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A matricial approach of fibre breakage in twin-screw extrusion of glass fibres reinforced thermoplastics

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A matricial approach of fibre breakage in twin-screw extrusion of glass fibres reinforced thermoplastics
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  A matricial approach of fibre breakage in twin-screw extrusion of glass fibresreinforced thermoplastics Audrey Durin a , Pascal De Micheli a , Julien Ville b,c , Funda Inceoglu d , Rudy Valette a , Bruno Vergnes a, ⇑ a MINES ParisTech, Centre de Mise en Forme des Matériaux (CEMEF), UMR CNRS 7635, BP 207, 06904 Sophia Antipolis, France b POLYTECHS SA, 76 450 Cany-Barville, France c Laboratoire d’Ingénierie des Matériaux de Bretagne, EA 4250, Université de Bretagne Occidentale, 29 238 Brest, France d  ARKEMA, CERDATO, 27 470 Serquigny, France a r t i c l e i n f o  Article history: Received 29 February 2012Received in revised form20 December 2012Accepted 31 December 2012Available online 19 January 2013 Keywords: A. Glass fibresC. Micro-mechanicsC. Computational modellingE. Extrusion a b s t r a c t Limitingfibrebreakageduringcompositeprocessingisacrucialissue. Thepurposeofthispaperistopre-dict the evolution of the fibre-length distribution along a twin-screw extruder. This approach relies onusing a fragmentation matrix to describe changes in the fibre-length distribution. The flow parametersin the screw elements are obtained using the simulation software Ludovic  . Evolution of an initialfibre-length distribution for several processing conditions was computed and the results were comparedwith experimental values. The computation gives satisfying results, even though more comparisons withexperiments would be necessary.  2013 Elsevier Ltd. All rights reserved. 1. Introduction A classical use for glass fibres in industry is thermoplastic poly-mers reinforcement, mostly for injected parts. Long-fibre compos-ites are known to offer better mechanical properties than short-fibre ones. Consequently, an important point is to preserve asmuch as possible long fibres during compounding, despite strongflow conditions, eventually leading to severe break-up.Fibreorientationfor long-fibrethermoplasticsinmouldingpro-cess has already been studied and modelled [1–4] and a quantita-tive model predicting changes in fibre-length distribution duringmould filling has been recently developed by Tucker et al. [5].However, prior to injection moulding, other processes are used tocompound glass fibres into polymer matrices. An important issueis thus to control fibre lengths in these processes to subsequentlyinject compounds exhibiting a suitable final length distribution.The most common of these processes is twin-screw extrusion, inwhich considerable fibre length degradation occurs [6–8]. Shon et al. [9] have been the firsts to develop an empirical modeldescribing the average fibre length evolution in different continu-ous processes, including twin-screw extrusion. More recently, thisapproachwas improvedto calculateaverage fibrelengthevolutionduring twin screw extrusion and Buss kneader compounding [10–11]. However, these methods do not provide information on thewhole fibre-length distribution. Therefore, the aim of the presentpaper is to propose a computational method to predict changesin the fibre-length distribution along a twin-screw extruder. 2. Theoretical model  2.1. Forgacs and Mason model Our model is based on the assumption that fibre breakage isonly due to flow-induced buckling, as described by Forgacs andMason [12]. According to this model, a rotating rigid fibre in a shear flow may break when oriented in the direction of compres-sive forces. Beyond a critical force, which depends on its mechan-ical properties and length, the fibre buckles and then breaks-up(Fig. 1). Breakage occurs because of the severe tensile stress  r s  in-duced on the external surface of the fibre when it is bending. Thisstress depends on the fibre radius  b , its Young modulus  E   and thelocal radius of curvature  R : r s ð  x Þ ¼   EbR ð  