COHESIVE CRACK PROPAGATION ANALYSIS USING A NON-LINEAR BOUNDARY ELEMENT FORMULATION

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1 Bluher Mehanal Engneerng Proeedngs May 2014, vol. 1, num. 1 COHESIVE CRACK PROPAGATION ANALYSIS USING A NON-LINEAR BOUNDARY ELEMENT FORMULATION H. L. Olvera 1, E.D. Leonel 2 1 Unversy o São Paulo, Shool o Engneerng o São Carlos, Deparmen o Sruural Engneerng. 2 Unversy o São Paulo, Shool o Engneerng o São Carlos, Deparmen o Sruural Engneerng (orrespondng auhor: edleonel@s.usp.br). Absra. Ths paper addresses o analyss o rak growh n quas-brle maerals usng he boundary elemen mehod (BEM) and ohesve models. BEM has been wdely used o solve many omplex engneerng problems, espeally hose where s mesh dmenson reduon nludes advanages on he modellng. The non-lnear ormulaons developed are based on he dual BEM, n whh sngular and hyper-sngular negral equaons are adoped. The rs ormulaon uses he onep o onsan operaor, n whh all orreons on he nonlnear sysem o equaons are perormed only by applyng approprae raons along he rak suraes. The seond proposed BEM ormulaon s an mpl ehnque based on he use o a angen operaor. Ths ormulaon s aurae, sable and always requres less eraons o reah he equlbrum whn a gven load nremen n omparson wh he lassal approah. Examples o problems o rak growh are shown o llusrae he perormane o hese wo ormulaons. Keywords: Cohesve rak growh, Non-lnear BEM ormulaon, Tangen operaor. 1. INTRODUCTION The rak growh phenomenon s one o he major aors aeng he sruural behavour o sruures omposed by mporan maerals suh as onree, bre-renored omposes and wood. These maerals belong o he group o quas-brle maerals or whh he proess o rakng an be desrbed usng ohesve rak approahes [1 4]. Due o he nheren omplex behavour o hese maerals, relable prognoses o serveably and ulmae saes o sruures made o quas-brle maerals requre robus ompuaonal nonlnear models. Compuaonal modellng o rak growh has been arred ou applyng he ne elemen mehod (FEM) [5,6]. However, s doman mesh requres remeshng proedures a eah rak lengh nremen, whh beame hs approah ompuaonally osly due he ranser o daa beween he deren meshes. Consequenly, makes hs approah less a-

2 rave. In he pas years, he exended ne elemen mehod (XFEM) and meshless mehods appeared also as arave alernaves o rak propagaon analyss [7,8]. These ehnques requre only he desrpon o he geomeral doman, and hus remeshng beomes unneessary. However, s sll dul o deal wh mulple rak growh and oalesene phenomena. The boundary elemen mehod (BEM) s an alernave numeral mehod wdely appled n rak propagaon smulaon. In raure mehans problems, BEM requres only o dsreze he boundary and he rak suraes. Consequenly, BEM requres less ompuaonal eor o generae new elemens n order o model rak growh. Some lassal BEM ormulaons were proposed n he leraure demonsrang he eeny and auray o hs mehod n raure problems. In hs onex, s worh o menon [9] and [10], where Green s unons and dsplaemen dsonnuy mehod were adoped, respevely. [11] developed he dual boundary elemen mehod (DBEM), n whh sngular and hyper-sngular negral equaons are wren or olloaon pons posoned a he oppose rak suraes. Consderng mulple rak propagaon, some BEM models an be ound n he leraure. The problems o mulple rak-hole neraon, [12,13] and mulple raks neraon [14-17], or nsane, have been suessully analysed. In hese applaons, however, non-lnear rak propagaon has no been onsdered. Non-lnear models wh BEM have been developed sne he 1970s. In he begnnng, he works were smple bu apable o demonsrae ha BEM an be appled o smulae hese knds o omplex problems [18]. The non-lnear soluon or he rs BEM ormulaons was based on he applaon o a orreng sress eld, keepng onsan all relevan mares and hereore leadng o a large number o eraons. Reenly, more relable soluon ehnques based on he use o angen operaors have been proposed [19,20]. They requre less eraons and are more sable and aurae han lassal approahes. Consderng non-lnear rak growh analyss usng BEM, he soluon ehnque adoped or he majory o he researhers s based on erave shemes ha nd raon values along rak suraes ha sasy an adoped reron [21]. Ths proess s smple and all relevan mares are kep onsan durng he proess. Agan, hs knd o ehnque requres a large number o eraons o aheve he equlbrum or a sngle load nremen. Moreover, or he ases o omplex paern o raks, or nsane wh a sold onanng many mro-raks, hs proess an be eher naurae or unsable. In hs work, a non-lnear BEM ormulaon appled o analyss o rak propagaon n sruures omposed by quas-brle maerals s presened. The proposed ormulaon, whh s based on ohesve rak modellng, uses he DBEM, n whh sngular (dsplaemen negral equaon) and hyper-sngular (raon negral equaon) negral equaons are adoped. The dsplaemen negral equaons are wren or he olloaon pons along one rak surae, whle raon negral equaons are used or he olloaon pons along he oppose rak surae. Ths ehnque avods sngulares o he resulng algebra sysem o equaons, despe he a ha he olloaon pons on he oppose rak lps have he same oordnaes. In order o deal properly wh he non-lnear problem, wo soluon ehnques were used. The rs approah s he lassal proedure, where he orreons on he ohesve

