Numerical investigation on the failure mechanisms of. particulate composite material under a various loading

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1 Fal report Numercal vestgato o the falure mechasms of partculate composte materal uder a varous loadg codtos usg a damage costtutve model P.I.: Hrosh Okada, Ph.D., Assocate Professor Address: Departmet of Nao Structure ad Advaced Materals Graduate School of Scece ad Egeerg, Kagoshma Uversty --40 Kormoto, Kagoshma , Japa Telephoe, Facsmle: , E-mal: okada@mech.kagoshma-u.ac.jp

2 Report Documetato Page Form Approved OMB No Publc reportg burde for the collecto of formato s estmated to average hour per respose, cludg the tme for revewg structos, searchg exstg data sources, gatherg ad matag the data eeded, ad completg ad revewg the collecto of formato. Sed commets regardg ths burde estmate or ay other aspect of ths collecto of formato, cludg suggestos for reducg ths burde, to Washgto Headquarters Servces, Drectorate for Iformato Operatos ad Reports, 5 Jefferso Davs Hghway, Sute 04, Arlgto A Respodets should be aware that otwthstadg ay other provso of law, o perso shall be subject to a pealty for falg to comply wth a collecto of formato f t does ot dsplay a curretly vald OMB cotrol umber.. REPORT DATE 0 SEP 006. REPORT TYPE Fal Report (Techcal). DATES COERED to TITLE AND SUBTITLE Numercal vestgato o the falure mechasms of partculate composte materals uder varous loadg codtos usg damage costtutve model 6. AUTHOR(S) Hrosh Okada 5a. CONTRACT NUMBER FA50904P07 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Kagoshma Uversty,--40 Kormoto,Kagoshma ,Japa,JP, SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) The US Resarch Labolatory, AOARD/AFOSR, Ut 4500, APO, AP, PERFORMING ORGANIZATION REPORT NUMBER 0. SPONSOR/MONITOR S ACRONYM(S) AOARD/AFOSR. SPONSOR/MONITOR S REPORT NUMBER(S) AOARD DISTRIBUTION/AAILABILITY STATEMENT Approved for publc release; dstrbuto ulmted. SUPPLEMENTARY NOTES 4. ABSTRACT Durg the preset research, the followgs have bee accomplshed ) The separate dlatatoal/devatorc damage costtutve model was proposed alog wth a method to extract ts materal parameters from oe-dmesoal stress-stra curve. The damage model was mplemeted the s-fem computer program. ) S-FEM formulato wth the cohesve zoe model has bee proposed ad has bee mplemeted the s-fem program. ) S-FEM aalyses o partculate composte materals, wth assumg the matrx damage ad dewettg betwee the partcles ad matrx materal, has bee carred out. Some characterstc behavor of damages the partculate composte materals has bee revealed. 5. SUBJECT TERMS Composte Fracture Mechacs, Numercal Smulato 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT a. REPORT uclassfed b. ABSTRACT uclassfed c. THIS PAGE uclassfed 8. NUMBER OF PAGES 77 9a. NAME OF RESPONSIBLE PERSON Stadard Form 98 (Rev. 8-98) Prescrbed by ANSI Std Z9-8

3 Cotets Summary 4. S-verso Fte Elemet Method (S-FEM) for the aalyss of partculate composte materals.5. Equato Formulatos 5. Numercal Implemetatos of s-fem (Evaluato of stffess matrces)..9.. Evaluato of elemet ad couplg stffess matrx 0.. Materal costats ad stra hstores. Damage costtutve model...5. Isotropc damage costtutve model.5. Separate Isotropc/devatorc damage model.8. Cohesve zoe model ad ts mplemetato the s-fem computer program. Cohesve zoe model.. Cohesve zoe wth a arbtrary oretato..6. S-FEM formulato wth the cohesve zoe model Formulatos Cohesve zoe elemet Materal damage evoluto partculate composte materals-damage costtutve laws Relatoshp betwee the effectve stress-lke parameters ad the damage parameters 4.. The sotropc damage model The separate Isotropc/devatorc damage model Materal damage evoluto partculate composte materals-cohesve zoe model (dewettg betwee the partcles ad

4 matrx materal) 4 5. Materal damage evoluto matrx materal uder the flueces of mechacal teractos of partcles- (Four partcle model wthout ambet pressure) Isotropc damage materal wthout ambet pressure Separate dlatatoal/devatorc damage materal wth α = 0. ad wthout ambet pressure Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wthout ambet pressure Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wthout ambet pressure Summary: Separate dlatatoal/devatorc damage materal wth α = 0., 0.99 ad 0.0, ad wthout ambet pressure Materal damage evoluto matrx materal uder the flueces of mechacal teractos of partcles- (Four partcle model wth ambet pressure) Isotropc damage materal wth ambet pressure Separate dlatatoal/devatorc damage materal wth α = 0. ad wth ambet pressure Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wth ambet pressure Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wth ambet pressure Summary: Separate dlatatoal/devatorc damage materal wth α = 0., 0.99 ad 0.0, ad wth the ambet pressure.5 5. Materal damage evoluto matrx materal uder the flueces of mechacal teractos of partcles- (Ne radomly dstrbuted partcle model wth ad wthout ambet pressure) Isotropc damage materal wthout ambet pressure (Ne radomly dstrbuted partcle model) Separate dlatatoal/devatorc damage materal wth α = 0. ad wthout ambet pressure (Ne radomly dstrbuted partcle model) 5

5 5.. Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wthout ambet pressure (Ne radomly dstrbuted partcle model) Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wthout ambet pressure (Ne radomly dstrbuted partcle model) Isotropc damage materal wth ambet pressure (Ne radomly dstrbuted partcle model) Separate dlatatoal/devatorc damage materal wth α = 0. ad wth the ambet pressure (Ne radomly dstrbuted partcle model) Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wth the ambet pressure (Ne radomly dstrbuted partcle model) Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wth the ambet pressure (Ne radomly dstrbuted partcle model) Summary: Separate dlatatoal/devatorc damage materal wth α = 0., 0.99 ad 0.0, ad wth ad wthout the ambet pressure (Ne radomly dstrbuted partcle model) Materal damage evoluto-dewettg betwee the partcles ad matrx materal (cohesve zoe model) Oe partcle model, wthout matrx damage (wthout the ambet pressure) Oe partcle model, wth matrx damage (wthout the ambet pressure) Four partcle model, wth matrx damage (wthout the ambet pressure) Coclusos...7 Refereces...74 Publcatos 74 Ackowledgemets..75

6 Summary I ths research, the s-verso fte elemet method (s-fem) s used to carry out aalyses o the partculate composte materals udergog progressve damage. Matrx materal s assumed to be a polymerc materal. Hard partcles are embedded matrx materal. S-FEM smplfes the modelg procedures for partculate composte materals because t allows us to buld fte elemet models for the structure as whole ad for partcles ad ther vctes, separately. They are called global ad local fte elemet models. Whe partculate composte materals are modeled, the local fte elemet model cotas a partcle ad ts mmedate surroudg rego. The local fte elemet models are superposed o the global model. By adoptg the s-fem, placg partcles matrx materal became a trval task. Matrx s cosdered to suffer from a materal damage due to the growth ad ucleato of mcrovods. Also, the composte expereces damage due to partcle-matrx dewettg. The former s accouted for by the use of a cotuum damage costtutve law. The later s by a cohesve zoe model. Two kds of cotuum damage models are used ths research. They are a sotropc ad a separate dlatatoal/devatorc damage costtutve law that accouts for the flueces of hydrostatc ad devatorc stresses separately. The costtutve ad the cohesve zoe models were mplemeted the s-fem computer program. Numercal aalyses were carred out to reveal the characterstc deformato behavor of partculate composte such as the flueces of hard partcles to matrx damage, the flueces of partcle-matrx dewettg to matrx damage, etc. The results revealed that the separate dlatatoal/devatorc damage costtutve model wth a small cotrbuto from the devatorc stresses was the most approprate model amog the models whch were tested ths study. Whe the cohesve zoe model s assumed at the terface betwee the partcles ad matrx materal, the stregth of bod of the cohesve zoe determes the damage mode that domates the other. Whe the bod s strog, the matrx damage s the major damage mode. Whe the bod s weak, the dewettg s the major damage mode. The outcomes of preset research revealed some characterstc behavor of progressve damage polymerc partculate compostes. 4

7 . S-verso Fte Elemet Method (S-FEM) for the aalyss of partculate composte materals. Equato Formulatos I ths secto, we dscuss about geeral formulatos of the s-verso fte elemet method (s-fem). It s assumed that partcles or fbers are dstrbuted the doma of aalyss as show Fgure. Though Fgure mples that the partcles or vods are sphercal ther shapes, there s o such restrctos the mathematcal formulato. The secod phase materal ca be fbrous or ay others ther shapes. Though the prevous formulatos of the s-fem (see Fsh [] for example) assumes oly oe overlad model to be superposed o the global model, we allow ay umbers of fte elemet models to be superposed. The, the overlad models are allowed to overlap each other, as depcted Fgure. Whe the shapes of the embedded secod phase materals are the same or smlar to each other, the same local fte elemet model ca be used repeatedly. Therefore, geeratg a model for the composte would be a smple task. I the followg dscussos, the regos of the global ad the p-th (p=,,,, M) G local fte elemet models are desgated to be Ω ad Ω Lp, as depcted Fgure. We assume that there are a total of M local model regos. The dsplacemets are defed based o the shape fuctos of elemets the global ad local models, depedetly [see refereces such as Bathe [] ad Hughes [] for the shape fuctos G G of fte elemets]. We wrte them to be u = u ( x) Ω G Lp Lp Lp ad u = u ( x) Ω, where x deotes the posto of a materal pot. At a pot whch s ot sde of ay local model regos, the dsplacemets are the same as the dsplacemet fuctos G u G of Ω. u u = ( x) () G u At a pot where some local fte elemet models overlap, the dsplacemet fuctos are gve by the sum of dsplacemet fuctos of the overlapped models. For u example, at a pot where the local models Lp Ω ad Lq Ω ( p, q M, p q ) overlap G o the global model Ω, the dsplacemets u are represeted by the sum of ther 5

