## Abstract

Earlier work which successfully modelled the kinetics of fibre breakage in unidirectional composites under monotonic tensile loading has been extended to quantify the kinetics of fibre failure during both monotonic and sustained tensile loading. In both cases, failure was seen to occur when a critical density of large clusters (more than 16 fibres are broken within the representative volume element) of fibre breaks developed. However, in monotonic loading failure occurred very quickly after the first development of these large clusters, whereas under sustained loading the composite could accommodate greater levels of large clusters because of the lower applied load.

This article is part of the themed issue ‘Multiscale modelling of the structural integrity of composite materials’.

## 1. Introduction

The objective of this study was to examine whether there exists a critical state of damage which provokes the failure of unidirectional carbon fibre reinforced composites when subjected to different types of tensile loading parallel to the fibre direction. As the fibres support approximately 99% of the applied load, it is their failure which is the most critical in this type of structure and is examined in this study. The final aim is to eventually quantify critical levels of damage in service for such structures as carbon fibre composite pressures vessels.

The difficulties of experimentally observing the accumulation of fibre breaks at the level of individual fibres are considerable. Even in a simple specimen, there are tens of thousands of fibres with diameters of around 7 μm and visualizing evolution of damage at this microscopic level is a daunting task. This paper has benefitted from the development of a multiscale model which takes into account the characteristics of the fibres, the matrix and also the fibre/matrix interface at the microscopic scale which is then used to accurately describe the kinetics of fibre breaks in composite specimens and structures such as filament wound pressure vessels. This model has already been favourably compared with experimental tests made on specimens using acoustic emission [1,2] and high-resolution tomography to monitor damage [3]. It has also been favourably compared to slow-burst tests made on carbon fibre composite pressure vessels [4]. This model has been extended so as to examine the critical states of damage leading to failure of the composite.

The study benefits from many previous studies over the last 60 years, but takes advantage of the latest computational means to model the kinetics of failure in much greater detail than has been hitherto possible. The simulation which has been used allows a detailed description of all types of carbon fibre composite structures (for instance, notched specimens [3], pressure vessels [4]). However, fundamental and representative results characterizing the material have to be obtained using the concept of the representative volume element (RVE). In the case of a structure, the results can be dependent on geometry which is not an issue concerning the fundamental failure process and this is the reason why, in this paper, the failure of unidirectional parallelepiped specimens will be considered. Failure is induced by the accumulation of fibre breaks which, in the simulation, induces a numerical instability: this instability is the basis of the definition of the failure point (§3b). With the model which is used, it is possible to determine, up to the failure point, the evolution of damage at the level of the fibres and follow the development of clusters of fibre breaks under both monotonic and steady tensile loading conditions. In the former case, the viscoelastic nature of the matrix has a negligible effect, but in the latter case, it is fundamental in determining the rate of increase of damage which can lead to failure. The effects of the viscoelastic nature of the resin matrix are then seen to be of fundamental importance in determining damage accumulation under sustained loading conditions. These effects have to be taken into account in quantifying safety factors for many composite structures.

This paper will show that an understanding of the development of clusters of fibre breaks, denoted as i-plets [5,6], allows the critical density (number/volume) of clusters of fibre breaks leading to failure of the composite to be quantified. It will also be shown that the nature of damage accumulation varies with the type of loading imposed on the composite. The basis of the model which is used here has been described in detail elsewhere [7–11].

