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Analysis and evaluation of the single-fibre fragmentation test
Composites Science and Technology 56 (1996) 893-909 0 1996 Published by Elsevier Science Limited Printed in Northern Ireland. All rights reserved ELSEVIER
PII:
ANALYSIS
AND
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EVALUATION OF THE SINGLE-FIBRE FRAGMENTATION TEST
T. J. Copponnex* Ecole Nationale Suptrieure
des Mines de Paris, Centre des Mathiaux
Pierre-Marie Fourt, Eury, France
(Received 29 June 1995; accepted 22 February 1996) Abstract
in the fibre coating as well as in the matrix are then evaluated. Interpretation of results is still controversial, however, partly because of the fact that common models used to relate fragment lengths to interfacial properties are based on simplistic assumptions and are mostly stress-based methods. The very well-known elastic-plastic Kelly and Tyson approach for instance, seeks to describe the stresses in the post-failure state and as such does not highlight the interfacial failure mechanisms taking place. Alternative energy-based methods, assuming that the energy required to debond the fibre is the failure-controlling mechanism, have been reported recently2-4 but these approaches have been developed mainly for other geometrical configuration tests such as pull-out.5,6 Related to the embedded single-fibre test, finite-element analyses7-9 to evaluate stress, strain and stored energy have been reported but it seems that little effort has been expended to improve existing analytical methods. A weak point with currently used energy-based approaches is that calculations are performed from one-dimensional stress analyses. In other words, only axial and shear stresses in the constituents are used to derive a debond criterion. Recently, Nairn” using a variational approach, and assuming perfect bonding between the fibre and the matrix, extracted an analytical expression for the bi-dimensional stress tensor in a fragmentation sample. He pointed out that radial and hoop stresses should also play an important role in interfacial properties. In the light of these recent improvements, it seems that a correct debonding criterion has to be derived from the knowledge of the complete stress tensor in the constituents. Unfortunately, Nairn’s expressions given for the case of perfect bonding cannot be used for this purpose. In the present study, the strain energy release rate principle is used to study the mechanism of interfacial failure in fragmentation tests on thermoplastic-based microcomposites. From Nairn’s approach an analytical expression for the complete bi-dimensional stress tensor for isotropic materials, including imperfect adhesion between constituents, is used to calculate the
The single-fibre fragmentation test has received attention as a preliminary test for studying interfacial properties in composite materials. Its merit as a powerful tool for measuring interfacial bonding at the jibre/matrix interface is not in doubt. ‘Simplistic’ assumptions generally used for the analytical treatment of experimental results seem nevertheless to hide the real mechanisms involved during the fragmentation process and especially with regard to the debonding following the jibre breakage. In this paper, an improved analysis is used of Nairn’s previous variational mechanics approach to the single-fibre fragmentation test, taking into account imperfect adhesion. The analytical expression for the stress tensor in the sample is used to derive an interfacial debonding criterion by means of an energy release rate analysis. It is shown that the extent of interfacial damage can be determined by this energy-based approach. Comparison of predictions with experiments shows that a critical interfacial fracture toughness can be used to predict the growth of debonding. 0 1996 Published by Elsevier Science Limited Keywords:
composite materials, single-fibre fragmentation test, variational mechanics, energy release rate, interfacial fracture toughness 1 INTRODUCTION The single-filament multi-fragmentation test, originally proposed by Kelly and Tyson,’ has received attention as a preliminary test for studying interfacial properties in composite materials. Under tension and above the maximum fibre elongation, a fragmentation process occurs. From the fibre fragment length distribution, properties of the interface are quantified by deriving an expression for the maximum interfacial shear stress acting at the fibre/matrix interface. Performance of chemical modifications incorporated *Present address: Shell Research, 1 avenue Jean Monnet, B1348 Louvain-la-Neuve, Belgium. 893
T. J. Copponnex
894
total energy release rate associated with the growth of interfacial debonding. Experimental results are given for E-glass fibres embedded in different polypropylene matrices. It is shown that a critical interfacial toughness can be used to predict the propagation of the debonding interface in these thermoplastic-based microcomposites. 2 THEORETICAL
ANALYSIS
2.1 Sample geometry
Consider geometry Cylinder fraction
the typical
single-fibre
fragmentation
test
in Fig. 1. Three cylinders are defined. 1 refers to the fibre of radius rr (fibre volume V,) and length L. Cylinders 2 and 3 are
respectively associated with the near-field matrix (matrix volume fraction V, = 1 - V,), where a fibre break disturbs stresses in the matrix, and far-field matrix, where these stresses are those that exist when the fibre remains unbroken.” The overall sample between two fibre breaks is submitted to tension cr,. This loading pattern introduces shear along the fibre/matrix interface and causes debonding. Two different regions are considered. In the central part of the fragment, the fibre is
-2
-P
-P+&,
0
P-6r
bonded to the surrounding matrix. At both fragment ends, failed interface is considered with asymmetric aspect. A set of dimensionless cylindrical coordinates (6, 8, 5) is used so that 5 = r/r* and i = z/rf correspond to the fibre axis. b runs from negative ( - p) on the left to positive (p) on the right, with p = L/2r, the fibre aspect ratio. Debonded lengths are limited to the -p + 6, coordinate on the left and p 6, on the right part. When the failed interface grows, -p + 6, and p - 6, move towards i= 0 at the expense of the central region. As pointed out by Nairn, the fibre breaks do not disturb the axisymmetric nature of the stresses and thus lead to a three-dimensional, axisymmetric stress analysis of the single-fibre fragment of length L. perfectly
2.2 Solution for the stress components 2.2.1 Nairn’s relationships
Following Nairn’s approach and under axisymmetric conditions, the shear stresses rVe and rze are zero. Stress components are then given by:
a
P
2’r0 = 0
(1)
r), = 0
(2)
ai = (&&a;,&)
(3)
where the superscript i = f in the fibre cylinder, m, in the near-field matrix cylinder and m, in the far-field matrix cylinder. We begin with Nairn’s relationships under the assumption that the axial stress, ok, in each cylinder is independent of the radial coordinate. Nairn stated the stress components in each cylinder in terms of a stress function, $t, its derivatives I,/#, I+?“,the total applied axial stress on the fibre cylinder and near-field stress matrix cylinder, a, (Fig. l), the temperature difference between the sample temperature and stress-free temperature, T, and the radial stress at the fibre/matrix interface, a, (5 = 1). The analytical expression for u, extracted by Nairn and reformulated for the case of isotropic materials is: + T(1 + vf)(q - cq,,)
(v, - v@ Cr,=
2v&-1 G
Fig. 1. A schematic illustration of Nairn’s three-cylinder model (with dimensionless coordinates) for a single fibre fragmentation test sample incorporating partial debonding. The inner cylinder (1) is the fibre. The surrounding matrix cylinder (2) is the near-field matrix cylinder. The outer cylinder (3) refers to the far-field matrix cylinder.
1+v,
(4)
E,
where v,,, and vr, are, respectively, the matrix and fibre Poisson ratios, cy,, arr the coefficients of dilatation, and E, and Ef, the Young’s moduli. a0 is obtained by equating total displacement between the near and far-field matrix cylinders. Since
Single-fibre fragmentation
explicit expressions for stress components are required prior to co determination, calculation and expression for the axial stress are reported in Appendix 1. The expression of the stress tensor in each cylinder can be given as: (Ti = B’tJl (5) with
test
895
In the near-field matrix cylinder (15 5 I V;“.5): Bmo = A,h-
A5h-
A$-
A4A+ AS/l+ A,h+
I
I
V,
(6)
‘P = (~0, T, G, G’, G”,a=) Introducing
the following coefficients: A =v~(l--‘)+l--y,+l+v, -
-
0
-6x,
V,Ef
(7)
VfEm
- 01
0
l-y, Em
(9)
-($+$)
As=
f
(10)
mm
(11)
VIII
l&E, A5=af-aCY, A4
V,,,A,
-- VmA, v,Ao
VYo
%A, T/,Ao
B’=
1
(13)
-- 5 2
0
0
0
0
where (3 + vf)t*+ II,b& =
Y,f
In the far-field matrix cylinder components are directly given by: gmi=a a Z fl:r=
-p+
6,,p-6,s
+ V,AIA-
([ 2 F”“),
(19) stress (20)
-fleZ7 am 5 c;=0
(21) (22)
55
p)
(23) r??(l) = WF(l) + ry where rY is the matrix shear yield stress. In a first approximation it will be evaluated from the tensile yield strength of the matrix, gYy,by using a simple von Equations (23) and Mises criterion (r, =g,,,/fi). (3)-(B) lead to a second-order differential equation for the stress function I,&~.+ where the subscript u refers, respectively, to the left debonded part (U = I) and right debonded part (U = r). Solving the system, the solution provided for the stress function in the fibre and matrix in the debonded regions I,&~~.+is:
2(1+ Ym)lnVf+ V,A, v __ m
kleb.,u(5) = JlL.(l
v,Ao
whereforp-
6,~
5”
- &eb.,&))
(24)
p:
16
(14) (1+ 3Yf)[2 + Y, - Yf +
b& =
(17)
(18) b”” = (1 + 3~,)(1 - {‘V,) - 2(1 + vm)lne2& + V,A{h+ 25 16V,
C’
0
0
0
In the debonded zones, stresses must still obey equilibrium and then follow the previous Nairn relationships. The stress transfer is governed by the Coulomb friction law. Assuming a constant coefficient of friction it follows that, at the interface (< = 1):
V,Ao
0
0o
2.2.2 Stress function +dCeb. in the debonded parts (-p 5
VmA3
___VA,
v,Ao
0
1):
Wo
-__VmA,
V,(l f A,A+)
- (“6) + 2(1+ v,)lnt”& 16V,
(12)
-_c_
b2
where
=
we obtain, for the fibre (0 5 5 5
0
,*+(1*&J (8)
_l_Yf -_-5
V,(l 4 A&j
; o---- &-l 0 XV,
15
2
b2
m
b”” = (3 + ~,)(l
A
0
+deb,,r(Q = &
[ue@-l)
- bea(p-i)]
(25)
2(1 + Y,)lnV, + &A, V,
and for -ps
Wo
16
ls
-p+S,:
(bdeb,,{([) = 5
(15)
[ae-b(pt’) - be--“(p+b)]
(26)
T. .I. Copponnex
896
fixed by the stress state previously extracted in debonded regions, is now used to determine *bond..
(27) and
_- 1 + 2
a=
V,A, + 4P2Gs VA f
The particular case that applies to the stresses far from the bonded/debonded boundary (which means &,&) = constant = $!&&) and therefore ignores debonding zones is first envisaged. Setting $!&,,,d.(~)= $&&) = 0, eqn (31) for rbond., constant apart, reduces to:
0 (28)
2/-&s
(29)
Since the coefficients a and b are not always positive, it may be necessary to adjust the coefficient of friction, p, for sticking or frictional effects in order to ensure compatible stresses between the debonded and the bonded parts. This effect should be particularly important with poorly bonded interfaces where extensive debonding can occur.
with 6 = 6, + 6,. It is minimised for
55
p-
p-8, I- bond.
=
P-fir
acvTdV
&uSudV +
I -p+%
i
(30)
aF a ____-___
=
$
a$hond.
Introducing
aF
ag
a2
+%n>nd.
+
2
aF
0
%f&
(37)
=
the following parameters:
3% - c44) q=g?A
P=
C,S
eqn (37) reduces equation:
to
(38)
5s
the
differential
fourth-order
-P+&
where S is the compliance tensor and (Yis the thermal expansion coefficient tensor. From eqn (5) this energy can be recast in dimensionless form as r bond.
