Alessandro Pegoretti
email: alessandro.pegoretti@unitn.it

Ivan Cristelli

Claudio Migliaresi

Experimental optimization of the impact energy absorption of epoxy-carbon laminates through controlled delamination

Keywords: interleaving, interlaminar fracture toughness, impact, carbon/epoxy composites

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Experimental optimization of the impact energy absorption of epoxy-carbon laminates through controlled delamination

Introduction

The potential for weight savings with fiber-reinforced composites renders these materials very attractive for several structural application areas, which include military, aircraft, space, automotive, marine and sporting goods applications [START_REF] Mallick | Fiber-reinforced composites: materials, manufacturing, and design[END_REF]. Many fibre-reinforced composite materials offer a combination of specific modulus and strength values that are either comparable to or better than many traditional metallic materials [START_REF] Mallick | Fiber-reinforced composites: materials, manufacturing, and design[END_REF]. For some specific components, such as bumpers, armours, helmets, disposable barrier shells, etc..., the most important material requirement is the capacity to absorb a large amount of mechanical energy under impact conditions before reaching complete fracture. Most composites are brittle and so can only absorb energy in elastic deformation and through damage mechanisms, and not via plastic deformation [START_REF] Richardson | Review of low-velocity impact properties of composite materials[END_REF]. There are five basic mechanical failure modes that can occur in a composite after the initial elastic deformation [START_REF] Hancox | An overview of the impact behavior of fibre-reinforced composites[END_REF]: i) fibre fracture, or, for aramids, defibrilation, ii) resin crazing, microcracking and gross fracture, iii) debonding between fibre and matrix, iv) fibre pull out from the matrix, and v) delamination of adjacent plies in a laminate. In general, each of the above mentioned mechanisms may contribute to the energy dissipation process under impact conditions. Delamination is the most important damage mechanism in impacted composite laminates [START_REF] Johnson | Influence of delamination on impact damage in composite structures[END_REF]. When a composite structure is impacted, delamination often occurs, which seriously reduces the load bearing capacity of the laminate especially under compressive loads [START_REF] Larsson | Damage tolerance of a stitched carbon/epoxy laminate[END_REF]. Therefore, several methods have been developed to improve the interlaminar strength and interlaminar fracture toughness of composite laminates, such as matrix toughening [START_REF] Kim | Drop-weight impact damage tolerance of CFRP with rubber modified epoxy matrix[END_REF][START_REF] Jang | Influence of processing method on the fracture toughness of thermoplastic-modified, carbon-fiber-reinforced epoxy composites[END_REF][START_REF] Sjogren | Toughening mechanisms in rubber-modified glass fiber/unsaturated polyester composites[END_REF][START_REF] Verrey | Interlaminar fracture toughness improvement in composites with hyperbranched polymer modified resin[END_REF], through-thickness stitching [START_REF] Jain | in the effect of stitching on mode-I delamination toughness of laminated composites[END_REF][START_REF] Jain | Determination of Mode II delamination toughness of stitched laminated composites[END_REF][START_REF] Dransfield | On the effects of stitching in CFRPs-I. Mode I delamination toughness[END_REF][START_REF] Jain | On the Effects of Stitching in CFRPs-II. Mode II Delamination Toughness[END_REF][START_REF] Watt | Thermal degradation of the mode I interlaminar fracture properties of stitched glass fibre vinyl ester composites[END_REF], Z-pinning [START_REF] Yan | Mode II delamination toughness of z-pinned laminates[END_REF][START_REF] Cartie | Delamination of Z-pinned carbon fibre reinforced laminates[END_REF], and interleaving [START_REF] Davidson | Effect of interlayer modulus on fracture mode ratio for interleaved composite laminates[END_REF][START_REF] Hillermeier | Interlayer toughening of resin transfer molding composites[END_REF][START_REF] Jiang | Loading rate dependence of Mode II fracture behavior in interleaved carbon fibre/epoxy composite laminates[END_REF][START_REF] Lee | Effect of interleaved non-woven carbon tissue on interlaminar fracture toughness of laminated composites: Part I -Mode II[END_REF][START_REF] Lee | Effect of interleaved non-woven carbon tissue on interlaminar fracture toughness of laminated composites: Part II -Mode I[END_REF][START_REF] Matsuda | Effect of ionomer thickness on mode I interlaminar fracture toughness for ionomer toughened CFRP[END_REF][START_REF] Yun | Toughening of carbon fiber/epoxy composite by inserting polysulfone film to form morphology spectrum[END_REF][START_REF] Gao | Mode II delamination and damage resistance of carbon/epoxy composite laminates interleaved with thermoplastic particles[END_REF][START_REF] Walker | Improving impact resistance of carbon-fibre composites through interlaminar reinforcement[END_REF] and short-fibre interlaminar reinforcement [START_REF] Walker | Improving impact resistance of carbon-fibre composites through interlaminar reinforcement[END_REF]. Most of the above mentioned methods have been reviewed by Kim and Mai [START_REF] Kim | Engineered interfaces in fibre-reinforced composites[END_REF], who also pointed out the key role played by the fibre-matrix interfacial shear strength in determining the interlaminar resistance of composite laminates. In fact, a marked improvement of the interlaminar resistance when the fibrematrix adhesion increases is reported in several studies [START_REF] Wang | Interfacial mobility and its effect on interlaminar fracture toughness in glass-fibre-reinforced epoxy laminates[END_REF][START_REF] Yeung | The effect of glass-resin interface strength on the impact strength of fiber reinforced plastics[END_REF][START_REF] Deng | Influence of fiber-matrix adhesion on mechanical properties of graphite/epoxy composites: I. Tensile, flexure, and fatigue properties[END_REF][START_REF] Madhukar | Fiber-matrix adhesion and its effect on composite mechancal properties: IV. Mode I and Mode II fracture toughness of graphite/epoxy composites[END_REF]. On the other hand, the improvement of damage resistance and tolerance in interlaminar fracture is at the expenses of other important mechanical properties, such as the the energy absorption under impact conditions [START_REF] Goan | The influence of interfacial bonding on the properties of carbon fiber composites[END_REF] and the fatigue resistance. In fact, as the interlaminar strength increases the delamination damage is hindered, and the composite laminates tend to fracture in a more brittle way under impact conditions. Hong and Liu [START_REF] Hong | On the relationship between impact energy and delamination area[END_REF] showed that there is an almost linear relationship between the impact energy and the total delaminated area in glass/epoxy laminates. As a consequence, an efficient way to improve the energy absorption capacity of laminate composites in the through-thickness direction is by promoting controlled delamination by weakening the interlaminar bond strength or interlaminar fracture toughness [START_REF] Kim | Engineered interfaces in fibre-reinforced composites[END_REF]. The transverse Charpy impact fracture energy of carbon fibre reinforced plastics (CFRPs) has been successfully increased by three times with embedded nylon sheets, at the expenses of some 25% reduction of the interlaminar shear strength [START_REF] Favre | Improving the fracture energy of carbon fiber reinforced plastics by delamination promoters[END_REF]. Perforated films have been proven to be more effective than unperforated ones because they could provide both the weak and strong bonding in the regions of film and perforation ("intermittent interlaminar bond concept" [START_REF] Mai | On fiber composites with intermittent interlminar bonding[END_REF]). In fact, using multi-layers of perforated Mylar films, Jea and Felbeck [START_REF] Jea | Increased fracture toughness of graphite-epoxy composites through intermittent interlaminar bonding[END_REF] reached a remarkable 500 % increase of the transverse fracture toughness (determined on reinforced modified compact tension speciemens) of CFRPs, while the tensile transverse strength dropped about 20 % and the Young's modulus remained the same. The transverse fracture toughness was determined by Jang et al. [START_REF] Jang | Controlled energy dissipation in fibrous composites, I Controlled delamination[END_REF] on fibrous composites with several other types of delamination promoters, including aluminium foils, bleached papers, polyester textile fabrics and polyimide Mylar. They found that the energy absorption mechanisms strongly depend on the loading direction and other testing parameters such as loading speed and span-to-depth ratio in bending. Moreover, the existence of an optimal value of the interlaminar fracture toughness corresponding to maximum energy absorption has been theoretically proven by Lear and Sankar [START_REF] Lear | Optimizing energy absorption in multi-layered materials through controlled delamination[END_REF], but not experimentally verified. Delamination, and therefore the energy absorbing capability, of composite laminates has been also successfully promoted by other techniques, such as the introduction of fibre orientation change between adjacent layers [START_REF] Hong | On the relationship between impact energy and delamination area[END_REF], the modification the target stiffness [START_REF] Schoeppner | Delamination threshold loads for low velocity impact on composite laminates[END_REF], or the introduction of ply grouping [START_REF] Fuoss | Effects of stacking sequence on the impact resistance in composite laminates -Part 1: parametric study Composite[END_REF].

Aiming at optimizing the energy absorption capability of carbon/epoxy laminates under impact conditions, in the present work the "intermittent interlaminar bond concept" has been further explored.

In particular, the possibility to selectively change the interlaminar fracture toughness by interleaving with poly(ethtlene-terephthalate) foils perforated with a variable number of holes per unit area (hole density), has been investigated, and its effects on the composites impact behaviour have been assessed.

Experimental

Composites manufacturing

By using a computer-controlled high-speed drill the PET foils were perforated with circular holes 1 mm in diameter (see Figure 1a). Holes were regularly spaced at a distance L, as schematized in Figure 1b.

As the distance L varied from 7 to 1.5 mm, the hole density changed from 2.0 up to 44.4 holes/cm 2 ), accordingly to Table 1. The interfacial contact area (ICA) is obviously related to the hole density and it can be estimated as follows:

ICA = π 4 D L () 2 (1)
The relationship between the parameter ICA and the hole density is reported in Table 1. Before to be used for composite preparation, both as-received and perforated PET foils were accurately washed with water and a detergent, and then rinsed, first in water and finally in n-heptane.

The prepreg laminae were cut in size of 150x150 mm 2 and non-interleaved or interleaved composites were realized by alternating prepreg laminae and PET foils according to the stacking sequences schematized in Figure 2. It is worth noting that an ICA value of 100% can be assigned to the noninterleaved composites, while a 0 % ICA value can be considered for the composites interleaved with non-perforated PET foils. By using perforated PET foils the interfacial contact area can be gradually varied in the range reported in Table 1. The number of stacked laminae was also changed depending on the type of test to perform (see following sections). All the prepreg fabrics were stacked with the same relative orientation.

