attic

J.

dge

5e 5

In the present paper, highly porous bre networks made of 316L bres, with dierent bre volume fractions, are characterized in

Highly porous materials, particularly those based on

faces favour bone anchoring by providing space for inltra-tion, rst by cells and ultimately by osseous tissue andvasculature.

cally designed to be highly porous (>50% porosity) are

they are periodic or stochastic (i.e. exhibiting random vari-ations in bre orientation distribution and spatial loca-tion). This can aect certain characteristics, particularlymechanical properties, since it inuences the incidence ofstress concentration eects. They can be isotropic or theymay be highly oriented (e.g. ropes) and therefore exhibit

Corresponding author. Tel.: +44 1223 766417; fax: +44 1223 332662.E-mail address: am253@cam.ac.uk (A.E. Markaki).

Available online at www.sciencedirect.com

ScienceDirect

Acta Materialia 66 (2014) 32633metals, are currently attracting a lot of research interest.Their high surface-to-mass ratio means that they can beattractive in terms of transport phenomena characteristics,leading to applications involving heat transfer, ltration,catalyst support, acoustic damping and also biomedicaldevices. Porous metals are used for hard (e.g. bone) tissuerepair and reconstruction. For instance, titanium brecoatings are commercially used in total hip prostheses suchas the Zimmer VerSys Epoch Full Coat Hip Prosthesis [1].The rationale behind this design is that porous implant sur-

often referred as foams, although it should be acknowl-edged that there is no universally accepted denition of afoam. It is important to recognize the distinction betweenopen-cell foams and bre networks. The latter are con-structed by assembling a set of slender members such asbres, wires or rods and bonding them at crossover points.While both are permeable to uid and abundant in nature(cancellous bone, sponge, sheeps wool and cotton), brenetworks do not exhibit a distinct cell structure. Fibrenetworks are sometimes classied according to whetherterms of network architecture, elastic constants and fracture energies. Elastic constants are measured using quasi-static mechanicaland modal vibration testing, yielding local and globally averaged properties, respectively. Dierences between quasi-static and dynamicelastic constants are attributed to through-thickness shear eects. Regardless of the method employed, networks show signs of materialinhomogeneity at high bre densities, in agreement with X-ray nanotomography results. Strong auxetic (or negative Poissons ratio)behaviour is observed in the through-thickness direction, which is attributed to bre kinking induced during processing. Measured frac-ture energies are compared with model predictions incorporating information about in-plane bre orientation distribution, bre volumefraction and single bre work of fracture. Experimental values are broadly consistent with model predictions, based on the assumptionthat this energy is primarily associated with plastic deformation of individual bres within a process zone of the same order as the inter-joint spacing. 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Elastic constants; Mechanical properties; Porosity; X-ray tomography; Fibre networks

1. Introduction Some porous materials are divided up into cells of somesort and are sometimes termed cellular. Those speci-Characterization and deformbre networks with auxe

S. Neelakantan, W. Bosbach,

Department of Engineering, University of Cambri

Received 9 July 2013; received in revised formAvailable onlin

Abstract1359-6454/$36.00 2013 Acta Materialia Inc. Published by Elsevier Ltd. Allhttp://dx.doi.org/10.1016/j.actamat.2013.11.020ion response of orthotropicout-of-plane behaviour

Woodhouse, A.E. Markaki

, Trumpington Street, Cambridge CB2 1PZ, UK

November 2013; accepted 5 November 2013December 2013

www.elsevier.com/locate/actamat

9rights reserved.

Mapronounced anisotropy in both structure and properties.When characterizing bre networks, the microstructure ofthe material constituting the bre is as important as thenetwork architecture (e.g. bre orientation distribution,bre segment aspect ratio (inter-joint distance to brediameter)). With the advent of digital imaging, the networkarchitecture is most eectively captured by computed X-raytomography [2,3].

There have been several previous experimental studies ofthe mechanical response of stochastic metallic bre net-works made of Cu [4], Ti [5], 20%Cr80%Ni [6], steel [79] and stainless steel [1022] bres (in particular 304[10,11], 316L [1218] and 446 [1922]). The majority ofthese studies focused on the eect of bre size and volumefraction on tensile [4,6,1014,17,19], compressive[5,7,13,17,18,2022], torsional [8,17] and impact [9]responses. Optimization of their performance clearlyrequires an understanding of the interplay between pro-cessing conditions, network architecture, bre microstruc-ture and mechanical response characteristics undervarious types of applied load.

It is important to note that existing studies use small testspecimens, which typically yield only local properties.However, the response of a structure to imposed loads isgoverned by the globally averaged properties. Fibre net-works are in many respects intermediate between materi-als and structures. Furthermore, when measuringelastic constants, conventional mechanical testing usuallyinvolves much higher applied loads than dynamic methods.This is potentially important, as it is well known that, inhighly porous materials, local plastic deformation occursreadily, even at low macroscopic strains (wherein the defor-mation is expected to be linear).

Auxeticity, i.e. negative Poissons ratio m, has been asso-ciated with enhancement of material properties related toPoissons ratio [23]. Approaches to designing auxetic brecomposites involve the use of auxetic constituents (usuallypolymeric bres) or selection of suitable stacking sequencesof highly anisotropic unidirectional bre-reinforced lami-nate (mostly ceramic bres in a polymeric matrix) [24].Negative Poissons ratios have been measured experimen-tally [14,17] and predicted theoretically [25,26] for stochas-tic metallic bre networks of the type investigated in thepresent study.