x Þ ð 1 Þ where  x  is the abscissa along the fibre principal axis. When thestress  r s  on the surface reaches the tensile strength value of the fi-bre  r c  , the fibre breaks-up. As the radius of curvature of the fibre islinked to the fibre deformation, the breakage phenomenon directlydepends on this deformation. In this work, it was assumed that,when buckling occurs, the fibre systematically breaks-up because 1359-835X/$ - see front matter  2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesa.2012.12.011 ⇑ Corresponding author. Tel.: +33 493 957 463; fax: +33 492 389 752. E-mail address:  bruno.vergnes@mines-paristech.fr (B. Vergnes).Composites: Part A 48 (2013) 47–56 Contents lists available at SciVerse ScienceDirect Composites: Part A journal homepage: www.elsevier.com/locate/compositesa  of the resulting huge deformation. This assumption is validated inSection 2.3.  2.2. Jeffery equation To determine when buckling (and then breakage) occurs, forcesapplied on the fibre as well as fibre orientation must be computed.Classically, the orientation  P  of a single ellipsoidal fibre of length2 a  and radius  b  in a shear flow is obtained by solving Jeffery equa-tion [13]: _ P  ¼ X  P þ  k ½ _ e  :  P  ð _ e  :  P  P Þ P  ð 2 Þ where  P  is the orientation vector of the fibre principal axis, X thevorticity tensor,  _ e  the strain rate tensor, and  k  a parameter relatedto the aspect ratio  b  = a/b : k  ¼  b 2  1 b 2  ð 3 Þ The orientation vector  P  describes the fibre orientation in thereference frame. In the case of simple shear flow, this frame is de-finedasdepictedinFig.2.Thisorientationcanalsobedescribedbythe angles  h  and  /  (which can be determined from  P ). As glass fi-bres are cylindrical, the ellipsoid aspect ratio  b  should be replacedin Eq. (3) with an equivalent aspect ratio  b e  for cylinders, theoret-ically determined by Burgers [14]: b e  ¼  0 : 74 b  ð 4 Þ Theorientationvectorwasthenusedintheforcescomputation.The shear inducedforce  F  B , integratedon a half-fibre, was given byBurgers[14], without furtherindicationonthe forcesdistribution  f  along the fibre: F  B  ¼  Z   0 a  f  ð  x Þ  dx  ¼  M  p g _ c a 2 ln ð 2 b Þ  1 : 75  ð 5 Þ where  g  is the viscosity and  _ c  the shear rate.  M   is defined as: M   ¼  sin 2 h sin / cos /  ð 6 Þ where  h  and  /  were obtained from the orientation vector  P . In amore convenient form, Eq. (5) can be written in terms of stress: r B  ¼  g _ c M   b 2 ln ð 2 b Þ  1 : 75  ð 7 Þ In order to determine when buckling occurred, forces were as-sumed to be punctually applied at fibre ends. Then, it was possibleto obtain the buckling threshold by applying Euler bucklingmethod.  2.3. Euler buckling method Inorder to confirmthat perfect (without anydefect) rigidfibrescannot break-up when simply bending below the buckling thresh-old (in the case of small deformations) and also always break-upwhenbuckling, thetensilestress r s  appliedontheexternal surfaceof the fibre when it bends (below the buckling threshold: smalldeformation) and when it buckles (over the buckling threshold:large deformation) must be computed. In this way, the tensilestress  r s  can be compared to the tensile strength  r c   in order tocheck if the fibre does break. Compressive forces were supposedto be only applied at fibre ends and along its principal direction.Thebendingmomentumbalanceforanon-deformedconfigurationgives: M  0 ð  x Þ ¼  F  B  y ð  x Þ ð 8 Þ where M  0 (  x ) isthebendingmomentumand  y (  x ) thefibredeflectionat point  x . From this equation, the deformation below the bucklingthreshold (assuming that there exists an initial deflection at rest)and beyondthe bucklingthresholdcanbe obtained. First, the buck-ling threshold was calculated under the assumption of ‘‘small’’deformations (Euler buckling method), in which the bendingmomentum  M  0  is approximated by: M  0  ¼  EI R    EIy 00  ð 9 Þ where I  isthemomentofinertiaand R theradiusofcurvature.Com-bining Eqs. (8) and (9) leads to the differential equation:  y 00  þ  k 2  y  ¼  0  ð 10 Þ with  k 2 ¼  4 b 2 r B E  . The only possible non trivial solution satisfyingthe homogenous boundary conditions (  y ( a )=0 and  y 0 (0)=0) is:  y  ¼  A cos ð kx Þ ð 11 Þ with  k ¼  p p 2 a  and  p  is a strictly positive integer. Considering thatthe fibre is brittle, and assuming that it breaks when it reaches its Fig. 1.  