Theoretical aspects of the fragmentation test

Composites Science and Technology 50 (1994) 265-279

T H E O R E T I C A L ASPECTS OF THE F R A G M E N T A T I O N TEST P. Feillard, G. D6sarmot* & J. P. Favre Materials Department, Office National d'Etudes et de Recherches Agrospatiales (ONERA), 29, avenue de la Division Leclerc, BP 72-92322 Chdtillon Cedex, France (Received 16 November 1992; revised version received 22 April 1993; accepted 6 May 1993) Mw

Abstract

This paper presents a review of the status of the fragmentation test. The processes occurring during the fragmentation test and especially the modelling of the test are analysed. The limits of one-dimensional models are defined and the quality of bidimensional models is outlined, one-dimensional numerical analysis of the fragmentation process being presented as a powerful tool for understanding the basic interracial phenomena, Finally, misconceptions in the analysis of some mechanical features of the test are emphasized. The most notable of these are that no analytical relationship between the critical fragment length and the mean fragment length can be established at present and that the ineffective length, which may allow the in-situ Weibull parameters of the fibre to be determined, has to be defined by using a probabilistic criterion. In conclusion, fragmentation phenomena still appear to be poorly understood, but the development of understanding is crucial for the micromechanical analysis of composites because the fragmentation test exhibits the basic damage modes which are present in multifibre composites,

Em Gf Gm l i lc 1~ It

Gamma function Applied strain Friction coefficient Matrix Poisson ratio Fibre strength Weibull scale parameter of the fibre Interracial shear strength Interracial friction stress Ability to transfer load Matrix yield stress

1 INTRODUCTION The fragmentation test is now commonly used to evaluate the interfacial properties of numerous fibre/matrix systems. Many studies have been conducted during the last 15 years dealing with the nature and the quality of the interfacial adhesion. Early investigations by Kelly and Tyson, ~ Fraser et al. 2 and Oshawa et al.3 provided tools for semi-empirical analysis of the test data from which the capacity of the fibre/matrix system to transfer load can then be found for a wide range of composites. The fragmentation test tends also to be employed to reproduce the microdamage occurring at the interface during loading 4-6 which is transposable to multifibre composites. Whereas the knowledge of the composites micromechanics is progressively gaining ground, some questions still arise about the analysis of the fragmentation test and the validity of the experimental data. Recently, it has been shown that the description of the fragmentation process by Cox's theory 7 is not satisfactory for some particular glass/epoxy systems and that more sophisticated models of load transfer are required for a better understanding of the

NOTATION

Ef

F e~ /u Vm Or o0 ~'d rf 17m Ty

Rf Rm

Keywords: fragmentation test, Weibull parameters, micromechanical analysis

df

u w

Weibull modulus of the fibre (or shape parameter) Fibre radius Matrix radius Radial displacement Longitudinal displacement

Fibre diameter Fibre elastic modulus Matrix elastic modulus Fibre shear modulus Matrix shear modulus Fibre length Mean fragment length Critical fragment length Ineffective length Load transfer length

*To whom correspondence should be addressed, Composites Science and Technology 0266-3538/93/$06.00 © 1993Elsevier Science Publishers Ltd. 265

266

P. Feillard, G. Ddsarmot, J. P. Favre

fragmentation process. ~ A recent bi-dimensional model s will be analysed and its limits defined, A computer simulation of the fragmentation test proposed by Jacques showed good agreement with experiment in the case of carbon/epoxy systems, 91° but a careful study of this simulation shows that some of the assumptions were not justified. It will be used in the following as a tool to underscore some aspects of the fragmentation process and, owing to the restrictions detailed below, the results must be analysed qualitatively rather than quantitatively. It is widely accepted by researchers that the fragment lengths at the saturation state are not uniformly distributed between 1c/2 and lc as assumed in the Kelly-Tyson model. Henstenburgh and Phoenix IJ have recently shown that the relationship between i and lc (1~=47/3) is not as simple as established by Oshawa et al.3 and actually depends on the Weibull modulus of the fibre. It will be shown in this paper that the relationship between lc and i is very complex and is unknown. Little attention has been paid to interracial friction, especially in the fragmentation test. The frictional stress is generally taken as constant and independent of the applied strain. 4~~2 It will be seen that this cannot be true in the case of the fragmentation test and that a friction p h e n o m e n o n ruled by Coulomb's law seems more appropriate although this has never been verified. The notion of ineffective length is required in order to determine the in-situ Weibull parameters of the fibre. 2"~3 It corresponds to the length where the fibre cannot break. Its definition is not clear. Questionable assumptions are often made and hence this notion must be rigorously investigated. It will be shown in this paper that the ineffective length could be defined by means of a probabilistic criterion. Taking into account the above restrictions, a reassessment of the fragmentation process is required in order to derive the scope and the value of this micromechanical test. Therefore, the objectives of the present paper are:

displays the basic phenomena which are present in multifibre composites (i.e. fibre fracture, interfacial debonding, friction, matrix plasticization, matrix cracking etc.). It will appear to the reader that no measurement of the interfacial characteristics such as Td and ~ can be made with the fragmentation test, by contrast with the micro-indentation or the pull-out test. But the fragmentation test shows how the varied characteristics of the composite components (fibre, matrix and interface) interact with each other. 2 STUDY OF THE FRAGMENTATION PROCESS As briefly outlined above, the fragmentation process is very complex with many parameters interacting with each other. Modelling of the test is necessary in order to define the major parameters and to take into account their influence on the process. Two distinct tools are required: (i) a load-transfer model which must describe correctly the interracial stress state, and (ii) a simple algorithm relating the multiple fibre failure process with the increase in applied load. The exact solution of the stress field in a composite system (fibre, matrix, interface) is still unknown, although it is a basic problem of composite materials. Few models have been developed and this analysis will be focused on analytical models. They are an approximation to the exact solution, but they are the only means of simulating the development of the stress state. Therefore, for each model, this section defines the basic assumptions and consequences for the validity of the model. 2.1 Load-transfer models 2.1.10ne-dimensionalmodels The first elastic load-transfer model was developed by Cox 7 to study the mechanical properties of fibrous materials. The main assumptions were:

- - t o present and examine various load transfer models for the fragmen':ation test; - - t o examine the current assumptions and the consequent misunderslandings associated with the fragmentation test.

(1) The lateral stiffnesses of the fibres and the matrix are the same and therefore only the load in the direction of the fibre axis is shared between the fibre and the matrix, which means that the matrix strain is homogeneous in the whole specimen and also along the interface.

Finally, some guidelines for future research are presented which may lead to a better understanding of mechanical load transfer between the fibre and the matrix, With this paper, the authors wish to demonstrate without ambiguity that the principle of the fragmentation test is not in question but rather that it is the analysis that is incomplete. The fragmentation test is essential for micromechanical analysis, because it

(2) The rate of load transfer from the matrix to the fibre and vice versa will depend on the relationship between the displacement u~ in the direction of the fibre length at a distance z from one end and v~, the displacement of the matrix at the same point in the absence of fibre. As a consequence of these restrictive assumptions, only Oz, the axial fibre stress and ~, the interracial shear stress, are determined. A simple calculation for

Theoretical aspects of the fragmentation test equilibrium gives: (

/z ) ) coshfl(~-z -----; cosh fl¼

o(z) =EfE~ 1

(1)

T ( z ) = E f e f l R f sinh fl 2

(~

- z

)

l cosh fl ~

where fl =

/.

