GRADIENT DAMAGE WITH VOLUMETRIC-DEVIATORIC DECOMPOSITION AND ONE STRAIN MEASURE

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1 Aam MECHANICS WOSAO AND CONROL RADIEN Vol. 30 No. 4 DAMAE 2011 WIH VOLUMERIC-DEVIAORIC DECOMPOSIION... Aam WOSAO * RADIEN DAMAE WIH VOLUMERIC-DEVIAORIC DECOMPOSIION AND ONE SRAIN MEASURE SUMMARY he paper presens a wo-fiel formulaion of he graien-enhance amage moel an is applicaion. his isoropic moel is characerize by wo amage parameers wih a volumeric-eviaoric ecomposiion. However, one srain measure governs he evelopmen of amage as for he scalar escripion. he heory is verifie by means of one-elemen benchmarks an also a more sophisicae simulaion, namely he spliing of concree cyliner in he razilian es is iscusse. eywors: isoropic amage, graien enhancemen, finie elemen meho RADIENOWY MODEL MECHANII USZODZEÑ Z ASJAOROWO-DEWIAOROW DEOMPOZYCJ I JEDN MIAR ODSZA CENIA Aryku³ przesawia wupolowe sformu³owanie graienowego moelu mechaniki uszkozeñ i jego zasosowanie. en izoropowy moel charakeryzuj¹ wa paramery uszkozenia z pozia³em aksjaorowo-ewiaorowym. Jenak- e jena miara oksza³cenia ecyuje o rozwoju uszkozenia jak w opisie skalarnym. eoria jes zweryfikowana za pomoc¹ esów jenoelemenowych, a ak e barziej zaawansowanej symulacji roz³upywania beonowego cylinra w zw. eœcie brazylijskim. S³owa kluczowe: izoropowy moel mechaniki uszkozeñ, moel graienowy, meoa elemenów skoñczonych 1. INRODUCION In he simples form of he amage heory one scalar parameer eermines he egraaion process. his efiniion of amage was firsly inrouce in (achanov, 1958). Alhough his amage concep was propose for isoropic coninuum, i can be generalize inroucing anisoropy. If amage irecions are isinguishe hen one scalar parameer is insufficien. In he lieraure amage vecors (rajcinovic an Fonseka 1981), secon-orer amage ensors (Murakami an Ohno 1981, een 1983, Liewka 1985) an fourh-orer amage ensors (Chaboche 1981, rajcinovic 1989, Lemaire an Chaboche 1990) have been efine. Differen escripions of he amage heory can be foun in (Skrzypek an anczarski 1999, Wu an Li 2008). However, numerical implemenaion of he heory wih fourh-orer amage ensors is complicae an a proper seup of maerial variables is ifficul. On he oher han, he scalar moel can be exene o wo amage parameers (Ju 1990) remaining wihin he isoropic escripion. Some auhors (Mazars an Pijauier-Cabo 1989, Comi 2001) employ he moificaion where amage is ecompose ino wo pars relae o ensile an compressive acions. However, he proposal iscusse in his paper resuls from a volumeric-eviaoric spli given for example in (Lubliner e al. 1989, Comi an Perego 2001). he firs amage parameer influences he bulk moulus an he secon one reuces he shear moulus. Hence wo ifferen amage growh funcions are isinguishe, bu one amage hisory parameer an one amage loaing funcion are assume. As a consequence, when he moel i is applie in a graien-enhance forma, one averaging equaion is sill he basis of a formulaion of finie elemen meho as for he scalar graien amage (Peerlings e al. 1996). he heory is briefly iscusse in Secions 2 an 3. he implemene moel is firs examine for one finie elemen, in Secion 4. he nex compuaional verificaion is performe in Secion 5 by he simulaion of he razilian cyliner spli es. Final remarks are given in Secion VOLUMERIC-DEVIAORIC SPLI he coninuum amage formulaion saisfies he isoropy coniion if wo amage parameers ω an ω for he volumeric an eviaoric pars respecively are consiere, cf. (Lubliner e al. 1989, Ju 1990, Comi an Perego 2001, Carol e al. 2002). he consiuive