Sebastian Heimbs
email: sebastian.heimbs@eads.net

Sebastian Schmeer

Middendorf

Martin Maier

Strain rate effects in phenolic composites and phenolic-impregnated honeycomb structures

Keywords: A. Structural materials, B. Impact behaviour, B. Mechanical properties, B. Stress/strain curves, Honeycomb

The influenc e of the loading rate on the material behaviour of glass fibre reinforced phenolic composites and phenolic resin-impregnated aramid paper (Nomex ® ) honeycomb structures was investigated experimentally. The composite specimens were made of woven fabric plies and loaded in tension and shear. Two types of Nomex ® honeycomb specimens (hexagonal and over-expanded) were loaded in uniaxial compression in all three material directions. Quasi-static test results were compared to dynamic test data obtained on a drop tower, where different strain rates from 10 s -1 to 300 s -1 were tested.

The glass/phenolic composite material showed a remarkable strain rate effect at higher loading rates with over 80% increase in tensile strength. Also for the Nomex ® honeycomb an increase of the stress level of up to 30% was observed. These material characteristics should be taken into account in case of dynamic analysis (e.g. crash, impact) of honeycomb sandwich panels for public transport applications, which are usually made from these phenolic materials.

Introduction

Fibre reinforced composites are characterised by high weight-specific stiffness and strength properties and are therefore preferred lightweight materials for the maritime, railway and aeronautical transportation industries . An even higher stiffness-to-weight ratio may be achieved by employing the sandwich construction principle: Separating two thin composite facesheets by a cellular core results in considerably higher bending stiffness compared to a monolithic structure of the same weight. Such lightweight sandwich structures are generally used for interior panels in ground and air transportation applications, where the requests for weight saving are as important as fire safety properties for occupant safety. For this reason phenolic resin-based glass fibre reinforced plastics (GFRP) or carbon fibre reinforced plastics (CFRP) are typically used as facesheet material. The core structure commonly consists of a phenolic resinimpregnated aramid paper honeycomb structure which is commercially known as Nomex ® honeycomb. Phenolic resins are polycondensation products of phenol and formaldehyde and, in contrast to other resin types, characterised by outstanding fire saf ety properties like low smoke emission, low toxicity and low heat release [START_REF] Gardziella | Phenolic resins[END_REF][START_REF] Hunter | Phenolic GRP and its recent applications[END_REF][START_REF] Nyden | Flammability properties of honeycomb composites and phenolformaldehyde resins. Fire and Polymers II: Materials and Tests for Hazard Prevention[END_REF][START_REF] Willson | An update on phenolic resins for use in mass transport, marine, off shore and construction applications[END_REF].

Although phenolic resins were among the first polymers developed and have been commercially available for about a hundred years, nowadays they have bec ome an almost exotic material, which is primarily due to delicate processing, the emission of volatile products during hardening and comparably low mechanical properties like interlaminar shear strength or sandwich peel strength. The majority of resins used for composite materials are epoxy, polyester and vinylester resins. Accordingly , m ost scientific papers cover the latter types of resins. Only few publications contain information about mechanics or material characteristics of phenolic composites [START_REF] Chandra | Glass/phenolic composites: properties and fracture features[END_REF][START_REF] Redjel | Mechanical properties and fracture toughness of phenolic resin[END_REF][START_REF] Charalambides | Fracture toughness characterization of phenolic resin and its composite[END_REF][START_REF] St John | Flexural and interlaminar shear properties of glass-reinforced phenolic composites[END_REF][START_REF] Morii | Influence of sizing and resin on the mechanical properties of unidirectional glass/phenolic composite[END_REF][START_REF] Morii | Interfacial effect on the mechanical properties of glass/phenolic composite[END_REF][START_REF] Tavakoli | Compatibility of commercial chopped-strand mats with a phenolic resin system for hand laminating[END_REF][START_REF] Barré | Comparative study of strain rate effects on mechanical properties of glass fibre-reinforced thermoset matrix composites[END_REF].

When designing sandwich panels for short-time dynamic loads, which can occur i n transportation applications by high decelerations or impact, it is necessary to have information about the influence of loading rate on the material properties of both facesheet and core. It is well known that in case of high loading rates an increase in material stiffness and strength compared to the static behaviour may occur, which is referred to as the strain rate effect. When this effect is neglected, dynamic finite element (FE) simulation results based on static material data often do not agree with experimental results, or constructions are too conservative in design, inhibiting potential weight savings. So far the strain rate effect of honeycomb structures has been primarily investigated for aluminium honeycomb. While in [START_REF] Maji | Mechanical properties of aluminium honeycomb impact limiters[END_REF] no effect was observed during dynamic flatwise compression testing (Charpy im pact test, strain rate 100 s -1 ), a 10% increase of crush strength compared to static val ues was reported in [START_REF] Koploy | GA-4/GA-9 Honeycomb impact limiter tests and analytical model[END_REF] (drop weight test, 200 s -1 ), a 15% increase in [START_REF] Mcfarland | The development of metal honeycomb energy absorbing elements[END_REF] (drop weight test), a 20% increase in [START_REF] Bandak | Honeycomb, a lightweight energy absorbing material[END_REF][START_REF] Shkolnikov | Honeycomb modeling for side impact moving deformable barrier[END_REF][START_REF] Kirk | Mechanical energy absorbers and aluminium honeycomb[END_REF] (drop weight test, 40 s -1 ), a 33% increase in [START_REF] Wu | Axial crush of metallic honeycombs[END_REF] (gas gun test, 1000 s -1 ), a 40% increase in [START_REF] Zhao | Crushing behaviour of aluminium honeycombs under impact loading[END_REF] (split Hopkinson pressure bar, 800 s -1 ) and a 50% increase in [START_REF] Baker | Static and dynamic properties of high-density metal honeycombs[END_REF][START_REF] Goldsmith | Energy absorption by sandwich plates: a topic in crashworthiness[END_REF][START_REF] Goldsmith | An experimental study of energy absorption in impact on sandwich plates[END_REF] (gas gun test, 2000 s -1 ).

The strain rate effect of Nomex ® h oneycomb was investigated in [START_REF] Goldsmith | Energy absorption by sandwich plates: a topic in crashworthiness[END_REF][START_REF] Goldsmith | An experimental study of energy absorption in impact on sandwich plates[END_REF]. Here flatwise compression tests were conducted under high loading rates (gas gun test) and a 10% increase in crush strength compared to static tests was reported. Also in [START_REF] Aitken | Damage due to soft body impact on composite sandwich aircraft panels[END_REF] a slight increase was observed, although dynamic testing was conducted only in the medium strain rate domain of 5 s -1 .

For material modelling of honeycomb structures, as for crash or impact simulations using commercial dynamic FE-codes, the nonlinear compressive stress-strain-relationship up to full compaction has to be defined not only in thickness direction but generally in all three material directions [START_REF] Heimbs | Honeycomb sandwich material modeling for dynamic simulations of aircraft interior components[END_REF]. In [START_REF] Gibson | Cellular solids: structure and properties[END_REF] the in-plane deformation behaviour of aluminium honeycomb is investigated analytically and phenomenologically. In-plane compression tests with regard to strain rate effects were conducted on aluminium honeycomb specimens in [START_REF] Gotoh | Crush behavior of honeycomb structure impacted by drop-hammer and its numerical analysis[END_REF] (drop weight test, strain rate 66 s -1 , 30% increase), [START_REF] Stronge | Dynamic crushing of elastoplastic cellular solids[END_REF] (strain rate 100 s -1 ) and [START_REF] Hönig | In-plane dynamic crushing of honeycomb[END_REF] (drop weig ht test, strain rate 500 s -1 ). Dynamic in-plane compression testing on polycarbonate honeycomb is reported in [START_REF] Papka | In-plane uniaxial and biaxial crushing of a polycarbonate honeycomb[END_REF], where an increase in crush strength was also observed. Dynamic in-plane compression testing of Nomex ® honeycomb has to the authors' best knowledge not been reported before.

Usually arbitrary assumptions are made concerning the in-plane structural behaviour due to nonexisting material data [START_REF] Bitzer | Useful analysis methods for sandwich structures[END_REF].

The aim of this paper is to fill the gap in lacking knowledge of the strain rate effect on the material properties of phenolic GFRP composites and phenolic resin -impregnated aramid paper honeycomb in all material directions . Therefore, results of dynamic tests on a drop tower facility are compared to static reference test data. The tensile and shear behaviour of GFRP composites as well as the in-plane and out-of-plane compressive behaviour of two types of Nomex ® honeycomb are investigated.

Specimen s

GFRP specimens

The GFRP specimens were made of satin-weave fabric E-glass fibre reinforced phenolic resin prepregs of the type Stesalit PHG 600-68-50 with a cured ply thickness of 0.33 mm. Two different GFRP plates were manufactured: One plate consisted of 6 equally oriented plies for the determination of the tensile behaviour in warp and weft direction, respectively, and one plate was made from 8 plies in a symmetric ±45°-configuration for the determination of the shear properties. Both plates were manufactured on a hot press at a temperature of 135 °C, a pressure of 2.0 bar and a curing cycle time of 90 min. Specimens were machined on a precision saw and equipped with GFRP doubler tabs on both sides. For each test series at least 5 specimens were tested. Tensile behavio ur was determined for both warp and weft direction of the fabric prepregs.

Fig. 1 shows a microscopic view of a cross-section of a 6-ply specimen. A large void content is clearly visible, which is the result of the polycondensation reaction of the phenolic resin. Water is the concurrent loss causing internal pressure and voids in the phenolic matrix and the surface of the composite. This is characteristic for phenolic resins and does not occur in epoxy resins, which are a product of polyaddition reactions .

