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Stress concentration factors around a broken fibre in a unidirectional carbon fibre-reinforced epoxy
Stress c o n c e n t r a t i o n f a c t o r s a r o u n d a b r o k e n f i b r e in a unidirectional carbon fibrereinforced epoxy M.R. NEDELE and M.R. WISNOM" (Institute of Structures and Design, Germany/'University of Bristol UK) Received 14 September 1993; revised 1 February 1994
An axisymmetric analysis was performed to investigate the case of a single broken fibre surrounded by matrix material and either perfect composite material or a system of concentric circular cylinders representing fibres and matrix separately, according to the assumed fibre volume fraction of the composite of 60%. The analysis shows that the stress concentration is dependent on the assumed fibre/ matrix bonding conditions and on the assumed matrix behaviour. In all cases investigated it is shown that the stress concentration in the adjacent fibres is much lower than the 1.104 predicted by Hedgepeth and van Dyke. The positively affected length where there is an increase in stress is only about half the ineffective length of the broken fibre. Further away from the break the axial stress in the adjacent fibres actually drops below the nominal axial stress. This results in a very small enhanced probability of failure in the adjacent fibres. Key w o r d s : stress concentration factors; axisymmetric analysis; unidirectional composites
The tensile load in the axial direction of a unidirectional (UD) composite is mainly carried by the stiff fibres. Axial failure of a unidirectional composite is therefore expected to be controlled by fibre properties. The axial failure of the fibres is controlled by flaws. An appropriate description of this influence is given by the Weibull distribution of strength for brittle materials. Single, isolated fibre breaks will occur in the composite according to the statistical distribution of the fibre strengths. Important parameters in the calculation of the strength of the composite are the stress concentration in the neighbouring fibres around the broken fibre and the ineffective length of the broken fibre. Both of these properties can be calculated analytically, but little finite element analysis has been done to investigate the influence of the assumptions that have to be made. The calculation of the ineffective length of a fibre in a composite was carried out by Cox 1. He applied the socalled shear-lag analysis and assumed that all constituents were elastic materials, Rosen 2 calculated the ineffective length of a broken fibre dependent on the fibre modulus, the shear modulus of the matrix and the fibre volume fraction (VF). He used this ineffective length to predict the unidirectional composite strength by means of a weakest link analogy. The
fibre strength was a statistical value defined by the twoparameter Weibull model. The stress concentration in neighbouring fibres was calculated by Hedgepeth and van Dyke 3 for three-dimensional fibre-matrix arrays assuming linear elastic material properties of the constituents. Two-dimensional stress concentration factors were also calculated for an ideal plastic matrix. The assumption of the calculation was that only the nearest neighbours were influenced. The differential equations were solved using an influence function technique. The integration was carried out numerically. An improvement to the statistical analysis of the axial failure of a unidirectional composite was presented by Zweben 4. He included the stress concentration around the broken fibre and with it the higher probability of failure of the adjacent fibres. A two-dimensional statistical theory of the strength of composites was presented by Scop and Argon 5. The theoretical model included the stress concentration in adjacent fibres. They showed the influence of the matrix on the axial strength of the composite. Another statistical theory of material strength was proposed by Zweben and Rosen 6. Two different failure types were considered, namely the fracture propagation mode and the cumulative weakening mode. The comparison
0010-4361/94/07/0549-09 © 1994 Butterworth-Heinemann Ltd COMPOSITES. VOLUME 25. NUMBER 7. 1994
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with experimental data showed good agreement for whisker-reinforced composites, while for continuous fibres the cumulative weakening mode represented an upper bound and the fracture propagation mode a lower bound. An analytical treatment of the stress concentration problem was presented by Lockett 7. The two-dimensional analysis took the possibility of different interface strengths into consideration, and Lockett showed that the shear-lag analysis yielded pessimistically high stress concentration factors, A more sophisticated failure prediction was presented by Menders et el. 8. They carried out a Monte Carlo simulation assuming the single-fibre strengths to follow the Weibull distribution. The simulation showed the critical influence of the ineffective length for establishing the failure stress. Batdorf 9 improved the failure modelling of an axially loaded unidirectional composite. He changed the approach of predicting the failure based on the failure of a single disc or cross-section of a composite. He calculated the probability of a further fibre break dependent on the applied stress and the number of given fibre breaks. He investigated a three-dimensional fibre array with a single fibre break being surrounded by four adjacent fibres with stress concentration factors as calculated by Hedgepeth and van Dyke 3. Ochiai and Osamura ~° performed a two-dimensional Monte Carlo simulation including the effect of the interface. They showed that the composite strength was very sensitive to the scatter of the fibre strengths and the interface strength. The fracture mode changed significandy from a non-cumulative to a cumulative failure mode with increasing scatter in fibre strength. They showed also that a decrease in the interface strength, especially below the yield stress of the matrix, lowered the composite strength, Two papers were presented by Bader ~,~2 using a new statistical model to explain the failure of a unidirectional composite under axial tension. Two important parameters of his model were the positively affected length and the stress concentration in the adjacent fibres. The influence of the matrix on the ultimate failure was shown as well. As also predicted by Ochiai and Osamura ~°, an increase in the interface strength led to an increase in the composite strength.
such as matrix plasticity and frictional slipping. These factors lead to a non-linear analysis which is computationally very time consuming. It is clear that the stress concentration factor in the fibres next to a broken fibre and the distance over which this acts are very important parameters for predicting tensile strength. The objectives of this investigation were to carry out a more accurate evaluation of the stress concentration factors and the positively affected length based on an axisymmetric micromechanical model. It is also clear that parameters like matrix plasticity and an assumed debonded interface will have a significant influence on these two properties. In contrast to macromechanical models, the fibre and matrix regions were modelled separately. The governing equations of the micromechanical model were solved using the finite element method. MICROMECHANICAL
The a x i s y m m e t r i c m o d e l s Two different cases were investigated. The first one is referred to as the simple case, and corresponds to a model with a broken fibre in the centre, surrounded by a matrix cylinder with an assumed fibre volume fraction of 60% which is embedded in a cylinder of perfect composite material. This set-up was chosen because of its geometrical simplicity, whilst allowing the investigation of matrix plasticity and interfacial slip. The model geometry in the radial and axial directions depends on the investigated microstructural factors. These factors include frictional slipping between the broken fibre and the matrix along the debonded part of the fibre/matrix interface, or ideal plasticity of the matrix material. A radial width of 50 gm, equivalent to 122 fibres, and an axial length of 350 p,m were deemed to be sufficient if the matrix material was linear elastic or ideal plastic. In the case where a debonded interface was assumed, it was necessary to extend the radial width to 70 gm and the axial length to 1000 gm. A sketch of this model is shown in Fig. 1. The element type is an eight-noded axisym-
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- 1000/~m
An improvement of Hedgepeth and van Dyke's 3 twodimensional inelastic analysis of the stress concentration in adjacent fibres was presented by Ochiai et al. 13. In contrast to earlier investigations they included the axial stiffness of the matrix in the shear-lag analysis, which is of relevance for metal matrix composites. They investigated the influence of the propagation of the fibre break in the adjacent matrix on the stress concentration in the adjacent fibres and showed that an increase of the stress concentration occurred. A three-dimensional linear elastic finite element analysis on a single fibre break was carried out by Nedele and Wisnom ~4. It was shown that the normally used stress concentration factors are much too high. The current work extends the previous analysis to include factors
550
COMPOSITES. N U M B E R 7 . 1994
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The area of the fibre rings is equal to the area o f the fibres represented. For example, the area of the first fibre ring is the same as the area of six adjacent fibres The fibre volume fraction of the composite rings is always the same, namely 60%
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metric element, called CAX8R, with a reduced integration scheme, and the finite element program used was ABAQUS ~5. The mesh size was altered according to the stress gradient, and the element size around the crack tip was such that six elements each were used in the radial direction to model both the fibre and the matrix. The axial size is equal to the radial size and rather small compared with the overall dimensions of the model, The second model is more complicated. This model stands between the simple axisymmetric model and a full three-dimensional model. It assumes a local fibre arrangement as shown in Fig. 2. A broken fibre is surrounded by six neighbouring fibres, the so-called hexagonal array. Further on, larger numbers of fibres, namely 12 and 18 surround the previous ones. A set of concentric circular cylinders represents the matrix and the fibres in this axisymmetric model. The areas of the individual
Boundary conditions The applied boundary conditions were identical for both models and are as follows. The matrix, adjacent fibre cylinders and the surrounding composite are fixed in the -direction along the axis of the fibre crack (radial axis, z = 0). In the case of a perfect interface, the coincident nodes of the fibre and the matrix at the crack tip are also released in the z direction. Far away from the fibre crack in the z direction (top face), a constant displacement in the z direction is fixed. The free edge in the radial direction (the outside of the composite) is constrained to a constant displacement in the radial direction. In order to suppress local numerical instability in the vicinity of the singularity at the fibre crack tip, all nodes along the z axis are fixed in the radial direction.