equaion becomes: σ = E ε (1) where: E = (1 ω ) ΠΠ + 2(1 ω ) Q (2) Here σ is he sress ensor an ε is he srain ensor. oh are consiere in a vecor form an ε = [ε 11 ε 22 ε 33 ε 12 ε 23 ε 13 ]. he amage parameer ω reuces bulk moulus an he parameer ω egraes shear moulus. he following relaions are inrouce: 1 3 Q = Q 0 ΠΠ (3) 1 3 where in hree imensions (3D): Π = [1, 1, 1, 0, 0, 0] an Q0 = iag 1, 1, 1,,, Qev = I ΠΠ (4) * Cracow Universiy of echnology, Faculy of Civil Engineering, Warszawska 24, Cracow, Polan; awosako@ L5.pk.eu.pl 254

2 MECHANICS AND CONROL Vol. 30 No Noe ha he srain an sress vecors are spli ino volumeric an eviaoric pars: 1 ε= θ ε ev 3 Π + (5) σ = Πp + ξ (6) an he variables on he righ-han sie are as follows: θ = Π ε is he ilaaion, ε ev = Qε is he eviaoric srain, 1 p = Π σ is he pressure an ξ = Q ev σ is he eviaoric 3 sress. he sress rae is obaine iffereniaing (1): &σ = Eε& ω& ÊΠΠ ε 2ω& Q ε (7) One amage hisory parameer κ is aope, i.e. κ = κ = κ, so ha one amage loaing funcion: f ( ε, κ ) =ε% ( ε ) κ = 0 (8) is assume. he uhn-ucker coniions are saisfie by he equivalen srain measure ε% an he amage hisory parameer κ. his means ha funcion ε% () ε can be efine analogically o he scalar amage moel. However, wo ifferen amage growh funcions are isinguishe: ω = ω (κ ) (9) ω = ω (κ ) (10) Since one hisory parameer κ governs he amage evoluion, he raes of amage parameers ω an ω uring loaing are respecively: ω κ ε% ω & = ε& κ ε% ε ω κ ε% ω & = ε& κ ε% ε (11) (12) is sill he basis of he wo-fiel formulaion like for he scalar graien amage (Peerlings e al. 1996). he above relaion involves he secon graiens of he average srain ε. he parameer c > 0 has a uni of lengh square an i is connece wih he inernal lengh scale l of 1 2 a maerial. he relaion c = l is erive for insance in 2 (Askes e al. 2000). Insea of he local equivalen srain ε% he average srain ε now governs he amage progress: f ( ε, κ ) = ε( ε% ( ε )) κ = 0 (14) he amage raes are compue as: ω κ ω & = ε & = ε & (15) κ ε ω κ ω & = ε & = ε & (16) κ ε he weak form of equilibrium equaions can be wrien as: δ ε σv = δ u bv + δu S (17) where u is he isplacemen fiel, b is he boy force vecor, is he racion vecor. A weak form of Equaion (13) is erive using reen s formula an he homogeneous naural bounary coniion: ( ε) ν =0: δε ε V + ( δε) c ε V = δε ε V % (18) he wo-fiel formulaion involves inepenen inerpolaions of isplacemens u an average srain measure ε in he semi-iscree linear sysem. he primary fiels are inerpolae in his way: an uring unloaing boh ω& an ω& are equal o 0. u = Na an ε=h e (19) 3. ISOROPIC RADIEN DAMAE I is by now common knowlege ha he graien enhance moel accoring o (Peerlings e al. 1996, eers 1997, Pamin, 2004) is nonlocal. During a failure process, from he onse of localizaion unil he oal loss of he siffness, he governing sysem of equaions remains ellipic an he regularizaion allows one o avoi a spurious mesh sensiiviy. his secion concerns he erivaion of isoropic amage wih a graien enhancemen. he equaions of he bounary value problem (VP) are almos he same as for scalar amage, bu he angen sress-srain relaion is efine in (7). Hence he averaging equaion: 2 ε c ε=ε% (13) where N an h conain suiable shape funcions. From he above inerpolaions he seconary fiels can be compue in he following way: ε = a an ε = g e (20) where = LN (L is he sanar ifferenial operaor marix) an g = h. he iscreize equaions mus hol for any amissible δa an δe, herefore: σv = N bv + N S ( hh + cgg ) ev = hεv % (21) (22) 255

