Sylvain Haupert
email: sylvain.haupert@upmc.fr

Sandra Gue ´rard

David Mitton

Pascal Laugier

Non Destructive Characterization of Cortical Bone Micro-Damage by Nonlinear Resonant Ultrasound Spectroscopy

The objective of the study was to evaluate the ability of a nonlinear ultrasound technique, the so-called nonlinear resonant ultrasound spectroscopy (NRUS) technique, for detecting early microdamage accumulation in cortical bone induced by four-point bending fatigue. Small parallelepiped beam-shaped human cortical bone specimens were subjected to cyclic four-point bending fatigue in several steps. The specimens were prepared to control damage localization during four-point bending fatigue cycling and to unambiguously identify resonant modes for NRUS measurements. NRUS measurements were achieved to follow the evolution of the nonlinear hysteretic elastic behavior during fatigue-induced damage. After each fatigue step, a small number of specimens was removed from the protocol and set apart to quantitatively assess the microcrack number density and length using synchrotron radiation micro-computed tomography (SR-mCT). The results showed a significant effect of damage steps on the nonlinear hysteretic elastic behavior. No significant change in the overall length of microcracks was observed in damaged regions compared to the load-free control regions. Only an increased number of shortest microcracks, those in the lowest quartile, was noticed. This was suggestive of newly formed microcracks during the early phases of damage accumulation. The variation of nonlinear hysteretic elastic behavior was significantly correlated to the variation of the density of short microcracks. Our results suggest that the nonlinear hysteretic elastic behavior is sensitive to early bone microdamage. Therefore NRUS technique can be used to monitor fatigue microdamage progression in in vitro experiments.

Introduction

Among various bone quality factor, bone microdamage is certainly the less understood, as in vivo microcracks detection is still challenging. Bone microdamage is a natural phenomenon caused by daily loading [START_REF] Frost | Presence of microscopic cracks in vivo in bone[END_REF]. Microdamage manifests as linear microcracks and diffuse damage in cortical bone [START_REF] Diab | Morphology, localization and accumulation of in vivo microdamage in human cortical bone[END_REF][START_REF] Donahue | Microdamage in bone: implications for fracture, repair, remodeling, and adaptation[END_REF]. The density of linear microcrack increases significantly with age in cortical bone [START_REF] Diab | Morphology, localization and accumulation of in vivo microdamage in human cortical bone[END_REF][START_REF] Schaffler | Aging and matrix microdamage accumulation in human compact bone[END_REF][START_REF] Norman | Microdamage of human cortical bone: Incidence and morphology in long bones[END_REF][START_REF] Zioupos | Accumulation of in-vivo fatigue microdamage and its relation to biomechanical properties in ageing human cortical bone[END_REF]. While microdamage is of little consequence under normal bone self-repair capability [START_REF] Taylor | Living with cracks: Damage and repair in human bone[END_REF], microdamage accumulation following impaired repair capabilities caused by disease, age or drug absorption [START_REF] Chapurlat | Bone microdamage: a clinical perspective[END_REF][START_REF] Saito | Collagen cross-links as a determinant of bone quality: a possible explanation for bone fragility in aging, osteoporosis, and diabetes mellitus[END_REF][START_REF] Allen | Bisphosphonate effects on bone turnover, microdamage, and mechanical properties: what we think we know and what we know that we don't know[END_REF] is suspected to reduce bone biomechanical competence, including toughness [START_REF] Yeni | Fatigue damage-fracture mechanics interaction in cortical bone[END_REF][START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF], stiffness [START_REF] Yeni | Fatigue damage-fracture mechanics interaction in cortical bone[END_REF][START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF][START_REF] Boyce | Damage type and strain mode associations in human compact bone bending fatigue[END_REF][START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF][START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF][START_REF] O'brien | Bone as a composite material: The role of osteons as barriers to crack growth in compact bone[END_REF] and ultimate load [START_REF] Yeni | Fatigue damage-fracture mechanics interaction in cortical bone[END_REF][START_REF] Danova | Degradation of bone structural properties by accumulation and coalescence of microcracks[END_REF]. Such alterations may lead ultimately to an increase in fracture risk [START_REF] Zioupos | Accumulation of in-vivo fatigue microdamage and its relation to biomechanical properties in ageing human cortical bone[END_REF]. Histomorphometry is the current gold standard to characterize microdamage in vitro [START_REF] Diab | Morphology, localization and accumulation of in vivo microdamage in human cortical bone[END_REF][START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF]. However, quantitative assessment of microcracks with histomorphometry entails serial sectioning and observer intervention, which is usually time consuming. Moreover, as microcracks are relatively scarce in bone 2-D cross-sections, the statistical validity remains challenging [START_REF] Martin | On sampling bones for microcracks[END_REF].

Recently, synchrotron radiation micro-computed tomography (SR-mCT) enabled the 3-D assessment of microcracks at a microscale resolution [START_REF] Nalla | Aspects of in vitro fatigue in human cortical bone: time and cycle dependent crack growth[END_REF][START_REF] Thurner | Timelapsed investigation of three-dimensional failure and damage accumulation in trabecular bone using synchrotron light[END_REF][START_REF] Voide | The importance of murine cortical bone microstructure for microcrack initiation and propagation[END_REF][START_REF] Larrue | Synchrotron Radiation Micro-CT at the Micrometer Scale for the Analysis of the Three-Dimensional Morphology of Microcracks in Human Trabecular Bone[END_REF]. These techniques are inherently destructive and cannot be used to study microdamage in vivo.

Several measurement modalities, including positron emission tomography (PET) [START_REF] Li | Imaging bone microdamage in vivo with positron emission tomography[END_REF][START_REF] Li | Small-animal PET/CT assessment of bone microdamage in ovariectomized rats[END_REF], nuclear magnetic resonance NMR T2 relaxation time [START_REF] Nicolella | Non-destructive characterization of microdamage in cortical bone using low field pulsed NMR[END_REF] and nonlinear ultrasound [START_REF] Hoff | Diagnosis of osteoporosis using nonlinear ultrasound[END_REF][START_REF] Engan | Nonlinear Ultrasound Detection of Osteoporosis IEEE[END_REF] are currently explored to non destructively assess microdamage in living bones.

Quantitative ultrasound is widely used to assess skeletal status [START_REF] Laugier | Instrumentation for in vivo ultrasonic characterization of bone strength[END_REF]. However, the linear elastic (speed of sound) and dissipative (attenuation) parameters derived in quantitative ultrasound are not sensitive to damage [START_REF] Bennell | Acute and subacute changes in the ultrasound measurements of the calcaneus following intense exercise[END_REF][START_REF] Nicholson | Quantitative ultrasound does not reflect mechanically induced damage in human cancellous bone[END_REF][START_REF] Yeni | The effect of yield damage on the viscoelastic properties of cortical bone tissue as measured by dynamic mechanical analysis[END_REF][START_REF] Wynnyckyj | Changes in bone fatigue resistance due to collagen degradation[END_REF]. Contrary to linear acoustics, in the framework of nonlinear acoustics, the propagation velocity and the attenuation (or dissipation) of acoustic waves are amplitude dependent. Those peculiarities give rise to various phenomena called nonlinear acoustical effects. Damaged materials have proved to exhibit a characteristic nonlinear behavior that can be used to infer material mechanical integrity. Elastic nonlinear parameters derived from dynamic wave studies were found to be far more sensitive than their linear counterparts to damage in a variety of materials [START_REF] Van Den Abeele | Inferring the degradation of pultruded composites from dynamic nonlinear resonance measurements[END_REF][START_REF] Van Den Abeele | Quantification of material nonlinearity in relation to microdamage density using nonlinear reverberation spectroscopy: Experimental and theoretical study[END_REF]. This has recently motivated several research groups to adapt ultrasound-based nonlinear dynamic elastic testing methods to assess the level of microdamage in cortical or cancellous bone using different techniques. These include nonlinear ultrasonic resonant spectroscopy (NRUS) [START_REF] Muller | Nonlinear resonant ultrasound spectroscopy (NRUS) applied to damage assessment in bone[END_REF][START_REF] Muller | Microdamage assessment using non-linear resonant ultrasound spectroscopy (NRUS) techniques: A feasibility study[END_REF][START_REF] Muller | Nonlinear ultrasound can detect accumulated damage in human bone[END_REF], dynamic acousto-elasticity testing (DAET) [START_REF] Renaud | Non-linear acoustic measurements to assess crack density in trabecular bone[END_REF][START_REF] Renaud | Exploration of trabecular bone nonlinear elasticity using time-of-flight modulation[END_REF][START_REF] Moreschi | Monitoring trabecular bone microdamage using a dynamic acousto-elastic testing method[END_REF], harmonic generation [START_REF] Hoff | Diagnosis of osteoporosis using nonlinear ultrasound[END_REF][START_REF] Engan | Nonlinear Ultrasound Detection of Osteoporosis IEEE[END_REF] and nonlinear wave modulation spectroscopy [START_REF] Ulrich | Application of nonlinear dynamics to monitoring progressive fatigue damage in human cortical bone[END_REF][START_REF] Zacharias | Microdamage evaluation in human trabecular bone based on nonlinear ultrasound vibro-modulation (NUVM)[END_REF]. The advantage of these nonlinear techniques is that they are inherently non destructive and can therefore potentially be implemented in vivo [START_REF] Hoff | Diagnosis of osteoporosis using nonlinear ultrasound[END_REF][START_REF] Engan | Nonlinear Ultrasound Detection of Osteoporosis IEEE[END_REF].

