S Behzadi

F R Jones
email: f.r.jones@shef.ac.uk

The effect of temperature on stress transfer between a broken fibre and the adjacent fibres in unidirectional fibre composites

Keywords: A. Polymer-matrix composites (PMCs), Matrix Yielding, B. Stress/strain curves, C. Finite element analysis (FEA), D. Stress concentrations

The effect of yielding of a polymer matrix on stress transfer between a broken fibre and an adjacent fibre has been studied using a three-dimensional (3D) finite element model. The full stress-strain curves of Araldite MY0510 epoxy resin have been measured as a function of temperature. The elasto-plastic behaviour of the epoxy resin has been included in a 3D multi-fibre FE model. The yielding of the matrix affects the axial strain development along both the fractured and the neighbouring fibre. The intensified axial strain at the intact neighbouring fibre has been found at the immediate radial distance from the broken fibre. Strain concentration Factor (SCF) of nearest neighbouring fibre decreased as a result of an increase in temperature.

Introduction

It is known [START_REF] Fukuda | On the strength distribution of unidirectional fibre composites[END_REF][START_REF] Curtis | A computer-model of the tensile failure process in unidirectional fibre composites[END_REF] that the "Rule Of Mixture" (ROM) overestimates the strength of composites containing high performance brittle fibres such as carbon. This method assumes a uniform distribution of strength along the fibres and simultaneous breakage of all fibres as well as a linear elastic matrix. The scatter in fibre strength along the fibres results in a sequential fibre fractures. This in turn induces an intensified stress on the adjacent intact fibres to a fibre-break. Therefore, the rule of mixtures is not reliable because a precise failure strength cannot be predicted [START_REF] Zweben | A bounding approach to the strength of composite materials[END_REF][START_REF] Wada | Approximate upper and lower bounds for the strength of unidirectional composites[END_REF]. Several studies [START_REF] Reifsnider | Modelling of the interphase in polymer-matrix composite material systems[END_REF][START_REF] Van Den Heuvel | Failure phenomena in multifibre model composites: Part 1. An experimental investigation into the influence of fibre spacing and fibre-matrix adhesion[END_REF][START_REF] Paipetis | Modelling the stress-transfer efficiency of carbon-epoxy interfaces[END_REF] have demonstrated that the durability and performance of fibre composites is determined by micromechanisms occurring at the fibre/matrix interface level. As the stored energy in the region around a fractured fibre-end is released, it is required to be redistributed among its intact neighbouring fibres to re-establish an equilibrium of load [START_REF] Hedgepeth | Local stress concentrations in imperfect filamentary composite materials[END_REF][START_REF] Landis | Stress concentrations in composites with interface sliding, matrix stiffness and uneven fibre spacing using shear lag theory[END_REF].

Depending on their distance from the fibre-break, the local intact fibres are introduced to a stress concentration factor. Curtis [START_REF] Curtis | A computer-model of the tensile failure process in unidirectional fibre composites[END_REF] demonstrated that a fibre-break in UD composite is most likely to be followed by propagating fibre fractures as a result of the local intensified stress. The intensified stress increases with the number of fractures and hence increases the probability of the failure of the intact fibres and modifies the mode of failure of composites [START_REF] Zweben | Failure analysis of unidirectional fibre composites under combined axial tension and shear[END_REF][START_REF] Rosen | Tensile failure criteria for fibre composite materials[END_REF]. The term of the stress concentration factor (SCF) is the ratio of the maximum (or local) to the applied (or undisturbed) stress of the intact fibre far away from the break. Hedgepeth and van Dyke [START_REF] Hedgepeth | Local stress concentrations in imperfect filamentary composite materials[END_REF] believed that the role of redistributed overload from a fibre-break is negligible compared to the axial load on the enormous number of fibres in a real composite. At some distant away from the broken fibre-end, the axial tensile stress recovers in the broken fibre and develops a length over which the stress rises to reach the applied stress. The distance over which the broken fibre is ineffective to support the applied load termed as "ineffective length" (IL) by

Rosen [START_REF] Rosen | Tensile failure of fibrous composites[END_REF] and shown schematically in Fig. 1. The transfer length or ineffective length determines the redistribution of the overload between neighbouring fibres [START_REF] Paipetis | Modelling the stress-transfer efficiency of carbon-epoxy interfaces[END_REF][START_REF] Goutianos | Comparative assessment of stress transfer efficiency in tension and compression[END_REF][START_REF] Van Den Heuvel | Failure phenomena in two-dimensional multi-fibre microcomposites. Part 4: a Raman spectroscopic study on the influence of the matrix yield stress on stress concentrations[END_REF]. It is believed [START_REF] Hedgepeth | Local stress concentrations in imperfect filamentary composite materials[END_REF][START_REF] Hedgepeth | Stress concentrations in filamentary structures[END_REF][START_REF] Eitan | Fibre interactions in two-dimensional composites[END_REF][START_REF] Wagner | Stress concentration factors in two-dimensional composites: Effects of material and geometrical parameters[END_REF][START_REF] Nedele | Stress concentration factors around a broken fibre in a unidirectional carbon fibre-reinforced epoxy[END_REF][START_REF] Zhou | Stress concentrations caused by fibre failure in twodimensional composites[END_REF] that the SCF of the neighbouring fibres together with ineffective length of the broken fibre are the main parameters to govern the ultimate failure mode of composites. In a UD composite, the load carried by a fibre is thrown onto the matrix when the fibre breaks. The strength of the interfacial bond also influences the stress transfer length because depending on shear stresses generated, either debonding or interphasal shear can occur. Therefore, when the interface remains intact, the matrix plays a significant role in determining the magnitude of the stress concentration in adjacent fibres.

However, the estimation of the stress concentration factor does not lend itself to experimental determination. In order to deal with the complicated 3D nature of real fibre composites, Finite Element Methods (FEM) were developed in the literature as the best alternative for obtaining stress distributions in the fibres. Nedele et al [START_REF] Nedele | Stress concentration factors around a broken fibre in a unidirectional carbon fibre-reinforced epoxy[END_REF][START_REF] Nedele | Three-Dimensional finite element analysis of the stress concentration at a single fibre break[END_REF] introduced a 3D FE model containing elastic fibres and matrix and reported that the stress concentration factor was not significance in the ultimate failure of a UD composite. In another approaches, Fiedler et al [START_REF] Fiedler | Stress concentrations in multiple fibre model composites[END_REF] and Lane et al [START_REF] Lane | Fibre/matrix stress transfer through a discrete interphase: 2. High volume fraction systems[END_REF] studied the load transfer between a fractured fibre and its surrounding intact fibres through linear elastic and linear elastic-ideal plastic matrices. They [START_REF] Fiedler | Stress concentrations in multiple fibre model composites[END_REF][START_REF] Lane | Fibre/matrix stress transfer through a discrete interphase: 2. High volume fraction systems[END_REF] strongly demonstrated the importance of matrix plasticity on the reinforcing efficiency of a broken fibre and on the SCF values.

