## Abstract

A parallel has been made between the design and the failure of some composite structures, such as pressure vessels, and that of a unidirectional carbon fibre-reinforced epoxy resin. This type of composite has many characteristics that make it attractive, however, long-term damage accumulation in it, when under load, is not understood in any quantitative manner. It is therefore necessary to identify the processes involved in the degradation of the composite when subjected to long-term loading. Acoustic emission (AE) has been used to detect damage in the composite material and to relate the microstructural damage processes to the activity recorded from composite specimens through the development of a detailed analytical model of the damage processes involved. A multiscale model beginning at the scale of individual fibres and the surrounding matrix has been developed by taking into account the accumulated effects of individual fibre breaks on the behaviour of the whole composite and therefore explains the AE detected during the tests.

## 1. Introduction

The processes leading to damage accumulation in composite materials and their eventual failure have been the objects of speculation ever since their initial development as structural materials. While some studies have dealt with the processes at the level of the fibres, most have preferred to examine macroscopic behaviour and only to examine the effects of macroscopic damage due to shock or delamination. High-performance carbon fibres first became available in the late 1960s and have taken an unassailable position as the reinforcement of choice for an ever-widening range of applications. Their high-performance elastic properties, which dominate the properties of any carbon fibre reinforced plastic (CFRP) composite, have led many commentators to conclude that, in the direction of the fibres, such composites behave as a purely elastic body, insensitive to time effects. This assumption is not justified at the microscopic level but is a reasonable approximation for short-term applications. However, the increasing use of these composites in structures for which lifetimes, under load, is measured in tens of years, has meant that a closer examination of time-dependent damage processes, at the microscopic level, leading to eventual failure, needs to be made and understood. In this way, reliable lifetime estimates of structures such as composite pressure vessels can be achieved and permit the development of large composite markets for this type of product. Computing power, now available, allows a multiscale approach to modelling damage accumulation in composites. This paper will demonstrate that it is possible to predict macroscopic behaviour of composite structures based on damage processes occurring at the microscopic level, and therefore to predict minimum residual lifetimes of some types of composite structures.

The most elementary level of any composite material is that of a fibre embedded in a matrix. The role of the matrix is to transfer load, through its deformation in shear, between the reinforcing fibres. If subjected to loads in the direction of fibre alignment, it is the response and possible failure of all such fibres in the composite and the consequences this has on fibres neighbouring the breaks that will govern macroscopic composite degradation. This degradation may not be detectable at the macroscopic level owing to the great numbers of fibres involved and the random nature of the failures, but an increasing number of breaks will eventually lead to instability in the composite when the density of fibre breaks reaches a critical threshold.

The simplest composite is one which is unidirectionally reinforced with continuous elastic fibres embedded in a resin and which is loaded in the fibre direction.

## 2. Existing theoretical models

Approaches to predicting the failure of unidirectional composites have generally been based on: the stochastic nature of fibre strength, often modelled by a Weibull distribution; the stress transfer coefficients on intact fibres neighbouring fibre breaks; and the development of a critical size of defect comprising broken fibres or the coalescence of regions of broken fibres so as to make the composite unstable.

Rosen (1964) proposed a model of unidirectional composite failure based on the effects of the scatter of fibre strengths. In this model, the average stress supported by the *m* fibres reinforcing the composite, before first fibre failure, was given the value, *σ*_{f0}. The shear of the matrix, around a fibre break, resulted in load being transferred back into the broken fibre, up to a given fraction *ϕ* of *σ*_{f0}. The composite was then divided into *N* links, analogous to links in a chain, the lengths of which were related to the critical load transfer length, *δ*, and resulted in the broken fibre continuing to support the applied load away from the point of failure. Within the link, all the remaining intact fibres were considered to take on the additional load originally supported by the broken fibre. No stress concentration effects were considered. The calculation of the parameters of Rosen's model was possible using Cox's analysis of a single short fibre embedded in a cylinder of matrix and which had introduced the concept of shear-lag, or load transfer, through the shear of the matrix around the fibre break (Cox 1951). The calculation of the failure stress of the composite, *σ*_{cR}, required the use of the function *G*_{m}(*σ*_{c}) (*σ*_{c} is the uniaxial stress existing in the composite), describing the distribution of fibre strengths in a bundle of *m* fibres of length *δ*. The model developed an expression for the most probable failure stress of the composite, which was *σ*_{cR}=*σ*_{0}(*δβe*)^{−1/β}, in which *β* and *σ*_{0} are the Weibull modulus and the scale factor, respectively. This model was found to overestimate composite failure strengths and considers far fewer fibres than that which is generally the case.