The present work involves a systematic investigation ofthe architecture, deformation and fracture of 316L austen-itic stainless steel bre networks with dierent bre volumefractions. Network architectural characteristics, such asbre orientation distributions and the mean bre segmentlength, were obtained from X-ray tomography. The in-plane Youngs moduli of the networks, and also their in-and out-of-plane Poissons ratios, were measured usingquasi-static mechanical (tensile and cantilever bend) anddynamic vibration (plate and beam) testing. Vibration test-ing automatically provided an estimate of the globally

S. Neelakantan et al. / Actaaveraged properties of the networks. A comparisonbetween the dierent testing methods is presented. Esti-mates of the nominal fracture energy of the networks wereobtained using a simple method based on analysis of tensilestressdisplacement plots. Experimental values were com-pared with predictions obtained using an analytical model,based on the assumption that this energy is primarily asso-ciated with plastic deformation of individual bres within aprocess zone neighbouring the fracture plane.

2. Materials and methods

2.1. Materials

The present study focuses on AISI 316L austenitic stain-less steel in the form of single bres and sintered bre net-works (N.V. Bekaert S.A., Belgium).

2.1.1. Single bres

The 316L bres were produced by a bundle-drawingprocess [27]. This process involves forming a bundle ofmetallic wires, each coated with a sacricial (separation)layer, encasing the bundle in a tube and drawing the tubeto smaller diameters. Bundle drawn bres exhibit a hexag-onal cross-sectional shape. Fig. 1a and b shows opticalmicrographs of transverse sections of 316L bres in as-received and as-heat-treated conditions, respectively. Itcan be seen that the diagonal width of the bres is40 lm (the side length is 20 lm).

2.1.2. Sintered bre networks

Fibre network plates were produced by solid-state sinter-ing at N.V. Bekaert S.A. (Belgium). In particular, loose feltsof bundle-drawn bres were stacked upon one another witha random planar orientation, and compressed to plates withthree dierent bre volume fractions f; 10, 15 and 20 vol.%.The plate dimensions were either 297 mm 210 mm 5 mm or 297 mm 210 mm 2 mm in the longitudinal(x), transverse (y) and through-thickness (z) directions,respectively. There were variations within the labelled brevolume fractions arising from processing. For the 5-mm-thick network sheets, themeasured bre fraction values were10.3% (0.003), 14.8% (0.002) and 20.2% (0.39). Thecorresponding values for the 2-mm-thick network sheetswere 9.2% (0.36), 14.7% (0.31) and 20.3% (0.36). It isof note that the 2 mm thick 10 vol.% network sheets hadinvariably a slightly lower bre volume fraction (9.2%).To avoid any edge eects, samples were cut at least 20 mmin from the edges of the sheets.

2.2. Material characterization

2.2.1. Electrolytic and tint etching for optical microscopy

Cross-sections of as-received and heat-treated breswere prepared, mounted, ground with a series of SiCpapers and afterwards further polished with 6 and 1 lmdiamond paste. Examination of bre cross sections was

terialia 66 (2014) 326339 327performed using a Leica DMLM microscope. Electrolyticetching was carried out using a 10% aqueous oxalic acid

nalw ono

Mareagent and a voltage of 6 V DC for 35 s. This reagentstrongly attacks the r phase after 6 s, outlines (and dis-solves) carbides after 1530 s and reveals grain boundaries

Fig. 1. Optical micrographs showing polished and tint-etched cross-sectioimages of (c) a sintered bond at an inter-bre crossing and (d) a plan vieelectrolytically etched cross sections of heat-treated bres (bres do not lie

328 S. Neelakantan et al. / Actaafter 4560 s [28].Tint etching with modied Berahas reagent (100 ml

water, 20 ml HCl, 2.4 g NH4FHF, 0.6 g K2S2O5) was usedto obtain information about the grain size and crystallo-graphic orientation. This reagent deposits a sulde-basedsurface lm that creates colour contrast through interfer-ence eects in bright-eld imaging mode. Fig. 1a and bshows the 316L bre microstructure in the as-receivedand heat-treated conditions, respectively. It can be seenthat the as-received bre (Fig. 1a) has an average grain sizeof 5 lm whereas, after heat treatment (Fig. 1b), substan-tial grain coarsening (grain size roughly equivalent to brediameter) and annealing twins are observed. From the nar-rowness of the colour range, it can be postulated that theas-received bres have a more preferred crystallographicorientation than the sintered ones.

2.2.2. Scanning electron microscopy

Top views and fracture surfaces were investigated byscanning electron microscopy (SEM), using a Zeiss EvoMA 15 instrument in secondary mode. Fig. 1c shows a typ-ical surface bre morphology and a sintered bond at a brecrossing of a network containing 10 vol.% of bres(f = 10%). It can also be seen that the bres have a coarserecrystallized grain structure (a bamboo structure) aftersintering. A typical plan view of the network is shown inFig. 1d.2.2.3. X-ray computed tomography

Tomographic data were obtained on specimens cut byelectro-discharge machining (EDM) from the network

views of (a) as received and (b) heat-treated (sintered) 316L bres. SEMf a 10 vol.% bre network. (e) Optical micrograph showing polished andrmal to the plane of the section).

terialia 66 (2014) 326339plates into 5 mm cubes. In total, six samples (two for eachbre volume fraction) of 316L were analysed.

2.2.3.1. Equipment. Tomography scans were carried out bya General Electric Phoenix X-ray Nanotom systemequipped with a sub-micron focal spot X-ray source. Thesource voltage and current were set at 120 kV and 40 lArespectively. The voxel size was 7.75 lm. A 0.2 mm copperlter was used. The scans were recorded at projectionangles between 0 and 360 in steps of 0.25. To increasethe signal-to-noise ratio of the images, 12 frames were cap-tured and averaged for each projection on a 5 megapixel12-bit detector array.

2.2.3.2. Architecture characterization. For extraction ofarchitectural parameters, a sub-volume of 4 4 4 mm3was analysed to avoid edge eects. The process involvedthree steps; image reconstruction, segmentation and topo-logical thinning. More details can be found elsewhere [3].