(36)
G3
tiond.):
6,)
The solution for the stress function, I&,~~~.,in the bonded part is a function of the fragment aspect ratio and debonded lengths. I,&~. is determined by means of minimisation of the complementary energy Ibond, in the bonded region. This energy is given by:
+ Cseucc + D,T
Close to the bondedldebonded boundary, we use the Euler equation for the fUnCtiOna F($!&nd., q!&,,d.,
2.2.3 Stress function q!thond. in the bonded part ( - p + S,S
C,,a,,
ICIDbO”d. = -
j--:_8~,(P’l’Kl~‘JJl + WW’l)d~ +
(31)
where
(39)
with the four boundary conditions $bmd.(
-
P
J/bond.(P
&,,d.(,
(32)
&md.( (33) In the symmetric matrix, C, integrations reveal that some terms are zero and finally 12 elements remain to be calculated. These coefficients, together with those of the D matrix, are given in Appendix 2. Expanding eqn (31), the complementary energy in the bonded part is then given by the integral of a functional F(‘i&lbond., &mnd., $%nd.):
+
&)
=
&eb.(l
-
6)
=
41/0drb.(1-
-
8,)
P +
=
-
6) = -
The solution extracted bonded zone is:
+bond.(<)
=
$Ohnnd.(l
-
-
4dcb..l(
-
&leb.,r(P
-
$&&'deb..r(~
&&dcb.,/(
P +
-
-
&))
6))
c41)
(42)
'%I
P +
t40)
6)
(43)
for the stress function in the
,<;,
%“(i))
P-6 r bond. =
I
F
-P+&
((Clbond.>&md.,
JlLd.W
(34)
Minimisation Of I&d. by means of the calculus of variations, taking into account boundary conditions
(45)
Single-fibre fragmentation
and a,&)
test
897
Since the continuity of the +” function is not assured at the boundary between bonded and debonded zones, eqn (49) has to be reformulated to separate both regions, previous to determination:
=
cosh[dr< - ( - l)s+r(p
- S,) - k(2p - S))] - kia’-”
r = An integration
rO+
rdeb. + bond.
(W
by parts of $J@”gives:
coshb(L’- ( - l)“+ - WI)}) c ( - l)%((a, k=P,-0
+ k)‘cosh[(a
- k)(2p - S)] - 2cwk) (46)
where
I
(Y= J-t+
The components
r deb.=
p=
+;-
+
(47) 1 I
In eqn (46), 6-e, and S,, are the Kronecker symbols. E,,k is the permutation operator on cy, k subscripts. Introducing eqn (46) in eqn (44) leads to an explicit formulation for the stress function in the bonded part of the sample. $bObond.(l) is expressed as a combination of &M.J( - P + a,>, Lb.,& &), &,.,,( - P + 6,) and #L.&J - &). Although it is not obvious that eqn (44) with eqn (46) reduces to the equation extracted by Nairn in the case of perfect interface, the reader can verify their identity by simplification of eqn (33), removing the failed interfacial parts. 2.3 Fibre/matrix
debonded
(48) Expression for the stored energy
First an expression for the stored energy in the sample has to be found. Expanding and simplifying the total complementary energy given in eqn (30) results in: r=r,+nl:
and
+ G#m
+ t&.
-P+&.
deb khb
+ &
mLl~:~; + c55
+ %,%m&w.)dS.
I
part
(53)
o
r bond. =
x6
[[(thxmd. + 2(cl,~O
+ c5+
+ &~))~&md~p-;s;61+
c55
(54) From eqns (25) and (26), ir@ratiOnS Of t,bdeb , t&,, $&.,, t,b&,. in the debonded parts are quickly achieved and explicitly given below for both left and right parts respectively:
criterion
In the previous section, expressions for the stress function have been extracted in the bonded and debonded parts of the sample. The stress tensor C? = (a;, a;, u;, r’,,) is thus determined in every location of the sample. The concepts of fracture mechanics can now be used to derive a debond criterion, The specific work of fracture of the interface GIrc in mode II, must equate to the differential thermoelastic strain energy stored in the sample with respect to an incremental debonding length LX?on both sides of the fibre fragment:
2.3.1
x ?f
[(@Ideb.+2(C15~0
/ +,
of the total energy are then:
p (c,,~,+2c,,(c/~‘+c,,(c1’2 I -P
- 2&J 4%X*.ti + 2W isgo + C560, + WW)d5 where
[b*(l - e-“‘u) - a’(1 - eCb6u)]j
iebO..cd+2d&)dl
= @~b.('u
(55)
+ (a _ ;)Q,
partu $1
_ e-zbsu)+
$(I
_ e-zas") - ?!?(I
- e-(a+wq
[
II (56) $&. OK=T:::y22 [i (1 -eeZbS4 Idebonded pen +:(I -e-2"du)-&(l -e2(.lh) -
2(a - b)(a2(1 - eubsu) - b2(1 - eFu”u))
(57)
debonded v&lb.( W5 = I part u
;f$y
[ g1
_ e-2ay
(49) + b(l - e-2bsu)} - 2
(1 - e-(a+b)Su)]
(58)
Integrations in the bonded part are much more tedious and to perform them the following coefficients
T. J. Copponnex
898 are introduced: p-6, Ack)
U,I
(59)
a(k)(c)d[ &I
=
I -p+&
and p-6, A(k)..
a(k)(&(k)(Qdl U,I VJ
=
UU,V
I -p+&
with (U = 1, r; u = evaluating Ack) U,I and reveal that:
(60)
1, r; i = 0, 1; j = 0, 1). In uv,ij coefficients, integrations Ack)
A uu,oo= Awoo A uu,,i= A”“,,
(61) i# j
VU # U,
(62)
and A uu,Oj All remaining
-
=
coefficients
to be calculated
A{+) ,7 A{:’,, for the calculus
(63)
Au,Oj
are then:
A:‘$, I A$),
of
By inserting eqns (.55)-(59) and eqns (64)-(67) in eqn (51) we obtain the explicit determination of the total strain energy. Energy changes in the system from initial to final debonding are then attainable and eqn (44) can be solved for interfacial fracture toughness evaluations.
3 EXPERIMENTS Single-fibre fragmentation test samples were prepared from E-glass fibres (Owens Corning) coated for modified unmodified and with application polypropylene (PP) matrices. Different PP matrices were used from the non-modified PP (SY 6100, Shell). Modifications were made, both by direct grafting of ASB molecule (3-azido-sulphonyl benzoic acid) on the PP backbone, and by indirect grafting by using maleic anhydride. The samples were made by placing glass fibres between two 0.25 mm thick sheets of PP which were melted together in a mould. The sample mould
p-6,
and
A&b,
Ah?,,
Ah’%,
Afb?,o,
for the determination
A%,
(4
Afi?1
of 141
Since A~~~ijparameters are more complicated than A$, they are given in an explicit form in Appendix 3. The following expressions for integrals in the bonded part are extracted: p-6,
I -p+6
ICltmndW
=
-
9”hont,.(
gAu,)
(2~
-
+
(6
ICl:eb.
+
c
%)I
Ad:::,,
lL=l,r
(64)
i=O,l P-s,
Ip-6
d4ondW~
A-A
,
:
,
._I
(b)
Fibre
Fig. 2. Sample
j=O,l u=I.r
1ClOdeb. &‘&.,,)
(67)
mould:
mould and detail with sheets and fibre in (a) sample mould geometry; (b) fibre and matrix arrangement.
Single-fibre fragmentation test
was made out of aluminium (Fig. bottom of this mould was placed a sheet of Teflon. Above this a sheet placed. Six E-glass fibres picked out
E 0 -:
g
2(a)). On the 0.25 mm thick of matrix was of a roving at
lo 8
0
0.025
0.500
0.075
0.100
Strain 12
random, were placed across the width of the mould with a mutual distance of 15 mm. The fibres were fixed to the mould with tape. Another sheet of matrix was then placed so that fibres were sandwiched between the two matrix sheets. A second 0.25 mm thick sheet of Teflon was placed on the top (Fig. 2(b)). The PP and Teflon sheets were fixed to the mould with tape. The mould was closed by a flat 5 mm thick aluminium plate. This package was placed in a heated press at 210°C under the same loading applied for 10 min. For each system, a part of the specimen was subjected to tensile strain in a Minimat tensile tester equipped with a microscope to follow the growth of decohesion on chosen fragments, with respect to the applied extension. Other specimens were tested on an Instron tensile tester equipped with acoustic emission apparatus (Module 920 Dunegan) to follow fragmentation kinetics. For all specimens, the minimum gauge length was 55 mm and the normal strain speed was O.OS%/min. To be certain that fibres were completely fragmented, which means that no more fibres would break if the specimen was strained further, an elongation of 9% was applied. 4 RESULTS
0
0.025
0.500
0.075
0.100
0.075
0.100
Strain
12
0
0.025
0.500
Strain Fig. 3. Acoustic emission results. Density of activated defects versus strain for (a) non-modified PP, (b) directly grafted PP, (c) indirectly grafted PP.
899
AND DISCUSSION
Figure 3 gives the fragmentation kinetics followed by acoustic emission for the three different systems. Adhesion improvement is clearly shown with modified matrices by a density of breaks at saturation, double (11.32 breaks/cm in the case of indirectly grafted PP matrix and 9.70 breaks/cm in the case of directly grafted PP) that obtained in the case of pure PP matrix ( = 5.92 breaks/cm). The first fibre breakage is obtained at 3.67% strain for the pure matrix whereas roughly 4% is needed for other matrices indicating a lower initial thermal compressive stress in the case of the unmodified matrix. The same behaviour in the case of the pure and directly grafted PP is obtained with a monotonic rate of fragmentation until the saturation is reached. In the case of the indirectly grafted matrix two different rates are highlighted. At the beginning of the fragmentation, the rate of breakage is high (5.245 breaks/% strain). This rate remains constant up to ~5.5% elongation. At this point a bend is observed, and the process is slowed down to a rate of 0.802 breaks/% strain. This change can be explained by the fact that when fibre fragments are short, fibre breakage can only occur in the middle part of the fragment whereas fracture can occur everywhere along the fragment length if the fibre is long. Finally, we indicate that since constriction of test samples was present at high applied strains, leading to reduced measured nominal loads, the lateral contraction and matrix Poisson effect were introduced into the results to calculate the real values of the applied load and strain. To a first approximation, the test
T. J. Copponnex
900
sample volume variation by:
Table 2. Linear coefficient of dilatation
during the process is given
Temperature
AV = r(1 - 2v,)
(68)
(Twxninal
and the real deformation &=
l-t.5 l+AV
(69)
~
W +
(70)
Fnominal)
Table 1. Acoustic emission results
Directly Indirectly
grafted
Crack density rate (def./% strain) 2.09
grafted
PP PP
10 15 21
the initial fibre length, after the first break, the axial stress in the matrix can be written as: (71)
4.1 Determination of the linear coefficient of dilatation of the matrix (Y, Stress calculations using the theoretical value of the coefficient of dilatation of pure PP (a, = 15 X lo-” “C’ in the temperature range of interest, Table 2) have given abnormally low axial stress values in the fibre far below those required for failure at the strain leading to the first break. Owing to its high viscosity, we can assume that during sample preparation, a perfect contact between the fibre and the matrix is not reached all along the length of the fibre. In addition, matrix properties may vary from the bulk polymer to the interphase region. We think that instead of taking a given value for (Y,, it is more appropriate to derive from experimental data an ‘average’ coefficient of dilatation for each different matrix used. For that purpose, we refer to a matrix plasticity model. Since at the crack tip of the fibre break the stress is limited by the matrix yield stress, after fibre breakage we impose the equality of the stress in the near-field matrix cylinder with the yield matrix stress, uY. If p,, is
Non-modified PP (Shell SY 6100)
K-‘)
is given by:
General acoustic results are given in Table 1. Prior to interfacial energy calculations, we now discuss the determination of two important parameters, namely the coefficient of dilatation of the matrix and the fibre volume fraction V,.
