MICROMECHANICAL MODELLING OF DAMAGE PROCESSES IN COMPOSITE MATERIALS

Transcription

1 HE 9 H NERNAONAL ONFERENE ON OMPOSE MAERALS MROMEHANAL MOELLN OF AMAE PROESSES N OMPOSE MAERALS. vancevc,. Smojver * epartment of Aeronautcal Engneerng, Faculty of Mechancal Engneerng and Naval Archtecture, Unversty of Zagreb, Zagreb, roata * orrespondng author (smojver@fsb.hr) Keywords: mcromechancal model, multscale analyss, composte materals, falure crtera ntroducton amage processes of heterogeneous materals are determned by processes at the mcromechancal level. As to nclude these effects n the numercal smulaton of engneerng problems, fnte element solvers need to be coupled to mcromechancal models. hs work features an example of the applcaton of the multscale method on the problem of falure and damage analyss of undrectonal composte materals. he smultaneous analyses n the presented methodology have been acheved by usng the Abaqus/Explct solver for the macro-scale problem, whle the Hgh Fdelty eneralzed Method of ells (HFM) s used to evaluate the stress and stran state at the mcromechancal level, as llustrated n Fg.. he mcro-scale analyss determnes the stress and stran feld wthn the unt cell, whch s a perodcally repeatng volume that characterzes the mcro-structure of the heterogeneous materal. Effectve mechancal propertes of the composte materal are calculated based on the unt cell morphology and the consttutve behavor of ts consttuents. As the stress dstrbuton wthn the unt cell s known, calculaton of falure crtera and consttutve response of the composte materal are performed on the mcro-level. Furthermore, degradaton of materal propertes due to damage effects s also beng modeled on the mcromechancal level by decreasng consttuent mechancal propertes at the subcell level. he descrbed methodology s stll n the development phase, n whch dfferent composte falure crtera and degradaton models are beng evaluated. hs work features the results of the comparson of mcromechancal falure crtera. Hgh Fdelty eneralzed Method of ells he dscretzaton scheme of the appled mcromechancal model s shown n Fg.. As the theoretcal background of the HFM model s relatvely complex, only the basc nformaton, whch are needed to understand ths paper, are gven here. More detals about HFM models and ther mplementaton nto ths methodology can be found n [-6]. Some more recent and nterestng applcatons of mcromechancal models based on the Method of ells model are provded n [7-8]. he representatve volume element of the composte materal s dscretzed usng Nβ Nγ rectangular subcells. he number of dfferent materal phases wthn the unt cell as well as the number of subcells s arbtrary. he partcular verson of the HFM model used n ths work s the reformulated HFM, whch has been ntroduced n [-4]. he reformulaton enhances the computatonal effectveness of the model and smplfes the unt cell modelng by departng of the generc cells concept of the orgnal HFM model []. Snce the appled model s a two-dmensonal mcromechancal model, t s sutable only for undrectonal composte materals. he doubly perodc unt cell s modeled wth fbers orented n the x drecton and arranged n a doubly perodc array n the x and x 3 drectons. he dsplacement feld wthn the unt cell s approxmated usng the Legendre-type polynomal expanson, after [,]: u ε x + W + y W + ( β) j j (00) (0) ( ) (, ) h γ βγ ( ) (, ) 3 (0) 3 β β βγ y W + y W(0) + 4 l 3 ( γ) γ y3 W (0), j,,3. 4 ()