8 dsplacemet fuctos, as: G Lp Lq = u ( x) + u ( x) u ( x) () u + To assure the cotutes of dsplacemets, those based o a local fte elemet model are set to be zero at ts outer boudary. We let: Lp Lp u = 0 at Ω () Lp Lp where Ω desgates the outer boudary of local model rego Ω. Stresses at a pot are wrtte terms of stras through Hooke s law [see Sokolkoff [4], for example]. σ j = Ejklε kl (4) where the elastc costats locato of a materal pot. E jkl may vary wth the sold ad are the fuctos of E = E ( x) (5) jkl jkl The statemet of prcple of vrtual work s wrtte to be: Ω G u x j E jk G ( Ω ) uk G G l dω = G Ω ubdω + G Ω utd t (6) t x l where u are the varatos of dsplacemets, b are the body force per ut volume, G t are the prescrbed tracto vector o the tracto prescrbed boudary Ω t. The varatos of dsplacemets are assumed the same maer as the dsplacemets u 6

9 u, by the superposto at materal pots where they overlap. Thus, G be u = ( x) u u are wrtte to where o local models overlap o the global model ad u G Lp Lq = u ( x) + u ( x) ( x) + u where local models Lp Ω ad Lq Ω ( p, q M, p q ) overlap o the global model. Lp of Ω. p u ( p M ) are set to be zero at the boudary Lp Ω Thus, the dsplacemets ad ther varatos are substtuted the statemet of vrtual work prcple. After some algebrac mapulatos, we arrve at: Ω G u x G j E jkl u x G k l dω G G Lp M u u k Lp G G + Ejkl Ω = Ω G u bdω + G Ω utd x x t p= ΩLp j l ( Ω ) d (7) G t Ω Lp u x Lp j E jkl u x G k l dω Lp + Ω Lp u x Lp j E jkl u x Lp k l dω Lp + M q= q p Lq Ω Lp u x Lp j E u Lq k jkl xl dω Lp Lq Lp = Ω Lp u bdω Lp (8) ( p =,,, L, M ) From the left had sdes of equatos (7) ad (8), varous stffess matrces are obtaed. Thus, a equato ca be wrtte a matrx form, as: K L K L K L K M LM K G G G G G K K K K G L K L L L L L M LM L K K K K G L L L K L L L M LM L K K K K G L L L L L K L M LM L L L L L O L K K K K G LM L LM L LM L LM K M LM u u u u M u G L L L LM F F F = F M F G L L L LM (9) s the ordary stffess matrx for the global fte elemet model whch arses = are those for the local G K Lp from the frst term of equato (7) ad K ( p,,, L, M ) G Lp fte elemet models arsg from the secod term of equato (8). K Lp G ( p =,,, L, M ) ad K ( p =,,, L, M ) are the couplg stffess matrces betwee the global ad local fte elemet models, arsg from the secod term of equato (7) ad the frst term of equato (8), respectvely. G Lp thrd term of equato (8). F ad F ( p,,,l, M ) Lp Lq p =,,,, M ; p q ) K (, q L are the couplg stffess matrces betwee the local fte elemet models arsg from the = are the odal force vectors 7

10 of exteral ad body forces. It s oted that the matrces have the propertes of Lp G G Lp T Lp Lq Lq Lp K = ( K ) ad K = ( K ) T had sde of equato (9) s symmetrc.. Therefore, the coeffcet matrx the left Ukow odal dsplacemets are obtaed by solvg the lear smultaeous equatos (9), ad the dsplacemet feld the composte s determed (see Okada, Lu, Nomya, Fuku ad Kumazawa [5] for the full detals of the soluto procedures). (a) Partculate com poste (b) Fbrous composte Fgure Schematc vews of composte materals Fgure A s-fem model for composte materal whch each fber/partcle ad ts mmedate vcty are modeled by a local fte elemet mesh 8

11 Fgure Local model regos ( Ω global model rego G Ω Lp ( p =,,, L,M )) whch are superposed o the. Numercal Implemetatos of s-fem (Evaluato of stffess matrces) A uque feature of the s-fem s that elemets global ad local fte elemet models overlap each other. There are may ways for elemets to overlap, as show Fgure 4. Elemets may overlap each other a arbtrary maer. Ths rases serous problems the evaluato of stffess matrces such that ) whe a couplg stffess matrx such as Lp Lq K s formed, elemets dfferet fte elemet models may partally overlap each other (two-dmesoal llustrato s preseted Fgure 5) ad ) more tha oe materal models or materal parameters that are specfed by the overlappg elemets may exst at a pot. Some specal care must be gve to overcome these ssues. I ths secto, how we ca overcome these ssues s-fem computer mplemetato s descrbed. 9

12 (a) (b) (c) (d) (e) (f) Fgure 4 May ways that two elemets overlap each other; (a)~(c): Two-dmesoal examples ad (d)~(f): Three-dmesoal examples Fgure 5 Overlappg rego LP Lqj Lp Ω of the regos Ω ad Ω Lqj of two elemets ( Lp ad Lpj).. Evaluato of elemet ad couplg stffess matrx We cosder a typcal scearo. Fgure 5 llustrates the fte elemets ad j of the local mesh regos p ad q. They are desgated to be elemets Lp ad Lqj, respectvely. They overlap each other ad, therefore, ther couplg stffess matrx s formed. The Lp Lqj couplg stffess matrx [ k ] ca be wrtte to be: 0

13 Lp w here [ ] Lp Lqj Lp T Lqj Lp Lqj [ ] = Lp Lqj [ B ] [ E][ B ] dω k (0) Lqj B ad [ ] Ω B are the stra-dsplacemet matrx for the elemets, ad [ E] Lp Lqj s the matrx represetg the elastc costats E l equato (5). Ω s jk the volume of overlappg rego. Whe the couplg stffess matrx s form ed, a umercal tegrato s performed for the overlappg volume Lp Lqj Ω. Lp Lqj However, the overlapped rego Ω may have a complex geometry. It s almost mpossble to explctly defe the geometry of the overlappg rego ad to apply a ordary umercal tegral scheme such as Gasuss quadrature. I ths study, we perform the tegral based o oe of the overlappg elemets. It ca be show to be: Lp Lqj Lp Lqj Lp T Lqj Lp [ ] = Ω Lp ( x)[ B ] [ E][ B ] dω k α () α Lp Lqj ( x) s a scalar fucto whose value s Lp Lqj Ω ad 0 outsde of Lp Lqj Ω, as dcated Fgure 5. Numercal tegral s performed based o elemet Lp. Therefore, the tegrad of equato () has a sever dscotuty sce the value of Lp Lqj α ( x) chages abruptly from to 0 or 0 to. Therefore, a ordary Gauss quadrature s uable to evaluate the tegral accurately. I order to crcumvet ths problem, we developed a elemet subdvso scheme. The elemet subdvso techque s llustrated Fgure 6 for two- ad three-dmesoal problems. As show Fgure 6 (a-), two-dmesoal elemets Lp ad Lqj partally overlap each other, ad the tegrato s performed based o Lp. Frst, elemet Lp s dvded to 4 sub-cells. The each subdvded-cell s checked f t tersects wth the edges of elemet Lqj. If a subdvded-cell tersects wth the edges of elemet Lqj, t s dvded to 4 sub-cells aga. Ths process s repeated utl the smallest sub-cell becomes small eough (typcally wth % of the volume of elemet Lp). The processes of creatg sub-cells are show Fgures 6 (a-)~(a-v). The, a ordary Gauss quadrature rule s appled each subdvded-cell. The same approach s adopted three-dmesoal problems, as show Fgure 6 (b-)-(b-v). However, t should be oted that proposed methodology s ot computatoally effcet ad takes much loger computatoal tme tha the ordary Gauss quadrature, because may

14 tegrato pots are used to carry out the umercal tegral. (a-) (a-) (a-) (a-v) (b-) (b-) (b-) (b-v) Fgure 6 Elemet subdvso techques for (a) two- ad (b) three- dmesoal problems.. Materal costats ad stra hstores I a ordary fte elemet method, the materal costats are assged to fte elemets ad the stra hstory formato, such as damage paramters are stored at the tegrato pots. I preset s-fem aalyss, the global ad local fte elemet models overlap each other ad materal data cludg tal stras ad stra hstory parameters are assged to global ad local fte elemet models depedetly. Ths meas that two or more sets of materal parameters exst wth a overlappg rego. Frst, prorty orders are assged to materal models the put data. Deformato hstory data s stored at ordary tegrato pots each elemet global ad local fte elemet model. Whe two or more fte elemets overlap at a pot, a materal model havg the hghest proprety order s chose frst. If there were two or more overlappg elemets havg the same materal model at a pot, deformato hstory data of the smallest elemet s used to form the stffess matrces. For example, Fgure 7, a two-dmesoal schematc llustrato s gve. There are two materal models A ad B that are assged to the elemets. At a pot wth the