3 Aam WOSAO RADIEN DAMAE WIH VOLUMERIC-DEVIAORIC DECOMPOSIION... he VP is linearize, hence a noal poins he incremens of he primary fiels from insan o insan + Δ are ecompose in such a way: a +Δ = a + Δa an e +Δ = e + Δe (23) Analogically, a inegraion poins we inrouce he ecomposiion of ε, σ an ε%. he equilibrium equaions hen become: ( σ +Δ σ)v = +Δ +Δ = N b V + N S an he averaging equaion has he form: ( hh + cgg )( e +Δ e)v = = h( ε % +Δε% )V (24) (25) he incremenal consiuive relaion is erive saring from Equaion (7): Δ σ = E Δa ΠΠ + 2Q ε h Δe where he amage incremens have been calculae as: (26) Δω = h Δe (27) Δω = h Δe (28) an he erivaives are eermine a insan.we can rewrie Equaion (24) in a marix form: +Δ aa Δa+ ae Δ e = fex f in (29) where: aa = E V (30) ae = [ ΠΠ Q] ε h V +Δ +Δ +Δ = V + (31) f ex N b N S (32) fin = σ V (33) In Equaion (25) he incremen of equivalen srain measure Δε% is compue from he inerpolae isplacemen incremen Δa: ε% Δε % = Δ = Δ ε s a ε an Equaion (25) can be formulae as follows: (34) eaδ a+ eeδ e = fε f e (35) he marices an vecors in Equaion (35) are similar o he scalar amage formulaion, cf. (Peerlings e al. 1996): ea = h s V (36) ee = ( hh + cgg )V (37) fε = h ε % V (38) fe = eee (39) Evenually, he following sysem of equaions is use: +Δ aa ae Δa f ex fin = ea ee Δe fε fe 4. ONE-ELEMEN ENCHMARS (40) Firsly wo benchmarks wih one finie elemen (FE) are compue. hese simple ess permi one o observe he behaviour of he isoropic amage a he poin level. eer unersaning of his moel is essenial o prouce more avance analyses ension in one irecion One hree-imensional finie elemen (FE) wih eigh noes is subjece o saic ension in one irecion. he aa for elasiciy are as follows: Young s moulus E = MPa, Poisson s raio ν = he normalize elasic energy release rae (Ju 1989) is employe in compuaions as he loaing funcion, however he Mazars efiniion (Mazars 1984) an he moifie von Mises efiniion (e Vree e al. 1995) for his kin of ension es are all equivalen, i.e. κ = ε 11.We emphasize ha amage hreshol κ o = influences one inepenen hisory parameer κ. he simple moificaion of scalar heory gives a possibiliy o aop of wo ifferen amage growh funcions, separaely for he egraaion of bulk moulus an shear moulus. All calculae cases are juxapose in able

4 MECHANICS AND CONROL Vol. 30 No here are presene symbols use o isinguish a given case an corresponing amage growh aa. During he amage evoluion ω can grow in ifferen ways. Linear sofening law (Peerlings e al. 1996): κu κ κo ωκ ( ) = κ κu κo (41) is inrouce in his example in orer o illusrae he properies of he isoropic amage moel. In he amage evoluion firsly κ excees hreshol κ o an nex ens o ulimae value κ u which correspons o oal amage. Here equal or ifferen ulimae values κ u are assume for he volumeric an eviaoric egraaion. Hence inex i =, permis one o isinguish he analyze processes. he cases wih ienical aa for boh amage growh funcions can be reae as reference (sanar scalar amage). lin lin, able 1 ension in one irecion compue cases Symbol of case Linear sofening Ulimae value of hisory parameer Volumeric κ u = κ u = Deviaoric κ u = κ u = Diagrams of srain ε 11 versus ω an ω in he final phase of amage are shown in Figure 1 for chosen cases. hey can help one o unersan how ifferen amage growh funcions influence he resuls. he non-monoonic funcion ω(ε 11 ) in Figure 1(b) resuls from snapback behaviour which is seen in Figure 2. In Figure 2 he cases wih linear sofening are presene. As expece, afer he peak, iagrams for lin, an lin, sar o ecline beween iagrams for lin an lin,&. However, he nonlinear characer of