2 Fg.. Mult-scale approach to damage modelng. he frst step of the mcromechancal model s calculaton of the subcell local stffness matrces based on the subcell mechancal propertes and dmensons. he stffness matrces are merged nto the global system of equatons by applcaton of dsplacement and tracton contnuty condtons between the subcells, and perodcty condtons n the x and x 3 drectons on the unt cell boundares. Addtonally, four dsplacement components are constraned n order to prevent unt cell rgd body moton as dsplayed by the four arrows at the unt cell borders n Fg.. he soluton of the global system of equatons enables calculaton of the stran concentraton tensor A whch relates the macroscopc stran tensor ε to the mcroscopc stran state of each subcell ε, after Eq.. ε A ε () Fg.. HFM unt cell dscretzaton and boundary condtons. 3 mplementaton nto Abaqus/Explct he mcromechancal model has been mplemented nto Abaqus/Explct usng the materal user subroutne VUMA. HFM s programmed as a subroutne whch s called at every ntegraton pont of the FE model. ntal undamaged composte materal propertes and ntal stran concentraton tensors are calculated at the start of the analyss. f the materal propertes are not degraded throughout the analyss, the stran concentraton tensors do not change ther values. hs enables calculaton of mcromechancal falure crtera wthout solvng of the complete system of equatons, as the stress dstrbuton wthn the unt cell s calculated usng the ntal stran concentraton tensors. Once the subcell materal propertes have been changed due to damage effects, the stran concentraton tensor has to be updated as to account for the modfed subcell elastcty propertes. n order to be able to evaluate dfferent mcromechancal falure and degradaton models, a standalone applcaton has been developed whch smulates nput from the Abaqus/Explct analyss to the HFM/VUMA materal model for varous macroscopc load cases. he results gven n ths work are obtaned usng ths standalone HFM model. As the structure of HFM subroutnes s the same for the standalone applcaton as for the mult-scale envronment, the secondary role of the standalone applcaton s the testng and debuggng of the HFM subroutnes. 4 Falure crtera and damage models he frst step n the process of mcromechancal damage modelng of composte materals s the applcaton of falure ntaton crtera. hese crtera, at the mcro-scale, ndcate ntaton of degradaton processes at the subcell level wthn the HFM model. As a subcell, n the smplest case of HFM, s occuped by ether fber or matrx materal, the stress or stran state wthn the unt cell s compared wth the consttuent stress or stran allowable values. n order to obtan the mcromechancal falure curves of the complete composte materal, fber and matrx falure has to be taken nto account smultaneously, as falure ntaton n the consttuents occurs under dfferent loadng condtons. hs work features the comparson of three falure ntaton models appled at the mcromechancal level: a) he frst falure model s taken from [9]. Falure ntaton n matrx subcells s calculated usng the 3 sa Hll crteron defned as

3 MROMEHANAL MOELLN OF AMAE PROESSES N OMPOSE MAERALS ( σ ) + ( σ ) + ( σ33 ) + Y σ σ σ σ σ σ Y ( σ ) + ( σ ) + ( σ ) 3 3 d, m + (3) where Y s the matrx transverse strength and s matrx shear strength. he value of the matrx strength used n Eq. 3 depends on the type of the appled transverse stress, after [9] Y Y σ < 0 Y c Yt σ > 0. (4) he crteron ndcates ntaton of falure processes when the rght-hand sde term d m takes values greater than. Falure ntaton n fber subcells s ndcated usng the maxmal stran crteron, whch s formulated as, U d f ε ft ε ε > 0. (5) ε denotes the subcell stran n the fber drecton U whle ε ft s the ultmate fber stran n the fber drecton whch can be calculated from tensle and compressve strength propertes provded n able. Smlarly to the matrx crteron, falure s ntated f the rght-hand sde term d reaches values above. b) he second falure model whch has been nvestgated n ths work employs the falure crtera appled by the Multcontnuum heory (M) [0]. he fber and matrx falure crtera have been used on the mcro-scale for the World Wde Falure Exercse (WWFE) wthn a fnte element mcromechancal model. As opposed to the orgnal M, the falure crtera n ths work have been appled to the HFM mcromechancal model, consequently