2Gm~

~EfR 2Iog(~) and Rf and Ef are, respectively, the fibre radius and Young's modulus; E m and G m a r e the matrix Young and shear moduli; e~ is the applied strain and R m is the mean separation of the fibres. In the case of single composite tests, Rm corresponds to the matrix cylinder which is under shear stress owing to the presence of the fibre, The determination of the value of R m is very critical and is discussed in Section 3.1.3. Galiotis et al. ~4"15 pointed out that good agreement was obtained between laser Raman spectroscopy experiments and Cox's theory when the ratio Rm/R f is about 10-15. An example of the stress state predicted by the Cox model in the fibre/matrix system is presented Fig. 1. Recently Sabat et al. 16 and Lhotellier et al. ]7"Is have presented a load-transfer model based on shear-lag theory (Cox's model) in which an interphase zone with given mechanical properties is placed between the fibre and the matrix. Many authors have proposed such a model and tried to prove that there is a cylindrical zone around the fibre which has mechanical properties different from those of the bulk matrix, 19~2~ but at present no method is available to characterize the interphase precisely and its mechanical behaviour is only assumed without experimental support, Lhotellier et al. 17.18 developed a model for glass/epoxy ~ (GPa) 2 I 1 ' [~ ~

"c (MPa) 60 ~

~ 30

0

~

~

0

-1 "20

~ - 3 0 100

200

300

400

501)60

Position along the fragment (~tm) Fig. 1. Stress profile predicted by Cox's model along a 500/~m long fragment,

267

composites and they introduced the possibility of matrix yielding at the interface, assuming a constant deformation rate beyond the yield threshold. As shown in Section 3.2, such assumptions about the matrix mechanical behaviour are questionable and should not be made prior to a thorough investigation of local and global plasticity in organic matrices. Another load-transfer model was proposed by Dow 22 which is based on more realistic assumptions than those of Cox's model. The matrix displacements are not assumed constant in the direction of the fibre axis. They vary with the distance from the fibre ends, though lateral displacements in the matrix are still not considered. From a consideration of the displacement field, it may be deduced for the fragmentation test, that: l/

o(z)

=

c°sh{A \2

///

Pm

/ Em] A~ +/Aim ~ ]

1

/ A I\ cosh/---! \dr2~

I !

(2)

/

where Pm and A m a r e the matrix load and the matrix cross-sectional area; and Af, df, Gf are the fibre cross-sectional area, diameter and shear modulus; and:

/

,-Gf(

AfEt-~

V2 ~ \1 + AmEm/ ).=2

__Gf( ~ A m + 2 - V ~ ) V~ - 1 + Gm

(3)

Although the continuity of w, the longitudinal displacement at the fibre ends, is not verified in the Cox and Dow models, the latter is not consistent with its own assumptions. The longitudinal displacements are supposed to vary in the matrix along the interface but one-dimensional models implicitly correspond to a plane strain problem. Such dependence implies that it is obligatory to take into account u, the radial displacement at the interface. Thus, the boundary conditions imposed by Dow cannot be verified with a one-dimensional model and a bi-dimensional model ought to be used. It can easily be shown from eqn (2) that the interfacial stress profile in Dow's model is more strongly dependent on the Rm value, the sheared matrix radius, and on the fragment length than in Cox's model. The comparison of these models is very revealing and outlines the limits of Dow's model. Despite its simplicity, Cox's model is quite a 'robust' model and is probably a good approximation to the stress state at the interface except at the ends of the fibre. 23 In contrast, Dow's model cannot be employed to simulate load transfer between the fibre and matrix. Hence, computations for fragmentation test developed by Ochiai and Osamura 24 from this

268

P. Feillard, G. Dgsarmot, J. P. Favre

model may need to be reassessed even though agreement was demonstrated between the model and

"cfz MPa

However, as already pointed out in a previous paper, 5 one-dimensional load-transfer models obviously possess some inadequacies and deficiencies in representing the real mechanical load transfer between fibre and matrix.

100 50

The solution presented by Nairn was derived by the variational method, and two of the four compatibility equations are not satisfied. The axisymmetric stress tensor is determined in the fibre as well as the matrix, It may be noted that the solution includes the thermal residual stress resulting from specimen manufacture. Figure 2 shows the development of some stress components in the fibre/matrix system predicted by Nairn. The mechanical constants are those for a glass/epoxy system and the Rm/Rf ratio in Cox's model has the same value as the computed results presented in Ref. 5 ( R J R t = 200). It is noted that the interfacial shear stress along the interface is zero at the fragment ends as a result of the boundary conditions, and a~, the radial stress in the matrix along the interface, is more compressive at the ends than at the middle.

i -50~ 0

1,000

,

-, - c ~

0 2.1.2 Bi-dimensional models More sophisticated models (bi-dimensional models) have been developed recently and they take into account the longitudinal and the radial displacements in the fibre/matrix system. The first was presented by Whitney and D r z a P and is based on the superposition of an exact far-field solution and an approximate local transient solution. The fibre fragment length is assumed to be semi-infinite and so the function giving the stress field is readily found. But this defined stress state is only verified for the long fragment lengths and is not valid for fragments that have an aspect ratio, l/df, lower than 50. In this model, the load transfer length, lt, i.e. the fragment part within which the fibre axial stress increases, does not depend on the fragment length whereas it should do so (Cox's theory predicts such a variation of the load-transfer area with the fragment length) and it is also confused with the critical fragment length (see Section 3.1). Correlation of the model with experimental data is poor and the computed values of rm are lower than those calculated from experiment with the aid of the Kelly-Tyson equation. A more complex and very attractive model has been recently proposed by Nairn. ~ After presenting the limitations of the shear-lag analysis, the axisymmetric problem of load transfer is solved for a semi-infinite fragment length and for a finite one. Concerning the fragmentation test, only the second solution need be considered,

(~,-* MPa

........

i

.... ~ 500

'

~

,

0 -500

100 200 300 400 Positionulongthetragment(v,m)

500

Fig. 2. Stress profile predicted by Nairn's model along a 500 #m long fragment. An equivalent matrix radius has to be defined because of the independence of the axial stress, o~, relative to the radial coordinate. This geometrical parameter, which is also present in the Cox model, has a prime importance in the Nairn model. Figure 3 shows the cumulative number of fragments versus the applied strain for various Rm/Rf ratios. All these curves are obtained from the Jacques algorithm~ by using Cox's and Nairn's models, for the case of a pure elastic loading mode, i.e. without interracial debonding. The number of fragments predicted by Nairn is clearly dependent on the Rm/Rf value. Several questions remain unanswered.

2.1.3 Nairn model's limitations Nairn proposes various ways of determining the R~ value, notably by measuring the matrix area within which the birefringence patterns are present which may correspond to the sheared matrix part. But in the class/epoxy systems with which we are dealing in the present study, measurement gives an Rm/Rf ratio of 10 which involves an overestimate of the fragmentation rate. Another possiblity is to remove from the

Numberoffibrebreaks 1

~

Fibrelength:L30ram is0

| ~ ~ _ _ ~r 1 t f 7

Cox~

Force(daN) ~ 120 _j_..~ , , 100 ~ 80

_v~ 100 i

R.~-~00 60 ~S~-400 40

so ~

Nairn 0 0

o.02

0.04

0.06

20 0 0.os

Appliedstraiu Fig. 3. Cumulative number of fibre ruptures versus applied strain. Comparison between Cox's theory and Nairn's model (Rm/Rf = 100, R m / R f = 200, R~/Rf = 400).