A fibre volume fraction of 48% was determined by the SEM-photomicrographic technique and resin burn-off method according to ASTM D2584.

Honeycomb specimens

Two different types of aerospace grade low density Nomex ® honeycomb were investigated in this study: a hexagonal and an over-expanded structure from the manufacturer Schütz

Cormaster. These honeycombs are manufactured in an expansion process, in which sheets of Nomex ® paper are equipped with adhesive strips and stacked with a shift. This block of Nomex ® paper sheets is then expanded to a hexagonal cell geometry and finally dipped in phenolic resin several times until the specific density is achieved. The over-expanded cell configuration is obtained by expanding the hexagonal cell until it forms a rectangle. It is bendable in the L-direction and used as sandwich core material for curved surfaces. Cell sizes and densities of both types of honeycomb are listed in Table 1.

All specimens were cut out of large honeycomb plates with a thickness o f 14.6 mm. The specimens for in-plane compression tests had a size of 200 mm x 50 mm (Fig. 2), for out-ofplane compression tests 50 mm x 50 mm. The latter tests were performed as 'bare compression' and 'stabilis ed compression' with GFRP facesheets glued to both sides of the specimens. The honeycomb in-plane material directions are illustrated in Fig. 2 and referred to as the Ldirection (ribbon direction), W-direction (direction perpendicular to the ribbon), and the out-ofplane direction as the T -direction (thickness direction). In this illustration it can also be seen that the cell walls oriented in the L-direction are twice as thick (=double cell walls) as the other cell walls (=single cell walls), which is a result of gluing the paper sheets in the manufacturing process.

Experimental setup

Static testing

Static testing of GFRP and honeycomb specimens was conducted on an Instron universal testing facility with a cross-head speed of 1 mm/min resulting in strain rates of 10 -4 -10 -3 s -1 .

Tensile tests on GFRP specimens were conducted according to DIN EN ISO 527-4. Two perpendicular strain gauges were used to measure the longitudinal and transverse deformation of the specimen. Shear behaviour was determined according to DIN EN 6031 using ±45°-tens ile specimens equipped with strain gauges parallel and transverse to the loading direction.

For the honeycomb compression tests according to DIN 53291 the cross-head displacement and force data were recorded in order to obtain force-displacement-curves and convert them into engineering stress-strain-curves . In addition to static testing the maximum cross-head speed of 300 mm/min of the Instron testing apparatus was used in order to conduct compression tests in the low strain rate domain of 0.1-0.5 s -1 . Compression testing was conducted in L-, W-and Tdirection.

Dynamic testing

A drop weight tower was used for dynamic testing at higher loading rates (10-300 s -1 ) for both GFRP and honeycomb specimens . While for the honeycomb compression tests the specimens were placed on an aluminium block connected to the load cell, a special apparatus was used for the GFRP tensile tests, in which the specimens were mounted vertically. The falling mass im pacted the mounting device on the lower end of the specimen inducing a tensile load, while the upper end of the specimen was connected to the load cell (Fig. 3).

A laser distance measurement system was used for displacement recording. For the GFRP specimens the use of strain gauges was impracticable at these loading rates , so an optical measurement of the longitudinal and transverse material deformation was incorporated: The specimens were marked with a high contrast colour grid, which was recorded with a VDS Vosskühler HCC-1000F high speed camera with a frame rate of 6832 fps during the dynamic test. The material deformation could be obtained by analysing these high speed films and measuring the relative displacement of specific pixels.

For both GFRP and honeycomb specimens two different strain rates were tested on the drop tower facility. The lower one was limited by the lowest possible drop height (in combination with the maximum possible mass in order to provide enough kinetic energy to destroy the specimen), the higher one was limited by the highest possible drop height (in combination with the minimum possible mass). The GFRP specimens were dynamically tested at strain rates of 10 s -1 and 50 s -1 , the honeycomb specimens at strain rates of 10 s -1 and 50 s -1 (in-plane) as well as 150 s -1 and 300 s -1 (out-of-plane).

SAE 300 and SAE 500 filters were used in order to filter out superposed high frequency oscillations. This was conducted to increase comparability of dynamic and static test data, although structure's oscillations are always associated with dynamic loads and particularly with impacts and they cannot be denoted as noise [START_REF] Shkolnikov | Strain rates in crashworthiness[END_REF].

Results and discussion

GFRP specimens

Static tensile testing of GFRP specimens was conducted in warp and weft direction to analyse the difference between the material directions of the woven fabric prepregs. The resulting stressstrain-curves of 5 specimens in each case (Fig. 4) are highly reproducible and show 8% higher stiffness (Table 2) and strength values (Table 3) for the warp direction. This is primarily due to the higher yarn density of 22 yarns per cm in warp direction compared to 21 yarns per cm in weft direction. In this case the Young's modulus was calculated as the slope of the strain interval from 0.05 to 0.25% according to the test standard.

In practice, for most constructions the fabric orientation is not specified and based on manufacturing criteria, design calculations are often carried out conservatively using the lower weft direction values. For this reason dynamic testing was primarily conducted in weft direction. The tensile stress-strain relationship for three different strain rates in weft direction is illustrated in Fig. 5. A remarkable increase of tensile strength is visible for higher loading rates.

For a strain rate of 50 s -1 the averaged increase is 88% compared to the static value (Table 4).

The relationship between strain rate and strength increase is illustrated in a strain rate diagram on a lo garithmic scale (Fig. 6). Even if no data for the medium strain rate domain is available, the typical curve with a distinctive plateau without strength increase for lower strain rates and a steep slope for higher strain rates can be plotted. The higher values for the warp direction compared to the weft direction are also visible in this diagram. The domain, in which a distinctive strain rate effect occurs, begins with strain rates exceeding approximately 1 s -1 . The failure strain increases with higher loading rates as well (Table 5). This effect is typical for glass fibre reinforced plastics, while the failure strain remains nearly constant for carbon fibre reinforced plastics and decreases for aramid fibre reinforced plastics [START_REF] Welsh | Effect of strain rate on the tensile failure of woven reinforced polyester resin composites[END_REF][START_REF] Yuanming | Constitutive equation for unidirectional composites under tensile impact[END_REF][START_REF] Behler | Effect of high loading rates on the mechanical properties and damage evolution of fibre reinforced plastics[END_REF]. The Young's modulus also seems to increase marginally according to Fig. 5. However, because even after data filtering some residual oscillations remained in the high strain rate curves, which made the calculation of a comparable slope impracticable, this effect was not quantified.

In case of shear behaviour, higher loading rates also lead to a remarkable increase of the stresscurves (Fig. 7). However, there is only a small difference between the two higher strain rates 10 s -1 and 50 s -1 . The increase of shear strength compared to the static case is 28% and 33% respectively (Table 6). The shear modulus also increases marginally, but this phenomenon was not quantified as in the case of tensile stiffness due to oscillations in the initial slope of the stress-curves. The maximum shear strain before rupture of the specimens is nearly constant for all strain rates and does not seem to be affected by loading rate.

A generally accepted explanation for the strain rate effect of composite materials is still not available and a complex issue due to the interaction of reinforcement fibres and matrix phase.

Different theories can be found in the according literature. One theory explaining the strength increase at higher loading rates is a change in failure mode with interface failure of fibre and matrix as an increasing source of damage for higher strain rates [START_REF] Barré | Comparative study of strain rate effects on mechanical properties of glass fibre-reinforced thermoset matrix composites[END_REF][START_REF] Sierakowski | Strain rate effects in composites[END_REF]. A second theory found in [START_REF] Barré | Comparative study of strain rate effects on mechanical properties of glass fibre-reinforced thermoset matrix composites[END_REF] is based on the influence of the elastic interaction between fibre and matrix, which means that woven fabric reinforced composites show a higher strain rate effect than unidirectional reinforced ones due to a larger interaction zone. Furthermore, in [START_REF] Barré | Comparative study of strain rate effects on mechanical properties of glass fibre-reinforced thermoset matrix composites[END_REF] it is stated that the strain rate effect of the composite material is caused by the viscoelasticity of the resin material, since the glass fibre sensitivity to strain rate is considered to be small or none. This is contradictory to [START_REF] Xia | A statistical model and experimental study of the strain-rate dependence of the strength of fibres[END_REF], where static and dynamic tests in the strain rate domain of 10 -4 to 1100 s -1

were conducted on pure E-glass fibre bundles and a rate dependency with an increase of strength and failure strain of almost 300% for the highes t loading rates was reported. Also in [START_REF] Rotem | Longitudinal strength of unidirectional fibrous composite under high rate of loading[END_REF] a considerable strain rate effect was obtained for E-glass fibre reinforced epoxy and pure Eglass fibre bundles, while pure epoxy specimens showed no strength increase for strain rates between 10 -6 and 30 s -1 . Here the rate sensitivity of the composite material is primarily ascribed to the strength increase of the fibre material. This is also presumed to be the explanation for the strain rate effect in the current investigation, which comprises a reinforcement of E-glass fibres as well. This theory is supported by the fact that the strain rate effect for shear loads is lower than for tensile loads, since the shear behaviour of composites is less fibre-dominated than the tensile behaviour (Table 5, Table 6).

Honeycomb specimens 4.2.1. Hexagonal honeycomb

Out-of-plane compression testing of hexagonal honeycomb specimens was conducted with and without facesheets (stabilised/bare compression). In case of bare compression the free ends of the honeycomb specimen have translational and rotational degrees of freedom and crushing is initiated at one of the free ends. By contrast, the stabilised compression test with facesheets represents the sandwich nature. In this case the cell wall edges are clamped and cell wall buckling and crushing occurs somewhere in the middle of the specimen. These results are much more useful for the designer [START_REF] Grimes | The adhesive-honeycomb relationship[END_REF].