takng nto account the stress varatons wthn the unt cell nstead of dealng wth averaged consttuent values. he expressons for the crtera have been derved from the quadratc nteracton falure crteron, by takng nto account assumptons on the falure modes for the fber and f matrx consttuents. Falure ntaton n the matrx s predcted f the stress state satsfes the relaton where K + K, (6) 3m 3 4m 4 K K 3m Sm + S33m 4m Sm, (7). (8) S m and S 33m are matrx strengths n the and 3 drectons, whch are dependent on the tensle/compressve character of the loadng n the drecton [0]. he shear strength of the matrx S m s not provded n [], therefore ts value s estmated to be half of the tensle strength value. hs estmaton s vald for epoxy matrces, as stated n []. Fber falure s predcted usng the equaton where and K + K, (9) K K f 4 f 4, (0) f S f. () 4 f S f he S f term denotes the strength n fber drecton, whch can be tensle or compressve, whle S f s the fber shear strength. he relatons n Equatons 6 and 9 nclude the transversally sotropc stress nvarants, expressed as [0] σ, σ + σ, 33 3 σ + σ33 + σ3 4 σ + σ3., () c) he last falure theory examned n ths work s the 3 Hashn-type stran based falure crteron for the matrx combned wth the maxmal strength crteron for the fber, as used n [3]. he damage 3

4 degradaton model ntroduced n [3] has also been ncorporated n the presented multscale procedure. he contnuum damage model presented n [3] accounts for multaxalty and progressve degradaton of subcell elastcty propertes. he falure crteron for the matrx subcells employs damage strans defned as ε γ 3 γ ε + +, Xε Rε Sε ε γ 3 γ ε + +, (3) Yε Qε Sε ε 33 γ 3 γ 3 ε3 + +, Zε Qε Rε where the X ε, Y ε and Z ε varables are the normal falure strans whle R ε, Q ε and S ε are engneerng shear falure strans. he damage strans defned n Eq. 3 have the same functon as the left-hand sdes of commonly used falure crtera for fber renforced composte materals, ndcatng ntaton of falure processes f ther values reach.0. Falure wthn fber subcells s predcted usng the maxmal stran crteron, defned as σ X tc,. (4) f fber subcells reach falure ntaton, mechancal propertes are nstantaneously set to a very low value (0.000 tmes the ntal value) thereby smulatng complete subcell falure. he damage progresson model s appled only for matrx subcells. he degradaton processes of matrx subcells are ntated f the damage strans reach values hgher than.0 or the maxmal obtaned value durng prevous load ncrements. amage evoluton n matrx materal s tracked usng sx dfferent damage varables, three for tenson and three for compresson. he ncremental changes of damage varables and assocated degradaton of mechancal propertes durng a load ncrement are calculated usng the relaton ' dε ( ) ε d k,, j,,3, (5) where dε s the ncrement of the damage stran, ' whle k s the modfed slope parameter calculated as k ' / B Ae ε,,,3, (6) wth A and B as the post-damage slope parameters. he damage model separates degradaton processes n tensle and compressve falure modes old + d for ε > 0, + d for ε < 0. old (7) he progressve degradaton model modfes only matrx subcell mechancal propertes. he elastcty propertes are degraded usng E de for,,3, (8) 0 ν dν for, j,,3. (9) 0 j j he d varables are calculated as b for σ > 0 d b for σ < 0,,,3 (0) where b parameters are the scalng parameters. he fnal falure of matrx subcells s predcted usng damage energy prncples. he stran energy densty s calculated usng the relaton dw { σ( ε + dε) ε( σ + dσ) },,,3. () able. onsttuent propertes as lsted n [] E-glass fber propertes E [ Pa] ν [ ] X [ MPa] X [ MPa] Epoxy matrx propertes E [ Pa] [ ] S ν X [ MPa] X [ MPa] X [ MPa] he fnal falure crteron s assocated wth the loadng modes - Mode (openng), Mode (nplane shear) and Mode (out of plane shear). he mode-specfc stran energy densty release rates, after [3], are