Zweben (1968) extended Rosen's model and introduced the stress concentrations, in fibres neighbouring fibre breaks, calculated by Hedgepeth (1961) and was able to calculate the effects of several adjacent fibre breaks on the probability of composite failure. This model introduced the concept of a critical defect size, or adjacent fibre breaks, which determined failure. The model, however, underestimated the strength of unidirectional composites.

Ochiai *et al*. (1991) examined the role of the axial rigidity and strength of the matrix and the effects of matrix cracking around any broken fibre by introducing three-dimensional shear-lag, as calculated by Hedgepeth (1961). Goree & Gross (1980) examined the effects of the number of broken fibres on load transfer coefficients to the remaining intact fibres. Harlow & Phoenix (1978*a*,*b*), inspired by Scop & Argon (1967, 1969), developed approximate expressions (Harlow & Phoenix 1981*a*,*b*) of the distribution functions for the failure of bundles of *m* fibres *G*_{m}(*σ*_{c}) and showed the effect of scale on the Weibull modulus. If the scale is very large, as with a composite structure, the value of the Weibull modulus was large and determinant in the failure of the composite. This situation was reached more rapidly if the effect of fibre breaks was concentrated.

The complexity of the physical processes involved encouraged Kong (1979) to propose a numerical approach. Kong introduced a Monte Carlo simulation in a Rosen-type model for calculating fibre failures in the composite and this was an important step in the understanding of composite failure, revealing that failure corresponded to a loss of stability in the calculation.

Batdorf (1982*a*,*b*) considered the probability of obtaining groups of i-fibres broken, called i-plets, analogous to the creation of a Griffith-type critical crack. For a given applied stress, there would exist a critical size of the i-plets leading to critical stress concentrations on intact fibres and composite instability. This implied that composite strength was a function of the confinement of the damage. The model linked the Weibull distribution of fibre strengths to that of the composite and showed how the composite strength became deterministic when the number of fibres was great. The model, which makes a number of major assumptions, nevertheless supplied a chronological description of damage processes. In the absence of macroscopic stress concentrations, random fibre failures in the composite led to the creation of damage zones, which grew with increasing stress until a critical size was reached, at which point failure occurred. The model revealed the effects of local mechanisms, in particular interfacial bonding on load transfer length, on the macroscopic strength of the composite.

Nedele & Wisnom (1994*a*,*b*) used a finite element (FE) approach to calculate the three-dimensional state of stress around fibre breaks. Their results were consistent with those using a shear-lag model and gave a stress concentration in intact fibres neighbouring a fibre break of 1.058, compared with 1.104 predicted by Hedgepeth & Van Dyke (1967). They also showed the dependence of stress concentrations on fibre volume fraction. Wisnom & Green (1995) later proposed a failure scenario based on the energy involved in the failure of a fibre and its localized effects on the matrix. Composite failure was seen to implicate the localized failure of between 4 and 15 fibres. This mechanism was related to the pullout energy of the fibres and allowed composite failure stresses to be calculated that were close to experimental values.

Baxevanakis (1994) used statistical data from multifragmentation experiments to characterize a microscopic scale of damage, which was incorporated into a more macroscopic two-dimensional FE analysis study of unidirectional composite behaviour. In this way, a representative volume element (RVE) of the composite material could be defined within which all the damage processes could be considered to occur and which defined the ultimate unidirectional composite strength.

Landis & McMeeking (1999) and Landis *et al*. (2000) pursued the three-dimensional resolution of the equations governing the shear-lag model. This allowed the simulation of a greater number of fibre breaks and confirmed the sequential nature of damage in composites, but remained limited by the shear-lag model and underestimated composite failure stresses.

The models described above, and by others, generally consider both the reinforcements and the matrix to be linearly elastic or perfectly plastic and do not consider the known viscoelastic nature of real resin matrix systems. However, Lifschitz & Rotem (1970) showed that a viscoelastic matrix would result in time-dependent failure of unidirectional composites, reinforced with elastic fibres and loaded in the fibre direction, based on Rosen's model. The mechanism proposed was the increase in the ineffective load transfer length from the broken ends of a fibre and therefore an increase of its effects on neighbouring intact fibres, due to the relaxation of the matrix shear stresses in the vicinity of the break.