cylinders representing the fibres are given by smoothing out the areas of the six, 12 and 18 fibres, as described in
Thermoelastic properties of the constituents
more detail in the Appendix (see also Fig. 3). The matrix material is represented by outer and inner cylinders of
The fibre is assumed to be transversely isotropic. The eight independent thermoelastic material properties of the fibre are given in Table 2. The plane of isotropy is the radial hoop plane. The matrix material is assumed to be either a linear elastic, isotropic material or a linear elastic, ideal plastic material with the von Mises criterion used to define yielding. The yield stress was evaluated using matrix compression test data provided by the University of Surrey ~7 and a reduction factor of 1.3. Reduction factors between 1.2 and 1.4 are given by HulP 8 for the tensile yield stress compared with the compressive yield stress, owing to the influence of a hydrostatic stress state on the yielding of epoxies. The interface is assumed to be perfect or partly debonded with frictional forces acting along the debonded part. The effect of the debonded length on the stress redistribution was investigated by considering five different debonded lengths, varying from 55 to 266 p.m. Three different coefficients of friction were used in the current investigation. They were determined at the British Aerospace Sowerby Research Centre using a single-droplet test, as described
the same area enclosing the fibre cylinder (dotted circles in Fig. 3). The total area of the two matrix cylinders is given by the assumed fibre volume fraction. The inner and outer radial distances of each of the three cylinders representing the adjacent fibres are given in Table 1. Finally, a large region of perfect composite material is modelled in the outer cylinder that extends up to a radius of 143 p.m (see Fig. 3). This large region was chosen on the assumption that the failure of a single fibre should not alter the overall stiffness of the composite significandy. The area covered in the radial hoop plane is identical to 1000 fibres in a composite with a fibre volume fraction of 60%. A single fibre break will therefore change the modulus by 0.1%, which is negligible, The length of the model in the axial direction is 400 p . m . The element type in this model is also an eight-noded axisymmetric element, called PLANE82, and the finite element program used was ANSYS 16.
C O M P O S I T E S . N U M B E R 7 . 1994
551
Table 1. Inner and o u t e r radial locations of t h e adjacent fibre rings for a hexagonal array and an assumed fibre v o l u m e f r a c t i o n of 6 0 %
Inner diameter (/zm) Outer diameter (pm)
First ring, six fibres
Second ring, 12 fibres
Third ring, 18 fibres
6.702 10.882
13.854 18.410
21.481 26.114
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Unit
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UD composite,
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that untreated fibres had a significantly lower coefficient compared with that measured for treated fibres, namely 0.6 compared with 1.5. The axial and transverse stiffnesses, Poisson's ratios and coefficients of thermal expansion of the surrounding composite material were calculated using a second finite element micromechanical modeP °. A quarter of a fibre and the surrounding matrix were modelled and appropriate boundary conditions applied to determine the thermoelastic properties of the composite based solely on the known thermoelastic
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Fig. 4 Stressrecoveryin the brokenfibre~linearelasticmodels
Applied load
RESULTS
Residual thermal stresses caused by the cooling down from the manufacturing temperature were included in the analysis. The temperature difference between the stress-free temperature and room temperature was chosen to be 130°C. The axial mechanical loading of the composite was applied by two different methods. The first one is the displacement method. An axial displacement is fixed at the far end of the model away from the crack (z direction) which is equivalent to an overall strain in the cornposite of 1.5%. This method was used for the simple case. The second method is the force method. Based on the known axial modulus of the composite and the overall strain in the composite of 1.5%, it is possible to calculate the average axial stress. This average stress and the area of the model perpendicular to the loading direction give an axial force that is applied at the far axial end. The stresses are normalized using the far field stresses in either the fibre rings (concentric cylinder model) or the composite (simple axisymmetric model), giving relative values of stress recovery and stress concentration. The two different methods were used for convenience, but should not produce any significant difference in results,
552
COMPOSITES . N U M B E R 7 . 1994
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Ineffective length of the broken fibre Fig. 4 shows the axial stress recovery in the broken fibre in the axial fibre direction for the Cox model, the simple case, the finite element (FE) concentric cylinder model, an analytical multiple-cylinder modeF 1 and a three-dimensional finite element model ~4. Two values of the ineffective length, defined as the length in the axial direction that is necessary to recover a certain percentage of the nominally carried axial stress, are also shown, based on assumed 90% and 95% relative stress recoveries. Fig. 5 shows the stress recovery in the broken fibre for the different parameters investigated with the simple model. All parameters investigated increase the ineffective length. In particular, a small coefficient of friction leads to quite a large ineffective length. The ineffective lengths predicted for the different cases can be more easily compared in Table 3. Shear stress in the matrix
The comparison of the shear stresses along the interface between the two finite element models, the Cox model and the newly developed analytical concentric cylinder modeF ° is given in Fig. 6. Additionally, the shear stresses
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Fig. 8 shows the relative stress concentration, defined as the axial stress divided by the nominal axial stress (taken as the stress at the far end away from the crack) in the middle of the adjacent fibre, for each of the two different axisymmetric finite element models, the three-dimensional finite element model and the analytical model. The stress concentration factors in the second and third fibre cylinders are also given. In Fig. 9 the stress concentration factors are plotted for the non-linear models. Matrix plasticity and frictional slipping along a partly debonded broken fibre are the factors of interest. The stress concentration factor is based on the finite element results in the middle of each fibre cylinder for the concentric cylinder model and at the distance where the centre of the adjacent fibres can be expected (hexagonal array). In the second case, the stress concentration is given by the ratio of the local composite stress to the nominal composite stress. The axial stress concentration at the first fibre cylinder/matrix interface is plotted in Fig. 10 together with that found in the centre of the first fibre cylinder. The positively affected length, defined as the axial distance from the crack as far as an axial stress in the adjacent fibre higher than the nominal one is found, is given in Table 3. The absolute distance where a stress disturbance in the adjacent fibres is found is also given.
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Table 3. Comparison of ineffective lengths, positively affected lengths and lengths of stress disturbance in adjacent fibres
Axisymmetric model Concentric cylinder Matrix plasticity F r i c t i o n p = 1.5 Friction p = 0.6
Ineffective length f o r 9 0 % stress recovery (gm)
Ineffective length f o r 9 5 % stress recovery (pm)
Positively affected length (gm)
Total length of stress disturbance (gm)
43.0 54.1 85.0 216.2 305.0
58.5 74.5 102.4 237.0 324.3
37.2 36.9 59.8 129.4 275.7
223.8 359.2 280.6 794.0 695.3
COMPOSITES
. NUMBER
7 . 1994
553
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dependent on the matrix plasticity, debonded length and coefficient of friction (COF)
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This distance includes the length of the adjacent fibre where an actual drop in stress below the nominal fibre stress is found, DISCUSSION The stress field The plots of the axial stress in the broken fibre along the fibre axis give a smooth variation of stress, indicating that the finite element mesh is fine enough. The difference between the two finite element models is more obvious for the ineffective length. The ineffective length is slightly longer for the concentric cylinder model,
The shear stress along the fibre/matrix interface is similar for all the different models. However, as Fig. 6 shows, the shear stress distribution along the first fibre cylinder/ matrix interface is quite different to the one found for the broken fibre. This is an important result which shows the limitations of the shear-lag analysis, The actual location where the stress concentration is evaluated in the adjacent fibre is open to discussion. The results presented are based on the values at the centre of the adjacent fibres. The stress concentration factor along the adjacent fibre/matrix interface is quite different, as can be seen in Fig. 10. However, if one takes the variation in axial stress in the radial direction into account, an average value of 1.061 is calculated. This value is in very good agreement with the value found in the middle of the fibres. Although the stress concentration factor is quite
554
COMPOSITES.