Briey, following X-ray acquisition, the data sets werelter back-projected (reconstructed) to produce a series oftwo-dimensional (2-D) raw greyscale images of the net-work. Owing to the stochastic nature of the network archi-tecture and the inherent limitations of the ltered back-projection technique, additional ltering was applied toimprove the resolution of the tomographic images. Imagesegmentation (separation of the bres from the inter-bre

Maspace (air)) can be carried out by applying a global thresh-old for the entire image (i.e. ignoring spatial dependence ofphases) or locally by applying dierent thresholds in dier-ent spatial regions. In this work, a local, spatially adaptivethresholding algorithm, known as indicator kriging [29],was employed.

The output of the segmentation procedure is a stack ofbinary images featuring only two phases, bres (black) andair (white). The segmentation algorithm was calibratedusing the bre volume fractions obtained from a simplevolumetric technique (weight and sample dimensions) anda density value of 7.9 Mg m3 for 316L. It is assumed thatthe porosity remains the same for any sub-section withinthe original sample volume. Following segmentation, thebinary images were subjected to standard post-segmenta-tion clean-up procedures, such as removal of small blobsand lling of holes. In the third step, the three-dimensional(3-D) Medial Axis (3DMA) algorithm (Nihon Visual Sci-ence Inc., Japan) [30,31] was employed to reduce the 3-Dreconstructed bres to their medial axes (skeletons), pre-serving their shape and topology. Once the medial axesof the bres were obtained, the local bre orientations, sizeand other architectural characteristics, such as the numberof inter-bre crossings and bre segment lengths, wereextracted.

2.3. Mechanical testing

2.3.1. Single bre tensile testing

Tensile testing was carried out using a screw-drivendesktop Instron testing machine, tted with a 250 N loadcell. The individual bres were mounted across 14 mmgauge length paper tabs, using Loctite superglue. Aftergripping each tabbed specimen, prior to testing, the tabedges were cut. The cross-head displacement was measuredusing an LVDT. All tests were conducted in displacementcontrol at a strain rate of 104 s1. Fibres were tested inthe as-received condition and after sintering. The singlebre work of fracture (in joules per metre) was measuredas the area under the loadstrain curve.

2.3.2. Network in-plane tensile testing

In-plane tensile testing of bre networks was conductedusing an Instron testing machine equipped with a 5 kNload cell. Dog-bone-shaped tensile specimens were elec-tro-discharge machined from the network plates accordingto ASTM E8-09 sub-size specimen standards. The gaugesections were 32 mm long, 6 mm wide and 5 mm thick. Inorder to prevent crushing in the grip sections, the ends ofthe specimens were impregnated with Loctite superglue.All tests were conducted in displacement control at a strainrate of 103 s1. Displacement was measured using a laserscanning extensometer with a 1 lm resolution. The in-plane Youngs moduli (Ex and Ey) were measured fromthe tangent slope of the unloading stressstrain curve

S. Neelakantan et al. / Acta(within the elastic regime). Estimation of the fracture ener-gies (in kilojoules per square metre) involved measuring thearea under the loaddisplacement curves and then dividingby the fracture cross section.

Digital image correlation (DIC) was employed [32,33] tomeasure in situ the 2-D surface strain elds (xy and xzplanes) in order to evaluate the in-plane and out-of-plane Poissons ratios. For this purpose, two independentbut synchronized high-resolution (2048 1536 pixels)PixeLink cameras were used to view the front (xy plane)and side face (xz plane) of the network samples. Consec-utive images were taken at a time interval of 500 ms. Track-ing of displacements from these series of images withsub-pixel resolution was performed using a previouslydeveloped code [34]. To acquire and process the data fromthese images, Matlab 7 with Optimization and Image Pro-cessing toolboxes was used. A region of 1000 (long) 200(wide) pixels within the gauge section, with a step size of20 pixels in each direction, was used for the correlation.

2.3.3. Network cantilever bend testing

The in-plane Youngs moduli of the bre networks weremeasured by cantilever bend testing using a laser scanningextensometer to monitor the beam deection d. Thisallowed measurement of deections with a resolution of5 lm. Beams 150 mm(L) 20 mm(b) 2 mm(h), cutparallel to the longitudinal and transverse (in-plane) direc-tions, were employed. Load was applied using small pre-weighed masses in the range 0.040.26 g. The response tounloading was also measured, to ensure that the beamswere still in the elastic regime. A correction to d wasapplied by measuring the apparent thickness of the beams,to account for errors arising from extraneous beamtwisting.

The deection d was measured at a distance x from theclamped end, while a load P was applied at a distance L(from the clamped end). The clamping region was impreg-nated with Loctite superglue to prevent crushing. TheYoungs moduli were calculated from

E PId

Lx2

2 x

3

6

1

where I is the second moment of area of the beam section(=bh3/12).

2.4. Vibration testing

2.4.1. Plate vibrations

The dynamic elastic constants were determined by excit-ing the low-frequency vibration modes of bre networkplates with freefree boundaries. The vibration modes werelocated and visualized by Chladni patterns: the natural fre-quencies correspond to the eigen-frequencies of the plates.Using classical thin plate bending theory, as described indetail elsewhere [35], an inverse calculation was performedto give a best match of the plates elastic constants to thepattern of measured natural frequencies. Within thin-plate

terialia 66 (2014) 326339 329theory, the potential energy functional U of a thin, at,orthotropic plate (i.e. with three mutually perpendicular

planes of symmetry) of thickness h, lying initially in the xyplane and vibrating with a middle surface transverse (out-of-plane) displacement wx; yeixt is given byU h

3

2

Z a0Z b

0

D1@2w@x2

2 D2 @

2w@x2

@2w@y2

D3 @2w

@y2

2 D4 @

2w@x@y

2" #dxdy 2

where a and b are the in-plane plate dimensions, x is the

specimen is supported over the loudspeaker on small blocksof soft polymeric foam. These foam supports are adjustedduring testing to lie precisely on nodal lines (i.e. lines wherethe amplitude of vibration is zero) of each tested mode. Thefrequency of the sine-wave is carefully tuned until thedesired Chladni pattern is observed, and the correspondingfrequency is then read from a frequency counter. The pat-tern is visualized by sprinkling powder over the surface of

330 S. Neelakantan et al. / Acta Materialia 66 (2014) 326339angular frequency, and t is the time. Constants D1 andD3 are associated with bending of the middle surface inthe xz and yz planes, respectively, D2 with Poissons ratiocoupling between the x and y directions, and D4 without-of-plane twisting of the middle surface. The four con-stants D1D4 are related to the in-plane engineering elasticconstants as follows:

D1 Ex121 mxymyx ; D3

Ey121 mxymyx

D2 mxyEy61 mxymyx

myxEx61 mxymyx

D4 Gxy=3

3

where Ex, Ey are Youngs moduli in the x and y directions,respectively, Gxy is the shear modulus in the xy plane, andmxy and myx are the in-plane Poissons ratios.

