FIBER-REINFORCED ELASTOMERS: MACROSCOPIC PROPERTIES, MICROSTRUCTURE EVOLUTION, AND STABILITY Oscar Lpez-Pamies State University f New Yrk, Stny Brk Department f Aerspace Engineering, Iwa State University February 9, 2 Ames, IA
Sft sids reinfrced with fibers Rubber reinfrced with carbn-back and fabric TEM f Cagen Fibris in Human Brain Arteries Thermpastic Eastmers (Sef-Assembed Nandmains) BCC HX-Cy. Gyrid Lamear ~ 5 nm f PS TEM f a tribck cpymer with cyindrica mrphgy
Issues in cnstitutive mdeing f FREs.6.4.2 S bi (MPa).8.6 Nninear cnstitutive matrix phase and fibers.4.2.2.4.6.8 2 Cmpex initia micrstructure Micrstructure evutin (gemetric nninearity) Devepment f instabiities
Prbem setting: Lagrangian frmuatin Kinematics O X Undefrmed W Lca cnstitutive behavir where W S = W F ( X, F) N r r ( XF, ) = c ( X) W ( F) å 2 ( ) ( ) r = x ( X) Defrmed W Defrmatin Gradient F x = X Right CG Defrmatin Tensr C = T F F Stred-energy functin f the matrix () ( F) W Stred-energy functin f the fibers W (2) ( F) Randm variabe characterizing the micrstructure ì ( r ) X Î r c ( X) = ï if phase í ïï therwise î
Prbem setting: Lagrangian frmuatin Kinematics O X Undefrmed W Lca cnstitutive behavir where W S = W F ( X, F) N r r ( XF, ) = c ( X) W ( F) å 2 ( ) ( ) r = x ( X) Defrmed W Istrpic matrix () () Defrmatin Gradient F x = X Right CG Defrmatin Tensr C = W ( F ) =Y ( I, I ) Transversey istrpic fibers where (2) (2) 2 T F F W ( F ) =Y ( I, I, I, I ) I = trc 2 4 5 I é 2 2 = C C ù 2 ê(tr ) -tr 2 ë ú û I = N CN I = N C N 4 5 2
Prbem setting: macrscpic respnse Definitin: reatin between the vume averages f S the stress and defrmatin gradient ver RVE = ò W W Variatina Characterizatin where and W S ( ) W = ( F) F S dx F = ò F dx W W F = min W ( X F) X FÎK( F) W ò, d W Undefrmed RVE L separatin f ength-scaes { = W = W } K( F F F x in, and x FX n )= Hi (972), Braides (985), Müer (987)
Existing anaytica appraches Phenmengica mdes Frmuated n the basis f invariants ( F ) = (, ) + (, ) W F I I G I I is 2 ani 4 5 Merdi, Hrgan, Ogden, Pence, Rivin, Saccmandi, amng thers Hmgenizatin/Micrmechanics mdes Incrprate direct infrmatin frm micrscpic prperties Estimates fr specia ading cnditins, and specia matrix and fiber cnstituents (He et a., 26; debttn et a., 26) Linear cmparisn variatina estimates fr genera ading cnditins and istrpic cnstituents (LP & Pnte Castañeda 26a,b)
Stabiity and faiure Casses f instabiities fiber faiure (ca) Materia fiber debnding (ca) matrix cavitatin (ca) shrt waveength (ca) Gemetrica ng waveength (gba) The ss f strng eipticity f the macrscpic respnse f the fiber-reinfrced eastmer, as characterized by the effective stred-energy functin W( F), dentes the nset f ng waveength instabiities ì 2 W ü B( F) = min ï vv F u u ï í ( ) = i k j ý u,v F F u = v = îï ij k ïþ Geymnat, Müer, Triantafyidis (993)