3 rak raons are perormed by applyng a non-equlbraed raons veor. The seond approah developed uses a angen operaor, whh has been derved, and he sysem o equaons s solved usng he Newon Raphson mehod. In hs ase, he non-lnear sysem o equaons s solved usng an mpl proedure, where he orreons on he ohesve raons are alulaed usng he angen operaor. These models are used o analyze ohesve rak propagaon problems where he sness reduon due he raks growh s a major problem. Besdes omparng he soluons obaned by usng he wo dsussed ehnque, whenever possble he resuls are ompared wh expermenal resuls. 2. COHESIVE CRACK MODEL Fraure mehans problems have reeved large neres by he sen ommuny, beause rak growh an explan he ollapse o sruures. Due o he load values or mposed dsplaemens beyond a ral level, mro-rak onenraon nreases as well as he sruural damage. The lnear elas raure mehans has been an mporan approah o solve many problems n sruural engneerng, parularly hose where he dsspaon zone surroundng raks are redued enough o be possble o negle s non-lnear ees. However, or quas-brle and also or some dule maerals, he damaged zone ahead o he rak p s large enough o produe non-lnear ees ha anno be negleed. The ohesve model s an approprae approah o ake no aoun hese ees. In hs model, he dsspaon phenomenon s assumed o our along he rak pah ahead o rak p, hereore redung by one he dmenson o he dsspaon zone. The rs models where he dsspaon zone was redued eher o a lne or 2D problems or o a surae or 3D problems are due o [3] and [1] who have proposed rak models o represen parularly he dule maeral behavour. The ohesve rak model approprae o quas-brle maerals s due o [4]. In hs work, he dsspaon proess, whh akes plae on a regon ahead o he rak p, s approxmaed by a smple soenng law, assumed along a ous rak. Ths law relaes he ous rak openng dsplaemen, u, o ensle surae ores or ohesve ores,, appled along he rak suraes. Fgure (1) llusraes he ohesve ores dsrbuon or he [4] model. As an be seen n hs gure, he ohesve ores ang along he ous rak o lengh equal o l dsappear aer a ral rak openng value, u. The rak sars openng when he ohesve ores reahes a ral ensle value. For values o he ensle sresses lesser han he rak remans losed. For values o he rak openng dsplaemen larger han, he ohesve ores are zero. u Fgure 1. Cohesve ores dsrbuon [4].

4 Several ohesve rak laws relang ohesve ores and rak openng dsplaemen have already been proposed n he leraure. Three o hem are oen adoped o arry ou rak growh analyss n quas-brle maerals. The smples law s gven by a lnear unon relang he ohesve ores o he ous rak openng dsplaemen smaller han he ral value, u. For ous rak openngs larger han u, ohesve ores are assumed equal o zero, Fgure (2a). The relaons ha represen he lnear ohesve law are gven by: E u u 1 0u u u u 0 u u (1) An alernave law relang ohesve ores and ous rak openng dsplaemen s he b-lnear model, Fgure (2b), whh s gven by he ollowng equaons: E '' u u 0 uu '' u u u u u u u '' '' '' '' 1 '' '' u u u u u 0 u u '' (2) For he b-lnear model, he varables expressons: '' '', u and u '' 3 0.8G '' u 3.6G u are dened by he ollowng (3) Fgure 2. Cohesve models. (a) lnear model; (b) b-lnear model, () exponenal model

