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Stress distribution along broken fibres in polymer-matrix composites
Composites Science and Technology 61 (2001) 1571–1580 www.elsevier.com/locate/compscitech
Stress distribution along broken fibres in polymer-matrix composites Alfredo Balaco´ de Morais* University of Aveiro, Mechanical Engineering Department, Campus Santiago, 3810 Aveiro, Portugal Received 26 April 2000; received in revised form 21 March 2001; accepted 19 April 2001
Abstract This paper presents a model for predicting the stress distribution along a broken fibre in a unidirectional composite. It is assumed that the matrix behaves in an elastic/perfectly-plastic manner, and that the interfacial shear strength is not lower than the matrix shear yield stress. Although ‘good’ interface bonding is assumed, the present analysis suggests that interface debonding may occur as a result of local matrix shear failure, resulting in the well-known splitting phenomenon observed in tensile tests. Along the debonded length, a decreasing interfacial shear stress is derived from Poisson contractions and Coulomb friction. The debond is followed by a matrix yielding zone, where the interfacial shear stress is assumed to be equal to the matrix shear yield stress. There is, finally, an elastic zone, where the interfacial shear stress follows a classical exponential law. The present model is in good agreement with a 3D finite-element model. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; B. Modelling; C. Stress transfer; C. Finite-element analysis; Ineffective length
1. Introduction The matrix-to-fibre stress transfer is an issue of great relevance for continuous fibre composites, as it affects the layer longitudinal tensile strength. The brittle nature of the reinforcing fibres results in a high degree of strength variability and size-dependence. During tensile loading of a unidirectional layer, a number of fibre failures will occur prior to overall collapse. The so-called ‘ineffective length’ i.e. the axial distance over which a broken fibre recovers its undisturbed remote stress value, plays a major role in the final failure stress [1–6]. In fact, strength prediction models are usually based on the assemblage of various single-fibre elements whose size equals the ineffective length, and often resort to elaborate Monte-Carlo simulation schemes [4–6]. The stress concentration in the fibres close to the broken fibre has been a major concern. Various analytical and numerical models [7–10] have been specifically devoted to this phenomenon. The 3D finiteelement modelling of a hexagonal fibre array by Nedele and Wisnom [8] showed, however, quite modest stressconcentration factors. This emphasises the need to focus on the stress transfer to the broken fibre so that the ineffective length can be accurately defined. * Tel.: +351-234-370830; fax: +351-234-370953. E-mail address: [email protected]
The early ‘shear-lag’ analysis by Cox [11] assumed matrix elasticity and perfect interface bonding. Those assumptions are clearly unrealistic for actual composites i.e. localised matrix cracking and yielding are inevitable if perfect bonding exists. Landis et al [12] have recently developed a numerical model to predict the stress transfer to a broken fibre surrounded by a yielding matrix. Other authors [2,3,13] have considered interface debonding the major micromechanical failure mechanism. Nevertheless, recent work on single-fibre and array-of-fibres specimens seems to show that localised matrix yielding is much more relevant than interface debonding [14–16]. In fragmentation testing of specimens prepared with untreated/unsized fibres, leading to poor interface properties, considerable debonds are observed [14]. However, when normally treated/sized fibres are used, interface debonds can be quite small, and instead, matrix cracking occurs [14]. In many studies involving single-fibre testing, the well-known Kelly–Tyson model [17] is used to compute the interfacial shear strength, whose value often correlates with the matrix shear yield stress [14– 16]. In a study involving specimens with 5 aligned fibres, finite element model predictions were compared with local stress measurements using Raman spectroscopy [14–16]. The results confirmed that matrix yielding was the fundamental stress transfer mechanism, while no significant interface debonding was observed.
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00058-6
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Nomenclature
G23
c df f m r
Gm H1i
rf rm tm u w wb wo z zd zp C D E1 E2 Ef Em G12
subscript used denote the composite fibre diameter subscript used denote the fibre subscript used denote the matrix radial coordinate of a cylindrical coordinate system fibre radius outer radius of the matrix cylinder thickness of the matrix cylinder radial displacement in a cylindrical coordinate system axial displacement in a cylindrical coordinate system axial displacement of the broken fibre axial displacement of the fibres closest to the broken one axial coordinate of a cylindrical coordinate system axial coordinate of the end of the debonded length axial coordinate of the end of the matrix yielding zone integration constant integration constant longitudinal modulus of a transversely isotropic solid transverse modulus of a transversely isotropic solid longitudinal fibre modulus matrix Young’s modulus longitudinal shear modulus of a transversely isotropic solid
It seems, therefore, that, in normal ‘well-bonded’ interfaces, the interfacial shear strength is not lower than the matrix shear yield stress. The longitudinal splitting sometimes observed in tensile tests suggests, however, that interface debonding could be a relevant failure mechanism. The stress transfer model presented below suggests that interface debonding may result from local matrix shear failure.
2. Model description 2.1. Preliminary qualitative analysis Under tensile loading, both parts of a broken fibre move away from each other. Therefore, matrix cracking, partial interface debonding, or perhaps a combination of both must obviously occur. Assuming good interface
H2i r m um 12 23 m r z f fd fo rz i ym
transverse shear modulus of a transversely isotropic solid matrix shear modulus function used in the equation for the interfacial shear stress function used in the equation for the interfacial shear stress function used in the equation for the axial fibre stress parameter used in the equation for the axial fibre stress normal radial strain normal circumferential strain matrix shear strain matrix ultimate shear strain interfacial friction coefficient longitudinal Poisson’s ratio of a transversely isotropic solid transverse Poisson’s ratio of a transversely isotropic solid matrix Poisson’s ratio circumferential coordinate of a cylindrical coordinate system normal radial stress normal circumferential stress normal axial stress normal axial fibre stress normal axial fibre stress at the end of the debonded length remote axial fibre stress axisymmetric shear stress interfacial shear stress matrix shear yield stress
bonding and since before fibre failure the matrix is under significant tensile loads, it is logical to expect that matrix cracking is the first local failure mechanism. The axial displacements of the broken fibre segments promote matrix shear strains high enough to cause yielding. In contrast with the single-fibre or array-of-fibres composite specimens, in actual unidirectional composites, there is a narrow matrix layer between the broken and the stiff neighbouring fibres that will be subjected to high shear strains. Brittle matrices may undergo local shear failure, giving rise to significant interface debonding. We will, therefore, consider the most general case, where the fibre break resulted in matrix cracking and in partial interface debonding (Fig. 1). The matrix crack is arrested somewhere between the broken and the surrounding fibres. The exact position of the crack front and the stress state in its vicinity are a complex fracture mechanics problem. The finite element model presented in 3. shows that the
A. Balaco´ de Morais / Composites Science and Technology 61 (2001) 1571–1580
actual size of the matrix crack does not influence the stress transfer to the broken fibre. A concentric cylinder model is used for analysis purposes (Fig. 1). The outer radius rm of the matrix layer is assumed tangent to the nearest neighbour fibres of the most representative hexagonal packing arrangement. The surrounding composite is treated as a hollow cylinder of very large outer radius. Axisymmetric stress analysis will be performed, neglecting fibre shear stresses. Force equilibrium of a fibre element in the longitudinal direction is then expressed by dz;f 4 i ¼ dz df
ð1Þ
where z;f is the axial fibre stress, df is the fibre diameter and i the interfacial shear stress. An elastic-perfectly plastic stress–strain behaviour is assumed for the matrix. It is also assumed that the matrix is in the elastic range at the instant of fibre fracture.