Matrix
q,, (lo-’
20-60 60-100 100-140
Using eqn (68), the real applied load is linked to the measured nominal load by: us =
(“C)
Crack density saturation (def./cm) 5.92
7,026
9.70
5.245 0.802
11.32
at
Just before the fibre failure we had ,n
=
flz
where ((I) derived from Equations fibre volume
ho
- V,b(Po>> = a,
(72)
VITI
is the average fibre strength at length po, experimental fibre strength distributions. (71) and (72) allow us to determine the fraction at the first break:
Vf(adb)= l -
b(d) (u(po))
+
(73)
(Ty _ a,
The coefficient of dilatation of the matrix is then given by solving numerically for IX,: &MXKLpl,(0) = M”))
(74)
Results of (Y, values are given in Table 3. 4.2 Determination of the fibre volume fraction, V, The determination of the zone where the matrix is affected by the presence of the fibre is an important physical parameter to be determined. Calculated Table 3. Properties
Matrix
Property E-glass fibre Young’s modulus (GPa) Coefficient of dilatation (xWhK) Poisson’s coefficient Shear modulus @Pa) fly (MPa) r, (MPa) Fibre diameter (pm)
of materials
72
5
0.35 29.5
Virgin PP
Directly grafted PP
Indirectly grafted PP
1.2
1.2
1.2
100.39
0.32
0,32
0.32
0.444
0.444
0.444
35 20,2 17.17
95.5
82.45
28 16.6
28 16.6
Single-jibre fragmentation test
stresses are highly dependent on the volume fraction of the fibre and it seems necessary to determine this precisely prior to interfacial energy determinations. Optical observations revealed that with systems currently used in this work, the matrix radius should be at least 10 times the fibre radius, giving us a first insight on the extent of this particular affected region. However, the fibre volume fraction is expected on one hand to decrease with applied strain and on the other hand to be a function of the fibre fragment length. A technique has then to be found to follow this development. Referring once again to the matrix plasticity model, we suggest the following way to determine Vf. Let us consider the fragmentation process at a given applied load, caa, with the corresponding mean fibre fragment length, p. On average at this stage, the effective stress state was sufficient to break all fibre fragments longer than 2p. With the mean fibre fracture strength at this length, the matrix plasticity model gives: i+,,2p
b@PN
j = 1-
b(2P))
(75)
+ ay - aa
From this calculated value of V, for a fragment of length 2p, &,ond.,p(O)the axial stress at 5 = 0 for the fibre fragment of length p is calculated and used in eqn (75) to find the final V, value of the fibre volume fraction:
v, = Vf(U,,P)
= 1-
&Xld.@(O) lCTbO”d.,p(0) + a, - @a
(76)
This process has been iterated with increasing applied loads to cover the whole fragmentation process from the first break up to the saturation. Results are given in Fig. 4 for the three systems.
om6 F 0
0
3
2
0.004
Directly grafted PP
-
&
~0.003 3 8
~0.002 -
2
Pure PP
I Indirectly grafted P
r
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
strain
Fig. 4. Evolution
of the fibre fragmentation
volume fraction process.
during
the
901
4.3 Interfacial energies In order to assess the interfacial micro-mechanisms taking place in the fragmentation test, it is common1’~12 to combine these mechanisms in a stochastic computer simulation of the process, to explain experimental results. Although these stressbased methods, which are often based on the shear-lag theory,13,14 give reasonably good agreement with experiment, inadequacies and deficiencies of these models have been already rep0rted.l’ Experimentally, acoustic emission curves (Fig. 3) give us the average fragment length at each stage of the fragmentation process. Optical observations allowed us to determine the average debonding length associated with this mean fragment length. From the expression for the energy stored in the sample, it is now possible to determine the total energy release rate associated with given decohesions, for different applied elongations. This is done by computing total energy release rates for a succession of small debonding lengths running incrementally from zero up to a chosen value. For each system we get predictions which can be used, in relation to the experimental average debonding lengths observed at each strain, to obtain the interfacial toughness of the system. We think that this kind of approach compared to the alternative computer-based simulations discussed above has the advantage of being simpler and faster. Figure 5 gives results for the different matrix-based systems. Debonding length is given in terms of an average dimensionless value m defined as: 6, i- s,
6
(77) m=2p=G It must be pointed out that the mathematical approach developed in the previous sections takes into account asymmetric decohesion at both ends of the fragment (6, # 6,). Stress states in the fragment are then determined for any damage length at each fibre fragment end. It is theoretically possible to calculate the energy release rate associated with any further increasing debonding from these particular situations. For simplicity however, symmetric situations (6, = 6,) were envisaged for calculations in the following examples. The incremental debonding length used in computations was fixed at Am = 0.005. To take into account yielding effects of the matrix, calculations were performed using the secant modulus. Each curve represents the total energy release rate versus m at a given strain. The different curves are constructed for increased strains with steps of 0.25%. Concerning the unmodified PP matrix (Fig. 5(a)) curves are given for strains from 3.75% up to 6.5%. For modified matrices, strains are going from 4% up to 6.25% for the directly grafted matrix (Fig. 5(b)) and 8.5% for the indirectly grafted matrix (Fig. 5(c)). The abacus of the unmodified PP matrix exhibits
T. J. Copponnex
902
become very close to each other for increased strains. The growth in debonding gets progressively lower as the strains increase. This behaviour is particularly obvious with indirectly grafted PP (Fig. 5(c)) vvherk
r Unmodified PP
0
0.25
0.5
0.75
1
Average Dimensionless Debonding Length m
Directly grafted PP
0
0.25
0.5
0.75
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.03
0.04
0.05
0.06 0.07 Strain
0.08
0.09
0.1
1
Average Dimensionless Debonding Length m (c)
600 _ _ Indirectly grafted PP
E
’
so.9 5 a 0.8
0
0.25
0.5
I
0.75
Average Dimensionless Debonding Length m Fig. 5. Interfacial
energies versus dimensionless length for increasing iterated strains.
debonded
Indirectly grafted PP
50.7 j +j 0.6 a 3 0.5 s .z 0.4 E aJ 0.3
E a 0.2
%I 2 0.1 2 more or less equally spaced curves indicating in that case, that the debonding process is directly related to the applied strain. For systems based on modified PP matrices, curves largely spaced for small strains
0 0.03
0.04
0.05
0.06 0.07 0.08 0.09 0.1 Strain Fig. 6. Average dimensionless debonding length for increasing strains (0, experimental debonding lengths; -, observed debonding lengths).
Single-fibre fragmentation test
debonding appears to be stabilised for high applied strains. This is a good illustration of the energy competition that exists between bonded and debonded regions for the debonding process.
4.5
. Pure PP 30 -
1
1
. Pure PP
2.5
2.0
2 -30 -
3 s ;;
903
454 -100
1.5
?? z
-75
-50
-25
0
25
50
75
100
50
75
100
Fibre length (l/d)
. Directly grafted
-100
-50
-75
-25
0
2.5
50
75
100
Fibre length (l/d) L
- Directly grafted PP
2.5 -
-45 -100
-75
-50
-25
0
25
1
Fibre length (l/d) 45
ot....1../11....1....I. -100
-75
-50
-25
0
.,\...I..,,
,,‘,,
25
50
75
I 100
Fibre length (l/d)
2.5
Indirectly
grafted PP
-100
-75
-50
-25
0
25
50
75
Fibre length (l/d) Fig. 8. Shear stress profile in a mean fibre fragment at saturation for the different systems.
-100
-75
-50
-25
0
25
50
75
100
Fibre length (l/d) Fig. 7.
Tensile stress profile in a mean fibre fragment at saturation
for the different
systems.
length
100
length
The main output of energy curves is to be used for modelling the average debonding length as a function of the applied strain to quantify interfacial of the systems. Figure 6 illustrates such
properties
results. We
T. J. Copponnex
904 0
Pure PP
-100
-75
-50
_ Pure PP
-25
0
25
50
75
-100
100
-75
-50
Fibre length (l/d)
1 m -15
-
2 z
-20
-
-ii s
-25
-
E -30 ‘;;r ‘G -35 d B E
2 -10? 2 v -15
B I ‘cj
-
-
-25
-
-30
-
-35
-
-40
-
-45
-
-45
-75
-50
’ -25
0
25
25
50
75
100
50
75
-
-20
t a, -40 z
-100
0
Directly grafted PP
-5
3 2 z g
_
-50 m,
-25
Fibre length (I/d)
-50 -100
100
-75
-50
( -25
0
25
Fibre length (l/d)
Fibre length (l/d) 0 -5 ‘;;i & -10 E
2 $
-25
L -30 -J 5‘G -35 2 -40 e
g
-25
i ._ 2
-30
s E
-40
-35
-45 -50 -100
,, ,,,,, -75
-50
l-25
0
251
50
75
100
Fibre length (l/d) Fig. 9. Radial stress profile in a mean fibre fragment at saturation for the different systems.
Fibre length (l/d) length
obtain a very good agreement between predictions and experimental data, although debonding seems to be underestimated at the end of the fragmentation process in the case of the indirectly grafted system.
Fig. 10. Hoop stress profile in a mean fibre fragment at saturation for the different systems.
length
4.4 The stress state at saturation We have seen that it is possible, from optical observations of debonding versus strain in a single-fibre fragmentation test to evaluate an interfa-
Single-fibre fragmentation
cial energy-based criterion for debonding. We emphasize now that the model presented is selfconsistent with regard to the stress state in a fibre fragment. By contrast simulations of the fragmentation process generally used are governed by a maximum stress criterion (i.e. the maximum stress exceeds the interface strength or the shear strength of the polymer matrix) and so require a stated value for this stress prior to computations. Here we do not impose such a restriction. Plots in Figs 7-10 give the different components of the stress profile along the fragment length for the average fibre fragment length at saturation for the three systems. It can be seen that the maximum shear stress (Fig. 8) in the case of the pure unmodified matrix system (zm,, = 19MPa) does not exceed the maximum yielding shear strength of the matrix, whereas the opposite is observed for modified matrices (r,,, = 38.6 MPa for the directly grafted PP and r,,, = 40.8 MPa for the indirectly grafted PP). Since matrix failure is never observed in all the tested samples, it should indicate that matrix yielding followed by consolidation occurs in modified matrices. This has been already reported for carbon/epoxy systems.‘” This result can be related to optical interface observations using a microscope where plasticization seems effectively to be present in front of the debonding lengths whereas it is not the case for the pure PP (see Figs 11 and 12). If we now look at the radial and hoop stress profiles (Figs 9 and lo), we can see that they show a similar shape. For all studied systems, the radial stress presents a similar compressive stress concentration at the fibre ends owing to fibre. At the Poisson contraction of the debonding/bonding boundary, we also observe such a compressive stress concentration, but here the value of the compressive stress is three times higher in the case of modified matrices than that which exists with
Fig. 11. Bondedldebonded
boundary
in the case of pure PP.
Fig. 12.
test
905
Two fibre ruptures
and matrix plasticity of the indirectly modified PP.
in the case
the unmodified PP. This strong compressive stress is expected to play an important role in the mechanism of debonding. It is highly probable that in the case of modified PP systems interfacial failure is strongly inhibited as a consequence of this effect. 5 CONCLUSION The bi-dimensional stress analysis of a single embedded fibre previously developed by Nairn for the case of a perfect interface (no interfacial debonding) has been improved by incorporating a damaged interface at the fibre ends. The overall stress tensor has been extracted and used to derive an interfacial energy debond criterion. Experimental single fibre fragmentation tests performed on different E-glass/PP matrix systems and followed by acoustic emission were used with the mathematical analysis developed to generate curves of interfacial energy versus applied strain. It is suggested that the use of these energy curves with the observed debonding lengths in single-fibre fragmentation tests enables one to quantify adhesion performance in composites. The experimental values of GnC obtained were of the order of 45 Jmp2 for pure PP whereas a value up to 183 Jme2 is predicted for modified PP (Table 4). It has also been shown that this energy-based approach is self-consistent with regard to the stress state in the fibre fragment. Plots of stress profiles for modified matrices indicate that yielding followed by consolidation may occur, whereas for unmodified matrix it may not. To summarise, we think that the present bidimensional energy-based approach is a very interesting and powerful alternative analysis to the existing computer-based simulations using the interfacial shear strength concept. Since extensive calculations are
T. J. Copponnex
906
Table 4. Fragmentation
Matrix
Non-modified PP (Shell SY 6100) Directly grafted PP Indirectly grafted PP
Mean fragment length at saturation (pm)
results
Interfacial energy (J mm’)
m (theo.) at saturation
m (exp.) at saturation
1690
0.72
0.72
45.3
1200 876
0.42 0.57
0.41 0.38
147.8 183
required to produce energy curves, a procedure is A copy of written in MathematicaTM for PC computers. the procedure is available on request. Computers equipped with a mathematical co-processor with 12 MB RAM are sufficiently powerful to give results within a reasonable time.