In previous NRUS studies, our group has reported that, under resonance conditions, the resonance frequency of damaged femoral diaphysis was down shifted with increasing vibration amplitude. We found a correlation of progressive fatigue of human bone samples to their nonlinear dynamical response [START_REF] Muller | Nonlinear resonant ultrasound spectroscopy (NRUS) applied to damage assessment in bone[END_REF][START_REF] Muller | Microdamage assessment using non-linear resonant ultrasound spectroscopy (NRUS) techniques: A feasibility study[END_REF][START_REF] Muller | Nonlinear ultrasound can detect accumulated damage in human bone[END_REF]. Such an effect can be interpreted as a softening of the material in presence of cracks when the wave amplitude increases gradually (i.e., the modulus of the material decreases with dynamical forcing). This softening effect rises as the elastic non linearity (i.e., the level of damage) of the material increases [START_REF] Van Den Abeele | Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage, Part II: Single-mode nonlinear resonance acoustic spectroscopy[END_REF]. With the possibility to non-invasively evaluate nonlinear properties assumed to be related to microdamage accumulation, NRUS is an attractive technique to evaluate microdamage in bone. In the above mentioned NRUS studies, nonlinear ultrasonic measurements were not validated by histology nor by high resolution mCT. The measured nonlinear elastic properties could not be correlated to microdamage characteristics. Therefore the goal of this study was to assess the relationships between the nonlinear elastic parameter and microdamage characteristics on human cortical bone specimens subjected to fatigue loading with a specific focus on assessment of the sensitivity of the technique to early phase of damage accumulation.

Materials and Methods

Specimen preparation and measuring protocol

Small parallelepiped beam-shaped human cortical bone specimens were subjected to cyclic four-point bending fatigue in several steps. The specimens were prepared to control damage localization during four-point bending fatigue cycling and to unambiguously identify resonant modes for NRUS measurements. Apparent dry density (r dry ) was evaluated by measuring the specimen volume and weight. Bone specimens were dried at 37uC during one night in a climate chamber (Memmert GmbH HCP 108, Schwabach, Germany) at relative humidity 15% in the presence of desiccators. Drying and rewetting procedure does not affect bone properties as the collagen molecular structure remains intact [START_REF] Currey | The effect of porosity and mineral content on the Young's modulus of elasticity of compact bone[END_REF][START_REF] Rho | Effects of drying on the mechanical properties of bovine femur measured by nanoindentation[END_REF].

The procedure for the NRUS and mechanical studies began with the initial NRUS measurements for all specimens to determine the initial nonlinearity of the material. The specimens were then taken through a damage step, consisting of cyclic fourpoint bending as described below, during which mechanical parameters were determined. NRUS measurements were repeated after each cycling session. Four damage steps were achieved. After each damage step, three or four specimens were removed for future 3-D SR-mCT investigations of microdamage characteristics. The measurement protocol is illustrated in Fig. 1.

NRUS measurements

The principles of NRUS measurements have been extensively described elsewhere [START_REF] Van Den Abeele | Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage, Part II: Single-mode nonlinear resonance acoustic spectroscopy[END_REF]. Briefly, a piezoceramic emitter (Fuji Ceramics Corporation, Yamamiya, Japan) glued on a backload (i.e. a heavy mass compared to the specimen) was bonded at one end of the specimens to ensure free-fix boundary conditions for NRUS measurements (Fig. 2). Each specimen was excited by a swept-sine (M2i.6012, Spectrum GmbH, Grosshansdorf, Germany) encompassing the first resonant modes of the cortical beam (assumed to be pure compression modes under asymmetric loading conditions). The dynamic strain amplitude e was calculated from the longitudinal particle displacement U at one

e~d U dx ~U Ã k~U Ã 2p 4L ð1Þ
where k is the wave number and L is the specimen length. The resonance peak frequency f of the first compression mode was derived for each voltage drive level from the strain amplitude measured as a function of frequency at the corresponding excitation level. From the resonance peak data, the nonlinear parameter a f can be calculated using the following equation [START_REF] Van Den Abeele | Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage, Part II: Single-mode nonlinear resonance acoustic spectroscopy[END_REF][START_REF] Johnson | Slow dynamics and anomalous nonlinear fast dynamics in diverse solids[END_REF]:

f {f 0 f 0 ~Df f 0 ~af 2 De ð2Þ
where f is the resonance frequency at increased strain level and f 0 its corresponding value at the lowest drive amplitude [START_REF] Johnson | Slow dynamics and anomalous nonlinear fast dynamics in diverse solids[END_REF]. Eq. 2 expresses that the frequency shift Df is proportional to the peak strain amplitude De via the nonlinear elastic a f parameter. This parameter, so-called nonlinear hysteretic elastic parameter, is typical of nonlinearities that appear for strains above approximately 10 25 [START_REF] Johnson | Slow dynamics and anomalous nonlinear fast dynamics in diverse solids[END_REF] in damaged materials. It conveys information about the amount of hysteretic nonlinearity directly linked to damage accumulation in a material. The widely used NRUS measurement protocol [START_REF] Van Den Abeele | Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage, Part II: Single-mode nonlinear resonance acoustic spectroscopy[END_REF] was optimized to achieve high sensitivity. The measurement of the reference resonance frequency f 0 used to compute a f was repeated before each excitation level. In doing so, the measurements become less affected by changes in environmental conditions such as temperature and yields more precise and stable a f estimates [START_REF] Haupert | Highaccuracy acoustic detection of nonclassical component of material nonlinearity[END_REF].

During the NRUS measurements, specimens were kept at fixed temperature (37uC 60.1uC) and relative humidity (15%65%) into a climate chamber (Fig. 2)

Biomechanical testing

The piezoceramic emitter attached to the specimen for NRUS measurements was removed and the specimen was rehydrated during 48 hours before each mechanical testing. All specimens were progressively damaged by cyclic four-point bending at 2 Hz under load control in a saline solution at 37uC( 61uC) using a hydraulic testing machine (INSTRON, 8802, High Wycombe, England) with a 1 kN loading cell (accuracy 0.5%) and the internal displacement transducer (accuracy 1%). The specific four-point bending assembly composed of 6.35 mm diameter roller-bearings with a 40 mm outer span and a pivoting 20 mm inner span minimizes the formation of grooves under the rollers [START_REF] Griffin | Artifactual nonlinearity due to wear grooves and friction in four-point bending experiments of cortical bone[END_REF][START_REF] Astm | Standard Test Method for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials by Four-Point Bending: D6272[END_REF]. In this configuration, damage is expected to occur specifically in the mid region of the sample [START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF], while the distal regions remaining intact may be used as control. Initial Young's modulus was determined during pre-cycling after 20 cycles by measuring the slope of the linear portion of the last load-displacement curve.

From the initial Young's modulus, the load (Fmax) corresponding to 5000me (i.e. an initial strain rate of 20000me/s) at the midspan was computed for all specimens [START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF]. The four-point bending fatigue was then applied between -10N and -Fmax. During the cycling session, the load and displacement curves were recorded to extract the linear elastic beam theory (LEBT) modulus (E LEBT )a s defined by Landrigan [START_REF] Landrigan | Systematic error in mechanical measures of damage during four-point bending fatigue of cortical bone[END_REF]. E LEBT is a combination of elastic (secant modulus) and plastic (residual strain) biomechanical parameters. After each damage step, the E LEBT modulus is normalized by the initial value measured for the first loading cycle of the first damage step. E LEBT has been shown to decrease as bone microdamage accumulates [START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF][START_REF] O'brien | Bone as a composite material: The role of osteons as barriers to crack growth in compact bone[END_REF][START_REF] Arlot | Microarchitecture influences microdamage accumulation in human vertebral trabecular bone[END_REF]. The progressive damage was performed in four steps (one step = one cycling session), each step was defined by multi-criteria: 10% decrease of the E LEBT or pre-determined number of cycles ( = 6000) or anomalous E LEBT decreasing speed. This multi-criteria definition of each step was chosen to achieve progressive damage accumulation and to avoid specimen failure before the end of the fourth step.

3-D synchrotron radiation mCT (SR-mCT)

At the end of each step, a subset of 3 or 4 bone specimens was measured by SR-mCT at the European Synchrotron Radiation Facility in Grenoble, France. Two different reconstructed volumes of interest (VOI) were investigated. VOI1, located in the load-free region at one distal end of the sample, outside of the roller-support, was assumed to be free of damage (except for pre-existing initial pre-fatigue damage). VOI2, located in the central portion of the beam, is the region where most microdamage was assumed to accumulate during mechanical fatigue. The photon energy was 25 KeV and the size of the VOI was 2.862.861.96mm 3 with a voxel size of 1.4 mm 3 . A set of 2500 projections were acquired at an angular step of 0.144u. The 20486204861400 3D images were reconstructed using a filtered back projection algorithm and the contrast was linearly rescaled to an 8-bits dynamic to save memory storage. For microdamage characterization, the size of the investigated volume of both VOIs was reduced to 2.262.061.96 mm 3 .

Microdamage is generally characterized on transverse or longitudinal 2-D sections by conventional or epifluorescence microscopy [START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF][START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF]. In this study, microdamage characteristics were quantitatively assessed in cortical bone volumes reconstructed from SR-mCT data. To this end, each VOI was sampled by eleven 2-D transverse cross-sections regularly spaced with an interval of 180mm (Fig. 3). Each cross-section was obtained by averaging a stack of 18 adjacent 1.4mm-thick slices in order to achieve transverse cross-section images with a depth of field of 25 mm equivalent to that achieved with epifluorescence microscopy. The averaging process has also the advantage to decrease the noise level and to improve to contrast between the microcracks and the bone matrix.

The surface microcrack density (Cr.Dn [#.mm 2 ]) and microcrack length (Cr.Le [mm]) were measured using the software ImageJ (NIH, USA) with the plugin NeuronJ [START_REF] Meijering | Neuron Tracing in Perspective[END_REF]. The bone surface was computed as the total area of bone section, including the pores (Haversian and Volkmann canals, and resorption cavities) as it is reported in the literature [START_REF] Chapurlat | Bone microdamage: a clinical perspective[END_REF][START_REF] Lee | Sequential labelling of microdamage in bone using chelating agents[END_REF]. The pores appeared as dark pixels and were clearly evidenced as one peak of the bimodal gray level histograms of the image. Thus they could easily be separated from bone tissue using a thresholding method to keep only low grey level pixels, with a threshold set to 100 (arbitrary unit) in the range [0-255] according to the histogram. The porosity value corresponds to the number of segmented pixels (pores) over the total surface of cortical bone including pores (in pixels).