Force equilibrium in the radial direction demands that, in each cylinder [18], ð2Þ
where r and denote the radial and circumferential directions, respectively. We assume a small variation of rz along the debonded length, so that r þ r
@r @r
ð3Þ
Letting u be the radial displacement, the relevant kinematic equations are u " ¼ r
@u @" ¼r þ " @r @r
ð5Þ
Using the more common numerical indices for the elastic properties of a transversely isotropic solid, the needed constitutive equation can be written as " ¼
1 12 ð 23 r Þ z E2 E1
ð6Þ
Inserting (3) into (6), and the resulting equation into (5), we obtain the differential equation r2
@2 ðrr Þ @ðrr Þ rr ¼ 0 þr @r2 @r
ð7Þ
where it has been assumed that the longitudinal stress z does not vary in the radial direction. The solution of (7) is r ¼ C þ
2.2. Stress transfer along the debonded length
@r r @ rz þ þ ¼0 @r r @z
"r ¼
1573
D r2
ð8Þ
where C and D are integration constants that must be determined from the boundary conditions. The circumferential stress (3) can then be expressed as D r2
¼ C
ð9Þ
We now analyse the cylinders representing the broken fibre, the surrounding matrix and the composite (Fig. 1). The values of the integration constants C and D differ for each case, and are, therefore, denoted with the corresponding cylinder index. In the fibre, the radial stress (8) must remain finite at r ¼ 0. We have thus Df ¼ 0 and constant radial and circumferential stresses i.e. r;f ðrf Þ ¼ ;f ðrf Þ ¼ Cf
ð10Þ
ð4Þ and from (6), ";f ðrf Þ ¼
12;f r;f ðrf Þ 1 23;f z;f E2;f E1;f
ð11Þ
As to the composite, at r ! 1 we must have r ! 0, since no external transverse loads are being applied. Using (8), (9) and (6),
r;c ðrm Þ ¼
Cc r2m
;c ðrm Þ ¼ r;c ðrm Þ ";c ðrm Þ ¼ Fig. 1. Scheme of the developed model.
12;c r;c ðrm Þ 1 þ 23;c z;c E2;c E1;c
ð12Þ ð13Þ ð14Þ
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We further assume a constant axial stress z;c E1;c
fo E1;f
We can now insert the interface shear stress (24) into (1), to obtain the differential equation ð15Þ
where fo is the remote longitudinal fibre stress. Finally, in the matrix cylinder we have r;m ¼ Cm þ
Dm r2
ð16Þ
Dm ;m ¼ Cm 2 ð17Þ r 1 Dm ";m ðrÞ ¼ Cm ð1 m Þ 2 ð1 þ m Þ m z;m Em r ð18Þ
df 4H2i 4H1i þ f ¼ dz df df
ð25Þ
where we have dropped the z-index, unnecessary at this stage. The solution is f ¼
H1i ð1 ez Þ H2i
ð26Þ
with ¼
4H2i df
ð27Þ
where, again, we assume a constant axial stress z;m Em
fo E1;f
ð19Þ
At the interfaces between the cylinders, the radial stresses must be continuous i.e. r;f ðrf Þ ¼ r;m ðrf Þ
ð20Þ
r;m ðrm Þ ¼ r;c ðrm Þ
ð21Þ
The interfacial radial displacements, and consequently the circumferential strains, must also be continuous. At the fibre/matrix interface, we can use Eqs. (11) and (18)– (21), to write ";f ðrf Þ ¼ ";m ðrf Þ as " # 1 m 1 23;f 1 þ m 1 23;f þ Cm Dm Em E2;f Em r2f E2;f r2f m fo 12;f z;f ¼ ð22Þ E1;f At the matrix/composite interface, Eqs. (14), (15) and (18)–(21), allow us to express ";m ðrm Þ ¼ ";c ðrm Þ as 1 m 1 þ 23;c 1 þ m 1 þ 23;c þ Cm Dm Em E2;c Em r2m E2;c r2m fo ¼ m 12;c ð23Þ E1;f Eqs. (22) and (23) can be easily solved for the constants Cm and Dm , which enable afterwards the determination of the fibre/matrix interfacial radial stress r;m ðrf Þ from (16). Finally, assuming Coulomb friction, the interfacial shear stress can be expressed as i ¼ r;m ðrf Þ ¼ H1i H2i z;f
Eq. (26) will apply from the broken fibre end (z ¼ 0) to some debonded length zd , which is defined in 2.4. 2.3. Stress transfer along the bonded length A matrix yielding zone immediately follows the interface debond. Obviously, the matrix is under a relatively complex stress state, including axial z , radial r and shear r;z stresses. An accurate analysis would result in a complex plasticity problem requiring numerical solution. We assume that, due to the much higher longitudinal fibre modulus, shear straining prevails in the matrix layer. Moreover, as in the simple Kelly–Tyson model [17], we consider matrix perfect plasticity and let the interfacial shear stress in the plastic zone be equal to the matrix shear yield stress ym . Although this assumption is clearly not valid near the end of the matrix plasticity zone, it is shown in (3) that it does not affect the accuracy of the predicted broken fibre stress distributions. The solution of (1) is then f ¼ fd þ
4 ym ð z zd Þ df
ð28Þ
valid for zd 4z4zp , fd being the fibre stress at the end of the debond. Finally, in the matrix elasticity zone, we assume that the matrix shear strain m is approximately constant in the radial direction. Neglecting radial displacements, m
@w wb wo @r rm rf
ð29Þ
ð24Þ
where H1i and H2i are functions of the friction coefficient , of the remote fibre stress fo and of the elastic moduli of the fibre, matrix and composite.