Lagoudas *et al*. (1989) modified the two-dimensional shear-lag model of Hedgepeth & Van Dyke (1967) so as to study the effects of a linearly viscoelastic matrix. By considering a creep rate of the matrix, which was described by a power law, they were able to show that the sizes of damage zones and the ineffective load transfer lengths increased with time. More recently, Beyerlein *et al*. (1998) developed a ‘viscous break interaction’ model based on the work of Lagoudas *et al*. (1989) but using the superposition principle used by Sastry & Phoenix (1993, 1994), for the elastic case, to take into account the random nature of fibre failures. However, these models are still limited, as shown by Goree & Gross (1980) and Nedele & Wisnom (1994*a*,*b*), to the case of an elastic matrix.

The above summary of existing theories considers some but not all of the processes involved in the accumulation of damage in unidirectional composites loaded in the fibre direction. None of the models discussed, however, is entirely satisfactory so that a three-dimensional FE model will be developed below to analyse more precisely the complex stress and strain fields that arise during the accumulation of fibre breaks in such a composite material. In order to make the analysis of any real composite structure manageable, given the large number of parameters involved, a multiscale approach has been adopted.

## 3. Multiscale modelling procedure

A RVE of the composite is the smallest volume of the composite, which can be considered to contain all the physical mechanisms that govern the composite behaviour. The material is here assumed to be periodic and considered to be made up of a regular network of elementary periodic cells that make up each RVE. At the microscopic scale, all of the different phases, fibres, matrix and interfaces, are considered and each is considered to be homogeneous. The scale at which the material is seen to be homogeneous is called the macroscopic scale. The calculation using a multiscale procedure consists first of applying the macroscopic stress or strain field to the RVE, at each point of the macroscopic structure, so as to determine any changes that occur within the RVE at the microscopic scale: this step is called the ‘localization step’. The next step is called the ‘homogenization step’: the response of the composite is calculated by summing the effects at each RVE. These two steps are then repeated until the end of the calculation. In our case, the calculation of the macroscopic stress or strain states will be carried out using FE method (FEM).

The analysis at the macroscopic level of the uniaxial tensile loading of a unidirectional composite, in the fibre direction, shows that the stress fields are rigorously uniaxial at the beginning of the loading (the composite is considered homogeneous and undamaged). During loading, microscopic phenomena such as fibre failures lead to degradation of the composite. Nevertheless, the macroscopic stress state is considered to be little affected by microscopic effects. Hence, we assume that the RVE is always subjected to uniaxial loading.

In this way, the localization step uses the smoothing of the results of a microscopic investigation, in which the RVE is uniaxially loaded so as to determine the effects of load transfer on intact fibres due to the breakages of fibres.

As such, the first part of the study is to evaluate the local stresses on the fibres within an RVE, for a given uniaxial macroscopic tensile stress using the FEM. In this way, as mentioned previously, it is possible to determine the variation of stress in intact fibres neighbouring a broken fibre by smoothing the values out that are calculated at points along the fibres. In this way, it becomes possible to construct a database for the localization step, which is directly useable in the multiscale analysis.

## 4. Summary of the microscopic analysis used in the localization step of the multiscale process

The results presented here are a summary of a more detailed investigation at the microscopic scale of fibre failure and its consequences (Blassiau 2005; Blassiau *et al*. 2006*a*,*b*). The approach used will be succinctly described as will the most important results obtained. These results will then be smoothed so as to use them in the multiscale process.

### (a) Modelling of the RVE for different damage states

This micromechanics study undertakes a three-dimensional analysis of the RVE loaded in the fibre direction. The two-dimensional RVE used by Baxevanakis (1994) that gave a constant average unidirectional composite failure stress consisted of six parallel fibres of 8 mm in length.

