosti modula elastičnosti: 1/7 i = 7/21, i dva odnosa CT (2 i.5). Na istom dijagramu, prikazane su vrednosti faktora nehomogenosti materijala za slučajeve bimaterijala bez zaostalih napona. Ovi uzorci nisu bili izloženi hlađenju, već samo mehaničkom opterećenju, čije povećavanje menja vrednosti zbog nehomogenosti u modulu elastičnosti. Može se primetiti da odnos CT veći od 1 ima negativan uticaj: za istu vrednost, faktor nehomogenosti napona ima veće vrednosti u poređenju sa materijalom bez zaostalih napona. Dakle, za slučaj / E 2 1 / 7, antishielding efekat zbog nehomogenosti E je intenziviran nehomogenošću CTE, dok je za / E 2 = 7 / 21 efekat shieldinga zbog nehomogenosti E smanjen (isprekidane linije na slici 4b). Važi i obrnuto, CTE odnos manji od 1 ima povoljan uticaj; za istu vrednost faktor nehomogenosti materijala ima niže vrednosti u poređenju sa materijalom bez zaostalih napona (pune linije na сlici 4b) 15 1 CT ΔT = -15 C L.5 mm ( 1/7) ( 1 GPa) ( = 7 GPa) ( = 7/21) a) L.5 mm, ΔT = -15 C puna linija: CT =.5 isprekidana linija: CT 1/7 = 7/21 bez zaostalih napona b) Slika 4: Bimaterijali sa spregnutom nehomogenošću po CTE i E; (a) Uticaj E na vrednosti ; (b) Poređenje bimaterijala sa i bez zaostalih napona za 1/7 i = 7/21 Figure 4: Bimaterials with coupled inhomogeneity for CTE and E, (a) Effects of E on values, (b) Comparison of the bimaterials with and without residual stresses for 1/7 and = 7/21 ZAVARIVANJE I ZAVARENE KONSTRUKCIJE (2/29), str
6 = 4 kj/m 2 = 8 kj/m 2 = 12 kj/m 2 = 16 kj/m 2 kj/m 2 4 kj/m 2 CT T 2 -T 1 = -15 C 1/ = 7/ L[mm] a) = 4 kj/m 2 = 8 kj/m 2 = 12 kj/m 2 = 16 kj/m 2 kj/m 2 4 kj/m 2 CT =.5 T 2 -T 1 = -15 C 1/7-2 = 7/ L[mm] b) Slika 5: Bimaterijali sa spregnutom nehomogenošću po CTE i E; u zavisnosti od L; (a) CT ; (b) CT =.5 Figure 5: Bimaterials with coupled inhomogeneity for CTE and E, in dependence of L, (a) CT ; (b) CT =.5 Uticaj rastojanja između vrha prsline i dodirne površine, L, na, za bimaterijale sa spregnutom nehomogenošću po CTE i E, je prikazan na Slici 5. Može se videti da je za male vrednosti L (manje od.5 mm) uticaj nehomogenosti značajan. Čak i pri nižim vrednostima spoljnog opterećenja (npr. = 8 kj/m 2 ), razlika između pozitivnih vrednosti (antishielding efekat) je velika (oko 14 kj/m 2, Slika 5). Takođe, može se videti da su i shielding i antishielding efekat izraženiji za pozitivne vrednosti L. To znači da su apsolutne vrednosti veće ukoliko se prslina nalazi u materijalu 1 nego odgovarajuće vrednosti u slučaju da se prslina nalazi u oba materijala (vrh prsline se nalazi u materijalu 2), iako ni ove vrednosti nisu zanemarljive. ZAKLJUČAK U ovom radu analiziran je uticaj zaostalih napona na silu rasta prsline u slučaju bimaterijala sa početnom prslinom čiji vrh se nalazi u blizini dodirne površine. Proučavani su linearno elastični bimaterijali sa nehomogenošću koeficijenta termičkog širenja, CTE, i modula elastičnosti, E. Pokazano je da J-integral na celoj oblasti,, ne može na zadovoljavajući način da opiše stvarnu otpornost prema širenju prsline ukoliko su u bimaterijalu prisutni zaostali naponi. Ova pojava je izraženija ukoliko je vrh prsline bliži dodirnoj površini i ukoliko se prslina nalazi samo u materijalu 1. Manje vrednosti odnosa CT i smanjuju vrednosti, poboljšavajući otpornost bimaterijalne strukture prema rastu prsline. 48 ZAVARIVANJE I ZAVARENE KONSTRUKCIJE (2/29), str