where wb and wo are the axial displacements of the broken and unbroken nearest neighbour fibres, respectively. Inserting (29) and Hooke’s law for the shear stress into (1), we obtain, after an additional derivation,
A. Balaco´ de Morais / Composites Science and Technology 61 (2001) 1571–1580
d2 f 4Gm dwb dwo ¼ dz2 df tm dz dz
ð30Þ
where tm ¼ rm rf is the thickness of the matrix cylinder. Since dwb f ¼ dz Ef
ð31Þ
and neglecting the stress concentration on the nearest neighbour fibre, dwo fo ¼ dz Ef
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where um is the matrix ultimate shear strain. Neglecting stress concentration, the axial displacement in the neighbouring fibres is wo
fo z Ef
ð39Þ
where it has been established that wo ðz ¼ 0Þ ¼ 0. The displacement of the broken fibre is, generally, ð f wb ¼ dx þ Cb ð40Þ Ef
Eq. (30) can be easily solved, giving the Cox [11] type equation
where the appropriate equation for f must be inserted, followed by the determination of the integration constant Cb . Along the matrix elasticity length, f is given by (36), and since we must have wb ! wo when z ! 1, then Cb ¼ 0 and the longitudinal displacement becomes
f ¼ fo þ Ce ez
wb ¼
ð32Þ
ð33Þ
fo 4 ym ðzp zÞ zþ e Ef Ef df 2
ð41Þ
with rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Gm ¼ df tm Ef
ð34Þ
The integration constant Ce can be evaluated from the condition that the interfacial shear stress, obtained by inserting (33) into (1), is equal to ym at z ¼ zp . We have finally i ¼ ym eðzp zÞ f ¼ fo
4 ym ðzp zÞ e df
ð42Þ
ð36Þ
wb ¼
ð37Þ
Having assumed that the interface shear stress is equal to the matrix shear yield stress, the interface debonding criterion is based on the maximum matrix shear strain. At the end of the interface debond, the matrix is assumed to be at the onset of shear failure, so that any subsequent increase in the remote stress results in debonding initiation or further growth. In view of (29), we have wb ðzp Þ wo ðzp Þ ¼ um tm
2 ym 2 4 ym fd z zd z þ z þ Cb E f df Ef df Ef
ð35Þ
The fibre stress must obviously be continuous at z ¼ zp . Using (28) and (36), we obtain 1 df þ ðfo fd Þ 4 ym
wb ¼
At zp , Eqs. (41) and (42) must give the same displacement. This continuity condition enables the determination of the integration constant Cb . In the matrix plasticity zone, we, therefore, have
2.4. The lengths of the interface debonding and yielding zones
zp ¼ zd
In the matrix plasticity zone, after inserting (28) into (40), we obtain
ð38Þ
2 ym 2 4 ym fd ðz z2p Þ zd ðz zp Þ þ ðz zp Þ E f df E f df Ef fo 4 ym zp þ ð43Þ þ Ef Ef df 2
We can now insert Eqs. (39) and (43) into Eq. (38) to obtain 2 ym 2 4 ym fo fd 2 ðz zp Þ zd þ ðzd zp Þ Ef df d Ef df Ef 4 ym þ ¼ um tm ð44Þ E f df 2 In view of the relation between zp and zd given by (37), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 ym 2 ym fd ¼ fo 2 Ef um tm ð45Þ df df 2 and, using (26), one finally obtains 1 H2i zd ¼ ln 1 fd H1i
ð46Þ
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3. Model evaluation The developed model was compared with a 3D finite element (FE) model implemented in the commercial code ABAQUS [19]. The model assumes hexagonal fibre packing (Fig. 1) and, in view of the symmetries, only a twelfth-section is considered (Fig. 2). In the inner zone, around the broken fibre, both constituents are modelled separately, while farther away the composite is modelled as a transversely isotropic homogeneous solid. Sufficiently large model dimensions were used to ensure representativity. The model was built with 6-node triangular prism and 8-node reduced integration brick elements (Fig. 3). Very fine meshes were used: the total number of elements was around 16,000. The tensile loading is applied by imposing a uniform axial displacement on the model upper face, while the lower z ¼ 0 face is a symmetry plane. In the cracked areas, however, no restrictions are imposed on the axial displacements of the corresponding nodes. Fig. 2 shows the contours of the two matrix cracks that were considered in the z ¼ 0 plane. Actually, the stress distribution along the broken fibre proved to be independent of the matrix crack pattern. Comparison with the developed model is shown here for the following set of properties. The fibre is assumed transversely isotropic with E1f =230 GPa, E1f =20 GPa, 12f =0.20, 23f =0.25 and G12f =20 GPa. For the isotropic matrix, we have used Em =3.5 GPa, m =0.35, and perfect plasticity with von Mises yield stress ym =70
Fig. 2. Schematic representation of the 3D FE model.
Fig. 3. Example of a FE mesh.
MPa. The elastic constants of the 60% fibre volume fraction composite were determined from finite element models presented in [20], giving E1c =139.4 GPa, E2c = 9.0 GPa, 12c =0.254, 23c =0.403 and G12c =4.2 GPa. When debonding was considered, a ¼ 0:5 Coulomb friction coefficient was used. One of the most relevant assumptions implicit in the model is the prevalence of shear straining in the yielding zone of the matrix layer around the broken fibre, and subsequent adoption of a constant interfacial shear stress equal to matrix shear yield stress. Normal stresses, which inevitably occur, are thereby neglected in the yield condition. A major implication of this assumption is that previous deformation history effects, which are in principle relevant in plasticity problems, are also neglected. In order to test the validity of this assumption, we have considered two loading histories. In the first case, the failure process was simulated by releasing the constraints on the nodes of the assumed cracked area after some remote stress value was achieved. In the second case, the same crack was assumed to exist from the initial unloaded state. Except for insignificant differences near the crack, the results were exactly the same. Figs. 4 and 5 show the interfacial shear stress distributions at 1% and 1.7% remote strain, respectively. The predictions of the
Fig. 4. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1% remote strain. Perfect interface bonding was assumed.
Fig. 5. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1.7% remote strain. Perfect interface bonding was assumed.