11.
12.
ACKNOWLEDGEMENT 13.
This work was supported by the Centre des MatCriaux Pierre-Marie Fourt of the Ecole SupBrieure des Mines de Paris and Shell Research (Louvain-la-Neuve, Belgium).
14.
15.
REFERENCES 16. 1. Kelly,
A. & Tyson, W. R., Tensile properties of fibre-reinforced metals: Copper/tungslen and copper/molybdenum. J. Mech. Phys. Solids, 13 (1965)
stresses around breaks in embedded fibers. Mech. Mater., 13 (1992) 131-157. Jacques, D., Transfert de charge entre fibre et matrice dans les composites carbone-rCsine. Comportement en traction d’un composite modkle monofilametaire. PhD thesis, Institut Polytechnique de Lorraine, Nancy, France, 1989. Ling, S. & Wagner, H. D., Relationship between fiber flaw spectra and the fragmentation process: A computer simulation investigation. Comp. Sci. Technol., 48 (1993) 35-46. Cox, H. L., The elasticity and strength of paper and other fibrous materials. Bit. J. Appl. Phys., 3 (1952) 72. Lacroix, R., Kennings, R., Desaeger, M. & Verpoest, I., A new data reduction scheme for the fragmentation testing of polymer composites. J. Muter. Sci., 30 (1995) 683-692. Feillard, P., DCsarmot, G. & Favre, J. P., Theoretical aspects of the fragmentation test. Comp. Sci. Technol., 50 (1994) 265-279. Desaeger, M., An integrated theoretical and experimental approach of the fibre-matrix interface in carbonepoxy composites. PhD thesis, KUL Leuven, Belgium, 1993.
329-350.
A. T., Measurement of the thermo2. DiBenedetto, mechanical stability of interphases by the embedded single fiber test. Conzp. Sci. Technot., 42 (1992) 3.
4.
5.
6.
7. 8.
9. 10.
103-123. Di Anselmo, A., Accorsi, M. L. & DiBenedetto, A. T., The effect of an interphase and energy distribution in the embedded single fibre test. Comp. Sci. Technol., 44 (1992) 215-225. Wagner, H. D. & Ling, S., An energy-based interpretation of interfacial adhesion from single fibre composite fragmentation testing. Adv. Comp. Lett., 2 (1993). Jiang, K. R. & Penn, L. S., Improved analysis and experimental evaluation of the single filament pull-out test. Comp. Sci. Technol., 45 (1992) 89-103. Zhou, L. M., Kim, J. K. & Mai, Y. W., Interfacial debonding and fibre pull-out stresses: II. A new model based on the fracture mechanics approach. J. Muter. Sci., 27 (1992) 3155-3166. MacLaughlin, T. F. & Barker, R. M., Effect of modulus ratio on stress near a discontinuous fiber. Experim. Mech., April (1972) 178-183. Carrara, A. S. & MacGarry, F. J., Matrix and interface stresses in a discontinuous fibre composite model. J. Comp. Mater., 2 (1968) 222-243. Ko, W. L., Finite element microscopic stress analysis of cracked composite systems. J. Camp. Mater., 12 (1978) 97-115. Nairn, J. A., A variational mechanics analysis of the
APPENDIX
1
Calculation of the total axial stress v0 a(, is calculated by equating total displacement of the near-field and far-field matrix cylinders. Integration of strains gives:
P
I
P
ET,~ dt
=
G” d5 i
-P
(78)
-6J
i.e. P I -P
Axial
strains
E,E$
in the matrix
d.$ = era
are given in cylinder
(79) i (i =
mf, m,) by:
E(‘) =
z
1
E m(
ap, _ Y&y
+ ay>>
(80)
Single-jibre fragmentation
Integration of for the stress parts, as well expression for ua +-
907
eqn (79) using eqn (80) and expressions function in the bonded and debonded as its derivatives yields the following crO:
2vmA, T+2v,l/t(l+~)U,+(;+~) A0
(92)
x i=,c. 4 li + ~,(G‘ I.. uo
test
+ b$
Z
4”
i=2,...,4
=
(93) 1
2vmA4
Vm
; (
Ao
+
?)I1
m
v,(b;;‘+
-
b$)f:
o
(81)
[v$(1+ -
3V,(l+
Y,)(S + 3Y,)
~,)(3 + Y,) + 6(5 + 3v,)] (95)
(96)
-
(1 - dd.,(S))‘x’di]
I d&parts APPENDIX
(85)
2 D1 = A$
Expressions
(97)
F+&(F)]]
for [C] and [Dj matrices C1,=L
~-- 1
A:V,
v, ( 2-&Vm
1 Ci3Z -~2&V,
(a,
f
Ao ) VmA3A4 %A,
D2=
(86)
- af)
(98)
0
-Cz2
(99)
,=(g-;)(a,-af)
(100)
(87) 4=~[(01,-ai)(~-(l+Y,)
C15 = 16~;,v,
[(1+ Y,)Z(l +T)
- (1 - y_)(l
- 7)
+ 2AoE,
C16= -‘-(Az+$)
- vmA%]
m
x (I
(88)
(89)
D6 =
V,A, A0
(
anI-
(Yf)
+yq) - y +
%lKl-
afvf
(101)
(102)
908
T. J. Copponnex
APPENDIX
3
Air.01= Determination of ALz,{i coefjicients
With &,ond.= 2p - 6, stating C[X] = cash [x&,~,,~.]and S[x] = sinh [xS bnd,], integrations give the following expressions for the A:$, coefficients: A II,00= kz;_P
[ %[
*
(c+,’
(y - k C [ Ea,k [ 2ak(cY + k)
(a” + 2a3k - 2a2k2
k=P,-P
+
2ak3 + k4)C[2a + k] m 12Cu2k2C[al
+s bond.
a3S[a]
kS[cy + Zk])]]
-q
(
(108)
+ 3kZ)
A!& = k=;_,J[ •~,&mt.(~=kz)(~
C[2a]
- (a” - kZ)aYY[2a] + 2ak(cY - k)‘C[a + k]) +
2a4 + 7a2k2 - k4 4a
+ (CY- k)*C[a + k] - (a” + k2)) + ak
S[2cx] + (a - k)4;k;ff;;
x S[2(a + k)] - 2u’y;;
3k) S[cz + k]]]
x
(103)
3a+k (a-k)4 S[2(a + k)] + 2ru2kxSS[a ( 8(cz - k)
+ k]
- 7a24+ k2 kS[Za])]]
&,,I = i&
[ ~~,p[&m~.(~a. - k)*Cb
a’-k2 - 2
+
kl A!;,& = kz;_,
C[2a] - a2 - k* + 5u24+u3k’ S[2a] >
- 2a(a2 + uk + 2k2) k(a + k) A Il.01= &zP x
(a + k)4 (
4
sb0n;a2k2 ((5k* - 3a2)C[a]
[ Q[
_ (a - k)’ C[a + 2
(Q - k)4 S[Z(cz + k)] + 8ak(a + k)
x (a” f Sla + k]]]
(
004)
k2)(11a2 f k*) S,(y] a-k
A(& =
C[2(a + k)] - 2a2(a2 + 3k2)C[a + k])
(c? - k2)[ (a - k)S[a + k] - d[2a]]]]
,&j
bk
[
2a’k2(2C[cu + k] Stxmt.ffk(Q2- k2)
2
uw
i- (a A lr.00= *=;_,
[ Q[F
((a + k)‘C[2a
- (a’ + k*)C[k])
2cr(cr + k) 1 g2”’
[ ~,k[++
((5a -
- (Q+ k)‘C[2Cy
-k])
+ k)
2a(a2 - k’)
x
i
“@;
w]]
(111)
+ 2a2k(3a a+k
007)
+ 2kl
+ &,o,,d.ak
k)*S[a + 2k] - k’s[a])]]
~Y2+
+ cuk(a - k)‘C[(-Y + k] +
S[LU+ 2k]
_ (5a2 + k2)(a2 + 3k’)
k)S[a + k])]]
((a - k)C[a
A);,‘, = *=F_, [ %3[ h”d.(
3k2)CPl
- (c~ -k)
x 2a3 - a2k + 2ak2 + k” 2ak(n
(106)
[ e,.p[z
+ 3(au’i,k’)C,a])
S[CX+ 2k]
+ khS[cl]]]
(kWa1
kl Ajf?, = k=&,
x 2a4 + 2a3k - 3a2k2 + 2ak’ + k4
A Ir.11= xz;_P
-
+ (CI -k)
- 2a6 - 2;;;:;
(110)
;) + &
- C[2cy] - 1) + +*y
=I) - 2($k)
+ (a + k)(a2 + 4uk - k’)S[cy + 2k])]] [ %P[ (a’ + kY(C[2011-
(109)
+ k)
3k2) k2(a2 - k’) 2
S[a + k] -
+ iTaLk;;
(112)
W;
S[2(a +
w4)
7k2) sr2cyl
W]]
(113)
Single-jibre fragmentation
test
Ai;& = k&
Aj;‘,, =
- (a” + k2)C[a]) + x - k)(a’-
_@
a
(a’ + k2)(a2 + 11k2)
(
4ffk - kz)S[a
+ 2k])]]
[ cqD[ b2k2( &md.($
_ k2(cy2’- k*)
(Y2 - k2
(114)
+ (3a2:
5k*) 4
aS[2cx] + iTaLk;;
Aj& = kzF_,
(a + k)
[ c+[
ak( ak(a2 +
+ (a ; k)4c[2(a -6 md~2
(a + k)
+ k)C[a
+ k])
A$;‘,,= kz_p [ E~,~[ +
S[a +
kl)]]
+
(115)
-
kZ)(CPd - ;)
- k’)( k’S[2a]
[ %P[&j
(2a2@
x ((Y’ + 2ak - k2)C[2a
C[cv,)
(CI - k)‘cz3S[a + 2k])]]
ffsta1
2(a. + k) x (Y(LY- k)S[cx + 2k]]]
(119)
(Ly2‘4 k2)3 + @’ ; k2)
(117)
+ 2cx2k( a-k)2(a+;)C[a+k])
+ ~ (a - k)4(a + 3k/2) sL2((y + k)] ( 4(m + k) -
-s bond, a2k2(a4 - a2k2 - k4)S[cx] +f
3Cr2k2 + 4k4)C[(u]
(116)
- k,
+ k] - s
(118)
2(a2 - k2 ) a4 + &y3k- 3a2k2 + 2ak3 + 2k4
x k4C[2a] = Jp
W)]]
+ 2k])
A!& = k=;_p [ E%J - Ld.( +k]))]]
Sta +
cx6 - 2a4k2 - 25a2k4 + 2k6
+ k)] - a2((r2 + 3k2)C[cx + k]
+ cx(cz - k)(cx +;)S[a
A&
((a4 -
+ (a! - k)2a2C[a
S[2(a + k)]
(a” + cyk + 2k2)
- 2a2
+ 2a2k - ak2 + 2k3)
(3a2 + k2)
C[2cr] - (Y((Y - k)‘(2a
( &mnd(~2 + k’)
+ (3a* + k2)(Cu2 + 5k2) cuS[a] + (a - k)(d
A{,?,&,= k&
[ G;D[ F
x (3a2 - 4k2)C[cr] f (c~ - k)*cx’C[cx + 2k])
S[Ql
a-k
909
cz4 - 7a2k2 - 2k4 4
SI2al
- 2a2kUL’,”k’ 3k)S[a
+ k])]]
(120)
PII:
ANALYSIS
AND
SO266-3538(96)00049-g
0266.3538/96/$15.00
EVALUATION OF THE SINGLE-FIBRE FRAGMENTATION TEST
T. J. Copponnex* Ecole Nationale Suptrieure
des Mines de Paris, Centre des Mathiaux
Pierre-Marie Fourt, Eury, France
(Received 29 June 1995; accepted 22 February 1996) Abstract
in the fibre coating as well as in the matrix are then evaluated. Interpretation of results is still controversial, however, partly because of the fact that common models used to relate fragment lengths to interfacial properties are based on simplistic assumptions and are mostly stress-based methods. The very well-known elastic-plastic Kelly and Tyson approach for instance, seeks to describe the stresses in the post-failure state and as such does not highlight the interfacial failure mechanisms taking place. Alternative energy-based methods, assuming that the energy required to debond the fibre is the failure-controlling mechanism, have been reported recently2-4 but these approaches have been developed mainly for other geometrical configuration tests such as pull-out.5,6 Related to the embedded single-fibre test, finite-element analyses7-9 to evaluate stress, strain and stored energy have been reported but it seems that little effort has been expended to improve existing analytical methods. A weak point with currently used energy-based approaches is that calculations are performed from one-dimensional stress analyses. In other words, only axial and shear stresses in the constituents are used to derive a debond criterion. Recently, Nairn” using a variational approach, and assuming perfect bonding between the fibre and the matrix, extracted an analytical expression for the bi-dimensional stress tensor in a fragmentation sample. He pointed out that radial and hoop stresses should also play an important role in interfacial properties. In the light of these recent improvements, it seems that a correct debonding criterion has to be derived from the knowledge of the complete stress tensor in the constituents. Unfortunately, Nairn’s expressions given for the case of perfect bonding cannot be used for this purpose. In the present study, the strain energy release rate principle is used to study the mechanism of interfacial failure in fragmentation tests on thermoplastic-based microcomposites. From Nairn’s approach an analytical expression for the complete bi-dimensional stress tensor for isotropic materials, including imperfect adhesion between constituents, is used to calculate the