Rotating fibre in a shear flow. Break-up occurs when the maximum flow-induced compressive force is high enough. Fig. 2.  Fibre end orbit in a simple shear flow. The shear plan frame (  x 0 ,  y 0 ,  z  0 ) istranslating with the fibre. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)48  A. Durin et al./Composites: Part A 48 (2013) 47–56   first buckling mode (  p  =1), leads to the expression of the maximalstress  r em  before buckling: r em  ¼  E   p 4 b   2 ð 12 Þ Comparing the stress  r B  induced by the flow on the fibre withthe maximal stress  r em  before buckling leads to a buckling param-eter  Bu : Bu  ¼  r B r em ¼   4 p   2 M  g _ c E  b 4 ln ð 2 b Þ  1 : 75  ð 13 Þ which is greater than 1 for a buckling fibre. Below the buckling threshold, a perfectly straight fibre cannotbedeformedapplyingonlycompressiveforces. Fibrescanhoweverexhibit an initial deflexion as they are not ideal objects. In theframework of small deformations, below the first buckling mode,it is reasonable to consider an initial deflexion of the fibre of theform:  y 0  ¼  h cos p  x 2 a  ð 14 Þ where  h  is the arbitrary fixed initial deflexion for  x  =0. Then, thedeformationbelowthebucklingthresholdcanbe computedreplac-ing  y  with Y   +  y 0  inEq. (8) and  y  with  Y   in Eq. (9). The followingdif-ferential equation was then obtained: Y  00  þ  k 2 Y   ¼  k 2 h cos p  x 2 a  ð 15 Þ Thesolutionofthisequationsatisfyingtheboundaryconditionsis: Y   ¼  h  11   Bu  cos p  x 2 a  ð 16 Þ From this expression and from Eq. (1), the expression of the lo-cal tensile stress  r s (  x ) applied on the external surface of the fibrebelow the buckling threshold was deduced under the hypothesisof small deformations  1 R ¼ Y  00   : r s ð  x Þ ¼  E hb p 2 b   2 11   Bu  cos p  x 2 a  ð 17 Þ On the other hand, to calculate the deformation beyond thebuckling threshold, the small deformations hypothesis can nolonger be used. The deformation was then supposed large enoughso that  y 0 is no more negligible with respect to 1. Eq. (9) then be-comes a non-linear second order differential equation: M  0  ¼  EI R  ¼  EI  y 00 ð 1 þ  y 0 2 Þ 3 = 2  ð 18 Þ From Eqs. (8) and (18), the following non-linear differentialequation was obtained:  y 00 ð 1 þ  y 0 2 Þ 3 = 2  þ  k 2  y  ¼  0  ð 19 Þ ThesolutionofEq.(19)withhomogeneousboundaryconditionsis [15]:  y  ¼  a 4  ffiffiffi 2 p  p  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bu   1 p   1  18 ð Bu   1 Þ   cos p  x 2 a  ð 20 Þ From Eqs. (1) and (20), the expression of the local tensile stress r s (  x ) applied on the external surface of the fibre over the bucklingthreshold was deduced: r s ð  x Þ ¼  E  p  ffiffi 2 p  b  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bu   1 p   1   18  ð Bu   1 Þ   cos p  x 2 a 1 þ 8 ð Bu   1 Þ  1   18  ð Bu   1 Þ   2 sin 2  p  x 2 a h i 32 ð 21 Þ Comparing these tensile stresses  r s (  x ) (Eqs. (17) and (21)) withthe tensile strength  r c   of the fibre, a break-up parameter wasobtained: Br   ¼  r s ð  x Þ r c  ð 22 Þ with Br   ¼  E  r c  hb ð p 2 b Þ 2  11   Bu  cos p  x 2 a  if   Bu  <  1  ð 23 Þ Br   ¼  E  r c  p  ffiffi 2 p  b  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bu   1 p   1   18  ð Bu   1 Þ   cos p  x 2 a 1 þ 8 ð Bu   1 Þ  1   18  ð Bu   1 Þ   2 sin 2  p  x 2 a h i 32 if   Bu  >  1  ð 24 Þ Br   depends on the bucklingparameter  Bu ¼  r B r em and is greater than 1for a theoretically perfect (i.e. without any defect) breaking fibre. Some  Bu  and  Br   values will now be computed, for a range of parameter whose order of magnitude corresponds with the caseofshortglassfibresandpolymerinatwin-screwextruder,inorderto better understand the breakage phenomenon. These results areonlyillustrativeandarenot tobecomparedwiththe experimentalresults that are shown further. Fig. 3 shows the change of thebreakage parameter  Br   as a function of the buckling