In brief, the procedure is to obtain rst estimates of theconstants D1, D3 and D4 from the measured resonant fre-quencies of three particular vibration modes, together withknowledge of the plate dimensions and density. Next, theaspect ratio of the plate is adjusted to make it eectivelysquare, and the frequencies of two modes termed the Omode (fo) and Xmode (fx) are measured. Their frequencyratio is particularly sensitive to Poissons ratio, and a chart-based method based on this ratio is used to give an estimateofD2. Finally, these rst estimates ofD1,D2,D3 andD4 canbe improved by an iterative process employing numericalcalculations of plate frequencies based on the RayleighRitzmethod. A virtue of this plate-based approach is that addi-tional natural frequencies can be measured to provide adegree of redundancy. If these additional frequencies arealso well predicted by the numerical calculations, the appli-cability of thin-plate theory is conrmed, and the user canhave more condence in the results.

The experimental setup used to measure the resonant fre-quencies involves a loudspeaker driven by a sine-wave gen-erator, mounted beneath a large at surface. The plate

Table 1Plate dimensions and densities used for vibration testing (Section 2.4.1).

Fibre volume fraction, f (%) Plate shape Length, a (mm)

10 Square 150.02Rectangular 175.05

15 Square 150.03Rectangular 175.0720 Square 150.03Rectangular 175.09the plate and observing the powder accumulating alongthe nodal lines. Sketches of some of the measured modeshapes are shown in Tables 6 and 7. Network plates werecut into square and rectangular shapes for these tests: theplate dimensions are given in Table 1.

2.4.2. Beam vibrations

A second vibration technique was also used to measurethe exural (in-plane) Youngs modulus of the bre net-work material. The technique involved excitation of150 mm(L) 20 mm(b) 2 mm(h) beams by means of alight external mechanical impulse. The beam was sup-ported at the two nodal points of the rst freefree vibra-tion mode, located 25% of the specimen length fromeither end. A microphone, connected to an amplier, waslocated directly underneath the sample. Following impul-sive excitation of the beam between the nodal points, theresponse signal was digitized and converted to the fre-quency domain using Fast Fourier Transform analysis.The peaks in the frequency spectrum show the locationsof natural frequencies of the specimen. The natural fre-quencies for dierent vibration modes, along with the beamdimensions and density, were used to give an estimate ofthe Youngs modulus of the networks in the direction par-allel to the length of the beam using the following equation

E x2nqAI

Lan

44

where q is the density, xn is the frequency of the nth mode(rad s1), and an is the modal constant, equal to 4.73 and7.85 for modes 1 and 2, respectively [36].

3. Results

3.1. Network architecture

Table 2 summarizes the network architecture character-istics obtained from X-ray tomography. As expected, the

Breadth, b (mm) Thickness, h (mm) Density (kg m3)

150.02 2.05 734.6140.04 2.05 703.4

150.03 2.03 1185.5140.06 2.03 1163.0150.03 2.01 1600.2140.03 2.01 1620.6

of

Maaverage number of bres and bre segments (sectionsbetween joints) and also the average number of bondsper bre increase with increasing bre volume fraction.Conversely, the mean segment length decreases withincreasing bre volume fraction. The bre tortuosity,which is the ratio between its geodesic length and theEuclidean distance between its two extremities, is higherfor the 20 vol.% networks. Since the bre length is similarfor all bre volume fractions, it can be postulated that, dur-ing processing, bres in the 20 vol.% networks bend moreover each other as the sheets are compressed.

Fibre segment orientation distributions are presented inthe form of histograms of the bre segment inclinationangle h to the through-thickness z direction (Fig. 2a) andstereographic projections on the xy plane (Fig. 2b and c).The vertical axis of the histograms is giving the probabilityof the bre segments Phi N hi=Dhi

Pni N hi lying at an

angle hi, where N hi is the number of bre segments fallinginto a bin of width of Dhi, centred at hi. As expected, theprobability of having segments inclined at low inclinationangles increases with increasing bre volume fraction, asbres are more likely to come into contact. Stereographicprojections, using the South Pole as the point of projection,are illustrated in Fig. 2b and c. In Fig. 2b, owing to thelarge number of points, each point on the stereogram rep-resents the in-plane orientation of 10 bre segments. Highorientation angles to the z axis would appear as points clus-tered near the periphery of the stereograms. It can be seenthat there is a clear bias towards the stereogram peripheryand that the number of points near the centre point of thestereogram (along the z axis) is increasing with bre vol-ume fraction. Fig. 2c shows the stereographic projectionswith density contours, produced using the counting gridapproach. A square grid (41 41 cells) is superimposedon the plot. The points inside each cell are totalled. Theresultant matrix of concentration values is then contoured.

Table 2Architectural characteristics extracted from the volumetric reconstructions

Fibre volumefraction, f (%)

No. ofbres ()

No. of bresegments ()

Mean segmentlength, L (lm)

10 2853 38 22,851 62 237 215 4623 297 39,264 2,215 186 520 6269 131 59,943 7 153 0

S. Neelakantan et al. / ActaIt can be seen that the bre inclination angles are not uni-formly distributed between the quadrants. This eectappears to become more pronounced in the 20 vol.% net-works; the right quadrants appear to contain a higher den-sity of points compared with those on the left.