15 overlappg rego, materal model A or B s chose accordg to ther assged prorty orders. Eve whe a ordary elemet stffess matrx s formed, abrupt chages materal costat may occur. The elemet subdvso scheme of secto.. s used for such cases. Next, we dscuss about treatmets for the stra hstory parameters. Whe the elemet subdvso techque s adopted, may tegrato pots just for umercal tegrato are geerated. If oe tres to store the stra hstory data at each oe of them, a large amout of computer memory wll be requred. Thus, preset s-fem program, each ordary tegrato pot carres the stra hstory formato. For example, the case of two-dmesoal lear fte elemet, there are four ordary tegrato pots, as depcted Fgure 8. Fgure 8 presets the postos of four tegrato pots ξ-η ormalzed coordate system. Each of the tegrato pots represets a quarter of area of the elemet as show Fgure 8. Whe the elemet subdvso techque s used, the ξ-η ormalzed coordate values of each geerated tegrato pot are evaluated ad the program chooses a ordary tegrato pot whose stra hstory parameters are used. For three-dmesoal case, the same strategy s adopted. Fgure 7 A rego of multply assgmed materal models ad the Gauss pots the overlappg elemets

16 Fgure 8 Gauss pots a elemet (two-dmesoal quadrlateral elemet) 4

17 . Damage costtutve model I ths research, we assume the ucleato ad growth of mcrovods polymerc materal ad the dewettg betwee the partcles ad polymerc matrx materal. Fgure 9 shows the schematc vew of damage modes. These materal damages reduce the effectve area of materal secto. Thus, the materal stffess s reduced gradually whle progressve materal damages take place. Two kds of damage costtutve laws were adopted. They are descrbed ths chapter. The sotropc damage model followg Smo ad Ju [6, 7] was exteded to separate dlatatoal/devatorc damage model that ca accout for the dlatatoal ad devatorc compoets of damages separately. The separate dlatatoal/devatorc damage model s especally powerful whe the deformato of the materal uder a ambet pressure s aalyzed. The dlatatoal part of materal damage s maly due to the ucleato ad growth of mcrovods. It s cosdered that uder egatve hydrostatc stress, the ucleato ad growth of mcrovods do ot occur. The dlatatoal damage model s assumed to be duced whe the hydrostatc stress s postve. Fgure 9 A schematc vew of two kds of damage modes; the growth of mcrovods matrx materal ad dewettg betwee the partcles ad matrx materal. Isotropc damage costtutve model 5

18 Frst, we descrbe the sotropc damage theory by followg Smo ad Ju [6]. I Smo ad Ju [6], the effectve stress cocept ad the hypothess of stra equvalece are descrbed. Whe they are appled to the case of elastc damage, the elastc potetal eergy of damaged materal, terms of the stras ε j ad the d s wrtte to be: ψ o ( ε kl d ) = ( d ) ψ ( ε kl, ) () where ψ o ( ε k l ) s the elastc-potetal fucto for vrg materal that s wrtte to be: ψ o ( ε kl ) = Cjklε jε kl () Just lke equvalet stress cocept the theory of plastcty, a scalar parameter τ that measures the magtude of stress/deformato s troduced. τ = ψ o ( ε kl ) (4) Crtera for the damage evoluto are wrtte as follows. Frst, the effectve stress eeds to reach the crtcal value. g ( τ ( ε ) r( d )) = τ ( ε ) r( d ) = 0 (5) kl kl where r( d ) s the fucto of the d. The relatoshp betwee r ( d ) ad d eeds to be determed based a expermetal data. The materal damage progresses whe the creases. Therefore, whe the damage s ogog, the tme dervatve of the d & must be postve. d & > 0 (6) d & s determed through the evoluto law, as: = rh & ( τ, d ) & τ ( ε ) H ( τ d ) (7) d& = k l, 6

19 The costat H ( τ, d ) characterzes the progress of damage wth respect to the effectve stress lke term. Equatos (5) ad (6) must be satsfed whe the progressve damage takes place. It s oted that the costat H ( τ, d ) s always postve. Therefore, the equvalet stress-lke term must crease to satsfy equato (7), whle the materal damage s progress. Sce the elastc potetal eergy s assumed the form of equato (), the stresses σ j are wrtte the followg form. = ( ) l (8) σ j d C jkl ε k The rates of stresses ca be expressed by dfferetatg both the sdes of equato (), as: o ( d ) C jk & ε k dc & jk ε k = ( d ) C jk & ε k d& σ j (9) & j = l l l l σ l l where o σ j ( = Cjklε kl ) are the stresses whe the vrg materal s assumed for the same stras. From equato (7), the rate of d ca be derved to be: d & d& o & σ kl H ( ) ( ) ( τ, d ) o = τ H τ,d = H τ,d & ε = σ & ε (0) τ kl τ kl kl H (, d ) τ characterzes the rate of d wth respect to the rate of the effectve stress-lke parameter Substtutg equato (0) equato (9), we arrve at: τ &. & σ j = ( ) & & o d C ε dσ = ( d ) ID = Djkl & ε kl jkl kl j H Cjkl & ε kl ( τ, d ) τ σ o j σ o & klε kl () where 7

20 D ID jkl H ( ) ( τ, d ) o o = d C σ σ () jkl τ j kl D ID jk l are the taget modul for the sotropc damage materal ad have the major ad ID ID ID ID mor symmetres,.e. D D, D D ad D D. jkl = ID k l j jkl = ID j l k jkl = jk l. Separate Isotropc/devatorc damage model The damage evoluto of some class of materals are more sestve to hydrostatc pressure stress tha shear stresses. A typcal scearo s a materal cotag mcrovods. Mcrovods grow uder appled postve-hydrostatc pressure stress. But they do ot grow uder egatve-hydrostatc pressure stress. The growths of the vods are assumed to be less sestve to the shear (devatorc) stresses tha postve-hydrostatc stress. Oe way to model such materal s to separate the cotrbutos of sotropc ad devatorc stresses to the damage growth. To do so wth a very smple model, we separate the d to the dlatatoal (volumetrc) ad devatorc parts. Hece, the elastc potetal eergy fucto [equato ()] s modfed ad s wrtte to be: o o ( ε, d, d ) = ( d ) ψ ( ε ) + ( d ) ψ ( ε ) ψ kl D kk D D j () where ε kk ad ε j are the volumetrc ad devatorc stras. The fuctos o ψ ad o ψ D are defed to be: o ψ = K( ε ) o ad ψ = µε ε (4) kk j j where K ad µ are the bulk ad shear modul. It s oted here that uder a costrat codto d = d D, the separate dlatatoal/devatorc damage model s the same as the sotropc damage model. We defe two kds of effectve stress-lke terms, as: 8

21 ( ε ) K( ε ) o ψ kk kk τ = = ad D = ψ D ( ε j ) = µε jε j o (5) τ Whe the damages are progress, the followg codtos must be satsfed. τ = r d ) ad σ > 0 for the dlatatoal damage (6) v ( v kk τ = r d ) for the devatorc damage (7) D v ( D The evoluto equatos for the s are wrtte, as: & d pp = & τ H ( τ, d ) = H ( τ, d ) K & ε = H ( τ, d ) & ε (8) kk Kε K ( ε ) qq kk d& D & µε = τ D H D µε ε kl ( τ D, d D ) = H D ( τ D, d D ) ε kl (9) pq The costats H ( τ, d ) ad H ( τ, d ) gover the evoluto laws for the damage D D D parameters. The costats are the fuctos of the effectve stress-lke terms τ ad τ ad of the s d ad d. From equato (), we ca wrte the D stresses, as: pq D ( ε kl, d, d D ) = ( d ) j Kε kk + ( d D ) µε j (0) ψ j = ε j σ By dfferetatg both the sdes of equato (0) ad makg use of equatos (8) ad (9), we ca wrte the rate form costtutve equato, as: ad, & σ j = = ( d ) K & ε + ( d ) ( d ) K & ε + ( d ) = D D jkl & ε j j kl kk kk D D µε& j µε& j d& j Kε H kk d& ( τ, d ) ( ) 4H ( τ d ) τ D µε j Kε & D D, kk ε ll τ D D j kl kl µ ε ε & ε () 9

22 D D jkl H = ( d ) K + ( d ) µ ( + ) kl jl ( τ,d ) ( ) H ( τ,d ) τ Kε j D 4 kk j kl D τ k D D D jk l j kl µ ε ε () D Djk l are the taget modul that relate the rate of stresses to those of stras. It s oted D D D D D that have the major ad mor symmetres,.e. D jk D j, D jk D k ad Djk l l = k l l = j l D D jkl = D D jk l. 0

23 . Cohesve zoe model ad ts mplemetato the s-fem computer program The cohesve zoe model was mplemeted the s-fem computer program. I preset research, the cohesve zoe model accouts for separato betwee the reforcg partcles ad matrx materal. I ths secto, the formulatos of the cohesve zoe model (Chadra [8], Foulk, Alle ad Helms [9], etc.) alog wth the s-fem are preseted.. Cohesve zoe model Iterface separato law s characterzed by the cohesve zoe model. The terface a sold s depcted Fgure 0. We assume that such terface separato takes place whle progressve dewettg betwee matrx ad a partcle occurs. A three-dmesoal llustrato s gve Fgure. Here, we defe two kds of coordate systems. Oe s the global coordate system ( x, x, x ) whch s fxed the space ad the other s local coordate system ( x, x, x ) whch x ad x are the plae of the terface ad x s perpedcular to the terface. We defe upper ad lower faces as depcted Fgure. The postve drecto of the x coordate ad the ormal drecto of the terface are defed as depcted Fgure. Ther drectos are defed from the lower to the upper surface. Relatve dsplacemets u are defed + as the dfferece of the dsplacemets u ad u of the upper ad the lower faces, respectvely. () dcate that the compoets of the vector are wrtte the ( x, x, x ) local coordate system. u = u u () + By usg the relatve dsplacemets u, we defe a dmesoless parameter λ whch characterzes the separato ad the slps of the terface, as: λ u u u = + + t t (4) where t ad are the legth parameters that characterzes the dstace of partal