The single-fibre fragmentation test has received attention as a preliminary test for studying interfacial properties in composite materials. Its merit as a powerful tool for measuring interfacial bonding at the jibre/matrix interface is not in doubt. ‘Simplistic’ assumptions generally used for the analytical treatment of experimental results seem nevertheless to hide the real mechanisms involved during the fragmentation process and especially with regard to the debonding following the jibre breakage. In this paper, an improved analysis is used of Nairn’s previous variational mechanics approach to the single-fibre fragmentation test, taking into account imperfect adhesion. The analytical expression for the stress tensor in the sample is used to derive an interfacial debonding criterion by means of an energy release rate analysis. It is shown that the extent of interfacial damage can be determined by this energy-based approach. Comparison of predictions with experiments shows that a critical interfacial fracture toughness can be used to predict the growth of debonding. 0 1996 Published by Elsevier Science Limited Keywords:
composite materials, single-fibre fragmentation test, variational mechanics, energy release rate, interfacial fracture toughness 1 INTRODUCTION The single-filament multi-fragmentation test, originally proposed by Kelly and Tyson,’ has received attention as a preliminary test for studying interfacial properties in composite materials. Under tension and above the maximum fibre elongation, a fragmentation process occurs. From the fibre fragment length distribution, properties of the interface are quantified by deriving an expression for the maximum interfacial shear stress acting at the fibre/matrix interface. Performance of chemical modifications incorporated *Present address: Shell Research, 1 avenue Jean Monnet, B1348 Louvain-la-Neuve, Belgium. 893
T. J. Copponnex
894
total energy release rate associated with the growth of interfacial debonding. Experimental results are given for E-glass fibres embedded in different polypropylene matrices. It is shown that a critical interfacial toughness can be used to predict the propagation of the debonding interface in these thermoplastic-based microcomposites. 2 THEORETICAL
ANALYSIS
2.1 Sample geometry
Consider geometry Cylinder fraction
the typical
single-fibre
fragmentation
test
in Fig. 1. Three cylinders are defined. 1 refers to the fibre of radius rr (fibre volume V,) and length L. Cylinders 2 and 3 are
respectively associated with the near-field matrix (matrix volume fraction V, = 1 - V,), where a fibre break disturbs stresses in the matrix, and far-field matrix, where these stresses are those that exist when the fibre remains unbroken.” The overall sample between two fibre breaks is submitted to tension cr,. This loading pattern introduces shear along the fibre/matrix interface and causes debonding. Two different regions are considered. In the central part of the fragment, the fibre is
-2
-P
-P+&,
0
P-6r
bonded to the surrounding matrix. At both fragment ends, failed interface is considered with asymmetric aspect. A set of dimensionless cylindrical coordinates (6, 8, 5) is used so that 5 = r/r* and i = z/rf correspond to the fibre axis. b runs from negative ( - p) on the left to positive (p) on the right, with p = L/2r, the fibre aspect ratio. Debonded lengths are limited to the -p + 6, coordinate on the left and p 6, on the right part. When the failed interface grows, -p + 6, and p - 6, move towards i= 0 at the expense of the central region. As pointed out by Nairn, the fibre breaks do not disturb the axisymmetric nature of the stresses and thus lead to a three-dimensional, axisymmetric stress analysis of the single-fibre fragment of length L. perfectly
2.2 Solution for the stress components 2.2.1 Nairn’s relationships
Following Nairn’s approach and under axisymmetric conditions, the shear stresses rVe and rze are zero. Stress components are then given by:
a
P
2’r0 = 0
(1)
r), = 0
(2)
ai = (&&a;,&)
(3)
where the superscript i = f in the fibre cylinder, m, in the near-field matrix cylinder and m, in the far-field matrix cylinder. We begin with Nairn’s relationships under the assumption that the axial stress, ok, in each cylinder is independent of the radial coordinate. Nairn stated the stress components in each cylinder in terms of a stress function, $t, its derivatives I,/#, I+?“,the total applied axial stress on the fibre cylinder and near-field stress matrix cylinder, a, (Fig. l), the temperature difference between the sample temperature and stress-free temperature, T, and the radial stress at the fibre/matrix interface, a, (5 = 1). The analytical expression for u, extracted by Nairn and reformulated for the case of isotropic materials is: + T(1 + vf)(q - cq,,)
(v, - v@ Cr,=
2v&-1 G
Fig. 1. A schematic illustration of Nairn’s three-cylinder model (with dimensionless coordinates) for a single fibre fragmentation test sample incorporating partial debonding. The inner cylinder (1) is the fibre. The surrounding matrix cylinder (2) is the near-field matrix cylinder. The outer cylinder (3) refers to the far-field matrix cylinder.
1+v,
(4)
E,
where v,,, and vr, are, respectively, the matrix and fibre Poisson ratios, cy,, arr the coefficients of dilatation, and E, and Ef, the Young’s moduli. a0 is obtained by equating total displacement between the near and far-field matrix cylinders. Since
Single-fibre fragmentation
explicit expressions for stress components are required prior to co determination, calculation and expression for the axial stress are reported in Appendix 1. The expression of the stress tensor in each cylinder can be given as: (Ti = B’tJl (5) with
test
895
In the near-field matrix cylinder (15 5 I V;“.5): Bmo = A,h-
A5h-
A$-
A4A+ AS/l+ A,h+
I
I
V,
(6)
‘P = (~0, T, G, G’, G”,a=) Introducing
the following coefficients: A =v~(l--‘)+l--y,+l+v, -
-
0
-6x,
V,Ef
(7)
VfEm
- 01
0
l-y, Em
(9)
-($+$)
As=
f
(10)
mm
(11)
VIII
l&E, A5=af-aCY, A4
V,,,A,
-- VmA, v,Ao
VYo
%A, T/,Ao
B’=
1
(13)
-- 5 2
0
0
0
0
where (3 + vf)t*+ II,b& =
Y,f
In the far-field matrix cylinder components are directly given by: gmi=a a Z fl:r=
-p+
6,,p-6,s
+ V,AIA-
([ 2 F”“),
(19) stress (20)
-fleZ7 am 5 c;=0
(21) (22)
55
p)
(23) r??(l) = WF(l) + ry where rY is the matrix shear yield stress. In a first approximation it will be evaluated from the tensile yield strength of the matrix, gYy,by using a simple von Equations (23) and Mises criterion (r, =g,,,/fi). (3)-(B) lead to a second-order differential equation for the stress function I,&~.+ where the subscript u refers, respectively, to the left debonded part (U = I) and right debonded part (U = r). Solving the system, the solution provided for the stress function in the fibre and matrix in the debonded regions I,&~~.+is:
2(1+ Ym)lnVf+ V,A, v __ m
kleb.,u(5) = JlL.(l
v,Ao
whereforp-
6,~
5”
- &eb.,&))
(24)
p:
16
(14) (1+ 3Yf)[2 + Y, - Yf +
b& =
(17)
(18) b”” = (1 + 3~,)(1 - {‘V,) - 2(1 + vm)lne2& + V,A{h+ 25 16V,
C’
0
0
0
In the debonded zones, stresses must still obey equilibrium and then follow the previous Nairn relationships. The stress transfer is governed by the Coulomb friction law. Assuming a constant coefficient of friction it follows that, at the interface (< = 1):
V,Ao
0
0o
2.2.2 Stress function +dCeb. in the debonded parts (-p 5
VmA3
___VA,
v,Ao
0
1):
Wo
-__VmA,
V,(l f A,A+)
- (“6) + 2(1+ v,)lnt”& 16V,
(12)
-_c_
b2
where
=
we obtain, for the fibre (0 5 5 5
0
,*+(1*&J (8)
_l_Yf -_-5
V,(l 4 A&j
; o---- &-l 0 XV,
15
2
b2
m
b”” = (3 + ~,)(l
A
0
+deb,,r(Q = &
[ue@-l)
- bea(p-i)]
(25)
2(1 + Y,)lnV, + &A, V,
and for -ps
Wo
16
ls
-p+S,:
(bdeb,,{([) = 5
(15)
[ae-b(pt’) - be--“(p+b)]
(26)
T. .I. Copponnex
896
fixed by the stress state previously extracted in debonded regions, is now used to determine *bond..
(27) and
_- 1 + 2
a=
V,A, + 4P2Gs VA f
The particular case that applies to the stresses far from the bonded/debonded boundary (which means &,&) = constant = $!&&) and therefore ignores debonding zones is first envisaged. Setting $!&,,,d.(~)= $&&) = 0, eqn (31) for rbond., constant apart, reduces to:
0 (28)
2/-&s
(29)
Since the coefficients a and b are not always positive, it may be necessary to adjust the coefficient of friction, p, for sticking or frictional effects in order to ensure compatible stresses between the debonded and the bonded parts. This effect should be particularly important with poorly bonded interfaces where extensive debonding can occur.
with 6 = 6, + 6,. It is minimised for
55
p-
p-8, I- bond.
=
P-fir
acvTdV
&uSudV +
I -p+%
i
(30)
aF a ____-___
=
$
a$hond.
Introducing
aF
ag
a2
+%n>nd.
+
2
aF
0
%f&
(37)
=
the following parameters:
3% - c44) q=g?A
P=
C,S
eqn (37) reduces equation:
to
(38)
5s
the
differential
fourth-order
-P+&
where S is the compliance tensor and (Yis the thermal expansion coefficient tensor. From eqn (5) this energy can be recast in dimensionless form as r bond.