Their values are in a good agreement with the analytical values reported by Wagner et al [START_REF] Wagner | Stress concentration factors in two-dimensional composites: Effects of material and geometrical parameters[END_REF]. Van den Heuvel et al [START_REF] Van Den Heuvel | Failure phenomena in fibrereinforced composites. Part 6: a finite element study of stress concentrations in unidirectional carbon fibre-reinforced composites[END_REF] introduced a 3D finite element model which included an element of full UD composite around a fibre-break similar to the model of Nedele et al [START_REF] Nedele | Three-Dimensional finite element analysis of the stress concentration at a single fibre break[END_REF] and obtained considerably lower stress concentration factor compared to their previous 2D planar model [START_REF] Van Den Heuvel | Failure phenomena in twodimensional multi-fibre model composites: 5. A finite element study[END_REF].

Epoxy resins are one of the most important thermosetting polymers which have been extensively used in aerospace composites materials over last few decades because of their high yield strengths, Young's moduli and glass transition temperatures. These materials exhibit brittle tensile behaviour in the bulk. However, in a real composite the large difference between the elastic modulus of the fibre and the thin layer of matrix results in interfacial shear yielding at a fibre-break [START_REF] Zweben | Failure analysis of unidirectional fibre composites under combined axial tension and shear[END_REF][START_REF] Iremonger | Effects of geometry on stresses in discontinuous composite materials[END_REF][START_REF] Agarwal | Plastic analysis of fibre interactions in discontinuous fibre composites[END_REF][START_REF] De Morais | Stress distribution along broken fibres in polymer-matrix composites[END_REF][START_REF] Gulino | Experiments on shear deformation, debonding and local transfer in a model graphite/glass/epoxy microcomposite[END_REF][START_REF] Nath | Finite element modelling of interfacial failure in model carbon fibre-epoxy composites[END_REF][START_REF] Tripathi | The effect of matrix plasticity on the stress fields in a single filament composite and the value of interfacial shear strength obtained from the fragmentation test[END_REF]. This develops a plastic region as the interfacial shear stress in the matrix exceeds the yield strength of the bulk epoxy resin. The measurement of fibre stress profiles using Raman spectroscopy [START_REF] Van Den Heuvel | Failure phenomena in two-dimensional multifibre microcomposites: 2. A Raman spectroscopic study of the influence of interfibre spacing on stress concentrations[END_REF] have confirmed that an extensive localised yield region occurs in the matrix at the vicinity of a fibre-break. Other studies [START_REF] De Morais | Stress distribution along broken fibres in polymer-matrix composites[END_REF][START_REF] Dibenedetto | The role of interphase debonding on cumulative fibre fractures in a continuous fibre-reinforced composite[END_REF][START_REF] Accorsi | Dynamic analysis of fibre breakage in single-and multiple-fibre composites[END_REF][START_REF] Jacobs | Finite element modelling of damage development during longitudinal tensile loading of coated fibre composites[END_REF] have demonstrated the occurrence of transverse matrix cracking or interfacial debonding, or mixed mode where both of these are coincident. Zhao et al [35] have demonstrated using photoelasticity that the sequence of events involves fibre fracture, which propagates into the matrix and delays the stress transfer back to the fibre through shear. The shear stress can then initiate debonding which can propagate back towards the fibre-break and away from it. Kettle et al [START_REF] Kettle | Plasma polymerisation for molecular engineering of carbon-fibre surfaces for optimised composites[END_REF] have demonstrated that all of these micromechanisms from debonding to transverse matrix cracking are a function of the strength of the interfacial bond.

Therefore, linking the experimental or modelled micromechanical results to the macroscopic mechanical properties of UD fibre reinforced composites is the main goal in characterisation of the composite materials. In this study, the elasto-plastic behaviour of a high performance resin matrix was identified as the principal matrix property which is a key mechanism for controlling the matrix-to-fibre stress transfer. In the literature, no attempt has been made to characterize the stress-strain behaviour of epoxy resins used in high performance fibre composites while most work is focused on crosslinked difunctional (DGEBA) epoxy resins [START_REF] Fiedler | Failure behaviour of an epoxy matrix under different kinds of static loading[END_REF][START_REF] Cook | Yielding behaviour in model epoxy thermosets-II. Temperature dependence[END_REF][START_REF] Kinloch | Fracture Behaviour of Polymers[END_REF][START_REF] Yamini | The mechanical properties of epoxy resins. Part 1: Mechanisms of plastic deformation[END_REF]. In general, the yield behaviour of the viscoelastic materials is highly dependent on the temperature. Therefore, the stressstrain behaviour of the epoxy resins was studied in a wide range of temperatures.

The elasto-plasticity behaviour of the epoxy resin is then used to predict the stress transfer from a broken fibre to the adjacent fibres and in particular stress concentration factor in the neighbouring fibres. Using statistical methods, the finding of this study has provided a predictive tool to evaluate the failure strain of an UD fibre composite [START_REF] Behzadi | Prediction of tensile failure strain of unidirectional fibre composites: Investigating the effect of matrix yielding[END_REF] and to optimize the composite performance through matrix selection.

Experimental

The epoxy resin used for this study is triglycidyl p-aminophenol (TGAP), Araldite Uniaxial compression testing of the specimens was performed with a Hounsfield universal testing instrument. The crossheads and the specimen ends were smeared with a touch of petroleum jelly before alignment to minimise the frictional forces at the junctions. Fig. 2 monitors the deformation of a typical epoxy resin specimen at different compressive strains over its corresponding true stress-strain curve. It can be seen that there is no visible barrelling in the specimen after the yield strain or even at higher strains before the fracture. Extensometers or strain gauges could not be mounted on the test specimens because of their small dimensions. Therefore, for obtaining the actual deformation of the specimen, the compliance of the testing machine had to be measured precisely and deducted from the total deformation monitored by the actuator. In the nonsample technique demonstrated by Kalidindi et al [START_REF] Kalidindi | Accurate characterization of machine compliance for simple compression testing[END_REF], the load-displacement relationship for the machine was directly measured with and without the specimen. This compliance correction [START_REF] Kalidindi | Accurate characterization of machine compliance for simple compression testing[END_REF] assumes that the specimen and loading parts are two springs in series. In this way, the functionality of the compliance displacement with applied load at a given crosshead speed can be obtained for the testing instrument. This data can be used to correct the recorded load-displacement data for any specimen tested under uniaxial compression by the testing machine at the same crosshead speed.

A fan-assisted oven mounted on the testing machine, delivered a range of temperatures from 22 °C to 180 °C during the uniaxial compression testing. The test was performed at a crosshead speeds of 0.1 mm/min which corresponds to initial strain rate of 1.67×10 -4 s -1 , considering the initial lengths of the test specimens. However, the true strain rate increases as the uniaxial compressive deformation of the specimen proceeds.