## 2. Bibliographical review

Unidirectional advanced composites, such as carbon fibre reinforced epoxy resin, when loaded in the fibre direction are often considered as having purely elastic mechanical behaviour. The carbon fibres which are used as reinforcements are elastic and their behaviour shows no time-dependent behaviour [12]. However, when embedded in a matrix, the behaviour of which is usually viscoelastic, progressive fibre failure has been shown to occur when the composite is under (any type of tensile) load, even though the macroscopic characteristics of the composite are dominated by the fibres [13,14]. Although such time-dependent phenomena are not easily apparent in monotonic tests of short duration, in the case of sustained loadings delayed fibre failure is clearly a potential issue for the long-term reliability of the composite. It has been shown that analogous behaviour is seen between that of unidirectional composites under sustained load and that of filament wound structures such as pressure vessels and pipes subjected to internal pressure. This time-dependent behaviour has been modelled and shown to be due to the viscoelastic behaviour of the matrix [8]. Experimental verification of delayed failure of unidirectional carbon fibre composites has been reported by Bunsell *et al.* [15] and the progressive accumulation of fibre breaks when unidirectional carbon fibre composites are subjected to monotonic tensile loading has been observed using high-resolution tomography by Scott *et al.* [3]. The increasing use of carbon fibre reinforced composite pressure vessels for gas storage under high pressures has highlighted the need for a better understanding of the failure processes in the composite and in particular the effects of prolonged loading on the reliability of the structures. Such pressure vessels and also filament wound pipes subjected to internal pressures are composed of rovings of continuous carbon fibres embedded in a viscoelastic matrix. In the case of a standard vessel subjected to internal pressure, the reinforcements in such filament wound structures are, locally, in tension so that an analogy with the behaviour of unidirectional composites can be made.

The failure processes in unidirectional composites have been studied by many researchers over many years. Some such studies, but by no means all, are: [5,6,16–44]. Very few studies considered the effects of time or prolonged loading but a few did such as Lipschitz & Rotem [45], who discussed how elastic fibres could nevertheless be subjected to differed failure by the relaxation of the resin around fibre breaks which led to increased loading and sometimes failure of neighbouring intact fibres. This was further discussed by Beyerlein *et al.* [46], who considered the evolution of the stress distribution around multiple fibre breaks in a composite. The work of Holmes *et al.* on fragmentation test of fibres should also be mentioned, the objective of which was to determine whether time affected the number of fibre breaks formed [47,48] and also to better understand the statistical special distribution of breaks along the fibre [49].

## 3. Methodology

### (a) Description of the material: i-plet populations and fibre breaks

The considered material is a standard unidirectional carbon/epoxy with a fibre volume fraction *V*_{f}=0.64. The properties of this material are similar to those that can be found in [50]. The description of a given i-plet population or of the fibre breaks population is given with a percentage which refers to the load applied as a percentage of monotonic breaking load and accumulated percentages of all possible breaks at different points during the tests (table 1). In the nomenclature used in the model [3,11,15,51], the populations of i-plets are: small-order i-plets (2-plets and 4-plets), medium order i-plets (8-plets and 16-plets) and high-order i-plets (32-plets).

### (b) Description of loadings in tension: failure point and clustering effect

Parallelepiped specimens loaded until failure have been numerically modelled. A uniform density of force, parallel to the longitudinal direction of the specimen, was applied to the cross-sectional surface of the specimen at the far ends so as to simulate a tensile test.

Two types of loading conditions were considered (figure 1):

— one was a monotonically increasing tensile loading at 1 MPa s

^{−1}so that the viscoelastic properties of the matrix could be neglected (the characteristic relaxation times of the viscoelastic behaviour which have a significant effect on the total strain of the unidirectional material are much higher than the duration of a test made a 1 MPa s^{−1}). This loading is denoted as ML loading. Numerically, the specimen was loaded until a numerical instability (predicted by the Aveston–Cooper–Kelly (ACK) model [52,53]) occurred indicating the (numerical) longitudinal failure stress*F*_{ML}of the specimen [51]. The average value of*F*_{ML}is denoted as 〈*F*_{ML}〉;— the other was a monotonically increasing tensile loading at 1 MPa s

^{−1}until a given loading*F*_{SL}(designated as the load level) was reached and then maintained for a simulated period of 20 years at this peak level. After that, the loading was once more increased at 1 MPa s^{−1}until a numerical instability occurred indicating the (numerical) longitudinal failure stress*F*_{R}of the specimen. The average value of*F*_{R}is denoted as 〈*F*_{R}〉. In this case of sustained loading, the viscoelastic character of the matrix can no longer be neglected. This type of loading is denoted as SL loading.