Micrstructure evutin Undefrmed F Defrmed W W Infrmatin n micrstructure evutin is imprtant t identify and understand the micrscpic mechanisms that gvern the macrscpic behavir. Fr that we need infrmatin abut the ca fieds F( X) LP (26)
New Apprach: Iterated Hmgenizatin
Iterated diute hmgenizatin Strategy: Cnstruct a particuate distributin f fibers ( X ) within a ( ) hypereastic materia fr which it is pssibe t cmpute exacty the effective stred-energy functin Step : iterated-diute hmgenizatin W r c ( ) diute phase Hmgenizatin Ad-infinitum... W c - H éw W Fù = W F = W c ê, ; ë ú, (,) û () (2) LP, J. App. Mech. (2)
Auxiiary diute prbem: sequentia aminates Step 2: sequentia aminates Rank- r simpe aminate Rank-2 aminate... ad-infinitum... incusin phase matrix phase When the matrix phase is diute: W HW é é ù () WFù () ê,, ú = W+ max -W ( F+ Ä ) n S ë û ò w x w x ( x)d w S ê ë F ú û distributina functin reated t the tw-pint statistics f fiber distributin in the undefrmed cnfiguratin Idiart, JMPS (28)
Iterated hmgenizatin framewrk The effective stred-energy functin W can be finay shwn t be given by the fwing Hamitn-Jacbi equatin W é W ù () c -W -max -W ( F+ Ä ) n S = c ò w x w x ( x) d ê F ú û w S ë subject t the initia cnditin W( F,) W ( F) = (2) the time variabe is t -nc (initia fiber cncentratin) the spatia variabe is F the Hamitnian is W HW é é ù () WFù () ê,, ú = W+ max -W ( F+ Ä ) n S ë û ò w x w x ( x)d w S ê ë F ú û LP & Idiart, J. Eng. Math. (2)
LP & Idiart, J. Eng. Math. (2) Iterated hmgenizatin framewrk: ca fieds Cnsider the fwing perturbed prbem fr W é W ù c t W t () - -max -W ( F+ Ä ) n S = t c ò w x w x ( x) d w S ê F ú ë û subject t the initia cnditin (2) W( F,) = W ( F) + tu( F) W t Then, the fwing identity is true ò W U ( FX ( )) dx= t (2) W W (2) c t t= U () can be any functin f interest, e.g., the defrmatin gradient
Remarks n the iterated hmgenizatin apprach The prpsed IH methd prvides sutins fr W in terms f W () and W (2) and the ne- and tw-pint statistics f the randm distributin f fibers In the imit f sma defrmatins as, the IH frmuatin reduces t the HS wer bund fr fiber-reinfrced randm media F I In the further imit f diute fiber cncentratin c, the IH frmuatin recvers the exact resut f Esheby fr a diute distributin f eipsdia fibers The prpsed IH methd prvides access t ca fieds, which in turn permits the study f the evutin f micrstructure and the nset f instabiities The cmputatins amunt t sving apprpriate Hamitn-Jacbi equatins, which are fairy tractabe
Appicatin t Fiber-Reinfrced Ne-Hkean Sids
Fiber-reinfrced Ne-Hkean sids Ne-Hkean matrix () () () m W ( F) =Y ( I ) = I -3 2 Stiffer Ne-Hkean fibers (2) ( ) (2) (2) m W ( F) =Y ( I ) = I -3 2 ( ) S 6 5 4 3 2 W randm, istrpic distributin N Macrscpic stred-energy functin.5 2 2.5 3 3.5 Hamitn-Jacbi equatin W( F, c ) =Y( I, I, I, I, c ) 2 4 5 m æ 2 () Y öæ ö () 2 Y c ( I ) m I I ç + -3 - Y + - - - = c ç I ç m I 4 2 () è è ø 2 ø subject t the initia cnditin Y ( I, I, I, I,) = Y (2) ( I, I ) 2 4 5 4 4