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developed model, given by i ¼ ym and by (35), are compared with FE results (the shear stress was averaged along the interfacial circumference). Due to numerical singularities at the matrix crack, the interfacial shear stress given by the FE analysis is locally much higher than the matrix shear yield stress. Farther away from the matrix crack, the FE interfacial shear stress slightly exceeds the matrix shear yield stress. The error is quite small (less than 2%) and is due to inevitable numerical disturbances associated with the master-slave surface contact modelling [19]. Although debonding was not considered in this case, it is important to define the interface as contacting fibre and matrix surfaces, since the common displacement interpolation based 3D finite elements do not ensure the shear stress continuity at the interfaces between different materials. As expected, the analytical model overestimates the interfacial shear stress in the last portion of the matrix yielding zone, because it ignores the normal stress in the matrix yielding criterion. Apparently, this discrepancy could result in overestimating the stress in the broken fibre, especially at high remote strain levels. However, the broken fibre stress distributions of Figs. 6 and 7 clearly show that the analytical model [Eqs. (28) and (36)] agrees quite well with the FE results. In addition, the analytical model actually underestimates the stress transfer to the broken fibre at 1.7%. This can be easily explained by
the larger mismatch of Poisson transverse effects between matrix and fibre in the yielding zone. Unfortunately, due to numerical singularities, the FE model gives unreliable overpredictions for the maximum matrix shear strain. This limitation reinforces the usefulness of the developed analysis, although it prevents its full validation. In the considered example, the maximum matrix shear strains predicted by the developed model are 32.5 and 91% at 1 and 1.7% remote strain, respectively. These high strain values suggest that interface debonding may be due to local matrix shear failure. This is consistent with the well-known splitting phenomenon usually observed in tensile testing of unidirectional brittle matrix composites [1]. The difficulty that immediately arises is the determination of the in situ matrix ultimate shear strain, since it is obviously highly questionable to apply bulk resin data to the very thin matrix layer around the broken fibre. For example, Chai [21] has shown that a brittle epoxy with an ultimate shear strain of about 10%, when in the form of a 25 mm adhesive layer, may have an ultimate shear strain more than 10 times larger. Bradley [22] has also shown that the shear strain at the tip of a crack in a composite may greatly exceed the bulk matrix ultimate shear strain. Clearly, this subject still requires considerable research. In the next examples, 240 mm interface debonds were assumed. Figs. 8 and 9 show the interfacial shear stress
Fig. 6. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1% remote strain. Perfect interface bonding was assumed.
Fig. 8. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1% remote strain. A 240 mm interface debond was assumed.
Fig. 7. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1.7% remote strain. Perfect interface bonding was assumed.
Fig. 9. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1.7% remote strain. A 240 mm interface debond was assumed.
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We can, therefore, conclude that the developed model is quite accurate at predicting the stress distribution along the broken fibre. The limitations of the FE analysis, however, do not enable the validation of the maximum matrix shear strain based interface debonding criterion. Further work should include the developed stress-transfer analysis in a longitudinal tensile strength prediction model, where it may provide the essential values of the ineffective length throughout the loading
distributions at 1 and 1.7% remote strain, respectively. The analysis [Eqs. (24), i ¼ ym and (35)] is in good agreement with the FE results, apart from the numerical singularity at the end of the debond and from the slight overestimation of the interfacial shear stress at the end of the matrix yielding length. The stress distributions along the broken fibre, depicted in Figs. 10 and 11, confirm the good performance of the developed model [Eqs. (26), (28) and (36)]. Another issue that has received considerable attention is the stress concentration in the fibres around the broken one. Figs. 12–14 show the stress concentration in the centre of the nearest neighbour fibre for the cases considered above. The present results do not differ significantly from those of Nedele and Wisnom [8] for matrix elasticity and perfect interface bonding. The highest stress concentration is relatively small, and it is only relevant along a relatively short length. If we define the ‘ineffective length’ as the length over which the broken fibre recovers 90 or 95% of the remote fibre stress (Table 1), the average stress in the nearest neighbour fibre is practically equal to the remote stress. Therefore, the role of stress concentration in the neighbour fibres seems to be practically negligible.
Fig. 12. Undimensional fibre stress f =fo (remote fibre stress) acting on the broken fibre’s nearest neighbour at 1 and 1.7% remote strain. Perfect interface bonding was assumed.
Fig. 10. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1% remote strain. A 240 mm interface debond was assumed.
Fig. 13. Undimensional fibre stress f =fo (remote fibre stress) acting on the broken fibre’s nearest neighbour at 1% remote strain. A 240 mm interface debond was assumed.
Fig. 11. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1.7% remote strain. A 240 mm interface debond was assumed.
Fig. 14. Undimensional fibre stress f =fo (remote fibre stress) acting on the broken fibre’s nearest neighbour at 1.7% remote strain. A 240 mm interface debond was assumed.
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Table 1 Evaluation of the ineffective length model predictions for the cases considered (see text for details) Interface bonding
Perfect
Remote strain (%)
1.0 1.7
Partial
1.0 1.7
a
Ineffective half-length
Nearest fibre average stressa
Fraction of remote stress (%)
FEM (mm)
Analytical (mm)
Error (%)
90 95 90 95
89.0 106.4 137.4 154.3
88.6 103.0 142.4 156.8
0.39 3.19 3.65 1.60
1.02 1.02 1.02 1.01
90 95 90 95
306.2 323.5 340.4 356.6
306.2 321.8 346.4 360.8
0.01 0.54 0.76 1.18
1.01 1.01 1.02 1.01
Averaged along the ineffective length and divided by the remote fibre stress.
process. There are, however, significant difficulties to be overcome. Besides the issue of the in situ matrix ultimate shear strain, already mentioned above, there are also uncertainties around the statistical parameters of the fibre strength distributions that are appropriate to the short ineffective lengths [5,6,14].
4. Conclusions This paper presents a model to predict the stress distribution along a broken fibre in unidirectional composites under longitudinal tensile load. In the most general case, the model considers an interface debond, where the stress transfer takes place by a decreasing interfacial Coulomb friction stress. Assuming matrix perfect plasticity, the debond is followed by a matrix yielding zone, where the interfacial shear stress is assumed equal to the matrix shear yield stress. There is, finally, an elastic zone, where the interfacial shear stress follows a classical exponential law. We have, therefore, assumed ‘good interface bonding’, i.e. that the interfacial shear strength is not lower than the matrix shear yield stress. Nevertheless, the analysis suggests that debonding may occur as a result of limited matrix ductility, since the stiff neighbouring fibres impose large shear strains on the matrix layer around the broken fibre. The present model was compared with a 3D finite element model. Unfortunately, numerical singularities in the latter affect the maximum matrix strains, precluding validation of the model interfacial failure criterion. Nevertheless, the excellent agreement obtained for the stress distributions along the broken fibre gives clear evidence of the accuracy of the present analysis. The present model can be easily incorporated in more-or-less sophisticated longitudinal tensile strength prediction models, where it may provide the essential values of the ‘ineffective length’ throughout the loading process. Considerable research efforts will be required to establish appropriate values for in situ matrix ultimate
shear strains and for statistical parameters of the fibre strength distributions.
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[15] van den Heuvel PWJ, Wubbolts MK, Young RJ, Peijs T. Failure phenomena in two-dimensional multi-fibre model composites: 5. A finite element study. Composites Part A 1998;29A:1121–35. [16] van den Heuvel PWJ, Peijs T, Young RJ. Failure phenomena in two-dimensional multi-fibre microcomposites: Part 4. A Raman spectroscopic study on the influence of the matrix yield stress on stress concentration. Composites Part A 2000;31A:161–71. [17] Kelly A, Tyson WR. Tensile properties of fibre reinforced metals: copper/tungsten and copper/molybdenum. J Mech Phys Sol 1965;13:329–50.