The loading curve (figure 2) defines two important points: the failure point (or instability point) denoted as *J* and the Start of INSTability point denoted as *I* [11,51]. The instantaneous extension on failure shown (figure 2) can be compared to analogous behaviour during damage accumulation in composites as described in the ACK model [52,53] and would not be experimentally observable. As has already been demonstrated [51,54], the first event of the clustering process is the random appearance of small order i-plets (in this step, clusters cannot really be said to exist). The second event is the random growth of small-order i-plets as medium or high-order i-plets (at this step clusters appears). The last event is the concentrated accumulation of 32-plets or in others words, clusters of clusters of 32 broken fibres (32-plets).

### (c) The statistical analysis of the results is based on a Monte Carlo process

Owing to the stochastic character of the fibre breaks, 250 calculations have been carried out for each of the loading configurations considered in this study. A simulation is defined as a set of 250 calculations of a given loading configuration. So as not to perturb the analysis and conclusions by the introduction of undesirable statistical aberrations, we proceeded as follows:

— 250 Monte Carlo runs were made: if

*N*_{G}was the total number of Gauss points of the mesh of the structure, each Monte Carlo run, denoted as*C*_{i}(*i*=1,…,250), gave*G*=5×*N*_{G}random values between 0 and 1. In this way, the Weibull fibre strength distribution gave a set of*G*values of fibre strength, denoted as*G*_{i}(*i*=1,…,250);— these 250 sets

*G*_{i}(*i*=1,…,250) of*N*_{G}values of fibre strength were then saved; and— then, one set

*G*_{i}(*i*=1,…,250) was used for one of the 250 calculations made for a given loading configuration.

In this way, a probability curve could be obtained for each of the simulations and described with the average and the standard deviation of the 250 results. Moreover, it has been seen that one among these 250 Monte Carlo realizations *G*_{i} (*i*=1,…,250) gave results which had a probability of approximately 0.5 (this is still the one which is statistically the most likely). This set has been denoted as *G*_{1/2}. For each of the loading configurations considered in this study, the analysis of the damage states was made with the calculation using *G*_{1/2}. In this way, undesirable statistical aberrations were avoided and a coherent critical damage state could be determinated.

The final point which has to be underlined is that the number of Gauss points was large enough (or, in other terms, the specimen was big enough) to ensure that there was no perturbation of the numerical results due to the size of the specimen: each set *G*_{i} (*i*=1,…,250) gave a good sample of the fibre strength distribution.

## 4. Critical damage state for the case in which the viscosity of the matrix could be ignored

### (a) Description of the loading

The density of force applied to the specimen had a monotonic linear dependence with time and the loading rate was 1 MPa s^{−1} (ML conditions). Two hundred and fifty simulations of the monotonic tensile loading were then carried out, each one differing from the others concerning its own data (in a way described in §3c). Consequently, the results obtained are different. For each calculation, a value of the failure stress of the specimen was obtained (, 1≤*i*≤250). Using these 250 values, a probability density of failure stress was obtained using a two-parameter Weibull function: 〈*F*_{ML}〉=2921 MPa and the standard deviation is approximately equal to 7 MPa.

### (b) Identification of a critical damage state under monotonic tensile loading

The analysis carried out was based on the results of the Monte Carlo calculation using *G*_{1/2}. Under ML conditions, the most important results already obtained were [11,51]:

— stable growth of random fibre breaks was predicted until the failure point

*J*;— the development of higher order i-plets (32-plets) reflects the second derivative of the (average) longitudinal strain 〈

*ε*_{11}〉, as a function of applied stress, in that, in the linear part of the loading curve, which demonstrates (quasi-) linear elasticity, very few 32-plets develop but begin to be formed when the curve deviates from linearity and increased dramatically during the final failure stage. This indicates that high-order i-plets was the key parameter associated with failure of the specimen;— numerical instability was predicted to initiate as the 32-plets began to develop, indicating that high-order i-plets damage was the key parameter associated with failure of the specimen;

— up to the Start of INSTability point

*I*(figure 2) the majority of fibre breaks consisted of small-order i-plets (2-plets, and to a lesser extent of 4-plets) randomly distributed. Very few medium order i-plets (8-plets and 16-plets) appeared until point*I*and more significantly they appeared and disappeared (being transformed to 32-plets) after point*I*. At this point, only around 4% of all possible fibre breaks were seen to have occurred, 80% of the all possible RVEs making up the specimen were not damaged and only around 2% of all possible 32-plets had developed (table 2);— at the failure point

*J*(figure 2), only around 8% of all possible fibres breaks were seen to have taken place, 70% of the all possible RVEs making up the specimen were not damaged and only around 6% of all possible 32-plets had developed (table 3); It was also seen that approximately 80% of the total number of fibre breaks developed were associated with the fibre breaks in the fibre failure clusters of 32-plets. This supports the view that the clustering of fibre breaks is the critical damage process controlling failure.

