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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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