Microcracks characteristics were determined first by including all microcracks. In a second step, microcracks fully embedded within the bone volume were processed separately from microcracks leading to the surface as they are mainly artifactual microcracks formed during the preparation process [START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF][START_REF] Arlot | Microarchitecture influences microdamage accumulation in human vertebral trabecular bone[END_REF][START_REF] Lee | Detecting microdamage in bone[END_REF].

Data analysis

Matlab 7.8 with statistics toolbox 7.6 (Mathworks, Natick, MA, USA) was used for statistical analyses. A non-parametric one-way analysis of variance (ANOVA) dedicated to repeated measure-ments (Friedman Test) followed by post hoc multiple comparison (Nemenyi test) was applied to test whether the levels of nonlinear elasticity achieved at each steps of the fatigue protocol were statistically significantly different. This analysis was performed on two subsets of the total set of specimens (N = 14). Group 1 (N = 8) includes all specimens having undergone the first three damage steps and Group 2 (N = 4) includes all specimens having undergone all the four damage steps. The effect of fatigue loading on microdamage characteristics (Cr.Dn and Cr.Le) in the control region and the damaged region was investigated with a nonparametric Wilcoxon signed rank test for all specimens (N = 14). The relationship of the nonlinear elastic parameter a f with microdamage characteristics was obtained by a Spearman correlation test taking into account all specimens (N = 14) as well as after removing the outlier (N = 13). P-values less than 0.05 were considered significant.

Results

Individual results for mechanical and NRUS testing are presented in Table 1. In Table 2, we report only the microdamage characteristics for fully embedded microcracks as the results were similar when all microcracks were included in the analysis or after excluding those leading to the specimen surface.

Biomechanical testing

The variability of E LEBT has been assessed on five dedicated specimens that were not included in the protocol. They went through 20 cycles (after system stabilization) of four-point bending test. The process was repeated 6 times with repositioning. The coefficient of variation (standard deviation/mean) was found to be 6.3% for E LEBT .

The mechanical characteristics (Table 1) were found to vary between the specimens. The average initial E LEBT modulus was 15.163.0 GPa the average apparent dry density r dry was 17926155 g.mm 23 and the average porosity was 12.065.2.

No significant trends could be observed between stopping criteria and the other measured variables (nonlinearity or microcracks characteristics).

Ultrasonic (NRUS) measurements

The measurement precision of NRUS, assessed by the coefficient of variation of three measurements with intermediate debonding of the piezo-electric source and repositioning, was found to be 8.5% for a f . The initial nonlinear values of a f ranged between 24.3 and 26.2, except for one highly nonlinear specimen (a f = 215.5). The average initial a f value was 25.862.9. After the last damage step (step 1 to 4 depending on the specimen), a f values ranged between 24.7 and 216.2.

On average, the nonlinear parameter a f increased with the number of fatigue steps. However, a disparity could be observed between the specimens.

A significant effect of fatigue on a f was measured for both groups (Group 1: p,0.05, no F value due to the number of steps less than four; Group 2, p = 0.01, F = 12.6). Group 1 including the eight specimens having undergone the first three stages of damage is represented in Fig. 4A. Group 2 including the four specimens having undergone all the four damage steps is represented in Fig. 4B. The result of the post-hoc comparison evidenced statistically significant variations of a f between damage steps for both groups except between the initial and first step and between the third and last step for specimens of the Group 2.

Finally, note that no correlation was found between the initial nonlinear elastic parameter a f or its variation Da f /a f and the initial elastic modulus E LEBT , the total number of cycles, the apparent density r dry nor the sample porosity.

Microtomography

The total number of microcracks found by pooling the data of all the specimens and both VOIs was 4106 with 1380 microcracks leading to the specimen surface and 2726 microcracks fully embedded within the bone volume.

The difference in microdamage characteristics between VOI1 (control region) (Fig. 5A-B) and VOI2 (damage region) (Fig. 5C-D In contrast, from the examination of the length distribution of microcracks, it appeared that fatigue cycling resulted in an increase of the density of the shortest microcracks, i.e., those in the first quartile (Q1), whereas the quantity of longer microcracks remained unchanged. The density Cr.Dn.Q1 of the shortest microcracks was significantly different between VOI1 and VOI2, both when all microcracks were included (VOI1: Cr.Dn.Q1 = 0.7760.38#/mm 2 and VOI2: Cr.Dn.Q1 = 1.0660.60#/mm 2 ; p,0.05) or when microcracks leading to the surface were excluded (VOI1: Cr.Dn.Q1 = 0.3860.26#/mm 2 and VOI2: Cr.Dn.Q1 = 0.6460.47#/mm 2 ;p,0.01) (Fig. 6B). The mean length Cr.Le.Q1 corresponding to the upper limit of the first quartile was found to be 32.668.2mm. Such a trend for short microcracks could not be observed for microcracks leading to the specimen surface (Fig. 6A).

The relative variation between VOI2 and VOI1 of density of short microcracks embedded within the bone matrix DCr.Dn.Q1/ Cr.Dn.Q1 is plotted against the relative variation of the nonlinear parameter Da f /a f in Fig. 7. No significant correlation was found taken into account all the specimen (N = 14). However, when the outlier (specimen #6 exhibiting the strongest a f variation) was

Discussion

This is the first study reporting the nonlinear elastic hysteretic behavior (assessed by the nonlinear elastic parameter a f ) and microdamage characteristics derived from SR-mCT, concurrently assessed in calibrated human cortical bone specimens during a four point-bending fatigue cycling protocol. By repeating the NRUS measurement protocol after each damage step, we were able to monitor the evolution of the nonlinear behavior during progressively induced mechanical damage, each specimen being its own control. We observed that the damage steps had a statistically significant effect on the nonlinear hysteretic parameter a f measured by NRUS. This suggests that the parameter a f is a sensitive marker to bone microdamage induced by ex vivo mechanical fatigue test. These results confirm the seminal observations made in previous studies by our group [START_REF] Muller | Nonlinear resonant ultrasound spectroscopy (NRUS) applied to damage assessment in bone[END_REF][START_REF] Muller | Microdamage assessment using non-linear resonant ultrasound spectroscopy (NRUS) techniques: A feasibility study[END_REF][START_REF] Muller | Nonlinear ultrasound can detect accumulated damage in human bone[END_REF]. The present study brings a new insight by unraveling the link existing between the nonlinear elastic behavior and some damage characteristics, the latter being derived from mCT volumetric imaging. Our result evidenced that the increase of elastic non linearity was related to an increase of the density of the shortest microcracks embedded within the bone matrix, i.e., those in the first quartile.

Origin of the elastic nonlinearity in bone

The nonlinear hysteretic elasticity (a f ) of human cortical bone samples was measured in this study, allowing the comparison with other materials. Here, the nonlinear hysteretic elasticity typical of damaged materials is measured, as opposed to the classical elastic non linear response which exists in most materials, including undamaged solids, due to intrinsic anharmonicity of, for instance, the crystalline lattice-atomic level vibration.

The initial (pre-fatigue) nonlinear value (a f = 25.862.9) is consistent with the value previously reported for undamaged cortical bovine bone (a f = 25.062.5) [START_REF] Haupert | Highaccuracy acoustic detection of nonclassical component of material nonlinearity[END_REF]. This value is weak (ten times lower than that of intact polycrystal metals [START_REF] Haupert | Highaccuracy acoustic detection of nonclassical component of material nonlinearity[END_REF] or hundred times lower than for rocks [START_REF] Johnson | Slow dynamics and anomalous nonlinear fast dynamics in diverse solids[END_REF] but not null, meaning that human cortical bone specimens exhibit low hysteretic elasticity behavior in pre-fatigue (native) configuration. Even after the progressive damage, the nonlinear behavior remains low when compared to other materials.

There are multiple sources of hysteretic nonlinearity in materials. Hysteresis in the dynamic strain-stress relationship is known to be produced by micro-friction, micro-adhesion and clapping due to presence of soft micro-structural features at different scales such as microcracks [START_REF] Guyer | Nonlinear mesoscopic elasticity: Evidence for a new class of materials[END_REF] or dislocations [START_REF] Granato | Theory of mechanical damping due to disclocations[END_REF]. Such process could be at the origin of the observed pre-and postdamage bone nonlinearities. The sources are not known and were beyond the scope of this study. As bone is a hierarchical material [START_REF] Weiner | The material bone: structure-mechanical function relations[END_REF], the process could take place at different level of the hierarchical structure:

(i) at the nano-scale level, debonding of collagen fibers [START_REF] Ritchie | Plasticity and toughness in bone[END_REF] could generate nonlinear effect as it is well known in fiber composite materials [START_REF] Van Den Abeele | Quantification of material nonlinearity in relation to microdamage density using nonlinear reverberation spectroscopy: Experimental and theoretical study[END_REF]. Stick-slip friction between collagen fibrils and nanocrystals [START_REF] Gupta | Nanoscale deformation mechanisms in bone[END_REF][START_REF] Tai | Nanogranular origins of the strength of bone[END_REF][START_REF] Buehler | Molecular nanomechanics of nascent bone: fibrillar toughening by mineralization[END_REF][START_REF] Gupta | Fracture of bone tissue: The 'hows' and the 'whys[END_REF] could also be a source of a hysteretic elastic behavior; (ii) at the micro-scale level, nonlinear behavior could have its origin in the cement line sliding or osteon pull-out [START_REF] Behiri | Crack velocity dependence of longitudinal fracture in bone[END_REF]; (iii) at the meso-scale level, micro and macrocracks might be the main structure generating nonlinear acoustic phenomena.