If we assume that the material is transversely isotropic, in the present three-dimensional study, we can suppose that the RVE consists of 6×6 parallel fibres. For reasons of convenience, an array of 32 fibres was chosen. For the periodic distribution assumed for the undamaged composite, in this study, the RVE was called CS-32 and the undamaged state was called E-*S*. The RVE of the undamaged material therefore contained 32 fibres of 8 mm in length, arranged in the form of a square. The degrees of damage were modelled considering six levels of failure in the RVE. The state of damage was described as E-*N* (*N*=32, 16, 8, 4 and 2). E-*N* represents the state of damage in the composite so that the number represents one fibre broken for that number of intact fibres. The associated periodic cell was given the abbreviation C-*N*. The failed state E-0 is represented by C-0. It will be shown that the state which represents isolated fibre failures, so that there is no interaction between breaks, is described by the damage state E-32 and gives unvarying failure stress for the composite. This justifies the choice of the CS-32 cell as the RVE to represent the three-dimensional state of the undamaged composite structure. It should be noted that the periodic nature adopted to describe the composite structure allows the interaction of damage sites to be modelled. The state of damage described as E-∞ represents one broken fibre in an otherwise undamaged composite. This is the state modelled by Nedele & Wisnom (1994*a*,*b*), who considered one broken fibre and its intact neighbour surrounded by a homogeneous medium with the average properties of the composite. The present model will be compared with the latter so as to validate the approach and we will compare the results obtained with the damage state E-32 and the C-32 cell.

### (b) Fibre and matrix behaviour

The fibre is considered to be always linearly elastic, homogeneous and isotropic. In contrast, two types of behaviour have been considered for the matrix. The first and the most reductive one considers the matrix to be linearly elastic, homogeneous and isotropic. The second one considers the matrix to show linear viscous behaviour described by the following equation: , where is the fourth-order relaxation tensor; _{0} is the initial rigidity tensor; and _{∞} is the final rigidity tensor after an infinite time. As the matrix is isotropic, each of the tensors is isotropic. This model allows the notion of time to be included in the behaviour of the composite and therefore permits deferred composite failure to be studied.

### (c) Debonding modelling

Debonding is modelled by the undoing of the boundary nodes of the matrix and the broken fibre. Outside this zone, the fibre/matrix bond is considered to be perfect. Owing to the periodic conditions, the debonded length is considered to be twice the length from the broken end of the fibre to the point where the fibre and the matrix are again adhered.

### (d) Definition of the load transfer coefficient

The goal of these calculations is to simulate the failure of a fibre and to observe the evolution of the axial stress in neighbouring intact fibres as a function of the damage state, time, distance from the plane in which the break lies and as a function of any debonding of the fibre/matrix interface. In order to do this, we define a coefficient of load transfer, *k*_{r}. This is obtained as the mean of two axial stresses, in the fibre considered, calculated between two consecutive normal sections of the fibre in the FE mesh compared with the same quantity in the undamaged state,where *C* is the studied cell; *L* is the debonding length; *t* is the time after the fibre break; *z* is the axial coordinate, *z*=0 is the axial coordinate of the plane of the break; *Z*_{i+1} and *Z*_{i} are the axial coordinates of the normal sections between those *k*_{r} is estimated; *Z*=(*Z*_{i+1}+*Z*_{i})/2; *S*_{F} is the current normal section of the considered fibre; *x* and *y* are the coordinates describing the plane of *S*_{F}; *σ*_{zz} is the axial stress in the fibre considered; and is the axial stress in the fibre considered in the undamaged material.

It has been verified that the discretization employed is that which results in convergence. In this way, the coefficient calculated is independent of the discretization as well as the length of the integration, where this length is the thickness of an element. The mesh is so fine that the stresses are nearly constant in each element.

Regardless of whether the interface between the broken fibre and the matrix is broken or not, and the matrix is viscoelastic or not, the definition for the coefficient *k*_{r} remains valid. However, so as to separate, quantitatively, in the coefficient that is due to the fibre break, the debonding or the effects of the viscoelasticity of the matrix (time), we write the following expression: , where *K*_{r} is the load transfer exclusively due to the failure of a fibre; is the additional part of the load transfer due exclusively to the debonding, with if this effect is neglected; and is due exclusively to the load transfer due to the viscoelastic properties of the matrix, with if this effect is neglected.

This coefficient will be evaluated for three different cases. Case 1 considers the C-∞ cell consisting of linearly elastic, homogeneous, isotropic components and made to evolve as the debond length increases from zero to a given length. Case 2 treats the different cells modelling increasing damage in the composite (C-32, …, C-2) considered as consisting of linearly elastic, homogeneous, isotropic components during the evolution of the debond length between zero and a given length. Case 3 reconsiders the above two cases but with a linearly viscoelastic, homogeneous and isotropic matrix. In all the cases, the fibre volume fraction is taken to be 40%.