5 The hrd ohesve rak model onsdered n hs work s represened by an exponenal law, Fgure (2). Equaon (4) gves he analyal expressons or hs ohesve model: E G 0 3. DUAL BOUNDARY ELEMENT METHOD (DBEM) u u e u (4) The boundary elemen mehod has been wdely appled n varous engneerng elds, suh as ona problems, ague and raure mehans, due o s hgh preson and robusness n modellng srong sress onenraon (.e. sngular sresses and dsplaemens). Consderng a wo-dmensonal homogeneous elas doman,, wh boundary,. The equlbrum equaon, wren n erms o dsplaemens, s gven by: 1 b u, jj uj, j 0 (5) 1 2 where s he shear modulus, s he Posson s rao, u are omponens o he dsplaemen eld, and b are body ores. Usng Be s heorem, he sngular negral represenaon, wren n erms o dsplaemens an be obaned, wh no body ores, as below: (6) u P u d P u d * * lk (, ) k ( ) lk (, ) k ( ) k ( ) lk (, ) where Pk and uk are raons and dsplaemens on he boundary, respevely, ndaes * he Cauhy prnpal value, he ree erm lk s equal o lk / 2 or smooh boundares, and P lk * and ulk are he undamenal soluons or raons and dsplaemens. Equaon (6) s suen o onsru he sysem o algebra equaons o analyse 2D elas domans. For solds wh raks, however, usng only hs equaon o assemble he sysem o equaons wll lead o a sngular marx as boh rak suraes are loaed along he same geomeral pah. Alhough possble usng only he sngular negral represenaon, Eq. (6) requres he denon o a ne gap beween he wo rak suraes and a very aurae negral sheme o ompue he negral along he quas sngular elemens. Several BEM ormulaons have been proposed n he leraure o properly deal wh rak problems, as dsussed n he nroduon par. The DBEM ormulaon s probably he mos popular BEM ormulaon o analyss o arbrary rak growh. In hs ormulaon, sngular negral represenaon, Eq. (6), s adoped o deermne he algebra represenaon relaed o he olloaon pons dened along one rak surae, whle hyper-sngular negral represenaon s hosen o oban he algebra represenaon relaed o he olloaon pons plaed along he oppose rak surae. Sngular represenaon s appled o he olloaon pons on he boundary, whh s suen o oban he requred algebra relaons. The hyper-sngular negral, wren n erms o raon, s obaned rom Eq. (6). Frs, hs equaon, wren or an nernal olloaon pon, s derenaed o oban he negral represenaon n erms o srans. Then, usng he Hooke s law, he sress negral represenaon s aheved. Fnally, he negral represenaon o sresses or a boundary olloaon

6 pon s obaned by arryng ou he relevan lms. The Cauhy ormula s appled o oban he raon represenaon as ollows: 1 Pj( ) k Skj(, u ) k( d ) k Dkj(, P ) k( d ) 2 (7) where ndaes he Hadamard ne par, and erms S kj and D kj onan he new kernels ompued rom P * lk and u * lk. Equaons (6) and (7) are, as usual, ransormed o algebra relaons by dvdng he boundary and he rak suraes no elemens along whh dsplaemens and raons are approxmaed. Besdes ha, one has o sele a onvenen number o olloaon pons o oban he algebra represenaons. The algebra equaons or boundary nodes are alulaed usng boundary olloaon pons eher a he elemen ends, hereore onden wh nodes, or along he elemen when dsplaemen and raon dsonnues have o be enored. Thus, usng he dsrezed orm o Eq. (6), appled only o boundary olloaon pons, he usual sysem o algebra equaons an be obaned, relang boundary values, as ollows: b b HU H U GP G P (8) b b b b b b where Ub and U are dsplaemens assgned o boundary (b) and o rak surae nodes (), P b gves he boundary raons, whlep represens he raons ang along he rak suraes; Hb, Hb, Gb and Gb are he orrespondng mares o ake no aoun dsplaemens b b and raons ees, he subsrp b ndaes ha he olloaon pon s on he boundary and he supersrps spey he boundary (b) or rak surae () values. On he rak suraes, boundary elemens, nodes and olloaon pons a eah rak lp have o be dened. Thus, or he rak suraes one has wo oppose olloaons pons ha orgnae our algebra ndependen relaons, orrespondng o our unknown rak surae values, wo dsplaemens and wo raons. Moreover, as he hyper-sngular represenaon Eq. (7) s onsdered, s onvenen o use olloaon pons dened along he elemen and no onden wh he dsrezaon nodes. The node values o rak dsplaemens and raons are kep a he elemen end. Thus, rom he dsrezed orms o Equaons (6) and (7), he se o algebra equaons below an be wren: b b HU HU GP GP (9) b b x x where he subsrp n he mares H andg ndaes equaon wren or olloaon pons along he rak surae. The DBEM ormulaon adoped n hs paper uses onnuous and dsonnuous lnear elemens along he exernal boundary and only dsonnuous lnear elemens along he rak suraes. The negrals n Eq. (6) are evaluaed by usng a Gauss Legendre numeral sheme aomplshed wh a sub-elemen ehnque. The negrals appearng n Eq. (7) are alulaed usng analyal expressons. Based on hese proedures, Equaons (6) and (7) are ransormed o algebra represenaons wh very low negraon error. Usng algebra equaons (8) and (9) ogeher wh he ohesve rak model desrbed n he prevous seon an approprae algorhm o analyze rak growh problems an be developed.