5 MROMEHANAL MOELLN OF AMAE PROESSES N OMPOSE MAERALS d dw dw, dw dw, 6 d+ d d dw dw, dw dw, 5 d+ d3 d d dw dw, dw dw,() dw 4 6 d + d3 d+ d d dw dw, dw dw, d+ d3 d 3 3 d + d3 dw. 4 he assocated stran energy release rates are l dw l dw b d b6 d l dw l dw b5 d b d l dw l dw b4 d b6 d l3 dw 3 l3 dw b33 d3 b53 d3 3 3 l3 dw b43 d3,,,,,,,,, (3) where b j denote scalng parameters, whle l s the characterstc materal length. omplete degradaton of matrx subcells propertes n tensle loadng modes s determned f the mode-specfc stran energy release rates reach the materal crtcal stran release rate., M,,. (4) M M For falure predcton n compressve loadng modes a crteron based on the total dsspated energy has been employed. he crteron has the form ( W ) + W + W V WS,,,3, (5) where V s the volume of the materal and W S s the crtcal compressve stran energy. An example of the propertes for the degradaton model s provded n able 3, after [3]. 5 Results Fg. 3 shows the vsualzaton of the matrx falure ntaton crtera n the σ σ33 plane. he mechancal propertes of the matrx for ths study are taken from [] for the MY750/HY97/Y063 epoxy matrx and are lsted n able. he dscontnutes n the 3 sa-hll and M curves arse from the dfference between the compressve and tensle strengths of the matrx. he comparson of the three falure curves shows that the 3 sa- Hll and the Hashn curve match reasonably well, whle the M curve devates from the other two curves n some regons of the stress space. hs devaton can be explaned by the fact that the M matrx falure crteron, defned n Eq. 6, has been derved wth the assumpton that the matrx s used wthn a fber renforced materal. he complete dervaton of the M falure crtera, along wth the modfed mechancal propertes for matrces used n the WWFE can be found n [0]. Fg 3. Falure curves n σ σ33 stress space for the MY750/HY97/Y063 epoxy matrx, n MPa. he nvestgated composte mcromechancal falure models have been valdated on a composte materal consstng of the Slenka E-glass fber and the MY750/HY97/Y063 epoxy matrx wth 60% fber volume fracton, as used n the WWFE []. Vsualzaton of the evaluated falure models for fber renforced composte materals n the homogenzed σ σ stress plane s shown n Fg. 4. he falure curves have been predcted usng a 40 x 40 HFM model wth a sngle fber at the unt cell centre. All results shown n ths work have been derved usng the standalone applcaton as descrbed n Secton 3. At the mcromechancal level the fber and matrx consttutve relatons can be approxmated by an 5

6 sotropc materal model whch s a common assumpton used n mcromechancal analyses as suggested n [9,3-5]. he homogenzed mechancal propertes calculated usng the HFM dffer from the composte ply propertes provded n [], as lsted n able. fferences between mcromechancally calculated and expermentally obtaned composte propertes are often encountered n the lterature e.g. [0, 3, 4]. hese dfferences are explaned by the assumpton that the consttuent propertes n a composte materal dffer from the orgnal consttuent propertes. herefore, mcromechancal analyses use modfed consttuent mechancal propertes, whch are selected as to match homogenzed propertes wth expermentally obtaned results. able. omposte lamna propertes Provded n [] 40 x 40 HFM E [ Pa ] E [ ] E Pa 3 [ Pa ] ν [-] ν [-] falure occurs. he 3 sa-hll crteron, on the other hand, predcts falure ntaton at sgnfcantly hgher load states. he falure curves n Fg. 4 show the loadng state at whch the HFM model ndcates falure of the frst subcell. As to get nsght nto the real dfferences between the evaluated falure models, Fg. 5 shows the dstrbuton of the value of matrx falure crtera at the tensle load state at whch the M crteron ndcates falure ntaton. Fg. 5. strbuton of the matrx falure crtera wthn the unt cell for the load state at whch the M falure crteron ndcates falure ntaton. he dstrbuton of the M crteron shows that the falure ntaton state s reached n a