### (e) Results for the case 1

To validate the assumptions made in the model, an initial comparison was made with the self-consistent case, already accepted in the literature, of one broken fibre surrounded by a homogeneous medium having the macroscopic properties of the intact composite. The convergence of the calculations that will be presented has been the subject of a detailed analysis presented elsewhere (Blassiau 2005; Blassiau *et al*. 2006*a*,*b*). The results obtained for the C-∞ cell, for the case in which the interfacial bond is maintained between the broken fibre and the matrix, are in good agreement with the results of Nedele & Wisnom (1994*a*,*b*) and Van den Heuvel *et al*. (1998, 2004). The self-consistent models inevitably underestimate the level of stresses supported by intact fibres near to the plane of failure, as the homogeneous equivalent material is more rigid than the matrix material. This leads to difficulties in understanding the subsequent behaviour of these fibres in a model that takes the effects of time into account. A fibre volume fraction of 40% has been chosen so as to consider the effects of interfacial debonding. Any effects due to friction between the broken and debonded fibre and matrix have been neglected. The effects of debonding have been calculated for discrete lengths. A comparison has been made with the results of Van den Heuvel *et al*. (2004), who concluded that a debonded length equivalent to approximately 10 fibre diameters (approx. 70 μm) from the point of failure gave rise to a load transfer coefficient considerably less than that calculated without debonding (Blassiau 2005; Blassiau *et al*. 2006*a*,*b*). This difference is due to the much greater fineness of the FE mesh used in the present study compared with those in earlier works. Less fine meshes do not reveal the increase.

### (f) Results for cases 2 and 3: analysis of the microscopic state as a function of composite damage

The overload of intact fibres neighbouring a broken fibre, *k*_{r}, within the RVE has been analysed for different states of composite damage (C-32, C-16, C-8, C-4 and C-2), with and without debonding. The matrix has been considered to be, successively, elastic and viscoelastic. The unrealistic result (Blassiau 2005; Blassiau *et al*. 2006*a*,*b*), that the value of *k*_{r} is less than unity for unbroken fibres, revealed by consideration of the C-∞ cell and in the Nedele & Wisnom (1994*a*,*b*) and Van den Heuvel *et al*. (2004) models is removed by the periodic nature of the model.

The results obtained for *k*_{r} in the plane of fibre failure (Blassiau 2005; Blassiau *et al*. 2006*a*,*b*), for an elastic matrix without debonding, confirm the choice of the CS-32 cell for the RVE. From the C-16 cell, the values of *k*_{r} for the nearest neighbours to the broken fibre are seen to be very close to those obtained for the C-∞ cell, with a reduction of *k*_{r} for fibres further removed. This demonstrates the existence of a limiting range of influence of the broken fibre that corresponds to the RVE chosen for the analysis and supports the concept of the concentration of damage postulated in the local load sharing (LLS) model, rather than being uniformly distributed among all the fibres in the section as supposed in the global load sharing (GLS) model.

The passage from an elastic to a viscoelastic matrix does not change the mechanisms of load transfer but introduces temporal effects. This, together with the statistics of fibre failure that can be obtained from single fibre tests, allows the effects of creep or relaxation of the matrix to be taken into account in determining delayed fibre failure. For instance, for the case of a C-8 cell, the stress in the nearest intact fibre neighbouring the break increases with time over a distance which also increases (Blassiau 2005, 2006*a*,*b*).

The analysis has allowed a detailed understanding of the load transfer mechanisms and the importance of the matrix behaviour and interfacial debonding to be more fully understood. It has allowed the evolution of the load transfer coefficients to be quantified in the cases of elastic and viscoelastic matrices, with and without debonding. These results will now be used to determine the effects at the microscopic level of localized damage on the global response of a unidirectional composite structure.

In order to reduce time of the multiscale calculations, the results obtained above are smoothed to obtain a continuous function. Hence, the localization step is analytically solved and will be directly useable in the multiscale process.