3.2. Single bre testing

Typical tensile stressstrain curves for as-received andheat-treated 316L bres are shown in Fig. 3. Table 3 sum-marizes the properties extracted from the curves. (It isworth noting that the values are comparable with thoseobtained for other variants of stainless steels bres[11,15,19].) It can be seen that, while both bres exhibitsimilar work hardening rates (Fig. 3), the sintering heattreatment induced a signicant reduction in strength andductility (Table 3). Consequently, the nominal work offracture Us for a single bre, measured as the area underthe loadstrain curve, was appreciably lower for the heat-treated bres than that for the as-received bres. The loweryield strength values may be attributed at least partly tograin coarsening, which occurred during the sintering pro-cess (Fig. 1a and b). Furthermore, the reduction in strengthand ductility may be attributable to precipitation of inter-metallic particles at the austenite grain boundaries. Electro-lytic etching (Fig. 1e) reveals the presence of r-phaseislands at the austenite grain boundaries after 6 s of etch-ing. It may be possible to reduce the r-phase embrittlementby annealing; precipitates can be dissolved on annealing[28].

3.3. Network in-plane tensile testing

3.3.1. Elastic properties (quasi-static)

The measured in-plane Youngs moduli and the Pois-sons ratios of the network plates are given in Table 4.The shear moduli Gxy, deduced from the Youngs modulusand Poissons ratio, are also tabulated. The table showsthat the in-plane Youngs moduli and the out-of-planePoissons ratios increase with bre volume fraction,whereas in-plane Poissons ratios are independent of bredensity. It can be seen that the 10 and 15 vol.% networksexhibit very similar moduli values in the longitudinal andtransverse directions, whereas a slight deviation in this isobserved for the 20 vol.% networks. However, in-planePoissons ratios mxy and myx are almost equal (in the range0.260.28), which would be expected if the plates were

316L bre networks.

Average no. of bondsper bre ()

Fibretortuosity ()

Mean bre inclinationangle, h ()

8.97 0.07 1.51 0.02 82.88 0.319.04 0.36 1.51 0.01 81.87 0.219.17 0.13 1.85 0.01 79.62 0.13

terialia 66 (2014) 326339 331transversely isotropic. Some dierences, though, areobserved between the out-of-plane Poissons ratios mxzand myz for the 10 and 20 vol.% networks.

A further point here, shown in Table 4 and also illus-trated in Fig. 4, is that the networks exhibit strong auxeticbehaviour in the out-of-plane direction, i.e. negative Pois-sons ratios. It should be noted that a transversely isotropicmaterial has no theoretical limit on the value of a negativePoissons ratio in the out-of-plane direction, in contrast tothe familiar limit of 1 for an isotropic solid. That limitarises from the relation between Youngs modulus,

Ma332 S. Neelakantan et al. / ActaPoissons ratio and shear modulus, but for a transverselyisotropic material these are independent elastic constants[37].

Evidently, auxeticity rises with bre density. Similar lev-els of auxeticity and dependence on bre density have beentheoretically predicted [25] and experimentally measured[14] for bre networks consisting of ner bres. Since thebres themselves are not auxetic, the origin of this behav-iour is attributed to the processing of these sheets. In cellu-lar solids, it is well established that a non-convex (re-entrant) cell shape gives a negative Poissons ratio [38],and that very high negative values can arise in some geo-metric congurations. Although bre networks do notexhibit a cell structure as such, bre kinking is intro-

(a)Fig. 2. Fibre segment orientation distributions for the three 316L networks sho(through-thickness) direction. Also shown are stereographic projections on thedensity contours.terialia 66 (2014) 326339duced during processing, as layers of randomly laid (loose)bre felts are compressed to plates with dierent bre vol-ume fractions and subsequently sintered. It is worth notingthat, while the bre tortuosity levels are higher for the20 vol.% networks, the 10 and 15 vol.% networks exhibitvery similar values (Table 2). It is also of note that the bresegment lengths (inter-joint distances) are decreasing withincreasing bre volume fraction, whereas the number ofbonds per bre is increasing. DIC images captured in situ(not shown) suggest that, even at very low strains, weakinter-bre bonding results in longitudinal cracking/split-ting. This is not surprising, as the width of the solid-statesintered necks can be quite small relative to the bre diam-eter. Bending of the bre bundle splits, resulting in outward

(b) (c)wing (a) the probability distributions of the bre inclination angle h to the zxy plane (b) with points representing the in-plane bre orientations and (c)

S. Neelakantan et al. / Acta Materialia 66 (2014) 326339 333Fig. 3. Typical tensile stressstrain curves for as-received and heat-treated316L bres.

Table 3Tensile properties of as-received and heat-treated single bres, asmeasured by single bre testing.

Condition r0.2%Yieldstrength(MPa)

Ultimatetensilestrength(MPa)

Strain tofailure(%)

Fractureenergy(J m1)

As-received 465 16 877 11 30.0 1 0.282 0.015Heat-treated 252 11 564 17 14.7 1 0.081 0.008bulging in the through-thickness direction. This can be seenin the post-fracture images shown in Fig. 5. It can be seenthat the cracks form a tortuous path as they pass throughbundles of bres. Of course, these post-fracture images donot relate directly to linear-elastic behaviour and Poissonsratio, but nevertheless the conspicuous cross-directionexpansion is indicative that the bre topology tends to pro-duce strong dilation in all regimes of deformation inresponse to uniaxial loading.

