Mechanics of Materials 75 (2014) 125–134

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Mechanics of Materials

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Effects of defects on the tensile strength of short-fibrecomposite materials

http://dx.doi.org/10.1016/j.mechmat.2014.04.0030167-6636/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +46 18 4713026.E-mail address: kristofer.gamstedt@angstrom.uu.se (E.K. Gamstedt).

Thomas Joffre a, Arttu Miettinen b, Erik L.G. Wernersson c, Per Isaksson a,E. Kristofer Gamstedt a,⇑a Uppsala University, Ångström Laboratory, Department of Engineering Sciences, Box 534, SE-751 21 Uppsala, Swedenb University of Jyväskylä, Department of Physics, Box 35, FI-40014 Jyväskylä, Finlandc Swedish University of Agricultural Sciences, Centre for Image Analysis, Box 337, SE-751 05 Uppsala, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 June 2013Received in revised form 21 March 2014Available online 18 April 2014

Keywords:StrengthAgglomerationX-ray microtomographyComposite materials

Heterogeneous materials tend to fail at the weakest cross-section, where the presence ofmicrostructural heterogeneities or defects controls the tensile strength. Short-fibre com-posites are an example of heterogeneous materials, where unwanted fibre agglomeratesare likely to initiate tensile failure. In this study, the dimensions and orientation of fibreagglomerates have been analysed from three-dimensional images obtained by X-ray mic-rotomography. The geometry of the specific agglomerate responsible for failure initiationhas been identified and correlated with the strength. At the plane of fracture, a defect inthe form of a large fibre agglomerate was almost inevitably found. These new experimentalfindings highlight a problem of some existing strength criteria, which are principally basedon a rule of mixture of the strengths of constituent phases, and not on the weakest link.Only a weak correlation was found between stress concentration induced by the criticalagglomerate and the strength. A strong correlation was however found between the stressintensity and the strength, which underlines the importance of the size of largest defects informulation of improved failure criteria for short-fibre composites. The increased use ofthree-dimensional imaging will facilitate the quantification of dimensions of the criticalflaws.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The micromechanical models used to predict tensilestrength of short-fibre composite materials, e.g. theFukuda–Chou model (1982) or the Cox model (1952), havebeen inspired by micromechanical models for elastic prop-erties, where the composite stiffness is determined fromstiffness contributions of the constituents and effects offibre orientation, fibre content and fibre length. Thesemodels may have surprisingly good predictive capabilities(Andersons and Joffe, 2011; Andersons et al., 2011; Virk

et al., 2012) although mechanism-based criterion wouldbe preferable, relying on the dimensions and propertiesof the most severe flaw initiating ultimate failure. Withthe advent and increased use of three-dimensional imag-ing techniques, most notably microfocus X-ray computedtomography, it is becoming easier to quantify these param-eters (Buffière et al., 1999; Molnár and Bojtár, 2012). Thepresent study is an attempt to investigate the correlationbetween composite strength and the physical features ofthe failure instigating flaw in an important class of brittleheterogeneous materials, namely short-fibre composites.

Although no previous studies have been found on theeffects of critical flaws in the specific form of fibre agglom-erates on strength, there are several studies on the relation

126 T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134

between voids and strength among other mechanical prop-erties of materials (Harper et al., 1987; Huang, 2004; Liuet al., 2006). The general consensus is that it is the mostsevere flaw or imperfection in the material that gives riseto brittle tensile failure. This is typically evidenced by frac-tography, where the examination of fracture surfaces byscanning electron microscopy can be traced back to thecritical flaw that initiated crack growth leading to finalfailure (Greenhalgh, 1993; Hull, 1999). X-ray computedtomography (XlCT) provides a useful non-destructivetechnique to quantify the microstructure of materials, notonly on the fracture surface, but also in three-dimensionsinside the material. To this end, XlCT has already beenused to measure crack growth and pores in concrete mate-rial under compressive loading (see e.g. Landis and Keane,1999; Landis et al., 2003). For fibre composite materials,XlCT has for instance been used to quantify internal dam-age (Schilling et al., 2005) and to observe crack openingdue to tensile loading (Moffat et al., 2008).