NUMBER 7. 1994
high at the fibre surface, it rapidly decreases away from there. Using the peak value would be too pessimistic, as the chance of finding a defect at the same location is small. Stellbrink 22 and Brokopf 23 have shown that for short fibre lengths, as relevant to the current investigation, internal flaws control fibre failure rather than surface flaws. It is therefore considered to be more realistic to use the mean value of the stress concentration. An initially surprising result is that the stress level in the adjacent fibres actually drops below the nominal stress. The length where an axial stress larger than the nominal axial stress is found is always shorter than the ineffective length of the broken fibre. It is of the order of half the ineffective length if a 95% stress recovery is assumed. Further along the adjacent fibre the axial stress actually drops below the nominal fibre stress. The drop comes from the need to maintain overall strain compatibility. The higher stress levels in the adjacent fibres produce an equivalently higher strain. However, the overall strain in the composite is not altered by the single fibre break. Therefore, a drop in the local fibre strain has to occur to fulfil the requirement of axial strain compatibility. As shown by the stress plot, the effect of the fibre crack on the surrounding area is restricted to about 100 ~tm from the actual crack in the axial direction and has hardly any influence on the composite beyond the first two adjacent fibre cylinders in the crack plane. Stress concentrations on fibres beyond these two cylinders are therefore negligible. The factors investigated, namely the plasticity of the matrix and the frictional stress transfer along the debonded interface, do alter the results significantly. It is evident that matrix plasticity affects the stress concentration factor in the adjacent fibre slightly, although the positively affected length is increased. The influence of debonding and friction is more severe. The distance away from the crack where the influence of the crack is visible is extended by a factor of about five compared with the linear elastic case. The stress concentration factor is also reduced, but not as much as expected based on the large positively affected length. It is also clear that there is a difference between the 100% surface-treated fibres and the untreated fibres. The much lower coefficient of friction measured for the untreated fibres (0.6 compared with 1.5) increases the positively affected length even further. The debonded length, assumed to be present in the cases where the influence of the coefficient of friction was investigated, is long enough to transfer the axial loading into the broken fibre by pure frictional shear forces for the high coefficient of friction. This can be most clearly seen in the shear stress plot. The singularity in stress that is present at the transition point between the broken and perfectly bonded interfaces disappears for the case with the high coefficient of friction. The results with the coefficient of friction of 1.3 in Fig. 9 show a clear trend with increasing debonded length. The results were obtained with a mesh that did not extend quite so far as the others. This may have some influence on the actual stress, but should not affect the trend. The initially assumed debonded lengths were arbitrarily chosen. One can imagine that a debonding occurs when a fibre break happens. We then need to know how far this
debonding will extend in a composite. Because of the numerical singularity at the transition point it is not possible to predict the debonded length with the current finite element analysis. The only length which can be predicted is the necessary debonded length to retransfer the axial loading by pure frictional forces, as was shown for the high coefficient of friction case. Once a suitable failure criterion for fibre/matrix interface debonding is developed which can be applied in numerical calculations, these debonding effects can be studied in more detail.
distribution along its length is calculated by integration using a Weibull approach. The necessary statistical data for the fibres were provided by the University of Surrey 24. The probability of survival for a fibre of a certain length is given by
Mean stress concentration in the adjacent fibres
length (62 mm) used to evaluate the two parameters, ~ and m.
A very important point is how these finite element results can be used for predicting the axial tensile failure of a unidirectional composite. The most well-known failure models2,4 6,s,10.t~ are all based on statistical methods. The strength of the individual fibres is not constant but given by a strength distribution. The probability of failure of each single fibre is increased with increasing stress levels, The first question is which value of the stress concentration in the adjacent fibre should be used in the statistical models. The second question is over what distance along the adjacent fibre should this stress concentration be assumed to apply. Both parameters are necessary for most statistical models. The maximum stress concentration factors (axial stresses averaged over the fibre cross-section) found to be present in the adjacent fibres are given in Table 4. All maximum stress concentration factors are much smaller than the one predicted by Hedgepeth and van Dyke of 1.104, or the more severe case of a local load sharing of 1.167. As already mentioned, the length where an axial stress larger than the average value is found in the adjacent fibres is only of the order of half the ineffective length of the broken fibre. Further along the adjacent fibre the axial stress actually drops beneath the nominal fibre stress. Two different methods are presented to account for the stress variation along the adjacent fibres. The simpler one is to take the average stress concentration over the length where a stress increase is found. This mean stress concentration factor does not take into account the stress decrease further away from the break. The calculated mean stress concentration factors are given in Table 4. A more accurate way to account for both effects, the stress increase and the stress decrease, is to calculate a statistical factor for the enhanced probability of failure of the adjacent fibres. This factor is defined by the ratio of the probability of failure of the adjacent fibre to the probability of failure of a fibre of the same length subjected to the nominal fibre stress. Based on the stress concentration curve along the axial length, the probability of failure of a fibre subjected to this varying stress
P(s) = exp[-(cr/cro)mL/Lo]
(1)
if the weakest link analogy is used, where P(s) is the probability of survival, cr is the applied stress, tr0 is the characteristic strength (3270 MPa), m is the shape parameter (5.3), L is the fibre length and L0 is the standard
The stress variation along the adjacent fibre is taken to be constant over small length increments of the order of the element size in the z direction and the probability of survival is calculated. The probability of survival for the whole fibre is given by the product of the probabilities for all the fibre length increments. The probability of failure is given by 1 - P(s). The probability of failure is only calculated over the length of the fibre where a variation in the axial stress is found. The different lengths used are given in the last column of Table 3. The ratios of the probabilities of failure for the different cases are given in Table 4. To compare these new results with the mean stress concentration factors normally used in statistical models, the following method was adopted. The mean stress concentration factors were assumed to be effective over a length of 35 lam on either side of the crack plane along the adjacent fibres, based on the positively affected length for the simple axisymmetric model. The mean stress concentration factor was based on the assumption that the reduction in stress is linear from the maximum value to the nominal value. Therefore, it was possible to predict the higher probability of failure for the adjacent fibres over a total length of 223 ~tm. The new statistical enhancement factors were calculated for the following cases: local load sharing with a maximum (average over the fibre cross-section) stress concentration factor (scF) of 1.167 and a mean of 1.0835; the Hedgepeth and van Dyke three-dimensional maximum scF of 1.104; and the two mean values of 1.022 and 1.028 for the simple axisymmetric model (SAM) and the concentric cylinder model (CCM), respectively. These values can be cornpared with the value from the simple axisymmetric model that takes into account the calculated variation of stress in the adjacent fibre. The following ratios of the probabilities of failure for the three different stress concentration factors were evaluated:
Table 4. Comparison of stress concentration factors
Axisymmetric model Concentric cylinder Matrix plasticity Friction/z = 1.5 Friction/z = 0.6
Maximum stress concentration factor
Mean stress concentration factor
Enhanced probability of failure
1.044 1.061
1.022 1.028
1.009 1.005
1.054
1.028
1.01 4
1.030 1.01 6
1.011 1.007
1.001 1.005
COMPOSITES. NUMBER 7 . 1994
555
Local load sharing (maximum SCF = 1.167) 1.083 Hedgepeth and van D y k e three-dimensional hexagonal array (maximum SCF = 1.104) 1.048 Stress concentration factor o f 1.022 (SAM) 1.019 Stress concentration factor o f 1.028 (CCM) 1.024 Calculated variation o f stress 1.009 As one can see, the stress concentration factors normally used in failure models give a m u c h higher probability o f failure than the factors calculated in this work. These results suggest that stress concentrations have a much smaller influence on overall tensile failure than generally believed. However, models ignoring the effect o f stress concentrations, such as the Rosen cumulative weakening model 2, generally predict too high a value for the composite strength. A possible explanation for this is the effect o f the interaction o f fibre breaks at different positions along the length o f the fibre, which is generally not considered. A new model based on this idea shows that realistic tensile strength predictions can be made
without the need for stress concentration factors2L CONCLUSIONS
The following conclusions can be drawn from the results
of this investigation. 1)
2)
The ineffective length o f the broken fibre was shown to be in g o o d agreement with the analytical solutions
for linear elastic behaviour of the matrix, The maximum stress concentrations in the adjacent fibres a r o u n d a single fibre break o f 1.044 for the simple case and 1.061 for the concentric cylinder model are smaller than the analytical result o f 1.104 from the shear-lag theory,
3)
4)
The distance away from the fibre break where a stress concentration in the adjacent fibre can be expected is only a b o u t half o f the ineffective length o f the broken fibre. Beyond this length the stress actually decreases below the nominal axial stress. The mean stress concentrations in the adjacent fibres averaged over the positively affected length are a b o u t 1.022 (simple model) and 1.028 (concentric cylinder model). Matrix plasticity does not alter the
stress concentration in the adjacent fibres signifi-
5)
cantly; however, a d e b o n d e d fibre/matrix interface decreases the stress concentration even further to values o f a r o u n d 1.01. The probability o f failure for the adjacent fibres because o f the varying stress distribution is only about 1% higher than for an equivalent length o f uniformly stressed fibre. This suggests that stress concentration at fibre breaks m a y play a m u c h smaller role in tensile failure than previously thought.
ACKNOWLEDGEMENTS The authors wish to acknowledge that the results presented in this paper arose from participation in the P R E D I C T project (Prediction o f D a m a g e Initiation and G r o w t h in Composite Materials), which is a collaboration between British Aerospace (Sowerby Research Centre), Ciba Composites, Rolls-Royce, Tenax Fibers, the National Physical L a b o r a t o r y , the University o f
556
COMPOSITES. NUMBER 7. 1994
Bristol and the University o f Surrey. The project is led by British Aerospace and is funded under the L I N K Structural Composites P r o g r a m m e o f the D T I ' s Research and Technology Initiative ( l E D grant R A 6/25/01). One o f the authors (M.R.N.) wishes also to acknowledge the fruitful discussions with his colleague A. J o h n s o n at D L R Stuttgart.