The curing cycle presented in Figure 3 was adopted for the composite consolidation.

Interlaminar fracture toughness test

Mode I interlaminar fracture toughness (G IC ) was measured on double cantilever beam (DCB) specimens whose geometry is reported in where P is the load, δ is the load point displacement, and the coefficient n is evaluated through a compliance calibration method. This method is based on the construction of a plot of log(δ i /P i ) versus log(a i ) using the visually observed delamination onset values and all the propagation values as represented in Figure 5a: n represents the slope of the least square linear regression line passing through the data. As reported in Figure 5b the obtained n values resulted to be practically independent of the interlaminar contact area with values oscillating around 2.5. curves.

Three point bending test

Bending modulus, E b , and bending strength, σ b , under impact conditions were calculated on the basis of Equations ( 3) and (4), respectively.

Results and discussion

Interlaminar fracture toughness

Optical micrographs of the cross section of composites, interleaved with both as-received and perforated PET foils, are reported in Figure 6. In Figure 6a it is very easy to identify the presence of the as-received (unperforated) PET interleaving foils separating the composite laminae. When perforated PET foils are used the typical composite structure is depicted in Figures 6b and6c

Quasi static mechanical tests

The mechanical behaviour under quasi static loading conditions has been investigated by three point bending tests whose results are summarized in Figure 10a. In this figure, the bending modulus, E b , and the bending strength, σ b , are plotted as a function of the interlaminar fracture toughness for crack initiation. Interestingly enough, the bending modulus is practically independent of the interlaminar fracture toughness, while the bending strength display a tendency to increase as G IC,NL increases.

The observed dependence of composite modulus and strength on the interlaminar fracture toughness closely resembles their dependence on the fibre-matrix adhesion. In fact, the observed behaviour is consistent with the existing literature on the effects of fibre-matrix adhesion on the quasi static tensile and flexural mechanical properties of epoxy/graphite unidirectional composites. As reported by Madhukar and Drzal [START_REF] Madhukar | Fiber-matrix adhesion and its effect on composite mechancal properties: II. Longitudinal (0°) and transverse (90°) tensile and flexure behavior of graphite/epoxy composites[END_REF] and by Deng and Le [START_REF] Deng | Influence of fiber-matrix adhesion on mechanical properties of graphite/epoxy composites: I. Tensile, flexure, and fatigue properties[END_REF] the fibre surface modification do not have much effect on the tensile and flexural moduli and on the fibre dominated properties. However, the strengths and the maximum strains, that are governed by the matrix and interface properties, are highly sensitive to the fibre surface modification. On the basis of our findings, similar considerations can be put forward on the effect of interlaminar fracture toughness on the flexural properties of carbon/epoxy fabric laminates.

Impact resistance

The bending modulus, E b , and the bending strength, σ b , have been also evaluated under impact conditions by means of instrumented Charpy tests. The obtained values, plotted as a function of the interlaminar fracture toughness for crack initiation, are reported in Figure 10b. It is interesting to note that, under impact conditions, both bending modulus and strength display a trend similar to that observed under quasi-static conditions. As expected, due to the viscoelastic behaviour of the PET interleaving foils and of the epoxy matrix, the stiffness and strength values of interleaved composites under impact conditions are higher than the corresponding data measured at a lower strain rate. processes (Figure 12b). For the case reported in Figure 12b, which refers to an interleaved composite with a low (4.9%) ICA value, the total impact fracture energy exceeds by a factor of about 1.8 the energy absorbed by non-interleaved composites.

For each specimen, the total fracture energy (E T ) could be evaluated by measuring the total area under the curve and normalising it to the specimen cross sectional area. E T is the sum of a crack initiation energy (E I ) and a crack propagation energy (E P ). The initiation energy is the energy measured up to the first load drop, while the propagation energy is the energy required from this point to break the specimen. Total fracture, initiation, and propagation impact energies are summarised in Figure 13 as a function of the interlaminar fracture toughness of the investigated laminates.

Interestingly enough, the interlaminar fracture toughness results to have an opposite influence on the initiation and propagation energies. In fact, while the initiation energy increases, the propagation energy decreases with the interlaminar fracture toughness. As a consequence, the total fracture energy passes through a maximum for a given G IC,NL value. The existence of an optimal value of the interlaminar fracture toughness, corresponding to maximum energy absorption, can be explained by considering the occurrence of two concurrent phenomena. In fact, as the interfacial adhesion decreases the delaminated area increases. At the same time, as the interlaminar fracture toughness decreases, less and less energy is absorbed for creating a given delaminated area. These two phenomena have a contrasting effect on the fracture energy absorption process, thus justifying the existence of an optimal interlaminar adhesion value. This result has been theoretically predicted by numerical simulation of a layered beam under statically applied three point bending loads [START_REF] Lear | Optimizing energy absorption in multi-layered materials through controlled delamination[END_REF].

As a matter of fact, quite limited, and in some cases contrasting, experimental information exist on the role of interlaminar resistance on the impact behaviour of composite laminates. Bader et al. [START_REF] Bader | The effect of fibre-matrix interface strength on the impact and fracture properties of carbon-fibre-reinforced epoxy resin composites[END_REF] observed that the principal effect of the interlaminar strength is to modify the mode of failure under impact conditions. On epoxy matrix unidirectional composites prepared with various both untreated and surface treated carbon fibres, they reported that at high levels of interlaminar strength a brittle failure occurred with relatively little absorbed energy. At low levels of strength they observed a multiple delamination, with an energy absorption about three times as high as in the brittle case, but with an extensive disintegration of the specimens. At intermediate values of interlaminar strength a progressive failure occurred which they consider to be the best practical compromise. Our results are in agreement with the above reported considerations. Also Yeung and Broutman [START_REF] Yeung | The effect of glass-resin interface strength on the impact strength of fiber reinforced plastics[END_REF] have shown that a non-monotonic correlation exists between the Charpy impact fracture energy and the interlaminar shear strength of epoxy and polyester -glass fabric laminates. The fibre-matrix interface adhesion was altered by surface treatments of the fabrics with silane coupling agents and with a silicone fluid mould release to achieve various interlaminar shear strengths (ILSS) in short-beam shear tests. The authors observed that with increasing ILSS the fracture initiation energy increased modestly, while the fracture propagation energy as well the total impact energy decreased exhibiting a minimum and levelling off at intermediate values.

Beaumont et al. [START_REF] Beaumont | Methods for improving the impact resistance of composite materials[END_REF] defined a dimensionless parameter called the ductility index (DI), which is found useful for ranking the impact performance of different materials under similar testing conditions. The DI is defined as the ratio between the propagation energy and the initiation energy, i.e.:

DI = E P E I
High values of DI would mean that most of the total energy is expended for crack propagation. The ductility index values of the investigated laminates are reported in Figure 14 as a function of the interlaminar fracture toughness. Even if the data are affected by some scattering, they clearly show that the ductility index markedly increases as the interlaminar fracture toughness decreases. In particular, for the lowest explored value of the interlaminar fracture toughness the fracture propagation energy is almost 4 time higher that the fracture initiation component.

Conclusions

The Mode I interlaminar fracture toughness of carbon-epoxy laminates has been selectively varied, in the range from 40 up to 260 J/m 2 , by interleaving with as-received and perforated PET foils. Its effect on quasi-static and impact mechanical response has been investigated by three point bending tests.

The following conclusions can be drawn:

i)

The bending modulus determined both under quasi-static and impact conditions is practically independent of the interlaminar fracture toughness.

ii)

The bending strength determined both under quasi-static and impact conditions markedly increase as the interlaminar fracture toughness increases.

iii)

The interlaminar fracture toughness has an opposite effect on the initiation and propagation energies under impact conditions. In fact, while the initiation energy increases, the propagation energy decreases with the interlaminar fracture toughness. As a consequence, the total fracture energy passes through a maximum for a given interlaminar fracture toughness value, for which the total impact energy increases is 1.8 times higher than the no PET foils 100.0 Stacking sequences of the non-interleaved and PET-interleaved laminates.

Captions for Figures

Figure 3.

Temperature and pressure profiles adopted for the composites curing cycle. Effect of interlaminar fracture toughness on the total fracture ( ), on the initiation ( ), and on the propagation ( ) energies.

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2. 1 .

 1 Materials Carbon fibre/epoxy resin cross-ply prepreg (CC 206 -ET443) was supplied by SEAL S.p.A. Legnano (MI), Italy. The carbon fabric (CC 206) consisted of high strength carbon yarns (HS 3K) in a twill 2/2 weave with a weight of 204 g/m 2 (102 g/m 2 in the warp and 102 g/m 2 in the weft directions). The matrix (ET443) was a transparent epoxy resin with a gel time of 9±3 min, a cured density of 1.2 g/cm 3 , and a fully cured glass transition temperature of about 145 °C. Mylar® (DuPont) poly(ethylene terephthalate) (PET) foils, 26 µm thick, were used as interleaving material. These foils displayed isotropic mechanical properties (with a tensile modulus of 4.5 GPa, and a yield strength of 97.2 MPa), a glass transition temperature of 105 °C, a melting temperature of 260 °C, and a crystallinity content of about 36 % [40].

Figure 4 .