3.3.2. Deformation and fracture

Typical loaddisplacement curves obtained from in-plane tensile testing of 316L networks with dierent brevolume fractions are presented in Fig. 6. A similar defor-

Table 4Elastic constants and fracture energies for 316L bre networks, as measured b

Elastic constants Fibre volume fraction, f (%)

10

Ex (GPa) 1.17 0.32Ey (GPa) 1.13 0.06mxy () 0.27 0.01myx () 0.27 0.01Gxy (GPa) 0.46mxz () 5.30 0.83myz () 4.45 0.85Fracture energy (kJ m2)Longitudinal 4.27 0.22Transverse 3.42 0.67mation response was observed in the in-plane directionsfor all three networks. All curves show an approximatelylinear response followed by a relatively short region ofplastic ow and strain hardening up to a peak load. There-after, accumulation of damage at the bres and joints even-tually initiates rupture, which is the source of the fall inload that precedes nal failure. The plastic ow and strainhardening region is relatively short for low-bre volumefractions compared with high-bre volume fractions. Thedisplacement to failure remains approximately the same,irrespective of bre volume fraction. Average fractureenergy values in the longitudinal and transverse directionswere approximately the same as shown in Table 4. Estima-tion of the fracture energy in units of kilojoules per squaremetre involves obtaining the area under the loaddisplace-ment curves and then dividing by the fracture cross section.

Fig. 4. Experimental data from in-plane (longitudinal) tensile testing of316L bre networks with dierent porosities, plotted in the form ofthrough-thickness expansion as a function of extension in the longitudinaldirection.As illustrated in Fig. 5, fracture is taking place in a tortu-ous path. A very rough estimate of the fracture area wasobtained by considering the initial gauge length and thick-ness to nd the diagonal length (equivalent to the hypote-nuse of the right angle triangle). Multiplying this length bythe initial gauge width of the samples provided an estimate

y in-plane tensile testing.

15 20

2.19 0.11 2.51 0.062.28 0.59 3.10 0.860.28 0.03 0.27 0.010.26 0.02 0.26 0.030.86 0.997.37 0.32 10.67 0.297.54 0.48 11.27 0.66

7.71 0.66 11.41 0.947.24 0.46 10.92 1.32

sid

334 S. Neelakantan et al. / Acta MaFig. 5. (a) Optical micrographs and (b) DIC images showingof the fracture area. Acknowledging that some gross sim-plications are being incorporated here, the estimated frac-ture area was approximately ve times larger than thecross-sectional area (based on nominal dimensions). Frac-ture energy values were found to increase with bre volumefraction in an approximately linear fashion, which is inagreement with a previously developed analytical model[16].

The aforementioned model [16] predicts the fractureenergy Gfr for this type of brous material, based on thework of fracture of a single bre, assuming that all bresfracture. This model leads to

Fig. 6. Typical tensile loaddisplacement curves for 316L bre networks,of dierent bre volume fractions, measured along the longitudinal (x)and transverse (y) directions.

Fig. 7. Schematic representation of the geometry used for prediction ofthe fracture energy of the bre networks.Gfr 2f sin/3

3

pa2

U fz 5

where a is the side length of the bre hexagonal cross sec-tion, z is the deformation zone and / is the in-plane breinclination angle to the x axis (Fig. 7). The fraction inbrackets represents the number of bres per unit sectionalarea.

From Eq. (5), it can be seen that, by raising the fractureenergy of the bre and controlling the deformation zonelength, this process can absorb large quantities of energy.In deriving this equation, it is assumed that all bre seg-ments lie at the same in-plane inclination angle /. How-ever, in reality, the in-plane orientation distribution ofthe bre segments is more complex than this. To approxi-mate this, an orientation probability distribution P(/) fora range of / values between 0 and p/2 was introduced inEq. (5).

Gfr 2fU fz

R p=20

sin/P /d/3

3

pa2

6

P(/) can be represented using histograms of the typeshown in Fig. 2a, derived using X-ray tomography, witheach measured in-plane angle allocated to a size bin. Dis-cretization of P(/) allows a probability P(/i) to be assignedto all bres allocated to a bin size of D/i, so Eq. (5) can berewritten as

Gfr 2fU fzPn

i sin/iP /iD/ip 7

e views of 316L bre networks after in-plane tensile testing.

terialia 66 (2014) 3263393 3a2

where P(/i) is given by

P/i 1

D/i N/iPn

i N/i8

N hi is the number of bres inclined at an angle /i (i.e. with-in a bin size of D/i, centred at /i).

A plot of the predicted fracture energies of the networksagainst bre volume fraction is shown in Fig. 8. Alsoshown are the experimentally estimated values of Gfr inthe longitudinal direction (Table 4). The experimental val-ues are in good agreement with predictions if the deforma-tion zones are taken to have lengths of 80, 110 and 130 lmfor 10, 15 and 20 vol.% networks, respectively. These val-ues are of the same order as the network bre segment

of the same volume fraction. Closed-cell foams are oftenvery poor, particularly under tensile loading, partlybecause of the presence of embrittling constituents in thecell walls [39,40].

3.4. Network cantilever bend testing

Fig. 8. Predicted dependence of the network fracture energy on the brevolume fraction. Also shown are the experimental data for bre networkswith dierent bre volume fractions.

Table 5In-plane elastic moduli for 316L networks, as measured by cantilever bendtesting.

Fibre volume fraction, f (%) In-plane elastic moduli (GPa)

Ex Ey

10 1.10 0.12 1.09 0.0815 1.69 0.03 1.78 0.0220 2.50 0.17 2.82 0.02

S. Neelakantan et al. / Acta Materialia 66 (2014) 326339 335lengths, as determined by X-ray tomography (Table 2).Evidently, the size of the deformation zone is increasingwith increasing bre density.

From images such as those shown in Fig. 9, it can beseen that failure occurs by failure at the sintered necks fol-lowed by plastic deformation of individual bres as theyalign along the loading direction. Grain rotationsbecome very obvious in a bamboo structure of this type,provided that very large plastic strains have been induced.In principle, plastic deformation and rupture of the breshas the potential for greater energy absorption than frac-ture of joints. There is considerable scope to maximize thiscontribution by tailoring the bre microstructure (forinstance by annealing: see Section 3.2) to obtain the desiredcombination of strength and toughness.