LP & Idiart, J. Eng. Math. (2) Overa stress-strain reatin Csed-frm sutin fr W where with m f( I ) = -3 2 ( I ) m = ( - c ) m () + cm (2) ( ) ( ) W( F ) =Y ( I, I ) = f I + g I 4 4 and and gi ( ) = 4 ( I 2)( I ) m - m + - 4 4 2 I Nte I: separabe functina frm Nte II: n dependence n the secnd I 2 nr fifth I 5 invariants Overa stress-strain reatin m () (2) ( - c ) m + ( + c ) m = m + () c m + - (2) ( ) ( c ) m W S = ( F) - pf = F+ ( - ) é -I ùfn Ä N-pF F ëê ûú -T -3/2 -T m m m 4 4 ()
Macrscpic Instabiities fiber faiure (ca) Materia fiber debnding (ca) Casses f matrix cavitatin (ca) instabiities shrt waveength (ca) Gemetrica ng waveength (gba) Ang an arbitrary ading path, the materia first becmes cr unstabe at a critica defrmatin F cr with I = F N F N 4 cr cr such that I æ m ö = - ç çè m ø cr 4 2/ 3 Nte: m cr 4 m I LP & Idiart, J. Eng. Math. (2)
Macrscpic Instabiities Observatins Macrscpic instabiities may ny ccur when the defrmatin in the fiber I 4 directin, as measured by, reaches a sufficienty arge cmpressive cr vaue I 4 The cnditin states that instabiities may devep whenever the cmpressive defrmatin ang the fiber directin reaches a critica vaue determined by the rati f hard-t-sft mdes f defrmatin m Hard mde m Sft mde W
Micrstructure evutin Undefrmed e 3 F Defrmed v 3 z - 2 = z - 2 = 3 e 2 z -2 z - 2 = 3 W e z - 2 = 2 W v z -2 2 v 2 The average shape and rientatin f the fibers in the defrmed cnfiguratin are characterized by the Euerian eipsid LP (26), LP & Idiart (2) { T } E = x x Z Zx Eigenvaues f where z, z, z ( )( (2) ) Z = I-N Ä N F - 2 3 Eigenvectrs f v,v,v 2 3 T Z Z T Z Z
Micrstructure evutin is the average defrmatin gradient in the fibers. In the IH framewrk, it is sutin f the pde F (2) c (2) (2) F F - Ä S = c F ò w x d S subject t the initia cnditin F (2) ( F,) = F Csed-frm sutin 2n -g F F FN N éf F N N ù 2n -g = é - Ä ù - - Ä + FN Ä u + FN Ä N (2) T T g - - ë û ê ú I ë û I 4 4 where u = ( I-N Ä N) F FN, and g = n + 2 T ( 2 ) I + n I I -I - I () 4 4 5 4 m, n = II - I + 2 I ( + c ) m + ( - c ) m 4 5 4 () (2) LP, Idiart, & Li (2)
Sampe Resuts
IH vs. FEM: in-pane stress-strain respnse 7 6 IH FE t S () m 5 4 3 2 c = 4. c = 2. = 2. Rigid fibers.5 2 2.5 3 3.5 Mraeda et a. (29)
Axisymmetric cmpressin at an ange φ Appied Lading F æ -2 / ö ç -2 / = ç çè ø Initia fiber rientatin N = csj e + sinj e 3 e 3 Ν F e 3 n j 2 / - 2 / - e j e W W
Axisymmetric cmpressin at an ange φ Appied Lading Current fiber rientatin F æ -2 / ö ç -2 / = ç çè ø e 3 j F n = csje + sinje é ê Arcs Ν n e 3 3 j = ê 2 3 2 cs j + cs j ú ë cs ù ú û j 2 / - 2 / - e j e W W
Axisymmetric cmpressin at an ange φ 5 4 Sma-defrmatin respnse () m = t = 5 c = 3. ds d 3 = 2 t = 2 2 / - 2 / - t = 5 2 4 6 8 j t m m ( 2) () hetergeneity cntrast