The true longitudinal strain, ε T was obtained by:

= t i T L L ln ε (eq. 1)
where L i and L t are the initial and instantaneous (corrected for the machine compliance) lengths. Because the continuous monitoring of the lateral expansion of the small test specimens was not practical, the instantaneous diameter of the specimens was obtained using the longitudinal true strain ( T ε = yy ε ) and Poisson's ratio. The schematic illustration of three-dimensional deformation of a cylinder under uniaxial compression is given in Fig. 3. The Poisson's ratio in uniaxial compression can be defined as:

yy zz yy xx ε ε ε ε ν - = - = (eq. 2)
where ν is the Poisson's ratio and xx ε and zz ε are the transverse strains which may be considered as (using eq. 1):

= = t i zz xx D D ln ε ε (eq. 3)
where i D and t D are initial and instant diameter of the test specimen. Eq. 3 can be rearranged, using eq. 2, to:

( )

yy i t D D νε - = exp (eq. 4)
In this way, the instant cross-sectional area of the specimen and hence true compressive stress can be obtained using the applied load. The yield was defined as the point where the load passed through a maximum or an apparent 'knee' in the stress-strain curve. The Poisson's ratio of thermoset polymers is a viscoelastic aspect of the material which is increased by temperature and strain rate [START_REF] Tschoegl | Poisson's ratio in linear viscoelasticity -A critical review[END_REF][START_REF] Maksimov | Comparative Studies on the Mechanical Properties of a Thermoset Polymer in Tension and Compression[END_REF]. A recent theoretical study by Foreman et al [START_REF] Foreman | Materials by Design-Matrix modelling for composite performance[END_REF] using Group Interaction Modelling (GIM), reported the variation in Poisson's ratio of the epoxy resin of interest over a range of temperatures. However, in this study, no significant change was observed in the true compressive stress-stress curves of the epoxy resin using the reported ranges of Poisson's ratio [START_REF] Tschoegl | Poisson's ratio in linear viscoelasticity -A critical review[END_REF][START_REF] Maksimov | Comparative Studies on the Mechanical Properties of a Thermoset Polymer in Tension and Compression[END_REF][START_REF] Foreman | Materials by Design-Matrix modelling for composite performance[END_REF] for the change in temperature and strain rate. Therefore, a constant Poisson's ratio of 0.35 is considered for the epoxy resin at all temperatures and strain rates.

Multi fibre finite element model

A 3D model of a square array of fibres was generated using ANSYS 8.1 and is shown in Fig. 4. The model is consisted of four quarter fibres of 100 µm long placed at the corners of a rectangular prism similar to the models of Fiedler et al [START_REF] Fiedler | Stress concentrations in multiple fibre model composites[END_REF] and Lane et al [START_REF] Lane | Fibre/matrix stress transfer through a discrete interphase: 2. High volume fraction systems[END_REF]. The properties of the transversely orthotropic carbon fibres (Table 1) used in this study were taken from elsewhere [START_REF] Nedele | Micromechanical modelling of unidirectional composites subjected to external and internal loadings[END_REF] and listed in Table 1. The properties of the carbon fibre were assumed to be temperature-independent. The fibres are enclosed in a volume However, in a composite, the epoxy resin matrix is uniformly distributed between the fibres and its thickness is not higher than 1 µm. Thus, it can yield under a tensile load between the stiff fibres. This is consistent with the recent study of Fiedler et al [START_REF] Hobbiebrunken | Experimental determination of the true epoxy resin strength using micro-scaled specimens[END_REF] on small volumes of an epoxy resin. They reported a large plastic deformation and necking for the resin fibre of a diameter of 2 mm. In this study, we assumed that the tensile stress-strain behaviour of such a small volume of the resin matrix can be considered identical to its compressive one. The true experimental compressive stress-strain curve of the epoxy resin tested at different temperatures was digitized and input into the model.

Using the von Mises yield criterion, a reliable calculation of the 3D matrix yielding properties can be obtained. Because of the large plastic strain at the fibre/matrix interface, post-yielding phenomena namely strain softening and strain hardening could not be ignored [START_REF] Van Den Heuvel | Failure phenomena in twodimensional multi-fibre model composites: 5. A finite element study[END_REF]. The resin matrix was considered as an isotropic material with

Poisson's ratio of 0.35. It is emphasized in Section 2 that the changes in Poisson's ratio within the range of temperatures and strain rates have not altered the true stress-strain curves of the epoxy resin. In order to ensure the accuracy of the FEM results, the corresponding Poisson's ratio [START_REF] Foreman | Materials by Design-Matrix modelling for composite performance[END_REF] was input into the model at each temperature.

However, no considerable variations in the findings of the model was observed over the range of temperature studied.

Results

Uniaxial compression test

The full compressive stress-strain curves of the epoxy resin as a function of temperature are shown in Fig. 5. These glassy polymers proved to be extensively ductile up to a compressive strain of 40 %. It is obvious that increasing the test temperature led to a significant decrease in the compressive yield strength as well as the yield strain.

Furthermore, although the range of test temperatures was selected so that the resin remained in a glassy state, the compressive modulus also showed a considerable reduction with temperature. The variations of the yield strength, yield strain and modulus of the epoxy resin with temperature are tabulated in Table 2. It has been reported elsewhere [START_REF] Behzadi | Yielding behaviour of the model epoxy matrices for fibre reinforced composites: Effect of strain rate and temperature[END_REF] that the yield properties decrease linearly with increase in test temperature.

Finite element modelling

Following the solution of the finite element model, the strain (or stress) in both the fibres and the matrix can be obtained. The effect of matrix yielding on the load transfer characteristics and particularly the SCF in the presence of a fibre-break has been studied as a function of the temperature. Fig. 6(a) shows the effect of ambient temperature on the development of the axial tensile strain along the broken fibre in the model composite containing the resin matrix MY0510/DDS. In this study, a strain analysis was carried out because of the known failure response of these composites. The distance from the fibrebreak is normalized along the fibres with the length of the FE model. It can be observed that there is an exponential increase in axial tensile strain in the broken fibre after a fibrebreak. However, at higher temperatures this exponential strain build-up becomes effectively linear. Fig. 6(a) also exhibits that because of the elasto-plastic behaviour of the matrix (Fig. 5), the applied strain cannot be fully recovered even at the far-field from the broken fibre so it falls below the applied strain of 1 %. Moreover, as the degree of plasticity of the matrix increased with temperature, the far-field strain decreased. The effect of a fibre-break on the adjacent fibre is studied by monitoring the development of axial tensile strain along the centre of the neighbouring fibre and shown in Fig. 6(b). It can be seen that the introduction of a break results in a considerable increase in the axial strain at the vicinity of the break. After passing through a maximum in the immediate vicinity of the fibre-break, the axial strain reduced along the neighbouring fibre to below the applied strain at the far-field. It can also be noted that as the degree of plasticity of the matrix increases with temperature, the maximum and far-field axial tensile strains decreased. Fig. 7 shows the variation in interfacial shear stress in the interface of neighbouring fibre/matrix as a function of temperature in a model composite containing the resin matrix. It indicates that the strain transfer from the matrix to the neighbouring fibre behind a fibre-break is caused by a local matrix yielding. As the temperature increases, the matrix becomes more plastic in nature so that a higher volume is involved in the strain transfer mechanism. Table 3 lists the SCF values in the neighbouring fibre to a broken fibre for a composite containing the resin matrix MY0510/DDS over a range of temperatures. In this study, the values of SCF are representative of the maximum strain concentration factors in-plane to the fibre-break along the centre of the neighbouring fibre. Therefore, these values are not dependent on the length of the FE model.