It is worth noting that, provided both bres and jointsare relatively strong and tough, metallic bre networkscan exhibit good mechanical properties under tensile load-ing compared with open- or even closed-cell metallic foamsFig. 9. SEM micrographs showing: (a) bre alignment along the loading direcexhibiting necking; (d) fracture at a sintered brebre joint; and (e) grain roThe in-plane Youngs moduli obtained from cantileverbend testing are listed in Table 5 for dierent bre volumefractions. Increasing moduli values were observed withincreasing bre volume fraction, similar to in-plane tensiletesting. In addition, similar to in-plane tensile testing, can-tilever bending yields dissimilar in-plane moduli values forthe 20 vol.% networks (Table 5).

3.5. Network vibration testing

3.5.1. Plate vibrations

As described in Section 2.4.1, the resonant frequenciesof the rst three mode shapes, shown in Table 6, were ini-tially measured. These frequencies led to rst estimates ofD1, D3 and D4. To estimate D2, the square plate (withadjusted aspect ratio) was used to observe the O (ring)and X vibration mode frequencies (mode nos. 2 and 3in Table 7). The accuracy of these rst estimates of D1,D2, D3 and D4, was improved using an iterative method,as described in Section 2.4.1. Since there should be notion; (b) necking of individual bres prior to fracture; (c) a fractured bretations at large plastic strains.

ad

me

MaTable 6Measured and predicted mode frequencies (in Hz) for rectangular plates m

Mode no. Mode shapes Fibre volu

1 101520

336 S. Neelakantan et al. / Actapreferred direction in any of the bre networks, this itera-tive process was carried out using an assumption of trans-verse isotropy. The rst estimates of elastic constants wereconsistent with this idea: the empirical values for D1 and D3were very similar for each network material.

Tables 6 and 7 show the measured natural frequenciesfor rectangular and square plates, with dierent bre vol-ume fractions, compared with the predictions using thenal iterated values of the four elastic constants. Identicalvalues of the material properties were used for the twoshapes of plate with each tested bre volume fraction: only

2 101520

3 101520

4 101520

5 101520

6 101520

7 101520

8 101520e of 316L networks with dierent bre volume fractions.

fraction, f (%) Frequency (Hz)

Measured Predicted

72 7270 7070 70

terialia 66 (2014) 326339the dimensions were changed. This results in considerableredundancy in the comparison: many more modes areexamined than would be needed simply to t the four val-ues. Irrespective of bre volume fraction, it can be seenthat, for the rst ve or six modes of each plate, there isgood agreement between measured and predicted frequen-cies. The disparity between measured and predicted fre-quencies increases as frequencies go up, with predictedfrequencies being systematically higher than measuredones. This eect seems to be more pronounced for the20 vol.% networks. This pattern of deviation is exactly

94 9388 8896 96

158 151146 146139 139

172 173157 168162 172

204 205185 198185 193

267 266241 254246 278

310 338288 326

319 344286 333290 333

of 3

me

MaTable 7Measured and predicted mode frequencies (in Hz) for square plates made

Mode no. Mode shapes Fibre volu

S. Neelakantan et al. / Actawhat one would expect from the inuence of through-thickness shear eects, as the theory used for the predic-tions ignores such shear deformation.

Table 7 also shows that all vibration modes that do notshow symmetry in the diagonal line of the square plate(modes 4, 5 and 6, 7) occur in pairs with approximately

1 101520

2 101520

3 101520

4 101520

5 101520

6 101520

7 101520

8 101520

9 10152016L networks with dierent bre volume fractions.

fraction, f (%) Frequency (Hz)

Measured Predicted

terialia 66 (2014) 326339 337the same resonant frequencies. The agreement is close forthe 10 and 15 vol.% networks, but less close for the20 vol.% network. This supports the assumption that theplates exhibit transverse isotropy. At low-bre volumefractions, there are no signicant deviations from the theo-retical predictions, while at high-bre volume fractions,

77 7874 7673 76

116 117112 112112 112

139 141136 136135 142

200 204192 197183 193

199 204187 197182 204

331 359314 348304 328

331 359314 344302 378

338 370308 359300 361

361 409336 394

ibra

ions

m

Mathere are some eects of material inhomogeneity, presum-ably attributable to the manufacturing process. This con-clusion is in general agreement with the quasi-statictensile and bend test results. Even with these eects of inho-mogeneity, the general pattern of frequencies ts the pre-dictions well enough that one can have some condencein the tted elastic constants, interpreted in an averagedsense over the areas of both plates and over all directionsof wave propagation. The iterated and adjusted D1D4 val-ues are related to the in-plane engineering elastic constants(Ex, Ey, mxy, myx and Gxy) via Eqs. (3). The values are sum-marized in Table 8.

3.5.2. Beam vibrations

The beams used for vibration testing gave most accurate

Table 8In-plane elastic constants for 316L bre networks, as measured by plate v

Fibre volume fraction, f (%) In-plane elastic moduli (GPa)

Ex Ey

10 1.29 1.2915 1.98 2.0220 2.55 3.40

Table 9In-plane Youngs moduli for 316L networks, as measured by beam vibrat

Fibre volume fraction, f (%) Specimen orientation Beam density (kg

10 Longitudinal 701.55Transverse 746.02

15 Longitudinal 1114.11Transverse 1193.32

20 Longitudinal 1569.43Transverse 1558.11

338 S. Neelakantan et al. / Actaresults for the lower modes of vibration (modes 1 and 2).The corresponding measured frequencies, which are anaverage of at least three measurements, and the beam den-sity values are given in Table 9. The obtained in-planeYoungs moduli of the networks are also listed in Table 9.Similar to other testing methods, the in-plane moduli val-ues increased with bre volume fraction, and deviationsdue to inhomogeneity are also observed for the 20 vol.%networks. As with the plate vibration test, when highermodes of the beam samples were examined, eects due tothrough-thickness shear were observed.