## 5. The localization step of the multiscale procedure: construction of the database with the microscopic analysis

The detailed understanding of the consequences of fibre failures allows the macroscopic response of each RVE to be determined as a function of different damage states and matrix behaviour. The values of *k*_{r} obtained at discrete points have to be written as a continuous function describing its value at any point within an RVE. This information will be used in the multiscale iterative calculation of the composite structure. The maximum value of the load transfer coefficient, , or, alternatively, its average value, , will be considered. As noted in §4*f*, and the expressions that are chosen for the smoothing of the results are the following:

The numerical values for the coefficients can be found in Blassiau (2005). The expression therefore takes account of the debonded length (*L*), the state of damage (*C*) and time (*t*), and implies that the effect of time is decoupled from the other phenomena so that any viscoelastic effect during initial loading is neglected. The effects of time on the coefficients are directly attributable to the viscoelastic properties of the matrix material used for the composite, as the properties of the fibre have been shown by Bunsell & Somer (1992) to be time independent. It has been observed, however, that tests on specimens of pure matrix do not give data that can be reliably used to model the behaviour of the same material reinforced with fibres (Schieffer 2002). The AE, recorded during constant loading tests on unidirectional specimens loaded to high fractions (more than 75%) of their known breaking stress, has been shown to originate from the failure of the reinforcing fibres (Fuwa *et al*. 1976). The rate of emission is therefore a reflection of the rate of load transfer to intact fibres in the vicinity of broken fibres. The values of the coefficients of smoothing have therefore been deduced from the rate of damage observed by the AE activity recorded during constant load tests. This means that in the multiscale approach to fit experimental results, the smoothing of the coefficient is adjusted so as to coincide the number of calculated fibre breaks with those recorded by AE during a constant load test at 75% of the failure stress.

## 6. The homogenization step of the multiscale procedure

Although the matrix is viscoelastic, no macroscopic viscoelastic effect is seen when a unidirectional CFRP is loaded in the direction of the fibres. For this reason, it was assumed for tensile tests that the homogeneous medium, equivalent to the composite material, behaved as a linearly elastic material. The failure of the reinforcements initially results in a fall in the longitudinal modulus of the equivalent homogeneous medium. The law of mixtures gives the following expression: , where *Q*_{11} is the rigidity of the equivalent damaged homogeneous material in the fibre direction; is the rigidity of the undamaged equivalent material; and *N*_{FC} is the number of intact fibres in the cell.

## 7. Macroscopic fibre failure criterion used in the multiscale procedure

Section 6 has allowed the data at the microscopic scale to be interpreted in a way that can now be used to give the localized values which are needed for the multiscale calculation of the composite structure. Locally, fibres will fail when the transferred additional stress, due to an earlier fibre break, results in the local critical stress for fibre failure being reached stress within an elementary cell. The criterion used at each integration point (IP) of the FE mesh is the following: the fibre is not broken until . The nature of fibre failure is, however, statistical, so that five failure stresses are associated with the RVE (CS-32 cell). These five failure stresses correspond to five stages that must be attained in passing from the undamaged state to that of the failed cell. The probability of carbon fibre failure has been shown to be described by Hitchon & Phillips (1979) using Weibull statistics and is given by the following expression: , where *σ* is the stress experienced by the fibre at a given point along its length and *σ*_{0} is a material parameter that has the dimensions of stress as the quotient is raised to the power *m*, which is known as the Weibull modulus, and multiplied by the fibre length, *L*. Random numbers between 0 and 1 are generated to represent the cumulated probability of fibre failure and used to calculate the failure stress of a fibre of length *L*, given by the following expression: . In order to generate the random stresses in the links, of which the fibre reinforcements are considered to be composed, it is necessary to define their lengths. Baxevanakis (1994) concluded that carbon fibres embedded in a matrix broke into lengths of approximately 500 μm. Hitchon & Phillips (1979) showed that the failure stress of carbon fibres varied little between lengths of 5 and 0.5 mm. The present analysis has taken the gauge length of the fibres to be 500 μm, so that the failure statistics can be defined from the scale factor *σ*_{0} extrapolated from a gauge length of 5 mm. The Weibull function used in this study has been the one extrapolated from results obtained with T600 carbon fibres with a gauge length of 5 mm and used for a length of 0.5 mm.

## 8. Multiscale three-dimensional calculation of the uniaxial loading of a unidirectional composite

The behaviour of a unidirectional CFRP subjected to a continuously increasing load in the direction of the fibres, as in a tensile test, was first considered. In this case, the test is of short duration and viscoelastic effects can be neglected so that the matrix is considered to be linearly elastic. The constant loading of such a specimen was then analysed and the viscoelastic behaviour of the matrix was taken into account.

The analysis considered the following three damage levels within the RVE:

level 1 corresponded to the reduction in composite rigidity due to fibre failures,

level 2 considered the load transfer between broken fibres and neighbouring intact fibres, and

level 3 included the effects of debonding.