 4 According to ASTM D 5528-01 standard the following dimensions were adopted: L = 135 mm; b = 25 mm; a 0 = 50 mm; h = 3.00-3.90 mm. Noninterleaved specimens were prepared by stacking 16 prepreg laminae, while interleaved specimens were prepared by alternating 16 prepreg laminae with 15 PET foils. A non adhesive fluoroethylenepropylene foil (Teflon® FEP -DuPont) with a thickness of about 25 µm was inserted at the midplane of the laminates during layup to form an initiation site for the delamination. After cutting the DCB specimens to the desired size, piano hinges were bonded by a room-temperature curable bicomponent epoxy resin. Delamination tests were performed at room temperature by an Instron 4502 universal testing machine at a cross-head speed of 2 mm/min recording the experiments by a digital video-camera. According to one of the data reduction methods proposed by the ASTM D5528 -01 standard, G IC value has been calculated as follows:

  According to ASTM D-790-03 standard, three point bending tests were performed on composite bars with a thickness of 1.7 mm, a width of 12.7 mm and a length of 110 mm. The test bars were machined out of composite plates consisting of 8 prepreg laminae (non-interleaved composites) or of 8 prepreg laminae interleaved with 7 PET foils. Load-deflection curves were obtained at room temperature by an Instron 4502 machine at a cross-head speed of 5 mm/min at a span length of 80 mm. At least 5 specimens were tested for each sample. According to ASTM D-790-03 standard, a bending modulus, E b , and a bending strength, σ b , were calculated by using the following formulae: the span length, b the width and d the thickness of the specimen, P max , the maximum measured load, and m the slope of the tangent to the initial straight-line portion of the load-deflection curve.2.5. Instrumented Charpy impact testInstrumented Charpy impact tests were performed on test bars of rectangular cross section, 80 mm long, 10 mm wide and 1.7 mm thick machined from the same laminates where specimens for three point bending test were obtained from. Tests were performed by a CEAST model 6549 instrumented impact pendulum. Specimens were supported to the machine anvils at a span length of 40 mm and broken by a single swing of the pendulum with the impact line midway between the supports in the flat wise direction. The striking nose of the pendulum was characterized by an included angle of 30° and a tip rounded to a radius of 2 mm. The striking hammer impacted the specimens at a speed of 2 m/s and with a kinetics energy of 1.91 J. Load-time data points were acquired at a sampling time of 0.01 ms. At least 10 specimens were tested for each sample.

  , in which the black arrows indicate the borders of a hole in the PET foils. In most cases, the epoxy matrix was able to flow through the holes in the interleaving foils during composite preparation, as represented in Figure6b, thus forming a resin bridge between two adjacent composite laminae. The number of these resin bridges obviously increases with the number of holes per unit area, and hence the interlaminar contact area (ICA), increases. In a few cases (upper and lower parts of Figure6c) the resin was not able to flow completely through the holes, and micro-voids were created in the laminate. The load-displacement curves for DCB delamination tests performed on composites with various interlaminar contact areas are compared in Figure7. It is interesting to observe that these curves tends to reach higher load values as the interlaminar contact area (and hence the number of resin bridges per unit area) increases. According to the data reduction procedures suggested by ASTM standard D5528 -01, an initiation G IC value can be determined using the load and deflection data measured at the point of deviation from linearity in the load-displacement curve (G IC,NL ), or at the point at which delamination is visually observed (G IC,VIS ) on the edge of the DCB specimen: have the same meaning as in Equation (2) with the subscripts NL and VIS referring to the point of deviation from linearity in the load-displacement curve and to the point at which delamination is visually observed, respectively. The initiation G IC values obtained using Equations (5) and (6) are reported in Figure8as a function of the interlaminar contact area of the laminates. As expected, the interlaminar toughness values for fracture initiation increase as the interlaminar contact area increase, due to the increasing numbers of interlaminar resin bridges. It is worth observing that, independently of the data reduction method, the initiation interlaminar fracture toughness of the composites interleaved with as-received PET foils (ICA = 0 %) assumes a value of about 40 J/m 2 . This indicates a relatively good adhesion level between the PET foils and the epoxy matrix. G IC,VIS and G IC,NL values are very similar up to interlaminar contact areas of about 40 %, while they markedly differ for the non-interleaved composites which display G IC,NL and G IC,VIS values of about 260 J/m 2 and 430 J/m 2 , respectively. In the continuation of this article we will refer to G IC,NL values since the underlying data reduction method is more straightforward. From the data reported in Figure9it is possible to observe that the interlaminar contact area also markedly affects the delamination resistance curves. In fact, as the interlaminar contact area decreases R curves tend to flatten and level off at progressively lower limiting values. For low ICA values the delamination resistance is practically independent of the crack length.At this point, before to present the results of quasi static and impact tests, it is important to underline that we are conscious that three point bending tests carried out on unnotched specimens are mostly characterized by a sliding mode (Mode II) rather than an opening mode (Mode I). Unfortunately, there is no European nor ASTM mode II standard at present, but the JIS group has a mode II test procedure based on the ENF specimen and an option to allow stabilising the test[START_REF] Davies | Mode II delamination, in Fracture mechanics testing methods for polymers, adhesives and composites[END_REF]. At the same time, it is worthwhile to note that most of the experimental data available in the scientific literature indicate that a monotonic relationship exists between Mode I and Mode II interlaminar fracture toughness. In other words, with only few exceptions[START_REF] Hojo | Modes I and II interlaminar fracture toughness and fatigue delamination of CF/epoxy laminates with self-same epoxy interleaf[END_REF][START_REF] Naik | Interlaminar fracture characterization for plain weave fabric composites[END_REF], when Mode I interlaminar fracture toughness increases, Mode II interlaminar fracture toughness generally does the same[START_REF] Lee | Effect of interleaved non-woven carbon tissue on interlaminar fracture toughness of laminated composites: Part I -Mode II[END_REF][START_REF] Lee | Effect of interleaved non-woven carbon tissue on interlaminar fracture toughness of laminated composites: Part II -Mode I[END_REF][START_REF] Alvarez | The Influence of Matrix Chemical Structure on the Mode I and II Interlaminar Fracture Toughness of GIass-Fiber/Epoxy Composites[END_REF][START_REF] Chen | Fracture behavior of interleaved fiber resin composites[END_REF][START_REF]Changing the toughness of graphite fiber/resin based composites by changing their internal structure[END_REF][START_REF] Mujika | Mechanical Properties of Carbon Woven Reinforced Epoxy Matrix Composites. A Study on the Influence of Matrix Modification With Polysulfone[END_REF][START_REF] Park | Roles of interfaces between carbon fibers and epoxy matrix on interlaminar fracture toughness of composites[END_REF][START_REF] Stevanovica | Mode I and mode II delamination properties of glass/vinyl-ester composite toughened by particulate modified interlayers[END_REF]. An estimation of the interlaminar shear strength (ISS) by the short beam shear test (ASTM D 2344) was also considered, but the small specimens dimensions required for this testing procedure posed a serious limitation on the usefulness of this test for the materials under investigation. In fact, according to ASTM D 2344 standard, specimen with span-to-thickness and a width-to-thickness ratios of 4 and 2, respectively are required. This would have lead to samples with as low as 1-2 resin bridges per lamina, in the case of composites interleaved with foils having the lowest holes densities. As a consequence, the specimens would have not been representative of the situation at a macro scale, like that encountered in quasistatic and impact three point bending tests. On the basis of the above considerations, G IC value has been preferred as a parameter representative of the interlaminar fracture toughness.

Figure 11 documents

 11 Figure 11 documents the conditions of some of the investigated laminates after the Charpy impact test. It clearly emerges that the fracture behaviour under low speed impact conditions is profoundly affected by the interlaminar fracture toughness of the tested composites. For non-interleaved composites, that display the highest G IC,NL values, the broken specimens show a typically brittle aspect, with relatively sharp fracture surfaces and no (or very little) delamination damages (see Figure11a). When the interlaminar fracture toughness decreases, the failure process of the impacted specimens is characterized by progressively increasing delamination damage, and more extended fracture surfaces. For the lowest interlaminar fracture toughness values a certain disintegration of the tested specimens is also observed. Depending on the different fracture mechanisms, the loaddisplacement and energy-displacement curves recorded during the Charpy instrumented tests also markedly change with the interlaminar fracture toughness of the investigated laminates. By way of examples, the typical impact curves of two different laminates are reported in Figure12. It is evident that the brittle failure mode of non-interleaved composites (Figure12a) results in a load-displacement curve characterized by a sudden load drop with practically no energy consumed for the crack propagation process. On the other hand, the load-displacement curve of interleaved composites displays a different trend, being a large amount of energy absorbed for damage delamination

Perforated

  PET foil: a) optical micrograph and b) schematic of the hole distribution.

Figure 2 .

 2 Figure 2.

Figure 4 .

 4 Figure 4. Schematic of the DCB specimens.

Figure 5 .

 5 Figure 5. a) typical compliance calibration plot and b) n exponent as a function of the interlaminar contact area.

Figure 6 .

 6 Figure 6. Optical micrographs of the cross section of composites interleaved with a) as-received, b) and c) perforated PET foils (ICA = 19.6%).

Figure 7 .

 7 Figure 7. Load-displacement curves for DCB delamination tests performed on composites with various interlaminar contact areas: a) ICA = 100%, b) ICA = 34.9 %, c) ICA = 12.6 %, d) ICA = 8.7 %, e) ICA = 3.1 %, f) ICA = 0 %.

Figure 8 .

 8 Figure 8. Mode I interlaminar fracture toughness evaluated at first deviation from linearity ( , G IC,NL ) and at visual onset of delamination ( , G IC,VIS ), as a function of the interlaminar contact area.

Figure 9 .

 9 Figure 9.Effect of the interlaminar contact area on the delamination resistance curve (R curve).

Figure 10 .

 10 Figure 10. Effect of the interlaminar fracture toughness (G IC,NL ) on the bending modulus ( ) and strength ( ) measured under a) quasi-static (5 mm/min) and b) impact (2 m/s) conditions.

Figure 11 .

 11 Figure 11. Pictures of specimens after the Charpy impact test: a) G IC,NL = 262 J/m 2 (ICA = 100%), b) G IC,NL = 155 J/m 2 (ICA = 34.9%), c) G IC,NL = 103 J/m 2 (ICA = 12.6%), and d) G IC,NL = 62 J/m 2 (ICA = 4.9%).

Figure 12 .

 12 Figure 12. Load-displacement and energy-displacement curves recorded during instrumented Charpy impact tests performed on composites with a) G IC,NL = 262 J/m 2 (ICA = 100%), and b) G IC,NL = 62 J/m 2 (ICA = 4.9%).

Figure 13 .

 13 Figure 13.

Figure 14 .

 14 Figure 14. Ductility index values as a function of the interlaminar fracture toughness.

  Figure 1

  Figure 3

  Figure 5

Table 1 .

 1 Relationship between the inter-hole distance (L), the hole density, and the interfacial contact area (ICA).

	Inter-hole distance	Hole density	Interfacial contact area
	L [mm]	[number of holes / cm 2 ]	ICA [%]
	no holes	0.0	0.0
	7.0	2.0	1.6
	6.0	2.8	2.2
	5.0	4.0	3.1
	4.0	6.3	4.9
	3.0	11.1	8.7
	2.5	16.0	12.6
	2.0	25.0	19.6
	1.5	44.4	34.9

Acknowledgments

The authors are grateful to Dr Martina Corasaniti of SEAL S.p.A. Legnano (MI), Italy for the selection and supply of the carbon-epoxy prepregs.

value observed for non-interleaved composites. The ductility index steadily increases from 0.2 up to 4.2 as the interlaminar fracture toughness decreases.