Discussion

On the occurrence of a fibre-break in a UD fibre composite, a large fraction of the load which used to be carried by the fibre is transferred to the matrix in the plane of the crack. This results in a high shear stress at the interface between the fibre and the matrix.

With a perfect interfacial bond, the overload is redistributed between the intact neighbouring fibres through the matrix shear yielding. The maximum axial tensile strain in the neighbouring fibre, SCF, demonstrates the increased probability of the failure of the intact neighbouring fibre. Most of the models [START_REF] Nedele | Stress concentration factors around a broken fibre in a unidirectional carbon fibre-reinforced epoxy[END_REF][START_REF] Nedele | Three-Dimensional finite element analysis of the stress concentration at a single fibre break[END_REF] for predicting the strain (or stress) concentration factor neglect the elasto-plasticity of the matrix. The assumption of a linear elastic material for the matrix limits the predictive accuracy of the models.

Hedgepeth and van Dyke [START_REF] Hedgepeth | Local stress concentrations in imperfect filamentary composite materials[END_REF] included the matrix plasticity in their work which resulted in a decrease in the stress concentration factor with increase in load. Ochiai et al [START_REF] Ochiai | Stress disturbances arising from cut fibre and matrix in unidirectional metal matrix composites calculated by means of a modified shear lag analysis[END_REF] also concluded that the inclusion of such local matrix yielding in the model decreases the SCF in the adjacent fibres and results in a better agreement between the experimental and analytical results. The elasto-plasticity of the matrix has been incorporated into finite element models by other authors [START_REF] Fiedler | Stress concentrations in multiple fibre model composites[END_REF][START_REF] Lane | Fibre/matrix stress transfer through a discrete interphase: 2. High volume fraction systems[END_REF] and led to a significantly lower strain concentration factor in comparison to the elastic case. In contrast, it has been shown elsewhere [START_REF] Van Den Heuvel | Failure phenomena in fibrereinforced composites. Part 6: a finite element study of stress concentrations in unidirectional carbon fibre-reinforced composites[END_REF] that matrix plasticity resulted in a slight decrease in the SCF and it was concluded that the SCF was mainly affected by the number of surrounding fibres.

It was also shown that the increase in temperature led to a decrease in axial tensile strain along the broken fibre. As the matrix yield strength decreases with increase in temperature, the exponential development of the axial tensile strain along the centre of the broken fibre was replaced by a linear behaviour. This results in a less efficient strain transfer back into the broken fibre since more of the stored energy released on fibre fracture is already absorbed by the matrix, which undergoes a larger plastic deformation.

Because the plasticity of the matrix increases with temperature, less strain is transferred back to the other end of the broken fibre. Thus, the broken fibre cannot recover the applied strain at its far-field. Table 4 shows the variation in far-field strain in the broken fibre as a function of temperature. It can be seen that the discrepancy between the far-field and applied strains becomes more pronounced at higher temperatures. This is consistent with a longer ineffective length of the broken fibre at higher temperatures.

Table 4 also lists the changes in the ineffective length in the broken fibre with temperature. In the literature, the ineffective length is known as the length of fibre over which 90% recovery in axial strain (or stress) occurs. However, in this study, the ineffective length is considered as the distance between the fibre-break and an axial strain plateau. As the modulus of the matrix decreases with an increase in temperature (Table 2), the mismatch between the moduli of the fibre and matrix increases and hence a larger volume of the matrix contributes to the shear yielding mechanism. Furthermore, it is shown in Fig. 7 that the interfacial shear yielded zone at the neighbouring fibre/matrix interface expands towards the middle of the fibre in the axial direction, with increase in the matrix plasticity. Thus, the stress-transfer capacity of the matrix is reduced more and more accordingly and as a result, a lower overload is redistributed between the neighbouring fibres. This is also associated with the decrease in maximum and far-field axial strains in the neighbouring fibre with increase in temperature as shown in Fig. 6(b).

Hence, the fibre adjacent to a broken fibre experiences a reduction in SCF as a result of a higher temperature (Table 3). In this study, the magnitude of the calculated SCF value obtained at room temperature (Table 3) is higher than those reported in other studies [START_REF] Hedgepeth | Local stress concentrations in imperfect filamentary composite materials[END_REF][START_REF] Fiedler | Stress concentrations in multiple fibre model composites[END_REF][START_REF] Lane | Fibre/matrix stress transfer through a discrete interphase: 2. High volume fraction systems[END_REF][START_REF] Van Den Heuvel | Failure phenomena in fibrereinforced composites. Part 6: a finite element study of stress concentrations in unidirectional carbon fibre-reinforced composites[END_REF] where matrix elasto-plasticity was included in the models. This can be explained by the redistribution of the overload from a broken fibre among a larger number of intact neighbouring fibres in those studies. On the other hand, the high modulus and yield strength of the high performance resin matrix used in the current FE model can be responsible for the increase in the SCF values in the neighbouring fibre.

Conclusion

The yield behaviour and properties of a cured epoxy resin which forms one of the main components of high performance commercial matrix systems have been investigated. The true compressive stress-strain curves of the resin have been measured

as a function of test temperature. A 3D FE model of UD fibre composite containing a fibre-break was introduced. The properties of the matrix were input into the model in the form of true compressive stress-strain curves of the epoxy resin. The relationship between the matrix yielding and the redistribution of the stress overload from the fibrebreak to the neighbouring fibres was studied. In high volume fraction composites, the yielding of the matrix governs the strain redistribution between the intact fibres at the vicinity of a broken fibre. Interfacial shear yielding of the matrix reduces the efficiency of load transfer and thus reduces the SCF in neighbouring fibres to the broken fibre. At a higher temperature, the reduced stress transfer efficiency results in a lower SCF in the neighbouring fibre. Thus, the probability of multiple in-plane fibre-breaks (and hence a flaw of critical dimensions) is reduced at higher temperatures. Therefore, the durability of an epoxy resin composite is strongly influenced by the elasto-plastic behaviour of the matrix.

Figures

Fig. 1. Schematic illustration of the strain build-up in a broken fibre along the ineffective length.