[18] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw-Hill, 1970. [19] ABAQUS 5-8 release. Hibbitt, Karlsson & Sorensen, 1999. [20] Morais AB. Transverse moduli of continuous fibre reinforced polymers. Comp Sci Tech 2000;60:997–1002. [21] Chai H. Observation of deformation and damage at the tip of cracks in adhesive bonds loaded in shear and assessment of a criterion for fracture. Int J Fract 1993;60:311–26. [22] Bradley WL. In: Pipes RB, editor. Thermoplastic composite materials. Amsterdam: Elsevier Science, 1989.
Stress distribution along broken fibres in polymer-matrix composites Alfredo Balaco´ de Morais* University of Aveiro, Mechanical Engineering Department, Campus Santiago, 3810 Aveiro, Portugal Received 26 April 2000; received in revised form 21 March 2001; accepted 19 April 2001
Abstract This paper presents a model for predicting the stress distribution along a broken fibre in a unidirectional composite. It is assumed that the matrix behaves in an elastic/perfectly-plastic manner, and that the interfacial shear strength is not lower than the matrix shear yield stress. Although ‘good’ interface bonding is assumed, the present analysis suggests that interface debonding may occur as a result of local matrix shear failure, resulting in the well-known splitting phenomenon observed in tensile tests. Along the debonded length, a decreasing interfacial shear stress is derived from Poisson contractions and Coulomb friction. The debond is followed by a matrix yielding zone, where the interfacial shear stress is assumed to be equal to the matrix shear yield stress. There is, finally, an elastic zone, where the interfacial shear stress follows a classical exponential law. The present model is in good agreement with a 3D finite-element model. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; B. Modelling; C. Stress transfer; C. Finite-element analysis; Ineffective length
1. Introduction The matrix-to-fibre stress transfer is an issue of great relevance for continuous fibre composites, as it affects the layer longitudinal tensile strength. The brittle nature of the reinforcing fibres results in a high degree of strength variability and size-dependence. During tensile loading of a unidirectional layer, a number of fibre failures will occur prior to overall collapse. The so-called ‘ineffective length’ i.e. the axial distance over which a broken fibre recovers its undisturbed remote stress value, plays a major role in the final failure stress [1–6]. In fact, strength prediction models are usually based on the assemblage of various single-fibre elements whose size equals the ineffective length, and often resort to elaborate Monte-Carlo simulation schemes [4–6]. The stress concentration in the fibres close to the broken fibre has been a major concern. Various analytical and numerical models [7–10] have been specifically devoted to this phenomenon. The 3D finiteelement modelling of a hexagonal fibre array by Nedele and Wisnom [8] showed, however, quite modest stressconcentration factors. This emphasises the need to focus on the stress transfer to the broken fibre so that the ineffective length can be accurately defined. * Tel.: +351-234-370830; fax: +351-234-370953. E-mail address: [email protected]
The early ‘shear-lag’ analysis by Cox [11] assumed matrix elasticity and perfect interface bonding. Those assumptions are clearly unrealistic for actual composites i.e. localised matrix cracking and yielding are inevitable if perfect bonding exists. Landis et al [12] have recently developed a numerical model to predict the stress transfer to a broken fibre surrounded by a yielding matrix. Other authors [2,3,13] have considered interface debonding the major micromechanical failure mechanism. Nevertheless, recent work on single-fibre and array-of-fibres specimens seems to show that localised matrix yielding is much more relevant than interface debonding [14–16]. In fragmentation testing of specimens prepared with untreated/unsized fibres, leading to poor interface properties, considerable debonds are observed [14]. However, when normally treated/sized fibres are used, interface debonds can be quite small, and instead, matrix cracking occurs [14]. In many studies involving single-fibre testing, the well-known Kelly–Tyson model [17] is used to compute the interfacial shear strength, whose value often correlates with the matrix shear yield stress [14– 16]. In a study involving specimens with 5 aligned fibres, finite element model predictions were compared with local stress measurements using Raman spectroscopy [14–16]. The results confirmed that matrix yielding was the fundamental stress transfer mechanism, while no significant interface debonding was observed.
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00058-6
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Nomenclature
G23
c df f m r
Gm H1i
rf rm tm u w wb wo z zd zp C D E1 E2 Ef Em G12
subscript used denote the composite fibre diameter subscript used denote the fibre subscript used denote the matrix radial coordinate of a cylindrical coordinate system fibre radius outer radius of the matrix cylinder thickness of the matrix cylinder radial displacement in a cylindrical coordinate system axial displacement in a cylindrical coordinate system axial displacement of the broken fibre axial displacement of the fibres closest to the broken one axial coordinate of a cylindrical coordinate system axial coordinate of the end of the debonded length axial coordinate of the end of the matrix yielding zone integration constant integration constant longitudinal modulus of a transversely isotropic solid transverse modulus of a transversely isotropic solid longitudinal fibre modulus matrix Young’s modulus longitudinal shear modulus of a transversely isotropic solid
It seems, therefore, that, in normal ‘well-bonded’ interfaces, the interfacial shear strength is not lower than the matrix shear yield stress. The longitudinal splitting sometimes observed in tensile tests suggests, however, that interface debonding could be a relevant failure mechanism. The stress transfer model presented below suggests that interface debonding may result from local matrix shear failure.
2. Model description 2.1. Preliminary qualitative analysis Under tensile loading, both parts of a broken fibre move away from each other. Therefore, matrix cracking, partial interface debonding, or perhaps a combination of both must obviously occur. Assuming good interface
H2i r m um 12 23 m r z f fd fo rz i ym
transverse shear modulus of a transversely isotropic solid matrix shear modulus function used in the equation for the interfacial shear stress function used in the equation for the interfacial shear stress function used in the equation for the axial fibre stress parameter used in the equation for the axial fibre stress normal radial strain normal circumferential strain matrix shear strain matrix ultimate shear strain interfacial friction coefficient longitudinal Poisson’s ratio of a transversely isotropic solid transverse Poisson’s ratio of a transversely isotropic solid matrix Poisson’s ratio circumferential coordinate of a cylindrical coordinate system normal radial stress normal circumferential stress normal axial stress normal axial fibre stress normal axial fibre stress at the end of the debonded length remote axial fibre stress axisymmetric shear stress interfacial shear stress matrix shear yield stress
bonding and since before fibre failure the matrix is under significant tensile loads, it is logical to expect that matrix cracking is the first local failure mechanism. The axial displacements of the broken fibre segments promote matrix shear strains high enough to cause yielding. In contrast with the single-fibre or array-of-fibres composite specimens, in actual unidirectional composites, there is a narrow matrix layer between the broken and the stiff neighbouring fibres that will be subjected to high shear strains. Brittle matrices may undergo local shear failure, giving rise to significant interface debonding. We will, therefore, consider the most general case, where the fibre break resulted in matrix cracking and in partial interface debonding (Fig. 1). The matrix crack is arrested somewhere between the broken and the surrounding fibres. The exact position of the crack front and the stress state in its vicinity are a complex fracture mechanics problem. The finite element model presented in 3. shows that the
A. Balaco´ de Morais / Composites Science and Technology 61 (2001) 1571–1580
actual size of the matrix crack does not influence the stress transfer to the broken fibre. A concentric cylinder model is used for analysis purposes (Fig. 1). The outer radius rm of the matrix layer is assumed tangent to the nearest neighbour fibres of the most representative hexagonal packing arrangement. The surrounding composite is treated as a hollow cylinder of very large outer radius. Axisymmetric stress analysis will be performed, neglecting fibre shear stresses. Force equilibrium of a fibre element in the longitudinal direction is then expressed by dz;f 4 i ¼ dz df
ð1Þ
where z;f is the axial fibre stress, df is the fibre diameter and i the interfacial shear stress. An elastic-perfectly plastic stress–strain behaviour is assumed for the matrix. It is also assumed that the matrix is in the elastic range at the instant of fibre fracture.