REFERENCES 1 Cox, H.L. "The elasticity and strength of paper and other fibrous materials' Br J Appl Phys 3 (1952) pp 72 79 2 Rosen, B.W. "Tensile failure of fibrous composites' AIAA J 2 No 11 (1964)pp 1985-1991 3 Hedgepeth, J.M. and van Dyke, P. 'Local stress concentration in imperfect filamentary composite materials' J Compos Mater 1 (1967) pp 294-309 4 Zweben, C. "Tensile failure of fibre composites' AIAA J 6 No 12 (1968) pp 2325-2331 5 Scop,P.M. and Argon, A.S. 'Statistical theory of strength of laminated composites II' J Compos Mater 3 (1969) pp 3047 6 Zweben, C. and Rosen, B.W. 'A statistical theory of material strength with application to composite materials" J Mech Phys Solids 18 (1970) pp 189 206 7 Loekett, F.J. 'Fibre stress-concentration in an imperfect twodimensional elastic composite' NPL Report: Maths 88 (National Physical Laboratory, Teddington, UK, 1970) 8 Manders,P.W., Bader, M.G. and Chou, T.-W. 'Monte Carlo simulation of the strength of composite fibre bundles' Fibre Sci Technol 17 (1982) pp 183 204 9 Batdorf,S.B. 'Tensile strength of unidirectionally reinforced composites' J ReinfPlast Compos 1 No 2 (1982) pp 153-164 10 Oehiai, S. and Osamura, K. 'Influences of interfacial bonding strength and scatter of fibre strength on tensile behaviour of unidirectional metal matrix composites" J Mater Sei 23 (1988) pp 886-893 11 Bader, M.G. 'The role of fibre strength and interactions with the matrix in the strength and failure of uniaxial composites in tension' in Composite 88, Patras, Greece (1988) 12 Bader, M.G. "Tensile strength of uniaxial composites" Sei Eng Compos Mater 1 (1988)ppl 11 13 Oehiai, S., Schulte, K. and Peters, P.W.M. 'Strain concentration factors for fibres and matrix in unidirectional composites' Compos Sci Techno141 (1991) pp 237 256 14 Nedele, M.R. and Wisnom, M.R. 'Three dimensional finite element analysis of the stress concentration at a single fibre break' Compos Sci Technol(in press) 15 ABAQUS User's Manual, Version 4.8 (Hibbitt, Karlsson and Sorensen, Providence, RI, 1989) 16 A N S Y S User's Manual, Revision 5.0 (Swanson Analysis Systems Inc., Houston, PA, 1992) 17 Matrix Properties (Composites Research Group, Department of Materials Science and Engineering, University of Surrey, 1992) 18 Hull, D. An Introduction to Composite Materials (Cambridge University Press, Cambridge, 1990) 19 Marshall, P. 'Coefficient of friction at the fibre matrix interface' Report PREDICT/MPR/BAE/920720/1 (British Aerospace, Bristol, 1992) 20 Nedele, M.R. and Wisnom, M.R.'Micromechanicalmodelling of a unidirectional carbon fibre-epoxy subjected to mechanical and thermal loading' in Proc 7th Teeh Conf. American Society Jot Composites (Technomic, Lancaster, PA, 1992) pp 328-338 21 McCartney, L.N. 'Stress transfer mechanics for multiple perfectly bonded concentric cylinder models of unidirectional composites' NPL Report (in preparation; National Physical Laboratory, Teddington, UK) 22 Stellbrink, K. 'On the probability of crack propagation in a unidirectional composite with a single fibre breakage' Dissertation (University of Stuttgart, 1977) (in German) 23 Brnkopf,S. 'Ober das versagen zugbelasteter kohlenstoffasern als einzelfaser und im verbund' Dissertation (University of Stuttgart, 1992) (in German) 24 Bader,M.G., Pickering, K.L., Buxton, A., Rezaifard, A. and Smith, P.A. 'Failure micro-mechanisms in continuous carbon-fibre/ epoxy resin composites' Report PREDICT/TR/UOS/MGB/
920915/1 (Department of Materials Science and Engineering,
University of Surrey, 1992) 25 Wisnom, M.R. anti Green, D. "Tensilefailure due to interaction of fibre breaks' (in preparation)
AUTHORS Martin R. Nedele is with the Institute of Structures and Design, D L R Stuttgart, 70569 Stuttgart, Germany. Micheal R. Wisnom, to whom correspondence should be addressed, is with the University of Bristol, Department of Aerospace Engineering, Bristol BS8 ITR, UK.
C A L C U L A T I O N OF THE RADIAL D I M E N S I O N S OF THE CONCENTRIC CYLINDERS The aim of the concentric cylinder model is to replace the composite by an axisymmetric model centred on the broken fibre with the same axial stiffness, preserving the distances between the broken fibre and the neighbouring fibres, The outer radius of the matrix ring surrounding the broken fibre is given by
( 1)
where rr is the radius of the fibre and vr is the assumed fibre volume fraction. The next composite cylinder (see Fig. 3) has an inner radius of rml and an unknown outer radius of rm3. The fibre volume fraction in this cylinder has to be vr. The known cross-sectional area of the six adjacent fibres will be represented by a concentric cylinder with the same cross-sectional area as these six fibres. The matrix material is evenly distributed in inner and outer concentric cylinders with equal cross-sectional areas placed around the fibre cylinder. With these assumptions it is possible to calculate the internal dimensions of the composite cylinder n(r~l - r2m0 = 3zrr~-(l - vf)/Vr
by the right-hand side of Equation (2). Equation (2) can be simplified by using the expression for rml given by Equation (I) to give rft = rrx/(4 - 3v0/Vr
(2)
(3)
The outer radius of the fibre ring is given by 7r(r{, -
Appendix
rml = rf/,]v,.
The left-hand side of Equation (2) represents the crosssectional area of the inner matrix cylinder, which has to equal half the area of the given matrix cross-sectional area fraction of the first composite ring. The volume fraction can be reduced to a cross-sectional area fraction in the current case. Half the matrix area fraction is given
d,) = 6=d
(4)
where the right-hand side is the area of the six fibres to be smeared out in the concentric cylinder representing the fibres. Equation (4) can be simplified using Equation (3) to give r~, = rfx/6 + (4 - 3vO/vr (5) The outer boundary of the cylinder assembly is given by zr(r~,,3 - r~,)= 3trr~(1 - vO/v,
(6)
which can be simplified to rm3 = r,-,,/6 + (7 - 6vr)/Vr
(7)
The calculation of the radii of the remaining cylinders follows the above method. It is important that the distance R, to the middle of the first fibre ring, given by R,. = (rr/2)[x/(4 - 3vr)/vr + x/6 + (4 - 3v,.)/vr]
(8)
compares well with the distance to the next neighbouring fibre for an assumed hexagonal packing geometry of R = rrx/2rc/x/~_vf
(9)
Using the assumed fibre volume fraction of 0.6, the ratio R~/R = 1.022, which means that the locations where the stress concentration is evaluated are nearly identical for both the three-dimensional model and the concentric cylinder model.