A Barroso

V Manti
email: mantic@esi.us.es

F París
email: paris@esi.us.es

lap joints for use in failure criteria

Keywords: A) Adhesive joints, C) Interfacial strength, C) Failure criterion, C) Stress singularity., ), ( ) 2 ( ), (, )

A procedure for the singularity characterization of anisotropic multimaterial corners which typically appear in adhesively bonded lap joints between metals and composites is presented and implemented in the present work. The characterization in terms of characteristic exponents (stress singularities), characteristic functions and generalized stress intensity factors allows the definition of an oriented test program in order to analyze the suitability of a singularity based failure criterion for structures of this type.

INTRODUCTION

Adhesively bonded joints between metallic sheets and composite laminas generate critical points, where the Linear Theory of Elasticity predicts unbounded stresses. These singularity stress fields are due to abrupt changes in geometry and/or abrupt changes in the elastic properties between the different materials, Sinclair [START_REF] Sinclair | Stress singularities in classical elasticity -I: Removal, interpretation and analysis[END_REF]. In the particular case of adhesively bonded single (or double) lap joints between an aluminium sheet and a CFRP laminate, these critical points can be easily identified in Fig. 1a.

Fig 1

With a polar coordinate system (r,θ) located at the multimaterial corner tip (Fig. 1b), where θ i-1 and θ i are the angles of the i-th material wedge (N wedges), the asymptotic singularity stress field in the vicinity of the corner can be assumed to have the following series expansion structure with separation of variables (some particular corner configurations, which will not be considered here, do not admit the separation of variables, see for example Joseph and Zhang [START_REF] Joseph | Multiple root solutions, wedge paradoxes and singular stress states that are not variable-separable[END_REF]).

where n is the number of terms considered in the series expansion, λ k are the characteristic exponents, leading to singular terms when 0< λ k <1 and regular terms if λ k 1, f ijk (θ) and g ik (θ) are the characteristic functions: smooth functions inside each material wedge which contain the angular dependency of the stress and displacement components respectively associated to the singularity mode λ k and K k are the Generalized Stress Intensity Factors (GSIFs) which represent the weight of each term in the asymptotic series expansion.

The characterization of the singular stress field is fundamental if the initiation of failure (an initial crack or localized damage whose propagation implies the complete failure of the joint) can be assumed to occur due to that singular stress field.

In the same way as Linear Elastic Fracture Mechanics (LEFM) accurately predicts the propagation of a crack when the SIF (K k ) reaches a certain allowable value (the fracture toughness of the material), it seems reasonable to consider the possibility of a GSIF based failure criterion (see Gradin & Groth [START_REF] Gradin | A fracture criterion for adhesive joints in terms of material induced singularities[END_REF], Groth [START_REF] Groth | Stress singularities and fracture at interface corners in bonded joints[END_REF], Hattori [START_REF] Hattori | A stress-singularity-parameter approach for evaluating the adhesive strength of single lap joints[END_REF], Reedy [START_REF] Reedy | Intensity of the stress singularities at the interface corner between a bonded elastic and rigid layer[END_REF],

Reedy & Guess [START_REF] Reedy | Butt tensile joint strength: interface corner stress intensity factor prediction[END_REF], Akisanya [START_REF] Akisanya | On the singular stress field near the edge of bonded joints[END_REF], Yang & Munz [START_REF] Yang | Stress singularities in a dissimilar materials joint with edge tractions under mechanical and thermal loadings[END_REF], Lefebvre & Dillard [START_REF] Lefebvre | A stress singularity approach for the predictions of fatigue crack initiation in adhesive bonds. Part 1: Theory[END_REF], Ishii et al [START_REF] Ishii | Evaluation of the fatigue strength of adhesively bonded CFRP/metal single and single-step double-lap joints[END_REF], Wang & Rose [START_REF] Wang | Compact solutions for the corner singularity in bonded lap joints[END_REF], Dunn et al. [START_REF] Dunn | Small scale geometric and material features at geometric discontinuities and their role in fracture analysis[END_REF], Penado [START_REF] Penado | Singular intensity factors at bimaterial anisotropic interfaces[END_REF], Lazzarin et al [START_REF] Lazzarin | A two terms stress function approach to evaluate stress distributions in bonded joints of different geometry[END_REF], Leguillon & Yosibash [START_REF] Leguillon | Crack onset at a v-notch. Influence of the notch tip radius[END_REF] and Quaresimin & Ricotta [START_REF] Quaresimin | Stress intensity factors and strain energy release rates in single lap bonded joints in composite materials[END_REF] among others). To analyze the suitability of such an approach using the concepts of the LEFM, it is necessary to have the tools to evaluate λ k , f ijk (θ), g ik (θ) and K k . Only after this careful characterization can experimental tests be carried out in an oriented way, changing in a controlled manner the values of the singularity parameters, in order to observe their influence (if it occurs) in a failure initiation.

In successive sections we will introduce the analytical procedure to determine the characteristic exponents and characteristic functions (Section 2), and the numerical models (in BEM) as well as the implemented procedure based on a least squares determination of the GSIFs (Section 3). The accuracy of the method is checked with some benchmark problems, and some examples regarding the characterization of adhesively bonded lap joints between metal and composites are presented (Section 4). With these tools, a preliminary experimental test program can be specifically designed to investigate the influence of the singularity parameters on failure initiation.

CHARACTERIZATION OF THE SINGULARITY MODES

The characteristic exponents λ k and the characteristic functions f ijk (θ) and g ik (θ) only depend on material elastic properties, local geometry and local boundary conditions.

Considering homogeneous linear elastic materials, perfect adhesion between the material wedges, homogeneous boundary conditions (at external faces) and plane states (plane stress, plane strain or generalized plane strain), at the present moment there is enough information and efficient analytical and computational tools for the evaluation of accurate values of λ k , see Dempsey and Sinclair [START_REF] Dempsey | On the stress singularities in the plane elasticity of the composite wedge[END_REF][START_REF] Dempsey | On the singular behaviour of a bi-material wedge[END_REF], Pageau et al. [START_REF] Pageau | The order of stress singularities for bonded and disbonded three-material junctions[END_REF], Ting [START_REF] Ting | Stress singularities at the tip of interfaces in polycristals. Damage and Failure of Interfaces[END_REF], Manti et al. [START_REF] Manti | Stress singularities in 2D orthotropic corners[END_REF] and Poonsawat et al. [START_REF] Poonsawat | Singular stress fields of an anisotropic composite wedge with a frictional interface[END_REF][START_REF] Poonsawat | Singular stress fields of angle-ply and monoclinic bimaterial wedges[END_REF] among others. In this work a semianalytical tool, developed by the authors, Barroso et al. [START_REF] Barroso | Singularity analysis of anisotropic multimaterial corners[END_REF], is employed.

Some of the main characteristics and improvements of the present tool in comparison with the previous works are explained briefly in the following.

All linear elastic material behaviour laws (from isotropic to fully anisotropic) can be included simultaneously in the analysis. Depending on the approach, it has often been observed in previous works that only material behaviour laws of one kind (only isotropic, or only orthotropic) have been considered in the analysis. When considering anisotropic materials and using the powerful and elegant Lekhnitskii-Eshelby-Stroh formalism for anisotropic elasticity: Lekhnitskii [START_REF] Lekhnitskii | Some cases of the elastic equilibrium of a homogeneous cylinder with arbitrary anisotropy[END_REF], Eshelby et al. [START_REF] Eshelby | Anisotropic elasticity with applications to dislocation theory[END_REF] and Stroh [START_REF] Stroh | Dislocations and cracks in anisotropic elasticity[END_REF][START_REF] Stroh | Steady state problems in anisotropic elasticity[END_REF], or simply, Stroh formalism in what follows, these limitations were usually associated to the fact that isotropic materials are, in the framework of the Stroh formalism, mathematically degenerate materials and the formalism becomes much more cumbersome, see Ting and Hwu [START_REF] Ting | Sextic formalism in anisotropic elasticity for almost non-semisimple matrix N[END_REF] and Wang and Ting [START_REF] Wang | The Stroh formalism for anisotropic materials that posses an almost extraordinary degenerate matrix N[END_REF].

A particular effort has been focused on obtaining explicit expressions of the Stroh matrices (A and B) for transversely isotropic materials, a material behaviour which can be assumed to represent the unidirectional long fibre composite laminas. These expressions have also been properly orthogonalized and normalized for the powerful orthogonality and closure relationships of the formalism to be fulfilled. These expressions, Barroso et al. [START_REF] Barroso | Computing stress singularities in transversely isotropic multimaterial corners by means of explicit expressions of the orthonormalized Stroh-eigenvectors[END_REF], will help to develop further application of the Stroh formalism to transversely isotropic materials. It is of major importance to notice that mathematically degenerate cases can appear when considering transversely isotropic materials (see Tanuma [START_REF] Tanuma | Surface impedance tensors of transversely isotropic materials[END_REF]) irrespective of the value of the elastic constants of the material and only depending on the relative orientation of the material, a fact that has also been considered in the program implementation.

All typical orthogonal homogeneous boundary conditions can be imposed at the external faces of the multimaterial corner (see Table 1), and closed corners (called also cross points, all materials bonded, no external faces) can also be analyzed. The matrices in Table 1 are defined in terms of the outward normal (n) and tangential vectors (s r ,s 3 ) (see Fig. 2) at the external faces (θ 0 ,θ N ) with 0 and I being respectively the null and identity matrices. Following an original idea by Ting [START_REF] Ting | Stress singularities at the tip of interfaces in polycristals. Damage and Failure of Interfaces[END_REF], the concept of transfer matrix has been further applied to degenerate and extraordinary degenerate materials, so that independently of the number and type of materials in the corner, the characteristic equation from which the characteristic exponents are obtained is derived in a simple manner, always given by the vanishing condition of a 3×3 determinant in the case of open corners (or 6×6 determinant for closed corners). The characteristic equation written as: are defined in Table 1 and

0 K = ) ( ˆ) 2 θ θ θ ϕ ϕ ϕ ϕ T u N N T N N T u N N u T N N u N D K D D K D D K D D K D K + + + + = (3
) (i N
K are the submatrices of N K which is the transfer matrix of the entire corner and is obtained as the product of the transfer matrices of the material wedges E N (N being the number of material wedges in the corner):

1 2 1 • •...• • E E E E K - = N N N . ( 4
)
The structure of a transfer matrix E i is detailed in Ting [START_REF] Ting | Stress singularities at the tip of interfaces in polycristals. Damage and Failure of Interfaces[END_REF] for non-degenerate cases and in Barroso et al. [START_REF] Barroso | Singularity analysis of anisotropic multimaterial corners[END_REF] for all cases of mathematical degeneracy.