sofening is surprising for he linear amage growh efine for cases lin, an lin,. Moreover, in case lin,, where he eviaoric par has larger κ u, arclengh conrol mus be use because of he snapback effec. herefore, i is enough o change he ulimae value κ u for given volumeric or eviaoric par in orer o obain a nonlinear escen of he sress-srain iagram. a) b) lin, κ u = κ u = lin,& exp exp, exp, Symbol of case exp, & κ u = Exponenial sofening Duciliy parameer Volumeric κ u = Deviaoric η = 1000 η = 1000 η = 750 η = 1000 η = 1000 η = 750 η = 750 η = 750 c) Exponenial sofening law (Mazars an Pijauier-Cabo 1989, Peerlings e al. 1998): κo η( κ κ ) ωκ ( ) = 1 1 α+αe o κ (42) ) is also analyze because his funcion is aope in furher compuaions. Here he parameers η an α are responsible for he rae of sofening an resiual sress which in one imension ens o (1 α)eκ o. In his analysis uciliy parameer η is varie. he amage evoluion is governe by ominaing volumeric or isorion failure. he same numerical sabiliy parameer α = 0.98 is assigne. I is also possible o aop wo ifferen amage laws uring he failure process, cf. (Wosako 2008). Fig. 1. Developmen of amage parameers in amage-srain iagrams 257

5 Aam WOSAO RADIEN DAMAE WIH VOLUMERIC-DEVIAORIC DECOMPOSIION... Fig. 2. Influence of ulimae u i κ (i =, ) in linear sofening Figure 3 epics he iagrams for amage growh funcions relae o exponenial sofening. he inerpreaion is easy since each sofening branch has an exponenial characer. For cases exp, an exp, wih ifferen uciliy parameers he iagrams run beween he exreme cases exp an exp,&. Analogically o he cases wih linear funcions a cerain regulariy can be noe. If he fracure energy for he eviaoric par is increase (compare cases exp an exp,), i gives a larger ifference in response han for larger fracure energy aope for he volumeric par. I is oubful o consier he concep of fracure energy only for a chosen par of he siffness, however in orer o simplify he explanaions his concep is use here. I is shown in Figure 4 how his parameer changes. In he case of linear sofening he value of ν ω rasically ens o a lower or upper limi. hese limis can be perceive as conroversial resuls. Such exreme behaviour in simulaion an as a consequence nonlinear relaion beween ε 11 an σ 11 seems o be unesirable. A complee egraaion for he volumeric par in case lin, gives finally ν ω equal o 1. On he oher han he zero shear siffness in case lin, leas o ν ω = 0.5 like for incompressible maerials, cf. (Carol e al. 2002). If exponenial sofening is use (see Figure 4, case exp,) a smooh rop o zero is observe, bu his is no always so. he saring value of ν ω an also he configuraion of he consiere es ecie on wheher ν ω becomes negaive. For quasibrile maerials like concree generally i is expece ha Poisson s raio ens o 0 uring he amage evoluion (Carol e al. 2002) Willam s es Fig. 4. Sensiiviy of Poisson s raio ν ω Fig. 3. Influence of uciliy parameer η i (i =, ) in exponenial sofening If ω ω he Poisson s raio is no consan. I can be compue as (Wosako 2008): ( ω) ( ω) ( ω ) + ( ω ) ν ω = (43) so ha Poisson s raio epening on he siffness egraaion is inrouce. o isinguish he Poisson s raio which is given as an elasic maerial parameer from he one efine in Equaion (43) he subscrip ω is use. Hence, his new parameer ν ω is compue uring he amage process. he ension-shear es calle also Willam s es was compue he firs ime in (Willam e al. 1987). I serves he purpose of verificaion of inelasic maerial moels a he poin level. he resuls of paricular moels can be ifferen even if hese moels in uniaxial