Theoretical aspects o f the fragmentation test

model the assumption that the matrix axial stress, o z z , is independent of the radial coordinate. Consequently, the notion of matrix volume disappears and a new solution for the stress function has to be determined, But, as Nairn points out, this complicates considerably the mechanical analysis of the problem. Nevertheless, his model gives stress states close to those derived from photoelastic model measurements 23'26-28 and those calculated by finite element methods. 28-3° The interracial debonding observed experimentally in polymer-matrix composites (PMCs) must be integrated into a simulation of the fragmentation process. Classical stress analysis may be carried out if the debonding criterion is assigned to the interfacial shear stress, i.e. beyond a certain stress value, elastic load transfer is replaced by a frictional loading mode. Jacques 9.H~ used such a criterion in his computation (see Section 2.2), but it cannot be employed in a bi-dimensional load-transfer model. Indeed, the maximum interfacial shear stress does not correspond with the fragment end (Fig. 2), where interfacial debonding appears experimentally. Another possibility is to calculate an equivalent stress, as is done for plasticity in metals (von Mises criterion), which could be compared to a threshold value determined experimentally with another micromechanical test. Kallas et al. 3j have recently used the equivalent stress to explain interracial failure in the case of the push-out test for thin specimens. However, the use of the equivalent stress at the interface as a debonding criterion implies that only the shear, through the deviatoric stress tensor, contributes to debonding, whereas, in fact, debonding initiation can also be induced by the spherical component of the stress tensor. Therefore, and as proposed by Nairn, interfacial debonding could be described more efficiently with an energy analysis. However, a classical critical energy release rate analysis, as commonly used with ceramic-matrix composites (CMCs), 32 is difficult with PMCs because it requires the experimental measurement of Gnc or, more simply, the development of debonded length with the applied strain. Steif and Dollar 33 have recently demonstrated the limits of stress or energy analysis, and proposed a semi-cohesive model which could be an interesting alternative. The debonding criterion appears to be a major micromechanical problem and it is not peculiar to the fragmentation test, but also to other micromechanical tests. 34 Another point is the mechanical behaviour of the matrix. Recently, the authors pointed out the complex interaction between lenticular matrix cracks and interfacial debonding around fibre breaks in some glass/epoxy systems. 5 DiAnselmo et al. 35 have proposed a criterion based on the predominant damage mode at the interface. They employed an

269

energy release rate analysis which may be coupled with Nairn's model. The complete solution of the load transfer between the fibre and the matrix still does not exist, but appears to be essential for accurate analysis of the fragmentation test. Obviously, the problem of the debonding criterion and the determination of Rm will not readily be solved and work will have to be done to understand properly the fragmentation process. In the following section, the computer simulation of the fragmentation process developed by Jacques will be examined. The load-transfer model which is employed is that of Cox. The limits of onedimensional models have been outlined above, but when Jacques developed his method four years ago, Cox's model was the most suitable one for such a study. Moreover, the similarity between Cox's model and that of Greszczuk, which is used for the analysis of pull-out tests, allows the debonding criterion to be readily chosen. 36

2.2 Simulation of the fragmentation process A simulation of the fragmentation process developed by Jacques 9 is based on the combination of two successive loading modes at the fragment ends, namely elastic stress transfer (Cox type) and friction. The interfacial shear strength, to, determined in a pull-out test made with the same system is taken as a debonding criterion defining transfer from one mode to the other. The computation is particularly suited to the prediction of the mean fragment length at the saturation stage. Relatively good agreement was found between the test data and the simulation in the case of carbon/epoxy systems. "~ It must be pointed out, however, that some simplifications were made which are not totally justified for most fibre/polymer matrix systems. (1) As with all models based on the shear-lag theory, Cox supposes that the matrix around the fibre is sheared only over a certain volume, which in the original problem corresponds to half the mean distance between the fibres. For the fragmentation case, the Rm value is taken as half the thickness of the sample. As outlined previously, and according to Nairn, the approximation for Rm is not suited to low fibre volume fractions, whereas Galiotis et al. judge the contrary.15 The determination of Rm is very difficult and seems to be the most questionable point about the shear-lag theory. But it must be pointed out that the influence of Rm/Rf is minor in Cox's theory if the Rrn/Rf ratio does not tend to 1 or oo. In the case of interfacial debonding, the influence of Rm/Rf is much lower than in the case of perfect bonding and is difficult to quantify.

270

P. Feillard, G. D ( s a r m o t , J. P. Favre

(2) The mechanical behaviour of the matrix is assumed to be linear elastic. Obviously, the common stress/strain curves (of epoxy resins for instance) do not match this assumption and a non-linear elastic law is more suitable. The authors introduced into Cox's model a macroscopic evolution of the matrix elastic modulus. However, this modification of Cox's model is conceptually false, 5 because the matrix strain along the interface is not homogeneous. A lowering of the elastic modulus with applied strain cannot be globally applied to the matrix owing to the lack of homogeneity, Consequently bi-dimensional models of the load transfer are required to take into account this essential mechanical aspect. Rigorously, the matrix state along the interface should be calculated through a complex bi-dimensional model, the strain increased locally (description of the fibre/matrix system with a mesh), and so a new loading profile is obtained at the interface. Many authors ~''w'373~ assume matrix yielding at the interface near the fibre ends because T.,, or rd values are higher than the macroscopic yield stress of the matrix, ry. Such yielding is difficult to observe and the results of some pull-out tests performed by D6sarmot and Favre 3" exhibit a very high value of interracial shear strength (120150MPa), which is twice the macroscopic yield stress of these resins. This tends to prove that yielding in epoxy resin may be followed by hardening as for metals 3'~ despite the fact that no macroscopic plasticity is observed on the stress/strain curves. Moreover, micromechanical studies conducted by Gulino et al. 4° show that in the case of carbon/epoxy systems, yielding appears at the interface near the fibre rupture before debonding takes place. This local plasticity shows that the mechanical behaviour law of the polymer, and especially of epoxy resins, cannot be settled by macroscopic tests such as the tensile and shear tests. More accurate studies of the local mechanical behaviour of the polymer matrix are urgently required at the interface as well as around the fibre fractures which are zones where most damage in composites is generated, (3) Jacques takes the coefficient of friction, H, as constant (whichever carbon fibre/matrix system is studied) and equal to 0.9. Such an assumption is not rigorous because tribological factors are expected to vary between fibres, e.g. highmodulus and high-strength carbon fibres. The measurement of friction coefficients carried out by Piggott et al 4~ for various fibre/matrix systems does not give 0.9 for carbon fibres but rather values between 0.35 and 0.6. The difference between the assumed value of # and the

experimental one can be explained throughout thc analysis of the frictional stress state established along the debonded fragment length. The friction phenomenon is assumed to follow Coulomb's law r~. . . . f~o,r

(4)

where Orr is a combination of the applied stress state and the residual thermal one. By using Cox's model for the elastic load transfer, the radial components vanish and consequently the radial stress observed on the friction area is less than is obtained with a bi-dimensional model. So the friction coefficient must be increased in order to obtain a realistic friction stress state. Moreover, the calculation of the residual thermal stresses assumes an infinitely long fibre (no strain gradient along the interface) which cannot be strictly applied to the case of a fibre end. Consequently, in spite of the afore-mentioned good agreement between computed and experimental results, the model proposed for the friction state is still questionable. (4) The tensile strength of very short fibres is required owing to the nature of the computation in which the fibre is arbitrarily sub-divided into 4-Hm segments. The Weibull modulus fl)r the fibre determined at large scale (four gauge lengths) '~4~ is usually used to extrapolate to small lengths. Some justification of this practice is to be found in the fragmentation onset which is well predicted by the simulation (criticism of the calculation of the residual thermal stresses are omitted owing to the large length of the first fragments), thus tending to prove the validity of the assumption. Moreover, the size of the segments (4Hm) that make up the tested fibre can be questioned and we may ask whether the size of segments is truly reprcsentative of the defect distribution on the fibre and whether it changes the fragmentation process. A calculation was carried out for E glass fibre (mechanical characteristics are given in Ref. 5) to determine the axial fibre stress that is required to create 100 fibre breaks whatever the segment size for a 30 mm fibre length (details of the calculation are given in the Appendix). Figure 4 presents the variation of the calculated stress versus the segment size. It is obvious that below a 10-ktm segment size, the required stress is constant and therefore the fragmentation process is unchanged. It must be noted that in the case of a lower Weibull modulus, the zone where the calculated stress is constant is increased and consequently so is the critical segment size. Therefore the distribution of defects on the tested fibre that is employed in this computation is valid. As a commentary on this section of the paper, it is