7 4. SOLUTION TECHNIQUE. TANGENT OPERATOR The analyss o quas-brle raured solds leads o he soluon o a non-lnear problem ha always requres he use o an erave proedure whn eah me nerval or load sep. In he onex o BEM, a ehnque based on he use o onsan mares s oen adoped, [21], wh sasaory resuls. A eah load nremen, he algebra equaons are kep onsan. The only modaon requred s o re-apply a non-equlbraed raon veor along he rak suraes o re-esablsh he equlbrum los when he onsuve model was mposed. Ths ehnque s smple, bu usually requres a large number o eraons o aheve he equlbrum whn a load nremen, whh mgh lead o unsable soluons. Non-lnear BEM ormulaons based on more elaboraed soluon ehnques have shown o be more aurae and sable. For nsane, non-lnear BEM ormulaons based on he use o angen operaors have demonsraed o lead o aurae and sable soluons when modellng omplex plas and damaged solds haraerzed by exhbng loalzaon and buraon phenomena [19,22]. To derve he ormulaon based on he use o angen operaor, Equaons (8) and (9) have o be moded. The raons and dsplaemens values on he rak suraes wll be separaed aordng s poson, a he rgh or le rak surae. b r b r H U H U H U G P G P G P (10) b b b r b b b b r b b r b r HU b H Ur H U GP b G Pr G P (11) where he subsrps r and are relaed o rgh and le rak suraes, respevely, as llusraed n Fgure (3). Fgure 3. Rgh and le surae varables (n and s ndae normal and parallel dreons o rak surae, respevely). Equaons (10) and (11) an be urher moded. Frs by ransormng he rak surae values, dsplaemens and raons, no loal oordnaes ns,, n whh n and s are oordnae axes perpendular and parallel o eah rak surae, respevely:

8 HU H U H U H U H U GP G P G P G P G P b rs rn s n b rs rn s n b b b rs b rn b s b n b b b rs b rn b s (12) b n HU H U H U H U H U GP G P G P G P G P b rs rn s n b rs rn s n b rs rn s n b rs rn s (13) n Aer hese modaons, he relave dsplaemens on he rak suraes an be nluded n he ormulaon. The rak relave dsplaemens on dreons parallel and perpendular o rak suraes, us and u n respevely, are used o replae he dsplaemen omponens along he le rak sde. us Us Urs U s us Urs (14) u U U U u U (15) n n rn n n rn Aer hese modaons, Equaons (12) and (13) an be rewren as ollow: b rs s rn n s n H U H H U H H U H u H u b b b b rs b b rn b s b n b rs rn s n GP b b Gb Prs Gb Prn Gb Ps Gb P (16) n H U H H U H H U H u H u b rs s rn n s n b rs rn s n b rs rn s n GP b G Prs G Prn G Ps G P (17) n Equaons (16) and (17) represen he equlbrum o sold onanng raks. For he ase o non-lnear raure problems, hese wo las equaons have o be wren and solved whn he onex o nremenal problems. Thus, Equaons (16) and (17) have o be onvenenly rewren n s nremenal orms. Frsly, hese equaons have o be wren n raes and hen ransormed no her nremenal orms by perormng he relevan me negrals over a ypal me nremen n 1 n. For he posulaed problem, he orrespondng nremenal orms o Equaons (16) and (17) s smple and obaned by replang all boundary and rak values x by her nremens x. For a gven load nremen, Equaons (16) and (17) an be urher moded by applyng he known boundary ondons. As usual n BEM ormulaons, all unknown boundary values are sored n a veor x and umulae he known boundary values ees no ndependen veors b and. Aer hese modaons, he unons below an be wren, o express he equlbrum o sold onanng raks: rs s rn n s n Y a x H H U H H U H u H u b b b b rs b b rn b s b n G G G G (18) rs s rn s b b b s b b n Y a x H H U H H U H u H u rs s rn n s n rs rn s n G G G G (19) rs s rn s s n In Equaons (18) and (19), he mares b a and a onan he oeens o mares reerred o unknown boundary dsplaemens and raons. These wo las equaons an be urher moded o emphasze he varables relaed o ohesve reron n he non-lnear proess. These varables are he rak relave dsplaemen and he raon, boh n n dreon. Then, he wo las equaons above an be rewren as:

9 Y A X H u F G G (20) n rn n b b b n b b b n Y A X H u F G G (21) n rn n n n In hese equaons, he mares A b and A onan he oeens o mares reerred o unknown boundary and rgh rak dsplaemens nremens ( Ub, U and rs U rn ), boundary raons ( P ) and he rak relave dsplaemen a he s dreon ( b u ). The s known boundary and rak values, hs las one only a he s dreon, are aken no aoun wh ndependen veors F and b F. Along he rak suraes, only he raon omponens n n dreon are onsdered n he non-lnear proess. The raon omponens n d- reon s are negleed aordng o he ohesve rak model. The non-lnear sysem o equaons gven n (20) and (21) an be solved by applyng Newon Raphson s sheme, or whh an erave proess may be requred o aheve he equlbrum. By lnearzng Equaons (20) and (21) and usng only he rs erm o Taylor s expanson one has: Yb Xk, unk,... Yb Xk, unk,... Ybunk Xk unk 0 (22) X u k,,...,,... Y Xk u nk Y Xk unk Y unk Xk unk 0 X u k The dervave erms n Equaons (22) and (23) gve he global angen operaor [C]. Thus, arryng ou all ndaed dervaes n hese wo las equaons, he angen operaor or he ase o ohesve rak an be obaned. Thereore: n rn n A b Hb Gb Gb n u n C (24) n rn n A H G G n u n where he dervae n u s obaned by derenang properly he ohesve rak laws n presened n seon 2. Thus, he orreons X k and u nk are obaned by solvng he lnearzed sysem represened by equaons (22) and (23): X 1 b nk k Y u C (25) u nk Y unk Whn a gven load nremen k he soluon s obaned by umulang he orreons alulaed usng Equaon (25): 1 X X X (26) k k k 1 nk nk nk nk nk (23) u u u (27) Aer solvng he mpl non-lnear sysem o equaons usng angen operaor n erms o X and k u all varables have o be updaed beore applyng he nex load nremen. The olerane o sop he erave proess whn an nremen o load s appled on he nk varaon o he rak openng dsplaemen orreons,.e, u u 1 olerane.

10 5. CRACK GROWTH SCHEME In order o deermne when he raks grow, he real sress sae a he rak p s ompared wh an ulmae sress sae, gven by an adoped reron. In hs paper, he ulmae sresses are alulaed usng he Rankne s model, whh has also been used by [21]. To aheve auraely he sress sae a he p, a polynomal nerpolaon proess has been adoped o ake no aoun he onnuy o he sresses along all pons surroundng he rak p. Several nernal pons are dened ahead o rak p, as shown n Fgure (4). Fgure 4. Dsrbuon o nernal pons ahead o he rak p The number o sem-rles and he nernal pons along eah one an be hosen aordng o he desred auray. The polynomal degree s hosen aordng he number o he sem-rles. For he ase llusraed n Fgure (4), he nerpolaon proess s perormed usng hrd degree polynomal. The nerpolaon s perormed or eah radal lne and hen he sress sae a he p s obaned by exrapolaon. The real sress sae a he p s hen obaned by averagng he values o all alulaed values. The dreon o he new rak lengh nremen s gven by he dreon perpendular o he maxmum rumerenal ensle sress,.e., maxmum rumerenal sress heory. Aordng hs reron, he rak growh dreon s gven by: xy p ArTan (28) 1 y where 1 s he maxmum ensle sress value. The new rak appears when he ensle sress a he rak p s larger han he ulmae ensle value gven by Rankne s reron. The rak lengh nremen s deermned