The question arises whether there is a critical damage state beyond which failure is inevitable. The evolution of the values considered, such as the population of i-plets and average distance to the nearest-neighbouring [11] break for the 32-plets, changed significantly at point *J* and allowed the critical damage state for failure to be ascertained. The calculation of point *J* therefore is pertinent and confirms that the point *J* is the failure (or instability) point. However, as mentioned by Thionnet *et al.* [11,51], it is considered that point *I* represents the practical limit of damage for the composite as the critical level at point *J* is too quickly attained to be observable. Then, by analysing the state of damage at the point *I*, a critical damage state in the case of ML loading can be defined as around 4% of all possible fibres breaks leading of the creation of around 2% of all possible 32-plets and about 20% of the all possible RVEs making up the specimen contains at least one fibre break.

## 5. Critical damage state for the case in which the viscosity of the matrix could not be ignored

### (a) Description of the loading

Under sustained tensile loading conditions, it becomes necessary to take into account the viscoelastic nature of the matrix material. The case considered will be that of a unidirectional composite loaded in the fibre direction up to a load, which is a proportion of 〈*F*_{ML}〉 (§4): *F*_{SL}=*X*%×〈*F*_{ML}〉. The load was then held constant for a simulated period of 20 years before being increased to induce composite failure (figure 1):

— the specimen was loaded from 0 to

*F*_{SL}at a speed 1 MPa s^{−1}(ML conditions). The instant that*F*_{SL}was reached was noted as*t*_{eol};— the specimen was then held at

*F*_{SL}for a simulated period of 20 years (SL conditions). The end of this loading period was noted*t*_{eos}so that*t*_{eos}−*t*_{eol}=20 years;— at

*t*_{eos}the load was then increased (ML conditions), without unloading, at a rate of 1 MPa s^{−1}until failure occurred at a time denoted as*t*_{f}. The failure stress is denoted as*F*_{R}and was determined in an exactly similar way as for*F*_{ML}.

Several values of the sustained load *F*_{SL} were examined such that *X*=1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 60, 70, 80, 82, 84, 86, 87, 88, 89, 89.5, 90%.

### (b) Identification of a critical state of damage under sustained loading

Many analytical models of composite failure have assumed that the matrix deformed plastically in shear by being taken beyond its elastic yield point. This greatly simplifies the analysis of stresses around fibre breaks, as the shear yield stress can be considered to be constant. It is undoubtedly justified for metal and some thermoplastic matrix composites. In the present case, however, the matrix was epoxy resin and examination of fracture surfaces did not suggest that the matrix sheared plastically around fibre breaks. For this reason, the load transfer through the matrix from a fibre break to neighbouring intact fibres was considered to be viscoelastic rather than plastic. As for the monotonic loading case, the analysis carried out was based on the results of the Monte Carlo calculation using *G*_{1/2}. Under SL conditions, the important results obtained were [11,54]:

— the fibre failure process is induced by the viscoelastic relaxation of the matrix. As a result, new fibre breaks could occur continuously during sustained tensile loading;

— the development of higher order i-plets (32-plets) occurred during sustained constant loading and increased dramatically, as did 〈

*ε*_{11}〉, during the final failure stage; this indicates that high-order i-plets was the key parameter associated with failure of the specimen. The appearance of 32-plets was much more evident than in the ML case. Very few were observed before and at the failure point*J*in the ML case, whereas in the SL case they represented around from 10 to 18% of the total fibre breaks;— at the same time, the i-plets of lesser order decreased (for the 2-plets) or only increased slightly (for the 4-plets) while these populations increased continuously in the ML case. This indicates that in the SL case the i-plets of lower order transform very quickly into higher order i-plets. For the i-plets of intermediate order, their appearance is similar to that seen for the ML case;