Force equilibrium in the radial direction demands that, in each cylinder [18], ð2Þ
where r and denote the radial and circumferential directions, respectively. We assume a small variation of rz along the debonded length, so that r þ r
@r @r
ð3Þ
Letting u be the radial displacement, the relevant kinematic equations are u " ¼ r
@u @" ¼r þ " @r @r
ð5Þ
Using the more common numerical indices for the elastic properties of a transversely isotropic solid, the needed constitutive equation can be written as " ¼
1 12 ð 23 r Þ z E2 E1
ð6Þ
Inserting (3) into (6), and the resulting equation into (5), we obtain the differential equation r2
@2 ðrr Þ @ðrr Þ rr ¼ 0 þr @r2 @r
ð7Þ
where it has been assumed that the longitudinal stress z does not vary in the radial direction. The solution of (7) is r ¼ C þ
2.2. Stress transfer along the debonded length
@r r @ rz þ þ ¼0 @r r @z
"r ¼
1573
D r2
ð8Þ
where C and D are integration constants that must be determined from the boundary conditions. The circumferential stress (3) can then be expressed as D r2
¼ C
ð9Þ
We now analyse the cylinders representing the broken fibre, the surrounding matrix and the composite (Fig. 1). The values of the integration constants C and D differ for each case, and are, therefore, denoted with the corresponding cylinder index. In the fibre, the radial stress (8) must remain finite at r ¼ 0. We have thus Df ¼ 0 and constant radial and circumferential stresses i.e. r;f ðrf Þ ¼ ;f ðrf Þ ¼ Cf
ð10Þ
ð4Þ and from (6), ";f ðrf Þ ¼
12;f r;f ðrf Þ 1 23;f z;f E2;f E1;f
ð11Þ
As to the composite, at r ! 1 we must have r ! 0, since no external transverse loads are being applied. Using (8), (9) and (6),
r;c ðrm Þ ¼
Cc r2m
;c ðrm Þ ¼ r;c ðrm Þ ";c ðrm Þ ¼ Fig. 1. Scheme of the developed model.
12;c r;c ðrm Þ 1 þ 23;c z;c E2;c E1;c
ð12Þ ð13Þ ð14Þ
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We further assume a constant axial stress z;c E1;c
fo E1;f
We can now insert the interface shear stress (24) into (1), to obtain the differential equation ð15Þ
where fo is the remote longitudinal fibre stress. Finally, in the matrix cylinder we have r;m ¼ Cm þ
Dm r2
ð16Þ
Dm ;m ¼ Cm 2 ð17Þ r 1 Dm ";m ðrÞ ¼ Cm ð1 m Þ 2 ð1 þ m Þ m z;m Em r ð18Þ
df 4H2i 4H1i þ f ¼ dz df df
ð25Þ
where we have dropped the z-index, unnecessary at this stage. The solution is f ¼
H1i ð1 ez Þ H2i
ð26Þ
with ¼
4H2i df
ð27Þ
where, again, we assume a constant axial stress z;m Em
fo E1;f
ð19Þ
At the interfaces between the cylinders, the radial stresses must be continuous i.e. r;f ðrf Þ ¼ r;m ðrf Þ
ð20Þ
r;m ðrm Þ ¼ r;c ðrm Þ
ð21Þ
The interfacial radial displacements, and consequently the circumferential strains, must also be continuous. At the fibre/matrix interface, we can use Eqs. (11) and (18)– (21), to write ";f ðrf Þ ¼ ";m ðrf Þ as " # 1 m 1 23;f 1 þ m 1 23;f þ Cm Dm Em E2;f Em r2f E2;f r2f m fo 12;f z;f ¼ ð22Þ E1;f At the matrix/composite interface, Eqs. (14), (15) and (18)–(21), allow us to express ";m ðrm Þ ¼ ";c ðrm Þ as 1 m 1 þ 23;c 1 þ m 1 þ 23;c þ Cm Dm Em E2;c Em r2m E2;c r2m fo ¼ m 12;c ð23Þ E1;f Eqs. (22) and (23) can be easily solved for the constants Cm and Dm , which enable afterwards the determination of the fibre/matrix interfacial radial stress r;m ðrf Þ from (16). Finally, assuming Coulomb friction, the interfacial shear stress can be expressed as i ¼ r;m ðrf Þ ¼ H1i H2i z;f
Eq. (26) will apply from the broken fibre end (z ¼ 0) to some debonded length zd , which is defined in 2.4. 2.3. Stress transfer along the bonded length A matrix yielding zone immediately follows the interface debond. Obviously, the matrix is under a relatively complex stress state, including axial z , radial r and shear r;z stresses. An accurate analysis would result in a complex plasticity problem requiring numerical solution. We assume that, due to the much higher longitudinal fibre modulus, shear straining prevails in the matrix layer. Moreover, as in the simple Kelly–Tyson model [17], we consider matrix perfect plasticity and let the interfacial shear stress in the plastic zone be equal to the matrix shear yield stress ym . Although this assumption is clearly not valid near the end of the matrix plasticity zone, it is shown in (3) that it does not affect the accuracy of the predicted broken fibre stress distributions. The solution of (1) is then f ¼ fd þ
4 ym ð z zd Þ df
ð28Þ
valid for zd 4z4zp , fd being the fibre stress at the end of the debond. Finally, in the matrix elasticity zone, we assume that the matrix shear strain m is approximately constant in the radial direction. Neglecting radial displacements, m
@w wb wo @r rm rf
ð29Þ
ð24Þ
where H1i and H2i are functions of the friction coefficient , of the remote fibre stress fo and of the elastic moduli of the fibre, matrix and composite.