Fig. 8 shows the sequence of cell wall deformation during flatwise compression loading. While Fig. 8a shows the initial configuration, in Fig. 8b a number of cell wall buckling waves are visible. Collapse of the cells occurs in the middle of the specimen due to loss of stability (Fig. 8c), where the folding process of the cell walls continues during displacement-controlled compression.

Fig. 9 and 10 show the respective stress-strain-diagrams. All of these and the following curves represent the mean data of 5 tests with very similar curves of the respective specimens. In both cases the curves consist of three areas : the elastic regime up to the compressive strength, the crushing regime at nearly constant plateau stress -which is refer red to as crush strength -and finally the densification regime, where the cellular structure is fully compacted resulting in a steep stress increase. In case of bare compression this densification point occurs at higher compressive strains (80%) compared to stabilised compression (70%), since the cell walls are not stabilised by adhesive fillets and can be folded more densely. Loading rate has no influence on the densification point. The compressive strength is almost equal for the bare and stabilis ed cases. It can be seen that dynamic loading leads to a significant increase of crush strength. In addition, the initial stiffness and compressive strength increase marginally, although the peak compressive strength is affected by the filter of the force data. This turned out to be a disadvantage of drop weight impact testing, because stress peaks cannot precisely be analysed due to data filtering as a result of oscillations. Therefore, in this investigation the increase of crush strength, i.e. the plateau stress, is primarily analysed. For the bare compression test t his is about 25% at a strain rate of 150 s -1 compared to the static curve. In case of stabilis ed compression, the increase is only about 10%. The explanation of this effect may be seen in the inertial response of the honeycomb structure, which will be addressed later in more detail.

The Young's modulus and plateau stress for in-plane compression behaviour are remarkably smaller than in case of flatwise compression. Expressed in terms of factors, they are 140 and 36

times smaller for the L-direction as well as 250 and 40 times smaller for the W-direction. In literature, the factor 1000 is sometimes assumed, which does not seem to be suitable in this case [START_REF] Anderson | Lightweight composite structures[END_REF].

The in -plane compression curves exhibit the same three areas as the flatwise compression curves (Fig. 11,12): an initial linear elastic regime, a nearly constant plateau stress, and the steep slope after compaction. However, a stress peak as in the case of flatwise compression does not occur. The shape of the curve in L-direction is rather the result of elastic cell wall bending (Fig. 13a). In the beginning the double cell walls initially oriented parallel to the loading direction remain parallel to the loading direction while the single cell walls deform elastically (linear elastic regime). This linear elastic region ends with the double cell walls folding laterally while the single cell walls are bent in an S-shape. This folding phenomenon continues (plateau stress) until the cell walls come into contact and the cellular structure is fully compacted (steep slope). This cell wall deformation scheme occurs homogeneous ly throughout the whole honeycomb specimen.

In contrast to this, the deformation of cell walls in W-direction is inhomogeneous (Fig. 13b).

After an initial elastic deformation of the cell walls (linear elastic regime) a local collapse of one cell wall at an arbitrary location occurs. The neighbouring cells are affected and weakened so that the whole cell row collapses while the remaining cells above and beneath keep their hexagonal shape. In the following the adjacent cell rows collapse under a constant load (plateau stress). Once all rows have collapsed the material is densified (steep slope).

For both L-and W-direction the stress-strain curves are influenced by loading rate with a higher increase for the L-direction. Only the plateau stress level is affected while initial slope and densification point are not influenced by loading rate. The explanation for the rate dependency lies rather in the structural design of the honeycomb than in material effects. In case of the Ldirection the plateau stress increases about 33% at a strain rate of 50 s -1 compared to the static curve. The local deformation of the cell walls is composed of a high level of bending, shifting and rotation giving inertial effects a large influence. In W-direction the increase is lower with about 10-20% because of the complete failure of single rows occurring successively one row after another and not accumulated.

Over-expanded honeycomb

Flatwise compression testing of over-expanded hon eycomb specimens was only conducted in stabilised configuration with facesheets bonded to the honeycomb. The stress-strain-curve in Fig. 14 is of a similar shape as for the hexagonal honeycomb of same density. The stress drop after compressive strength is even more significant with a respective lower crush strength.

Loading rate has a minor influence on crush strength with an increase of 10-15% for high strain rates.

The in-plane compressive stress-strain-diagram of over -expanded honeycomb specimens in Ldirection (Fig. 15) differs from the diagram in W-direction (Fig. 16): There is no plateau stress but a linear elastic stress slope up to a steep increase in the strain region of about 50%. This can be explained by means of the in-plane cell deformation in Fig. 17a. The double cell walls are oriented parallel to the load direction and the load transfer occurs by elastic bending deformation of the single cell walls in an S-shape. Here the double cell walls do not turn sideward in the following as in the case of hexagonal honeycomb, resulting in a larger linear elastic regime. In the strain region of about 50% the cells are deformed in a way that the double cell walls come into contact, which leads to the steep increase. Further compressive deformation leads to a crushing of the double cell walls. This densification point occurs much earlier than in case of the other specimens. The strain rate-affected increase of the stress-curve is about 10% at higher strain rates due to small local deformation levels at the cell walls comprising mainly of bending.

The in-plane compressive stress-strain-diagram in W-direction consists of a distinctive plateau stress as in case of hexagonal honeycomb. However, the cell deformation phenomena are completely different (Fig. 17b). Since the cell walls are continuously oriented in load direction (L-direction: discontinuously), the linear elastic slope is remarkably higher than for the hexagonal honeycomb. The result of this compressive stress state is buckling of the single cell walls at arbitrary locations. Several cells laterally buckle deforming the neighbouring cells. This initiates an irregular asymmetrical cell deformation of the specimen progressing under a constant stress level. Up to a compressive strain of 48%, almost undeformed cells can still be found besides others that are highly deformed, justifying this constant stress level. In case of hexagonal honeycombs loaded in W-direction, the cell walls are ideally folded and densification occurs at strains of approximately 80% (Fig. 12). By contrast, for over-expanded specimens the cell interference occurs much earlier so that higher loads are necessary in order to further deform this irregular structure. Therefore, the stress increase begins in a strain region of 50-60% (Fig. 16). For high loading rates the formation of a stress peak following the linear elastic regime is noticeable, which is assumed to result from a higher stability of the single cell walls.

The single cell walls are supported laterally by double cell walls from alternating sides. Up to a strain of about 3% this structure can bear rising loads with increasing lateral movement of the double cell walls inducing irreg ularities in the force transferring single cell walls at quasi-static strain rate. This leads to a stiffness reduction in stress-strain-relation and a smooth approach to the stress plateau. With higher strain rates the irregularities and the stiffness reduction are kept smaller due to inertial effects of the cell walls. As a result, the structure can bear higher loads for a small time leading to a peak stress with a sudden decrease to the stress plateau level after stability loss of the single cess walls. Even the plateau stress is affected by strain rate due to large local deformations with inertial effects and additionally due to friction between the inhomogeneous ly and chaotic ally folding cell walls. The stress plateau increases about 23% at a strain rate of 50 s -1 .

3. Summary

For hexagonal and over-expanded honeycomb cell configurations various crushing and collapse phenomena were observed and explained in transverse and in-plane material directions.

The orientation of the double and single cell walls to the load direction has a key influence on deformation behaviour (Fig. 18). While the compressive cell deformation of hexagonal and over -expanded honeycomb in L -direction is regular and homogeneous, it is localised and inhomogeneous in W-direction. All of the stress-strain-curves turned out to be affected by loading rate. The relationship between strain rate and increase of plateau stress is illustrated in Fig. 19. Except for two contradicting data points for hexagonal honeycomb all curves are monotonically increasing towards higher strain rates. The absolute value of increase is higher for the hexagonal honeycomb than for the over-expanded honeycomb of the same density. In [START_REF] Zhao | Crushing behaviour of aluminium honeycombs under impact loading[END_REF][START_REF] Reid | Dynamic compression of cellular structures and materials[END_REF][START_REF] Kobayashi | Dynamic and quasi-static deformation of aluminium honeycomb sandwich panel in three point bending[END_REF][START_REF] Harrigan | Inertia effects in impact energy absorbing materials and structures[END_REF] it is depicted for aluminium honeycomb, that micro-inertial effects of the cellular structure are responsible for the strain rate increase, since the aluminium sheet material has no significant rate dependence in the respective domain and constitutive strain rate effects can therefore be ruled out. Other explanations for the rate effect are a more complicated crushing pattern of the cells under dynamic loading as well as the influence of trapped air in case of stabilised compression [START_REF] Shkolnikov | Honeycomb modeling for side impact moving deformable barrier[END_REF][START_REF] Lassat De Pressigny | Numerical simulation of ground impact after airdrop[END_REF][START_REF] Zhou | Characterization of aluminium honeycomb material failure in large deformation compression, shear, and tearing[END_REF]. However, in this experimental investigation, no trapped air effect could be observed, because in that case the stabilised compression specimens should have shown a higher rate dependency than the bare compression specimens, assuming that the air can escape from the upper and lower opening of the cells in the latter case. Quite contrary to that, the bare specimens showed a higher strain rate dependency. Therefore, the reason for the strain rate effect in this investigation is primarily ascribed to inertial effects rather than trapped air or rate effects of the cell material. This theory is supported by the fact that the stress increase is most pronounced for collapse phenomena involving large local deformations in an evenly distributed manner, where inertial effects dominate.