MY0510 ® cured with 4 - 4 '

 44 diaminodiphenyl sulphone, DDS hardener. While thermosetting resins fracture at very low tensile strains, they undergo considerable plastic deformation under uniaxial compressive load. The cured cylindrical test specimens were machined to achieve an accurate Length/Diameter ratio of 1:1 (~ 10 mm/10 mm).

  of the resin matrix. The fibre volume fraction was chosen to be 50 %, which is realistic of actual material. The coinciding nodes all along the interfaces in the FE model ensured perfect interfacial bonding between the fibres and matrix. The 8-noded solid brick structural SOLID45 element was employed to mesh both the fibres and the matrix. In order to introduce a fibre-break the top layer of the elements can be removed from the fibre-end as shown in Fig.4. It is also shown that the concentration of the elements was increased towards the front face of the model where the fibre-break occurred. An axial displacement of 1 % was applied on the front face of the model in z-direction while the back face was constrained with z = 0. The applied strain of 1 % is practically around a strain level, at which a fibre fractures. The model is constrained from motion on its other four faces regarding the symmetry boundary conditions to satisfy the model's stability.The tensile failure of a bulk cured epoxy resin is dominated by the concentration of flaws. This results in premature failure of the resin specimens at very low tensile strains.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Photographs of deformation of an epoxy resin MY0510/DDS specimen under uniaxial compression load at different strains shown on the corresponding true stress-strain curve at room temperature and a strain rate of 1.67 × 10 -3 s -1 .

Fig. 4 .

 4 Fig. 4. Illustration of the geometry and meshing of the 3D FE model of a unidirectional high volume fraction composite used in this study. The first row of elements was removed to represent a fibre-break on the broken fibre. Neighbouring fibre is the immediate adjacent fibre to the broken fibre.

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. True compressive strain-stress curves of epoxy resin MY0510/DDS at a range of temperatures and the initial strain rate of 1.67 × 10 -4 s -1 .

Table 1

 1 Mechanical properties of the carbon fibre employed in the FE model taken from elsewhere[START_REF] Nedele | Micromechanical modelling of unidirectional composites subjected to external and internal loadings[END_REF]

	Tables
	Type	HTA5131
	Diameter (µ µ µ µm)	7.0
	Elongation to break (%)	1.43
	Young's modulus (GPa)	E z E x = E y	235 13.8
		ν ν ν ν yx	0.35
	Poisson's ratio	ν ν ν ν zy	0.2
		ν ν ν ν zx	0.2
	Shear modulus (GPa)	G xy G yz = G xz	5.11 18

Table 2

 2 Compressive yield strength (σ cy ), yield strain (ε cy ) and compressive modulus (E c ) of epoxy resin MY0510/DDS over a range of temperatures and the strain rate of 1.67 × 10 -4 s -1 (± standard error).

	Temperature (°C)	σ σ σ σ cy (MPa)	ε ε ε ε cy	E c (GPa)
	22	167.4 ± 0.4	0.09	3.82 ± 0.01
	70	127.3 ± 0.3	0.07	3.42 ± 0.01
	100	106.8 ± 0.1	0.06	3.33 ± 0.00
	140	88.7 ± 0.1	0.04	2.94 ± 0.03
	180	67.1 ± 0.2	0.03	2.49 ± 0.02

Table 3

 3 The strain concentration factors (SCF) in the neighbouring fibre to a broken fibre in a UD composite containing the resin matrix MY0510/DDS as a function of temperature.

	Temperature (°C)	22	70	100	140	180
	SCF	1.193	1.191	1.183	1.172	1.146

Table 4

 4 The variation in the fractional ineffective length and far-field strain in the broken fibre with the changes in the matrix yield strength as a result of the temperature in a UD composite containing the resin matrix MY0510/DDS.

	Temperature	Matrix yield	Fractional ineffective	Far-field strain (%)
	(°C)	strength (MPa)	length	[Max = 1 %]
	22	167.4	0.85	0.89
	70	127.3	0.89	0.87
	100	106.8	0.92	0.85
	140	88.7	0.94	0.82
	180	67.1	0.96	0.73

Acknowledgement

This project is funded by an EPSRC (UK) portfolio grant. S Behzadi is grateful to UK universities for an Overseas Research Scholarship (ORS). We also acknowledge Dr Simon Hayes and Dr Bradley Wynne of the University of Sheffield for the valuable discussions on finite element modelling and mechanical properties, respectively.


R Craven
email: r.craven06@imperial.ac.uk

P Sztefek

R Olsson

Investigation of Impact Damage in Multi-Directional Tape Laminates and its Effect on Local Tensile Stiffness

Keywords: A. Carbon Fibres, A. Laminate, B. Mechanical Properties, C. Finite Element Analysis (FEA) D. Inverse Method

This paper examines the effect of impact damage on local tensile stiffness of multidirectional tape laminates. A finite element model is used for parametric studies of the effect of various patterns of delaminations and cracks across the fibres. Regular and random crack patterns are considered for eight and sixteen ply quasi-isotropic laminates.

The effect on local stiffness is evaluated by a recently developed inverse numerical approach. It is found that fibre cracks control the loss of stiffness but that delaminations are instrumental in extending the zone influenced by these cracks. Fractographic observations are finally used to generate a detailed model of the damage in a real laminate. A close agreement was found between the predicted and measured strain fields and the corresponding stiffness distributions obtained with the inverse method.

Introduction

Impact damage is a constant concern in the design of structures of composite materials, and frequently controls the allowable strains [START_REF] Davies | Impact on composite structures[END_REF]. The largest effects are normally seen in compression, primarily due to the interaction of delaminations and local buckling.

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Tensile stiffness reductions have mainly been linked to fibre fracture and some authors have suggested that the region with fibre fracture may be regarded as a hole [START_REF] Chamis | Fiber composite structural durability and damage tolerance: simplified predictive methods[END_REF]. Others have suggested modelling the fibre damage zone as a soft inclusion [START_REF] Cairns | Residual strength of graphite/epoxy and kevlar/epoxy laminates with impact damage[END_REF]. The latter paper used laminate theory to calculate the inclusion stiffness, assuming zero stiffness of plies with broken fibres. An elementary finite element (FE) model of an impact damage in an 18 ply laminate under compression was presented in [START_REF] Pavier | Finite element prediction of the post-impact compressive strength of carbon fibre composites[END_REF], where the damage was modelled as a single rectangular delamination and a number of cracks perpendicular to the loading direction. However, the appropriate stiffness distribution in the damage zone remains unclear. It should be noted that a detailed description of the stiffness gradients is crucial for determination of the stress concentrations and strength reductions caused by impact damage.

An early experimental study of fibre damage in impact damage zones was done by observing the tensile strength variation of fibre bundles within the damage zone [START_REF] Elber | Failure mechanics in low-velocity impacts on thin composite plates (NASA TP-2152)[END_REF].

Tensile tests on coupons taken from different parts of the damage zone have indicated that the stiffness reductions are non-uniform, non-zero and concentrated to a small central damage region with fibre fracture, [START_REF] Sjögren | Experimental determination of elastic properties of impact damaged in carbon fibre/epoxy laminates[END_REF]. A recent study used an inverse numerical method to evaluate experimentally measured displacement fields in impacted specimens, [START_REF] Sztefek | Tensile stiffness distribution in impacted composite laminates determined by an inverse method[END_REF]. The predicted stiffness distribution confirmed the observations in [START_REF] Sjögren | Experimental determination of elastic properties of impact damaged in carbon fibre/epoxy laminates[END_REF].