11 adjusng he sze o he elemen n suh a way ha he sress sae a he new rak p, alulaed by he maxmum ensle sress reron, s equal o he ulmae ensle. 6. APPLICATIONS Three examples were hosen o llusrae he eeny and robusness o he proposed BEM ormulaon n modellng non-lnear ohesve rak propagaon analyss. The rs example presens he analyss o rak growh n a hree pon bendng onree beam. A our pon bendng onree sruure s analyzed n he seond applaon, whle a our pon bendng mul-raured onree sruure s onsdered as las applaon Conree hree pon bendng beam The sruure onsdered n hs example s llusraed n Fgure (5). I s a hree pon bendng onree sruure whh has 800 mm o lengh, 200 mm o hegh and onans a enral noh o 50 mm. The expermenal resuls o hs example are gven by [21], rom whh he ollowng maeral parameers were hosen: ulmae ensle sress 3.0MPa; Young's modulus E=30,000 MPa; Posson's ν=0.15; and raure energy G =75 N/m. Fgure 5. Three pon bendng beam The analyss o hs example was perormed onsderng he hree ohesve laws dsussed earler (lnear, b-lnear and exponenal). The load has been appled n 25 load nremens and he onvergene has been vered wh a olerane o Fgure (6) shows he omparave among expermenal resul obaned by [21] and numeral responses aheved by he BEM ormulaons desrbed earler. In hs gure, he sux CTO denes he resuls obaned usng angen operaor. Aordng he resuls shown n Fgure (6) good agreemen s observed among he expermenal and numeral resuls. Fgure (7) desrbes he rak growh pah observed n hs analyss. The number o eraons requred by eah model and ohesve rak law o aheve he onvergene s llusraed n Fgure (8). Aordng hs gure, he ormulaon based on angen operaor has demonsraed be aser han he lassal proedure (onsan operaor).

12 The angen operaor ormulaon s aser even n he begnnng o he analyss, when he non-lnear ohesve rak lengh s small. Ths behavour ours beause he angen operaor s onsan when lnear and b-lnear ohesve laws are adoped. Then, he non-lnear proess may aheve he onvergene usng only one eraon. Ths suaon s observed when no hanges our n he ohesve zone,.e, none node leaves he ohesve zone rom one eraon o he nex. Fgure 6. Load dsplaemen urves. Fgure 7. Crak growh pah. Inal noh and nal rak pah.

13 Fgure 8. Number o eraons requred o reah he onvergene Conree hree pon bendng beam The our pon beam onsdered n hs example s shown n Fgure (9). The geomery s gven by s lengh o 675mm, hegh o 150mm and enral noh 75mm deep. The maeral properes were aken rom [23], who have perormed a laboraory es: ensle srengh G 3.0 MPa, Young s modulus E MPa, Posson s rao 0.20 and raure energy 69 N m. Fgure 9. Four pon bendng beam. Dmensons n mm. For he presen analyss hree ohesve laws were used: lnear, b-lnear and exponenal. The load was appled or all ases n 24 nremens and he adoped olerane whn eah 5 nremen was 10. The wo non-lnear sysem soluon ehnques dsussed prevously were esed: (a) usng onsan operaor (b) usng angen operaor. The resuls obaned are gven n

14 Fgure (10) where he angen operaor soluons are dened by he symbol CTO. The oher urves are obaned by usng he onsan operaor. Fgure 10. Load x Dsplaemen urves. Alhough all obaned resuls are n aordane wh he expermenal values, seems ha he soluons obaned by he angen operaor are more aurae. Moreover, he angen operaor gves always more sable soluon requrng a redued number o eraon a eah load nremen. I an be seen ha he desenden branh obaned by usng onsan operaor s slghly deren due o he umulaed errors omng rom he large number o pos-pk eraons. Fgure (11) llusraes he rak growh pah durng he beam loadng, leadng o rupure suraes smlar han he ones expermenally obaned. To emphasze he large derenes beween hese wo sysem soluon shemes, he eraon numbers o reah he equlbrum a some spe load nremens are gven n Fgure (12). I s mporan o observe ha he onsan operaor sheme requres a very large number o eraons aer pk, and hs may lead o less aurae soluons. Fgure 11. Crak growh pah durng he loadng proess.