relatvely large area of the unt cell at the same load ncrement. he maxmal values of the other crtera for ths loadng state are and 0.95 for the 3 sa-hll and Hashn crteron, respectvely. Fg. 6 shows the comparson of the matrx falure crtera for the load state at whch the 3 sa-hll reaches the falure state. he damage degradaton effects have been neglected for ths comparson. he results show that at ths load state falure ntaton has been predcted n a large number of subcells for the M and the Hashn crtera. Fg. 4 Falure curves n the macro-level σ σ plane, n MPa. he dscontnutes of matrx falure curves, observable n Fg. 3, are also present n the results for the composte materal. he three evaluated falure curves compare reasonably well to each other, especally for the fber falure loadng modes (compressve loadng n the fber drecton). he largest dfferences occur n the σ loadng curves, where the M falure theory s the most conservatve by predctng lower values at whch Fg. 6. omparson of falure crtera dstrbuton for the load state at whch the 3 sa-hll matrx falure crteron reaches the crtcal value. he results of the progressve damage degradaton model are shown n Fg. 7. For ths analyss the consttuent mechancal propertes have been modfed as to be able to compare the mplementaton of the model wth the results

7 MROMEHANAL MOELLN OF AMAE PROESSES N OMPOSE MAERALS provded n [3]. he modfed mechancal propertes are lsted n able 3. he fnal falure crteron of the complete unt cell s not specfed n [3]. he fnal falure curve n Fg. 7 has been constructed by settng the complete degradaton of the unt cell to occur when 0% of the total number of matrx subcells fals. he unt cell falure crteron s very mportant snce the falure of the unt cell ndcates the falure of the materal pont for the case n whch the HFM s appled wthn the multscale envronment, as explaned n Secton 3. herefore, ths parameter controls the element deleton crteron n the FE analyss. he ntal subcell falure curve s very close to the fnal unt cell falure curve, thus only the fnal unt cell curve s shown n Fg. 7. he HFM model used for ths analyss s dscretzed usng 30x30 subcells. able 3. Modfed consttuent propertes and parameters of the degradaton model as specfed n [3] Slenka E-glass propertes E [ Pa] ν [ ] X [ MPa] X [ MPa] MPa. he frst subcells n ths load case fal at the σ loadng of 36.4 MPa, and the contour of the faled subcells compares well to the ε dstrbuton shown n Fg. 5 and Fg. 6. he 0% matrx falure condton s reached at 36.6 MPa. Fg. 7: ntal and fnal falure curves of the E-glass epoxy composte wth the falure and degradaton model provded n [3]. MY750/HY97/Y063 epoxy propertes E [ Pa] ν [ ] X ε [ ] X ε [ ] [ ] S ε Post-damage slope parameters A 0.7 A.0 B 0.8 B 0.96 b Scalng parameters b.3 b4 b5 b rtcal energy release rates Property Symbol Value Mode SERR Mode and SERR rtcal compressve stran energy Materal length 800 J/m 400 J/m 6 W S.86 0 J 5 l m Fg. 8 shows the evoluton of the subcell falure process for loadng n the tensle σ drecton. he damage stran n the drecton ndcates ntaton of degradaton processes n the frst subcells at 33.5 Fg. 8. Subcell falng process n the σ loadng drecton. 6 onclusons he three evaluated falure crtera compare reasonably well to each other. urrently, a sophstcated damage progresson model has been mplemented nto the model. he lterature survey on mcromechancal falure models shows the need for progressve degradaton models. As ndcated n e.g. [6], the sudden loss of mechancal propertes s a too conservatve approach to mcromechancal damage modelng and the progressve nature of damage has to be taken nto account as to be able to replcate expermental results. he progressve damage model wthn the HFM framework employs contnuum damage prncples at the subcell level, whch enables the varaton of matrx mechancal propertes wthn the unt cell. 7