Conclusions

Dynamic testing was conducted on phenolic GFRP and phenolic-impregnated aramid paper honeycomb specimens in order to analyse the effect of loading rate on the mechanical behavio ur. The strain rate effect for the composite material is remarkable and leads to increasing Young's modulus, tensile strength, tensile failure strain, shear modulus and shear strength. This strain rate effect was found to be negligible for the strain rate domain below 1 s -1 .

Also for the honeycomb specimens higher loading rates lead to an increase of the stress-curves.

In this study the plateau stress was primarily analysed and compared, since the densification point was not influenced by strain rate and the peak stress could not finally be evaluated because of the strong influence of data filters. The increase of the plateau stress is about 10-30% for the honeycomb specimens in the strain rate domain from 50 s -1 to 300 s -1 depending on the material direction. The hexagonal cell configuration turned out to be more influenced by loading rate than the over-expanded honeycomb. The rate dependency seems to be affected more severely by the structural design of the honeycomb than by pure Nomex ® material behaviour and is ascribed to inertial effects of the buckling and folding cell walls. The data presented in this paper can be used for rate-dependent material modelling for dynamic F E-simulations of phenolic GFRP/Nomex ® sandwich structures.

Figures:

Fig.

Tables:

Table 1 Honeycomb types investigated in this study

1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .Fig. 17 .Fig. 18 .Fig. 19 .

 12345678910111213141516171819 Fig. 1. Cross-section of 6-ply woven fabric GFRP specimen Fig. 2. Nomex ® honeycomb specimens for in -plane compression tests: a) hexagonal , b) over-expanded cell configuration Fig. 3. Drop weight test apparatus for a) tensile tests of GFRP specimens, b) compression tests of honeycomb specimens Fig. 4. GFRP tensile test results: static stress-strain-curves for warp and weft direction Fig. 5. GFRP tensile test results: stress-strain -curves at different strain rates (weft direction) Fig. 6. GFRP strain rate diagram: strain rate dependent tensile strength in warp and weft direction Fig. 7. GFRP ±45° tensile test results: shear stress-shear strain-curves at different strain rates Fig. 8. Cell wall buckling deformation of stabilised hexagonal Nomex ® honeycomb under compressive load in Tdirection with percentage compressive strain Fig. 9. Hexagonal honeycomb bare compression test results in T-direction: stress-strain-curves at different strain rates Fig. 10. Hexagonal honeycomb stabilised compression test results in T-direction: stress-strain-curves at different strain rates Fig. 11. Hexagonal honeycomb in-plane compression test results in L-direction: stress-strain-curves at different strain rates Fig. 12. Hexagonal honeycomb in-plane compression test results in W-direction: stress-strain -curves at different strain rates Fig. 13. In-plane cell deformation of hexagonal Nomex ® honeycomb under compressive load in a) L-direction, b) Wdirection with percentage compressive strain Fig. 14. Over-expanded honeycomb stabilised compression test results in T -direction: stress-strain-curves at different strain rates Fig. 15. Over-expanded honeycomb in-plane compression test results in L-direction: stress-strain-curves at different strain rates Fig. 16. Over-expanded honeycomb in-plane compression test results in W -direction: stress-strain-curves at different strain rates Fig. 17. In-plane cell deformation of over-expanded Nomex ® honeycomb under compressive load in a) L-direction, b) W-direction with percentage compressive strain Fig. 18. Orientation of single and double cell walls to the load direction Fig. 19. Nomex ® honeycomb strain rate diagram: strain rate dependent increase of plateau stress for hexagonal and over-expanded honeycomb

Fig 1 :

 1 Fig 1:

Fig 2 :

 2 Fig 2:

Fig 19 :

 19 Fig 19:

Table 2

 2 Static Young's modulus of phenolic GFRP in warp and weft direction Table3Static tensile strength of phenolic GFRP in warp and weft directionTable 4 Strain rate dependent increase of tensile strength for phenolic GFRP in weft direction Table5Strain rate dependent increase of tensile failure strain for phenolic GFRP in weft direction

Table 6

 6 Strain rate dependent increase of shear strength for phenolic GFRP

Table 1 :Table 2 :Table 3 :Table 4 :Table 5 :Table 6 :

 123456

Fig 3:

Fig 4:

Fig 5:

Fig 6:

Fig 7:

Fig 8:

Fig 9:

Fig 10:

Fig 11:

Fig 12:

Fig 13:

Fig 14:

Fig 15:

Fig 16:

Fig 17:

Fig 18:

Acknowledgement

This investigation was accomplished in framework of the 2 nd aeronautical research programme of the Free and Hanseatic City of Hamburg. Sincere thanks are given to E.

Wittmann (EADS, Munich) and S. Gabriel (IVW, Kaiserslautern) for performing the material testing.


P P Camanho
email: pcamanho@fe.up.pt

P Maimí

C G Dávila

Prediction of size effects in notched laminates using continuum damage mechanics

Keywords: Size effect, Continuum Damage Mechanics, Fracture Mechanics

This paper examines the use of a continuum damage model to predict strength and size effects in notched carbon-epoxy laminates. The effects of size and the development of a fracture process zone before final failure are identified in an experimental program. The continuum damage model is described and the resulting predictions of size effects are compared with alternative approaches: the point stress and the inherent flaw models, the Linear-Elastic Fracture Mechanics approach, and the strength of materials approach. The results indicate that the continuum damage model is the most accurate technique to predict size effects in composites. Furthermore, the continuum damage model does not require any calibration and it is applicable to general geometries and boundary conditions.

Introduction

The introduction of advanced composite materials in new applications relies on the development of accurate analytical and computational tools that are able to predict the thermo-mechanical response of composites under general loading conditions and geometries. In the absence of accurate analytical models, the design process has to rely on costly matrices of mechanical tests based on large numbers of test specimens [START_REF]Military Handbook[END_REF] and empirical knockdown factors [START_REF]Going to extremes: meeting the emerging demand for durable polymer matrix composites[END_REF].

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The prediction of ultimate strength remains the main challenge in the simulation of the mechanical response of composite materials [START_REF] Hinton | Predicting failure in composite laminates: the background to the exercise[END_REF]. The simulation of size effects on the strength of composites is of particular interest and relevance [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF]- [START_REF] Bažant | Size Effect on Flexural Strength of Fiber-Composite Laminates[END_REF]: reliable analytical and numerical models must represent the decrease of the ultimate strength when the structural dimensions increase [START_REF] Bažant | Size effect[END_REF].

Size effects in laminated composites occur at different material and structural levels. At the meso-mechanical level, it is observed that the transverse tensile and in-plane shear strengths of a ply constrained by sub-laminates depend on the ply thickness [START_REF] Parvizi | Constrained cracking in glass fibrereinforced epoxy cross-ply laminates[END_REF]. This size effect is normally called the "in-situ" effect and can be accounted for in the prediction of matrix cracking onset using the "in-situ" strengths in appropriate failure criteria. The "in-situ" strengths can be calculated from analytical closed-form solutions using ply elastic properties and fracture energies [START_REF] Dvorak | Analysis of first ply failure in composite laminates[END_REF], [START_REF] Camanho | Prediction of in-situ strengths and matrix cracking in composites under transverse tension and in-plane shear[END_REF].

Size effects also occur at the macro-mechanical level. For example, it is shown in [START_REF] Waddoups | Macroscopic fracture mechanics of advanced composite materials[END_REF] that the strength of notched quasi-isotropic composite laminates decreases for increasing notch sizes when thin plies are used. This effect, usually known as the "hole size effect", is caused by the development and propagation of non-critical ply-level damage mechanisms that occur in the vicinity of the hole before the final collapse of the laminate. The exact nature of the non-critical damage mechanisms has been reported by several authors. Using Moiré interferometry in notched [0/ + 45/90/ -45] s laminates, Mollenhauer et al. [START_REF] Mollenhauer | Examination of ply cracking in composite laminates with open holes: a Moiré interferomic and numerical study[END_REF] observed a strain redistribution as a result of matrix-fiber splitting in the 0 • surface ply and sub-surface ply cracking. Green et al. [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF] reported fiber splitting in the 0 • plies, matrix cracking in the off-axis plies, and delamination in [+45 m /90 m / -45 m /0 m ] ns carbon-epoxy laminates with a central circular hole.

The observed ply-level damage mechanisms can be regarded as a fracture process zone that develops before final failure of the laminate. For very small specimens, the fracture process zone affects the entire width of the laminate. On the other hand, the size of the fracture process zone in large specimens is negligible when compared with the characteristic dimensions of the specimen. The relative dimension of the fracture process zone with respect to the specimen size justifies the different strengths observed in small and large specimens. Therefore, to predict the hole size effect in quasi-brittle materials with general dimensions, methods that account for the energy dissipated by the propagation of non-critical damage mechanisms are required [START_REF] Bažant | Scaling of structural failure[END_REF].

While the strength of notched multidirectional laminates manufactured using thin plies generally decreases with hole diameter, Green et al. [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF] reported an opposite trend for laminates with plies with the same fiber orientation blocked together (ply-level scaling): for a 4 mm thick [45 4 /90 4 / -45 4 /0 4 ] s carbon fiber reinforced plastic (CFRP) laminate, increasing the hole size from 3.2 mm to ACCEPTED MANUSCRIPT 25.4 mm increased the strength by 51%. This new finding was attributed to the formation of delaminations at the edge of the hole [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF]. Ply-blocked specimens exhibit a delamination type of failure, and for small hole diameters the size of the delamination is relatively large and growths unstably.