ension exhibi a quie similar behaviour, as i was noice in (Winnicki an Cichoñ 1996), so he simulaion of ension-shear loaing process a he poin level complees he numerical analysis of a given moel. One finie elemen wih four noes in plane sress is subjece o loaing in wo phases: I. Uniaxial horizonal ension wih verical conracion ue o he Poisson s effec, accoring o he relaion beween he srain incremens: Δε 11 : Δε 22 : Δγ 12 = 1 : ν : 0. hese coniions are obeye unil he ensile srengh is aaine. II. Immeiaely afer he ensile srengh is reache he change of configuraion is enforce. he proporions for he srain incremens are arrange as follows: Δε 11 : Δε 22 : Δγ 12 = 0.5 : 0.75 : 1. his relaion inuces ension in wo irecions an aiionally a shear srain. As a consequence a roaion of principal srain axes occurs, bu he ension regime is preserve. 258

6 MECHANICS AND CONROL Vol. 30 No he es is passe if he maximum principal sress is lower han or a mos equal o he given uniaxial ensile srengh, cf. (Pivonka e al. 2004). he secon coniion is ha finally all sress componens shoul converge o zero. he se of aa is base on Pivonka e al. (2004), namely Young s moulus E = MPa, Poisson s raio ν = 0.20, an he remaining parameers are une o uniaxial ensile srengh f = 3 MPa, uniaxial compressive srengh f c = 38.3 MPa an ensile fracure energy f = 0.11 N/mm. Moifie von Mises efiniion (e Vree e al. 1995) is use, so he raio beween compressive an ensile srengh is equal o k = f c /f he hreshol is calculae as he quoien of he ensile srengh an Young s moulus κ o = f /E = In his es exponenial sofening is consiere. he firs parameer α is equal o 1.0, which means ha a complee loss of siffness is amie. As previously four cases wih ifferen prescribe uciliy parameers are consiere. he basic case exp is for pure scalar amage wih η = η = Such a value seems o be unrealisically huge, bu resuls in a fas maerial failure. Nex wo cases wih ifferen uciliies for he volumeric an eviaoric amage are calculae. he case wih more ucile exponenial sofening for he eviaoric par wih parameers η = 4000 an η = 2000 an as an effec larger volumeric amage ω is calle exp,. he opposie aa η = 2000 an η = 4000 are use for he case calle exp,, where amage for he eviaoric par governs he soluion. o complee he resuls of Willam s es he case exp,& wih boh smaller parameers η = η = 2000 is also analyze. In Figure 5 iagrams for cases exp an exp,& eermine bouns for cases wih ifferen uciliy parameers. he sress-srain iagrams for exp, an exp, are foun in exchange posiions comparing wih he example of uniaxial ension. I seems surprising, bu he Willam es eals wih a ifferen loaing hisory. In Figure 5(b) shear relaion γ 12 σ 12 is epice. A full agreemen beween iagrams for exp an exp,, an also beween iagrams for exp,& an exp, is characerisic for his es. I is connece wih he fac ha hese respecive cases have he same parameer for he eviaoric par. In Figure 6 he componens of sress ensor ogeher wih principal sresses are epice versus srain ε 11. he figure shows he resuls only for he isoropic moel aking ino accoun opions wih ifferen uciliy parameers. I is noice ha for case exp, he secon principal sress has negaive values for large ε 11. he final enency is ha all componens converge o zero. Figure 7 is ploe for all consiere cases. As i is epice in enlarge secor he zero value is finally reache. If he parameer α is less han 1.0, he resiual values remain, see (Wosako 2008). a) b) a) b) Diagrams of relaions γ 12 σ 12 Fig. 5. Willam s es influence of uciliy parameer η i (i =, ) Fig. 6. Willam s es comparison of sress componens 259