This list is not exhaustive but opens up about the multiplicity of factors behind the nonlinear hysteretic elastic behavior of cortical bone. As it stands, we cannot draw any conclusion about the origin of the native (here, ''native'' means ''prior to any damage step'') nonlinear behavior of cortical bone. It may have its origin in damage, microcrack or diffuse type, native or induced by sample preparation. Such damage is observed by measuring the control regions located at the ends of the specimens. As for other potential sources of damage (dislocation, delamination, slip osteons, etc.), it would be necessary to assess their magnitude and explore them by dedicated experiments. However, as each sample is its own control, we can attribute unambiguously the increase in the nonlinear hysteretic behavior after fatigue to damage accumulation.

Besides, water saturation, capillarity effects and fluid flow pressure may also play a role as in rocks [START_REF] Van Den Abeele | Influence of water saturation on the nonlinear elastic mesoscopic response in Earth materials and the implications to the mechanism of nonlinearity[END_REF] and concrete [START_REF] Payan | Effect of water saturation and porosity on the nonlinear elastic response of concrete[END_REF] by modifying bone nonlinear elasticity, viscoelasticity and relaxation properties. For this reason, a particular attention was given to keep the samples at the same relative humidity (e.g. 15%65%) during each NRUS experiment.

Pre-existing microdamage (VOI1)

The density of microcracks (Cr.Dn) embedded within undamaged bone volumes (1.9861.15#/mm 2 in VOI1) is one order of magnitude larger compared to the values found in most studies on human cortical femur (0.2160.16#/mm 2 [START_REF] Norman | Microdamage of human cortical bone: Incidence and morphology in long bones[END_REF], 0.2160.21#/mm 2 [START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF], 0.1560.16#/mm 2 [START_REF] Wasserman | In vivo linear microcracks of human femoral cortical bone remain parallel to osteons during aging[END_REF], 0.160.06#/mm 2 [START_REF] Landrigan | Contrast-enhanced micro-computed tomography of fatigue microdamage accumulation in human cortical bone[END_REF]). Only two studies have shown comparable Cr.Dn ranging between 1#/mm 2 to 5#/mm 2 for women older than 45 years old and higher than 1.5#/mm 2 for human femur older than 70 years old [START_REF] Schaffler | Aging and matrix microdamage accumulation in human compact bone[END_REF][START_REF] Zioupos | Accumulation of in-vivo fatigue microdamage and its relation to biomechanical properties in ageing human cortical bone[END_REF]. However, one has to remind that microcracks density increases with age [START_REF] Diab | Morphology, localization and accumulation of in vivo microdamage in human cortical bone[END_REF][START_REF] Schaffler | Aging and matrix microdamage accumulation in human compact bone[END_REF][START_REF] Norman | Microdamage of human cortical bone: Incidence and morphology in long bones[END_REF][START_REF] Zioupos | Accumulation of in-vivo fatigue microdamage and its relation to biomechanical properties in ageing human cortical bone[END_REF][START_REF] Wasserman | In vivo linear microcracks of human femoral cortical bone remain parallel to osteons during aging[END_REF] and that the age of the donors in our study (88.569.8) is generally higher than in the above mention studies. This being said, even if we discarded microcracks leading to the surface, we cannot exclude that crack density can be artificially augmented by some artifactual microcracks formed during the preparation process of the specimens, especially those located 500mm from cutting edge [START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF][START_REF] Arlot | Microarchitecture influences microdamage accumulation in human vertebral trabecular bone[END_REF].

Factors other than age of donors and artifactual microcracks, however, can explain the difference between studies. Indeed, microcrack density may depend on the technique used to assess microdamage characteristics [START_REF] Schaffler | Examination of compact bone microdamage using back-scattered electron microscopy[END_REF][START_REF] Huja | Development of a fluorescent light technique for evaluating microdamage in bone subjected to fatigue loading[END_REF]. For example, the number of microcracks counted by epifluorescence microscopy [START_REF] Huja | Development of a fluorescent light technique for evaluating microdamage in bone subjected to fatigue loading[END_REF][START_REF] Lee | Fluorescence-aided detection of microdamage in compact bone[END_REF] or by backscatter scanning electron (BSE) microscopy [START_REF] Schaffler | Examination of compact bone microdamage using back-scattered electron microscopy[END_REF] could be up to twice the number measured by conventional microscopy based on basic Fuchsin dye. In this study, cortical bone microdamage quantitative assessment was done by SR-mCT. The contrast, resolution and depth of field of SR-mCT images differ from those of conventional optical microscopy-based histomorphometry approaches, which may affect differently the detection of microcracks and their characteristics. A face-to-face comparison between the different techniques would be warranted to provide an answer to this issue.

Induced microdamage (VOI2)

When we considered all the detected microcracks, leading to the surface of the specimens and embedded within the bone matrix, we found no significant change of microcrack density and length. This is at odds with several previous studies, although once again, the difference in technologies used in different studies to quantitatively assess microdamage should be emphasized, namely, X-ray micro-tomography in our study versus optical-based microscopy in previous reports. Several studies report that three or four-point bending fatigue tests on calibrated cortical bone specimens induced the progression of microdamage by increasing microcrack density and length in human [START_REF] Boyce | Damage type and strain mode associations in human compact bone bending fatigue[END_REF][START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF] or bovine bone [START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF]. However there is no consensus on bone microdamage induced by in vitro mechanical fatigue tests. For example, a recent experiment dedicated to four-point bending test on bovine cortical bone, showed no microcrack density variation but only an increase of their average length (control: Cr.Le = 41mm622mm/loaded: Cr.Le = 108mm663mm) [START_REF] Landrigan | Systematic error in mechanical measures of damage during four-point bending fatigue of cortical bone[END_REF]. In another study on whole canine femur with comparable experimental protocol [START_REF] Burr | Does microdamage accumulation affect the mechanical properties of bone?[END_REF], a modulus loss threshold was observed, i.e. microcracks accumulation started when loading modulus loss exceeds 15%. Moreover, the average microcrack length was not significantly different between control and cyclically loaded specimens, as in the present study. was suggested that damage initiates at tissue level as nanodamage before being visible as microcracks [START_REF] Burr | Does microdamage accumulation affect the mechanical properties of bone?[END_REF]. However, numerous studies suggest that rather than a continuum between diffuse damage and microcracks, both types of damage are different events. Diffuse damage is generally found to be mainly created in response to tensile stress [START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF][START_REF] Boyce | Damage type and strain mode associations in human compact bone bending fatigue[END_REF] and in young bone [START_REF] Diab | Age-related change in the damage morphology of human cortical bone and its role in bone fragility[END_REF] whereas microcracking occurs preferentially under compressive stress [START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF] and in old bone [START_REF] Diab | Age-related change in the damage morphology of human cortical bone and its role in bone fragility[END_REF].

When the microcracks leading to the specimen surface were analyzed separately, the density of short microcracks was not statistically significantly different between VOI1 (control region) and VOI2 (damage region). This is in contrast with the results obtained when we considered fully embedded microcracks only. The reason for this discrepancy remains unclear. One possibility is that pre-existing and/or preparation microcracks leading to the specimen surface release stress concentration, thus preventing significant growth of microdamage at the periphery of the specimens. On the contrary, microcracking process, as an essential mechanism inside bone specimen to release stress concentration, may explain why small microcracks density increases. Further experiments are required to elucidate this issue.

Microcracking process is well known in fiber-reinforced composite materials as the so-called three-phase modulus degradation curve [START_REF] Reifsnider | Fracture of fatigue-loaded composite laminates[END_REF]. The evolution of bone microdamage characteristics (density and length) throughout modulus degradation was observed for the first time in bovine tibia during tension fatigue cycling [START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF][START_REF] O'brien | Bone as a composite material: The role of osteons as barriers to crack growth in compact bone[END_REF]. O'Brien observed that the formation of new microcracks was initiated during an early phase of damage, which confirmed the above mentioned hypothesis. New microcracks grew up to reach a maximal length of about 100mm mainly due to cement lines acting as barriers [START_REF] Presbitero | Distribution of microcrack lengths in bone in vivo and in vitro[END_REF], while only 6% of the native microcrack propagated. Thus, the level of damage depends on the phase of the modulus degradation curve. In early stages of fatigue, damage first manifests by short microcracks, confined to interstitial bone tissue, releasing and redistributing local stress in order to enhance fatigue life, as it was suggested by Sobelman [START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF].

Our results seem to be in line with the progressive growing of microdamage described in [START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF][START_REF] O'brien | Bone as a composite material: The role of osteons as barriers to crack growth in compact bone[END_REF]. Indeed, we observed a doubling of the shortest microcracks density Cr.Dn.Q1 (those in the first quartile) without an increase of the global microcracks density, neither an expansion of the average length of the preexisting microcracks. It is likely that our fatigue cycling protocol was not sufficiently strong to increase the average length of the pre-existing microcracks. Nevertheless it was sufficient to lead to the formation of new short microcracks, revealed by the increase of Cr.Dn.Q1. This suggests that the final damage state of our specimens remained low compared to the previous studies [START_REF] Diab | Effects of damage morphology on cortical bone fragility[END_REF][START_REF] Boyce | Damage type and strain mode associations in human compact bone bending fatigue[END_REF][START_REF] Sobelman | Do microcracks decrease or increase fatigue resistance in cortical bone?[END_REF][START_REF] Landrigan | Systematic error in mechanical measures of damage during four-point bending fatigue of cortical bone[END_REF] The significant correlation found between the variation of a f and the variation of Cr.Dn.Q1 is suggestive of the sensibility of the nonlinear hysteretic elastic parameter to newly formed microcracks in early phases of bone damage. However we cannot reject an effect of early diffuse or nanodamage to hysteretic nonlinearity as this damage was not investigated, neither the initiation and growing of a single macrocrack (i.e. known to produce large nonlinear elastic behavior) that was not contained by the microcracking process.