Green et al. also performed tests on thickness-scaled CFRP laminates [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF]. A decrease of the ultimate strength with test specimen thickness was reported for both ply-level and sublaminate-level scaled laminates, where the laminate thickness is increased by increasing the number of sublaminates while keeping the ply thickness constant. When increasing the thickness from 1 mm to 8 mm strength reductions of 16.5% and 64.4% were measured for the sublaminate level and ply-level scaled specimens, respectively. The strength reduction was attributed to the higher energy release rate at the interfaces of the plylevel scaled specimens, which promotes delamination, and to the higher stress concentration relief that occurs as a result of damage in the surface plies of sublaminate-level scaled specimens.

The calculation of macro-mechanical size effects is often based on semi-empirical methods that require calibration such as the point stress and average stress models proposed by Whitney and Nuismer [START_REF] Whitney | Stress fracture criteria for laminated composites containing stress concentrations[END_REF]. The point stress model assumes that final failure occurs when the stress at a characteristic distance from the notch reaches the unnotched strength of the laminate. In the average stress model it is assumed that final failure occurs when the laminate stress averaged over a characteristic distance is equal to the unnotched strength of the laminate. Modifications of the point stress and average stress models using ply strengths have been proposed to predict the strength of laminates with open and loaded holes [START_REF] Chang | Strength of mechanically fastened composite joints[END_REF]- [START_REF] Camanho | A design methodology for mechanically fastened joints in laminated composite materials[END_REF]. The advantage of using ply properties rather than laminate properties is that the need to measure laminate strengths for every layup is avoided. However, the measurement of the characteristic distances is still required for each lay-up and geometry [START_REF] Camanho | A design methodology for mechanically fastened joints in laminated composite materials[END_REF].

On the other hand, models based on continuum damage mechanics do not require calibration, so they potentially provide the means for a truly predictive methodology for the strength prediction of composite laminates. Continuum damage models are defined in the framework of the thermodynamics of irreversible processes. Generally speaking, the formulation of continuum damage models starts by the definition of a potential (e.g. the complementary free energy) as a function of one or more damage variables that is the basis for establishing the relation between the stress and the strain tensors. It is also required to define the damage activation functions, i.e. the conditions that lead to the onset of inelastic response, and the damage evolution functions. Some of the models proposed in the literature are exclusively based on thermodynamic restrictions of the constitutive model and on some adjusting functions for damage onset and evolution. Other models, besides satisfying the thermodynamic restrictions, are based on the failure mechanisms [START_REF] Maimí | A thermodynamically consistent damage model for advanced composites[END_REF], i.e. the damage

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activation functions are related to the physics of the different failure mechanisms, and the damage variables are related to the orientation of the ply failure planes experimentally observed. Mechanism-based continuum damage models can predict damage onset and the extent and type of non-critical damage mechanisms. Furthermore, continuum damage models that relate the damage variables to the normal components of the stress tensor are able to simulate the effect of crack closure under load reversal cycles. Therefore, such models be used to predict the strength under non-monotonic loading including load reversals.

The objective of this paper is to investigate the use of a continuum damage model for the prediction of size effects in notched carbon-epoxy laminates loaded in tension. An experimental program is conducted to measure the relevant material properties and to identify size effects occurring in laminates with different hole sizes. The recently proposed continuum damage model is described and analysis of open hole specimens subjected to tension loads are presented. The analyses results are compared with the experimental data and with predictions obtained using a strength of materials approach, Linear-Elastic Fracture Mechanics, and the point stress model.

Experimental program

Material selection and characterization

The material selected for the present study is Hexcel's IM7-8552 carbon epoxy unidirectional tape with a nominal ply thickness of 0.131 mm. The material was cured according to the manufacturer's specifications, with temperature stages of 110 • C for one hour, followed by 180 • C for two hours. A pressure of 7 bar was applied for the duration of the cure cycle.

The fiber volume fraction was measured using image processing techniques resulting in an average value of 59.1%. The coefficients of thermal expansion were measured using a dilatometer and the resulting values are α 11 = -5.5 × 10 -6 / • C for the longitudinal direction, and α 22 = 25.8 × 10 -6 / • C for the transverse direction. The elastic properties and strengths were measured using ASTM test standards [START_REF]Standard test method for tensile properties of polymer matrix composite materials[END_REF]- [START_REF]Standard test method for in-plane shear response of polymer matrix composite materials by test of a ±45 • laminate[END_REF]. Five specimens were used for each test performed.

The mean measured values of the ply elastic properties are shown in Table 1. E 1 and E 2 are the longitudinal and transverse Young modulus respectively, G 12 is the shear modulus, and υ 12 is the major Poisson's ratio. Table 1 also presents the standard used in each test, the standard deviation (STDV), and

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the coefficient of variation (CV).

[Table 1 about here.]

The measured ply strengths are shown in Table 2. X T and Y ud T are the longitudinal and transverse tensile strengths, respectively. X C and Y C are the longitudinal and transverse compressive strengths, respectively. S ud L is the inplane shear strength.

[Table 2

about here.]

The values of the transverse tensile strength (Y ud T ) and of the in-plane shear strength (S ud L ) measured in the test specimens correspond to the strengths of unconstrained unidirectional plies. The transverse tensile and shear strengths of constrained plies (in-situ strengths) are higher than the ones of an unidirectional ply [START_REF] Parvizi | Constrained cracking in glass fibrereinforced epoxy cross-ply laminates[END_REF] and decrease when increasing the ply thickness. The in-situ strengths are calculated using models previously proposed by the authors, which are based on the mode I fracture toughness, G 2+ , and on the mode II fracture toughness, G 6 [START_REF] Camanho | Prediction of in-situ strengths and matrix cracking in composites under transverse tension and in-plane shear[END_REF]. These models use the simplifying assumption that the in-situ strengths are not a function of the elastic properties and geometry of the neighboring layers.

To measure the components of the fracture toughness, double cantilever beam (DCB) [START_REF]Standard test methods for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites[END_REF] and 4-point bending end notched flexure (4-ENF) [START_REF] Martin | Characterization of mode II delamination using the 4ENF[END_REF] tests were performed. The measured components of the fracture toughness are shown in Table 3.

[Table 3

about here.]

The in-situ strengths are calculated as functions of the fracture toughness and ply elastic properties using the models described in [START_REF] Camanho | Prediction of in-situ strengths and matrix cracking in composites under transverse tension and in-plane shear[END_REF] with a shear response factor β = 2.98 × 10 -8 MPa -3 . The calculated in-situ strengths are shown in Table 4.

[Table 4

about here.]

The shear strength in the transverse direction is calculated as [START_REF] Dávila | Failure criteria for FRP laminates[END_REF], [START_REF] Pinho | Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity[END_REF]:

S T = Y C cos α 0 sin α 0 + cos α 0 tan 2α 0 (1)
where α 0 is the fracture angle of a ply under pure transverse compression [START_REF] Puck | Failure analysis of FRP laminates by means of physically based phenomenological models[END_REF].

For a fracture angle α 0 = 53 • , the shear strength in the transverse direction is calculated as S T = 75.3 MPa.

The continuum damage model also requires the fracture energies per unit sur-

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face for longitudinal failure. These energies were measured using the Compact Tension (CT) and Compact Compression (CC) tests in cross-ply laminates proposed by Pinho et al. [START_REF] Pinho | Modelling failure of laminated composites using physically-based failure models[END_REF]- [START_REF] Pinho | Fracture toughness of the tensile and compressive fibre failure modes in laminated composites[END_REF]. The fracture energies per unit surface measured are shown in Table 5.

[Table 5 about here.]

Notched laminates

Tests of notched composite laminates were performed to quantify the size effect and to obtain empirical data to validate the numerical model. Quasi-isotropic laminates were manufactured in Hexcel IM7-8552 CFRP with a stacking sequence of [90/0/ ± 45] 3s .

The unnotched tensile strength of the laminate, X L T , was measured using five test specimens and the average value obtained was 845.1 MPa. The average value of the failure strain, 12900 µε, was measured in the five test specimens using strain gages.

The notched test specimens were machined using a procedure that prevents delaminations in the regions close to the insertion point and the exit of the drill bit. Sacrificial frontal and backing plates were used to clamp the specimens during the drilling process. All test specimens were machined to class 1 hole quality used in aerospace [START_REF] Sawicki | The influence of fastener clearance upon the failure of compression-loaded composite bolted joints. Composite Structures: Theory and Practice[END_REF]. No damage was observed in a sample of test specimens inspected using X-rays.

Specimens with five different hole diameters, d=2 mm, 4 mm, 6 mm, 8 mm, 10 mm and with a width-to-diameter ratio w/d equal to 6 were tested in a MTS servo-hydraulic machine following the ASTM D-5766 standard [START_REF]Open hole tensile strength of polymer composite laminates[END_REF] according to the test matrix shown in Table 6. Five specimens were tested for each geometry.

[Table 6 about here.] The specimens labeled OHT3, OHT6 and OHT9 were instrumented with two strain gages in the positions schematically shown in Figure 1.

[Fig. 1

about here.]

The distances d s shown in Figure 1 are respectively 10.5 mm, 13.5 mm, and 12.5 mm for the test specimens OHT3, OHT6 and OHT9. The specimens OHT10 and OHT11 were not instrumented. Acoustic emission (AE) sensors were used in one test specimen for each size.

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Figure 2 shows the applied load and the cumulative number of AE signals as a function of time for one OHT3 test specimen.

[Fig. 2

about here.]