(36)
G3
tiond.):
6,)
The solution for the stress function, I&,~~~.,in the bonded part is a function of the fragment aspect ratio and debonded lengths. I,&~. is determined by means of minimisation of the complementary energy Ibond, in the bonded region. This energy is given by:
+ Cseucc + D,T
Close to the bondedldebonded boundary, we use the Euler equation for the fUnCtiOna F($!&nd., q!&,,d.,
2.2.3 Stress function q!thond. in the bonded part ( - p + S,S
C,,a,,
ICIDbO”d. = -
j--:_8~,(P’l’Kl~‘JJl + WW’l)d~ +
(31)
where
(39)
with the four boundary conditions $bmd.(
-
P
J/bond.(P
&,,d.(,
(32)
&md.( (33) In the symmetric matrix, C, integrations reveal that some terms are zero and finally 12 elements remain to be calculated. These coefficients, together with those of the D matrix, are given in Appendix 2. Expanding eqn (31), the complementary energy in the bonded part is then given by the integral of a functional F(‘i&lbond., &mnd., $%nd.):
+
&)
=
&eb.(l
-
6)
=
41/0drb.(1-
-
8,)
P +
=
-
6) = -
The solution extracted bonded zone is:
+bond.(<)
=
$Ohnnd.(l
-
-
4dcb..l(
-
&leb.,r(P
-
$&&'deb..r(~
&&dcb.,/(
P +
-
-
&))
6))
c41)
(42)
'%I
P +
t40)
6)
(43)
for the stress function in the
,<;,
%“(i))
P-6 r bond. =
I
F
-P+&
((Clbond.>&md.,
JlLd.W
(34)
Minimisation Of I&d. by means of the calculus of variations, taking into account boundary conditions
(45)
Single-fibre fragmentation
and a,&)
test
897
Since the continuity of the +” function is not assured at the boundary between bonded and debonded zones, eqn (49) has to be reformulated to separate both regions, previous to determination:
=
cosh[dr< - ( - l)s+r(p
- S,) - k(2p - S))] - kia’-”
r = An integration
rO+
rdeb. + bond.
(W
by parts of $J@”gives:
coshb(L’- ( - l)“+ - WI)}) c ( - l)%((a, k=P,-0
+ k)‘cosh[(a
- k)(2p - S)] - 2cwk) (46)
where
I
(Y= J-t+
The components
r deb.=
p=
+;-
+
(47) 1 I
In eqn (46), 6-e, and S,, are the Kronecker symbols. E,,k is the permutation operator on cy, k subscripts. Introducing eqn (46) in eqn (44) leads to an explicit formulation for the stress function in the bonded part of the sample. $bObond.(l) is expressed as a combination of &M.J( - P + a,>, Lb.,& &), &,.,,( - P + 6,) and #L.&J - &). Although it is not obvious that eqn (44) with eqn (46) reduces to the equation extracted by Nairn in the case of perfect interface, the reader can verify their identity by simplification of eqn (33), removing the failed interfacial parts. 2.3 Fibre/matrix
debonded
(48) Expression for the stored energy
First an expression for the stored energy in the sample has to be found. Expanding and simplifying the total complementary energy given in eqn (30) results in: r=r,+nl:
and
+ G#m
+ t&.
-P+&.
deb khb
+ &
mLl~:~; + c55
+ %,%m&w.)dS.
I
part
(53)
o
r bond. =
x6
[[(thxmd. + 2(cl,~O
+ c5+
+ &~))~&md~p-;s;61+
c55
(54) From eqns (25) and (26), ir@ratiOnS Of t,bdeb , t&,, $&.,, t,b&,. in the debonded parts are quickly achieved and explicitly given below for both left and right parts respectively:
criterion
In the previous section, expressions for the stress function have been extracted in the bonded and debonded parts of the sample. The stress tensor C? = (a;, a;, u;, r’,,) is thus determined in every location of the sample. The concepts of fracture mechanics can now be used to derive a debond criterion, The specific work of fracture of the interface GIrc in mode II, must equate to the differential thermoelastic strain energy stored in the sample with respect to an incremental debonding length LX?on both sides of the fibre fragment:
2.3.1
x ?f
[(@Ideb.+2(C15~0
/ +,
of the total energy are then:
p (c,,~,+2c,,(c/~‘+c,,(c1’2 I -P
- 2&J 4%X*.ti + 2W isgo + C560, + WW)d5 where
[b*(l - e-“‘u) - a’(1 - eCb6u)]j
iebO..cd+2d&)dl
= @~b.('u
(55)
+ (a _ ;)Q,
partu $1
_ e-zbsu)+
$(I
_ e-zas") - ?!?(I
- e-(a+wq
[
II (56) $&. OK=T:::y22 [i (1 -eeZbS4 Idebonded pen +:(I -e-2"du)-&(l -e2(.lh) -
2(a - b)(a2(1 - eubsu) - b2(1 - eFu”u))
(57)
debonded v&lb.( W5 = I part u
;f$y
[ g1
_ e-2ay
(49) + b(l - e-2bsu)} - 2
(1 - e-(a+b)Su)]
(58)
Integrations in the bonded part are much more tedious and to perform them the following coefficients
T. J. Copponnex
898 are introduced: p-6, Ack)
U,I
(59)
a(k)(c)d[ &I
=
I -p+&
and p-6, A(k)..
a(k)(&(k)(Qdl U,I VJ
=
UU,V
I -p+&
with (U = 1, r; u = evaluating Ack) U,I and reveal that:
(60)
1, r; i = 0, 1; j = 0, 1). In uv,ij coefficients, integrations Ack)
A uu,oo= Awoo A uu,,i= A”“,,
(61) i# j
VU # U,
(62)
and A uu,Oj All remaining
-
=
coefficients
to be calculated
A{+) ,7 A{:’,, for the calculus
(63)
Au,Oj
are then:
A:‘$, I A$),
of
By inserting eqns (.55)-(59) and eqns (64)-(67) in eqn (51) we obtain the explicit determination of the total strain energy. Energy changes in the system from initial to final debonding are then attainable and eqn (44) can be solved for interfacial fracture toughness evaluations.
3 EXPERIMENTS Single-fibre fragmentation test samples were prepared from E-glass fibres (Owens Corning) coated for modified unmodified and with application polypropylene (PP) matrices. Different PP matrices were used from the non-modified PP (SY 6100, Shell). Modifications were made, both by direct grafting of ASB molecule (3-azido-sulphonyl benzoic acid) on the PP backbone, and by indirect grafting by using maleic anhydride. The samples were made by placing glass fibres between two 0.25 mm thick sheets of PP which were melted together in a mould. The sample mould
p-6,
and
A&b,
Ah?,,
Ah’%,
Afb?,o,
for the determination
A%,
(4
Afi?1
of 141
Since A~~~ijparameters are more complicated than A$, they are given in an explicit form in Appendix 3. The following expressions for integrals in the bonded part are extracted: p-6,
I -p+6
ICltmndW
=
-
9”hont,.(
gAu,)
(2~
-
+
(6
ICl:eb.
+
c
%)I
Ad:::,,
lL=l,r
(64)
i=O,l P-s,
Ip-6
d4ondW~
A-A
,
:
,
._I
(b)
Fibre
Fig. 2. Sample
j=O,l u=I.r
1ClOdeb. &‘&.,,)
(67)
mould:
mould and detail with sheets and fibre in (a) sample mould geometry; (b) fibre and matrix arrangement.
Single-fibre fragmentation test
was made out of aluminium (Fig. bottom of this mould was placed a sheet of Teflon. Above this a sheet placed. Six E-glass fibres picked out
E 0 -:
g
2(a)). On the 0.25 mm thick of matrix was of a roving at
lo 8
0
0.025
0.500
0.075
0.100
Strain 12
random, were placed across the width of the mould with a mutual distance of 15 mm. The fibres were fixed to the mould with tape. Another sheet of matrix was then placed so that fibres were sandwiched between the two matrix sheets. A second 0.25 mm thick sheet of Teflon was placed on the top (Fig. 2(b)). The PP and Teflon sheets were fixed to the mould with tape. The mould was closed by a flat 5 mm thick aluminium plate. This package was placed in a heated press at 210°C under the same loading applied for 10 min. For each system, a part of the specimen was subjected to tensile strain in a Minimat tensile tester equipped with a microscope to follow the growth of decohesion on chosen fragments, with respect to the applied extension. Other specimens were tested on an Instron tensile tester equipped with acoustic emission apparatus (Module 920 Dunegan) to follow fragmentation kinetics. For all specimens, the minimum gauge length was 55 mm and the normal strain speed was O.OS%/min. To be certain that fibres were completely fragmented, which means that no more fibres would break if the specimen was strained further, an elongation of 9% was applied. 4 RESULTS
0
0.025
0.500
0.075
0.100
0.075
0.100
Strain
12
0
0.025
0.500
Strain Fig. 3. Acoustic emission results. Density of activated defects versus strain for (a) non-modified PP, (b) directly grafted PP, (c) indirectly grafted PP.
899
AND DISCUSSION
Figure 3 gives the fragmentation kinetics followed by acoustic emission for the three different systems. Adhesion improvement is clearly shown with modified matrices by a density of breaks at saturation, double (11.32 breaks/cm in the case of indirectly grafted PP matrix and 9.70 breaks/cm in the case of directly grafted PP) that obtained in the case of pure PP matrix ( = 5.92 breaks/cm). The first fibre breakage is obtained at 3.67% strain for the pure matrix whereas roughly 4% is needed for other matrices indicating a lower initial thermal compressive stress in the case of the unmodified matrix. The same behaviour in the case of the pure and directly grafted PP is obtained with a monotonic rate of fragmentation until the saturation is reached. In the case of the indirectly grafted matrix two different rates are highlighted. At the beginning of the fragmentation, the rate of breakage is high (5.245 breaks/% strain). This rate remains constant up to ~5.5% elongation. At this point a bend is observed, and the process is slowed down to a rate of 0.802 breaks/% strain. This change can be explained by the fact that when fibre fragments are short, fibre breakage can only occur in the middle part of the fragment whereas fracture can occur everywhere along the fragment length if the fibre is long. Finally, we indicate that since constriction of test samples was present at high applied strains, leading to reduced measured nominal loads, the lateral contraction and matrix Poisson effect were introduced into the results to calculate the real values of the applied load and strain. To a first approximation, the test
T. J. Copponnex
900
sample volume variation by:
Table 2. Linear coefficient of dilatation
during the process is given
Temperature
AV = r(1 - 2v,)
(68)
(Twxninal
and the real deformation &=
l-t.5 l+AV
(69)
~
W +
(70)
Fnominal)
Table 1. Acoustic emission results
Directly Indirectly
grafted
Crack density rate (def./% strain) 2.09
grafted
PP PP
10 15 21
the initial fibre length, after the first break, the axial stress in the matrix can be written as: (71)
4.1 Determination of the linear coefficient of dilatation of the matrix (Y, Stress calculations using the theoretical value of the coefficient of dilatation of pure PP (a, = 15 X lo-” “C’ in the temperature range of interest, Table 2) have given abnormally low axial stress values in the fibre far below those required for failure at the strain leading to the first break. Owing to its high viscosity, we can assume that during sample preparation, a perfect contact between the fibre and the matrix is not reached all along the length of the fibre. In addition, matrix properties may vary from the bulk polymer to the interphase region. We think that instead of taking a given value for (Y,, it is more appropriate to derive from experimental data an ‘average’ coefficient of dilatation for each different matrix used. For that purpose, we refer to a matrix plasticity model. Since at the crack tip of the fibre break the stress is limited by the matrix yield stress, after fibre breakage we impose the equality of the stress in the near-field matrix cylinder with the yield matrix stress, uY. If p,, is
Non-modified PP (Shell SY 6100)
K-‘)
is given by:
General acoustic results are given in Table 1. Prior to interfacial energy calculations, we now discuss the determination of two important parameters, namely the coefficient of dilatation of the matrix and the fibre volume fraction V,.