The current paper presents a numerical study of the influence of impact damage on the local tensile stiffness of multi-directional laminates. Different patterns of fibre cracks and delaminations are included in a detailed Finite Element (FE) model of uniaxially loaded laminates. The corresponding local stiffness is evaluated using the inverse numerical approach in [START_REF] Sztefek | Tensile stiffness distribution in impacted composite laminates determined by an inverse method[END_REF]. The models are validated by comparing the predicted strain fields with measurements on impacted laminates.

Initial Parametric Studies

The initial studies were inspired by the investigations of Olsson et al. [START_REF] Olsson | Experiments and analysis of laminates with artificial damage[END_REF] who conducted tensile stiffness evaluation of specimens with artificial impact damage. That gave a starting point for the study and motivated the use of the inverse method as verification of the FE results. The star crack pattern in [START_REF] Olsson | Experiments and analysis of laminates with artificial damage[END_REF] was generated by creating a cut perpendicular to the fibres in each ply prior to lay-up, producing a star crack pattern upon lay-up. This pattern was expected to give a large stiffness reduction in tension but the experimental results did not show this. This was felt to be a good starting point for a study into the effects of impact on tensile stiffness reduction. A parametric study of these star fibre fracture cracks was conducted using FE modelling to determine the parameters that controlled the stiffness reduction. These were constructed of 8 ply (90/-45/+45/0) s laminates 100 x 20 x 2 mm with single 18 mm fibre fracture cracks perpendicular to the fibres in each ply. They were modelled using brick elements with each ply being one brick thick. The study looked at an undamaged laminate, a laminate with fibre fracture cracks only, a laminate with delaminations only and a laminate with fibre fracture and delaminations. It was found that delaminations alone had no effect on the stiffness of the laminate in tension and that fibre fractures alone caused minimal average reduction in stiffness (12%), whereas delaminations combined with fibre fracture resulted in significant average stiffness reduction (55%). From this it was concluded that delaminations and fibre fracture are required to be present for stiffness reduction to occur in tension. This model of delamination and fibre fracture was also tested for eight different lay-ups (0/-45/+45/90) s , (0/90/+45/-45) s , (90/0/-45/+45) s , (90/-45/0/+45) s , (+45/-45/90/0) s , (+45/90/-45/0) s , (-45/90/0/+45) s and (-45/0/90/+45) s . No difference between stiffness reductions in lay-ups with the same face ply orientation

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was observed and very little variation was observed between laminates with different face ply fibre orientations.

These results are confirmed by the experimental findings in [START_REF] Olsson | Experiments and analysis of laminates with artificial damage[END_REF], where quasi-isotropic laminates with artificial star shaped fibre damage were tested. The resulting strain concentrations were found to be negligible at low applied strains, although strain concentrations appeared in the vicinity of the cracks as they opened and delaminations developed.

Full Scale Damage Models

For the full scale models combinations of different delamination patterns and fibre crack patterns, Fig. 1, were considered. In order to alleviate any problems with edge effects or hour-glassing of elements due to large deformation between the fibre fractures and the edge of the laminate, the laminate size was increased to 100 x 80 mm, with the damage located centrally. The laminate is an 8 ply quasi-isotropic balanced laminate with (90/-45/+45/0) s lay-up, nominally 2 mm thick. Having proven the stiffness reduction to be independent of lay-up sequence this lay-up was used because the fibre fracture cracks on the 90º plies do not open during loading. This provides a surface free from strain discontinuities from which the strain field can be measured, which is desirable to obtain good results from the inverse method.

There are four different delamination patterns as seen in Fig. 1a. These were chosen from the literature to investigate the differences in stiffness reduction caused by each, the circular delamination is the most basic and widely used [START_REF] Abrate | Impact on laminated composite materials[END_REF], the twin ellipse pattern was proposed by Davidson [START_REF] Davidson | On modelling the residual strength of impact damaged compression loaded laminates[END_REF], the peanut or lemniscates by Cairns [START_REF] Cairns | Impact and post impact response of graphite/epoxy and kevlar/epoxy structures (TELAC Report 87-15[END_REF], and finally the split peanut suggested by Hull and Shi [START_REF] Hull | Damage mechanism characterisation in composite damage tolerance investigations[END_REF].

The crack patterns in Fig. 1b include some patterns suggested in the literature as well as some random patterns generated in the present study. The star pattern from [START_REF] Olsson | Experiments and analysis of laminates with artificial damage[END_REF] assumes the cracks occur perpendicular to the fibres in each ply. The line cracks, which only occur in 0º plies, represent the model of stiffness reduction in [START_REF] Cairns | Residual strength of graphite/epoxy and kevlar/epoxy laminates with impact damage[END_REF], where it is assumed that the stiffness reduction can be modelled by failure in the 0º plies only. The three random distributions of cracks were generated using a Matlab ® script, which randomly assigns a fibre fracture crack length between 1 mm and 15 mm positioned perpendicular to the fibre orientation inside an 18 mm diameter circle, and places between one and three cracks per ply. To compare the crack patterns a dimensionless parameter of average crack density (Avg. CD) has been created based upon the parameter crack density defined in [START_REF] Shen | A numerical simulation for the effective elastic moduli of plates with various distributions and sizes of cracks[END_REF], Eq. (1.1), where the sum of the half length L/2 of each crack in a ply is summed and divided by an area A. In the present case A is the area of the 18 mm diameter circle that can be seen in Fig. 1b surrounding each of the crack patterns.

The average crack density (Avg. CD) was obtained by averaging the crack density across all n plies in the laminate

( ) = = = = m i i j n j j L A where n CD Avg 1 2 1 2 1 1 . ρ ρ (1.1)
The relative average crack densities for all crack patterns can be seen in the comparison in Fig. 2.

In order to verify the findings of these models, a model of a real 16 ply laminate impacted at 7J [START_REF] Sztefek | Tensile stiffness distribution in impacted composite laminates determined by an inverse method[END_REF] was created based on fractography of the real laminate. The fractography for this study was taken from [START_REF] Gamelin | Observation of impact damage zones[END_REF] although the And-scan ® images of delaminations were too poor to determine delamination shapes so peanut shapes of the correct diameter were used instead. Furthermore, some of the fibre fracture cracks were missed in the original fractographic study in [START_REF] Gamelin | Observation of impact damage zones[END_REF] and had to be determined from reexamination of the actual de-plied laminate. To replicate the experimental loading precisely, the boundary conditions were the far field displacements observed in the experiment by using optical measurements based on Digital Image Correlation (DIC).

A second FE model was also produced using the same peanut delaminations. The real fibre fracture was, however, replaced by a star crack pattern to demonstrate that reasonable results can still be obtained by simplifying the damage, to a complexity that can be easily implemented in a model.