Damage to the composite, due to the accumulation of fibre failures, was included in this FE analysis by the introduction of the statistics of fibre failure. Failure of the structure was determined by the instability of the calculation, as has been shown by Kong (1979). In order to carry out this analysis, the conditions leading to the convergence of the calculations had to be studied. In order to compare the numerical results with those obtained experimentally, the sizes of meshes for both the statistical fibre failure and the FE analysis were chosen as 0.10×0.10×15 mm, so that one element covered four RVEs within a specimen with the following dimensions: 5×0.9×150 mm. All three levels of damage were studied. For each damage level, 15 simulations were carried out with 15 different randomly chosen fibre strengths. The calculations are carried out for and .

### (a) Study of convergence

The RVE of the undamaged material was the CS-32 cell, which was a parallelepiped with a square section of 50×50 μm and a length of 8 mm. However, it was not feasible to model the whole specimen with such fineness so that the use of larger elements was necessary, but the size of the elements was chosen so that the results remained physically meaningful. The study of convergence of the calculations as a function of size of element was therefore necessary. First, the meshes for the statistical data and the FE study were merged and the sizes of the elements varied. Second, the statistical mesh was kept fixed and the number of elements varied. So as to limit the study of convergence, only the first level of the analysis was considered so as to result in a loss of 5% of macroscopic stiffness of the composite specimen. The first step was to vary the length of the parallelepiped (15, 10 and 6 mm) and the sides of the square section (1, 0.5, 0.3 and 0.15), which gave 12 distinct meshes for the tensile specimen. The number of degrees of freedom passed from 726 to 40 000 resulting in calculation times varying from 35 to 8000 s. Fifteen simulations were carried out for each size of mesh. Using the average macroscopic strengths obtained for the meshes, the average composite strength and standard deviations for each simulation, as a function of the fineness of the mesh, were calculated. Beyond a certain mesh size, the behaviour was asymptotic to a value of number of broken fibres, suggesting that the strength of the composite approximated to a determinist value with a standard deviation tending to zero. In the second procedure, the size of the statistical mesh was fixed, as well as the failure stresses of the fibres. The size chosen for the statistical mesh was that of 36 RVEs, which was 10×0.30×0.30 mm. This size was chosen as the mesh size allowed reasonable calculation times even with large specimens. For the statistical mesh, three FE levels of mesh were examined: mesh 1 for which the statistical and FE meshes were merged, the elements were linear consisting of eight IPs for each element; mesh 2 was identical to mesh 1, except that the elements were quadratic consisting of 20 IPs for each element; and mesh 3 for which the statistic mesh grouped eight elements. The results of these calculations revealed that the numerical solution did not depend on the fineness of the statistical mesh so that, for the rest of the study, the statistical and FE meshes were merged.

### (b) Results and comparison with the AE curve for a uniaxial tensile test

A uniaxial tensile test, to failure, of a unidirectional composite material was simulated using a three-dimensional FE mesh. Figure 1 shows that the proposed model faithfully reflected the experimental curve obtained. Table 1 shows the results obtained. It can be seen that the fineness of the analysis allowed to be used in the calculation and this resulted in a reduction in the failure stress of the composite. Progressing from levels 1 to 2 resulted in a reduction of 6% in strength and 9% when level 3 was considered. At level 3, an increase in debonded length had no effect on the failure strength. The same calculations based on showed that the coefficients of load transfer were less affected so that the composite failure stress was little altered. Failure stress was reduced with increasing level of damage; however, the number of broken fibres and the macroscopic rigidity of the specimen for levels 2 and 3 were similar. The distribution of fibre failures in each section of the tensile specimen is shown in figure 3 at the instant of failure of the composite specimen. A fall in failure stress can be seen due to a concentration of damage. The effect of the load transfer tends to reduce the number of intact fibres eventually provoking local instability and specimen failure.

The model gives a tensile curve for each level of refinement and for the failure statistics as a Weibull function for fibres of 5 mm in length as well as the accumulated fibre failures during loading. This allows a comparison with the recorded AE originating in a region of 100 mm in length between two transducers. Given the size chosen for the FE mesh, equivalent to four RVEs, the number calculated must be multiplied by four to represent the number of fibres broken in this region. Having constructed this curve, it can be compared with those obtained experimentally using AE, which related to fibre failures (figure 2). Other sources of emissions are ignored in this analysis of a unidirectional composite. The threshold chosen for the AE was 40 dB.