Matrix
q,, (lo-’
20-60 60-100 100-140
Using eqn (68), the real applied load is linked to the measured nominal load by: us =
(“C)
Crack density saturation (def./cm) 5.92
7,026
9.70
5.245 0.802
11.32
at
Just before the fibre failure we had ,n
=
flz
where ((I) derived from Equations fibre volume
ho
- V,b(Po>> = a,
(72)
VITI
is the average fibre strength at length po, experimental fibre strength distributions. (71) and (72) allow us to determine the fraction at the first break:
Vf(adb)= l -
b(d) (u(po))
+
(73)
(Ty _ a,
The coefficient of dilatation of the matrix is then given by solving numerically for IX,: &MXKLpl,(0) = M”))
(74)
Results of (Y, values are given in Table 3. 4.2 Determination of the fibre volume fraction, V, The determination of the zone where the matrix is affected by the presence of the fibre is an important physical parameter to be determined. Calculated Table 3. Properties
Matrix
Property E-glass fibre Young’s modulus (GPa) Coefficient of dilatation (xWhK) Poisson’s coefficient Shear modulus @Pa) fly (MPa) r, (MPa) Fibre diameter (pm)
of materials
72
5
0.35 29.5
Virgin PP
Directly grafted PP
Indirectly grafted PP
1.2
1.2
1.2
100.39
0.32
0,32
0.32
0.444
0.444
0.444
35 20,2 17.17
95.5
82.45
28 16.6
28 16.6
Single-jibre fragmentation test
stresses are highly dependent on the volume fraction of the fibre and it seems necessary to determine this precisely prior to interfacial energy determinations. Optical observations revealed that with systems currently used in this work, the matrix radius should be at least 10 times the fibre radius, giving us a first insight on the extent of this particular affected region. However, the fibre volume fraction is expected on one hand to decrease with applied strain and on the other hand to be a function of the fibre fragment length. A technique has then to be found to follow this development. Referring once again to the matrix plasticity model, we suggest the following way to determine Vf. Let us consider the fragmentation process at a given applied load, caa, with the corresponding mean fibre fragment length, p. On average at this stage, the effective stress state was sufficient to break all fibre fragments longer than 2p. With the mean fibre fracture strength at this length, the matrix plasticity model gives: i+,,2p
b@PN
j = 1-
b(2P))
(75)
+ ay - aa
From this calculated value of V, for a fragment of length 2p, &,ond.,p(O)the axial stress at 5 = 0 for the fibre fragment of length p is calculated and used in eqn (75) to find the final V, value of the fibre volume fraction:
v, = Vf(U,,P)
= 1-
&Xld.@(O) lCTbO”d.,p(0) + a, - @a
(76)
This process has been iterated with increasing applied loads to cover the whole fragmentation process from the first break up to the saturation. Results are given in Fig. 4 for the three systems.
om6 F 0
0
3
2
0.004
Directly grafted PP
-
&
~0.003 3 8
~0.002 -
2
Pure PP
I Indirectly grafted P
r
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
strain
Fig. 4. Evolution
of the fibre fragmentation
volume fraction process.
during
the
901
4.3 Interfacial energies In order to assess the interfacial micro-mechanisms taking place in the fragmentation test, it is common1’~12 to combine these mechanisms in a stochastic computer simulation of the process, to explain experimental results. Although these stressbased methods, which are often based on the shear-lag theory,13,14 give reasonably good agreement with experiment, inadequacies and deficiencies of these models have been already rep0rted.l’ Experimentally, acoustic emission curves (Fig. 3) give us the average fragment length at each stage of the fragmentation process. Optical observations allowed us to determine the average debonding length associated with this mean fragment length. From the expression for the energy stored in the sample, it is now possible to determine the total energy release rate associated with given decohesions, for different applied elongations. This is done by computing total energy release rates for a succession of small debonding lengths running incrementally from zero up to a chosen value. For each system we get predictions which can be used, in relation to the experimental average debonding lengths observed at each strain, to obtain the interfacial toughness of the system. We think that this kind of approach compared to the alternative computer-based simulations discussed above has the advantage of being simpler and faster. Figure 5 gives results for the different matrix-based systems. Debonding length is given in terms of an average dimensionless value m defined as: 6, i- s,
6
(77) m=2p=G It must be pointed out that the mathematical approach developed in the previous sections takes into account asymmetric decohesion at both ends of the fragment (6, # 6,). Stress states in the fragment are then determined for any damage length at each fibre fragment end. It is theoretically possible to calculate the energy release rate associated with any further increasing debonding from these particular situations. For simplicity however, symmetric situations (6, = 6,) were envisaged for calculations in the following examples. The incremental debonding length used in computations was fixed at Am = 0.005. To take into account yielding effects of the matrix, calculations were performed using the secant modulus. Each curve represents the total energy release rate versus m at a given strain. The different curves are constructed for increased strains with steps of 0.25%. Concerning the unmodified PP matrix (Fig. 5(a)) curves are given for strains from 3.75% up to 6.5%. For modified matrices, strains are going from 4% up to 6.25% for the directly grafted matrix (Fig. 5(b)) and 8.5% for the indirectly grafted matrix (Fig. 5(c)). The abacus of the unmodified PP matrix exhibits
T. J. Copponnex
902
become very close to each other for increased strains. The growth in debonding gets progressively lower as the strains increase. This behaviour is particularly obvious with indirectly grafted PP (Fig. 5(c)) vvherk
r Unmodified PP
0
0.25
0.5
0.75
1
Average Dimensionless Debonding Length m
Directly grafted PP
0
0.25
0.5
0.75
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.03
0.04
0.05
0.06 0.07 Strain
0.08
0.09
0.1
1
Average Dimensionless Debonding Length m (c)
600 _ _ Indirectly grafted PP
E
’
so.9 5 a 0.8
0
0.25
0.5
I
0.75
Average Dimensionless Debonding Length m Fig. 5. Interfacial
energies versus dimensionless length for increasing iterated strains.
debonded
Indirectly grafted PP
50.7 j +j 0.6 a 3 0.5 s .z 0.4 E aJ 0.3
E a 0.2
%I 2 0.1 2 more or less equally spaced curves indicating in that case, that the debonding process is directly related to the applied strain. For systems based on modified PP matrices, curves largely spaced for small strains
0 0.03
0.04
0.05
0.06 0.07 0.08 0.09 0.1 Strain Fig. 6. Average dimensionless debonding length for increasing strains (0, experimental debonding lengths; -, observed debonding lengths).
Single-fibre fragmentation test
debonding appears to be stabilised for high applied strains. This is a good illustration of the energy competition that exists between bonded and debonded regions for the debonding process.
4.5
. Pure PP 30 -
1
1
. Pure PP
2.5
2.0
2 -30 -
3 s ;;
903
454 -100
1.5
?? z
-75
-50
-25
0
25
50
75
100
50
75
100
Fibre length (l/d)
. Directly grafted
-100
-50
-75
-25
0
2.5
50
75
100
Fibre length (l/d) L
- Directly grafted PP
2.5 -
-45 -100
-75
-50
-25
0
25
1
Fibre length (l/d) 45
ot....1../11....1....I. -100
-75
-50
-25
0
.,\...I..,,
,,‘,,
25
50
75
I 100
Fibre length (l/d)
2.5
Indirectly
grafted PP
-100
-75
-50
-25
0
25
50
75
Fibre length (l/d) Fig. 8. Shear stress profile in a mean fibre fragment at saturation for the different systems.
-100
-75
-50
-25
0
25
50
75
100
Fibre length (l/d) Fig. 7.
Tensile stress profile in a mean fibre fragment at saturation
for the different
systems.
length
100
length
The main output of energy curves is to be used for modelling the average debonding length as a function of the applied strain to quantify interfacial of the systems. Figure 6 illustrates such
properties
results. We
T. J. Copponnex
904 0
Pure PP
-100
-75
-50
_ Pure PP
-25
0
25
50
75
-100
100
-75
-50
Fibre length (l/d)
1 m -15
-
2 z
-20
-
-ii s
-25
-
E -30 ‘;;r ‘G -35 d B E
2 -10? 2 v -15
B I ‘cj
-
-
-25
-
-30
-
-35
-
-40
-
-45
-
-45
-75
-50
’ -25
0
25
25
50
75
100
50
75
-
-20
t a, -40 z
-100
0
Directly grafted PP
-5
3 2 z g
_
-50 m,
-25
Fibre length (I/d)
-50 -100
100
-75
-50
( -25
0
25
Fibre length (l/d)
Fibre length (l/d) 0 -5 ‘;;i & -10 E
2 $
-25
L -30 -J 5‘G -35 2 -40 e
g
-25
i ._ 2
-30
s E
-40
-35
-45 -50 -100
,, ,,,,, -75
-50
l-25
0
251
50
75
100
Fibre length (l/d) Fig. 9. Radial stress profile in a mean fibre fragment at saturation for the different systems.
Fibre length (l/d) length
obtain a very good agreement between predictions and experimental data, although debonding seems to be underestimated at the end of the fragmentation process in the case of the indirectly grafted system.
Fig. 10. Hoop stress profile in a mean fibre fragment at saturation for the different systems.
length
4.4 The stress state at saturation We have seen that it is possible, from optical observations of debonding versus strain in a single-fibre fragmentation test to evaluate an interfa-
Single-fibre fragmentation
cial energy-based criterion for debonding. We emphasize now that the model presented is selfconsistent with regard to the stress state in a fibre fragment. By contrast simulations of the fragmentation process generally used are governed by a maximum stress criterion (i.e. the maximum stress exceeds the interface strength or the shear strength of the polymer matrix) and so require a stated value for this stress prior to computations. Here we do not impose such a restriction. Plots in Figs 7-10 give the different components of the stress profile along the fragment length for the average fibre fragment length at saturation for the three systems. It can be seen that the maximum shear stress (Fig. 8) in the case of the pure unmodified matrix system (zm,, = 19MPa) does not exceed the maximum yielding shear strength of the matrix, whereas the opposite is observed for modified matrices (r,,, = 38.6 MPa for the directly grafted PP and r,,, = 40.8 MPa for the indirectly grafted PP). Since matrix failure is never observed in all the tested samples, it should indicate that matrix yielding followed by consolidation occurs in modified matrices. This has been already reported for carbon/epoxy systems.‘” This result can be related to optical interface observations using a microscope where plasticization seems effectively to be present in front of the debonding lengths whereas it is not the case for the pure PP (see Figs 11 and 12). If we now look at the radial and hoop stress profiles (Figs 9 and lo), we can see that they show a similar shape. For all studied systems, the radial stress presents a similar compressive stress concentration at the fibre ends owing to fibre. At the Poisson contraction of the debonding/bonding boundary, we also observe such a compressive stress concentration, but here the value of the compressive stress is three times higher in the case of modified matrices than that which exists with
Fig. 11. Bondedldebonded
boundary
in the case of pure PP.
Fig. 12.
test
905
Two fibre ruptures
and matrix plasticity of the indirectly modified PP.
in the case
the unmodified PP. This strong compressive stress is expected to play an important role in the mechanism of debonding. It is highly probable that in the case of modified PP systems interfacial failure is strongly inhibited as a consequence of this effect. 5 CONCLUSION The bi-dimensional stress analysis of a single embedded fibre previously developed by Nairn for the case of a perfect interface (no interfacial debonding) has been improved by incorporating a damaged interface at the fibre ends. The overall stress tensor has been extracted and used to derive an interfacial energy debond criterion. Experimental single fibre fragmentation tests performed on different E-glass/PP matrix systems and followed by acoustic emission were used with the mathematical analysis developed to generate curves of interfacial energy versus applied strain. It is suggested that the use of these energy curves with the observed debonding lengths in single-fibre fragmentation tests enables one to quantify adhesion performance in composites. The experimental values of GnC obtained were of the order of 45 Jmp2 for pure PP whereas a value up to 183 Jme2 is predicted for modified PP (Table 4). It has also been shown that this energy-based approach is self-consistent with regard to the stress state in the fibre fragment. Plots of stress profiles for modified matrices indicate that yielding followed by consolidation may occur, whereas for unmodified matrix it may not. To summarise, we think that the present bidimensional energy-based approach is a very interesting and powerful alternative analysis to the existing computer-based simulations using the interfacial shear strength concept. Since extensive calculations are
T. J. Copponnex
906
Table 4. Fragmentation
Matrix
Non-modified PP (Shell SY 6100) Directly grafted PP Indirectly grafted PP
Mean fragment length at saturation (pm)
results
Interfacial energy (J mm’)
m (theo.) at saturation
m (exp.) at saturation
1690
0.72
0.72
45.3
1200 876
0.42 0.57
0.41 0.38
147.8 183
required to produce energy curves, a procedure is A copy of written in MathematicaTM for PC computers. the procedure is available on request. Computers equipped with a mathematical co-processor with 12 MB RAM are sufficiently powerful to give results within a reasonable time.