The characteristic functions f ijk (θ), g ik (θ) associated to λ k can also be numerically evaluated, always bearing in mind that GSIFs have to be defined together with these characteristic functions for the stress field to be completely and coherently defined, see Yosibash & Szabo [START_REF] Yosibash | A note on numerically computed eigenfunctions and generalized stress intensity factors associated with singular points[END_REF] and also Pageau et al. [START_REF] Pageau | Standardized complex and logarithmic eigensolutions for n-material wedges and junctions[END_REF].

In short, the tool presented in this section permits an analytical knowledge of the characteristic exponents and functions, which will be considered to be known in the next section.

GENERALIZED STRES INTENSITY FACTORS

In the evaluation of the GSIFs, the global geometry and far field loading are now necessary, thus numerical models or experimental tests (e.g. using photoelasticity) have to be used. Additionally, in most cases, a post processing of the data is necessary, the accuracy of the results being finally dependent on the "quality" of both processes. For example, due to the asymptotic character of the elastic representation in (1), GSIFs obtained from the solution far from the corner tip can not be representative of the real elastic solution, while GSIFs obtained extremely near the corner tip can be affected by numerical errors associated to the model discretization.

There exist techniques which overcome these difficulties by means of path-independent integrals evaluated in the neighbourhood of the corners, see for example Qian and Akisanya [START_REF] Qian | Wedge corner stress behaviour of bonded dissimilar materials[END_REF], Wu [START_REF] Wu | Near-tip field and the associated path-independent integrals for anisotropic composite wedge[END_REF], Banks-Sills and Sherer [START_REF] Banks-Sills | A conservative integral for determining stress intensity factors of a bimaterial notch[END_REF] and Cisilino & Ortiz [START_REF] Cisilino | Boundary element analysis of three-dimensional mixed-mode cracks via the interaction integral[END_REF] among others. Further information on these different techniques can be found in Helsing and Jonsson [START_REF] Helsing | On the computation of stress fields on polygonal domains with V-notches[END_REF] and Sinclair [START_REF] Sinclair | Stress singularities in classical elasticity -I: Removal, interpretation and analysis[END_REF].

Unlike the typically high accuracy found in the literature regarding the evaluation of characteristic exponents, the accuracy in the evaluation of GSIFs is substantially worse. More reliable results for benchmark problems are needed which could be used as reference values in new evaluation methods. Some interesting comments regarding the validity of published numerical results in the literature can be found in Helsing and Jonsson [START_REF] Helsing | On the accuracy of benchmark tables and graphical results in the applied mechanics literature[END_REF].

In the present work, with the previous knowledge of accurate values of the characteristic exponents and characteristic functions, a simple procedure has been implemented based on a least squares method in terms of displacements and/or stresses. The numerical models have been performed using a BEM code developed by Graciani [START_REF] Graciani | Formulación e implementación del Método de los Elementos de Contorno para problemas axisimétricos con contacto[END_REF]. BEM models are more suitable (when compared with FEM models) due to their higher accuracy in solving problems involving singular stresses.

Extending the approach developed by Munz and Yang [START_REF] Munz | Stresses near the edge of bonded dissimilar materials described by two stress intensity factors[END_REF] using FEM, the present work uses a least squares method to evaluate the GSIF values. An error function J (5a) is defined as summing squares of differences between the displacements, J u (5b), and/or tractions, J t (5c), obtained from the numerical model (u BEM and T BEM ) and the corresponding values obtained from the asymptotic series expansion in (1) (u series and T series ). The function J depends on the values of a and b, the usual choices being ( )

) 0 , ( , ,..., 0 1 ≥ + = b a bJ aJ K K K J t u q n (5a) [ ] 2 1 0 1 0 1 ) , ,..., , , ( ) , ( = = = - = A B j M n q n j n series j n BEM u K K K r u r u J α α α θ θ (5b) [ ] 2 1 0 1 1 ) ,..., , , ( ) , ( = = = - = A B j M n n j n series j n BEM t K K r T r T J α α α θ θ (5c)
The only unknowns in J are the GSIFs, K k (k=1,...,n) and the rigid body translations K 0q (if necessary). In the J expression three summatories are considered: Α represents the number of displacement components (A=1 for u r , u θ or u z only, A=2 for u r and u θ , u r and u z or u θ and u z and A=3 for u r , u θ and u z ) B defines the number of edges used in the evaluation of the error (with N material wedges, there exist N+1 edges where the BEM model gives direct results, without internal points) and finally M indicates the number of nodes along an edge used in the evaluation of J. An approximation for K k (and K 0q ) is obtained from the solution of the following linear equation system:

0 ) ,..., ( 1 = ∂ ∂ k n K K K J , (k=1,...,n) 0 ) , ,..., ( 0 0 1 = ∂ ∂ q q n K K K K J , (q=x,y,z) (6)
The authors have successfully checked the procedure, defined by ( 5) and ( 6), using displacements and stresses as well. In the next section the benchmark problems present corners with free-free boundary conditions, where only displacements have been used in the evaluation of the GSIFs, and, for comparison purposes, the rest of the problems analyzed have been evaluated using only displacements (a,b)=(1,0).

BENCHMARK PROBLEMS AND APPLICATIONS

To analyze the accuracy of the implemented method, two simple problems have been selected, a 90º notch with tensile loadings and an isotropic free-free bimaterial corner.

Although the method is independent of the complexity of the problem (number and type of materials, boundary conditions, etc.), it seemed reasonable to check it at least in these two problems. Nevertheless, the lack of benchmark problems in the literature to check with is evident. As mentioned before, the values of the GSIFs depend on of the definition of the characteristic functions (Yosibash and Szabo [START_REF] Yosibash | A note on numerically computed eigenfunctions and generalized stress intensity factors associated with singular points[END_REF]), and many authors do not present both of them, the difficulty of such comparisons also being due to the lack of uniformity and information in the normalization/standardization of the presented results as indicated by Pageau et al [START_REF] Pageau | Standardized complex and logarithmic eigensolutions for n-material wedges and junctions[END_REF].

Single-edge 90º notched specimen subjected to tensile loading.

This simple problem (Fig. 3) has been analyzed with a/w=0.5 and h/w=1, with a=5cm.

The BEM model has 709 nodes and 709 linear elements, with a size along the outer edges of 0.1 cm. Along the edges converging at the corner the element adjacent to the tip has a size of 10 -7 cm, growing with a factor of 1.5 until a size of 0.1 cm is reached, with a total of 100 elements along these faces.

Fig 3

This symmetric load configuration, which generates a mode I (opening) in the neighbourhood of the corner, has been analyzed in detail by Helsing and Jonsson [START_REF] Helsing | On the computation of stress fields on polygonal domains with V-notches[END_REF] from where the reference value of K I =4.295886967699 has been taken.

Using u r and u θ (A=2) and both edges (N=2) for the evaluation of the error J in (5), Fig. 4 represents the absolute value of the relative error (up to 5%) between the obtained solution and the reference considering: a) three and b) seven terms in the asymptotic series expansion in [START_REF] Sinclair | Stress singularities in classical elasticity -I: Removal, interpretation and analysis[END_REF]. It is important to notice that horizontal axes do not represent distances to the notch tip but only node numbers, and although element length far from the tip is constant, element lengths near to the notch tip change gradually.

All possible combinations resulting from a consecutive group of nodes have been considered, the right hand side axis being the initial node number and the left hand side axis the final node number (the bottom half part of plots in Fig. 4 then being meaningless). It is clear that major errors are associated to:

• A small group of nodes far away from the notch tip (the peak at the left hand side of Fig. 4) where the more regular terms of the asymptotic expansion and also possibly the external boundary obviously affect the solution. • A small group of nodes very close to the notch tip (the peak at the right hand side of Fig. 4), where numerical errors associated to the discretization affect the solution.

Fig 4

The more terms of the asymptotic expansion are considered in the analysis, the lower the errors associated to the far group of nodes are achieved. The previous consideration is clear when comparing Fig. 4a) and 4b). In both cases, it can be observed that, from an engineering point of view, for most of the "reasonable" group of nodes (avoiding small groups of nodes far away and very close to the notch tip) results are accurate enough (<3%).

For the same configuration, and using a path-independent technique, Ortiz et al. [START_REF] Ortiz | A domain-independent integral for computation of Stress Intensity Factors along three-dimensional crack front and edges by BEM[END_REF], the errors obtained are always below 1%.

Isotropic bimaterial corner.

The tools developed in Sections 2 and 3 have also been used to analyze the bimaterial corner in Fig. 5, (with h=1) from Qian and Akisanya [START_REF] Qian | Wedge corner stress behaviour of bonded dissimilar materials[END_REF]. The BEM model used in the present work has linear elements with a size of 0.025h far away from the corner and 10 -8 h at the corner tip. The characteristic exponents obtained using the semi-analytical tool developed in Barroso et al. [START_REF] Barroso | Singularity analysis of anisotropic multimaterial corners[END_REF] were λ 1 =0.673473, λ 2 =1.167477, λ 3 =1.589147, very similar to those obtained by Qian and Akisanya. The evaluation of the GSIFs was also performed using ( 5) and ( 6) with u r and u θ (A=2), the three edges which converge at the corner (N=3) and an average value from the flattest part of reasonable groups of nodes (the central top part of the corresponding representation, similar to Fig 4b ), avoiding far and close nodes to the corner tip.

Considering the first three terms of the asymptotic representation, the obtained value was a 1 =0.673688, which differs 6.92% from the result by Qian and Akisanya. Ortiz et al. [START_REF] Ortiz | A domain-independent integral for computation of Stress Intensity Factors along three-dimensional crack front and edges by BEM[END_REF] for the same problem and a path-independent technique reported an average value of a 1 =0.67829, which differs 7.65% from Qian and Akisanya and 0.68% from the present results.