15 Fgure 12. Requred number o eraons or he esed soluon ehnques Mul-raured onree sruure In hs applaon, he sruure presened n Fgure (13) s analyzed. Ths s a onree beam subjeed o a our bendng es. Ths sruure has 1.5 m o lengh and 0.50 m o hgh as shown n he same gure. The maeral properes adoped or hs applaon were: ensle srengh 3.0 MPa, Young s modulus E 30,000 MPa, Posson s rao 0.20 and raure energyg 75 N m. In hs analyss, eleven raks were dsrbued along he lower sruural boundary as llusraed n Fgure (14). In he same gure s also presened he BEM mesh onsdered or hs analyss. Thereore, he perormane o he proposed non-lnear ormulaon n he ase o mulple raks s evaluaed. Only wo ohesve rak laws were onsdered n hs example: lnear and b-lnear. Boh ohesve laws were oupled wh angen operaor ormulaon. The 5 load was appled n 25 nremens and he adoped olerane whn eah nremen was10. F/2 F/2 0,5 m 1,5 m Fgure 13. Dmensons and boundary ondons or he sruure.

16 The load dsplaemen urves obaned n he analyss are shown n Fgure (15). Aordng hs gure, an be observed very smlar resuls rom hose llusraed n Fgure (10),.e, he sruural behavour aheved by he lnear CTO model s more rgd han hose obaned by b-lnear CTO model. In boh ases he soenng branh was vered, aer equal resuls or pre-pk sruural behavour. Fgure 14. Crak dsrbuon and BEM mesh adoped. Fgure 15. Load x Dsplaemen urves. Fgure (16) llusraes he rak growh pah observed unl he las load sep appled. Aordng hs gure, an be observed ha only ve o eleven raks have propagaed. Fgure 16. Crak growh pah.

17 7. CONCLUSION In hs paper, rak propagaon proess n quas-brle maerals has been suded. The omplex sruural behavour arsng rom hese maerals an be modelled by solvng a non-lnear sysem o equaons whh appears due he dependeny beween rak openng dsplaemen and raons along he normal dreon o he rak lps. To smulae hs nonlnear sruural problem, BEM has shown o be an aurae and een alernave. Two non-lnear BEM ormulaons have been developed and mplemened n hs paper. The rs apples a onsan operaor, where he non-lnear problem s solved by keepng onsan all relevan mares and alulang, a eah load sep, he non-equlbraed veor ore. The seond approah s developed by usng a angen operaor. In hs ase, he dervave se o he non-lnear equaons s used and he problem s aser solved. In hs ase he problem s solved mplly usng he angen operaor expresson. The ormulaon based on he angen operaor has shown o be more sable and lead o more aurae resuls n omparson wh he lassal proedure (onsan operaor). The use o angen operaor has shown o be always reommended o analyze rak propagaon problems, parularly or he ases where he aer pk regon s reahed. Ths knd o ormulaon may be appled n oher non-lnear problems n he uure. Cona problems and rak growh n renored sruures, or nsane, an be modelled usng hs een approah. Aknowledgemens Sponsorshp o hs researh proje by he São Paulo Sae Foundaon or Researh FAPESP - s grealy appreaed. 8. REFERENCES [1] Barenbla, G.I. The mahemaal heory o equlbrum raks n brle raure. Advanes n Appled Mehans. 7:55 129, [2]Carpner, A. Pos-peak and pos-buraon analyss o ohesve rak propagaon. Engneerng Fraure Mehans. 32: , [3]Dugdale, D.S. Yeldng o seel shees onanng sls. Journal o Mehans and Physs o Solds. 8: , [4] Hllerborg, A; Modeer, M; Peerson, P.E. Analyss o rak ormaon and rak growh n onree by mean o alure mehans and ne elemens. Cemen Conree Researh. 6: , 1976.

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