11.
12.
ACKNOWLEDGEMENT 13.
This work was supported by the Centre des MatCriaux Pierre-Marie Fourt of the Ecole SupBrieure des Mines de Paris and Shell Research (Louvain-la-Neuve, Belgium).
14.
15.
REFERENCES 16. 1. Kelly,
A. & Tyson, W. R., Tensile properties of fibre-reinforced metals: Copper/tungslen and copper/molybdenum. J. Mech. Phys. Solids, 13 (1965)
stresses around breaks in embedded fibers. Mech. Mater., 13 (1992) 131-157. Jacques, D., Transfert de charge entre fibre et matrice dans les composites carbone-rCsine. Comportement en traction d’un composite modkle monofilametaire. PhD thesis, Institut Polytechnique de Lorraine, Nancy, France, 1989. Ling, S. & Wagner, H. D., Relationship between fiber flaw spectra and the fragmentation process: A computer simulation investigation. Comp. Sci. Technol., 48 (1993) 35-46. Cox, H. L., The elasticity and strength of paper and other fibrous materials. Bit. J. Appl. Phys., 3 (1952) 72. Lacroix, R., Kennings, R., Desaeger, M. & Verpoest, I., A new data reduction scheme for the fragmentation testing of polymer composites. J. Muter. Sci., 30 (1995) 683-692. Feillard, P., DCsarmot, G. & Favre, J. P., Theoretical aspects of the fragmentation test. Comp. Sci. Technol., 50 (1994) 265-279. Desaeger, M., An integrated theoretical and experimental approach of the fibre-matrix interface in carbonepoxy composites. PhD thesis, KUL Leuven, Belgium, 1993.
329-350.
A. T., Measurement of the thermo2. DiBenedetto, mechanical stability of interphases by the embedded single fiber test. Conzp. Sci. Technot., 42 (1992) 3.
4.
5.
6.
7. 8.
9. 10.
103-123. Di Anselmo, A., Accorsi, M. L. & DiBenedetto, A. T., The effect of an interphase and energy distribution in the embedded single fibre test. Comp. Sci. Technol., 44 (1992) 215-225. Wagner, H. D. & Ling, S., An energy-based interpretation of interfacial adhesion from single fibre composite fragmentation testing. Adv. Comp. Lett., 2 (1993). Jiang, K. R. & Penn, L. S., Improved analysis and experimental evaluation of the single filament pull-out test. Comp. Sci. Technol., 45 (1992) 89-103. Zhou, L. M., Kim, J. K. & Mai, Y. W., Interfacial debonding and fibre pull-out stresses: II. A new model based on the fracture mechanics approach. J. Muter. Sci., 27 (1992) 3155-3166. MacLaughlin, T. F. & Barker, R. M., Effect of modulus ratio on stress near a discontinuous fiber. Experim. Mech., April (1972) 178-183. Carrara, A. S. & MacGarry, F. J., Matrix and interface stresses in a discontinuous fibre composite model. J. Comp. Mater., 2 (1968) 222-243. Ko, W. L., Finite element microscopic stress analysis of cracked composite systems. J. Camp. Mater., 12 (1978) 97-115. Nairn, J. A., A variational mechanics analysis of the
APPENDIX
1
Calculation of the total axial stress v0 a(, is calculated by equating total displacement of the near-field and far-field matrix cylinders. Integration of strains gives:
P
I
P
ET,~ dt
=
G” d5 i
-P
(78)
-6J
i.e. P I -P
Axial
strains
E,E$
in the matrix
d.$ = era
are given in cylinder
(79) i (i =
mf, m,) by:
E(‘) =
z
1
E m(
ap, _ Y&y
+ ay>>
(80)
Single-jibre fragmentation
Integration of for the stress parts, as well expression for ua +-
907
eqn (79) using eqn (80) and expressions function in the bonded and debonded as its derivatives yields the following crO:
2vmA, T+2v,l/t(l+~)U,+(;+~) A0
(92)
x i=,c. 4 li + ~,(G‘ I.. uo
test
+ b$
Z
4”
i=2,...,4
=
(93) 1
2vmA4
Vm
; (
Ao
+
?)I1
m
v,(b;;‘+
-
b$)f:
o
(81)
[v$(1+ -
3V,(l+
Y,)(S + 3Y,)
~,)(3 + Y,) + 6(5 + 3v,)] (95)
(96)
-
(1 - dd.,(S))‘x’di]
I d&parts APPENDIX
(85)
2 D1 = A$
Expressions
(97)
F+&(F)]]
for [C] and [Dj matrices C1,=L
~-- 1
A:V,
v, ( 2-&Vm
1 Ci3Z -~2&V,
(a,
f
Ao ) VmA3A4 %A,
D2=
(86)
- af)
(98)
0
-Cz2
(99)
,=(g-;)(a,-af)
(100)
(87) 4=~[(01,-ai)(~-(l+Y,)
C15 = 16~;,v,
[(1+ Y,)Z(l +T)
- (1 - y_)(l
- 7)
+ 2AoE,
C16= -‘-(Az+$)
- vmA%]
m
x (I
(88)
(89)
D6 =
V,A, A0
(
anI-
(Yf)
+yq) - y +
%lKl-
afvf
(101)
(102)
908
T. J. Copponnex
APPENDIX
3
Air.01= Determination of ALz,{i coefjicients
With &,ond.= 2p - 6, stating C[X] = cash [x&,~,,~.]and S[x] = sinh [xS bnd,], integrations give the following expressions for the A:$, coefficients: A II,00= kz;_P
[ %[
*
(c+,’
(y - k C [ Ea,k [ 2ak(cY + k)
(a” + 2a3k - 2a2k2
k=P,-P
+
2ak3 + k4)C[2a + k] m 12Cu2k2C[al
+s bond.
a3S[a]
kS[cy + Zk])]]
-q
(
(108)
+ 3kZ)
A!& = k=;_,J[ •~,&mt.(~=kz)(~
C[2a]
- (a” - kZ)aYY[2a] + 2ak(cY - k)‘C[a + k]) +
2a4 + 7a2k2 - k4 4a
+ (CY- k)*C[a + k] - (a” + k2)) + ak
S[2cx] + (a - k)4;k;ff;;
x S[2(a + k)] - 2u’y;;
3k) S[cz + k]]]
x
(103)
3a+k (a-k)4 S[2(a + k)] + 2ru2kxSS[a ( 8(cz - k)
+ k]
- 7a24+ k2 kS[Za])]]
&,,I = i&
[ ~~,p[&m~.(~a. - k)*Cb
a’-k2 - 2
+
kl A!;,& = kz;_,
C[2a] - a2 - k* + 5u24+u3k’ S[2a] >
- 2a(a2 + uk + 2k2) k(a + k) A Il.01= &zP x
(a + k)4 (
4
sb0n;a2k2 ((5k* - 3a2)C[a]
[ Q[
_ (a - k)’ C[a + 2
(Q - k)4 S[Z(cz + k)] + 8ak(a + k)
x (a” f Sla + k]]]
(
004)
k2)(11a2 f k*) S,(y] a-k
A(& =
C[2(a + k)] - 2a2(a2 + 3k2)C[a + k])
(c? - k2)[ (a - k)S[a + k] - d[2a]]]]
,&j
bk
[
2a’k2(2C[cu + k] Stxmt.ffk(Q2- k2)
2
uw
i- (a A lr.00= *=;_,
[ Q[F
((a + k)‘C[2a
- (a’ + k*)C[k])
2cr(cr + k) 1 g2”’
[ ~,k[++
((5a -
- (Q+ k)‘C[2Cy
-k])
+ k)
2a(a2 - k’)
x
i
“@;
w]]
(111)
+ 2a2k(3a a+k
007)
+ 2kl
+ &,o,,d.ak
k)*S[a + 2k] - k’s[a])]]
~Y2+
+ cuk(a - k)‘C[(-Y + k] +
S[LU+ 2k]
_ (5a2 + k2)(a2 + 3k’)
k)S[a + k])]]
((a - k)C[a
A);,‘, = *=F_, [ %3[ h”d.(
3k2)CPl
- (c~ -k)
x 2a3 - a2k + 2ak2 + k” 2ak(n
(106)
[ e,.p[z
+ 3(au’i,k’)C,a])
S[CX+ 2k]
+ khS[cl]]]
(kWa1
kl Ajf?, = k=&,
x 2a4 + 2a3k - 3a2k2 + 2ak’ + k4
A Ir.11= xz;_P
-
+ (CI -k)
- 2a6 - 2;;;:;
(110)
;) + &
- C[2cy] - 1) + +*y
=I) - 2($k)
+ (a + k)(a2 + 4uk - k’)S[cy + 2k])]] [ %P[ (a’ + kY(C[2011-
(109)
+ k)
3k2) k2(a2 - k’) 2
S[a + k] -
+ iTaLk;;
(112)
W;
S[2(a +
w4)
7k2) sr2cyl
W]]
(113)
Single-jibre fragmentation
test
Ai;& = k&
Aj;‘,, =
- (a” + k2)C[a]) + x - k)(a’-
_@
a
(a’ + k2)(a2 + 11k2)
(
4ffk - kz)S[a
+ 2k])]]
[ cqD[ b2k2( &md.($
_ k2(cy2’- k*)
(Y2 - k2
(114)
+ (3a2:
5k*) 4
aS[2cx] + iTaLk;;
Aj& = kzF_,
(a + k)
[ c+[
ak( ak(a2 +
+ (a ; k)4c[2(a -6 md~2
(a + k)
+ k)C[a
+ k])
A$;‘,,= kz_p [ E~,~[ +
S[a +
kl)]]
+
(115)
-
kZ)(CPd - ;)
- k’)( k’S[2a]
[ %P[&j
(2a2@
x ((Y’ + 2ak - k2)C[2a
C[cv,)
(CI - k)‘cz3S[a + 2k])]]
ffsta1
2(a. + k) x (Y(LY- k)S[cx + 2k]]]
(119)
(Ly2‘4 k2)3 + @’ ; k2)
(117)
+ 2cx2k( a-k)2(a+;)C[a+k])
+ ~ (a - k)4(a + 3k/2) sL2((y + k)] ( 4(m + k) -
-s bond, a2k2(a4 - a2k2 - k4)S[cx] +f
3Cr2k2 + 4k4)C[(u]
(116)
- k,
+ k] - s
(118)
2(a2 - k2 ) a4 + &y3k- 3a2k2 + 2ak3 + 2k4
x k4C[2a] = Jp
W)]]
+ 2k])
A!& = k=;_p [ E%J - Ld.( +k]))]]
Sta +
cx6 - 2a4k2 - 25a2k4 + 2k6
+ k)] - a2((r2 + 3k2)C[cx + k]
+ cx(cz - k)(cx +;)S[a
A&
((a4 -
+ (a! - k)2a2C[a
S[2(a + k)]
(a” + cyk + 2k2)
- 2a2
+ 2a2k - ak2 + 2k3)
(3a2 + k2)
C[2cr] - (Y((Y - k)‘(2a
( &mnd(~2 + k’)
+ (3a* + k2)(Cu2 + 5k2) cuS[a] + (a - k)(d
A{,?,&,= k&
[ G;D[ F
x (3a2 - 4k2)C[cr] f (c~ - k)*cx’C[cx + 2k])
S[Ql
a-k
909
cz4 - 7a2k2 - 2k4 4
SI2al
- 2a2kUL’,”k’ 3k)S[a
+ k])]]
(120)