8 A lmtaton of the presented damage modelng procedure s that t s not applcable to a localzed damage effects wthn the composte materal due to the perodcty condtons on the unt cell boundares. As a result, the model ndcates that the calculated damage repeats tself n the unt cell plane whch contradcts the physcal behavor of damage n composte materals. A further drawback of the appled damage model s the strong dependence of the load ncrement on damage progresson. he falure curves obtaned wth 30x30 unt cell model are very close to fner unt cells, ndcatng that damage processes of smple unt cells can be effcently modeled usng relatvely coarse unt cell models. Although mcromechancal falure crtera predct the onset of damagng processes n only a sngle subcell, the obtaned results show that a large number of the subcells satsfy the falure ntaton crteron at the same load state. onsequently, damage processes are ntated n a large part of the unt cell for the same macroscopc loadng condton. References [] J. Aboud, M.J. Pndera and S.M. Arnold Hgherorder theory for perodc multphase materals wth nelastc phases. nternatonal Journal of Plastcty, Vol. 9, pp , 003. [] Y. Bansal and M.J. Pndera Effcent reformulaton of the thermoelastc Hgher-Order heory for FMs. NASA/R , 00. [3] Y. Bansal and M.J. Pndera A second look at the Hgher-Order heory for perodc multphase materals. Journal of Appled Mechancs, Vol. 7, pp 77-95, 005. [4] Y. Bansal and M.J. Pndera Fnte-Volume rect Averagng Mcromechancs of heterogeneous materals wth elastc-plastc phases. nternatonal Journal of Plastcty, Vol., pp , 006. [5]. vancevc and. Smojver Multscale damage analyss of lamnated composte structures usng eneralzed Method of ells theory and Abaqus. Proceedngs of the 5th European onference on omposte Materals, Vence, taly, 0. [6]. vancevc and. Smojver Multscale analyss of aeronautcal composte structures usng Hgh Fdelty eneralzed Method of ells. Proceedngs of the European ongress on omputatonal Methods n Appled Scences and Engneerng (EOMAS 0), Venna, Austra, 0. [7] B. Kurnatowsk, A. Matzenmller oupled twoscale analyss of fber renforced composte structures wth mcroscopc damage evoluton. nternatonal Journal of Solds and Structures, Vol. 49, pp , 0. [8] R. Haj-Al and J. Aboud A new and general formulaton of the parametrc HFM mcromechancal method for two and three-dmensonal mult-phase compostes. nternatonal Journal of Solds and Structures, Vol. 50, pp , 03. [9] M.J. Pneda, A.M. Waas and B.A. Bednarcyk Multscale model for progressve damage and falure of lamnated compostes usng an explct fnte element method. Proceedngs of the 50th AAA/ASME/ASE/AHS/AS Structures, Structural ynamcs, and Materals onference, Palm Sprngs, alforna, USA, 009. [0] J.S. Mayes and A.. Hansen omposte lamnate falure analyss usng Multcontnuum heory. ompostes Scence and echnology, Vol. 64, pp , 004. [] M.J. Hnton, A.S. Kaddour and P.. Soden A further assessment of the predctve capabltes of current falure theores for composte lamnates: omparson wth expermental evdence. ompostes Scence and echnology, Vol. 64., pp , 004. [].W. Ehrensten Faserverbund-Kunststoffe. arl Hanser Verlag München Wen, 006. [3] B.A. Bednarcyk, J. Aboud, and S.M. Arnold Mcromechancs modelng of compostes subjected to multaxal progressve damage n the consttuents. AAA Journal, Vol.48, No. 7, pp , 00. [4] E.J. Pneda, B.A. Bednarcyk, A.M. Waas and S.M. Arnold mplementaton of a smeared crack band model n a mcromechancs framework. NASA/M , 0. [5] Z. ang and B. Zhang Predcton of baxal falure envelopes for composte lamnates based on eneralzed Method of ells. ompostes: Part B, Vol. 43, pp 94-95, 0. [6] A.M. Moncada, A. hattopadhyay, B.A. Bednarcyk and S.M. Arnold Mcromechancs-based progressve falure analyss of composte lamnates usng dfferent consttuent falure theores. Proceedngs of the 49th AAA/ASME/ ASE/AHS/AS Structures, Structural ynamcs, and Materals onference, pp