From the AE signals shown in Figure 2, it can be concluded that non-critical damage mechanisms accumulate well before final failure of the specimen, creating a fracture process zone (FPZ). Similar results are observed in the OHT6 and OHT9 specimens, as well as in other experimental investigations [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF], [START_REF] Mollenhauer | Examination of ply cracking in composite laminates with open holes: a Moiré interferomic and numerical study[END_REF].

The remote failure stress is defined using the failure load measured in the tests ( P ) and the measured values of the specimen thickness (t L ) and width (w) as: σ ∞ = P wt L . The remote failure stresses obtained for the different geometries are summarized in Table 7.

[Table 7 about here.] The failure mode observed in all specimens is net-section tension, as shown in Figure 3. Figure 4 shows the relation between the remote stress and the strain measured by strain gages SG3 for one test specimen of each of the three different geometries.

[Fig. 3 The experimental results presented in Table 7 clearly identify a size effect: an increase in the hole diameter from 2 mm to 10 mm results in a 32.8% reduction in the strength. The observed size effect is caused by the development of the fracture process zone identified in the AE results, which re-distributes the stresses and dissipates energy. In small specimens, the fracture process zone extends towards the edges of the specimen and the average stress at the fracture plane tends to the unnotched strength of the laminate.

The effect of size on the strength can be explained using a simple example based on the cohesive crack model, which is well-suited to simulate fracture of quasi-brittle materials [START_REF] Turon | A damage model for the simulation of delamination in advanced composites under variable-mode loading[END_REF]. Consider that the fracture process zone is represented by a cohesive crack with the simple constitutive relation shown in Figure 5 a).

[Fig. 5

about here.]

The cohesive constitutive model relates the laminate cohesive stress, σ, to the crack opening, w, and must satisfy the following condition: ∞ 0 σ(w)dw = G C . Structural collapse occurs when a point along the fracture plane reaches the critical opening, w c , and the corresponding length of the fracture process zone ACCEPTED MANUSCRIPT can be estimated using the Irwin model as [START_REF] Turon | An engineering solution for using coarse meshes in the simulation of delamination with cohesive zone models[END_REF]. Based on the constitutive law shown in Figure 5 a), it is possible to schematically represent the stress distribution at failure along the fracture planes of specimens with different sizes, as shown in Figure 5 b). It is observed that in small specimens the fracture process zone extends towards the edges, whereas in large specimens the fracture process zone is confined to the vicinity of the hole. As a consequence, the average stress acting on the fracture plane, and hence the strength, are larger for small specimens.

l FPZ ≈ EG C π(X L T ) 2 [33]-
3 Simulation of the effect of size on strength Strength prediction methods uniquely based on stress or strain failure criteria are unable to predict the size effects observed in notched specimens. Consider for example a calculation of the final failure of a specimen with a central hole using the value of the longitudinal stress in the fiber direction (maximum stress criterion). The distribution of the longitudinal stress in the critical plies, the 0 • plies along the fracture plane, defined by θ = 90 • in Figure 1, can be calculated using an approximate closed-form solution as [START_REF] Tan | Stress concentrations in laminated composites[END_REF]:

σ 11 = σ xx (0, y)(Q 11 a * 11 + Q 12 a * 12 ) (2)
where a * ij are the components of the laminate compliance matrix defined as [START_REF] Herakovich | Mechanics of fibrous composites[END_REF]:

[a * ] = t L [A] -1 ( 3
)
where the matrix [A] relates the in-plane forces per unit length to the midplane strains. Q ij are the components of the plane stress transformed reduced stiffness matrix of the 0 • plies [START_REF] Camanho | A design methodology for mechanically fastened joints in laminated composite materials[END_REF], and t L is the thickness of the laminate.

The through-the-thickness averaged normal stress in the fracture plane for a quasi-isotropic laminate is calculated by Tan [START_REF] Tan | Stress concentrations in laminated composites[END_REF] as:

σ xx (0, y) = 2 + (1 -d/w) 3 6(1 -d/w)   2 + d 2y 2 + 3 d 2y 4   σ ∞ xx , y ≥ d/2 (4)
where σ ∞ xx is the remote tensile stress.

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From equations ( 2) and ( 4) it is clear that for the same material and stacking sequence the stress concentration factor, and hence the maximum longitudinal stress in the 0 • ply, depends on the ratio between the specimen hole diameter and width. Applying the maximum stress criterion and using equations ( 2) and ( 4):

σ 11 X T = 1 ⇒ σ ∞ = (1 -d/w)X T [2 + (1 -d/w) 3 ](Q 11 a * 11 + Q 12 a * 12 ) (5)
Equation ( 5) demonstrates that the application of the maximum stress criterion results in the same strength prediction for different hole diameters when the d/w ratio is held constant. The lack of size effect on the predicted strength clearly contradicts the experimental observations.

Linear-Elastic Fracture Mechanics

There are two approaches that can be used with Linear Elastic Fracture Mechanics (LEFM) to calculate the effect of size on the strength of notched composite laminates. In the first approach, it is assumed that the length a of a pre-existing crack in the laminate is scaled in the same proportion of the hole diameter and specimen width, and that the critical value of the laminate's stress intensity factor, K Ic , is independent of the crack length. Consider two specimens with hole diameters d 1 and d 2 . The stress intensity factor at failure is:

K Ic = σ ∞ 1 F w 1 d 1 , a 1 d 1 √ πa 1 = σ ∞ 2 F w 2 d 2 , a 2 d 2 √ πa 2 (6)
Taking into account the fact that the crack length is proportional to the hole diameter and that the finite width correction factors, F (w/d, a/d), are equal for scaled geometries, the failure stress of a specimen with a hole diameter d 2 can be calculated from the failure stress of the specimen with a hole diameter d 1 :

σ ∞ 2 = σ ∞ 1 d 1 d 2 (7)
The second approach to predict size effects using LEFM is the inherent flaw model (IFM) proposed by Waddoups et al. [START_REF] Waddoups | Macroscopic fracture mechanics of advanced composite materials[END_REF]. It is considered that the noncritical damage mechanisms occurring before ultimate failure of a composite

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laminate can be lumped into a constant "region of intense energy", or "inherent flaw", of length a. The critical value of the stress intensity factor of a plate with a hole of radius R is given by:

K Ic = f (a, R) σ ∞ √ πa (8)
where f (a, R) is Bowie's solution for the calculation of the stress intensity factor of two cracks emanating from a circular hole, given as [START_REF] Bowie | Analysis of an infinite plate containing radial cracks, originating at the boundaries of an internal circular hole[END_REF]- [START_REF] Tada | The stress analysis of cracks handbook[END_REF]:

f (a, R) = 0.5 3 - a d/2 + a   1 + 1.243 1 - a d/2 + a 3   (9)
Waddoups et al. [START_REF] Waddoups | Macroscopic fracture mechanics of advanced composite materials[END_REF] considered that the strength of an unnotched specimen can be predicted by taking into account that the hole radius tends to zero, in which case the function f (a, R) tends to one, leaving:

K Ic = X L T √ πa ( 10
)
where X L T is the tensile strength of the unnotched laminate.

From ( 8) and ( 10) the equation proposed by Waddoups et al. [START_REF] Waddoups | Macroscopic fracture mechanics of advanced composite materials[END_REF] is obtained:

σ ∞ = X L T /f (a, R) (11)
The strength of the laminate containing an open-hole is predicted using two parameters: the length of the inherent flaw, a, that needs to be calculated from a baseline specimen, and the unnotched tensile strength of the laminate, X L T .

Point-stress model

The point-stress model (PSM) proposed by Whitney and Nuismer [START_REF] Whitney | Stress fracture criteria for laminated composites containing stress concentrations[END_REF], considers that ultimate failure occurs when the stress at a given distance from the hole boundary, r ot , reaches the unnotched strength of the laminate, X L T . An alternative version of the point stress model uses the ply stresses and strengths, so that it is not necessary to measure the strength for every different laminate.

Using equations ( 2) and (4), the strength predicted using the PSM is:

ACCEPTED MANUSCRIPT σ ∞ = X T    2 + (1 -d w ) 3 6(1 -d w )   2 + d d + 2r ot 2 + 3 d d + 2r ot 4   (Q 11 a * 11 + Q 12 a * 12 )    -1 (12)
Failure is predicted using two parameters: the characteristic distance in tension r ot , and the longitudinal tensile strength of the ply, X T .

Continuum damage model

Continuum Damage Mechanics is a methodology well suited for the simulation of damage evolution and ultimate failure of composites under general loads and boundary conditions for which no analytical solution is available.

The continuum damage model used here is based on previous work by the authors [START_REF] Maimí | A thermodynamically consistent damage model for advanced composites[END_REF], [START_REF] Maimí | A continuum damage model for composite laminates: part I -constitutive model[END_REF], [START_REF] Maimí | A continuum damage model for composite laminates: part II -computational implementation and validation[END_REF]. The main aspects of the continuum damage model are presented in the following sections. The full details of the model can be found in references [START_REF] Maimí | A thermodynamically consistent damage model for advanced composites[END_REF], [START_REF] Maimí | A continuum damage model for composite laminates: part I -constitutive model[END_REF] and [START_REF] Maimí | A continuum damage model for composite laminates: part II -computational implementation and validation[END_REF].

Constitutive model

The proposed definition for the complementary free energy density of a ply is: where the damage variable d 1 is associated with longitudinal (fiber) failure, d 2 is the damage variable associated with transverse matrix cracking, and d 6 is the damage variable associated with longitudinal and transverse cracks. β 11 and β 22 are the coefficients of hygroscopic expansion in the longitudinal and transverse directions, respectively. ∆T and ∆M are the differences of temperature and moisture content with respect to the corresponding reference values. The coefficients of thermal expansion of a ply are also affected by the failure mechanisms. The exact dependence of the coefficients of thermal expansion with damage can be obtained for simple laminates in the absence of stress gradients [START_REF] Joffe | COD-based simulation of transverse cracking and stiffness reduction in (S/90n)s laminates[END_REF]. These conditions are not met by the laminate under investigation here and the effects of damage on the coefficients of thermal expansion are neglected.