COMPOSITES
. NUMBER
7 . 1994
557
An axisymmetric analysis was performed to investigate the case of a single broken fibre surrounded by matrix material and either perfect composite material or a system of concentric circular cylinders representing fibres and matrix separately, according to the assumed fibre volume fraction of the composite of 60%. The analysis shows that the stress concentration is dependent on the assumed fibre/ matrix bonding conditions and on the assumed matrix behaviour. In all cases investigated it is shown that the stress concentration in the adjacent fibres is much lower than the 1.104 predicted by Hedgepeth and van Dyke. The positively affected length where there is an increase in stress is only about half the ineffective length of the broken fibre. Further away from the break the axial stress in the adjacent fibres actually drops below the nominal axial stress. This results in a very small enhanced probability of failure in the adjacent fibres. Key w o r d s : stress concentration factors; axisymmetric analysis; unidirectional composites
The tensile load in the axial direction of a unidirectional (UD) composite is mainly carried by the stiff fibres. Axial failure of a unidirectional composite is therefore expected to be controlled by fibre properties. The axial failure of the fibres is controlled by flaws. An appropriate description of this influence is given by the Weibull distribution of strength for brittle materials. Single, isolated fibre breaks will occur in the composite according to the statistical distribution of the fibre strengths. Important parameters in the calculation of the strength of the composite are the stress concentration in the neighbouring fibres around the broken fibre and the ineffective length of the broken fibre. Both of these properties can be calculated analytically, but little finite element analysis has been done to investigate the influence of the assumptions that have to be made. The calculation of the ineffective length of a fibre in a composite was carried out by Cox 1. He applied the socalled shear-lag analysis and assumed that all constituents were elastic materials, Rosen 2 calculated the ineffective length of a broken fibre dependent on the fibre modulus, the shear modulus of the matrix and the fibre volume fraction (VF). He used this ineffective length to predict the unidirectional composite strength by means of a weakest link analogy. The
fibre strength was a statistical value defined by the twoparameter Weibull model. The stress concentration in neighbouring fibres was calculated by Hedgepeth and van Dyke 3 for three-dimensional fibre-matrix arrays assuming linear elastic material properties of the constituents. Two-dimensional stress concentration factors were also calculated for an ideal plastic matrix. The assumption of the calculation was that only the nearest neighbours were influenced. The differential equations were solved using an influence function technique. The integration was carried out numerically. An improvement to the statistical analysis of the axial failure of a unidirectional composite was presented by Zweben 4. He included the stress concentration around the broken fibre and with it the higher probability of failure of the adjacent fibres. A two-dimensional statistical theory of the strength of composites was presented by Scop and Argon 5. The theoretical model included the stress concentration in adjacent fibres. They showed the influence of the matrix on the axial strength of the composite. Another statistical theory of material strength was proposed by Zweben and Rosen 6. Two different failure types were considered, namely the fracture propagation mode and the cumulative weakening mode. The comparison
0010-4361/94/07/0549-09 © 1994 Butterworth-Heinemann Ltd COMPOSITES. VOLUME 25. NUMBER 7. 1994
549
with experimental data showed good agreement for whisker-reinforced composites, while for continuous fibres the cumulative weakening mode represented an upper bound and the fracture propagation mode a lower bound. An analytical treatment of the stress concentration problem was presented by Lockett 7. The two-dimensional analysis took the possibility of different interface strengths into consideration, and Lockett showed that the shear-lag analysis yielded pessimistically high stress concentration factors, A more sophisticated failure prediction was presented by Menders et el. 8. They carried out a Monte Carlo simulation assuming the single-fibre strengths to follow the Weibull distribution. The simulation showed the critical influence of the ineffective length for establishing the failure stress. Batdorf 9 improved the failure modelling of an axially loaded unidirectional composite. He changed the approach of predicting the failure based on the failure of a single disc or cross-section of a composite. He calculated the probability of a further fibre break dependent on the applied stress and the number of given fibre breaks. He investigated a three-dimensional fibre array with a single fibre break being surrounded by four adjacent fibres with stress concentration factors as calculated by Hedgepeth and van Dyke 3. Ochiai and Osamura ~° performed a two-dimensional Monte Carlo simulation including the effect of the interface. They showed that the composite strength was very sensitive to the scatter of the fibre strengths and the interface strength. The fracture mode changed significandy from a non-cumulative to a cumulative failure mode with increasing scatter in fibre strength. They showed also that a decrease in the interface strength, especially below the yield stress of the matrix, lowered the composite strength, Two papers were presented by Bader ~,~2 using a new statistical model to explain the failure of a unidirectional composite under axial tension. Two important parameters of his model were the positively affected length and the stress concentration in the adjacent fibres. The influence of the matrix on the ultimate failure was shown as well. As also predicted by Ochiai and Osamura ~°, an increase in the interface strength led to an increase in the composite strength.
such as matrix plasticity and frictional slipping. These factors lead to a non-linear analysis which is computationally very time consuming. It is clear that the stress concentration factor in the fibres next to a broken fibre and the distance over which this acts are very important parameters for predicting tensile strength. The objectives of this investigation were to carry out a more accurate evaluation of the stress concentration factors and the positively affected length based on an axisymmetric micromechanical model. It is also clear that parameters like matrix plasticity and an assumed debonded interface will have a significant influence on these two properties. In contrast to macromechanical models, the fibre and matrix regions were modelled separately. The governing equations of the micromechanical model were solved using the finite element method. MICROMECHANICAL
The a x i s y m m e t r i c m o d e l s Two different cases were investigated. The first one is referred to as the simple case, and corresponds to a model with a broken fibre in the centre, surrounded by a matrix cylinder with an assumed fibre volume fraction of 60% which is embedded in a cylinder of perfect composite material. This set-up was chosen because of its geometrical simplicity, whilst allowing the investigation of matrix plasticity and interfacial slip. The model geometry in the radial and axial directions depends on the investigated microstructural factors. These factors include frictional slipping between the broken fibre and the matrix along the debonded part of the fibre/matrix interface, or ideal plasticity of the matrix material. A radial width of 50 gm, equivalent to 122 fibres, and an axial length of 350 p,m were deemed to be sufficient if the matrix material was linear elastic or ideal plastic. In the case where a debonded interface was assumed, it was necessary to extend the radial width to 70 gm and the axial length to 1000 gm. A sketch of this model is shown in Fig. 1. The element type is an eight-noded axisym-
z '
- 1000/~m
An improvement of Hedgepeth and van Dyke's 3 twodimensional inelastic analysis of the stress concentration in adjacent fibres was presented by Ochiai et al. 13. In contrast to earlier investigations they included the axial stiffness of the matrix in the shear-lag analysis, which is of relevance for metal matrix composites. They investigated the influence of the propagation of the fibre break in the adjacent matrix on the stress concentration in the adjacent fibres and showed that an increase of the stress concentration occurred. A three-dimensional linear elastic finite element analysis on a single fibre break was carried out by Nedele and Wisnom ~4. It was shown that the normally used stress concentration factors are much too high. The current work extends the previous analysis to include factors
550
COMPOSITES. N U M B E R 7 . 1994
MODELLING
./. ~ 3.5~m 4.~lt~/~m
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l
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Fig. 2 Assumedfibre arrangementfor the concentric cylinder finite element model (not to scale)
rml ~ ~ ~ rm3 ~ ~ ~ rm5
The area of the fibre rings is equal to the area o f the fibres represented. For example, the area of the first fibre ring is the same as the area of six adjacent fibres The fibre volume fraction of the composite rings is always the same, namely 60%
Fig. 3 Concentric cylinder model (not to scale)
metric element, called CAX8R, with a reduced integration scheme, and the finite element program used was ABAQUS ~5. The mesh size was altered according to the stress gradient, and the element size around the crack tip was such that six elements each were used in the radial direction to model both the fibre and the matrix. The axial size is equal to the radial size and rather small compared with the overall dimensions of the model, The second model is more complicated. This model stands between the simple axisymmetric model and a full three-dimensional model. It assumes a local fibre arrangement as shown in Fig. 2. A broken fibre is surrounded by six neighbouring fibres, the so-called hexagonal array. Further on, larger numbers of fibres, namely 12 and 18 surround the previous ones. A set of concentric circular cylinders represents the matrix and the fibres in this axisymmetric model. The areas of the individual
Boundary conditions The applied boundary conditions were identical for both models and are as follows. The matrix, adjacent fibre cylinders and the surrounding composite are fixed in the -direction along the axis of the fibre crack (radial axis, z = 0). In the case of a perfect interface, the coincident nodes of the fibre and the matrix at the crack tip are also released in the z direction. Far away from the fibre crack in the z direction (top face), a constant displacement in the z direction is fixed. The free edge in the radial direction (the outside of the composite) is constrained to a constant displacement in the radial direction. In order to suppress local numerical instability in the vicinity of the singularity at the fibre crack tip, all nodes along the z axis are fixed in the radial direction.
cylinders representing the fibres are given by smoothing out the areas of the six, 12 and 18 fibres, as described in
Thermoelastic properties of the constituents
more detail in the Appendix (see also Fig. 3). The matrix material is represented by outer and inner cylinders of
The fibre is assumed to be transversely isotropic. The eight independent thermoelastic material properties of the fibre are given in Table 2. The plane of isotropy is the radial hoop plane. The matrix material is assumed to be either a linear elastic, isotropic material or a linear elastic, ideal plastic material with the von Mises criterion used to define yielding. The yield stress was evaluated using matrix compression test data provided by the University of Surrey ~7 and a reduction factor of 1.3. Reduction factors between 1.2 and 1.4 are given by HulP 8 for the tensile yield stress compared with the compressive yield stress, owing to the influence of a hydrostatic stress state on the yielding of epoxies. The interface is assumed to be perfect or partly debonded with frictional forces acting along the debonded part. The effect of the debonded length on the stress redistribution was investigated by considering five different debonded lengths, varying from 55 to 266 p.m. Three different coefficients of friction were used in the current investigation. They were determined at the British Aerospace Sowerby Research Centre using a single-droplet test, as described
the same area enclosing the fibre cylinder (dotted circles in Fig. 3). The total area of the two matrix cylinders is given by the assumed fibre volume fraction. The inner and outer radial distances of each of the three cylinders representing the adjacent fibres are given in Table 1. Finally, a large region of perfect composite material is modelled in the outer cylinder that extends up to a radius of 143 p.m (see Fig. 3). This large region was chosen on the assumption that the failure of a single fibre should not alter the overall stiffness of the composite significandy. The area covered in the radial hoop plane is identical to 1000 fibres in a composite with a fibre volume fraction of 60%. A single fibre break will therefore change the modulus by 0.1%, which is negligible, The length of the model in the axial direction is 400 p . m . The element type in this model is also an eight-noded axisymmetric element, called PLANE82, and the finite element program used was ANSYS 16.