Theoretical aspects of the fragmentation test

271

3 MISCONCEPTIONS IN FRAGMENTATION ANALYSIS

3.95

,~, 3.9

In this section, the common assumptions of the fragmentation test will be analysed and discussed. It will be seen that most of these ought to be justified and, therefore, present some difficulties when comparing results from different papers.

3.85 ,, 3.8

I~ 3.75

j

3.7

1E-09

IE-08

lE-07

1E-06

lE-05 0.001 0.| 0.0001 0.01 Segment length (~m)

1

10

100

~000

Fig. 4. Longitudinal fibre stress required to create 100 f i b r e breaks versus the segment size.

concluded that the improvement of the computer simulation of the fragmentation process requires a modified load-transfer model that gives information on the multiaxial stress state currently present along the interface. A bi-dimensional model (such as Nairn's) would be appropriate with some major modifications, particularly to the debonding criterion, The local mechanical-behaviour law of the matrix is also very important because, from the microscopic examination evidence, matrix plasticization seems to precede interracial debonding. It is therefore necessary to consider mechanical hardening of the matrix which cannot be explained by the usual macroscopic behaviour of the matrix. This apsect is not easy to solve because no means are presently available to characterize locally the matrix and the relationship between the microscopic level and, moreover, the macroscopic one is still unknown. These problems (load-transfer model, debonding criterion, local behaviour of the matrix) may appear formidable and even intractable in the short term. However, the authors would like to point out that they are not specific to the fragmentation test. They are also common to other micromechanical tests such as the pull-out and microindentation tests and so their thorough investigation is necessary. Although the analysis of the fragmentation process appears partly imperfect, the preferred approach seems to describe correctly the observed phenomena (fibre rupture, interracial debonding). This simulation9 will be used in the following section of the paper as a tool to examine thoroughly the rupture process and to highlight the dominant parameters of the fragmentation test. Hence, the limits of this simulation must be kept in mind and the results obtained by this means must not be analysed quantitatively. We are only interested in the general features that will be observed,

3.1 Critical length versus mean length Kelly and Tyson defined the critical length at the saturation stage as the largest fragment which cannot break or, inversely, the smallest fragment which is able to break. So a fragment of length lc(1 + e) always breaks whereas one of length lc(1- e) does not. Owing to this definition, all the fragments are uniformly distributed in the domain [lJ2,1~] around the i value at the saturation onset, j The critical fragment length is defined as

lc-

or(lc)df 2~-

(5)

where or(It) is the mean breaking stress for a fragment of length, lc, df is the fibre diameter, and ry is the mean interracial shear stress which equals the yield stress in the original problem. The fragmentation test with PMCs first conducted by Fraser et al. 2 does not have the same objective. This test is aimed at determining the load-transfer ability of the system under consideration, rm, rather than the critical fragment length which is given by the test. Thus the problem of determining lc arises. Oshawa et al. 3 proposed a simple relationship between lc and the mean fragment length, l, as: /A\

l~=~3)l

(6)

The fragment length distribution is implicitly symmetric. Up to now this equation, coupled with eqn (5), is widely used to characterize many kinds of fibre/matrix systems (eqn (7)) and qualifies the adhesion. or(It)dr 17m= ~ / a S (7) 2~i However, such a uniform distribution of fragment lengths around i and bounded by I J 2 and l~, has never been observed and some authors did not accept this analysis.4'6'43 Drzal et al. 43 proposed another definition of critical fragment length. The fragment length distribution at the saturation stage is modelled by a two-parameter Weibull distribution, or is the shape parameter, fl the scale parameter, and these do not correspond to those for fibre tensile strength. For Drzal et al., the most probable fragment length, l*, calculated from eqn (8), equals the critical fragment

P. FeiUard, G. D~sarmot, J. P. Favre

272

length and a mean value of the interfacial shear stress, r, is determined. lc* =

fl

ProbabUityOe.~,y Mw=10 ; g=0,6

Ta=150MPa

o.4

r(i1)

(8) 0.2

However, along a fragment of length l*, the maximum fibre axial stress is unknown and cannot be assumed to be the fibre tensile strength, whereas along the longest fragment of the distribution, It, it can be taken to be the fibre strength because a fragment of length 1c(1 + e) breaks before saturation. The only definition of the critical fragment length is the largest fragment that did not break or the highest value of the fragment length distribution at saturation. Strictly speaking, the r values determined are of low significance. " Henstenburg and Phoenix ~l have proposed a modified version of the Kelly-Tyson equation in which the Weibull modulus of the fibre is integrated•

= ( 2 ~M*+I/Mw1~

\A-~M~!

l

(9)

where AM~ is parameter depending on the fibre Weibull modulus and is determined by using a reference curve extracted from the simulation. But the chosen load-transfer model (the Kelly-Tyson model) is not realistic and computation has to be changed to consider an elastic loading mode (Cox model). To complete this study, some calculations with Jacques' simulation have been made to illustrate the development of the fragment length distribution at the saturation stage as a function of several critical parameters (Mw, ra,/~). Note that each curve represents a sample of 1000-1500 fragments and the x axis scale is the same for all curves•

0.1 LI

'

o 0~

A

l

/ \

/

1

x,~ 50

\

/

~

2

3

........................................... 5 6

4

Fragmentlength(mm)

Fig. 6. Distribution of fragment lengths at the saturation stage for Mw = 10, ~ = 0.6 and various interracial shear strengths. First, Fig. 5 shows the fragment length distribution for three Weibull moduli. Its influence is clearly observed. The higher the modulus, the narrower the distribution• For a high shape parameter (i.e. Mw > 50), the fragment length tends to be distributed approximately between lm~x and lmax/3, but not between lJ2 and 1~ as expected. Second, for Mw = 10, the influence of the rd value was examined (Fig. 6) and the higher rd, the narrower the fragment length distribution. Another set of simulations have been run with various/~ values (Fig. 7). A higher friction coefficient leads to a narrower distribution. Finally, Fig. 8 also presents the influence of the Weibull modulus but as can be observed its influence is small when # and rd are low. The distribution of the fragment lengths for Mw = 5 and Mw -- 10 are almost identical. It arises from these computations that for a high Weibull modulus, the influence of rd and/~ are small

Probabilitydensity 0.6 _

.

]

ProbaMty demity

ii

Mw=lO ; 'l~.~lO0 M P a

,

~,=ISOMPa ; g=0,6

•

.......... ~ Mw=50

•

0.4

i

]

.....

i

0.1 0,2. 0

0

I

2

3

4

5

6

Fragment length from) 0

1

2

4

Fragment length (am) Fig. 5. Distribution of fragment lengths at the saturation

stage for rd = 150 MPa, /~ = 0.6 and various Weibull moduli,

Fig. 7. Distribution of fragment lengths at the saturation stage for Mw = 10, zd = 100MPa and various interfacial friction coefficients.