Conclusion

Altogether our results evidence:

i)

an increased number of short microcracks in damaged regions compared to the load-free (control) regions. These shortest microcracks, with length in the first quartile, are suspected to be newly formed microcracks as a result of fatigue cycling [START_REF] O'brien | Microcrack accumulation at different intervals during fatigue testing of compact bone[END_REF]; ii) a significant effect of damage steps on the nonlinear hysteretic elastic parameter a f . iii) a significant relationship between the relative variation of nonlinear elasticity and the relative variation of the density of newly formed microcracks.

The hysteretic nonlinear parameter (a f ) is sensitive to early bone microdamage. Our results suggest that NRUS can be used to monitor fatigue microdamage in in vitro experiments. The ability to non invasively quantitatively assess microdamage accumulation in living bones would represent an important step to improve our understanding of skeletal status. However, several scientific and technical problems have to be solved first, such as the adaptation of the nonlinear ultrasound techniques to in vivo measurement requirements.

  Fourteen human cortical bone specimens were prepared from the femoral mid-diaphysis of four female donors (78, 80, 98, 98 years old). Ethical approval for the collection of samples was granted by the Human Ethics Committee of the Centre du don des Corps at the University Paris Descartes (Paris, France). The tissue donors or their legal guardians provided informed written consent to give their tissue for investigation, in accord with legal clauses stated in the French Code of Public Health. The specimens were wet machined (Isomet 4000, Buehler GmbH, Du ¨sseldorf, Germany) as parallelepiped beams (50*4*2 mm), defatted[START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF] and stored at 220uC until experiments.

Figure 1 .

 1 Figure 1. Diagram illustrating the experimental protocol. doi:10.1371/journal.pone.0083599.g001

Figure 2 .

 2 Figure 2. NRUS experimental setup. Bone specimen bonded on a piezoceramic emitter is placed in a climate chamber. doi:10.1371/journal.pone.0083599.g002

Figure 3 .

 3 Figure 3. Diagram illustrating the process leading to an equivalent histomorphometric 2-D transverse cross-section image from 3-D reconstructed bone volumes acquired by SR-mCT. doi:10.1371/journal.pone.0083599.g003

  ) did not reach statistical significance when all microcracks were included (VOI1: Cr.Le = 71.7624.0mm; Cr.Dn = 3.1161.51 #/ mm 2 and VOI2: Cr.Le = 69.6625.7mm; Cr.Dn = 3.3361.58#/ mm 2 ) nor after excluding from the analysis, microcracks leading to the specimen surface (VOI1: Cr.Le = 72.6613.0mm; Cr.Dn = 1.9861.15#/mm 2 and VOI2: Cr.Le = 65.7610.8mm; Cr.Dn = 2.3061.22#/mm 2 ). Damage characteristics of microcracks fully embedded in bone are summarized in Table 2. Note the large inter-specimen variability of Cr.Dn ranging between 0.23#/mm 2 to 3.88#/mm 2 (VOI1) and between 0.25#/mm 2 to 4.59#/mm 2 (VOI2).

Figure 4 .

 4 Figure 4. Box plot of the nonlinear elastic parameter a f after each damage step. (A) Group 1 (N = 8) specimens having undergone the first three stages of damage; (B) Group 2 (N = 4) specimens having undergone the four damage steps. (!) means no significant effect of fatigue on parameter a f between two steps (p.0.05). doi:10.1371/journal.pone.0083599.g004

Figure 5 .Figure 6 .

 56 Figure 5. Example of a 2-D transverse cross-section extracted from the (A-B) unloaded region (VOI1) and (C-D) loaded region (VOI2) of the specimen #13. Red arrows point microcracks. Figures B and D are a zoom of the figures A and C respectively. doi:10.1371/journal.pone.0083599.g005

Figure 7 .

 7 Figure 7. Correlation between the relative variation Da f /a f of the nonlinear elastic parameter (a f represents the difference between the initial value and the value measured after the last damage step) and the relative variation of short microcracks density DCr.Dn.Q1/Cr.Dn.Q1 between VOI1 and VOI2. One specimen (#6; DCr.Dn.Q1/Cr.Dn.Q1 = 0.36; Da f /a f = 2.48) exhibiting the strongest a f variation is not represented doi:10.1371/journal.pone.0083599.g007

Table 1 .

 1 Characteristics of the human cortical bone specimens: density (r dry) , initial mechanical modulus LEBT (E LEBT ), number of damage steps, number of cycles, initial, intermediate and final nonlinear elastic parameters (a f ).Total num of cycles a f initial a f step 1 a f step 2 a f step 3 a f step 4

		Architectural		Mechanical (wet)		Nonlinear ultrasonic parameters (dry)
				initial E LEBT	final E LEBT
	Specimens	r dry [g/cm3] Porosity [%]	[KPa]	[KPa]
	#1	1753	11.4	15215	nm	155	26	26.4
	#2	1897	13.2	17069	nm	120	24.8	24.7
	#3	1870	6.3	17294	16027	906	215.5	-15.3
	#4	1865	10.3	17382	12583	2100	24.6	25.9	25.8

Table 2 .

 2 Characteristics of microcracks embedded within the bone matrix in the control (VOI1) and fatigue-loaded (VOI2) volumes.

		Cr.Dn [#/mm 2 ]		Cr.Le [mm]		Cr.Dn.Q1 [#/mm 2 ]	Cr.Le.Q1 [mm]
	Specimens	VOI1	VOI2	VOI1	VOI2	VOI1	VOI2	VOI1	VOI2
	#1	3.88	3.03	80.4	68.1	0.73	0.98	35.8	34.8
	#2	3.16	2.76	64.1	65.4	0.72	0.6	27.8	34
	#3	1.53	1.58	58.4	56.2	0.19	0.41	23.3	25.6
	#4	1.97	1.77	69.7	69	0.35	0.53	31.5	31.8
	#5	0.9	1.84	76.8	62.9	0.19	0.88	45.2	34.2
	#6	3.17	2.84	85.9	70.6	0.67	0.91	38	36.3
	#7	2.13	4.14	78.5	57.1	0.38	1.96	41.4	32.6
	#8	0.36	0.44	99.5	72.5	0.08	0.18	46.7	42.5
	#9	3.23	3.11	77.2	72.9	0.65	0.74	32.2	30.8
	#10	2.6	4.59	73.1	95.5	0.43	0.45	31.3	31.6
	#11	2.49	2.01	55.4	60.2	0.61	0.41	26.2	27.6
	#12	0.23	0.25	51.1	54.4	0.02	0	17.3	-
	#13	1.06	1.9	80.4	53.5	0.12	0.33	28.8	24.4
	#14	1.08	1.99	66	61.6	0.16	0.61	30.3	27.9
	Mean	1.98	2.3	72.6	65.7	0.38	0.64	32.6	31.8
	Std Dev	1.15	1.22	13	10.8	0.26	0.47	8.2	4.9
	Median	2.05	2	74.9	64.1	0.36	0.56	31.4	31.8

Cr.Dn and Cr.Le correspond to the microcracks density and their average length respectively. Cr.Dn.Q1 and Cr.Le.Q1 corresponds to the microcracks density and the average length of short microcracks, i.e. with length in the first quartile of each sample. doi:10.1371/journal.pone.0083599.t002

PLOS ONE | www.plosone.org

January 2014 | Volume 9 | Issue 1 | e83599

Acknowledgments

The authors are grateful to G Renaud and J Rivie `re (Laboratoire d'Imagerie Parame ´trique, Paris, France) for their helpful comments and suggestions during the preparation of the experiments, as well as to PA Johnson (Los Alamos National Laboratory, NM, USA) for fruitful discussions. The authors acknowledge the support of the ESRF for providing beamtime on beamline ID19 for synchrotron micro-CT imaging in the context of the Long Term Project (LTP) MD431, and to E Boller, M Zuluaga and M Langer for help during the experiments.

This research was supported by the Agence Nationale pour la Recherche (ANR), France (Grant BONUS_07BLAN0197). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Author Contributions


Karthik Ram Ramakrishnan
email: karthik.ramakrishnan@adfa.edu.au

Sandra Guérard

Philippe Viot

Krishna Shankar

Effect of block copolymer nano-reinforcements on the low velocity impact response of sandwich structures

Keywords:

Sandwich composites with fibre reinforced plastic (FRP) facesheets have emerged as a major class of lightweight structural materials in a wide range of engineering fields including aerospace, automotive and marine structures. This is due to attractive mechanical properties such as high specific stiffness and high strength. However, sandwich structures are susceptible to damage caused by impact. The objective of this paper is to evaluate the dynamic response of sandwich composites based on Kevlar fibre reinforced epoxy and Rohacell Ò foam. The improvement in impact performance of these sandwich structures that can be achieved by the addition of nanoparticles in the resin matrix is investigated. Nanostrength Ò , an acrylate triblock copolymer that self-assembles in the nanometer scale is added to the epoxy matrix. The effect of the nano-reinforcements on flat sandwich plates under low velocity impact is investigated at different scales. An instrumented drop tower setup is used for the low velocity impact tests of the sandwich plates with neat or nano-reinforced epoxy matrix, at different energies. The macroscopic response of the sandwich structure and the microscopic phenomena involved in dissipating the impact energy are identified and compared for sandwich plates with and without nanoparticles.