C O M P O S I T E S . N U M B E R 7 . 1994
551
Table 1. Inner and o u t e r radial locations of t h e adjacent fibre rings for a hexagonal array and an assumed fibre v o l u m e f r a c t i o n of 6 0 %
Inner diameter (/zm) Outer diameter (pm)
First ring, six fibres
Second ring, 12 fibres
Third ring, 18 fibres
6.702 10.882
13.854 18.410
21.481 26.114
Table 2. T h e r m o p l a s t i c properties of t h e c o n s t i t u e n t s Property E~ E,
Unit
Carbon fibre,
Epoxy matrix,
UD composite,
Akzo HTA
Ciba-Geigy F922
V F = 60%
G Pa G Pa G Pa GPa 10-6K 1 10-6K 1
G~, G, v~, v,
C~a ~,
232.00 13.80 18.00 5.11 0.22 0.35 -0.10 10.00
elsewhere ~9. The first one (1.3) is the average value based
3.89 3.89 1.42 1.42 0.37 0.37 58.00 58.00
i®~ ............ .
on all test data, whilst a closer look at the results showed
.;~+
~
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that untreated fibres had a significantly lower coefficient compared with that measured for treated fibres, namely 0.6 compared with 1.5. The axial and transverse stiffnesses, Poisson's ratios and coefficients of thermal expansion of the surrounding composite material were calculated using a second finite element micromechanical modeP °. A quarter of a fibre and the surrounding matrix were modelled and appropriate boundary conditions applied to determine the thermoelastic properties of the composite based solely on the known thermoelastic
80~o ~ /o , ~ 60~ ~ ~t ~ ,o~ . ~t
properties of the constituents.
Fig. 4 Stressrecoveryin the brokenfibre~linearelasticmodels
Applied load
RESULTS
Residual thermal stresses caused by the cooling down from the manufacturing temperature were included in the analysis. The temperature difference between the stress-free temperature and room temperature was chosen to be 130°C. The axial mechanical loading of the composite was applied by two different methods. The first one is the displacement method. An axial displacement is fixed at the far end of the model away from the crack (z direction) which is equivalent to an overall strain in the cornposite of 1.5%. This method was used for the simple case. The second method is the force method. Based on the known axial modulus of the composite and the overall strain in the composite of 1.5%, it is possible to calculate the average axial stress. This average stress and the area of the model perpendicular to the loading direction give an axial force that is applied at the far axial end. The stresses are normalized using the far field stresses in either the fibre rings (concentric cylinder model) or the composite (simple axisymmetric model), giving relative values of stress recovery and stress concentration. The two different methods were used for convenience, but should not produce any significant difference in results,
552
COMPOSITES . N U M B E R 7 . 1994
~°~ o~
. ~ I I
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Ineffective length of the broken fibre Fig. 4 shows the axial stress recovery in the broken fibre in the axial fibre direction for the Cox model, the simple case, the finite element (FE) concentric cylinder model, an analytical multiple-cylinder modeF 1 and a three-dimensional finite element model ~4. Two values of the ineffective length, defined as the length in the axial direction that is necessary to recover a certain percentage of the nominally carried axial stress, are also shown, based on assumed 90% and 95% relative stress recoveries. Fig. 5 shows the stress recovery in the broken fibre for the different parameters investigated with the simple model. All parameters investigated increase the ineffective length. In particular, a small coefficient of friction leads to quite a large ineffective length. The ineffective lengths predicted for the different cases can be more easily compared in Table 3. Shear stress in the matrix
The comparison of the shear stresses along the interface between the two finite element models, the Cox model and the newly developed analytical concentric cylinder modeF ° is given in Fig. 6. Additionally, the shear stresses
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Fig. 8 Stress concentration factors around a single fibre break-linear elastic models
Fig. 8 shows the relative stress concentration, defined as the axial stress divided by the nominal axial stress (taken as the stress at the far end away from the crack) in the middle of the adjacent fibre, for each of the two different axisymmetric finite element models, the three-dimensional finite element model and the analytical model. The stress concentration factors in the second and third fibre cylinders are also given. In Fig. 9 the stress concentration factors are plotted for the non-linear models. Matrix plasticity and frictional slipping along a partly debonded broken fibre are the factors of interest. The stress concentration factor is based on the finite element results in the middle of each fibre cylinder for the concentric cylinder model and at the distance where the centre of the adjacent fibres can be expected (hexagonal array). In the second case, the stress concentration is given by the ratio of the local composite stress to the nominal composite stress. The axial stress concentration at the first fibre cylinder/matrix interface is plotted in Fig. 10 together with that found in the centre of the first fibre cylinder. The positively affected length, defined as the axial distance from the crack as far as an axial stress in the adjacent fibre higher than the nominal one is found, is given in Table 3. The absolute distance where a stress disturbance in the adjacent fibres is found is also given.
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.
Concentric C~llnder FE ;
at two more locations are given for the finite element concentric cylinder model. These locations are the inner and outer fibre cylinder/matrix interfaces of the fibre cylinder surrounding the broken fibre. In Fig. 7 the shear stress along the fibre/matrix interface is plotted for a partly debonded fibre and for the matrix showing ideal plastic behaviour. The different debonded lengths and coefficients of friction (COF) are again detailed and the results plotted.
Shear stress along the interface--linear elastic models
150
•
[
50
Con,~,,,oC~.~o,
160
120
Di. . . . . fromtheCrack/~m
.-. .
i
1st Ring Inner Boundary rfl
, .....................,
.
First Ring Concentric C~iinder IrE Second Ring
'
•
0~ -50 i
~
~
MatrixPlasticlty oy = 1_30MPa
~o.Mo,°, ~,~mo,,c,o~
i
.
0.98
Fig. 5 Stress recovery in the broken fibre dependent on the matrix plasticity, debonded length and coefficient of friction (COF)
~,0 i~k
C.....iC~nd.~~ ,
1.04
k~ 102
Debonded Length 266~m ;
.
3 dim. FE Model
8
COF!3
I
0% ~ 0
~
: Debonded Length 22q~m i
*
'~
DebondedLength165#m
"
/
: ,.
Simple Case I 06
COF''3
."
.....
Axisymmetric Model
Debonded Length 110~m
~, ~'
Debonded Length 55urn ,~
~
400
c°D' oo~o,~o, Le.gt,~m COF 0.6 I .I ~ 501 Plasticity oy=l~MPa /
Shear stress along the interface dependent on the matrix
plasticity, debonded length and coefficient of friction (COF)
Table 3. Comparison of ineffective lengths, positively affected lengths and lengths of stress disturbance in adjacent fibres
Axisymmetric model Concentric cylinder Matrix plasticity F r i c t i o n p = 1.5 Friction p = 0.6
Ineffective length f o r 9 0 % stress recovery (gm)
Ineffective length f o r 9 5 % stress recovery (pm)
Positively affected length (gm)
Total length of stress disturbance (gm)
43.0 54.1 85.0 216.2 305.0
58.5 74.5 102.4 237.0 324.3
37.2 36.9 59.8 129.4 275.7
223.8 359.2 280.6 794.0 695.3
COMPOSITES
. NUMBER
7 . 1994
553
t.~ .... 1.04
I i
/ [
~t,,hs~,. c°r'~3-
: , t ........... t
O.boo~,.*h COF 1.3 ,,~..
!
~ Length16Stem Debonded
,." .~
,r,
I~1.02
~ ~ ~
~
_c°F"~_L DebondedLength22~m
~ ~
~
coF13
DcbondedLength2/a6~m
1.00
coFl.~
Debonded Length2~J.m 0,98
0
1oo
2o0
~
~0
~
~..~,om,,.~.~.,,
ooro.~ P~,.,~
~-1~0,~.
Fig. 9 Stress concentration factor around a single fibre b r e a k
dependent on the matrix plasticity, debonded length and coefficient of friction (COF)
i 1"227
1'2Ii0 1.25
: Cent,eAdjacent
' ---'
t..
! ,.,0 1.05
FibreRing !
i InterfaceMatrlx --_FibreR~
~ ..... 062
!