Theoretical aspects of the fragmentation test Probability demity

0.3 0.25 0.2

Mw=S0

X~-S0MVa;Lt---0,2 '

l0 ~

0.is 0a

-

j

~

0.0s 0

0

1

3

4

s

6

Fragmentlength(tara)

Fig. 8. Distribution of fragment lengths at the saturation stage for ra = 50 MPa, # = 0.2 and various Weibull moduli, and, in the same way, in the case of a low Weibull modulus, the influence of the friction coefficient is weak for high values of Zd (i.e. rd > 80-100 MPa). It is obvious from these curves that the fragments are not distributed between 1c/2 and lc even for high parameter values. The authors agree with the results of Henstenburg and Phoenix on the dependence of the critical fragment length on the Weibull modulus and go further by concluding: --an analytical relationship between lc and i is unknown at present, --there is a complex interaction between three parameters which are, in order of importance, Mw, rd and #. Thus, eqn (6) does not give a correct description of what 1c is and, accordingly, the rm values calculated from eqn (7) are not representative of the relative fibre/matrix systems and cannot be strictly compared.

3.1.1 A new approach to the critical fragment length The critical fragment length is defined as the highest value of the fragment length distribution at saturation, The fracture of the fibre is nonetheless a statistical phenomenon and, owing to the nature of the Weibull model, no upper limit for the distribution can be defined. Consequently, it is theoretically possible to observe very long fragments which have a low occurrence probability and, practically, the critical fragment length value varies with the population size of the distribution. W h a t , then, happens to the notion of critical length? Perhaps another definition may be proposed, Let it be supposed that for a large sample we determine the Weibull parameter of the distribution and the fragment length, ll, such that

( _ (l,~Mw~ Pr(l-> l l ) = p = 1 - exp

\~/

/

(10)

The median length of [l~, ~] domain is IM which

273

depends on p. If p =0-05 or 0.01, IM defines respectively 2.5 and 1% of the longest fragments. IM(0"01) or IM(0.05) could be taken as lc because the probability of finding a longer fragment is 0.5% and 2.5%. This lc value does not depend on the sample size and would be even more suitable if IM(0.01) is close to IM(0.05). Accordingly, the definition of lc corresponds to the choice of a threshold (p value) related to the confidence wished for in the result. Nevertheless, a more detailed study may be required to determine the accuracy of the measurement. In summary, the critical fragment length is really an interfacial parameter and improvement of the measurement of l~ would obviously increase the validity of the rm value. The comparison between different fibre/matrix systems could then be more realistic. 3.2 gf is not strain independent Friction which occurs along the debonded fragment area is sometimes assumed to be constant as the applied strain increases. However the friction shear stress is not equivalent to the macroscopic yield shear stress which is truly constant for a purely plastic matrix. Friction can be considered as governed by Coulomb's law. In the case of a totally embedded fibre with perfect interfacial bonding, the stress profile along the interface near the fibre end described by Nairn's model is presented in Fig. 2. It appears that Orr develops along the interface and moreover increases with the applied strain. In the case of interracial debonding, the stress profile is not known but it can be assumed that the radial stress in the matrix develops in the same way as for perfect bonding. Therefore, the frictional shear stress cannot be assumed to be constant along the interface as a function of the applied strain. Accordingly, the friction coefficient has to be known in order to simulate this phenomenon correctly. Some authors have supposed rf to be constant 4"1l'12 and they do not question the consequences of this assumption which may lead to erroneous conclusions. Lacroix et al. 4 have recently proposed a new approach to the fragmentation test in the spirit of the work of Favre et at. 9'1°'44 (a combination of an elastic stress loading mode Cox's theory--and a frictional mode-Coulomb's law). They determine the interracial shear strength, rd, and the frictional shear stress, rf, from the critical fragment length and the mean debonded length versus the applied strain. The main result of the work of Lacroix et al. 4 is presented in Fig. 9. This is the development of the critical fragment length (note: not the mean fragment length) versus the applied strain. For a given set of values of rd, rf, e= (the applied strain) and OR (the fibre tensile stress which equals the fibre tensile

274

P. Feillard, G. Ddsarmot, J. P. Favre 260

-)

3

240

I ~N,

~

j

/

\

/

~

/

°

220

\

200

"

o ~ 180 •~ "1:;'

3

~

'"-~-~---~P"

~'---~--"

I'
I

I"
160

/

II

140

""

,

Increasing bond s t r e n g t h

0.02 0.025 0.03 0.035 0.04 0.045 0.0S 0.0SS 0.06 Applied Strain

I a

Fig. 9. Critical aspect ratio versus applied strain: 1, high r~,; 2, critical length predicted by the Kelly-Tyson equation; 3, low r~,. (After Lacroix et al. ~) strength), the corresponding critical length is calculated for a given strain from the following equation: 1 [ ,/ 2Ta ~ sc = 2nrf(eo~) ~nEfe~ + 2rf(e~) a c o s m\ n ( E f e ~ ---- OJSc))}J - 2ra coth asinh L

,,~ • . ~-, \ntt~fe~-oAscl)/JJ

(11)

where n = r(1 + vm)In and s~ = lc/dj., This is a state curve and not a behaviour curve. So each point of the curve corresponds to a saturation stage. Curve 1 presents a minimum which is very surprising and, of course, cannot be observed experimentally, whereas curve 3 is common. Lacroix et al.4 judge that such a minimum is not realistic and the critical length that is found experimentally corresponds to this lower value. Thus l,. defined by the Kelly-Tyson equation (curve 2) does not match the calculated critical length and Lacroix et al. concluded that the 'critical' fragment length can, in some particular cases (high values of rd), be much lower then predicted by the Kelly-Tyson model. The minimum on curve 1 can be explained by considering a fragment of length l, for a high ra value. From the fragmentation onset, and within a few per cent strain, there is no interfacial debonding and the load-transfer slope is steep (Fig. 10(a)). So tile critical length corresponding to this stress state decreases rapidly, explaining the first part of curve 1. When debonding takes place (Fig. 10(c)) as a consequence of the constant and low r~ value, load transfer is not as efficient as previously (Fig. 10(b)). The maximum axial stress in the fibre for the same length decreases (Fig. 10(c)). In order to determine

r
c

l">r d

Fig. 10. Common stress profile development during the fragmentation process when the friction is ruled by Coulomb's law (I) and when rf is constant (II): 1, axial fibre stress; 2, interracial shear stress; 3, fibre strength. the critical length (i.e. the maximum axial stress is equal to the tensile fibre strength), the fragment length corresponding to this mixed loading mode has to be of higher value (Fig. 10(d)). This is the only possibility for the axial stress to reach the fibre failure stress. A minimum is obtained on the curve, the magnitude of which is strongly related to the % and r t va,ues As rf is taken as strain independent, the mean shear stress, tin, is fixed because, beyond the saturation stage, rm corresponds to the mean friction shear stress along the interface when total debonding occurs. From eqn (7), the critical fragment length is thus automatically determined. This implies that two fibre/matrix systems which have the same friction shear stress, whatever the interfacial shear strength, exhibit the same critical length, which is very surprising. From the authors' experience of the fragmentation test, it is felt that two distinct fibre/matrix systems may exhibit the same mean fragment length at saturation, but they do not experience the same process and do not possess the same fragment length distribution. If rt is not constant, no critical length can be predicted. Only at saturation, an apparent steady mechanical state along the interface is reached and a critical fragment length determined. It appears that the hypothesis of rf being strain independent must be rejected because it eliminates one degree of freedom from the system and leads to some unrealistic conclusions. Recently Dollar and Steif 45 confirmed that the interfacial shear stress determined from Coulomb's law cannot be taken as constant. To conclude, it is asserted that: --1~ is an intrinsic property of the fibre/matrix system and not of the interface; and