— this indicates that the mode of evolution of fibre breaks before failure is a creation of clusters in the case of SL conditions while it is predominantly random in the ML case. In addition, the clustering effect appears early in the loading of the specimen, at the beginning of the period of steady loading;

— the numbers of broken fibres and 32-plets at the failure point

*J*in case of the SL conditions were approximately double those calculated for the ML conditions; and— it was also observed that the proportion of broken fibres shown as 32-plets was always very high as a percentage of all the fibre breaks (80%). This confirms the central role of the high-order i-plets in determining the failure process.

A complete synthesis of the results for all the loads investigated is shown in tables 2 and 3. The damage states for both ML and SL conditions can be found for the Start of INSTability point *I* in table 3 and for the failure point *J* in table 2. The critical damage state in case of SL conditions is approximately double as for the ML conditions.

## 6. Conclusion

This study has extended the knowledge on the kinetics of damage accumulation in unidirectional carbon fibre composites. In the framework of the RVE of unidirectional composites, monotonic tensile loading leads initially to random single fibre failures throughout the composite. The points of fibre failure are determined by variations in strength along the fibres due to local irregularities or defects induced by the fabrication process or handling of the fibre. The stochastic nature of fibre failure is therefore a controlling characteristic in determining the kinetics of damage accumulation in the composites.

In the case of monotonically increasing tensile loading, a point is reached at which clusters of breaks occur around earlier breaks. The kinetics of the development of these clusters of fibre breaks has been quantified. When larger clusters of breaks occur, associating typically 16–32 broken fibres, damage rapidly develops and tensile failure takes place soon afterwards. Significantly just before failure, at *I*, the last experimentally observable point, only around 4% of all the fibres in the composite are broken. The failure point *J* occurs when about 8% are broken and failure follows immediately afterwards.

In monotonically increasing tensile loading, the viscoelastic nature of the matrix does not play a significant role but becomes the controlling mechanism when the load is applied for long periods. In this study, steady loading has been considered but earlier experimental work on the effects of constant amplitude cyclic loading of unidirectional carbon fibre composites revealed very similar behaviour [13]. Under these conditions, the steady load is obviously lower than that which causes failure in the monotonic tests and the composite is able to support greater degrees of damage without rapidly failing. However, the accumulation of damage can lead to delayed failure of the composite. It has been shown, and quantified, that a higher level of large clusters (32-plets) of fibre breaks can be supported by the composite under sustained loading and do not immediately lead to failure. Indeed, randomly distributed clusters (medium or high i-plets) are created early in the sustained period of loading however when the instability point is reached, either under extended sustained loading or by loading to failure after lengthy sustained loading, failure will occur just as described for monotonic loading. In contrast to the monotonic loading case sustained loading leads to a much more diffuse damage so that when failure occurs at one point there are many other points throughout the composite which are almost just as vulnerable. The damage level at point *I* and failure point *J* under sustained loading is approximately twice that which is calculated for monotonic loading, irrespective of the sustained loading level. The role of the viscoelastic nature of the matrix is clearly demonstrated under long-term loading. In this way, it becomes possible to evaluate in a quantitative manner the critical damage levels under different loading conditions which should allow the intrinsic safety factors for composite structures to be determined.

## Data accessibility

The authors are willing to provide any data requested concerning this study as long as it has been published in the open literature.

## Authors' contributions

A.R.B. conceived and coordinated the study, provided the background information, defined the research programme based on the physical mechanisms governing failure, contributed to the interpretation of the data and drafted the manuscript. A.T. developed the computer simulation of the damage processes and collaborated in interpreting the data. Both authors gave final approval for publication.

## Competing interests

The authors declare that they have no competing interests.

## Funding

We received no funding for this study.

## Footnotes

One contribution of 22 to a Theo Murphy meeting issue ‘Multiscale modelling of the structural integrity of composite materials’.

- Accepted March 2, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.