parameter Bu  for different values of relative shear stress  g _ c E   (from 2  10  8 to 3  10  6 ) and aspect ratio  b  (from 10 to 200). Other parameterswere kept constant:  x / a  =0 (centre of the fibre),  h / b  =0.01 (initialrelative deflexion),  /  =  p /4,  h  = p /2 (fibre position in the flow), E  / r c   =30 (fibre flexibility). In area (1), the fibre has not reachedthe buckling threshold nor the break-up threshold, so it remainsstraight and does not break. In area (2), only the break-up thresh-old is reached, so the fibre breaks-up instantaneously withoutbuckling (weak fibre). In area (3), only the buckling threshold isreached, so the fibre buckles without breaking (flexible fibre). Fi-nally, in area (4), both thresholds are reached, so the fibrebreaks-up during buckling.Theeffectoffibreorientation(angle h )isshowninFig.4forvar-ious aspect ratios  b  (from 10 to 600), the other parameters beingconstant:  x / a  =0 (centre of the fibre),  h / b  =0.01 (initial relativedeflexion),  /  =  p /4 (fibre position in the flow),  g _ c = E  ¼ 10  7 (rela-tive shear stress),  E  / r c   =30 (fibre flexibility). In the tested condi-tions, it was noticed that fibres were almost always breaking-upwhen buckling, justifying the proposed approximation that buck-ling is equivalent to break-up. It was thus possible to directly use Fig. 3.  Correlation between buckling parameter  Bu  and breakage parameter  Br   fordifferentaspectratios b . s : b  =10; d : b  =50; h : b  =100; j : b  =150; 4 : b  =200(thearrows indicate the evolution when changing the relative shear stress  g _ c E   from2  10  8 to 300  10  8 , other parameters constant).  A. Durin et al./Composites: Part A 48 (2013) 47–56   49  the buckling parameter  Bu  (Eq. (10)) to determine at which shearstress  g _ c  a perfect fibre would break for a given length.It was also observed that fibres were breaking-up even whenthe angle  h  between them and the  z  0 axis is smaller than  p /2 (fibreoutoftheshearplan,seeFig.2).Moreover,fibresspendalongtimealong the flow direction during a Jeffery rotation period, and theangle  h  at this moment is maximal. Consequently, the angle be-tween most of the fibres and the  z  0 axis is almost all the time closeto  p /2. Therefore, the parameter  h  was set to  p /2 in the computa-tions, supposing that even if some of the fibres were slightly outof the plan, they would also break-up. 3. Modelling approach In order to turn these theoretical models in a computationalmethod, specific tools, such as the probability distribution andthe fragmentation matrix have to be used.  3.1. Probability distribution Because the minimal radius of curvature is located in the mid-dle of a bending fibre (Eq. (1)), a perfect fibre will always break-up at this point. However, a real fibre always presents defectswhich can be considered as weak points, leading to a possiblebreak-up below the critical stress value. Therefore, introducing astatistical distribution of the breakage probability along the fibrewas necessary in order to mimic the presence of defects. The usualbreakage probability law for glass fibre reinforcement is the Wei-bull distribution [16]: P  ð r Þ ¼  1  exp    r  r u r 0   m    ð 25 Þ where  r  is the stress,  r u  the minimal stress for breakage,  r 0  a scalefactor, and  m  a shape parameter. It expresses the probability for afibre to break at the stress  r . The problem is that this law is onlyvalid for a homogenous stress along the fibre, i.e. for a fibre loadedintension.Consequently,thebreakageprobability(Eq.