24 separato before the fal oe. The, a eergy desty fucto s defed terms of λ. W = W ( λ) (5) W = W ( λ) s the eergy potetal ad has propertes of: W W ( λ = 0) = ( λ ) 0 = W Separato (6) W s the eergy requred to ope ut area cohesve zoe. Separato Sce W = W ( λ) represets the eergy, we ca express: W W W& = α ( uα ) ( u ) ( λ ) u& + u& ( α =,) (7) Therefore, cohesve tractos T are wrtte to be: Tα W ( u ) = ( =, ) α α, W T = (8) ( u ) Thus, we ca express T, as: W dw dw u dw u λ Tα = (9) T α α = = = ( u ) dλ ( u ) dλ λ dλ λ α W α = = = ( u ) dλ ( u ) dλ λ dλ λ dw dw u t dw u λ = (40) t It s assumed here that W has the followg form: W ( λ) = σ f ( λ) (4) Here, σ s the maxmum ormal stress ad the maxmum value of f ( λ) s oe. f ( λ ) s the dmesoless fucto of λ. The fucto f ( λ) takes ts values f ( λ = 0 ) = 0 ad f ( λ ) = 0. By usg equato (4), the tractos ca be wrtte, as:

25 W dw Tα = α uα T = ( ) dλ u ( u ) = σ t W = = σ t λ α ( λ) df λ dλ = ( ) dλ u ( u ) u dw ( λ) df λ dλ λ dw = dλ dw = dλ uα t u = σ λ = σ λ ( λ) df dλ ( λ) df dλ uα λ t u λ (4) (4) I may lteratures such as [8] ad [9], equatos (4) ad (4) are terpreted to be: Tα T where, u α = ασ F( λ) (44) t u σ F( λ) = (45) α = (46) t Or we ca wrte: Tα T u α = σ F( λ) (47) t t u σ F( λ) = (48) I order to obta rate form taget modul, we further dfferetate equatos (47) ad (48) wth respect to tme. & Tα = σ t ( λ) F t ( λ) u& α df uα u + λ dλ t t u& + t uα u t u& uα u + t u& (49) Lkewse,

26 ( ) { ( ) = t t u u u u u u u u u F u F T d d & & & & & λ λ λ λ σ (50) I a matrx form, we ca wrte: ( ) ( ) + = t t t t t t t t t t t t t t t t t t t t t t u u u u u u u u u u u u u u u u u u u u u F F T T T d d & & & & & & λ λ λ λ σ (5) Therefore, the taget matrx the rate formulato s symmetrc. Fucto ( ) λ F have some choces as descrbed Chadra [8]. I preset aalyses, followg Foulk, Alle ad Helms [9], we let ( ) λ F be: ( ) ( 4 7 λ λ λ + = F ) (5) ad therefore, ( ) ( ) ( ) d d = + = λ λ λ λ F I order to show the sgfcace of ths choce of fucto ( ) λ F, we cosder the eergy per ut area of the terface due to a opeg mode. For the absece of u ad u compoets, we ca compute the eergy, as: Q ( ) ( ) T u T Q σ λ λ λ λ σ λ 48 7 d d d = + = = = (5) where 4

27 ad u 7 4 T = σ F( λ) = σ F( λ) λ = σ ( - λ + λ )λ (54) ( u ) = dλ d (55) From the expresso of (54), the maxmum value of s for T σ λ =. From equatos (5), (54) ad (55), oe ca state: σ ad determes the characterstcs ormal separato (eergy ad tal stffess of the terface). σ characterzes the tal stffess of the terface. I addto, by examg equato (50), oe ca fd that characterzes the t stffess rato betwee the ormal ad the tagetal separatos of the terface. Fgure 0 Progressve dewettg/terface separato betwee a partcle ad matrx materal 5

28 Fgure The cohesve zoe ad assocated coordate systems ( ( x,x, x ) global ad ( x,x, x ) local coordate systems). Cohesve zoe wth a arbtrary oretato I the prevous secto, the cohesve zoe model was descrbed the ( x, x x ) local, coordate system. I ths secto, detaled dscussos o how the coordate traslatos from the local to the global coordate system are carred out are gve. We assume a rate form formulato ad the rates of stresses o the terface are wrtte terms of the rates of relatve dsplacemets. & T kj u& j = (56) where the compoets of k j are foud from equato (5), as: 6

29 k k k k k k k k = σ k F ( λ) t 0 0 t 0 0 u u t t t df( λ) u u + λ dλ t t t u u t t 0 0 u u t t t u u t t t u u t t t t u u t u u t u u (57) Whe the terface s oreted a arbtrary drecto, matrx k j the ( x, x, x ) local coordate system must be trasformed to the c global coordate system. The drectos of x ad x coordate axes are the plae of the terface. x s perpedcular to the terface. A ter-elemet face costtutes a cohesve zoe elemet. The local ormalzed coordates ( ξ η), the local coordates ( x, x x ) ad the global coordates ( x ),, x, x are set as show Fgure. Ut bass vectors wth respect to the ( x, x x ) ad x x ) coordate systems are desgated to be ( e, e e ) ad, ( e, e, e ) (,, x, respectvely., Frst, we let the drectos of ξ ad x coordate axes be the same, as show Fgure. Thus, the bass vector e s show to be: x x ξ ξ = e e (58) e The, we defe aother ut vector ê, ad e ad e are calculated to be: x x = e e η η e ˆ, ad e ( ) e eˆ e eˆ =, e e e = (59) 7

30 The compoets global coordate system, as: k j of tagetal stffess of the cohesve zoe are trasformed to the k j ( k e e ) e = k ( e e )( e e ) = e kl k l j kl k l j (60) k show the stffess of the cohesve zoe the ( x, x x ) global coordate system. j, Fgure The ( x,x, x ) local coordate system ad ts ut bass vectors ( e, e, ), ad ( ξ η) ormalzed coordates e. S-FEM formulato wth the cohesve zoe model.. Formulatos The cohesve zoe model s mplemeted the s-fem computer program. It s assumed that the cohesve zoes are oly mplemeted the local fte elemet models. The global fte elemet model s ot resposble for the cohesve zoe model. I the s-fem formulato, we start wth the prcple of vrtual work, as stated below. 8

31 Ω G u x j uk Ejkl xl G dω = Ω + S G ubdω + Ω G + G t + + t& u& d G utd( Ωt ) + ( S ) + t& u& d( S S ) (6) + + where S ad S are both the faces of the cohesve zoe. t& ad t & + are the rate of tractos o S ad S, respectvely, that are duced by the + opeg of the cohesve zoe. ad are the varatos of the u& dsplacemet rates o + S ad S. They are the state of equlbrum ad therefore we ca wrte: u& + t = t& & (6) Therefore, we have: S t& u& d + ( S ) + S t& u& d( S ) = + t& ( u& u& ) d( S ) S (6) The tracto rates u & + t& are wrtte terms of the rates of relatve dsplacemets o the cohesve zoe, as show equato (56). The relatve dsplacemets are wrtte terms of jumps the dsplacemets across the cohesve zoe. Thus, are wrtte to be: + + ( u& u& ) + t& t & = k (64) j j j + Where u& ad u & are the veloctes at both the faces of the cohesve zoe. j j kj are the taget stffess of the cohesve zoe, as descrbed prevous secto. The rght had sde of equato (64) has a egatve sg because the postve drecto of the surface s opposte from that prevous secto. By substtutg equato (64) ad by usg the superposto of dsplacemets based o the global ad local fte elemet models, we ca wrte: Ω G G u x j G u M k G Ejkl dω + xl p= Ω Lp G u x j Lp uk Lp G G G Ejkl dω = G Ω u bdω + G Ω u Ω t td t xl ( ) (65) 9

32 Ω Lp + S Lp G Lp Lp u M uk Lp u Ω + u Lp E k jkl d Lp Ω Ejkl dω + Ω x j xl x j xl q= q p + ( ( Lp) ( Lp) + ) ( Lp) ( Lp) + u& u& kj u& j u& j d S Lp Lq Lp Lq u uk Lp Lq Ejkl dω x j xl + ( ) ( ( Lp) Lp ) = Lp u b dω Ω Lp (66) ( p =,,, L, M ) Here, we assume that there are M local models that are superposed o the global model. The local models may overlap each other ad ther couplg terms appear equato (66). It s oted that though the cohesve zoes of dfferet local model regos may tersect each other, ther couplg terms do ot appear equato (66). That s because two cohesve zoes make a le of tersecto whe they tersect ad the le has a zero area... Cohesve zoe elemet Whe the cohesve zoe s mplemeted the s-fem computer program, we troduce a cocept of cohesve zoe elemet. The cohesve zoe elemet behaves lke a terface elemet. Fgure shows the cohesve zoe elemet. For a llustratve purpose, the cohesve zoe elemet Fgure a small thckess. I actual model, the thckess s zero. The terms that are related wth the cohesve zoe elemet equato (66) as expaded further, as: S + = S S + ( Lp ( ) ( Lp) + ) ( Lp) ( Lp u& ) u& kj u& j u& j d S + ( Lp) + ( Lp) + ( Lp) + + u& j + k u& j d S S u& ( Lp) + ( Lp) + ( Lp) + u& k u& d S + + u& j j + ( ) ( ( Lp) ) ( ) ( Lp) ( Lp) + ( ( Lp k ) ju& j d S ) ( ) ( Lp) ( Lp) + ( ( Lp k u& d S )) S j j (67) Dscretzato procedures for the frst term the rght had sde of equato (67) are descrbed, as a example. Elemet stffess matrx for a cohesve zoe elemet + ( Lp) + S s derved as follows. The dsplacemets wth ( Lp S ) are expressed by usg the shape fuctos I =,,, where the elemet s assumed to be lear quadrlateral elemet. = { } [ ] ( ~4 N u ) N I ( 4) u& (68) 0