G = σ 2 11 2 (1 -d 1 ) E 1 + σ 2 22 2 (1 -d 2 ) E 2 - ν
The strain tensor is equal to the derivative of the complementary free energy

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density with respect to the stress tensor:

ε = ∂G ∂σ = H : σ + α∆T + β∆M (14)
The lamina compliance tensor can be represented as:

H = ∂ 2 G ∂σ 2 =          1 (1 -d 1 ) E 1 - υ 12 E 1 0 - υ 12 E 1 1 (1 -d 2 ) E 2 0 0 0 1 (1 -d 6 ) G 12          (15)
The closure of transverse cracks under load reversal is taken into account by defining four damage variables associated with longitudinal and transverse damage. To distinguish between the active and the passive damage variables, it is necessary to define the longitudinal and transverse damage modes as follows:

d 1 = d 1+ σ 11 |σ 11 | + d 1- -σ 11 |σ 11 | d 2 = d 2+ σ 22 |σ 22 | + d 2- -σ 22 |σ 22 | (16)
where x is the McCauley operator defined as x := (x + |x|) /2.

Damage activation functions

The determination of the domain of elastic response under complex stress states is an essential component of an accurate damage model. It is assumed that the elastic domain is enclosed by four surfaces, each of them accounting for one damage mechanism: longitudinal and transverse fracture under tension and compression. Those surfaces are formulated by the damage activation functions based on the LaRC04 failure criteria [START_REF] Pinho | Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity[END_REF].

The four damage activation functions, F N , associated with damage in the longitudinal (N = 1+, 1-) and transverse (N = 2+, 2-) directions represented in Figure 6, are defined as:

F 1+ = φ 1+ -r 1+ ≤ 0 ; F 1-= φ 1--r 1-≤ 0 F 2+ = φ 2+ -r 2+ ≤ 0 ; F 2-= φ 2--r 2-≤ 0 (17)
where the loading functions φ N (N = 1+, 1-, 2+, 2-) depend on the strain tensor and material constants (elastic and strength properties). The elastic domain thresholds r N (N = 1+, 1-, 2+, 2-) take an initial value of 1 when the material is undamaged, and they increase with damage. The elastic domain thresholds are related to the damage variables d M (M = 1+, 1-, 2+, 2-, 6) by the damage evolution laws.

[Fig. 6

about here.]

The current values of the elastic domain thresholds r N are obtained using the loading functions φ N according to the following equations [START_REF] Maimí | A thermodynamically consistent damage model for advanced composites[END_REF], [START_REF] Maimí | A continuum damage model for composite laminates: part I -constitutive model[END_REF], [START_REF] Maimí | A continuum damage model for composite laminates: part II -computational implementation and validation[END_REF]:

r 1+ = max 1, max s=0,t φ s 1+ , max s=0,t φ s 1- r 1-= max 1, max s=0,t φ s 1- r 2+ = max 1, max s=0,t φ s 2-, max s=0,t φ s 2+ r 2-= max 1, max s=0,t φ s 2- (18)
Longitudinal tensile fracture

The LaRC04 criterion for fiber tension is defined as:

φ 1+ = E 1 X T ε 11 = σ11 -υ 12 σ22 X T ( 19
)
where the effective stress tensor σ is computed as σ = H 0 -1 : ε. H 0 is the undamaged compliance tensor.

Longitudinal compressive fracture

The damage activation function used to predict damage under longitudinal compression (σ 11 < 0) and in-plane shear (fiber kinking) is established as a function of the components of the stress tensor σ(m) in a coordinate system (m) representing the fiber misalignment:

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)
where the coefficient of longitudinal influence can be approximated as [START_REF] Pinho | Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity[END_REF]:

η L ≈ - S L cos (2α 0 ) Y C cos 2 α 0 (21)
with α 0 = 53 • [START_REF] Puck | Failure analysis of FRP laminates by means of physically based phenomenological models[END_REF]. The components of the effective stress tensor in the coordinate system associated with the rotation of the fibers are calculated as:

σm 22 = σ11 sin 2 ϕ C + σ22 cos 2 ϕ C -2 |σ 12 | sin ϕ C cos ϕ C σm 12 = (σ 22 -σ11 ) sin ϕ C cos ϕ C + |σ 12 | cos 2 ϕ C -sin 2 ϕ C (22)
The misalignment angle (ϕ C ) is determined using standard shear and longitudinal compression strengths, S L and X C , respectively [START_REF] Pinho | Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity[END_REF]:

ϕ C = arctan     1 -1 -4 S L X C + η L S L X C 2 S L X C + η L     (23)
Transverse fracture perpendicular to the mid-plane of the ply Transverse matrix cracks perpendicular to the mid-plane of the ply, i.e. with α 0 = 0 • , are created by a combination of in-plane shear stresses and transverse tensile stresses, or in-plane shear stresses and small transverse compressive stresses. These conditions are represented by the following failure criteria:

φ 2+ =          (1 -g) σ22 Y T + g σ22 Y T 2 + σ12 S L 2 if σ22 ≥ 0 1 S L |σ 12 | + η L σ22 if σ22 < 0 ( 24
)
where g is the fracture toughness ratio defined as: g = G 2+ G 6 .

Transverse compressive fracture

The matrix failure criterion for transverse compressive stresses consists of a quadratic interaction between the effective shear stresses acting on the fracture plane:

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φ 2-= τ T eff S T 2 + τ L eff S L 2 if σ22 < 0 ( 25
)
where the effective stresses τ T eff and τ L eff are computed as [START_REF] Pinho | Failure models and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity[END_REF]:

τ T eff = -σ 22 cos (α 0 ) sin (α 0 ) -η T cos (α 0 ) cos (θ s ) τ L eff = cos (α 0 ) |σ 12 | + η L σ22 cos (α 0 ) sin (θ s ) ( 26
)
with η T = -1 tan(2α 0 ) and θ s = arctan -|σ 12 | σ22 sin(α 0 ) .

Damage evolution laws and numerical implementation

Strain-softening constitutive models that do not take into account the finite element discretization produce results that are mesh-dependent, i.e. the solution is non-objective with respect to the mesh refinement and the computed energy dissipated decreases with a reduction of the element size [START_REF] Bažant | Crack band theory for fracture of concrete[END_REF]- [START_REF] Bažant | Nonlocal integral formulations of plasticity and damage: survey of progress[END_REF]. An effective solution to assure objective solutions consists of using a characteristic length of the finite elements (l * ) in the definition of the constitutive model [START_REF] Bažant | Crack band theory for fracture of concrete[END_REF]. As schematically shown in Figure 7, the post-peak response of the material is scaled as a function of the element size to keep the computed energy dissipation independent of the size of the element, and equal to the material fracture energy.

[Fig. 7 about here.]

The energetic regularization of the model proposed requires the fracture energies associated with the four fracture planes shown in Figure 6. These fracture energies were measured in the experimental programe and are used in the damage evolution laws.

The exponential damage evolution laws proposed by the authors [START_REF] Maimí | A thermodynamically consistent damage model for advanced composites[END_REF], [START_REF] Maimí | A continuum damage model for composite laminates: part I -constitutive model[END_REF], [START_REF] Maimí | Modelización constitutiva y computacional del daño y la fractura de materiales compuestos[END_REF] are expressed in the following general form:

d M = 1 - 1 f N (r N ) exp {A M [1 -f N (r N )]} f (r K ) (27)
where the function f N (r N ) is selected to force the softening of the constitutive relation and it is taken as being independent of the material. The term f (r K ) represents the coupling factor between damage laws and elastic threshold domains. The specific damage evolution laws for each damage variable are presented in [START_REF] Maimí | A thermodynamically consistent damage model for advanced composites[END_REF], [START_REF] Maimí | A continuum damage model for composite laminates: part I -constitutive model[END_REF]- [START_REF] Maimí | Modelización constitutiva y computacional del daño y la fractura de materiales compuestos[END_REF].

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The regularization of the energy dissipated is performed by integrating the rate of energy dissipation for each failure mode. The energy dissipated in each failure mode must be independent of the element size, and must be equal to the fracture energy measured in the experiments:

∞ 1 ∂G ∂d M ∂d M ∂r M dr M = G M l * , M = 1+, 1-, 2+, 2-, 6 (28)
Using ( 27) in [START_REF] Pinho | Modelling failure of laminated composites using physically-based failure models[END_REF], it is possible to numerically integrate the resulting equation and calculate the parameters A M that assure a mesh-independent solution [START_REF] Maimí | A thermodynamically consistent damage model for advanced composites[END_REF].

The constitutive model was implemented in the ABAQUS Finite Element (FE) code [START_REF]ABAQUS 6.5 User's Manual[END_REF] as a user-written UMAT subroutine.

Mesh objectivity and unidirectional notched specimen

The mesh objectivity of the model proposed is illustrated by simulating the response of a notched [90] 24 CFRP laminate loaded in tension. The specimen simulated is 150 mm long, 12 mm wide, 3 mm thick, and contains a central circular notch with a diameter of 6 mm. The properties used are reported in Tables 1 to 3.