Axisymmetric cmpressin at an ange φ S 35 3 25 2 5 j = 6 Large-defrmatin stress-strain respnse j = 8 ( ) m = c = 3. t = 2 j = 9 j 8 6 4 Evutin f fiber rientatin j = 89. 9 j = 89 j = 9 j = 8 j = 6 j = 35. 3 5 j = 35. 3 j =.2.3.4.5.6.7.8.9 2 j = j =.2.4.6.8 Nte: Macrscpic instabiity at LP, Idiart, & Li (2)
Axisymmetric cmpressin at an ange φ Onset f macrscpic instabiities at cr Effect f fiber rientatin Effect f fiber-t matrix cntrast cr.9.8.7.6.5 t = 5 t = 2 cr.9.8.7 c = 5. c =. c = 3..4 t = 5.3 c = 3. ( ) m =.2 3 4 5 6 7 8 9 j.6.5 j = 9 ( ) m = 2 4 6 8 t
Axisymmetric cmpressin at an ange φ Onset f macrscpic instabiities at cr Effect f fiber rientatin Evutin f fiber rientatin.9 j = 89. 9 j = 9 cr.8.7.6 t = 5 t = 2 j 8 6 j = 89 j = 8 j = 6.5.4 t = 5.3 c = 3. ( ) m =.2 3 4 5 6 7 8 9 j 4 2 j = j = 35. 3 j =.2.4.6.8
Remarks The rtatin f the fibers which depends criticay n the reative rientatin between the ading axes and the fiber directin can act as a dminant gemetric sftening mechanism. It was fund that the ng axes f the fibers rtate away frm the axis f maximum cmpressive ading twards the axis f maximum tensin. Ladings with predminant cmpressin ang the fibers ead t arger rtatin f the fibers, which in turn ead t arger gemetric sftening f the cnstitutive respnse, and in sme cases when the hetergeneity cntrast between the matrix and the fibers is sufficienty high as t the ss f macrscpic stabiity.
Remarks The resuts f this wrk can hep understanding the behavir f many ther sids with riented micrstructures Reinfrced Eastmers (Wang and Mark 99) S 2 m Pystyrene Eipsids Liquid crysta eastmers (Nishikawa and Finkemann999) Transparent Opaque
Cntact with earier wrk with aminates -? -» - - Fr the aigned pane-strain ading f a aminate the nset f macrscpic instabiities ccurs at where Lam cr æ L m ö /4 = - ç m çè ø m = ( - c) m () + cm (2) and æ L c c ö m - = + ç () (2) çè m m ø - Rsen (965); Triantafyidis and Maker (985)
Cntact with earier wrk with aminates Aigned pane-strain ading cnditins - - Effect f fiber cncentratin c.95 t Laminate ( 2) ( ) = m / m = 2 Effect f cntrast.95 Laminate t = m / m ( 2) ( ) cr.9.85.8 Fibers cr.9.85 Fibers.75.8 c = 3..7.75.2.4.6.8 2 4 6 8 ( 2) ( ) c t = m / m
Cntact with experiments Matrix: Siicne Yung s Mduus E () = 2.9 MPa Fibers: Spaghetti Yung s Mduus E (2) = 69 MPa Vume fractin c = 3% Experimenta setup IH predictin Jef & Feck (992)
Fina remarks An iterated hmgenizatin apprach in finite easticity has been prpsed t cnstruct exact (reaizabe) cnstitutive mdes fr fiber-reinfrced hypereastic sids Because the prpsed frmuatin grants access t ca fieds, it can be used t thrughy study the nset f faiure and the evutin f micrstructure in fiber-reinfrced sft sids with randm micrstructures The required anaysis reduces t the study f tractabe Hamitn-Jacbi equatins As a first appicatin, csed-frm resuts were derived fr fiber-reinfrced Ne-Hkean eastmers These ideas can be generaized t mre cmpex systems f sft hetergeneus media with randm micrstructures