The good agreement reported by Qian and Akisanya (using FEM) in the range 10 -4 h and 10 -1 h is improved by the present results, see Fig. 6 (for internal points at r=0.1h) with a BEM model. The range used with the least squares method in the present work is 5.818585•10 -6 h<r<0.6224868h, which leads to a closer agreement between numerical and analytical results.

Fig 6

Once the tool has been demonstrated to be accurate in these particular problems, it will be used to characterize the multimaterial corners which typically appear in adhesively bonded joints between metals and composites.

Note that only stresses (a,b)=(0,1) have also been used in a least squares expression (5), instead of only displacements (a,b)=(1,0), obtaining similar results with differences below 2%. In this particular case, as the corner faces are stress free boundaries, only the common interface (at 0º) could be used to evaluate K k by the least squares using σ θθ .

Adhesively bonded joint between a metal and a composite.

Different test configurations can be specifically designed to analyze the suitability of a singularity based failure initiation criterion by means of previous knowledge of the singularity parameters. For example, as GSIF dimensions depend on the associated characteristic exponents λ k , only GSIFs associated to the same value of λ k can be strictly compared. Accordingly, the following four different joint configurations (Fig. 7, where L o is the overlap length and the numbers between brackets at the caption are the thicknesses in mm) with the same local corner configuration can be analyzed (only the overlap zone is detailed) in which the inner adherent is the metallic sheet and the first outer material of the composite adherent is a 0º layer:

To show the capabilities of the previously introduced tools, the numerical analysis of configuration in Fig. 7a) is presented. The bonded double-lap joint is loaded in shear by means of a tensile force (see Fig. 8, where the units of the axes are the dimensions of the model in mm). The symmetry of the double-lap joint configuration is used to model half of the geometry. Both the undeformed and the deformed geometry (x20) are shown in Fig. 8. Linear elements have been used and a progressive refinement towards the inner corner has been applied up to a final size of 10 -8 mm at the corner tip, a total of 1484 nodes having been used in the BEM model. ºC -1 ). The applied tensile load is 400 N (200 N at each laminate at the right hand side).

Fig 8

Using the semi-analytical tool previously introduced (Barroso et al. [START_REF] Barroso | Singularity analysis of anisotropic multimaterial corners[END_REF]) the characteristic exponents (λ k ) and functions (f(θ) and g(θ)) have been obtained. The GSIFs associated to both mechanical and thermal loads have been evaluated using expressions ( 5) and ( 6) once the BEM model has been solved.

Results for the inner closed corner (corner "b" in Fig. 8 with 90º wedge of [0º] CFRP and 270º wedge of adhesive) and the outer corner at the left hand side of the adhesive spew fillet (corner "a" in Fig. 8 with 180º aluminium wedge and 23º adhesive wedge) are shown in Table 2 (K g associated to the characteristic exponent λ k =1 representing the rigid body rotation).

Table 2

The values of K k are standardized following Pageau et al. [START_REF] Pageau | Standardized complex and logarithmic eigensolutions for n-material wedges and junctions[END_REF] in such a way that σ θθ | θ=0º =K/(2πr) 1-λ . In the particular case of the closed corner, and due to the low thickness of the adhesive layer (≈0.1 mm), the least squares adjustment has been performed between r=10 -6 and r=0.03 mm (measured from the corner tip) to avoid the nearest nodes to the corner tip (between 10 -8 and 10 -6 mm) and those which could be far away from the asymptotic representation (>1/3 of the adhesive thickness).

The results for r=0.0194 mm are plotted in Fig. 9, where circles are the BEM solution (at internal points) and the angle is measured from the horizontal interface between the CFRP laminate and the adhesive. Separate contributions of each term (K 1 , K 2 and K 3 ) are plotted separately, together with results considering two and three terms.

Fig 9

It can be seen in Fig. 9 that three terms (the two singular and the first regular terms) are enough to accurately fit BEM results. It also has to be pointed out that the K k used for this representation have been obtained from the nodes at the interfaces and the comparison performed in Fig. 9 is with internal points not used for the least squares adjustment, which is an additional verification of the correctness and accuracy of the whole procedure.

The representation using the series expansion approach also gives useful information regarding the behaviour of each stress and displacement component of each term included in the representation. For example, in Fig. 10 With the present procedure, all configurations in Fig. 7 can be analyzed in detail, the values for each K k at each corner being comparable as they are associated to the same characteristic exponent. With this singularity characterization, the definition of an experimental test program has a higher added value as particular configurations can be found in which, for example, the value of K 1 (or K 2 ) changes in a controlled manner.

CONCLUSIONS AND FUTURE DEVELOPMENTS

In the present work a procedure for the calculation of Generalized Stress Intensity Factors (GSIFs) has been implemented based on a least squares procedure using the numerical results of BEM models.

Asymptotic stress and displacement representation by means of a series expansion allows the shape angular functions associated to each one of the terms of the series expansion to be determined. This gives a general validity to the representation, as the presence of an analyzed corner in another geometry configuration, or subjected to other load conditions, will only change the values of the associated GSIFs (the characteristic exponents and functions will be the same for the same local corner configuration). Thus, although the final stress or displacement fields will change, they will be obtained by means of the known shape angular functions multiplied by different weights (GSIFs).

The procedure described in this paper, together with a previous work by the same authors, Barroso et al. [START_REF] Barroso | Singularity analysis of anisotropic multimaterial corners[END_REF], has allowed the singularity characterization of several problems to be carried out. First, the method has proved to be robust and very accurate when applied to well known benchmark problems existing in the literature. Second, the procedure has been applied to study anisotropic multimaterial corners which typically appear in adhesively bonded lap joints between metals and composites. In particular the corners appearing in a double lap joint between aluminum and a CFRP laminate have been fully characterized, the local fields of displacements and stresses having been obtained.

The tool developed allows the definition of an oriented experimental test program to analyze the suitability of a GSIF based failure criterion in this kind of structural configuration. Different joints having locally the same corner configurations will allow the analysis of the role of the GSIFs in failure to be performed. In any case, it has to be stressed that GSIFs associated to different characteristic exponents have different dimensions and can therefore not be directly compared.

i N θ 0 θ N θ i θ i-1 x 1 x 2 1 i N θ 0 θ N θ i θ i-1

  Fig 2

  (a,b)=(0,1), (a,b)=(1,0) or (a,b)=(l -2 ,σ -2 ) where l and σ are some characteristic length and stress values.

Fig 5

 5 Fig 5

Fig 7

 7 Fig 7

  we have the angular behaviour of σ θθ stress component around the "90º sector of [0º] CFRP -270º sector of adhesive" corner, where the dashed circle indicates the zero value, with tractions outside the circle and compressions inside it.

Fig 10 With

 10 Fig 10

Fig. 1 -Fig. 3 - 4 -Fig. 5 -

 1345 Fig. 3-Single-edge notched specimen.

Fig. 6 -Fig. 8 -

 68 Fig. 6-Comparison of stresses in the bimaterial problem.

  Fig. 9-σ σ σ σ θθ θθ θθ θθ in corner "b" (90º wedge [0º] CFRP -270º wedge of adhesive).

Table 2

 2

	.-Values of λ λ λ λ and K for the analyzed corners.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support of the Spanish Ministry of Science and Technology (MAT2003-03315).


J P Casas-Rodriguez

I A Ashcroft

V V Silberschmidt

DAMAGE IN ADHESIVELY BONDED CFRP JOINTS: SINUSOIDAL AND IMPACT FATIGUE

The main aim of this paper is to investigate the behaviour of adhesively bonded CFRP joints subjected to cyclic low-velocity impacts and to compare this with fracture in specimens tested in standard fatigue (i.e. non-impacting, constant amplitude, sinusoidal fatigue). It is seen that the accumulated energy associated with damage in impact-fatigue is significantly lower than that associated with similar damage in standard fatigue and that the mechanisms of failure are very different for the two loading regimes. For both types of loading, fracture initiates in the adhesive layer and then propagates into the 0º ply of the composite adjacent to the adhesive layer. However, the fracture surfaces after impact-fatigue are generally less uniform and exhibit more signs of high rate/brittle fracture than seen in the fracture surfaces after standard fatigue testing. Various parameters are proposed to characterise damage in standard and impact-fatigue and it is shown that crack velocity, accumulated absorbed energy and normalised maximum force are all useful parameters for characterising damage evolution.

Introduction

The use of carbon fibre reinforced polymer (CRFP) in aerospace, automotive and other high performance applications has created the necessity to study the behaviour of this material in fatigue. Records in the time-load histories have shown that these kinds of structures support a variety of cyclic loads that vary through the structure. In some cases, repeated low energy impacts appear in the load spectrum. This phenomenon is known as "impact-fatigue" and it has already been shown that this type of loading can be far more damaging than "standard" fatigue [START_REF] Casas-Rodriguez | Damage evolution in adhesive joints subjected to impact fatigue[END_REF] (defined as nonimpacting, constant amplitude fatigue for the purposes of this report).

The main joining methods used in CFRP structures are mechanical fastening and adhesive bonding and the advantages and disadvantages of these techniques have been discussed in [START_REF] Schön | Fatigue of joints in composites structures[END_REF]. The advantages of adhesive joints include low weight and the potential reduction in stress concentrations. However, concerns with long-term performance under sustained complex loading and the effects of environmental ageing remain.

In the last few decades, researchers have identified the importance of studying the behaviour of fibre reinforced polymeric composites (FRPC) at high strain rates in order to observe the damage and mechanisms of failure in impact conditions. In the laboratory, impact has been commonly simulated using three types of tests. These are: tests with pendulum type machines, where tests rates are below 5 m/s; drop-weight tests, with rates up to 10 m/s and the split Hopkinson pressure bar (SHPB) for rates up to 100 m/s [START_REF] Bezemer | New impact specimen for adhesive: optimization of high-speed-loaded adhesive joints[END_REF].