275

Theoretical aspects o f the fragmentation test

--rm is just a characteristic of an mechanical steady state,

apparent

By definition, the critical fragment length can never be lower than the value calculated from the Kelly-Tyson equation, 3.3 The ineffective length Many authors employ the fragmentation test to determine the in-situ Weibull parameters of the fibre, Rosen 46 demonstrated that the cumulative damage of thin, multifibre glass/epoxy tapes cannot be predicted by a model based on the Weibull parameters of the fibre measured ex-situ by the use of single fibre tests. The reason is the presence of an ineffective length that was introduced later by Fraser et alfl This length represents the fibre portion where the axial stress is built up and so is less than the maximum stress occuring theoretically in the middle of the fragment, Fibres cannot break within this 'forbidden zone' and have to be withdrawn from the real fragment length for the calculation of the Weibull parameters. Fraser et al. and many other a u t h o r s 14A5"47-53 propose that the ineffective length, l~, equals the critical fragment length measured in the fragmentation test. They argue that 1¢ is the shortest fragment length that can be broken and, accordingly, represents the fibre part where no rupture can occur. However, lc corresponds to the ineffective length only when debonding is completed, and defines only an apparent mechanical steady state. It does not describe previous stress states and lc must not be used to define a 'forbidden zone'. El Asloun et al. 5° also reduced the ineffective length, which depends on the fibre/matrix properties and the stress state, to the critical fragment length which indicates the ability of the interface to transfer load. A relationship is proposed between lc and the fibre/matrix Young's moduli ratio in the case of perfect interfacial bonding and so the critical fragment length can be known without doing any fragmentation tests. But the critical fragment length can only be defined at the saturation stage which is never achieved in the case of perfect bonding 5 and, moreover, this equation cannot show the influence of the fibre treatment and/or size which is not necessary to define, Such an analysis is very dangerous because it over-simplifies the fragmentation process which is a very complex phenomenon where many parameters interact with each other. The ineffective length and the critical fragment length represent two distinct features of the fragmentation test and are equal only when total interfacial debonding has occurred. Before the saturation stage, lc is undefined and cannot be used for stress states different from the saturation stage, The ineffective length can also be defined as the

portion of the fibre where the fibre axial stress is lower than a certain percentage of the maximum axial stress (for instance 99% or 95%). The ineffective length includes the debonded length and a part of the elastic load transfer area. As the fragmentation process continues, the debonded length increases and so reduces the elastic load transfer. The ineffective length then tends to equal the debonded length. So an accurate definition of the ineffective length does take into account the development of the interfaciai stress state with the applied strain. Curtin 12 has presented a similar definition of the 1~ extension but he assumes that the frictional shear stress is strain-independent, an assumption that has been shown above to be invalid. The main consequence is that the frictional load transfer is underestimated and the ineffective length overestimated. The comparison of this model with the simulation conducted by Gulino and Phoenix 54 shows that the model works well but still does not fit the experimental data. In order to verify the validity of such a definition of ineffective length, some calculations with the simulation of Jacques have been made until the saturation stage is achieved. At each fibre break the ratio of breaking stress to maximum axial stress is determined. Figure 11 presents the distribution of the value of this ratio on a semi-logarithmic scale and corresponds to a sample of more than one thousand ruptures. It appears that the distribution of the fibre breaking stresses depends on the input values of the system (Mw, rd,#). For instance, in system 1 there are no breaks below 97%, while in system 2 some breaks appear below 70% of the maximum axial fbre stress. A simple stress threshold cannot be used, then, for all systems. The two extreme systems taken as examples may be not realistic but prove that the ineffective length cannot be defined by a stress criterion. The ineffective length can also be defined with a probabilistic criterion. Ling and Wagner s3 have recently determined such a probability but with linear loading modes. They also confused the critical length with the ineffective length. If the axial stress is taken to be constant in each segment of the fragment (this can be done because the segments are short), the hyperbolic stress profile can be approximated by a crenellated profile and the failure probabiliy of the broken fragment can be calculated from the simple equation: Pr(o)= 1 - e x p

( - l° x l

] (12) \ oo / / A set of calculations has again been made. At each fibre break, the stress in the broken segment (the fibre is arbitrarly divided in 7500 segments of 4 #m length) is noted and the corresponding failure probability calculated. Figure 12 gives the frequency of these

276

P. Feillard, G. Ddsarmot, J. P. Favre 1.

I --

0.1:

°

I

i

I

!

I

I

1

100

150

1

2 3

5 5

15g ,"

1 0,2

~1~

I

0.01.

"

.

70

73 71.5

II Ibltlllil

iitlilliIiIll!llll lLili

76 79 82 85 88 91 94 97 100 7 4 . 5 77.5 8 0 . 5 8 3 . 5 86.5 89.5 92.5 9 5 . 5 98.5

Stress ratio Fig. 11. Normalized frequency distribution of the ratio (breaking stress/maximum axial stress).

failure probabilities and the shape of the distribution is very similar for all systems. This suggests that such a criterion can be used for all kinds of systems. This value depends on the confidence that is considered necessary. For instance, with confidence limits of 95%, the criterion will be 0.4: all the segments having a failure probability smaller than 0-4, will be within the ineffective length, The determination of the failure probability can be refined by considering the axial stress profile in a

I

2 3

Mw

Zd

kt

5 5

150

1 0,2

oA.

00,.

segment as non-uniform (either as linear or hyperbolic--Cox's loading mode--variation). Such an analysis with varying stress in the modelled volume was developed by Oh and Finnie 55 and was applied to the prediction of debonded lengths in ceramic matrix compositesY '56 Calculation of the failure probability can be carried out in the first case (i.e. linear variation of the axial stress) and gives similar expressions for the probability to those presented by Thouless and Evans. 56 But in the case of hyperbolic load transfer,

[

I I.

1

2

,

I

3 _

i

0.001:

0.0001

O' ' O A ' '0,2' '0.3' '0.4' ~0.5" '0.6' '0.7' '0.8' '0.9' '1 ' 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

Failure probability Fig. 12. Normalized frequency distribution of the failure probability of the fragment.

Theoretical aspects of the fragmentation test the analytical solution is not obvious and the problem must be solved numerically. 57 The main point of this section is that the ineffective length needs to be exactly defined and can absolutely not be taken as the critical fragment length. The calculation of the in-situ Weibull parameters for the fibre requires the probabilistic definition of the ineffective length. Some authors have found a decrease in the Weibuil parameters 58 and others did not observe any changes. 59 Gulino and Phoenix 54 showed that in the case of a model microcomposite the in situ Weibull parameters of the fibre cannot b e determined. Much work has to be done in order to define the ineffective length accurately so that, within a small error, any fibre break occurs within this length,

4 CONCLUSION The present state of knowledge about the fragmentation process has been reviewed. The Cox one-dimensional load-transfer model gives quite good approximations to the stress state remote from the fibre break but one-dimensional models are too simple to give information on the multiaxial stress state around the site of fibre rupture. Moreover, they are not able to take into account the non-linear mechanical behaviour of the matrix. Bi-dimensional load-transfer models are more appropriate and exhibit good agreement with the stress state obtained from finite element analysis. However, interfacial debonding cannot be introduced at present in these models owing to the lack of a multiaxial debonding criterion, From comparison between experiment and cornputation, and from optical examination, the matrix mechanical behaviour cannot even be considered to be linear elastic. The microsopic and macroscopic mechanical levels are not comparable. A thorough investigation of the local mechanical behaviour of the matrix appears essential for a better understanding of the micromechanics. Some current misunderstandings in the analysis of the fragmentation test have been outlined. It has been demonstrated here that the critical fragment length depends both on the Weibull modulus and on the intrinsic parameters of the interface ( r d , # ) Consequently, the common relationship between i and lc is not valid and is unknown at present. A new method is envisaged to determine this critical fragment length. In the same way, the frictional shear stress along the debonded fragment length cannot be taken as constant and strain-independent. Such an assumption leads to erroneous conclusions about the fragmentation process. Finally the ineffective length, li, has to be defined by using a probabilistic criterion.