Introduction

The sandwich structure is based on a simple construction of two thin high strength facesheets bonded to either side of a thick low density core such as foams and honeycombs. This provides a lightweight structure with high stiffness. The skins are designed to resist tensile and compressive stresses and are usually made of aluminium or fibre reinforced polymers. The core is designed to resist compression and shear stresses and is usually made of wood, polymer foams, or expanded metal or polymer honeycombs. Sandwich composites with fibre reinforced plastic (FRP) facesheets and foam cores have emerged as a major class of lightweight structural materials in a wide range of engineering fields including aerospace, automotive and marine structures. One of the main drawbacks of these high performance structures is their relatively poor resistance to impact loading [START_REF] Bhuiyan | Low-velocity impact response of sandwich composites with nanophased foam core and biaxial braided face sheets[END_REF]. Impact damage in sandwich structures can be caused by tool drops, runway debris, bird strikes, hailstorms or ballistic loading. The damage caused by low-velocity impact may result in drastic reduction of stiffness and residual strength of the sandwich composite [START_REF] Wang | Experimental and numerical study on the lowvelocity impact behavior of foam-core sandwich panels[END_REF]. Ballistic impacts cause localized damage which is clearly visible on inspection while low-velocity impacts involve long contact time between impactor and target which result in global structural deformation with internal damage at points far from the contact region [START_REF] Bhuiyan | Low-velocity impact response of sandwich composites with nanophased foam core and biaxial braided face sheets[END_REF]. Extensive delamination and core damage were observed in specimens with no visible surface damage [START_REF] Bernard | Impact resistance of composite sandwich plates[END_REF], indicating the importance of studying the low velocity impact response of sandwich structures.

Richardson and Wisheart [START_REF] Richardson | Review of low-velocity impact properties of composite materials[END_REF] identified low-velocity impact as impacts where the contact duration is long enough for the entire structure to respond to the impact load. The upper limit of what constitutes a low velocity impact has been defined by different researchers as either under 10 m/s or impact speeds up to 100 m/s [START_REF] Richardson | Review of low-velocity impact properties of composite materials[END_REF]. The low-velocity impact damage of sandwich plates is typically tested using a drop tower test facility [START_REF] Anderson | Experimental investigation of low-velocity impact characteristics of sandwich composites[END_REF]. Many researchers have conducted low velocity impact tests on sandwich panels composed of different facesheets and core materials [START_REF] Gustin | Low velocity impact of sandwich composite plates[END_REF][START_REF] Cantwell | A comparative study of the mechanical properties of sandwich materials for nautical construction[END_REF][START_REF] Hazizan | The low velocity impact response of foam-based sandwich structures[END_REF][START_REF] Mines | Low velocity perforation behaviour of polymer composite sandwich panels[END_REF][START_REF] Xia | Work on low-velocity impact properties of foam sandwich composites with various face sheets[END_REF]. There has been considerable effort to improve the impact resistance of sandwich structures [START_REF] Dvorak | Protection of sandwich plates from low-velocity impact[END_REF][START_REF] Hosur | Processing of nanoclay filled sandwich composites and their response to low-velocity impact loading[END_REF][START_REF] Avila | Nano-structured sandwich composites response to low-velocity impact[END_REF][START_REF] Reis | Impact response of sandwich composites with nano-enhanced epoxy resin[END_REF]. Dvorak and Suvorov [START_REF] Dvorak | Protection of sandwich plates from low-velocity impact[END_REF] investigated the effect of placing a ductile interlayer between the facesheet and the foam core. The brittle nature of the epoxy matrix and its lack of resistance to crack growth is one of the limiting factors on the impact performance of the FRP sandwich structures [START_REF] Bagheri | Rubber-toughened epoxies: a critical review[END_REF]. Epoxy resins are one of the most commonly used resins because of their properties, such as thermal stability, mechanical response and low density [START_REF] Denneulin | The influence of acrylate triblock copolymer embedded in matrix on composite structures responses to lowvelocity impacts[END_REF]. Epoxy resins are the matrix material for glass-, Kevlar-and other fibre reinforced composites in many applications. Shih et al. [START_REF] Shih | Instrumented impact testing of composite sandwich panels[END_REF] reported that the impact failure mechanism of sandwich panels containing less tough facesheets was found to change from facesheet-dominated to foamcore-dominated behaviour. Attempts to reduce matrix damage and improve the interlaminar fracture toughness of thermoset resins has involved incorporating plasticizing modifiers, or adding rubber or thermoplastic particles to the resin. Many researchers have investigated various methods to toughen the matrix with additives such as rubber [START_REF] Bagheri | Rubber-toughened epoxies: a critical review[END_REF][START_REF] Garg | Failure mechanisms in toughened epoxy resins -a review[END_REF]. However, the drawback of this approach is that it sacrifices the stiffness of epoxy significantly and decreases the glass transition temperature. One of the proposed solutions is the modification of the thermoset resins with nano-sized organic and inorganic particles. Carbon nanotubes (CNT), carbon nanofibres and nanoclays were identified as potential nano-scale materials for the reinforcement of epoxy [START_REF] Njuguna | Epoxy-based fibre reinforced nanocomposites[END_REF][START_REF] Reis | Impact response of Kevlar composites with nanoclay enhanced epoxy matrix[END_REF][START_REF] Hosur | Processing of nanoclay filled sandwich composites and their response to low-velocity impact loading[END_REF][START_REF] Avila | Nano-structured sandwich composites response to low-velocity impact[END_REF][START_REF] Reis | Impact response of sandwich composites with nano-enhanced epoxy resin[END_REF]. Hosur et al. [START_REF] Hosur | Processing of nanoclay filled sandwich composites and their response to low-velocity impact loading[END_REF] studied the improvement in low velocity impact response of sandwich plates due to the addition of nanoclays both in the facesheets as well as the polyurethane foam core. They identified different failure mechanisms in the sandwich panels with nanoclay and concluded that it was possible to sustain higher loads, reduce the damage size during impact like events and lower the reduction in mechanical properties by adopting nanophased sandwich construction. Avila et al. [START_REF] Avila | Nano-structured sandwich composites response to low-velocity impact[END_REF] studied the effect of adding nanoclay to fibre glass/epoxy system and its use as facesheet in a sandwich structure. The authors varied the percentage of nanoclay present in the epoxy resin and studied the low velocity impact response of a GFRP/PS Foam sandwich plate. Their preliminary results however suggest that the improvement in the stiffness and impact resistance of sandwich structures due to the nanoclay is minimal.

While most of the research has been focussed on carbon nanotubes (CNT) or nanoclay and silicate nanocomposites, a new method to synthesize block copolymers which self-assemble in the nanoscale was reported by Barsotti [21]. This would appreciably reduce the problems associated with dispersion of the nanoparticles. Nanostructuration is induced by the strong repulsions between the side and middle block. The thermodynamic miscibility of the block copolymer leads to a homogeneous and reproducible dispersion on a nanometer scale. The author claimed that using triblock polymers consisting of polymethylmethacrylatepolybutylacrylate-polymethylmethacrylate (MAM) and styreneb-butadiene-b-polymethylmethacrylate (SBM) families enhanced the fracture toughness and impact performance with minimal sacrifice of thermal properties [START_REF] Barsotti | Nanostrength block copolymers for epoxy toughening[END_REF]. A systematic study on the effect of these nano reinforcements on the mechanical performance of FRPs made with these nano-modified resins, especially to impact loading, is lacking. Denneulin et al. [START_REF] Denneulin | The influence of acrylate triblock copolymer embedded in matrix on composite structures responses to lowvelocity impacts[END_REF][START_REF] Denneulin | Influence of nanoparticles embedded in matrix on composite structures response to low velocity impacts[END_REF] identified the dearth of literature on composites with nano-elastomers of block copolymer and studied the influence of Nanostrength embedded in the matrix, on the low velocity impact response of Kevlar fibre reinforced composite structures. Denneulin [START_REF] Denneulin | Influence of nanoparticles embedded in matrix on composite structures response to low velocity impacts[END_REF] reported that while the carbon nanotubes did not have significant effect, the addition of the nanoparticles of block copolymer to the resin improved the impact resistance of the Kevlar FRP and prevented catastrophic failure due to fibre breakage and perforation seen in the sample without the nanoparticles.

The paper aims to compare the low velocity impact response of sandwich structures with and without Nanostrength in the epoxy matrix. The sandwich structures chosen for the study have facesheets made of Kevlar fibre reinforced epoxy laminates and a core made of closed cell PMI foam. A series of low velocity impact tests are performed using an instrumented drop tower. The forcedisplacement curves of the sandwich panels, the energy absorbed during impact and post-mortem observations are employed to compare the impact resistance of sandwich structures.

Manufacturing of samples

Material

The sandwich panels with the Kevlar fibre reinforced epoxy skins were manufactured using a wet layup process. Kevlar129 (Saatilar Style 802; Taffeta 190 g/m 2 ; thickness: 260 lm) was chosen for this study because of its very high tensile toughness (r r = 3.4 GPa, r = 3.5%). Sandwich composites with Kevlar fabric facesheets were shown to possess the best impact resistance and the least extent of damage compared to glass, carbon and carbon/Kevlar hybrid facesheets [START_REF] Xia | Work on low-velocity impact properties of foam sandwich composites with various face sheets[END_REF]. Three layers of plain woven fabric with the ply orientation [0/90] were used. Closed cell Rohacell Ò PolyMethacryImide (PMI) foam with a density of 51 g/cm 3 was chosen as the core material for the sandwich panels.

The DGEBA thermoset epoxy resin Epolam and hardener supplied by Axson Technologies was used. The hardener was used in the ratio 0.345 (w/w) (34.5 g of hardener for 100 g of resin) as recommended by supplier. For the resin with nano-reinforcements, 10 g of triblocks copolymer M52N Nanostrength Ò supplied by Arkema was added to 100 g of Epolam resin. Denneulin et al. have shown from tests on Kevlar composites with three different formulations of block copolymer (M22, M42, and M52N) that 10% M52N Nanostrength Ò in the epoxy resin system provided the best performance with regard to perforation resistance [START_REF] Denneulin | The influence of acrylate triblock copolymer embedded in matrix on composite structures responses to lowvelocity impacts[END_REF]. Nanostrength Ò which is in powder form is added to the resin by mixing, using a mechanical stirrer at 290 rpm at 110 °C for a duration of 2 h. Transmission Electronic Microscopy (TEM) was used by Denneulin et al. [START_REF] Denneulin | The influence of acrylate triblock copolymer embedded in matrix on composite structures responses to lowvelocity impacts[END_REF] to check the self-assembling process of the block copolymer nanoparticles.