,.,~
,,0o .~;,:,~,~_:,~:: - ~ ........ + ........ ~ ........ 0.95 [ . _ _ ~ 0
] 50
] I00
] 150
Distanoe fromthe Crack~m
Fig. 10 C o m p a r i s o n radial location
stress
] --
260
--
250
concentration factors dependent on the
This distance includes the length of the adjacent fibre where an actual drop in stress below the nominal fibre stress is found, DISCUSSION The stress field The plots of the axial stress in the broken fibre along the fibre axis give a smooth variation of stress, indicating that the finite element mesh is fine enough. The difference between the two finite element models is more obvious for the ineffective length. The ineffective length is slightly longer for the concentric cylinder model,
The shear stress along the fibre/matrix interface is similar for all the different models. However, as Fig. 6 shows, the shear stress distribution along the first fibre cylinder/ matrix interface is quite different to the one found for the broken fibre. This is an important result which shows the limitations of the shear-lag analysis, The actual location where the stress concentration is evaluated in the adjacent fibre is open to discussion. The results presented are based on the values at the centre of the adjacent fibres. The stress concentration factor along the adjacent fibre/matrix interface is quite different, as can be seen in Fig. 10. However, if one takes the variation in axial stress in the radial direction into account, an average value of 1.061 is calculated. This value is in very good agreement with the value found in the middle of the fibres. Although the stress concentration factor is quite
554
COMPOSITES.
NUMBER 7. 1994
high at the fibre surface, it rapidly decreases away from there. Using the peak value would be too pessimistic, as the chance of finding a defect at the same location is small. Stellbrink 22 and Brokopf 23 have shown that for short fibre lengths, as relevant to the current investigation, internal flaws control fibre failure rather than surface flaws. It is therefore considered to be more realistic to use the mean value of the stress concentration. An initially surprising result is that the stress level in the adjacent fibres actually drops below the nominal stress. The length where an axial stress larger than the nominal axial stress is found is always shorter than the ineffective length of the broken fibre. It is of the order of half the ineffective length if a 95% stress recovery is assumed. Further along the adjacent fibre the axial stress actually drops below the nominal fibre stress. The drop comes from the need to maintain overall strain compatibility. The higher stress levels in the adjacent fibres produce an equivalently higher strain. However, the overall strain in the composite is not altered by the single fibre break. Therefore, a drop in the local fibre strain has to occur to fulfil the requirement of axial strain compatibility. As shown by the stress plot, the effect of the fibre crack on the surrounding area is restricted to about 100 ~tm from the actual crack in the axial direction and has hardly any influence on the composite beyond the first two adjacent fibre cylinders in the crack plane. Stress concentrations on fibres beyond these two cylinders are therefore negligible. The factors investigated, namely the plasticity of the matrix and the frictional stress transfer along the debonded interface, do alter the results significantly. It is evident that matrix plasticity affects the stress concentration factor in the adjacent fibre slightly, although the positively affected length is increased. The influence of debonding and friction is more severe. The distance away from the crack where the influence of the crack is visible is extended by a factor of about five compared with the linear elastic case. The stress concentration factor is also reduced, but not as much as expected based on the large positively affected length. It is also clear that there is a difference between the 100% surface-treated fibres and the untreated fibres. The much lower coefficient of friction measured for the untreated fibres (0.6 compared with 1.5) increases the positively affected length even further. The debonded length, assumed to be present in the cases where the influence of the coefficient of friction was investigated, is long enough to transfer the axial loading into the broken fibre by pure frictional shear forces for the high coefficient of friction. This can be most clearly seen in the shear stress plot. The singularity in stress that is present at the transition point between the broken and perfectly bonded interfaces disappears for the case with the high coefficient of friction. The results with the coefficient of friction of 1.3 in Fig. 9 show a clear trend with increasing debonded length. The results were obtained with a mesh that did not extend quite so far as the others. This may have some influence on the actual stress, but should not affect the trend. The initially assumed debonded lengths were arbitrarily chosen. One can imagine that a debonding occurs when a fibre break happens. We then need to know how far this
debonding will extend in a composite. Because of the numerical singularity at the transition point it is not possible to predict the debonded length with the current finite element analysis. The only length which can be predicted is the necessary debonded length to retransfer the axial loading by pure frictional forces, as was shown for the high coefficient of friction case. Once a suitable failure criterion for fibre/matrix interface debonding is developed which can be applied in numerical calculations, these debonding effects can be studied in more detail.
distribution along its length is calculated by integration using a Weibull approach. The necessary statistical data for the fibres were provided by the University of Surrey 24. The probability of survival for a fibre of a certain length is given by
Mean stress concentration in the adjacent fibres
length (62 mm) used to evaluate the two parameters, ~ and m.
A very important point is how these finite element results can be used for predicting the axial tensile failure of a unidirectional composite. The most well-known failure models2,4 6,s,10.t~ are all based on statistical methods. The strength of the individual fibres is not constant but given by a strength distribution. The probability of failure of each single fibre is increased with increasing stress levels, The first question is which value of the stress concentration in the adjacent fibre should be used in the statistical models. The second question is over what distance along the adjacent fibre should this stress concentration be assumed to apply. Both parameters are necessary for most statistical models. The maximum stress concentration factors (axial stresses averaged over the fibre cross-section) found to be present in the adjacent fibres are given in Table 4. All maximum stress concentration factors are much smaller than the one predicted by Hedgepeth and van Dyke of 1.104, or the more severe case of a local load sharing of 1.167. As already mentioned, the length where an axial stress larger than the average value is found in the adjacent fibres is only of the order of half the ineffective length of the broken fibre. Further along the adjacent fibre the axial stress actually drops beneath the nominal fibre stress. Two different methods are presented to account for the stress variation along the adjacent fibres. The simpler one is to take the average stress concentration over the length where a stress increase is found. This mean stress concentration factor does not take into account the stress decrease further away from the break. The calculated mean stress concentration factors are given in Table 4. A more accurate way to account for both effects, the stress increase and the stress decrease, is to calculate a statistical factor for the enhanced probability of failure of the adjacent fibres. This factor is defined by the ratio of the probability of failure of the adjacent fibre to the probability of failure of a fibre of the same length subjected to the nominal fibre stress. Based on the stress concentration curve along the axial length, the probability of failure of a fibre subjected to this varying stress
P(s) = exp[-(cr/cro)mL/Lo]
(1)
if the weakest link analogy is used, where P(s) is the probability of survival, cr is the applied stress, tr0 is the characteristic strength (3270 MPa), m is the shape parameter (5.3), L is the fibre length and L0 is the standard
The stress variation along the adjacent fibre is taken to be constant over small length increments of the order of the element size in the z direction and the probability of survival is calculated. The probability of survival for the whole fibre is given by the product of the probabilities for all the fibre length increments. The probability of failure is given by 1 - P(s). The probability of failure is only calculated over the length of the fibre where a variation in the axial stress is found. The different lengths used are given in the last column of Table 3. The ratios of the probabilities of failure for the different cases are given in Table 4. To compare these new results with the mean stress concentration factors normally used in statistical models, the following method was adopted. The mean stress concentration factors were assumed to be effective over a length of 35 lam on either side of the crack plane along the adjacent fibres, based on the positively affected length for the simple axisymmetric model. The mean stress concentration factor was based on the assumption that the reduction in stress is linear from the maximum value to the nominal value. Therefore, it was possible to predict the higher probability of failure for the adjacent fibres over a total length of 223 ~tm. The new statistical enhancement factors were calculated for the following cases: local load sharing with a maximum (average over the fibre cross-section) stress concentration factor (scF) of 1.167 and a mean of 1.0835; the Hedgepeth and van Dyke three-dimensional maximum scF of 1.104; and the two mean values of 1.022 and 1.028 for the simple axisymmetric model (SAM) and the concentric cylinder model (CCM), respectively. These values can be cornpared with the value from the simple axisymmetric model that takes into account the calculated variation of stress in the adjacent fibre. The following ratios of the probabilities of failure for the three different stress concentration factors were evaluated:
Table 4. Comparison of stress concentration factors
Axisymmetric model Concentric cylinder Matrix plasticity Friction/z = 1.5 Friction/z = 0.6
Maximum stress concentration factor
Mean stress concentration factor
Enhanced probability of failure
1.044 1.061
1.022 1.028
1.009 1.005
1.054
1.028
1.01 4
1.030 1.01 6
1.011 1.007
1.001 1.005
COMPOSITES. NUMBER 7 . 1994
555
Local load sharing (maximum SCF = 1.167) 1.083 Hedgepeth and van D y k e three-dimensional hexagonal array (maximum SCF = 1.104) 1.048 Stress concentration factor o f 1.022 (SAM) 1.019 Stress concentration factor o f 1.028 (CCM) 1.024 Calculated variation o f stress 1.009 As one can see, the stress concentration factors normally used in failure models give a m u c h higher probability o f failure than the factors calculated in this work. These results suggest that stress concentrations have a much smaller influence on overall tensile failure than generally believed. However, models ignoring the effect o f stress concentrations, such as the Rosen cumulative weakening model 2, generally predict too high a value for the composite strength. A possible explanation for this is the effect o f the interaction o f fibre breaks at different positions along the length o f the fibre, which is generally not considered. A new model based on this idea shows that realistic tensile strength predictions can be made
without the need for stress concentration factors2L CONCLUSIONS
The following conclusions can be drawn from the results
of this investigation. 1)
2)
The ineffective length o f the broken fibre was shown to be in g o o d agreement with the analytical solutions
for linear elastic behaviour of the matrix, The maximum stress concentrations in the adjacent fibres a r o u n d a single fibre break o f 1.044 for the simple case and 1.061 for the concentric cylinder model are smaller than the analytical result o f 1.104 from the shear-lag theory,
3)
4)
The distance away from the fibre break where a stress concentration in the adjacent fibre can be expected is only a b o u t half o f the ineffective length o f the broken fibre. Beyond this length the stress actually decreases below the nominal axial stress. The mean stress concentrations in the adjacent fibres averaged over the positively affected length are a b o u t 1.022 (simple model) and 1.028 (concentric cylinder model). Matrix plasticity does not alter the
stress concentration in the adjacent fibres signifi-
5)
cantly; however, a d e b o n d e d fibre/matrix interface decreases the stress concentration even further to values o f a r o u n d 1.01. The probability o f failure for the adjacent fibres because o f the varying stress distribution is only about 1% higher than for an equivalent length o f uniformly stressed fibre. This suggests that stress concentration at fibre breaks m a y play a m u c h smaller role in tensile failure than previously thought.