33 The detal of equato (68) ca be wrtte to be: u& N u& = 0 u& 0 0 N N N N N N N N 4 N N 0 u& u& u& u& u& 0 u& 0 4 u& N u& u& 4 u& 4 u& 4 u& (69) The superscrpts equato (69) desgate the odal umbers (,, or 4). The varatos of the veloctes ca also be wrtte the same maer. The stffess of the ca be wrtte a matrx from, as: k k k k = k k k (70) k k k [ ] Thus, we have: S + u& + ( Lp) + ( Lp & ) + d ( Lp k u S ) j j ( ) ( ~4 { ) T } [ ( )]{ ( ~4 u& + Lp = K u& )} { ( ~4) T } [ ] [ ][ ] ( ( )){ ( ~4 u& T + Lp = N k N d u& ) + S } S (7) The other terms are expressed the same maer, as: S + u& + ( Lp) ( Lp & ) + d ( Lp k u S ) j j ( ) ( ~4 { ) T } [ ( )]{ ( 5~8 u& + Lp = K u& )} { ( ~4) T } [ ] [ ][ ] ( ( )){ ( 5~8 u& T + Lp = N k N d u& ) + S } S (7)

34 S + u& ( ) ( 5~8 { ) T + } [ ( Lp) ]{ ( ~4 K u& )} { ( 5~8) T } [ ] [ ][ ] ( ( )){ ( ~4 u& T + Lp = N k N d u& ) + S } ( Lp) + ( Lp & ) + d ( Lp k u S ) = u& j j S (7) S + u& ( Lp) & + ( Lp) d ( Lp k u S ) j j ( ) ( 5~8 { ) T } [ ( )]{ ( 5~8 u& + Lp = K u& ) } { ( 5~8) T } [ ] [ ][ ] ( ( )){ ( 5~8 u& T + Lp = N k N d u& ) + S } S (74) Thus, the term of the elemet ca be wrtte as ts fal form, as: S + ( & + ( Lp) & ( Lp) ) & + ( Lp) & ( Lp u u k u u ) d S = = + ( ) ( ( Lp) j j j ) { ( ~4) T } [ ( )]{ ( ~4 ) } { ( ~4) } [ ( )]{ ( 5~8 u& + Lp T u& u& + Lp K K u& ) } { ( 5~8) T } [ ( )]{ ( ~4 ) } { ( ~4) } [ ( )]{ ( ~4 u& + Lp T u& u& + Lp K + K u& ) } ( ) ( ) { ( ~8) T + Lp + Lp } { ( ~8 u& K K u& ) } + ( Lp) + ( Lp) K K (75) where + [ ( Lp) T + ] = [ ] [ ][ ] ( ( Lp K ) + N k N d S ) (76) S Fgure A schematc vew of a cohesve zoe elemet

35 4. Materal damage evoluto partculate composte materals: damage costtutve laws I ths chapter, we dscuss about the mechacal teractos betwee eghborg partcles ad the dstrbuto ad the evoluto of materal damage. We assume four kds of materals; () the sotropc damage materal ad the separate dlatatoal/devatorc damage materal wth the costat α beg () 0., () 0.99 ad (4) 0.0. For the separate dlatatoal/devatorc damage model, the dstrbutos of the dlatatoal ad devatorc damages are examed separately. Matrx materal s assumed to udergo the damage. The sources of damage evolutos are due to the ucleato ad the growth of mcrovods. The stress-stra curve of matrx materal s postulated to be that of Kwo ad Lu [0]. Although the uaxal stress-stra curve of Kwo ad Lu [0] was measured for a polymerc composte materal, t s used to model matrx materal ths vestgato sce there s o other avalable materal behavor. I ths chapter, we dscuss about () how the relatoshp betwee the effectve stress-lke parameters ad the s are obtaed, () the results of aalyses for ut cell models are descrbed. 4. Relatoshp betwee the effectve stress-lke parameters ad the damage parameters. 4.. The sotropc damage model The damage evoluto law of equato (9) has a scalar fucto H ( τ,d ) of τ ad d. H (, d ) τ characterzes the damage evoluto behavor ad should be derved from a set of expermetal data. I ths secto, how the scalar fucto H ( τ,d ) s determed s descrbed. Frst, we assume that we have a stress-stra curve whch was measured a expermet. Let us assume that we have a uaxal stress-stra curve, as show Fgure 4 Stress s expressed by: σ = ( d ) Eε (77)

36 Therefore, the d s expressed by the stress ad stra, as: σ d = (78) Eε The value of the effectve stress-lke parameter τ ca be wrtte terms of the uaxal stress ad stra, as: τ σ σ o kl o = Cjk l ε jε kl = σ klε kl = ε kl = ε = σ ε = Eε (79) d d Therefore, oce the uaxal stress-stra curve s gve, relatoshp betwee the effectve stress-lke ad the ca be obtaed. I preset vestgato, a uaxal stress-stra curve s approxmated by a seres of pecewse straght les, as depcted Fgure 5. I Fgure 5, pots o the stress-stra curve ( σ,ε ), ( σ,ε ), ( σ,ε ),, ( σ, ε ), ( σ +, ε + ), coect straght le segmets. From a set of data ( σ, ε), ( σ,ε ), ( σ,ε ),, ( σ, ε ), ( σ +, ε + ),, we ca compute ( τ, d ), ( τ, d ), ( τ, d ),, ( τ, d ), ( τ +, d+ ), by usg equatos (78) ad (60). Thus, the relatoshp betwee τ ad d s also approxmated by a seres of pecewse straght les. The fucto H ( τ, d ) whch govers the damage evoluto law of equato (0) s approxmated by: H d& d d + ( τ d ) = ( d < d < d, τ < τ < τ ) (80), & + + τ τ + τ The fucto H ( τ, d ) whose value s determed by equato (80) s used the stress-stra relatoshp of equato (). We the extract a seres of data pots ( σ,ε ), ( σ,ε ), ( σ,ε ),, ( σ 7, ε 7 ), from the expermetal stress-stra curve of Kwo ad Lu [0], as show Fgure 6. As a example, the relatoshp betwee the uaxal stra ad the for the case of sotropc damage s depcted Fgure 7. Thus, the uaxal stress-stra curve s recostructed by a seres of lear approxmatos, as show Fgure 8. 4

37 Fg. 4 Uaxal stress-stra curve Fg. 5 Pecewse lear approxmato for uaxal stress-stra curve Fgure 6 Uaxal stress-stra curve of polymerc partculate composte materal that was gve Kwo ad Lu [0] 5

38 Damage arable Damage varable Stra Fgure 7 Relatoshp betwee the stra ad the for the sotropc damage costtutve model, whch was extracted from the stress-stra curve of Fgure 6. Stress (kpa) Gve data Ft by the damage model Stra Fgure 8 Uaxal stress-stra curve that was recostructed after the relatoshp betwee the effectve stress-lke ad the s s establshed by the proposed procedures. 4.. The separate Isotropc/devatorc damage model I ths secto, the damage evoluto law for the separate sotropc/devatorc damage 6

39 model s dealt wth. As see equatos (8) ad (9), there are two dfferet damage evoluto laws ad two damage varables H ( τ, d ) ad H D ( τ D, d D ). I order to fully determe H ( τ, d ) ad H ( τ, d ), we eed at least two sets of stress-stra curves. D D D Whe oly oe stress-stra curve s avalable, we eed to troduce at least oe addtoal codto. We assume that a uaxal stress-stra curve, as show Fgure 5 or 6 s avalable. As a addtoal codto to uquely determe the damage evoluto law, we postulate that the rato ( d d ) betwee the dlatatoal ad devatorc damage D parameters remas to be the same durg the uaxal deformato. We wrte: d D = αd ( 8) Here, α s a postve costat ( 0 α ). Whe α equals zero, oly the dlatatoal dam age s preset. Whe α =, the magtudes of dlatatoal ad devatorc parts equal each other ad, therefore, ths case s very smlar to that of the sotropc damage. Whe α s ftely large, the dlatatoal s zero. Therefore, materal udergoes devatorc damage oly. d We assume uaxal deformato x drecto. We ca wrte the followg statemets. σ ε 0 0, ε ( ukow) ; σ = σ = σ = σ = σ = 0 ( kow) = ε = ε = ( kow) ; ε = ( ukow) 0 ε (8) Therefore, by substtutg equatos (6) ad (6) equato (8), we have: σ σ σ = = ( ) K( ε + ε + ε ) + ( αd ) µ ( ε ε ε = ( d K ε + ε + ε + αd µ ε ε ε ( d ) K( ε + ε + ε ) + ( αd ) µ ( ε ε ε ) d ) (8) ) ( ) ( ) ( ) From the secod ad the thrd of equato (8) ad ε = ε, oe ca obta: 7

40 0 = d µ ( d ) K( ε + ε + ε ) + ( α ) ( ε ε ) (84) The frst of equato (8) ad equato (84) lead to: σ ε + ε = d ( α ) µ (85) By substtutg equato (85) the frst of equato (8), we ca establsh a relatoshp betwee σ ad ε, as: ( αd )( d ) µ Kε ( αd ) µ + ( d ) K σ = 0 (86) By rearragg equato (86), we ca establsh: αd αµ + K σ 9µ K ε + ( + α ) d + = 0 (87) σ Eε Therefore, for a gve set of varables σ, ε ad α, we ca solve for d. Two specal cases ( α = 0 ad α = ) are preseted frst. Whe α = 0, we have: ad, 0 d 0 µ + K 9µ K σ + 0 d ε σ + = Eε + ( ) 0 (88) σ d = ( ε σ = 0 K µ α ) (89) Whe α =, we have: ad, µ + K σ σ d + d + = 0 9 µ K ε Eε (90) 8