Method of fabrication

The sandwich samples were manufactured using a wet lay-up process. The different steps involved in the fabrication of the sandwich plate are shown in the Fig. 1. The manufacturing process was similar to the method explained in [START_REF] Denneulin | The influence of acrylate triblock copolymer embedded in matrix on composite structures responses to lowvelocity impacts[END_REF]. In the first step, each layer of Kevlar fabric cut to desired dimensions was impregnated with the resin-hardener mix manually (with a brush). The three impregnated layers of fabrics were placed between two sheets of baking paper and compressed in the press for 5 min at room temperature and a pressure of 1.5 bars. This step ensures that excess resin is ejected from the facesheet. The step was repeated for both the top and bottom facesheet of the sandwich. The two facesheet layers of ½0=90 3 Kevlar fibre were then bonded on each side of the core material and the sandwich panel was co-cured in a hot press at 90°for 90 min. No additional adhesive was introduced between the face sheets and the core. Co-curing the facesheets ensured good adhesion with the core. Finally, the sandwich composites were post-cured in an oven at 80 °C for 2 h. Square plates of length 200 mm were fabricated using this method. The nominal thickness of the cured sandwich plates was 20 mm.

Experimental setup

Low velocity impact testing was accomplished using a drop tower. The drop tower setup consists of an instrumented impactor that is secured to a carriage that falls along guideposts and collides with the plate. An electromagnet holds the carriage lifted to a predetermined height and the impact event is initiated by switching off the electromagnet and letting the carriage fall freely under the action of gravity. The impactor head transfers the impact energy to the test specimen. The drop height is varied to give a range of impact energies while the mass of the impactor is kept constant. The maximum impact energy that can be obtained is limited by the weight of the impactor carriage and the maximum height of the tower. The drop tower setup used for the low velocity impact testing is shown in Fig. 2.

The drop tower consists of two rectified columns attached to a metallic gantry and these two columns guide the falling carriage, on which different impactor geometries can be fixed. A winch with an electromagnet was used to lift the projectile to the desired impact height. During the test, the impactor carriage is released by the electromagnet, freefalls, and strikes the structure. A reboundcatcher has been implemented to avoid a second impact, which could further damage the structure and prevent a post-mortem analysis of the damage and residual strain. The force response of the sandwich structure during impact was measured using a piezoelectric force sensor (Kistler force transducer, force range of 5 kN). The signals of both sensors were recorded by a National Instruments card acquisition at a frequency of 30 kHz. In addition to these sensors, two high-speed video cameras (Photron FASTCAM-APX RS and Photron SA3) were used to measure displacement of the impactor and to observe the top facesheet, respectively. The initial velocity was obtained by a linear regression of the displacement before contact.

Initial conditions

The sandwich plates without the nanoparticles in the facesheets are designated KR sandwich plate (to denote Kevlar-Rohacell sandwich) while the plates with Nanostrength embedded in the epoxy matrix are designated KNR sandwich plate (Kevlar-Nanostrength-Rohacell sandwich). The sandwich panels were tested for three drop heights of 0.5 m, 0.75 m and 1 m corresponding to impact energies 8 J, 12 J and 16 J. The initial of the impact was calculated from the mass of the impactor (1.8 kg) and initial impact velocity found from the slope of the displacement before contact. The 16 mm diameter hemispherical headed impactor strikes the centre of the square target plate which is clamped with square aluminium plates with a circular hole of diameter 70 mm. The bolts were tightened to a torque of 5 N m using a torque wrench, in order to provide consistent clamping from experiment to experiment.

A typical force and displacement-time history of low velocity impact is shown in Fig. 3. The curves shown correspond to 8 J impact on KR sandwich plate. Time t 0 and t 3 corresponds to the beginning and end of impact, respectively. It can be seen that the total duration of the impact is about 10 m s. The contact force of the sandwich composite exhibits a linear increase with respect to time as the indenter comes into contact with the specimen and is followed by a sudden drop in the load after it reached the first peak value at t 1 . The maximum displacement of the impactor occurs at t 2 and marks the beginning of the unloading phase. The unloading is finally followed by the rebound, which is represented by the contact force value reaching zero i.e. the impactor is no longer in contact with the sandwich plate (t 3 ).

Results of drop tower test

The measurements of time-history of force and displacement made it possible to plot the force vs. displacement curve for each type of sandwich plate tested. In order to verify the repeatability of the test, three samples were tested at each energy. Fig. 4 shows the force-displacement curves for KR sandwich plate impacted with an initial energy of 8 J. It can be observed from these curves that only a weak dispersion of material behaviour exists between the different samples. The initial linear portion of the curve is robust while the large oscillations that marked the beginning of fibre breakage are not reproducible. However, it is safe to assume that the tests are repeatable and only a representative sample from each energy is used for comparison with other samples henceforth.

Figs. 567show the force-displacement plot for the Kevlar-Rohacell sandwich with neat resin and with 10% M52N Nanostrength at different impact energies. The samples shown correspond to impact energies of 8 J, 12 J and 16 J (drop height of 0.5 m, 0.75 m and 1 m) for KR and KNR sandwich plates respectively. The force-displacement curve of KR and KNR sandwich plates are compared for the three different impact energies. It is observed that the force-displacement curve of the sandwich panels show a quasi-linear behaviour for low values of indentation, followed by a non-linear regime. It can also be observed that the stiffness of the sandwich panel during the elastic bending phase, i.e. slope of the linear portion of the curve, the elastic stiffness is reduced by the addition of copolymers, which is likely due to the more compliant elastomeric phase. However, the stiffness has not changed dramatically for the sandwich plate with the nanoparticles. This is particularly important as the toughening of the resin with rubber particles resulted in a considerable loss of stiffness. The behaviour of the KNR sandwich is distinctly different from the KR sandwich, including the quantitative values of peak force and maximum displacement. For the sandwich with neat resin (KR), a sharp drop in force is observed which corresponds to the failure of facesheets through fibre breakage and perforation. The behaviour after this peak exhibits a core-dominant behaviour and follows regions of plastic plateau in which the cellular structure collapse initiate by buckling of the cell walls and edges and densification of the core. The second peak seen for sandwich plates impacted at the higher energy shows that the lower facesheet has begun to play a role in the deformation.

Typical images of the top facesheet post-impact are also shown in the Figs. 567. There is considerable difference in the failure observed in the KR and KNR sandwich plates. It can be seen that even for the lower energy, there is beginning of fibre failure and perforation in the sandwich plates with neat resin. There is complete penetration of the top facesheet in the neat resin sandwich impacted at higher energy. The sandwich plates with nano-modified resin does not show any perforation but some failure in the fibre directions, i.e. [0-90], is clearly visible. It can also be observed from visual examination of the top facesheet that the load is spread over a larger area for the samples with nano-modified resin while the sandwich plate with neat resin shows more local damage. It can be concluded that the KNR sandwich plate exhibits a higher strength and an absence of sudden drop in rigidity associated with fibre breakage and perforation of the KR sandwich plate.

Fig. 8 shows the images from the second high speed camera used to observe the top facesheet. The images, corresponding to an impact test at 16 J, begin 6 m s before the beginning of contact and shows the impactor indent the top facesheet. It can be seen that there is complete perforation of the top facesheet in the case of the impact of KR sandwich plate and prolonged contact whereas the KNR sandwich is able to resist the impact event.

An additional test was carried out at an initial energy of 20 J (drop height 1.25 m) to observe if the KNR sandwich had perforation at this higher energy. Fig. 9 shows the force-displacement curve of the 20 J impact on KR and KNR sandwich plate and the top and bottom facesheet of the sandwich sample. The second peak observed for KR sandwich is the result of loading on the bottom facesheet. Visual examination of the facesheets after impact show that there is considerable failure in the bottom facesheet of the KR sandwich plate. It can be observed that the 20 J impact caused penetration failure in the top facesheet of the KNR sandwich plate whereas the bottom skin shows little damage.

The evolution of peak force and maximum displacement vs. the initial energy were plotted in Fig. 10. It can be observed that the peak force values for KNR sandwich show a steady increase up to a maximum of 3180 N followed by a drop to 2480 N for the 20 J impact. This can be explained by the perforation of the top facesheet observed in the 20 J impact. A corresponding increase in the maximum displacement is noted. The peak force for KR sandwich plate does not show an increasing trend as the sandwich plate with nano-modified resin. There is a drop in the peak force for the 20 J impact but there is a second peak corresponding to the deflection of bottom facesheet. The maximum displacement continues to show an increasing trend to a maximum of 26.3 mm for the 20 J impact.

Energy absorbed

Denneulin et al. [START_REF] Denneulin | The influence of acrylate triblock copolymer embedded in matrix on composite structures responses to lowvelocity impacts[END_REF] defined the energy until the initiation of the perforation, computed as the integral of the force-displacement curve up to the displacement corresponding to the peak force. This energy represents the elastic energy of the plate, the energy dissipated by matrix damage, the energy dissipated by friction and by wave propagation in the experimental setup. The energy up to the peak force clearly represents the energy required to generate sufficient damage to reduce the stiffness of the plate and affect its structural integrity severely, and was calculated from the integral of the force-displacement curve. The energy required to initiate failure for the KR and KNR sandwich plates is compared in Fig. 11. It is observed that the elastic energy has similar trend to the results presented by Reis et al. [START_REF] Reis | Impact response of Kevlar composites with nanoclay enhanced epoxy matrix[END_REF] for the addition of nanoclay particles. The KNR sandwich with nanoparticles has comparable energy absorbed at the lower energy of impact but at higher energies, the sandwich with nanoparticles clearly absorbs more energy. This is due to increasing peak force before the initiation of damage. According to Denneulin et al. [START_REF] Denneulin | The influence of acrylate triblock copolymer embedded in matrix on composite structures responses to lowvelocity impacts[END_REF], a ranking regarding perforation resistance can be performed with this criterion. A polynomial fit was not attempted given the limited number of impact energies tested, but the trends are shown as dotted lines and it is evident that the addition of Nanostrength Ò increases the penetration threshold for the Kevlar sandwich plates.