It is well know [4][START_REF] Hosur | Experimental studies on the punch shear characterization of satin weave graphite/epoxy composite at room and elevated temperatures[END_REF][START_REF] Hiley | Effect of strain rate on the fracture process in interlaminar shear specimens of carbon fibre-reinforced laminates[END_REF][START_REF] Lifshitz | Response of fibre-reinforced polymers to high strain-rate loading in interlaminar tension and combined tension/shear[END_REF] that the interlaminar shear strength of carbon fibre/epoxy composites increases with strain rate. Hosur et al. [4] suggested that this behaviour can be attributed to the time-dependent deformation of the material, which is a product of the viscoelastic behaviour of the matrix in the composite. In addition, they found that even in specimens that do not exhibit failure, the stress-strain behaviour is different after the impact. This has been attributed to the heat generated by the high strains affecting the structure of the matrix material. Further work [START_REF] Hosur | Experimental studies on the punch shear characterization of satin weave graphite/epoxy composite at room and elevated temperatures[END_REF] has shown that the maximum stress supported by the material in one punch impact decreases considerably with an increase in the temperature. Hiley et al. [START_REF] Hiley | Effect of strain rate on the fracture process in interlaminar shear specimens of carbon fibre-reinforced laminates[END_REF], however, working with CFRP single lap shear specimens, found no significant differences in the failure surface of samples subjected to quasi-static and impact loads.

Analysis of impact-fatigue in FRPCs has been principally aimed at characterising the reduction in fatigue life as the load is increased. Ray et al. [START_REF] Ray | Impact fatigue behaviour of vinylester resin matrix composites reinforced with alkali treated jute fibres[END_REF], working with a jute/vinyl-ester composite in a cyclic charpy test, saw increases in the endurance of the material as the impact energy decreased. Some researchers [START_REF] Khan | Low velocity impact fatigue studies on glass epoxy composite laminates with varied material and test parameters -effect of incident energy and fibre volume fraction[END_REF][START_REF] Sinmmazçelik | Impact-fatigue behaviour of unidirectional carbon fibre reinforced polyetherimide (PEI) composites[END_REF] have also identified a threshold energy, below which no visible delamination is seen. Yuan et al. [START_REF] Yuan | Low-energy charpy impact of interleaved CF/EP laminates[END_REF] also observed that the response to impact loading depends on the orientation of the fibres in CFRP.

Gomoa et al. [START_REF] Gomaa | Subcritical crack propagation under cyclic stress impulse[END_REF] compared the performance of PMMA in impact and standard fatigue by testing samples in four-point bending using a SHPB machine. They found that the PMMA was more resistant to crack propagation under impact-fatigue than under standard fatigue with the same maximum force and at quasi-similar strain rates.

They also proposed that the critical stress intensity factor is a function of the type of load as well as the load level. Ding et al. [START_REF] Ding | Effect of impact and fatigue loads on the strength of plain weave carbon-epoxy composites[END_REF] investigated the fatigue life of CFRP laminates when combining both sinusoidal loads and an impact. They found that the fatigue strength of the CFRP was affected by the sequence, being more critical in the case when the sinusoidal followed the impact than vice versa.

The behaviour of adhesives under impact has also been studied. Harris and Adams [START_REF] Harris | An assessment of the impact performance of bonded joints for use in high energy absorbing structures[END_REF], working with a single lap joint in a pendulum impact machine, found little difference between joints tested under impact and quasi-static conditions. In contrast, Beevers and Ellis [START_REF] Beevers | Impact behaviour of bonded mild steel lap joints[END_REF] observed higher strengths in impact loading and suggested this was associated with the strain rate dependence of the adherends. Kihara et al. [START_REF] Kihara | A study and evaluation of the shear strength of adhesive layers subjected to impact loads[END_REF] designed a test to study the response of a thick adherend shear joint subjected to various impact stress waves and observed that the type of fracture was associated with the level of the incident stress. Adams and Harris [START_REF] Adams | A critical assessment of the block impact test for measuring the impact strength of adhesive bonds[END_REF] analysed the block-impact test and concluded that the stress condition in the test is uncertain because of the complex dynamic effects generated by uncertainties in the contact interface between the block and hammer and suggested that the impact-wedge test should be used to measure the impact properties of an adhesive. Blackman et al [START_REF] Blackman | The impact wedge peel performance of structural adhesives[END_REF] also used the impact-wedge test and found that results are highly dependent on the environmental conditions. Yokoyama and co-workers [START_REF] Yokoyama | Experimental determination of impact tensile properties of adhesive butt joints with the split Hopkinson bar[END_REF][START_REF] Yokoyama | Evaluation of impact shear strength of adhesive joints with the split Hopkinson bar[END_REF] looked at the impact properties of adhesive joints using SHPB test and found that the tensile strength increased considerably with the loading rate and that this was dependent on the type of adherend used. In addition, they identified an optimum adhesive thickness that was independent of the adherend.

Bezemer et al [START_REF] Bezemer | New impact specimen for adhesive: optimization of high-speed-loaded adhesive joints[END_REF], however, showed that increases in energy absorption at higher strain rates were only seen in some adhesives.

In contrast to the vast body of research into the impact loading of adhesive joints, impact-fatigue has received very little attention to date. In many cases the analysis has been limited to a relatively short series of impacts. Usui and Sakata [START_REF] Usui | Impact-fatigue strength of adhesive joints[END_REF] analysed impact-fatigue in GFRP single lap joints bonded with an epoxy adhesive using a dropweight test. Their results showed that the impact-fatigue strength of the joints was dependent on the magnitude of stress and the loading time. A recent study by Casas-Rodriguez et al. [START_REF] Casas-Rodriguez | Damage evolution in adhesive joints subjected to impact fatigue[END_REF] using single lap joints found significant decreases in the fatigue life when specimens were tested in impact-fatigue, at relatively modest maximum input forces and relatively few cycles, in comparison with standard fatigue. They defined two models to characterise damage in impact-fatigue the modified load-time model and the normalized load-time model.

The current state of research into the impact-fatigue behaviour of adhesive joints using CFRP composites as adherends is characterised by a lack of experimental studies of the many facets of this phenomenon. The main aim of this paper is to investigate the behaviour of bonded CFRP lap-strap joints subjected to cyclic lowvelocity impacts, and to compare this to behaviour in standard fatigue. Various characteristics of the joint response to such loading are presented and some new parameters to characterise the process are introduced. Results from standard fatigue tests are used as a basis for evaluating the danger of neglecting impact-fatigue in the durability analysis of adhesive joints in composites structures.

Experimental setup

Sample preparation

Samples were manufactured by adhesive bonding cured panels of CFRP. The composite used throughout this study was T800/5245C supplied by Cytec Ltd. The matrix, Rigidite 5245C, is a modified bismaleimide/epoxy system and is reinforced with T800 fibres supplied by Toray Industries Ltd. The composite panels were layedup from unidirectional (UD) pre-preg with a volume fraction of 0.6 and thickness of 0.125 mm. A multidirectional (MD) lay-up scheme of [(0/-45/+45/0) 2 ] S was used and the panels were cured for 2 hours at 182 ºC with an initial autoclave pressure of approximately 600 KN/m 2 . The cured panels were ultrasonically scanned for defects.

The material properties for the MD panels are given in Table 1. There were calculated from the UD properties using laminate analysis. The adhesive used was Hysol Dexter's EA9628, which was supplied as a 0.2 mm thick film. This adhesive is based on diglycidyl ether of bisphenol A with a primary amine curing agent. A reactive liquid polymer, based upon carboxyl terminated butadiene acrylonitrile rubber, was used as a toughening agent.

The lap-strap joints (LSJ) were manufactured using the pre-cured CFRP laminate.

The CFRP panels were grit blasted and acetone cleaned prior to bonding. Assembled joints of adhesive and CFRP were cured in an autoclave for 60 min at 120ºC. The fatigue samples were cut from the bonded panels using a diamond saw. End tabs were bonded to the samples to aid grip in the fatigue tests and to provide load alignment.

Holes were drilled in the impact fatigue specimens using 3 different diameters of drill to minimise problems of delamination in the composite.

Impact-Fatigue Tests

Impact-fatigue was carried out using a modified CEAST RESIL impactor, as described in detail in [START_REF] Casas-Rodriguez | Damage evolution in adhesive joints subjected to impact fatigue[END_REF]. The basis of this method is that the specimen is fixed at one end to an instrumented vice and a special impact block is attached to the free end (as shown in

Load direction

Load direction

In the impact-fatigue test, the pendulum is released from a pre-selected initial angle, which in this case corresponded to a potential energy of 1.07 J and impacts the specimen at velocity of 1.50 m/s. These parameters are kept constant during the impact-fatigue test. Changes in the electrical resistance of the piezo-electric sensor are recorded, with a pre-selected sampling frequency of 1000 kHz and up to 3000 data points per cycle. In order to decrease the data noise, a 1 kHz filter was used. The amplified and filtered data was downloaded to a computer as magnitudes of force and time. Typical records of force versus time for two distinct cycles are presented in figure 3. Velocity and energy were calculated from force and time data using equations 1 and 2 respectively. The initial value of velocity was obtained by assuming the conservation of lineal momentum between the hammer and impact block (Fig. 2).

1

.

i i o o v v F dt m = - (1)
. .

i i o E F v dt = (2)

Standard Fatigue

A servo-hydraulic fatigue testing machine, utilising digital control and data logging, was used in the standard fatigue tests. Testing was in displacement control using a sinusoidal wave, R-ratio (minimum-to-maximum displacement) of 0.1 and frequency of 5 Hz. All testing was in ambient laboratory environmental conditions.

Temperature and relative humidity were monitored during the tests and varied between 18-25ºC and 50-60%, respectively. The maximum displacement was selected to generate an initial force that was 56% of the quasi-static failure load.

Fractography

After testing, both edges of the sample and the fracture surfaces were examined with an optical microscope. In order to retain good resolution over large areas, six pictures were taken over the fracture surface and then merged using Photostitch V 3.1 software. Scanning electron microscopy was used for higher magnification examination of selected fracture surfaces. Specimens were extracted using a diamond saw and gold coated prior to examination.

Results

Standard fatigue

Analysis of the failure surfaces of specimens loaded in standard fatigue identified three different fracture regions (as shown in Fig. 4). The first region (region I in Fig. 4) corresponds to failure in the adhesive layer. A second region (region II in Fig. 4) is a transition region, in which a mix of failure in the adhesive and in the 0º ply of the CFRP adjacent to the adhesive is seen. In region III, failure is dominated by fracture in the CFRP ply adjacent to the adhesive. The observed fracture paths are in agreement with previous studies [START_REF] Ashcroft | A simple model to predict crack growth in bonded joints and laminates under variable-amplitude fatigue[END_REF].

Fig. 4 Failure surface of sample tested in standard fatigue

It should be noted that the lap-strap joint test results in a mixed mode fracture.