Modelling Technique

Finite element modelling was conducted using ABAQUS ® 6.6. The laminates were created using linear 3D brick elements (C3D8R) with one brick per ply thickness. The ply properties were entered as engineering material properties and the material (fibre) orientation for each ply was then assigned separately. The material properties used in the parametric studies correspond to the carbon/epoxy system HTA/6376C as reported in [START_REF] Sjögren | Fractographic characterization of impact damage in carbon fibre/epoxy laminates[END_REF]. They are E 11 = 137 GPa, E 22 = E 33 = 10.4 GPa, G 12 = G 13 = 5.19 GPa, G 23 = 3.9 GPa, 12 = 13 = 0.3 and 23 = 0.51. For the experimental model, the material properties used corresponded to the material properties of the experimental laminate, taken from [START_REF] Sztefek | Tensile stiffness distribution in impacted composite laminates determined by an inverse method[END_REF]. These are E 11 = 141 GPa, E 22 = E 33 = 8.9 GPa, G 12 = G 13 = 4.73 GPa, G 23 = 2.9 GPa, 12 = 13 = 0.3 and 23 = 0.50.

The delamination shape is created as a sketch on the surface and then extruded through the ply so that delaminations for both faces of the ply appeared on both faces, which allows a uniform mesh through the thickness of the ply. Ideally the fibre cracks within ACCEPTED MANUSCRIPT the ply should be described by two disconnected surfaces along a straight line. In models with one ply this was achieved by cutting the ply into two parts and using the tie command to join the ply parts on either side of the crack. In multiply models this was not feasible as the nodes along the joined edges would have been used in two separate tie commands which is not permitted within ABAQUS ® . For this reason the ply cracks in multiply models were modelled as highly extended ellipses, having a crack length of 1-15 mm and a crack width of 0.001 mm, by using the cut out tool available in ABAQUS 6.6® CAE. The laminate is then created by tying the plies together. All sections on the face of a ply, apart from the delaminated area are defined as a surface and then adjacent surfaces are tied together using the tie surface command within ABAQUS ® . Studies using single ply models showed very little difference in the strain field between the elliptical cracks and the ideal line cracks, making elliptical cracks acceptable. The same problem with common nodes prevented shell elements from being used to model individual plies as the nodes were common to both surfaces and could not be used to tie both sides of the ply into the laminate.

No contact constraints have been used between the delaminations in these models, although the delaminated areas were observed to move and intersect in the out of plane direction during the solution. Comparing a laminate without contact constraints and a laminate with contact constraints revealed only a 1% difference in stiffness reduction but a six fold increase in solution time for the laminate with contact constraints.

Consequently contact constraints were considered unnecessary.

Inverse Method

The homogenised constitutive parameters of the laminates represented by FE models as introduced above were sought using the indirect inverse method based on iterative updating of the material properties in an FE model using shell elements with a number of piecewise homogeneous zones [START_REF] Sztefek | Tensile stiffness distribution in impacted composite laminates determined by an inverse method[END_REF]. The strategy is to minimise the difference between the displacements numerically predicted in the piecewise homogeneous FE model and the displacement fields obtained by solving the FE models with damage and stacked plies. As the inverse method is usually coupled with full field measurements obtained by Digital Image Correlation (DIC), the input (reference) displacements from the FE laminates with discrete damage features are considered only on the surface of the models. The iterative FE analyses of the piecewise homogeneous model, performed with the commercial software ABAQUS ® , provide an updated displacement field resulting from a specific set of material parameters. The difference of these two displacement fields is denoted as displacement error and this error is minimised by employing The Steepest Descent optimization method. The basic principle of the inverse method is illustrated in Fig. 3.

The method has earlier been validated on numerical test cases using an isotropic FE model with artificial impact damage [START_REF] Sztefek | Tensile stiffness distribution in impacted composite laminates determined by an inverse method[END_REF]. The Young's modulus and the Poisson's ratio were determined accurately when considering an "academic" displacement field without noise and were in a very good agreement when adding white noise to the reference displacement field. The method was then applied to DIC measurements on experimentally impacted laminates and the material properties were determined.

Stiffness Evaluation

The inverse method was applied to the FE models with an artificial damage created by coupling 5 different types of fibre fracture (star and line pattern, and random light, medium and severe cracks) together with 6 delamination types (circular delamination with radius of 10, 20 and 30 mm, twin ellipse, peanut and split peanut shapes having outer radius of 30 mm), see Fig. 1. The delaminations were orientated so that the major axis was aligned along the fibre direction of the lower ply, seen in Fig. 1c, as observed experimentally when examining impacted laminates [START_REF] Sjögren | Fractographic characterization of impact damage in carbon fibre/epoxy laminates[END_REF]. The FE models were subjected to tensile load resulting in an applied strain of 0.3%, which had been selected not to exceed the elastic range of the material. Having different types of delaminations with outer radii 10, 20 and 30 mm, and fibre fracture concentrated within an 9 mm radius circle, it was decided to model the piecewise homogeneous FE model with the damage discretised into 6 sections to evaluate stiffness distribution with a reasonable resolution. As seen from previous studies [START_REF] Sjögren | Experimental determination of elastic properties of impact damaged in carbon fibre/epoxy laminates[END_REF] and [START_REF] Olsson | Experiments and analysis of laminates with artificial damage[END_REF], as well as from in-house experiments, the damage in quasi-isotropic laminates is fairly circular and therefore 5 mm wide concentric rings, positioned in the centre of the model were considered, Fig. 4. The outer radius of the damage region was then 30 mm. As the lay-up of the FE model is quasi-isotropic, the constitutive parameters for each section are limited to the Young's modulus and the Poisson's ratio. Both properties are sought as dimensionless relative values with respect to the undamaged material. A full convergence to the results being presented in this paper is achieved at typically about 50 iterations (about 2.5 hours using a Pentium ® 4 CPU, 2.8 GHz). The elastic constants determined are plotted as normalised values (with respect to the undamaged material) across the width of the investigated sample through the centre of damage region, e.g. Fig. 5.

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The results of peak and average stiffness for the different combinations of delamination shape and fibre fracture crack patterns are shown in Table 1. Not all delamination shapes were modelled with all fibre fracture crack distributions so where a model was not produced there is no entry in the table. Where a particular result is of interest it is identified with roman numerals I -XIII and are referred to in the following text.