ACKNOWLEDGEMENTS The authors wish to acknowledge that the results presented in this paper arose from participation in the P R E D I C T project (Prediction o f D a m a g e Initiation and G r o w t h in Composite Materials), which is a collaboration between British Aerospace (Sowerby Research Centre), Ciba Composites, Rolls-Royce, Tenax Fibers, the National Physical L a b o r a t o r y , the University o f
556
COMPOSITES. NUMBER 7. 1994
Bristol and the University o f Surrey. The project is led by British Aerospace and is funded under the L I N K Structural Composites P r o g r a m m e o f the D T I ' s Research and Technology Initiative ( l E D grant R A 6/25/01). One o f the authors (M.R.N.) wishes also to acknowledge the fruitful discussions with his colleague A. J o h n s o n at D L R Stuttgart.
REFERENCES 1 Cox, H.L. "The elasticity and strength of paper and other fibrous materials' Br J Appl Phys 3 (1952) pp 72 79 2 Rosen, B.W. "Tensile failure of fibrous composites' AIAA J 2 No 11 (1964)pp 1985-1991 3 Hedgepeth, J.M. and van Dyke, P. 'Local stress concentration in imperfect filamentary composite materials' J Compos Mater 1 (1967) pp 294-309 4 Zweben, C. "Tensile failure of fibre composites' AIAA J 6 No 12 (1968) pp 2325-2331 5 Scop,P.M. and Argon, A.S. 'Statistical theory of strength of laminated composites II' J Compos Mater 3 (1969) pp 3047 6 Zweben, C. and Rosen, B.W. 'A statistical theory of material strength with application to composite materials" J Mech Phys Solids 18 (1970) pp 189 206 7 Loekett, F.J. 'Fibre stress-concentration in an imperfect twodimensional elastic composite' NPL Report: Maths 88 (National Physical Laboratory, Teddington, UK, 1970) 8 Manders,P.W., Bader, M.G. and Chou, T.-W. 'Monte Carlo simulation of the strength of composite fibre bundles' Fibre Sci Technol 17 (1982) pp 183 204 9 Batdorf,S.B. 'Tensile strength of unidirectionally reinforced composites' J ReinfPlast Compos 1 No 2 (1982) pp 153-164 10 Oehiai, S. and Osamura, K. 'Influences of interfacial bonding strength and scatter of fibre strength on tensile behaviour of unidirectional metal matrix composites" J Mater Sei 23 (1988) pp 886-893 11 Bader, M.G. 'The role of fibre strength and interactions with the matrix in the strength and failure of uniaxial composites in tension' in Composite 88, Patras, Greece (1988) 12 Bader, M.G. "Tensile strength of uniaxial composites" Sei Eng Compos Mater 1 (1988)ppl 11 13 Oehiai, S., Schulte, K. and Peters, P.W.M. 'Strain concentration factors for fibres and matrix in unidirectional composites' Compos Sci Techno141 (1991) pp 237 256 14 Nedele, M.R. and Wisnom, M.R. 'Three dimensional finite element analysis of the stress concentration at a single fibre break' Compos Sci Technol(in press) 15 ABAQUS User's Manual, Version 4.8 (Hibbitt, Karlsson and Sorensen, Providence, RI, 1989) 16 A N S Y S User's Manual, Revision 5.0 (Swanson Analysis Systems Inc., Houston, PA, 1992) 17 Matrix Properties (Composites Research Group, Department of Materials Science and Engineering, University of Surrey, 1992) 18 Hull, D. An Introduction to Composite Materials (Cambridge University Press, Cambridge, 1990) 19 Marshall, P. 'Coefficient of friction at the fibre matrix interface' Report PREDICT/MPR/BAE/920720/1 (British Aerospace, Bristol, 1992) 20 Nedele, M.R. and Wisnom, M.R.'Micromechanicalmodelling of a unidirectional carbon fibre-epoxy subjected to mechanical and thermal loading' in Proc 7th Teeh Conf. American Society Jot Composites (Technomic, Lancaster, PA, 1992) pp 328-338 21 McCartney, L.N. 'Stress transfer mechanics for multiple perfectly bonded concentric cylinder models of unidirectional composites' NPL Report (in preparation; National Physical Laboratory, Teddington, UK) 22 Stellbrink, K. 'On the probability of crack propagation in a unidirectional composite with a single fibre breakage' Dissertation (University of Stuttgart, 1977) (in German) 23 Brnkopf,S. 'Ober das versagen zugbelasteter kohlenstoffasern als einzelfaser und im verbund' Dissertation (University of Stuttgart, 1992) (in German) 24 Bader,M.G., Pickering, K.L., Buxton, A., Rezaifard, A. and Smith, P.A. 'Failure micro-mechanisms in continuous carbon-fibre/ epoxy resin composites' Report PREDICT/TR/UOS/MGB/
920915/1 (Department of Materials Science and Engineering,
University of Surrey, 1992) 25 Wisnom, M.R. anti Green, D. "Tensilefailure due to interaction of fibre breaks' (in preparation)
AUTHORS Martin R. Nedele is with the Institute of Structures and Design, D L R Stuttgart, 70569 Stuttgart, Germany. Micheal R. Wisnom, to whom correspondence should be addressed, is with the University of Bristol, Department of Aerospace Engineering, Bristol BS8 ITR, UK.
C A L C U L A T I O N OF THE RADIAL D I M E N S I O N S OF THE CONCENTRIC CYLINDERS The aim of the concentric cylinder model is to replace the composite by an axisymmetric model centred on the broken fibre with the same axial stiffness, preserving the distances between the broken fibre and the neighbouring fibres, The outer radius of the matrix ring surrounding the broken fibre is given by
( 1)
where rr is the radius of the fibre and vr is the assumed fibre volume fraction. The next composite cylinder (see Fig. 3) has an inner radius of rml and an unknown outer radius of rm3. The fibre volume fraction in this cylinder has to be vr. The known cross-sectional area of the six adjacent fibres will be represented by a concentric cylinder with the same cross-sectional area as these six fibres. The matrix material is evenly distributed in inner and outer concentric cylinders with equal cross-sectional areas placed around the fibre cylinder. With these assumptions it is possible to calculate the internal dimensions of the composite cylinder n(r~l - r2m0 = 3zrr~-(l - vf)/Vr
by the right-hand side of Equation (2). Equation (2) can be simplified by using the expression for rml given by Equation (I) to give rft = rrx/(4 - 3v0/Vr
(2)
(3)
The outer radius of the fibre ring is given by 7r(r{, -
Appendix
rml = rf/,]v,.
The left-hand side of Equation (2) represents the crosssectional area of the inner matrix cylinder, which has to equal half the area of the given matrix cross-sectional area fraction of the first composite ring. The volume fraction can be reduced to a cross-sectional area fraction in the current case. Half the matrix area fraction is given
d,) = 6=d
(4)
where the right-hand side is the area of the six fibres to be smeared out in the concentric cylinder representing the fibres. Equation (4) can be simplified using Equation (3) to give r~, = rfx/6 + (4 - 3vO/vr (5) The outer boundary of the cylinder assembly is given by zr(r~,,3 - r~,)= 3trr~(1 - vO/v,
(6)
which can be simplified to rm3 = r,-,,/6 + (7 - 6vr)/Vr
(7)
The calculation of the radii of the remaining cylinders follows the above method. It is important that the distance R, to the middle of the first fibre ring, given by R,. = (rr/2)[x/(4 - 3vr)/vr + x/6 + (4 - 3v,.)/vr]
(8)
compares well with the distance to the next neighbouring fibre for an assumed hexagonal packing geometry of R = rrx/2rc/x/~_vf
(9)
Using the assumed fibre volume fraction of 0.6, the ratio R~/R = 1.022, which means that the locations where the stress concentration is evaluated are nearly identical for both the three-dimensional model and the concentric cylinder model.
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. NUMBER
7 . 1994
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