7 Aam WOSAO RADIEN DAMAE WIH VOLUMERIC-DEVIAORIC DECOMPOSIION... Fig. 7. Willam s es evoluion of principal sresses for isoropic moel are confrone, in (Meschke e al. 1998) muliirecional kinemaic sofening amage-plasiciy an fixe crack moels are ese. Due o a ouble symmery only a quarer of he omain (wih raius equal o 40 mm) is consiere. he general geomery aa are base on (Winnicki e al. 2001), bu plane srain coniions are assume. he loa is applie o he specimen inirecly via a siff plaen (wih 5 mm, heigh 2.5 mm). he plaen is perfecly connece wih he specimen. In hese compuaions only one mesh shown in Figure 9 is employe, in orer o focus on he response of isoropic version of graien amage. However, mesh insensiiviy for scalar graien amage is wiely iscusse in (Wosako 2008). he loa acs ownwars a he op of he plaen. he maerial aa are juxapose in ables 2 an 3. I is shown here ha no only he inernal lengh parameer in he graien moel ecies abou he resuls of he es, bu also oher parameers, for insance he choice of he amage growh funcion, in paricular for he volumeric or eviaoric par of amage. Four opions of exponenial sofening are analyze, where ifferen combinaions of values of uciliy parameer η ecie wheher he amage process is more or less brile. he parameer α equals 0.99 for each case. A snapback response is possible so he es is compue applying he arc lengh meho. Fig. 8. Willam s es sensiiviy of Poisson s raio ν ω. Differen relaions beween parameers α an η i (i =, ) If α is equal o 1 he lower or upper limi of ν ω is reache, cf. Figure 8. he value of ν ω can reurn o he iniial one, bu for α smaller han 1.0. For quasi-brile maerials a ecreasing of Poisson s raio is expece (see also Carol e al. 2002), bu negaive values are raher non-physical an unesirable. Aenion is focuse on case exp, where he values of ν ω are less or a mos equal o given in aa. I is possible o seup boh parameers α an η separaely for volumeric an eviaoric par in such way ha negaive Poisson s raio is prevene, see (Wosako 2008). Fig. 9. Mesh for razilian es able 2 razilian es maerial moel aa 5. RAZILIAN ES he spliing effec is use o esablish he ensile srengh in quasi-brile maerials, because he compression beween he loaing plaens inuces he perpenicular ensile force acion in he mile. his phenomenon an he snapback response in he razilian es is no easy o reprouce in numerical compuaions, because uner he plaen a plasic zone of slip can appear or amage can localize in a small region. In fac, ifferen mechanical moels have been verifie numerically using his es, for example in (Chen an Chang 1978) plasiciy heories are analyze, in (Feensra 1993) roaing crack an plasiciy moels Specimen: Young s moulus: Poisson s raio: Equivalen srain measure: Fracure energy: Inernal lengh scale: hreshol: Plaen: Young s moulus: Poisson s raio: amaging E c = MPa ν = 0.15 moifie von Mises, k = 10 f = N/mm l = 6 mm, i.e. c = 18.0 κ o = ?5 elasic E s = 10 E c ν =

8 MECHANICS AND CONROL Vol. 30 No able 3 razilian es compue cases Symbol of case η η Damage growh exp more inensive his means ha he eviaoric amage is more imporan in he siffness egraaion an ecies abou he proper behaviour in he razilian es. Conour plos in Figures are epice for he peak an he final sae, he respecive poins A an are marke in Figure 10. exp, ω < ω exp, ω > ω exp,& less inensive a) b) c) ) Fig. 10. razilian es influence of uciliy parameer ηi (i =, ), loa-isplacemen iagrams he loa-isplacemen iagrams in Figure 10 are race for he four consiere cases. I is noice ha for cases exp, an exp he