It can be seen that the numerical simulation of the uniaxial tensile loading of a unidirectional CFRP composite gives a reliable prediction of the composite failure stress and shows that the physical processes considered do control the failure. The model allows a good description of the processes that initiate the AE which are recorded during the tensile test. From figure 1, it can be seen that the average failure stresses calculated for each level of damage refinement agree with the experimental results. The failure of the unidirectional composite is therefore controlled by all the processes considered rather than any single one. The dispersion obtained with the numerical simulation can be measured quantitatively in terms of fibre failures. Depending on the localized zone, a standard deviation of more than 2500 breaks can be obtained. For each level of damage refinement considered, the average cumulated number of fibre failures remains constant. This number decreases as the damage refinement increases, illustrating the effect of damage concentration. However, it is found experimentally that the number of fibre failures per unit surface increases, with the reduction of the distance between the two AE transducers, and tends to a value close to that calculated numerically. This observation is thought to be due to the limitations of the AE technique and is illustrated by the large experimental dispersion obtained, which is not seen with the numerical simulation. The model suggests that fibre failures occur at lower stress levels than 70% of failure at which point AE is usually considered to be initiated.

### (c) Results and comparison with the AE curve for a uniaxial constant loading test

The viscoelastic behaviour of the matrix must be taken into account in the numerical simulation of the uniaxial constant loading of a unidirectional composite. A series of 15 simulations for loading conditions corresponding to 75, 80, 85 and 90% of composite failure stress have been carried out (Blassiau 2005). Figure 4 shows the average results obtained for an applied steady stress equal to 85% of failure stress using the three degrees of damage refinement and based on and . The scatter for the simulated curves can be seen to correspond well to the observed experimental scatter. Independent of the degree of damage refinement, an accumulation of fibre failures occurs due to the relaxation of shear stresses in the matrix and progressive load transfer to intact fibres neighbouring fibre breaks. The dispersion in the experimental results is reproduced in the numerical simulation and is seen to be due to the influence of the statistics of fibre failure and the evolution of load transfer, as a function of time. The model predicts an accumulation of fibre failures analogous to that found experimentally by the AE technique. This observation is independent of the degree of damage refinement that is considered.

## 9. Conclusions

The model accurately reconstructs damage processes occurring within a loaded unidirectional elastic fibre-reinforced composite. Uniaxial loading of the composite is shown to induce random fibre failures that affect neighbouring fibres. Tensile loading produces an increasing number of fibre breaks, controlled by the random nature of defects on or in the fibres. The damage rate and ultimate failure are influenced by several mechanisms at the level of individual fibres around fibre breaks. The load originally supported by the broken fibre is distributed to its neighbours, which locally experience an increase in load. The broken fibre debonds from the matrix over a length that determines the limits of the effects of the break. The composite fails when a sufficient concentration of fibre breaks occurs in a particular section. The model has been able to calculate the failure stresses of unidirectional carbon fibre composites to a ±2% accuracy and also to simulate the AE curves obtained. These latter observations confirm that the failure of fibres is the dominant damage mechanism. The viscoelastic properties of the matrix have been shown to govern the behaviour of the composites during a constant load test. The load transfer from the broken fibres to neighbouring intact fibres leads to additional stresses on them. The length over which this occurs varies with time, due to the relaxation of the matrix leading to an increasing number of broken fibres. If this activity continues, it will eventually lead to the failure of the composite. The results of this study have important implications for the use of advanced composite materials, which are usually not considered to evolve in the fibre direction when under a steady load. Even under these conditions, the composite is not seen to be purely elastic at the microscopic scale, even though the fibres are perfectly elastic and no discernable creep is recorded at the macroscopic scale. The relaxation of the resin at the microscopic scale induces a continuing fibre failure process, which is akin to a natural ageing of the composite when it is under load. Composite structures, such as filament-wound pressure vessels, behave in a very similar manner to that seen with unidirectional specimens, when their reinforcing fibres are subjected to tensile forces only. Such vessels are being increasingly used for storing natural gas and in the future will be used for a hydrogen-based economy. It can be seen therefore that the composite pressure vessels will continue to age under load and this must be considered in the evaluation of residual lifetimes in such structures.

## Footnotes

- Received April 18, 2006.
- Accepted January 4, 2007.

- © 2007 The Royal Society