4. Discussion and conclusions

An experimental investigation was carried out into thedeformation response of bre networks, made by solid-state sintering of 316L austenitic stainless steel bres. Net-works with 10, 15 and 20 vol.% bre volume fractions havebeen characterized in terms of network architecture andtheir response to quasi-static mechanical and dynamic test-ing. The following conclusions can be drawn from thiswork.1. Fibre segment lengths were found to decrease withincreasing bre volume fraction. The highest bre tortu-osity levels were observed in the 20 vol.% networks,which may be associated with the way in which thesheets have been processed; an external pressure isapplied to randomly laid bre felts in order to producesheets of specic porosity. As expected, the majority ofthe bre segments lie at inclination angles (to the verti-cal) >75. However, the probability of having bre seg-ments inclined at such inclination angles increases asbre density falls. Stereographic projections on the xyplane (in-plane directions) suggest that the 10 and15 vol.% networks are transversely isotropic, whereasthere is some indication that the 20 vol.% networks areinhomogeneous in terms of the spatial arrangement of

tions.

Poissons ratios () Shear modulus (GPa)

mxy myx Gxy

0.26 0.26 0.470.27 0.28 0.760.24 0.32 1.07

.

3) Mode frequencies (Hz) In-plane elastic moduli (GPa)

Mode 1 Mode 2

117.19 0.00 332.03 23.28 1.08 0.07109.86 4.88 330.40 32.51 1.11 0.10104.17 11.28 292.97 0.00 1.49 0.13100.91 5.04 277.62 5.22 1.39 0.06100.91 5.04 287.39 13.64 1.94 0.18117.19 0.00 312.50 0.00 2.39 0.00

terialia 66 (2014) 326339the bres.2. A range of quasi-static and dynamic techniques were

employed to determine the elastic constants of the net-work sheets. The main corollary, in agreement withthe tomography results, is that the 20 vol.% sheets showsigns of material inhomogeneity. This eect is mademost evident in the plate vibration results (Tables 6and 7). When comparing the in-plane stiness valuesobtained from in-plane tension with those measuredby cantilever bending, and plate and beam vibrations,it can be seen that all techniques yield very similar sti-ness values for the 10 vol.% networks, but dierencesarise among the dierent methods with increasing bredensity. This may be attributed to the fact that, as thebre volume fraction increases, there is progressive evi-dence of shear eects in the cantilever and vibration test-ing. This could be attributed to the fact that the bresegment length decreases (Table 2), causing the shearcomponent arising from the individual struts to have amore dominant eect on the network deformation,resulting in lower stiness values. It should also be notedthat bend tests, whether static or dynamic, are inu-

enced mainly by the material in the outer layers of thenetwork sheet, whereas tensile testing gives equal weightto the material throughout the thickness.

3. The networks were found to expand in the through-thickness direction when stretched in-plane, resultingin Poissons ratios lying between 5 and 11. Whilethe deformation mechanisms that give rise to suchstrong auxetic behaviour cannot be easily identied withthis type of testing, this behaviour was attributed to out-

[6] Kostornov AG, Shevchuk MS, Gorb ML. Sov Powder Metall MetCeram 1972;11:326.

[7] Liu P, He G, Wu LH. Mater Sci Eng, A 2008;489:21.[8] Liu P, He G, Wu L. Mater Des 2009;30:2264.[9] Liu P, He G, Wu LH. Mater Charact 2009;60:900.[10] Golosnoy IO, Tan JC, Clyne TW. Adv Eng Mater 2008;10:192.[11] Tan JC, Clyne TW. Adv Eng Mater 2008;10:201.[12] Liu P, He G, Wu L. Mater Sci Eng, A 2009;509:69.[13] Ducheyne P, Aernoudt E, De Meester P. J Mater Sci 1978;13:2650.

S. Neelakantan et al. / Acta Materialia 66 (2014) 326339 339ward bending of kinked bres (kinking was induced dur-ing processing, owing to the applied pressure), as thenetworks are stretched in-plane. Such a mechanismwould be similar to that studied in detail for 2-D honey-comb structures by Gibson and Ashby [41]. SynchrotronX-ray tomography experiments are currently beingplanned to validate the above observations.

4. Fracture energies of the order of several kilojoules persquare metre were obtained experimentally. These val-ues are consistent with predictions based on the assump-tion that this energy is primarily associated with plasticdeformation of individual bres within a process zone ofthe order of inter-joint spacing. It is worth noting that,by raising the fracture energy of the bre (e.g. by anneal-ing), this process can make a signicant contribution tothe fracture energy of the networks.

Acknowledgements

This research was supported by the European ResearchCouncil (Grant No. 240446). The authors wish to thankMiss Erika Oberg, of the Materials Science Departmentat Cambridge University, for help with cantilever testing.Acknowledgement is also due to Mr. Karthikeyan Kan-dan, of the Engineering Department at Cambridge Univer-sity, for his help with DIC measurements.

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Characterization and deformation response of orthotropic fibre networks with auxetic out-of-plane behaviour1 Introduction2 Materials and methods2.1 Materials2.1.1 Single fibres2.1.2 Sintered fibre networks

2.2 Material characterization2.2.1 Electrolytic and tint etching for optical microscopy2.2.2 Scanning electron microscopy2.2.3 X-ray computed tomography2.2.3.1 Equipment2.2.3.2 Architecture characterization

2.3 Mechanical testing2.3.1 Single fibre tensile testing2.3.2 Network in-plane tensile testing2.3.3 Network cantilever bend testing

2.4 Vibration testing2.4.1 Plate vibrations2.4.2 Beam vibrations

3 Results3.1 Network architecture3.2 Single fibre testing3.3 Network in-plane tensile testing3.3.1 Elastic properties (quasi-static)3.3.2 Deformation and fracture

3.4 Network cantilever bend testing3.5 Network vibration testing3.5.1 Plate vibrations3.5.2 Beam vibrations

4 Discussion and conclusionsAcknowledgementsReferences