7 Zahvalnost MR, OK i FDF zahvaljuju na finansijskoj podršci: Österreichische Forschungsförderungs-gesellschaft mbh, pokrajini Styria, Steirische Wirtschaftsförderungsgesellschaft mbh i gradskoj upravi Leoben, u okviru Austrijskog Kplus Programa (proj. SP18-WP3). BM i MR zahvaljuju na finansijskoj podršci Ministarstvu za nauku Republike Srbije (proj. OI 14427). LITERATURA [1] Pippan, R., Flechsig, K., Reimelmoser, F.O., Fatigue crack propagation behaviour in the vicinity of an interface between materials with different yield stresses, Materials Science and Engineering A, 283 (2) pp [2] Laz, P.J., Chan, K.S., McClung, R.C., Leverant G.R., Effects of CTEinduced residual stresses around hard alpha particles on fatigue crack growth in Ti 6Al 4V, Fatig. & Fract. Eng. Mat. Struct., 26 (23) pp [3] Lee, W., Myoung, J.M., Yoo, Y.H., Shin, H., Effect of thermal misfit stress on crack deflection at planar interfaces in layered systems, Compos. Sci. Technol., 66 (26) pp [4] Suresh, S., Sugimura, Y., Tschegg, E., The growth of a fatigue crack approaching a perpendicularly-oriented bimaterial interface, Scripta Metallurgica 27 (1992) pp [5] Kolednik, O., Int. Journ. Solids Struct., The yield-stress gradient effect in inhomogeneous materials, 37 (2) pp [6] Maugin, G.A., Material Inhomogeneities in Elasticity, Chapman and Hall, London, [7] Gurtin, M.E., Configurational Forces as Basic Concepts of Continuum Physics, Springer, Berlin, 2. [8] Simha, N.K., Fischer, F.D., Kolednik, O., Chen, C.R., Inhomogeneity effects on the crack driving force in elastic and elastic-plastic materials, Journ. Mech. Phys. Solids., 51 (23) pp [9] Simha, N.K., Predan, J., Kolednik, O., Shan, G.X., Fischer, F.D., J- integral and Crack Driving Force in Elastic-Plastic Materials, Journ. Mech. Phys. Solids, submitted. [1] Rakin, M., Kolednik, O., Simha, N.H., Medjo, B., Fischer, F.D., The effect of residual stresses on bimaterial structure on bimaterial structure with initial crack located near interface, Proceedings of the 3 rd International Conference: Deformation Processing and Structure of Materials, Belgrade, 27, pp [11] Fischer, F.D., Predan, J., Kolednik, O., Simha, N.K., Application of material forces to fracture of inhomogeneous materials: ilustrative examples, Archive of Applied Mechanics, 77 (27) pp [12] Kolednik, O., Predan, J., Shan, G.X., Simha, N.K., Fischer, F.D., On the fracture behaviour of inhomogeneous materials a case study for elastically inhomogeneous bimaterials, Int. Journ. Solids Struct., 42 (25) pp [13] Simha, N.K., Kolednik, O., Fischer, F.D., Material force models for cracks influences of eigenstrains, thermal strains & residual stresses, Proceedings of the 11 th International Conference on Fracture, Turin, 25, on CD [14] Rakin, M., Kolednik, O., Simha, N.K., Fischer, F.D., Influence of residual stresses on the crack driving force in bimaterials with sharp interface, Proceedings of the 11 th International Conference on Fracture, Turin, 25, on CD [15] ESIS Procedure for determining the fracture behavior of materials, European Structural Integrity Society, ESIS P2-92, ZAVARIVANJE I ZAVARENE KONSTRUKCIJE (2/29), str
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