41 d σ = ( α = 0 ) (9) Eε I s oted that there are two solutos that satsfy equato (90) ad they are σ = ad. The secod oe takes a costat value ad, therefore, s ot a d d = Eε feasble soluto. For more geeral case ( α 0, ), lettg: we solve equato (87) for d, by d + Ad + B = 0 α (9) αµ + K 9µ K σ ε A = ( + α ) ad σ B = Eε (9) Therefore, the value d s determed to be: d A ± = A 4αB α (94) I equato (94), the sg assocated wth A 4αB eeds to be determed. To determe the sg, we check a specal case ( α = ). By lettg α =, equato (94), we have: σ σ = ± Eε Eε d (95) Whe we take the postve sg, equato (95) results : d σ σ = + = Eε Eε (96) Ths s a costat valu e ( d = ) ad, therefore, s approprate. Whe the egatve sg s assumed, we obta: d σ σ σ = = Eε Eε Eε (97) 9

42 Ths result s the same as the case of sotropc damage materal (equato (78)). Therefore, we choose the egatve sg equato (94). d A = A 4αB α (98) Thus, the devatorc d D s also determed to be: d D = αd A = α A 4αB α (99) Oce uaxal stress-stra curve as depcted Fgure 4 s gve, we correct a seres of pots ( σ,ε ), ( σ,ε ), (σ,ε ),, ( σ, ε ), ( σ +, ε + ),, o the curve, as show Fgure 5. Damage parameters ( d ad d D ) correspodg to the stress ad stra ( σ, ε ) are obtaed. Thus, by usg equatos (8) ad (8), ukow stras ε ad ε ( ε = ε ) are derved. Therefore, we ca compute the effectve stress -lke terms τ ( = Kε ) ad ( µ ε ε ) ( ) τ =. Thus, we have a seres of data d, d D, τ, τ D, ( d ), d D, τ, τ D ), (d d D, τ τ,, D,, ( d τ, τ ) d ), D D, d,, + D τ + τ + D,,, ( d + ad H ( τ, ) are derved to be: D D d D kk D j j. The scalar fuctos, H ( τ, d ) H H D d d d + ( τ, d ) = ( d < d < d, τ < τ < τ ) d& d D d D + D ( τ, d ) = ( d < d < d, τ < τ < τ ) D D & & τ & D τ τ τ + τ D + τ D D D + D + D D + D + (00) 40

43 5. Materal damage evoluto partculate composte materals-cohesve zoe model (dewettg betwee the partcles ad matrx materal) 5. Materal damage evoluto matrx materal uder the flueces of mechacal teractos of partcles- (Four partcle model wthout ambet pressure) I ths secto, we try to preset how the damage zoe develop whe partculate composte materals are loaded. Frst, we cosder a smple problem such that a block cotag four partcles s subject to teso, wth or wthout ambet pressure. I Fgure 9, a model that cotas four partcles s preseted. The fte elemet models for the s-fem aalyss are also llustrated Fgure 9. As preseted prevous chapter, the stress-stra curve of Kwo ad Lu [0] s postulated ad the relatoshp betwee the effectve stress-lke parameters ad the s are obtaed. From the results of ths smple problem, we ca clarfy the flueces of partcle arragemets ad of ambet pressure to the evoluto of matrx damage. E* = 0 E M Local Model: 98 elemets 007 odes Fgure 9 The problem of four partcles wth ad wthout the ambet pressure 5.. Isotropc damage materal wthout ambet pressure 4

44 The dstrbutos of stresses ad are preseted Fgure 0, for the levels of overall stra ε beg 0.06 ad 0.. It s see that, for ε = 0. 06, both tesle ad trasverse stresses cocetrate at the top ad bottom of the partcles. The magtude of the cocetratos seem to be slghtly severer the gaps betwee the partcles tha the other locatos. Both tesle ad trasverse stresses are postve, resulstg a large value of hydrostatc stress. Dmage zoes start to develop at the top ad bottom of the partcles. I the gaps betwee the partcles, the s slghtly larger tha the other locatos. That s very smlar to the case of stresses. ε, the major treds are smlar to the case of ε = However, materal For = 0. damage takes place almost throughout the rego of matrx materal. (a-) Tesle stress (a-) Trasverse stress (a-) Damage parameter (a) ε = (b-) Tesle stress (b-) Trasverse stress (b-) Damage parameter (b) ε = Fgure 0 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of sotropc damage wthout ambet pressure Separate dlatatoal/devatorc damage materal wth α = 0. ad wthout ambet pressure 4

45 I the case of the separate dlatatoal/devatorc damage model, the dlatatoal damage s set to domate the other. I Fgure, t s see that, for ε = 0. 06, the dstrbutos of the tesle ad trasverse stresses are smlar to those of the sotropc damage case. The dstrbutos of the trasverse stress look dfferetly. That s because the rages are dfferet. The dlatatoal ad devatorc damage zoes develop at the top ad bottom of the partcles. They coect the partcles vertcal drecto. No damage develops horzotal drectos from the partcles. As expected, the dlatatoal s much larger tha the devatorc oe. Whe the level of overall stra ε creases to 0., the damage zoes elarge ad the most of part of matrx materal suffer from the materal damages. The dlatatoal s much larger tha the devatorc oe. The partcles costra the deformato of matrx materal ad the damage varables at both the sdes of the partcles are almost zero. 4

46 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a) ε = 0.06 (a-4) Devatorc (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b) ε = 0. (b-4) Devatorc Fgure The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0. ad wthout ambet pressure 5.. Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wthout ambet pressure I ths case, parameter α s set to be The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are depcted Fgure. The dstrbutos of the stresses are smlar to afore metoed cases. Materal damages develop at the top ad bottom of the partcles where stresses are large. As show Fgures (a-), (a-4), (b-) ad (b-4), the levels of the dlatatoal ad devatorc s are smlar. Ther patters of dstrbutos are slghtly dfferet. The dlatatoal damage varable at the sdes of partcles s almost zero ad such regos coect the partcles horzotal drectos. 44

47 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal α = 0.99 (b) ε = 0. (b-4) Devatorc Fgure The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth ad wthout ambet pressure 5..4 Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wthout ambet pressure I ths case, the parameter α s set to be 0. It s assumed that the devatorc damage mode domates the dlatatoal oe. I Fgure, the dstrbutos of the stresses ad the s are show. The dstrbutos of stresses are smlar to afore metoed cases. The level of the devatorc s much larger tha the dlatatoal oe. 45

48 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal α = 0.0 (b) ε = 0. (b-4) Devatorc Fgure The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth ad wthout ambet pressure 5..5 Summary: Separate dlatatoal/devatorc damage materal wth α = 0., 0.99 ad 0.0, ad wthout ambet pressure The dstrbutos of the tes le ad trasverse stresses ad the dlatatoal ad devatorc s are descrbed secto 5..~5..4. The stresses are large at the top ad bottom of the partcles. The regos of hgh stress coect the partcles the vertcal drecto. At the sdes of the partcles, the partcles costra matrx materal from deformato. Therefore, the s are small at the sdes of the partcles. The overall treds of dstrbutos of the dlatatoal ad devatorc damage parameters are very smlar. Therefore, t s summarzed that the choce of costtutve models (the sotropc ad the separate dlatatoal/devatorc damage models wth dfferet costat α ) does ot make much dfferece the results whe o ambet pressure loadg s appled to the block. 46

49 5. Materal damage evoluto matrx materal uder the flueces of mechacal teractos of partcles- (Four partcle model wth ambet pressure) I ths secto, the same four partcle problems are solved wth ambet pressure that s appled as dstrbuted force from the sdes of the block. The ambet pressure s 400kPa. Results (the dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s) are vsualzed for overall tesle stra ε beg 0. ad 0.5. Sce the ambet pressure s appled, Posso s rato effect produces egatve tesle stress. Thus, overall tesle stra ε s set to be 0. or 0.5 so that the tesle stress the block s almost the same as the cases of prevous secto (wthout ambet pressure). 5.. Isotropc damage materal wth ambet pressure I Fgure 4, the dstrbutos of tesle ad trasverse stre sses ad the sotropc are depcted. Major treds the dstrbutos of tesle stress are smlar to those wthout the ambet pressure. Also, major treds the dstrbutos of trasverse stress are also aalogous to those wthout the ambet pressure, except for that the value of stress s about -400 kpa at dstat pots from the partcles whereas t s about zero for the case wthout the ambet pressure. The dstrbutos of the sotropc are qute dfferet from those wthout the ambet pressure. Whe the block s subject to teso wthout the ambet pressure, the damage zoes develop at the top ad bottom of the partcles ad do ot appear at the sdes of the partcles. However, wth the ambet pressure, the damage zoes develop at the sdes of the partcles. The cocetrato of egatve stress due to ambet pressure occurs at the sdes of the partcles. That s why the damage zoes develop at the sdes of the partcles. Developmet of damage zoe due to the growth ad ucleato of mcrovods uder egatve stress are ot lkely to occur. Therefore, the use of sotropc damage costtutve model may ot be sutable for preset vestgato. 47