Result I from Table 1, is the control case with undamaged material, no delaminations and no fibre fracture cracks. Results II and III illustrate the stiffness reduction for line and star fibre fracture crack distributions without delamination; this stiffness reduction was found to be very localised. Results IV and V demonstrate that delamination alone causes no stiffness reduction in tension. Fig. 5 shows results VI -VIII which indicate that fibre fracture crack distributions combined with circular delaminations cause significant stiffness reduction. However, the predicted stiffness distribution only reflects the variation within the top ply, as the large circular delamination uncouples the displacements in the top ply from the remaining plies within the entire region where the local stiffness is evaluated. Consequently a delamination pattern which retains the coupling between the plies-as observed in real specimens and seen in the literature-is necessary. In Fig. 6a the stiffness reduction can be seen for the star and line fibre fracture crack distributions without delaminations (II, III). Fig. 6b shows the same fibre fracture crack distributions for peanut delaminations (IX, X), clearly showing that delaminations result in a more widespread stiffness reduction. Fig. 7 proves that the stiffness reduction for the twin ellipse (XI), peanut (XII) and split peanut (XIII) models are very similar. Finally in Fig. 8 the comparison of the peak and average stiffness reductions for all different fibre fracture distributions is shown for the peanut delamination pattern. The trend is that as crack density increases the stiffness decreases although this is not true for the moderate damage case. This, however, is a result of the cracks being more evenly distributed and not clustered in the centre. This creates a stiffer central region which is detected by the inverse method but offers no increase in structural integrity as it is surrounded by damaged material, resulting in an artificially high prediction of structural stiffness.

Experimental Comparison

The experimental comparisons in Fig. 9 illustrate that a replica model of the observed damage can reproduce the average stiffness reduction seen in the experiment to within 21%. This shows that a combination of delaminations and fibre fractures is the dominant failure mode in tension. The stiffness in Fig. 9 for the idealised model of peanut shaped delaminations with star pattern fibre fracture cracks is within 12% of the experimental results. This idealised model validates the work done in the parametric studies and demonstrates that such a model can be used to replicate impact damage in tension. It can easily be implemented for any stacking sequence or number of plies, based upon the delamination diameter obtained from C-Scan.

Discussion

The results in Table 1 clearly illustrate that delaminations without fibre fracture do not cause any tensile stiffness reduction. When fibre cracks are present the region with stiffness reduction is very small unless the fibre cracks are accompanied by delaminations. This observation can be explained by considering the shear lag effect, which causes a gradual recovery of stresses at some distance from a stress free surface.

The stress recovery due to intralaminar (in-plane) shear stresses is relatively slow, as the characteristic decay length is of the order of the fibre crack length. In contrast, the recovery due to interlaminar (out-of-plane) shear stresses is very fast as the characteristic length is of the order of a ply thickness. This is illustrated by Fig. 10, which shows that the strain concentration is very local when fibre cracks are not accompanied by delaminations. Similar observations have been made experimentally in [START_REF] Olsson | Experiments and analysis of laminates with artificial damage[END_REF]. The presence of delaminations eliminates the influence of the interlaminar stresses in the entire delaminated area so that only the slow shear lag effect from intralaminar stresses remains. This explains how delaminations increase the range of influence of fibre cracks.

The plots in Fig. 6b also demonstrate that for quasi isotropic lay-ups the model in [START_REF] Cairns | Residual strength of graphite/epoxy and kevlar/epoxy laminates with impact damage[END_REF] for modelling tensile stiffness reduction, which only considers fibre fracture in zero degree plies, is incorrect. The stiffness reduction for the star distribution is significantly higher than for line distribution where only the 0º fibres have fibre fractures, which shows that fibre fracture in plies of all orientations is significant. The model from [START_REF] Cairns | Residual strength of graphite/epoxy and kevlar/epoxy laminates with impact damage[END_REF] would probably still be valid for laminates with a high percentage of 0º plies.

The peanut model was chosen as its delamination pattern is the closest match to fractography of delaminations for carbon fibre epoxy laminates observed in the literature, [START_REF] Levin | Characterisation of delamination and fibre fractures in carbon fibre reinforced plastics induced from impact[END_REF] and [START_REF] Hull | Damage mechanism characterisation in composite damage tolerance investigations[END_REF]. The other delamination patterns, such as the split peanut and double ellipse, are still relevant and could be more suitable for laminates made from different fibre or matrix materials.

The effect of crack placement on limiting stiffness reduction for the moderate damage in Fig. 5 is one that should be noted. It demonstrates that the distribution of the stiffness reduction must be studied and that the peak and average stiffness values cannot be relied upon alone. A similar effect, where a region of less damaged material occurs in the centre, can also be seen in the split peanut and ellipse models, Fig. 7. Although this region is less damaged it does not contribute to the structural integrity of the laminate because it is surrounded by damaged material. Similar phenomena have been observed experimentally; see Figs. 17 and 18 in [START_REF] Sztefek | Tensile stiffness distribution in impacted composite laminates determined by an inverse method[END_REF].

The comparison between the experiments, the replica model and the idealised model shows that the fundamental parameters causing stiffness reduction in tension for impact damage have been captured. It is probable that the replica could have achieved a closer match to the experiment had better fractographic images of the delamination patterns been available, or if matrix cracks and initial imperfections in the laminate had been included. These factors contribute to the stiffness reduction although the effect is likely to be small. It is important to remember that the experiment is based on results from only one sample due to limited available data, and more importantly that the aim was to produce a method of modelling impact damage under tension load rather than replicating one particular case exactly. To this end a successful idealised model has been produced that replicates the impact damage stiffness reduction. It can easily be applied to any lay-up or thickness only requiring knowledge of the lay-up and the diameter of the delamination area determined from C-scan, which allows the size of the peanuts to be determined. The length of the fibre fracture cracks for the star pattern can be assumed to be approximately 50% of the delamination diameter [START_REF] Sjögren | Fractographic characterization of impact damage in carbon fibre/epoxy laminates[END_REF]. This approach can be tailored to suit a particular laminate with further investigation using fractographic techniques or with prior knowledge of the typical delamination shape and the extent of the fibre fracture cracks in a particular laminate.

Conclusion

In conclusion, the important parameters of impact damage under tensile loading have been identified as fibre fracture cracks and delaminations in combination rather than simply fibre fracture cracks. Fibre fracture cracks in plies of all fibre alignments have been demonstrated to be important rather than just cracks in the zero degree plies, as previously thought.

An idealised model based on peanut shaped delaminations with a star pattern distribution of cracks has been developed and has been shown to provide a good approximation of the behaviour of impact damage observed in experiments. This model can be produced with knowledge of only lay-up and delamination diameter obtained from C-scan, although better results would be obtained by tailoring the delamination shapes to suit the laminate being modelled and with detailed fractographic information about the extent of fibre fracture cracks within the impact damaged laminate.

Figure 1 aFigure 2 Figure 3 Figure 4 Figure 5 Figure 7 Figure 8 Figure 9

 12345789 Figure 1 a) Delamination patterns, b) Fibre fracture patterns and average crack

Acknowledgements

This project is funded by EPSRC in UK under grants GR/T18783/01 and EP/C531590/1 with CASE support from Airbus UK. The authors would like to thank Dr. Brian Falzon, Department of Aeronautics at Imperial College London and Dr. Matt Jevons, industrial supervisor of Airbus UK, for their help and support with this research.

Tables

Table 1 Stiffness reductions for all crack patterns and delamination shape combinations studied