(25))andthelocal breakage probability distribution along a fibre (Eq. (26)) wereassumed to be uncorrelated. The fibre breakage probability is (Fig. 5): P  ð Bu Þ ¼  1  exp ð Bu Þ 1  exp ð 1 Þ  if   Bu  <  1  ð 25a Þ P  ð Bu Þ ¼  1 if   Bu  >  1  ð 25b Þ This probability law allows a fibre to break below the bucklingthreshold because of the defects. The fibre necessarily breaks-uponcethisvalueis reached,complyingwiththeprevioustheoreticalobservations (Fig. 3).The local breakage probability distribution along the fibre waschosenas aWeibull-likedistribution(Eq. (26)), but it remainspos-sible to choose another type of distribution, for example a normaldistribution [5].The local breakage probability distribution on the fibre is(Fig. 6): P  l ð  x Þ ¼ 1  exp    1  ð  xa  Þ 2   m h iR  1  1  1  exp    1   xa   2   m h i  d  xa   ð 26 Þ where  m  allows to change the distribution shape. The breakageprobability is maximal at the middle of the fibre (  x  =0). In this firstapproach,Weibullexponent m wassetto3,whichislargerthananyexperimentally fitted value found in the literature. Doing so, a lowenough ‘‘breakage polydispersity’’ was introduced in the model toensure dispersion around the fibre centre position failure. A lowervalue for  m  did not change significantly the time-dependent distri-bution, as shown by a parameter study on  m , for representativetwin-screw extrusion conditions (not presented in the paper). Thevalue m  =3shouldthenbe consideredas a minimalvaluetoensurevariability in breakage position along the fibre. Fig. 4.  Correlation between buckling parameter  Bu  and breakage parameter  Br   fordifferent angles  h  (with the  z  0 axis perpendicular to the shear plan). s :  h  = p /8; d : h  = p /4; h :  h  =3 p /8;  j :  h  = p /2 (the arrows indicate the evolution when changingthe aspect ratio  b  from 10 to 600, other parameters constant). Fig. 5.  Fibre breakage probability as function of buckling parameter  Bu . Fig. 6.  Local breakage probability distribution along the fibre ( m  =3).50  A. Durin et al./Composites: Part A 48 (2013) 47–56    3.2. Fragmentation matrix Inordertodescribetheentiredistribution, afragmentationma-trix, based on the mass conservation [17] was used. Fibres weredistributed into  n  classes according to their length. The minimalaccessible length  L n  was set to the minimal length below which afibre cannot buckle for any applied stress ( b =2.88, Eq. (13)). Then,the mass transfer from long fibre classes to shorter ones (Fig. 7)was expressed using the set of Eq. (27), and solved with a Dor-mand–Prince method [18] using Matlab  .The mass transfer set of equations is: dm i  ¼ X n j ¼ 1 ð s  j P  ij M  ij  dt  Þ  s i m i  dt   8 i ;  1  <  i  <  n  ð 27 Þ where  m i  is the mass ratio represented by the fibres of length  L i ,  P  ij the probability for breaking-up a  L  j  length fibre to generate a  L i length one,  M  ij  (Eq. (28)) the mass transfer from the  L  j  length fibreclass to the  L i  length one (if a fibre breaks at the point required togenerate a  L i  length fibre), and  s  j  the  L  j  length fibre breakage rate,that was considered to be equivalent with the  L  j  fibre length break-age probability per time unit. The quantities  P  ij ,  M  ij  and  m i  aredimensionless and the breakage rate is expressed in s  1 . When a L  j  length fibre breaks-up, generating a  L i  length fibre, a part of itsmassistransferredtothe L i  lengthfibreclass. Themasstransferbe-tween classes is then  M  ij : M  ij  ¼  L i L  j m  j  ð 28 Þ As the local breakage probability distribution on a fibre is sym-metricrelativelytoitscentre(Fig.8),theprobability P  ij  forabreak-ing-up  L  j  length fibre to generate a  L i  length one is twice theprobability  P  l  (Eq. (26)) to break at one of the two possible points,leading to: P  ij  ¼  2 P  l ð  x i Þ ð 29 Þ The L  j  lengthfibrebreakagerate s  j  wasconsideredasequivalentto the probability for a  L  j  lengthfibre to break-up during a rotationperiod t  r   dividedbythisperiod.Thebreakageprobability P  tr   duringa period was computed by adding the breakage probabilities  P   (Eq.(25)) for each successive orientation  P  along the fibre orbit  C  : P  tr  ð L  j Þ ¼ Z  orbitC  P  ð r B ð P ÞÞ  1 Ct  r  dC   ð 30 Þ Fig. 7.  Mass transfer from long fibre classes to shorter ones from the  L i  length class point of view. Fig. 8.  Exampleofshearrateandcumulativeresidence-timealongthetwin-screwextrudercomputedusingLudovic  software.(Forinterpretationofthereferencestocolourin this figure legend, the reader is referred to the web version of this article.)  A. Durin et al./Composites: Part A 48 (2013) 47–56   51
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