50 (a-) Tesle stress (a-) Trasverse stress (a-) Damage parameter (a) ε = 0. (b-) Tesle stress (b-) Trasverse stress (b-) Damage parameter (b) ε = 0.5 Fgure 4 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of sotropc damage wth ambet pressure 5.. Separate dlatatoal/devatorc damage materal wth α = 0. ad wth ambet pressure The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are depcted Fgure 5. The dstrbutos of the stresses are smlar to those of the sotropc damage costtutve law. The dstrbutos of the s are qute terestg. The dlatatoal s very small compared wth the case wthout the ambet pressure. The dstrbutos of devatorc are almost uform wth matrx materal. The level of the devatorc s small. 48

51 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = 0. (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b) ε = 0.5 Fgure 5 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0. ad wth the ambet pressure (b-4) Devatorc 5.. Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wth ambet pressure The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are depcted Fgure 6. It s see that the dlatatoal damage parameter s much smaller tha the devatorc oe. The dstrbutos of the devatorc are aalogous to the cases of the sotropc damage costtutve law wth the ambet pressure. Ths mples that the developmet of damage the case of the sotropc damage law s govered by the devatorc stresses. 49

52 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = 0. (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal Fgure 6 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0.99 (b) ε = 0.5 ad wth the ambet pressure (b-4) Devatorc 5..4 Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wth ambet pressure The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are depcted Fgure 7. The dstrbutos of the stresses are smlar to those of the sotropc ad the separate dlatatoal/devatorc damage model wth the other values of parameter α. As the parameter α s set to be 0.0, the dlatatoal s much smaller tha the devatorc oe. The dstrbutos of the devatorc look smlar to prevous cases that are of the sotropc ad the dlatatoal/devatorc damage wth α = 0. ad α =

53 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = 0. (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal Fgure 7 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0.0 (b) ε = Summary: Separate dlatatoal/devatorc damage materal wth α = 0., 0.99 ad 0.0, ad wth the ambet pressure ad wth the ambet pressure (b-4) Devatorc Results wth the ambet pressure revealed that whe the sotropc damage model s used, the progressve materal damage occurs eve uder egatve hydrostatc stress. Ths cotradcts wth our basc postulato that the damage matrx materal s due to the growth ad ucleato of mcrovods. These should ot be sestve to the devatorc stresses. Therefore, t s plausble to use the separate dlatatoal/devatorc damage model wth a small value of costat α. From preset sets of aalyses we are uable to suggest the most sutable value for α. 5. Materal damage evoluto matrx materal uder the flueces of mechacal teractos of partcles- (Ne radomly dstrbuted partcle model wth ad wthout ambet pressure) 5

54 Followg the four partcle problems, we solved a problem wth e radomly dstrbuted partcles. The problem cofgurato s depcted Fgure 8. Aga, we adopt the sotropc damage model ad the separate dlatatoal/devatorc damage costtutve model wth α = 0., α = ad α = 0. 0 ad wthout the ambet pressure.. Aalyses are carred out wth E* = 0 E M Local Model: 98 elemets 007 odes Fgure 8 The problem of oe radomly dstrbuted partcles wth ad wthout the ambet pressure 5.. Isotropc damage materal wthout dstrbuted partcle model) ambet pressure (Ne radomly The dstrbutos of tesle ad trasverse stresses ad the are preseted Fgure 9. Treds are smlar to the case of four regularly dstrbuted partcles. It s see Fgures 9 (a-) ad (b-) that the zoe of large value of the damage parameter coects the partcles whe they are close to each other. The stresses cocetrate at the top ad bottom of the partcles. The trasverse stress at the sdes of the partcles s egatve ad the zoes of egatve stress coect the partcles whe the dstaces betwee partcles are small. 5

55 (a-) Tesle stress (a-) Trasverse stress (a-) Damage parameter (a) ε = (b-) Tesle stress (b-) Trasverse stress (b-) Damage parameter (b) ε = 0. Fgure 9 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of sotropc damage wthout ambet pressure (Ne radomly dstrbuted partcle model) 5.. Separate dlatatoal/devatorc damage materal wth α = 0. ad wthout ambet pressure (Ne radomly dstrbuted partcle model) The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are preseted Fgure 0. The overall treds are smlar to that of four regularly dstrbuted partcle case. Aga, the cocetrato of the trasverse stress s ot very clear. The value of the dlatatoal s much larger tha that of the devatorc oe. 5

56 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b-4) Devatorc α = (b) ε = 0. Fgure 0 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth 0. ad wthout the ambet pressure (Ne radomly dstrbuted partcle model) 5.. Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wthout ambet pressure (Ne radomly dstrbuted partcle model) The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are depcted Fgure. The cocetratos of the trasverse stress at the top ad bottom of the partcles appear more clearly tha the case of α = 0.. The tesle stress cocetrates at the top ad bottom of the partcles as see all the other cases. The values of the dlatatoal ad devatorc s are smlar. 54

57 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b-4) Devatorc (b) ε = 0. Fgure The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0.99 ad wthout the ambet pressure (Ne radomly dstrbuted partcle model) 5..4 Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wthout ambet pressure (Ne radomly dstrbuted partcle model) The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are depcted Fgure. As see Fgures (a-) ad (b-), the cocetratos of the trasverse stress at the top ad bottom of the partcles are clearly see. The value of the dlatatoal s much smaller tha that of the devatorc oe. 55

58 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a-4) Devatorc (a) ε = (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b-4) Devatorc (b) ε = 0. Fgure The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0.0 ad wthout the ambet pressure (Ne radomly dstrbuted partcle model) 5..5 Isotropc damage materal wth ambet pressure (Ne radomly dstrbuted partcle model) The dstrbutos of tesle ad trasverse stresses ad the are depcted Fgure. As see the case of regularly placed four partcle problem, progressve materal damage at the sdes of the partcles are see. They are due to egatve trasverse stress. 56

59 (a-) Tesle stress (a-) Trasverse stress (a-) Damage parameter (a) ε = 0. (b-) Tesle stress (b-) Trasverse stress (b-) Damage parameter (b) ε = 0. 5 Fgure The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of sotropc damage wth the ambet pressure (Ne radomly dstrbuted partcle model) 5..6 Separate dlatatoal/devatorc damage materal wth α = 0. ad wth the ambet pressure (Ne radomly dstrbuted partcle model) The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are preseted Fgure 4. The dstrbutos of the dlatatoal ad devatorc s are qute dfferet from that of the of the sotropc damage model. Also, they look dfferetly from the case wthout the ambet pressure. It seems that the ambet pressure weakes the magtude of hydrostatc stress ad prevets the dlatatoal damage zoe from elargg matrx materal. The devatorc looks more uform tha the case wthout the ambet pressure. 57

60 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a) ε = 0. (a-4) Devatorc (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b) ε = 0. 5 (b-4) Devatorc Fgure 4 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0. ad wth the ambet pressure (Ne radomly dstrbuted partcle model) 5..7 Separate dlatatoal/devatorc damage materal wth α = 0.99 ad wth the ambet pressure (Ne radomly dstrbuted partcle model) The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are preseted Fgure 5. The levels of the dlatatoal damage parameter are lower tha the case of α = 0.. I the dstrbutos of the devatorc, the cocetratos of the are see ot oly at the top ad bottom of but also at both the sdes of the partcles. It seems that the compressve stress due to the ambet pressure duces the evoluto of the devatorc damage. 58

61 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a) ε = 0. (a-4) Devatorc (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b-4) Devatorc Fgure 5 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0.99 (b) ε = 0. 5 ad wth the ambet pressure (Ne radomly dstrbuted partcle model) 5..8 Separate dlatatoal/devatorc damage materal wth α = 0.0 ad wth the ambet pressure (Ne radomly dstrbuted partcle model) The dstrbutos of tesle ad trasverse stresses ad the dlatatoal ad devatorc s are preseted Fgure 6. The major treds are the same as the case of α = The value of the devatorc s slghtly larger tha that of α = The value of the dlatatoal s much smaller tha those of α = 0. 0 ad α =

62 (a-) Tesle stress (a-) Trasverse stress (a-) Dlatatoal (a) ε = 0. (a-4) Devatorc (b-) Tesle stress (b-) Trasverse stress (b-) Dlatatoal (b-4) Devatorc Fgure 6 The dstrbutos of the tesle ad trasverse stresses ad the damage parameter for the case of the separate dlatatoal/devatorc damage model wth α = 0.0 (b) ε = 0. 5 ad wth the ambet pressure (Ne radomly dstrbuted partcle model) 5..9 Summary: Separate dlatatoal/devatorc damage materal wth α = 0., 0.99 ad 0.0, ad wth ad wthout the ambet pressure (Ne radomly dstrbuted partcle model) From the sets of aalyses preseted secto 5., t s clearly see that the major treds the case of radomly dstrbuted partcles are the same as those the four partcle problem. However, t s see that the damage zoes coect the partcles depedg o ther dstaces. The most mportat fdg secto 5. s as follows. The results seem to be somewhat reasoable wth all the costtutve models,.e., the sotropc ad the separate dlatatoal/devatorc damage model wth α = 0., α = ad α = 0. 0 whe the ambet pressure s ot appled. However, whe the ambet pressure s appled to the block, the devatorc damage zoes develop at both the sdes of the partcles, where the hydrostatc stress s egatve. I preset study, we assume that materal damage s due to the growth ad ucleato of mcrovods matrx materal. Such materal damage s ot cosdered to take place uder egatve hydrostatc stress. 60