A second criterion can also be defined as the total energy dissipated during the impact. This corresponds to the total area under the force-displacement curve from the beginning of impact until the end of unloading phase when the force reaches zero again. This energy represents the residual elastic energy of the plate. The total energy dissipated is plot against the initial kinetic energy of the impactor (shown in Fig. 12). The 45°line is the total available energy, i.e. for initial impact energy of 8 J, the maximum energy that can be dissipated during the impact is 8 J and so on. It can be seen that penetration occurs when the dissipated energy curve approaches the 45°line, i.e. samples that undergo penetration absorbs a large portion of the kinetic energy of the impactor in damage of the different components of the sandwich plate. The KR sandwich has higher total energy absorbed because of the perforation of the facesheet and extensive core crushing. The maximum displacement reached for the 16 J impact is almost 22 mm. This core dominated region is involved in dissipating a large portion of energy. Even though the KNR sandwich plates have a higher peak force value than the KR sandwich counterparts, there is sufficient recovery of the samples in the unloading phase, that it has a lower total energy dissipated. It is understood that higher values of dissipated energy relates to lower elastic recovery and consequently major damage while the lower values of the sandwich plate with nanoparticles implies that the KNR sandwich plate can withstand impact loading of energy upwards of 16 J before failing by perforation. It is also interesting to note that for the impact at 20 J, the KR sandwich plate has reached a saturation limit and the higher energies does not cause an increase of this value. The KNR sandwich continues to have an increasing trend in the energies tested and has not reached the saturation limit.

Post-mortem damage analysis

The impacted sandwich sample is cut in the middle in order to observe the section view. The sandwich sample is mounted with a coloured resin before cutting to preserve the shape of the damage of the facesheets as well as the core. For the sandwich plates where there is no perforation of the top facesheet, a small hole is drilled on the top facesheet, through which the coloured resin is injected. A room temperature curing epoxy resin was used along with a red dye for this purpose. The sandwich plate was cut using a diamondtoothed saw and the section of interest is embedded in a clear Mecaprex resin under vacuum. The mounted samples are then polished under progressively finer abrasive papers. The section view of the KR and KNR sandwich plates for impact energies of 8, 12, 16 and 20 J are shown in Fig. 13. Visual examination of the impacted samples highlight that for many cases with considerable damage in the top facesheet, the lower facing of the sandwich structure was undamaged. There was no damage in the bottom facesheet unless there was penetration of the top facesheet and extensive core crushing. Energy absorption is maximized through the sandwich construction, as penetration through all the three components (top facesheet, core, and bottom facesheet) is required for the complete failure of the sandwich structure. The KNR sandwich impacted at 20 J for instance, has perforation in the top facesheet but the bottom skin is intact which would allow for more energy to be dissipated before complete failure of the sandwich plate.

It can also be observed in samples impacted at 20 J (image (d) and (iv)), that the penetration of the top facesheet resulted in extensive damage in the core region immediately below the point of impact. This resulted in debonding between the bottom facesheet and core, where the red coloured resin can be clearly observed. The visual examination also confirms that the bottom facesheet of the KR sandwich plate has considerable fibre failure under 20 J impact.

Three damage parameters are defined from the sections of the impacted specimen as shown in Fig. 14. They are:

1. Depth of indentation of the top facesheet (d 1 ). 2. Depth of indentation of the core (d 2 ). 3. Width of debonding between facesheet and core (l 1 ).

The section view of the KR sandwich plates shows the extensive core crushing just below the point of impact (displayed as the redcoloured resin). There is also some debonding between the top facesheet and the core where the red resin has penetrated. Debonding occurs because the laminated facesheet is able to recover it's shape during the unloading phase of the impact while the crushable foam cannot. It can be observed that the damage is concentrated in the local area around the point of impact. There is no perforation in the top facesheet of the KNR sandwich except at the highest energy level. Since the resin is injected through a tiny hole, it does not penetrate the region between the facesheet and the core as well. However, the area of damage is clearly distinguishable as a different colour. There is considerable damage to the core and the debonding between the facesheet and core is spread over a larger area.

Fig. 15 shows the comparison of the different damage parameters for the KR and KNR sandwich plates. It can be seen for KR sandwich plates that both the depth of indentation in facesheet and core (d 1 and d 2 ) exhibit an increasing trend with initial energy while the KNR sandwich shows an almost constant value. The higher values observed for the KR sandwich plates are due to the more local damage observed in them. The extent of damage l 1 which corresponds to the amount of debonding between the top facesheet and the core have higher values for KNR sandwich plates which confirm that the load is spread over a larger area in the sandwich plate with nano-reinforcements.

Microscopic observations of the sandwich plates

The section of the sandwich plate is also observed under an digital optical microscope. Microscopy reveals three prominent modes of failure in the top facesheet; matrix cracking, delamination, fibre breakage and perforation. The KR sandwich plate after an impact of 12 J is shown in Fig. 16. It can be seen that the damage in the KR sandwich plate was concentrated in the indentation zone corresponding to the dimensions of the impactor. Fibre breakages are clearly visible close to the point of impact. In the second zone, a gap can be detected between the three layers of the Kevlar fabric, corresponding to the delamination failure. Finally, matrix cracking is observed in the 90°fibre yarns.

KNR sandwich plate after an impact of 16 J is shown in Fig. 17.A high density of matrix cracks between 90°fibre yarns is observed in the sandwich plates with nanoparticles. These matrix cracks are mainly responsible for the dissipated energy. It is hypothesised that these matrix cracks and the resulting delamination are responsible for the increase in dissipation of the impact energy before initiation of failure. It is also interesting to note that the matrix cracks are not limited to the impact zone but are present farther from the point of impact as well.

Fig. 18 shows the uncrushed and crushed cells of the Rohacell Ò foam core for 12 J impact. Buckling of the cell walls can be clearly observed in the crushed core. The foam core plays an important role in the energy absorbed by the sandwich structure as a large amount of energy is dissipated in the core crushing.

Conclusions and recommendations

The objective of this research was to evaluate the effect of including block copolymer nanoparticles to the matrix on the low velocity impact behaviour of sandwich composites based on Kevlar fibre reinforced epoxy and PMI foam. A 10% M52N Nanostrength Ò triblock copolymer was chosen for the study. Kevlar fabric was chosen for their good properties under dynamic loading and their application in impact resistant structures. A drop weight tower was used to conduct low-velocity impact tests on the two types of sandwich plates at different initial energies. The effect of the nano-reinforcements on the sandwich plates under lowvelocity impact was firstly studied at the macroscopic level. A comparison of sandwich panels in terms of the energy absorbed by the panel was undertaken. The KNR sandwich plates required higher energy to fail and also did not exhibit perforation failure of the facesheets as the sandwich plates without the nanoreinforcements. The damage in the foam core was also not localised in the impact zone but was spread across a significant part of the structure. Different damage parameters were described to compare the two types of sandwich structures. While the KR sandwich plate failed by perforation, no perforation was detected in the top facesheet for the sandwich plate with nanoreinforcements for the same impact energy. Perforation of top facesheet was observed in KR sandwich plates for impact at 12 J whereas KNR sandwich plates did not have perforation even at 16 J impact. Optical microscopy was undertaken to observe the different failure mechanisms in the Kevlar facesheet. It is observed that the damage phenomena in the KNR sandwich plate consist mainly of matrix cracking and it is proposed that this may explain the improvement in impact resistance. It can be concluded that the M52N elastomeric nanoparticles significantly improved the resistance of Kevlar FRP sandwich structures under low velocity impact.

Fig. 1 .

 1 Fig. 1. Steps involved in the fabrication of sandwich samples.

Fig. 2 .

 2 Fig.2. Drop tower setup for low velocity impact testing.

Fig. 3 .

 3 Fig.3. Typical force and displacement-time history of low velocity impact on sandwich panels (8 J).

Fig. 4 .

 4 Fig.4. Force-displacement curve of low velocity impact of KR sandwich panels (8 J).

Fig. 5 .

 5 Fig.5. Force-displacement curves of KR and KNR sandwich plates after 8 J impact and image of top facesheet after impact.

Fig. 6 .

 6 Fig.6. Force-displacement curves of KR and KNR sandwich plates after 12 J impact and image of top facesheet after impact.

Fig. 7 .

 7 Fig. 7. Force-displacement curves of KR and KNR sandwich plates after 16 J impact and image of top facesheet after impact.

Fig. 8 .

 8 Fig. 8. Comparison of propagation of impact in KR (top) and KNR (bottom) sandwich plates for impact of 16 J.

Fig. 9 .

 9 Fig. 9. Force-displacement curves of KR and KNR sandwich plates after 20 J impact and image of top and bottom facesheet after impact.

Fig. 10 .

 10 Fig. 10. Evolution of peak force and maximum displacement.

Fig. 11 .

 11 Fig. 11. Energy up to peak force.

Fig. 12 .

 12 Fig.12. Total energy dissipated in the impact.

Fig. 13 .

 13 Fig. 13. Comparison of cross-section of sandwich plate (a-d) KR sandwich, (i-iv) KNR sandwich.

Fig. 14 .

 14 Fig. 14. Defining the damage parameters.

Fig. 15 .

 15 Fig. 15. Different damage parameters measured from section of the sandwich sample.

Fig. 16 .

 16 Fig. 16. Microscopic observations of section of KR sandwich (12 J impact).

Fig. 17 .

 17 Fig. 17. Microscopic observations of section of KN sandwich (16 J impact).

Fig. 18 .

 18 Fig. 18. Uncrushed and crushed cells of PMI foam (impact of 12 J).