Two FE models with different mesh refinements and using the damage model outlined in the previous sections were created. Models 1 and 2 use, respectively, 6 and 20 elements along the fracture plane. Only one-half of the specimen width is modeled. The details of two meshes are shown in Figure 8.

[Fig. 8 about here.]

Figure 9 shows the load-displacement relation predicted using the constitutive model proposed. It is observed that the solution is independent of the mesh refinement.

[Fig. 9 about here.]

In order to demonstrate the error introduced by not accounting for element size, two analyses with different levels of mesh refinement were also conducted with a constitutive model that is not adjusted using equations [START_REF] Pinho | Modelling failure of laminated composites using physically-based failure models[END_REF]. Instead, a constant softening parameter A 2+ = 1.5 is used, independently of the mesh refinement. The load-displacement relation predicted by this model is shown in Figure 10. It is clear from this figure that the maximum load and energy dissipation predicted are a function of the refinement of the mesh.

[Fig. 10 about here.]

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Quasi-isotropic open hole tension specimens

Finite element models of all OHT specimens shown in the test matrix presented in Section 2 were created using ABAQUS [START_REF]ABAQUS 6.5 User's Manual[END_REF] four-node S4 shell elements. The difference between the working and reference temperatures used to calculate the residual thermal stresses was -155 • C. An implicit dynamic analysis was subsequently performed, and the loading rate used in the experiments, 2mm/min, was also applied to the numerical models. The use of an implicit dynamic finite element model enables the prediction of the load drop that occurs when the specimens fail catastrophically. The material properties used are presented in Tables 1 to 5.

Delamination is not simulated by the model. As explained by Green at al. [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF], delamination is the driving failure mechanism for ply-blocked laminates, but not for sublaminate-level scaled laminates, such as the ones used in this work.

The simulation of delamination is required for ply-blocked laminates, and can be performed using cohesive elements connecting several shell elements that represent the layers [START_REF] Dávila | Decohesion elements for shell analysis[END_REF].

The models simulate the fracture process from the onset of damage up to structural collapse. Figure 11 shows the evolution of fiber fractures predicted in the top 0 • ply, as well the plane of localized deformation (fracture plane) for specimen configuration OHT9.

[Fig. 11 about here.]

Figure 12 shows the relation between the applied remote stress and the longitudinal deformation measured using the strain gages and the corresponding numerical predictions in the specimen OHT6. The numerical results correspond to the averaged strain calculated in the group of elements whose position and total area correspond approximately to the area where the strain gages were bonded to the specimen. The location of the different strain gages is shown in Figure 1.

[Fig. 12 about here.]

The remote failure stresses measured in the experimental programe and predicted by the numerical model are shown in Table 8.

[Table 8 about here.]

From the comparison between the experimental and numerical results, both in terms of stress-strain relations and failure stresses, it can be concluded that the model is capable of predicting with good accuracy the response of all OHT specimens that were tested.

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Comparison of approaches

The four methods previously described, i.e. strength of materials, LEFMscaled, LEFM-inherent flaw model, point stress model, and continuum damage model were applied to predict the size effect for the specimens described in Section 2.2. Equation ( 7) provides the LEFM-scaled prediction for the notched strength of the laminate when all the in-plane dimensions are scaled. The average failure stress measured in the specimens with a hole diameter of 6 mm was used in the LEFM model to predict the strength of the specimens with different geometries.

Equation [START_REF] Dvorak | Analysis of first ply failure in composite laminates[END_REF] provides the LEFM-inherent flaw model prediction of the notched strength. The specimen with a 6 mm hole diameter is used to calculate the length of the inherent flaw. Using the measured mean failure stress in equation [START_REF] Dvorak | Analysis of first ply failure in composite laminates[END_REF], the length of the inherent flaw is calculated as a = 1.28 mm.

The point-stress prediction of the size effect is performed using equation [START_REF] Camanho | Prediction of in-situ strengths and matrix cracking in composites under transverse tension and in-plane shear[END_REF]. The characteristic distance of 0.75 mm was obtained by using the measured mean failure stress in the specimen with a 6 mm hole diameter. This value of the characteristic distance is used to predict the strength of the other specimens.

The predictions of the normalized strength as a function of the hole diameter obtained using the different models are shown in Figure 13.

[Fig. 13 about here.] It can be observed that both the point stress and LEFM-IFM models can predict with reasonable accuracy the size effect law of notched composite laminates. The point stress and inherent flaw models are particularly accurate for specimens with hole diameters close to the one used to calculate the characteristic distance (PSM) and length of inherent flaw. For specimens with small hole diameters, the predictions lose accuracy. Therefore, to accurately predict the notched strength of laminates these models require the calculation of the characteristic distance and length of inherent flaw for different geometries, and the definition of an extrapolation procedure to define the values of these parameters for other geometries [START_REF] Camanho | A design methodology for mechanically fastened joints in laminated composite materials[END_REF]. It should also be noted that the basic equation used in the inherent flaw model, equation [START_REF] Dvorak | Analysis of first ply failure in composite laminates[END_REF], is only valid when finite width effects are negligible, which is the case of the specimens tested.

For smaller ratios between the specimen width and hole diameter, the inherent flaw model should be modified.

The continuum damage model can predict the size effect law observed in the

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experiments, especially for specimens with hole diameters smaller than 6 mm. Unlike the point stress and inherent flaw models, the continuum damage model does not require any adjustment parameter and only uses material properties that are measured at ply level as well as the fracture energies.

Figure 13 indicates that the use of the LEFM-scaled model results in accurate predictions for hole sizes comprised between 6 mm and 10 mm. However, the strength is overpredicted for small hole diameters. For small specimens, the damaged region in the vicinity of the hole cannot be considered as negligible when compared with the characteristic dimensions of the specimen, and LEFM is not applicable.

LEFM-scaled predictions are also inaccurate for large specimens because the notched strengths of those specimens tend to a constant value [START_REF] Green | An experimental investigation into the tensile strength scaling of notched composites[END_REF]. Bažant [START_REF] Bažant | Scaling of structural failure[END_REF] relates this asymptotic structural response to the invariance of the size of the fracture process zone when the characteristic dimensions of large specimens are increased. It should also be noted that the LEFM predictions based on scaled specimens always result in a line with a -1/2 slope that passes through the baseline point (Figure 13). This means that the use of a small hole diameter as the baseline point would result in severe underpredictions of the notched strength of larger specimens.

The maximum stress criterion for longitudinal failure is unable to predict size effects and always underpredicts the strength of notched laminates. For a hole diameter of 2 mm, the application of the maximum stress criterion results in an error of -47.4%. The error associated with the strength of materials approach is even larger when using a failure criterion for transverse (matrix) cracking, which occurs before fiber fracture, or failure criteria that are unable to distinguish fiber and matrix failure modes.

Conclusions

The size effect in notched IM7-8552 CFRP was detected and quantified in an experimental program. The acoustic emission results show that final fracture is preceded by a process of accumulation of non-critical damage mechanics.

By comparing the experimental data with the different models that are commonly used for the strength prediction of composites, it can be concluded that fiber-based failure criteria (strength of materials approach) cannot predict size effects. In addition, the strength of materials approach always underpredicts the strength of notched composites, with errors as high as -49.1% for a specimen with a 2 mm hole diameter.

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The Linear Elastic Fracture Mechanics approach using an hole diameter of 6 mm for calibration predicts the size effect accurately for specimens with hole diameters comprised between 6 mm and 10 mm. However, Linear Elastic Fracture Mechanics should not be used for the strength prediction of specimens with hole diameters equal to or lower than 2 mm, or for larger specimens whose failure stresses tend to a constant value.

The point stress and inherent flaw models are simple approaches that do not require complex FE implementations yet provide reasonable predictions for the range of hole diameters tested. However, the accuracy of these models relies upon the measurement of the characteristic distance and length of the inherent flaw for each lay-up and stacking sequence.

The continuum damage model proposed can predict with good accuracy hole size effects in composite laminates subjected to tension. The model requires material properties that are measured at the ply level and fracture energies that are measured using both standard test methods and novel compact tension and compact compression test methods. The continuum damage models provides not only the final failure load, but also information concerning the integrity of the material during the load history. Furthermore, the finite element-based damage model can be applied to structures and components of arbitrary configurations where analytical solutions could not be developed.

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List of Figures

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  about here.] [Fig. 4 about here.]

Fig. 1 .

 1 Fig. 1. Position of strain gages.

Fig. 2 .

 2 Fig. 2. Applied load and AE signals as a function of time for the specimen with a 6 mm diameter hole.

Fig. 3 .

 3 Fig. 3. Net-section tension failures in specimens with w/d = 6.

Fig. 5 .

 5 Fig. 5. Cohesive crack constitutive law and fracture process zone.

21 Fig. 6 .Fig. 8 .Fig. 11 .

 216811 Fig. 6. Fracture surfaces and corresponding internal variables.

Table 1

 1 Measured ply elastic properties for IM7-8552.

	Property	Standard Mean value STDV CV (%)
	E 1 (GPa)	Ref. [20] 171.42	2.38	1.39
	E 2 (GPa)	Ref. [20] 9.08	0.09	1.03
	G 12 (GPa) Ref. [22] 5.29	0.13	2.53
	υ 12	Ref. [20] 0.32	0.02	6.18

Fig. 7. Scaling of constitutive model for different element sizes.

Acknowledgements

The financial support of the Portuguese Foundation for Science and Technology (FCT) under the project PDCTE/50354/EME/2003 is acknowledged by the first author. Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-06-1-3072. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposed notwithstanding any copyright notation thereon.

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Fig. 9. Load-displacement relation predicted using the model proposed.

List of Tables 1 Measured ply elastic properties for IM7-8552.

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