where wb and wo are the axial displacements of the broken and unbroken nearest neighbour fibres, respectively. Inserting (29) and Hooke’s law for the shear stress into (1), we obtain, after an additional derivation,
A. Balaco´ de Morais / Composites Science and Technology 61 (2001) 1571–1580
d2 f 4Gm dwb dwo ¼ dz2 df tm dz dz
ð30Þ
where tm ¼ rm rf is the thickness of the matrix cylinder. Since dwb f ¼ dz Ef
ð31Þ
and neglecting the stress concentration on the nearest neighbour fibre, dwo fo ¼ dz Ef
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where um is the matrix ultimate shear strain. Neglecting stress concentration, the axial displacement in the neighbouring fibres is wo
fo z Ef
ð39Þ
where it has been established that wo ðz ¼ 0Þ ¼ 0. The displacement of the broken fibre is, generally, ð f wb ¼ dx þ Cb ð40Þ Ef
Eq. (30) can be easily solved, giving the Cox [11] type equation
where the appropriate equation for f must be inserted, followed by the determination of the integration constant Cb . Along the matrix elasticity length, f is given by (36), and since we must have wb ! wo when z ! 1, then Cb ¼ 0 and the longitudinal displacement becomes
f ¼ fo þ Ce ez
wb ¼
ð32Þ
ð33Þ
fo 4 ym ðzp zÞ zþ e Ef Ef df 2
ð41Þ
with rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Gm ¼ df tm Ef
ð34Þ
The integration constant Ce can be evaluated from the condition that the interfacial shear stress, obtained by inserting (33) into (1), is equal to ym at z ¼ zp . We have finally i ¼ ym eðzp zÞ f ¼ fo
4 ym ðzp zÞ e df
ð42Þ
ð36Þ
wb ¼
ð37Þ
Having assumed that the interface shear stress is equal to the matrix shear yield stress, the interface debonding criterion is based on the maximum matrix shear strain. At the end of the interface debond, the matrix is assumed to be at the onset of shear failure, so that any subsequent increase in the remote stress results in debonding initiation or further growth. In view of (29), we have wb ðzp Þ wo ðzp Þ ¼ um tm
2 ym 2 4 ym fd z zd z þ z þ Cb E f df Ef df Ef
ð35Þ
The fibre stress must obviously be continuous at z ¼ zp . Using (28) and (36), we obtain 1 df þ ðfo fd Þ 4 ym
wb ¼
At zp , Eqs. (41) and (42) must give the same displacement. This continuity condition enables the determination of the integration constant Cb . In the matrix plasticity zone, we, therefore, have
2.4. The lengths of the interface debonding and yielding zones
zp ¼ zd
In the matrix plasticity zone, after inserting (28) into (40), we obtain
ð38Þ
2 ym 2 4 ym fd ðz z2p Þ zd ðz zp Þ þ ðz zp Þ E f df E f df Ef fo 4 ym zp þ ð43Þ þ Ef Ef df 2
We can now insert Eqs. (39) and (43) into Eq. (38) to obtain 2 ym 2 4 ym fo fd 2 ðz zp Þ zd þ ðzd zp Þ Ef df d Ef df Ef 4 ym þ ¼ um tm ð44Þ E f df 2 In view of the relation between zp and zd given by (37), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 ym 2 ym fd ¼ fo 2 Ef um tm ð45Þ df df 2 and, using (26), one finally obtains 1 H2i zd ¼ ln 1 fd H1i
ð46Þ
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3. Model evaluation The developed model was compared with a 3D finite element (FE) model implemented in the commercial code ABAQUS [19]. The model assumes hexagonal fibre packing (Fig. 1) and, in view of the symmetries, only a twelfth-section is considered (Fig. 2). In the inner zone, around the broken fibre, both constituents are modelled separately, while farther away the composite is modelled as a transversely isotropic homogeneous solid. Sufficiently large model dimensions were used to ensure representativity. The model was built with 6-node triangular prism and 8-node reduced integration brick elements (Fig. 3). Very fine meshes were used: the total number of elements was around 16,000. The tensile loading is applied by imposing a uniform axial displacement on the model upper face, while the lower z ¼ 0 face is a symmetry plane. In the cracked areas, however, no restrictions are imposed on the axial displacements of the corresponding nodes. Fig. 2 shows the contours of the two matrix cracks that were considered in the z ¼ 0 plane. Actually, the stress distribution along the broken fibre proved to be independent of the matrix crack pattern. Comparison with the developed model is shown here for the following set of properties. The fibre is assumed transversely isotropic with E1f =230 GPa, E1f =20 GPa, 12f =0.20, 23f =0.25 and G12f =20 GPa. For the isotropic matrix, we have used Em =3.5 GPa, m =0.35, and perfect plasticity with von Mises yield stress ym =70
Fig. 2. Schematic representation of the 3D FE model.
Fig. 3. Example of a FE mesh.
MPa. The elastic constants of the 60% fibre volume fraction composite were determined from finite element models presented in [20], giving E1c =139.4 GPa, E2c = 9.0 GPa, 12c =0.254, 23c =0.403 and G12c =4.2 GPa. When debonding was considered, a ¼ 0:5 Coulomb friction coefficient was used. One of the most relevant assumptions implicit in the model is the prevalence of shear straining in the yielding zone of the matrix layer around the broken fibre, and subsequent adoption of a constant interfacial shear stress equal to matrix shear yield stress. Normal stresses, which inevitably occur, are thereby neglected in the yield condition. A major implication of this assumption is that previous deformation history effects, which are in principle relevant in plasticity problems, are also neglected. In order to test the validity of this assumption, we have considered two loading histories. In the first case, the failure process was simulated by releasing the constraints on the nodes of the assumed cracked area after some remote stress value was achieved. In the second case, the same crack was assumed to exist from the initial unloaded state. Except for insignificant differences near the crack, the results were exactly the same. Figs. 4 and 5 show the interfacial shear stress distributions at 1% and 1.7% remote strain, respectively. The predictions of the
Fig. 4. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1% remote strain. Perfect interface bonding was assumed.
Fig. 5. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1.7% remote strain. Perfect interface bonding was assumed.