sofening pahs are monoonic an wihou any snapback. he same value of parameer η = 1200 leas o similar soluions. On he oher han, for cases exp, an exp,& he snapback response is simulae. a) e) g) f) h) b) Fig. 12. Damage paerns in razilian es Fig. 11. razilian es average srain ε he spliing is obaine only for he cases which correspon o he larger value of fracure energy f for he eviaoric par, i.e. he uciliy η = 600. I is confirme by means of Figure 11, where he isribuions of average srain are ploe for he wo sages poins A an. herefore he ineracion beween he compressive loaing an he ensile response seems o be ransferre via he eviaoric characerisics in he moel. Figure 12 shows he 261

9 Aam WOSAO RADIEN DAMAE WIH VOLUMERIC-DEVIAORIC DECOMPOSIION... amage paerns for cases exp, an exp,. Accoring o he assumpions inclue in able 3, he ominaion of amage ω in case exp, an inversely ω in case exp, is noice. he spliing effec observe for case exp, is expece in he razilian es an he coincience wih he ecrease of ν ω uring he process is suiable for concree. 6. FINAL REMARS Isoropic amage wih he volumeric-eviaoric spli is enhance by averaging equaion wih secon orer graien erm. wo ifferen amage growh funcions are assume, however one average srain measure is hol. his approach wih one srain measure involves one amage loaing funcion. Hence wo-fiel finie elemen formulaion as for he scalar graien amage is obaine. I is shown even in one FE ension es ha one of he feaures of he isoropic moels is ha evolving Poisson s raio is simulae, which is characerisic for egraing quasi-brile maerials. I is however quesionable wheher is negaive values shoul be amie in he simulaion process. In his moel his is no assure, hence only by means of appropriae values for he parameers of he moel negaive Poisson s raio can be avoie. However we can overcome his problem, if we apply he moel wih a resricion similar o propose in (anczarski an arwacz 2004). he resuls for all he opions of he moel saisfy he coniions o pass he Willam es. he case wih exp, seems o be he mos promising, because for his case he behaviour similar o quasi-brile maerials is obaine. A oal separaion of he volumeric an eviaoric pars is possible. In a more general approach wo amage loaing funcions are inrouce, wo ifferen averaging equaions an hree-fiel formulaion are erive. A similar approach wih oal separaion of amage parameers is escribe in (Carol e al. 2002), where wo amage variables influence wo loaing funcions. Such moel is calle here biissipaive isoropic moel. Acknowlegmens Valuable iscussions wih Prof. Jerzy Pamin an Prof. Anrzej Winnicki from Cracow Universiy of echnology are graefully acknowlege. he compuaions were performe using he evelopmen version of he FEAP program of Prof. R.L. aylor. References Askes H., Pamin J., e ors R. 2000, Dispersion analysis an elemen-free alerkin soluions of secon- an fourh-orer graien-enhance amage moels. In. J. Numer. Meh. Engng, 49, pp een J. 1983, Damage ensors in coninuum mechanics. J. Méc. héor. Appl., 2(1), pp Carol I., Rizzi E., Willam. 2002, An exene volumeric/eviaoric formulaion of anisoropic amage base on a pseuo-log rae. Eur. J. Mech. A/Solis, 21(5), pp Chaboche J.-L. 1981, Coninuous amage mechanics: a ool o escribe phenomena before crack iniiaion. Nuclear Engng. an Design, 64(2), pp Chen W.F., Chang.Y.P. 1978, Plasiciy soluions for concree spliing ess. ASCE J. Eng. Mech. Div., 104(EM3), pp Comi C. 2001, A non-local moel wih ension an compression amage mechanisms. Eur. J. Mech. 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