277

li must be known to calculate the in-situ Weibull parameters of the fibre and it is often taken erroneously as the critical fragment length. This paper shows that the analysis of the fragmentation test is sometimes not rigorous. Many mechanical events appear simultaneously and interact with each other during the test, so that simplistic assumptions are not valid. Although the micromechanical features of the fragmentation test are globally understood, some experimental facts cannot be taken into account in the present models of the test. Obviously, this test is not the only tool for studying the micromechanics. The authors feel that some points (debonding criterion, friction) can be analysed intensively with the pull-out test. In contrast, the fragmentation test seems well adapted for the study of matrix plasticization and load-transfer models. For 15 years, the fragmentation test has been employed successfully to qualify interfacial adhesion. Especially, it can be used at present as a powerful tool for the study of the basic damage modes that are present in multifibre composites.

ACKNOWLEDGEMENTS The authors would like to thank Mrs Dominique Frugier and Messrs Pascal Chartier and Eric Dallies from Saint-Gobain Recherche for their scientific, technical and financial support.

REFERENCES 1. Kelly, A. & Tyson, W. R., Tensile properties of fibre-reinforced metals: copper/tungsten and copper/molybdenum. J. Mechanics and Physics in Solids, 13 (1965)329-50. 2. Fraser, W. A., Ancker, F. H. & DiBenedetto, A. T., A computer modeled single filament technique for measuring coupling and sizing agents effects in fibre reinforced composites. In Proceedings of the 30th Annual Technical Conference, Reinforced Plastics/Composites Institute, The Society of the Plastics Industry, WA, 1975, Section 22-A, pp. 1-13. 3. Oshawa, T., Nakyama, A., Miwa, M. & Hasegawa, A., Temperature dependence of the critical fibre length for the glass fibre-reinforced thermosetting resins. J. Applied Polymer Science, 22 (1978) 3203-12. 4. Lacroix, T., Tiimans, B., Keunings, R., Desaeger, M. & Verpoest, I., Modeling of critical fibre length and interracial debonding in the fragmentation test of polymer composites. Composites Science and Technology, 43 (1992) 379-87. 5. Feillard, P., D6sarmot, G. & Favre, J. P., A critical assessment of the fragmentation test with glass/epoxy systems. Comp. Sci. Technol., 49 (1993) 109-19. 6. Figueroa, J. C., Carney, T. E., Schadler, L. S. & Laird, C., Micromechanics of single filament composites. Comp. Sci. Technol., 42 (1991)77-101. 7. Cox, H. L., The elasticity and strength of paper and other fibrous materials. British Journal of Applied Physics, 3 (1952) 72-9.

278

P. Feillard, G. Ddsarmot, J. P. Favre

8. Nairn, J., A variational mechanics analysis of the stresses around breaks in embedded fibres. Mechanics of Materials, 13 (1992) 131-57. 9. Jacques, D., Transfert de charge entre fibre et matrice dans les composites carbone-r6sine. Comportement en traction d'un composite module monofilamentaire. PhD Thesis, Institut Polytechnique de Lorraine, Nancy, France, 1989. 10. Favre, J.-P., Sigety, P. & Jacques, D., Stress transfer by shear in carbon fibre model composites Part 2: Computer simulation of the fragmentation test. J. Materials Science, 26 (1991)189-95. 11. Henstenburg, R. B. & Phoenix, S. L., lnterfacial shear strength studies using single filament composite test Part 2: A probabilisty model and Monte-Carlo simulation, Polymer Composites, 10 (5) (1989) 389-408. 12. Curtin, W., Exact theory of fibre fragmentation in a single fibre composite. J. Materials Science, 26 (1991) 5239-53. 13. Fraser, W. A., Ancker, F. H., DiBenedetto, A. T. & Elbirli, B., Evaluation of surface treatments for fibers in composite materials. Polymer Composites, 4 (4) (1983) 238-48. 14. Galiotis, C., Robinson, M., Young, R. J., Smith, B . J . E. & Batchelder, D. N., The study of model polydiacetylene/epoxy composites. J. Materials Science, 19 (1984)3640-8. 15. Galiotis, C., Interfacial studies on model composites by laser raman spectroscopy. Composites Science and Technology, 42 (1991)125-30. 16. Sabat, P. J., Dwight, D. W. & Brinson, H . F . , Evaluation of fibre-matrix interfacial shear strength in fibre reinforced plastics. Virginia Tech. Center for Adhesion Science Report VPI/CAS/ESM-86-1, Virginia Polytechnic Institute and State University, March 1986. 17. Lhotellier, F. C. & Ward, T. C., Matrix-fibre stress transfer in composite materials elasto-plastic model with an interphase layer. Virginia Tech. Center for Adhesion Science Report VPI-E-87-27/CAS/ESM-87-12, Virginia Polytechnic Institute and State University, December 1987. 18. Lhotellier, F. C. & Brinson, H. F., Matrix-fiber stress transfer in composite materials: elasto-plastic model with an interphase layer. Composite Structures, 10 (1988) 281-301. 19. Williams, J. G., Donellan, M. E., James, M. R. & Morris, W. L., Properties of the interphase in organic matrix composites. Materials Science and Engineering, A126(1990) 305-12. 20. Tsai, H. C., Arocho, A. & Gause, L., Prediction of fibre/matrix interphase properties and their influence on interface stress, displacement and matrix toughness of composites materials. Materials Science and Engineering, A126 (1990)295-304. 21. DiBenedetto, A. T. & Lex, P. J., Evaluation of surface treatments for glass fibres in composites materials. Polymer Engineering and Science, 29 (8) (1989) 543-55. 22. Dow, N. F., Study of stresses near a discontinuity in a filament-reinforced composite metal. Technical report, no. R63SD61, Space Sciences Laboratory, Missile and Space Division, General Electric Co., 1963. 23. Tyson, W. & Davies, G., A photoelastic study of the shear stresses associated with the transfer of stress during fibre reinforced metals. British Journal of Applied Physics, 16 (1965) 199-205. 24. Ochiai, S. & Osamura, K., Stress distribution of a

25.

26. 27. 28. 29. 30. 31.

32. 33. 34.

35.

36. 37. 38.

39. 40.