Johnson [START_REF] Johnson | Stress analysis of the cracked lap shear specimen: An ASTM Round-Robin[END_REF] described a typical mode ratio (G I /G II ) between 0.2 and 0.3 for the LSJ.

The mechanism of failure initiation of this type of joint tested in fatigue has been studied previously [START_REF] Ashcroft | The effect of environmental on the fatigue of bonded composite joints Part1: testing and fractography[END_REF], using both experimental work and finite element analysis.

I II III

Crack propagation Fillet

The results indicate that the crack initiates in the adhesive at a point near to the embedded corner of the lap adherend (as indicated in Fig. 5). This coincides with the position of a theoretical stress singularity. The 3-dimensional evolution of damage in bonded lap joints is further elucidated in [START_REF] Hua | Modelling environmental in EA9321-bonded joints using a progressive damage failure model[END_REF]. Here it is shown that cracks will propagate from the singularity in two directions, across the fillet to the free surface and through the adhesive layer towards the adherend/interface and then along the bondline. Furthermore, they found that initiation and propagation is a 3-D mechanism with the crack initiated away from the sample edges and progressing across the sample width until a full through-width crack has developed. This through-width crack is often the first sign of cracking seen in optical examination of damage in the joint and is also the type of crack modelled in 2D analyses. Analysis of principal stresses in the adhesive fillet region suggests that after crack initiation, propagation through the fillet is at an angle of approximately 45º to the bondline [START_REF] Johnson | Stress analysis of the cracked lap shear specimen: An ASTM Round-Robin[END_REF]. However, an alternative crack path in which the cracks propagates up the fillet/lap interface for a short distance before propagating through the fillet, is commonly seen in practice, as shown in Fig. 5. the presence of flakes that are partly orientated with respect to the direction of the crack growth. Voids with diameter between 1 to 5 m are distributed over the adhesive fracture surface. Previous studies have related these voids with failure in rubber-toughened epoxy adhesives [START_REF] Ashcroft | The effect of environmental on the fatigue of bonded composite joints Part1: testing and fractography[END_REF] and suggest that these are generated by cavitation of the rubber particles in the strain field ahead of a crack. [START_REF] Little | The durability of structural adhesive joints[END_REF]. Fig. 6(b) shows the transition region (Region II), in which failure both in the adhesive and in the CFRP can be seen. Figs. 6(c) and (d) show region III fracture surfaces, in which failure is predominantly in the 0º ply of the CFRP adjacent to the adhesive. It can be seen that these is a mix of failure in the matrix and fibre debonding. Shear cusps can be seen in the areas of matrix failure, which are identified with failure mode II fracture [START_REF] Hiley | Effect of strain rate on the fracture process in interlaminar shear specimens of carbon fibre-reinforced laminates[END_REF]. Some fibre breakage is also seen in the fracture surface; however, the main crack front does not break through the fibres and hence remains in the surface of the ply adjacent to the adhesive. Fig. 6 (d parameter is undefined when the crack area is zero, but can be used when the crack is established. It can be seen in Fig. 7 that * T / cr E A increases in the early period of the crack propagation that corresponds to region I in Fig. 3. However, at approximately 20 mm crack growth, there is a sharp discontinuity in the plot. This coincides with the transition from crack growth in the adhesive layer to its growth in the composite. It can also be seen that * T / cr E A is lower for crack growth in the composite than in the adhesive.

Impact-fatigue

On initial optical examination, apparently similar patterns of failure to those described for standard fatigue are also seen in specimens subjected to impact-fatigue (Fig. 9). Again, three regions can be defined. The first region (region A) is defined when crack growth is in the adhesive, a second region (region B) is a transition zone and the third region (region C) is defined when the crack growth is the 0º ply of the composite adjacent to the adhesive layer. It can be seen that the region A fracture surface from impact fatigue differ significantly from the region I fracture surface seen after standard fatigue. The fracture surface is less uniform and there are signs of multiple damage initiation and termination sites. In some areas of the fracture surface the impression of fibres can be seen indicating that some of the damage is occurring close to or in the composite. The voids and flakes seen in region I of the standard fatigue fracture surface are also less evident. The lack of voids, which have been attributed to cavitation of the rubber toughening particles, is in agreement with the work of Takeshi [START_REF] Takeshi | The fatigue behaviour of toughened epoxy polymers[END_REF]. He observed that the generation of voids in the adhesive was dependent on the crack speed, relating fast crack growth with low void generation. This phenomenon was related to the relationship between high crack growth and low plastic deformation. Micrographs from region C of the impact-fatigue fracture surface can be seen in Figs. 10(c) and (d). It can be seen that fracture of fibres is far more common than in the standard fatigue fracture surface. Also, as with the adhesive A B C 10 mm Crack Propagation failure in impact-fatigue, the fracture surface is less uniform than that for standard fatigue and shows signs of multiple damage events. Fracture in the composite matrix can be more clearly observed in Figs. 10(e) and (f). As in standard fatigue, shear cusps can be seen; however, these are more randomly distributed and no longer appear to be related to the crack propagation rate. Also, the matrix fracture exhibits more brittle behaviour, as seen in Fig. 10(f). Overall, it can be said then that fracture surfaces after impact-fatigue are less uniform than those after standard fatigue, with multiple crack fronts, varying crack depth and non-uniform crack front development.

They also exhibit more features associated with brittle and mode I fracture. As with standard fatigue, various parameters can be calculated from the data collected during the impact-fatigue tests in order to characterise the failure process. One parameter that can be used to characterise impact-fatigue is the maximum force reached in the n th cycles, max n F . This can be normalised with respect to the maximum force measured in the 10 th cycle in order to obtain a criterion that can be used to detect crack initiation. The 10 th cycle is selected rather the 1 st cycle as the normalising parameter in order to avoid possible problems with misalignment in the initial stages of the fatigue life. In Fig. 11 it can be seen that stable behaviour was detected for the first 1000 cycles, after which a dramatic change was detected. This change is identified as a phase of crack initiation in the joint. After the 3000 th cycle the normalized maximum force becomes practically constant, albeit with a large degree of scatter that is consistent with the non-uniform fracture surfaces. This part of the plot corresponds to region C of the fracture surface, as identified in Fig. 9. An alternative parameter that can be used to identify changes in the crack growth mechanism is the accumulated energy over the n th cycle, * n E , as already defined in Eqn. 2 for impact and Eqn. 3 for standard conditions. In the case of impact-fatigue this can be normalized with respect to the initial potential energy of the hammer, 0 E . This is plotted as a function of cycles in Fig. 12. It can be seen that below 1000 cycles there is quasi-constant behaviour. However, Fig. 12 Changes in absorbed energy during crack growth in impact-fatigue

Comparison of impact-fatigue and standard fatigue

Differences in crack area evolution in standard and impact fatigue are compared in Fig. 13, using the number of cycles to failure as the index of comparison. The results

show that impact-fatigue is characterized by two principal tendencies; fast crack growth in the early fatigue life, i.e. less than 10.000 cycles, followed by slow crack propagation. Unfortunately, precise details of crack growth in the fast fracture stage have not been captured to date. Standard fatigue shows an initiation period followed by an acceleration phase. After this there is quasi-constant propagation until a second acceleration phase is reached. Initial measurements indicate crack growth in the early stages of impact-fatigue in excess of

Conclusions

The behaviour of adhesively bonded composite lap joints subjected to repetitive low velocity impacts (impact-fatigue) and standard constant amplitude fatigue has been analysed and compared in this paper. Macroscopic analysis of the crack growth has shown that for both types of loading, failure initiates in the adhesive layer and then propagates into the 0º ply of the CFRP adjacent to the adhesive layer. However, higher-magnification examination of the fracture surfaces shows that the failure mechanisms involved in impact-fatigue are very different to those in standard fatigue.

In the adhesive failure region there is extensive voiding and the formation of regular flakes oriented with respect to the direction of crack growth after standard fatigue.

However, in the corresponding region after impact-fatigue the flakes are less regular, the voids are not present, the crack depth varies more and there are signs of multiple damage initiation and arrest events. In standard fatigue, fracture in the composite failure region is dominated by regular shear cusps and matrix rollers in the matrix failure areas and fibre debonding. After impact-fatigue the composite failure region is less uniform, with higher incidence of fibre breakage and a more brittle appearance to the matrix failure areas.

It has been shown that the maximum force reached in the n th cycle ( ) max n F and the accumulated energy over the n th cycle ( )

* n E can be used as criteria for damage characterisation. The scatter seen in these parameters for crack growth in impactfatigue can be related to the non-uniform appearance of the fracture surfaces.

It has been shown that crack propagation rates tend to be higher in impact-fatigue and that damage occurs at significantly lower energy levels than that required for similar damage in standard fatigue. This reinforces the view proposed previously [START_REF] Casas-Rodriguez | Damage evolution in adhesive joints subjected to impact fatigue[END_REF] that impact-fatigue is a potentially dangerous form of loading, both for adhesives and polymer composite materials

Fig. 1

 1 Fig. 1 Dimensions of lap strap joint specimens: (a) impact-fatigue specimen, (b) standard fatigue specimen.

  Fig 2).

Fig. 2

 2 Fig. 2 Sample set-up for impact-fatigue. (a) Plan view, (b) side view.

Fig. 3

 3 Fig. 3 Evolution of force in the 10 th and 40 000 th impact.

Fig. 5

 5 Fig. 5 Crack initiation and propagation in standard fatigue

Fig. 6

 6 Fig. 6 Scanning electron micrographs of fracture surfaces from samples tested in standard fatigue. (a) Region I, (b) region II, (c) and (d) region III.

Fig. 7 .Fig. 8

 78 Fig. 7. Changes in the accumulated absorbed energy per unit area with crack length in standard fatigue.

Fig. 9

 9 Fig. 9 Failure surface of a sample tested in impact-fatigue

Fig. 10

 10 Fig. 10 Scanning electron micrographs of fracture surfaces from samples tested in impact-fatigue: (a) and (b) region A, (c) -(f) region C.

Fig. 11

 11 Fig. 11 Changes in the normalised maximum force during crack growth in impact-fatigue

3 2 . 3 1

 23 4 10 - × mm/cycle. In contrast, the maximum crack speed measured for the initial 20 mm of crack growth in standard fatigue was approximately .4 10 - × mm/cycle.

Fig. 13 Fig 14

 1314 Fig. 13 Crack propagation in impact-fatigue and standard fatigue