63 Therefore, t s approprate to use the separate dlatatoal/devatorc damage costtutve model wth a small umber of costat α. Whe the costat α s small, the cotrbuto of the devatorc damage s small relatve to that of the dlatatoal damage. 5.4 Materal damage evoluto-dewettg betwee the partcles ad matrx m ateral (cohesve zoe model) I ths secto, some prelmary results are preseted. Due to the softeg ature of the cohesve zoe model, s-fem aalyses ted to become ustable. Thus, we could ot use teratve solver to solve for the system of lear smultaeous equatos. Thus, we used a drect solver (skyle solver) the aalyses of secto 5.4, although t s ot very effcet terms of ts memory usage. It s oted that for all the other aalyses, a teratve equato solver (Cojugate Gradet Method) s used. Because of ths problem, we could place as may as four partcles. Sce the costats, t ad σ the cohesve zoe model are ot kow, postulated values are used the aalyses. We let ad t be 0.0 whch s much smaller tha the sze of the partcle out problem. σ s vared ad from small to large values relatve to the olear stress-stra curve of Fgure 6,.e., σ =50 kpa, 500 kpa, 000 kpa ad 000 kpa Oe partcle model, wthout matrx damage (wthout the ambet pressure) I ths secto, the results of oe partcle problem that s show Fgure 7 are preseted. The partcle s placed the block made of elastc materal. The costats for the cohesve zoe model are postulated to be: = 0.0, = 0. 0, σ =50 kpa, 500 kpa, 000 kpa or 000 kpa (0) t I Fgure 8, the tesle stress dstrbutos the cetral secto of the block are show for the case of σ =50 kpa. I Fgure 8 (a), the stress dstrbuto show s stress cocetratos at the top ad bottom of the partcle. The dstrbuto s very smlar to that wthout the cohesve zoe model. Ths meas that, at ths step, the cohesve zoe s almost rgd. I Fgure 8 (b), t s see that the stress cocetratos at the top ad bottom of the partcle are weaker tha those Fgure 7 (a). The cohesve zoe opes 6

64 ad the load trasmttg capablty of the cohesve zoe starts decreasg. I Fgure 8, the dstrbutos of the tesle stress for the case of The treds are smlar to those of σ σ σ =500 are show. =50. I Fgures 9 ad 40, the dstrbutos are depcted for =000 ad σ =000, respectvely. Except for Fgure 9 (b), the σ treds are very smlar to those of =50 ad =500. Due to the softeg ature of the cohesve zoe model, the aalyss suddely became ustable. That s why the stress dstrbuto Fgure 9 (b) looks asymmetrc. σ E* = 0 E M Local Model: 98 elemets 007 odes Fgure 7 Oe partcle problem wth the cohesve zoe model (a) Overall stra=0.05 Fgure 7 The dstrbutos of tesle stress for (b) Overall stra=0.074 σ =50 kpa 6

65 (a) Overall stra=0.0 Fgure 8 The dstrbutos of tesle stress for (b) Overall stra= σ =500 kpa (a) Overall stra=0.0 Fgure 9 The dstrbutos of tesle stress for (b) Overall stra=0. σ =000 kpa (a) Overall stra=0.0 Fgure 40 The dstrbutos of tesle stress for (b) Overall stra=0.95 σ =000 kpa 5.4. Oe partcle model, wth matrx damage (wthout the ambet pressure) I secto 5.4., addto to the cohesve zoe model, we assume the evoluto of 6

66 materal damage matrx materal. As see precedg sectos, the separate dlatatoal/devatorc damage model wth a small value of costat α has bee recommeded to approprately model the matrx damage due to the growth ad ucleato of mcrovods. Thus, ths secto, α s set to be 0.. The stress-stra curve of Fgure 6 s assumed to model the behavor of matrx mater al., a costat the cohesve zoe model s postulated to be 50 kpa, 500 kpa, 000 kpa or 000 kpa. σ I Fgure 4, the dstrbutos of tesle stress ad of the dlatatoal for = 50 kpa are preseted. It s cosdered that = 50 kpa s small σ compared wth the level of stress-stra curve of Fgure 6. Accordg to Fgure 6, the olear deformato of matrx materal tates at 500 kpa. The value = 50 kpa s smaller tha the stress at the tato of damage. The tesle stress dstrbuto of Fgure 4 (b) dcates that the stress cocetrato weakes due to the opeg of the cohesve zoe at the top ad bottom of the partcle. Fgure 4 (c) shows that the dlatatoal s very sm all. Therefore, whe the value of σ s small compared wth that of damage tato, the opeg of cohesve zoe occurs before the materal damage of matrx materal takes place. Therefore, the major part of the materal damage s dewettg betwee the partcle ad matrx materal. σ σ I Fgure 4, the dstrbutos of tesle stress ad the dlatatoal are show for σ = 500 kpa. I ths case, the value of σ s comparable to the stress value at the tato of matrx damage. I Fgures 4 (a) ad (c), the dstrbutos of tesle stress ad the dlatatoal at overall stra are depcted. At ths stage, the damage zoe has ot developed much. The tesle stress dstrbuto s smlar to that of elastc problem. I Fgures 4 (b) ad (d), the dstrbutos of tesle stress ad the dlatatoal damage parameter are preseted for overall stra beg The stress dstrbuto dcates that the dewettg betwee the partcle ad matrx materal s takg place. Fgure 4 (d) shows that the dlatatoal damage zoes are developg at the top ad bottom of the partcle. I ths case, both the damage modes.e., matrx damage ad dewettg betwee the partcle ad matrx take place. I Fgure 4, the dstrbutos of tesle stress ad the dlatatoal are preseted for σ = 000 kpa. From the fgures, t s see that the cohesve zoe does 64

67 ot fluece the dstrbutos of stress ad the. The cohesve zoe s so stff that t does ot ope ad materal damage due to the dewettg does ot occur. I Fgure 44, the dstrbutos are preseted for detcal to those of σ = 000 kpa. σ = 000 kpa. They are almost (a) Tesle stress at overall stra=0.05 (b) Tesle stress at overall stra=0.07 (c) Dlatatoal at overall stra=0.07 Fgure 4 The dstrbutos of tesle stress ad of the dlatatoal damage parameter for σ =50 kpa. 65

68 (a) Tesle stress at overall stra=0.04 (b) Tesle stress at overall stra=0.06 (c) Dlatatoal at overall stra=0.04 (d) Dlatatoal at overall stra=0.06 Fgure 4 The dstrbutos of tesle stress ad of the dlatatoal damage parameter for =500 kpa. σ 66

69 (a) Tesle stress at overall stra=0.06 (b) Tesle stress at overall stra=0. (c) Dlatatoal at overall stra=0.06 (d) Dlatatoal at overall stra=0. Fgure 4 The dstrbutos of tesle stress ad of the dlatatoal damage p arameter for =000 kpa. σ 67

70 (a) Tesle stress at overall stra=0.06 (b) Tesle stress at overall stra=0. (c) Dlatatoal at overall stra=0.06 Fgure 44 The dstrbutos of tesle stress ad of the dlatatoal damage parameter for =000 kpa. σ (d) Dlatatoal at overall stra= Four partcle model, wth matrx damage (wthout the ambet pressure) I ths secto, the results of aalyses o the four partcle model are preseted. The same damage costtutve law as that s used to solve the oe partcle problems s adopted (the separate dlatatoal/devatorc model wth α = 0. ). I Fgure 45, the dstrbutos of tesle stress ad the dlatatoal for σ =50 kpa are preseted. It s see the fgures that the value of the damage parameter s small although the cohesve zoes start separatg ad that the stress cocetrato at the top ad bottom of the partcles weakes due to the softeg of the cohesve zoe,.e., dewettg. I Fgure 46, the dstrbutos of tesle stress ad he dlatatoal for σ =500 kpa are depcted. I ths case, the stress at the maxmum stress at the cohesve zoe ad that at the tato of matrx damage are comparable. It s see Fgure 46 (b) that the stress cocetrato at the top ad bottom of the partcles weakes 68

71 due to both the damage mechasms. The damage zoes coect the partcles alged the vertcal drecto, as see Fgures 46 (c) ad (d). I Fgures 47 ad 48, the dstrbutos of tesle stress ad the dlatatoal damage parameter are preseted for =000 kpa ad =000 kpa, respectvely. The dstrbutos Fgures 47 ad 48 are very smlar, although the dfferet values are gve to the parameter σ σ. I both the cases of Fgures 47 ad 48, the cohesve zoes do ot gve ay flueces to the deformatos of the compostes. The cohesve zoes are stff eough that they do ot ope. σ As summary, we ca coclude that depedg o the combato of stresses at the tato of matrx damage ad at the separato of cohesve zoe. I preset aalyses, the stress-stra curve of matrx materal s postulated to be what was gve Kwa ad Lu [0] ad the parameter of cohesve zoe model that govers the stress at the separato of cohesve zoe s vared. Whe σ s set to be 50 kpa, the stress at the separato of cohesve zoe s smaller tha that at the tato of matrx damage. I such a ca se, terface separato at the cohesve zoe s the major damage mechasm of the composte. Whe s 500 kpa, the stress at the separato of cohesve zoe s comparable to that at the progress of matrx damage. Both the mechasms of materal damage take place. Whe σ s 000 kpa or 000 kpa, the terface separato at the cohesve zoe do ot occur at all. The matrx damage s the major damage mechasm. σ σ 69

72 (a) Tesle stress at overall stra=0.05 (b) Tesle stress at overall stra=0.04 (c) Dlatatoal at overall stra=0.04 Fgure 45 The dstrbutos of tesle stress ad of the dlatatoal damage parameter for =50 kpa (four-partcle problem). σ (a) Tesle stress at overall stra=0.04 (b) Tesle stress at overall stra=0.06 (c) Dlatatoal at overall stra=0.04 (d) Dlatatoal at overall stra=0.06 Fgure 46 The dstrbutos of tesle stress ad of the dlatatoal damage parameter for =500 kpa (four-partcle problem) σ 70