41.

segmented fibre in loaded single fibre metal matrix composites. Z. Memllkunde, 77 (1986)249-54. Whitney, J. M. & Drzal, L. T., Axisymmetric stress distribution around an isolated fibre fragment. Toughnened composites ASTM STP 937, (ed. N. J. Johnston). American Society for Testing Materials, Philadelphia, 1987, pp. 179-96. Allison, I. & Hollaway, L., Stresses in fibre-reinforced materials. British Journal of Applied Physics, 18 (1967) 979-89. Iremonger, M. J. & Wood, W. G., Effects of geometry on stresses in discontinuous composite materials. J. Strain Analysis, 4 (2) (1969) 121-4. MacLaughlin, T. F. & Barker, R. M., Effect of modulus ratio on stress near a discontinuous fiber. Experimental Mechanics, (April 1972) 178-83. Carrara, A. S. & MacGarry, F. J., Matrix and interface stresses in a discontinuous fibre composite model. J. Composite Materials, 2 (1968) 222-43. Ko, W. L., Finite element microscopic stress analysis of cracked composite systems. J. Composites Materials. 12 (1978) 97-115. Kallas, M. N., Koss, D. A., Hahn, H. T. & Hellmann, J . R . , Interfacial stress state present in a 'thin-slice' fibre push-out test. J. Materials Science. 27 (1992) 3821-6. Evans, A. G. & Marshall, D. B., The mechanical behaviour of ceramic matrix composites. Acta Metallurgica et Materialia, 37 (10) (1989) 2567-83. Steif, P. S. & Dollar, A., Models of fibre-matrix interfacial debonding. J. American Ceramic Society, 75 (6) (1992) 1694-6. Feillard, P., D6sarmot, G. & Favre, J. P., Analyse critique du test de fragmentation appliqu6 aux syst~mes verre/6poxy. Comptes-rendus des 86mes Journ6es Nationales sur les Composites (JNC-8), Ecole Polytechnique, Palaiseau-France, 16-18 November 1992. Publi6s par O. Allix, J. P. Favre & P. Ladev~ze. Edition AMAC, Paris, 1992. DiAnselmo, A., Accorsi, M. L. & DiBenedetto, A. T., The effect of an interphase on the stress and the energy distribution in the embedded single fibre test. Composites Science and Technology, 44 (1992) 215-25. D6sarmot, G. & Favre, J. P., Advances in pull-out testing and data analysis. Composites Science and Technology, 42 (1991) 151-87. Bascom, W. D. & Jensen, R. M., Stress transfer in single fibre/resin tensile tests. J. Adhesion, 19 (1986) 219-39. Marmonier, M. F., D6sarmot, G., Barbier, B. & Letalenet, J. M., A study of the pull-out test by a finite element method. J. Mdcanique Thdorique & Appliqude, 7 (6)(1988)741-65. Glad, M. D., Microdeformation and network structure in epoxies. PhD Thesis, Cornell University, Ithaca, June 1986. Gulino, R., Schwartz, P. & Phoenix, S. L., Experiments on shear deformation, debonding and local load transfer in a model graphite/glass/epoxy microcomposite. J. Materials Science, 26 (1991) 6655-72. Piggott, M. R., Sanadi, A., Chua, P. S. & Andison, D., Mechanical interactions in the interphasial region of fibre reinforced thermosets. In Proceedings of the First

International Conference of Composites and Interfaces (ICCI-I) (ed. H. Ishida & J. L. Koenig). North-Holland, New York, May 1986. 42. El Asloun, M., Donnet, J. B., Guilpain, G., Nardin,

Theoretical aspects of the fragmentation test

43.

44.

45. 46.

47.

48.

49.

50.

51. 52.

M. & Schultz, J., On the estimation of the tensile strength of carbon fibres at short lengths. J. Materials Science, 24 (1989) 3504-10. Drzal, L. T., Rich, M. J., Camping, J. D. & Park, W. J., Interfacial shear strength and failure mechanisms in graphite fibre composites. In Proceedings of the 35th Annual Technical Conference, 1980. Reinforced Plastics/Composites Institute, The Society of the Plastics Industry, New York, Section 20-c, pp. 1-7. Favre, J.-P. & Jacques, D., Stress transfer by shear in carbon fibre model composites Part 1: Results of single-fibre fragmentation tests with thermosetting resins. J. Materials Science, 25 (1990) 1373-80. Dollar, A. & Steif, P. S., Load transfer in composites with a coulomb friction interface, lnternational J. S o l i d s and Structures, 24 (8) (1988) 789-803. Rosen, B. W., Strength of uniaxial fibrous composites. Mechanics of composites materials (ed. F. W. Wendt, H. Liebowitz & N. Perrone). Proceeding of the 5th Symposium in Naval Structural Mechanics, Pergamon Press, New York, 1967, pp. 621-50. DiLandro, L., DiBenedetto, A. T. & Groeger, J., The effect of fibre/matrix stress transfer on the strength of fibre-reinforced composite materials. Polymer Composites, 9 (3) (1988) 209-21. Wagner, H. D. & Eitan, A., Interpretation of the fragmentation phenomenon in single-filament composite experiments. Applied Physical Letters, 56 (20) (1990) 1965-7. Yavin, B., Gallis, H. E., Scherf, J., Eitan, A. & Wagner, H. D., Continuous monitoring of the fragmentation phenomenon in single fiber composite materials. Polymer Composites, 12 (6) (1991) 436-46. El Asloun, M., Nardin, M. & Schultz, J., Stress transfer ,n composites: e.ec, a es on e,as, c modulus of the fibre and matrix, and polymer chain mobility. J. Materials Science, 24 (1989) 1835-44. Baxevanakis, C., Jeulin, D. & Valentin, D., Fracture statistics of single fibre composites specimens. Comp. Sci. Technol., 48 (1993) 47-56. Monette, L., Anderson, M. P., Ling, S. & Grest, G . S . , Effect of modulus and cohesive energy on critical fibre

n i reeorce om ose

Science, 2"/(1992) 4393-405. 53. Ling, S. & Wagner, H. D., Relationship between fiber flaw spectra and fragmentation process: a computer simulation investigation. Comp. Sci. Technol., 48 (1993) 35-46. 54. Gulino, R. & Phoenix, S. L., Weibull strength statistics for graphite fibres measured from the break progression

55. 56. 57.

58.

59.

279

in a model graphite/glass/epoxy microcomposite. J. Materials Science, 26 (1991) 3107-18. Oh, H. L. & Finnie, I., On the location of fracture in brittle solids-I: Due to static loading, International J. Fracture Mechanics, 6 (3) (1970) 287-300. Thouless, M. D. & Evans, A. G., Effects of pull-out on the mechanical properties of ceramic-matrix composites. Acta Metallurgica, 36 (3) (1988) 517-22. Rouby, D., Navarre, G. & R'Mili, M., Mecanismes d'extraction de fibres dans les composites a matrices c6ramiques. Mod61isation th6orique et analyse exp6rimentale. In Proceedings of Mat~riau.r Composites pour Applications ~ Hautes Tempdratures. (ed. R. Naslain, J. Lamalle, J. L. Zulian). AMAC/ CODEMAC. Bordeaux, France, March 1990, pp. 69-80. Bader, M. & Baillie, C., Determination of the strength of single carbon fibres embedded in resin. In Fifth European Conference on Composites Materials,ECCM-5 (ed. A. D. Bunsell, J. F. Jamet & A. Massiah). EACM, Bordeaux, France, April 1992, pp. 339-44. Waterbury, M. C. & Drzal, L. T., On the determination of fiber strengths by in situ fiber strength testing. J. Composites Technology & Research, 13 (1) (1991) 22-8.

APPENDIX To determine the validity of the 4 ~ m segment size choice, the fibre axial stress under which N fibre ruptures are obtained is calculated. From the Weibull statistical model, this becomes: Pr = 1 - exp \ oo /

_1

Let us define Nbseg as the total n u m b e r of segments and Nbe~cm the total n u m b e r of elements. The stress level is:

o0(n(

1

Nbclem//

The present case is for an E glass fibre and we have taken: Mw = 28; oo = 3-1 G P a for a 10-ram gauge length;Nbd~m = 100; and fibre length = 30 mm.