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developed model, given by i ¼ ym and by (35), are compared with FE results (the shear stress was averaged along the interfacial circumference). Due to numerical singularities at the matrix crack, the interfacial shear stress given by the FE analysis is locally much higher than the matrix shear yield stress. Farther away from the matrix crack, the FE interfacial shear stress slightly exceeds the matrix shear yield stress. The error is quite small (less than 2%) and is due to inevitable numerical disturbances associated with the master-slave surface contact modelling [19]. Although debonding was not considered in this case, it is important to define the interface as contacting fibre and matrix surfaces, since the common displacement interpolation based 3D finite elements do not ensure the shear stress continuity at the interfaces between different materials. As expected, the analytical model overestimates the interfacial shear stress in the last portion of the matrix yielding zone, because it ignores the normal stress in the matrix yielding criterion. Apparently, this discrepancy could result in overestimating the stress in the broken fibre, especially at high remote strain levels. However, the broken fibre stress distributions of Figs. 6 and 7 clearly show that the analytical model [Eqs. (28) and (36)] agrees quite well with the FE results. In addition, the analytical model actually underestimates the stress transfer to the broken fibre at 1.7%. This can be easily explained by
the larger mismatch of Poisson transverse effects between matrix and fibre in the yielding zone. Unfortunately, due to numerical singularities, the FE model gives unreliable overpredictions for the maximum matrix shear strain. This limitation reinforces the usefulness of the developed analysis, although it prevents its full validation. In the considered example, the maximum matrix shear strains predicted by the developed model are 32.5 and 91% at 1 and 1.7% remote strain, respectively. These high strain values suggest that interface debonding may be due to local matrix shear failure. This is consistent with the well-known splitting phenomenon usually observed in tensile testing of unidirectional brittle matrix composites [1]. The difficulty that immediately arises is the determination of the in situ matrix ultimate shear strain, since it is obviously highly questionable to apply bulk resin data to the very thin matrix layer around the broken fibre. For example, Chai [21] has shown that a brittle epoxy with an ultimate shear strain of about 10%, when in the form of a 25 mm adhesive layer, may have an ultimate shear strain more than 10 times larger. Bradley [22] has also shown that the shear strain at the tip of a crack in a composite may greatly exceed the bulk matrix ultimate shear strain. Clearly, this subject still requires considerable research. In the next examples, 240 mm interface debonds were assumed. Figs. 8 and 9 show the interfacial shear stress
Fig. 6. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1% remote strain. Perfect interface bonding was assumed.
Fig. 8. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1% remote strain. A 240 mm interface debond was assumed.
Fig. 7. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1.7% remote strain. Perfect interface bonding was assumed.
Fig. 9. Distribution of the undimensional interfacial shear stress i = ym (matrix shear yield stress) at 1.7% remote strain. A 240 mm interface debond was assumed.
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We can, therefore, conclude that the developed model is quite accurate at predicting the stress distribution along the broken fibre. The limitations of the FE analysis, however, do not enable the validation of the maximum matrix shear strain based interface debonding criterion. Further work should include the developed stress-transfer analysis in a longitudinal tensile strength prediction model, where it may provide the essential values of the ineffective length throughout the loading
distributions at 1 and 1.7% remote strain, respectively. The analysis [Eqs. (24), i ¼ ym and (35)] is in good agreement with the FE results, apart from the numerical singularity at the end of the debond and from the slight overestimation of the interfacial shear stress at the end of the matrix yielding length. The stress distributions along the broken fibre, depicted in Figs. 10 and 11, confirm the good performance of the developed model [Eqs. (26), (28) and (36)]. Another issue that has received considerable attention is the stress concentration in the fibres around the broken one. Figs. 12–14 show the stress concentration in the centre of the nearest neighbour fibre for the cases considered above. The present results do not differ significantly from those of Nedele and Wisnom [8] for matrix elasticity and perfect interface bonding. The highest stress concentration is relatively small, and it is only relevant along a relatively short length. If we define the ‘ineffective length’ as the length over which the broken fibre recovers 90 or 95% of the remote fibre stress (Table 1), the average stress in the nearest neighbour fibre is practically equal to the remote stress. Therefore, the role of stress concentration in the neighbour fibres seems to be practically negligible.
Fig. 12. Undimensional fibre stress f =fo (remote fibre stress) acting on the broken fibre’s nearest neighbour at 1 and 1.7% remote strain. Perfect interface bonding was assumed.
Fig. 10. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1% remote strain. A 240 mm interface debond was assumed.
Fig. 13. Undimensional fibre stress f =fo (remote fibre stress) acting on the broken fibre’s nearest neighbour at 1% remote strain. A 240 mm interface debond was assumed.
Fig. 11. Distribution of the undimensional broken fibre stress f =fo (remote fibre stress) at 1.7% remote strain. A 240 mm interface debond was assumed.
Fig. 14. Undimensional fibre stress f =fo (remote fibre stress) acting on the broken fibre’s nearest neighbour at 1.7% remote strain. A 240 mm interface debond was assumed.
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Table 1 Evaluation of the ineffective length model predictions for the cases considered (see text for details) Interface bonding
Perfect
Remote strain (%)
1.0 1.7
Partial
1.0 1.7
a
Ineffective half-length
Nearest fibre average stressa
Fraction of remote stress (%)
FEM (mm)
Analytical (mm)
Error (%)
90 95 90 95
89.0 106.4 137.4 154.3
88.6 103.0 142.4 156.8
0.39 3.19 3.65 1.60
1.02 1.02 1.02 1.01
90 95 90 95
306.2 323.5 340.4 356.6
306.2 321.8 346.4 360.8
0.01 0.54 0.76 1.18
1.01 1.01 1.02 1.01
Averaged along the ineffective length and divided by the remote fibre stress.
process. There are, however, significant difficulties to be overcome. Besides the issue of the in situ matrix ultimate shear strain, already mentioned above, there are also uncertainties around the statistical parameters of the fibre strength distributions that are appropriate to the short ineffective lengths [5,6,14].
4. Conclusions This paper presents a model to predict the stress distribution along a broken fibre in unidirectional composites under longitudinal tensile load. In the most general case, the model considers an interface debond, where the stress transfer takes place by a decreasing interfacial Coulomb friction stress. Assuming matrix perfect plasticity, the debond is followed by a matrix yielding zone, where the interfacial shear stress is assumed equal to the matrix shear yield stress. There is, finally, an elastic zone, where the interfacial shear stress follows a classical exponential law. We have, therefore, assumed ‘good interface bonding’, i.e. that the interfacial shear strength is not lower than the matrix shear yield stress. Nevertheless, the analysis suggests that debonding may occur as a result of limited matrix ductility, since the stiff neighbouring fibres impose large shear strains on the matrix layer around the broken fibre. The present model was compared with a 3D finite element model. Unfortunately, numerical singularities in the latter affect the maximum matrix strains, precluding validation of the model interfacial failure criterion. Nevertheless, the excellent agreement obtained for the stress distributions along the broken fibre gives clear evidence of the accuracy of the present analysis. The present model can be easily incorporated in more-or-less sophisticated longitudinal tensile strength prediction models, where it may provide the essential values of the ‘ineffective length’ throughout the loading process. Considerable research efforts will be required to establish appropriate values for in situ matrix ultimate
shear strains and for statistical parameters of the fibre strength distributions.
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