In the present study, the samples were scanned beforeand after mechanical testing, allowing to identify the spe-cific microstructural features that caused the failure. Ashort-fibre composite was chosen as model for a heteroge-neous material, whose microstructure can be characterizedby XlCT. Wood fibres were dispersed in a polylactide (PLA)matrix, forming a renewable composite material suitablefor packaging applications. The cellulosic fibres tend toagglomerate in the injection-moulding process (Chinga-Carrasco et al., 2012), and thus form natural defects whichcould be related to the macroscopic strength.

In the following, the modelling approach is presented.Two strength models are outlined: (i) one accounting forthe stress concentration factor caused by the largestdetected flaw, (ii) one based on a fracture mechanicsapproach, where the size of the largest flaw is considered.Both of these models account the largest defect, in contrastto rule-of-mixture based criteria. The stress concentrationfactor approach accounts for the shape of the largestdefect, and the stress intensity factor approach accountsalso for the size of the largest defect. Thereafter, experi-mental details are presented, followed by the results andcomparison of the correlation between the experimentalresults and the two models. Finally, some concludingremarks are given concerning developments of mechanismbased models for strength predictions of heterogeneouscomposite materials.

2. Stress analysis

2.1. Assumptions

For the present model material, scanning electronmicroscopy revealed that the fibres were agglomeratedinto clusters of roughly ellipsoidal shape. Thus, three-dimensional general ellipsoids were considered versatileenough to describe the shape of the agglomerates for mod-elling purposes. Although it is a major idealization that thefibre agglomerates take an ellipsoidal shape with regard tothe surrounding stress field, some hypothesis has to bemade in order to make a quantitative analysis possible.

We argue that this assumption is acceptable, since theaim at this stage is only to compare stress concentrationeffects with size-dependent stress-intensity effects onstrength. Furthermore, the mechanical behaviour of thematerial is assumed to be elastic, since the macroscopictensile behaviour was observed to be linearly elastic. Fur-thermore, the material was observed to fail in a brittlemanner.

Fractographic studies of the material and other similarwood-fibre composites (Chinga-Carrasco et al., 2010), haveshown that fibre agglomerates easily debond from the sur-rounding thermoplastic matrix. Oksman et al. (2009)noticed that the strength of cellulose fibre compositeswas unaffected by the type of fibre when the fibre–matrixinterface is poor, as is commonly the case for melt-processed cellulose fibre reinforced thermoplastics.Nyström et al. (2007) showed that the aspect ratio, i.e.the shape of the reinforcing units, had a large impact oncomposite strength, whereas the strength of the reinforce-ment has negligible influence on the measured strength forcellulose-fibre composites. Again, this was linked to pooradhesion and interfacial stress transfer. No residual resinwas found on the fibre surfaces as would be expected incohesive failure with a stronger interfacial bond. At theinstance of failure, it is assumed that the largest agglomer-ate is completely debonded from the matrix, and that theagglomerate does not carry any load. Even if the fibreagglomerates would be bonded to the matrix, the stiffnessof the fibre in the transverse direction is much inferior totypical polymer matrix systems (Bergander and Salmén,2000; Joffre et al., 2014). If effects of lumen and other voidsare taken into account, the transverse stiffness of the fibreagglomerates would be even lower.

Since the agglomerates tend to debond from the matrixand have relatively low transverse stiffness, they areregarded as ellipsoidal cavities. Thus the stiffness of theellipsoidal inclusions is effectively set to zero. The dimen-sions, distributions, and orientation of defects inside thematerial have been characterized by XlCT. Since it is anon-destructive technique, the samples have been scannedbefore and after tensile testing. The three-dimensionalimages obtained have been used to identify the specificagglomerate responsible for the failure of the sample dur-ing mechanical loading.

The first step is to estimate the stress state around theellipsoidal cavity surrounded by a resin rich region. Accu-rate analytical solutions exist for the stress state aroundan inclusion in an infinite matrix (Mura, 1987). The stressperturbation with regard to the far-field applied stress inthe composite is different from the one caused by inclusionin the hypothetical infinite matrix. If we assume that theagglomerate is embedded in a matrix surrounded by thecomposite material, the strain continuity at the matrix/composite interface gives a relationship between thefar-field applied stress in the composite case, r1ij , and thefictitious applied stress in the infinite matrix case, r0

ij,

Scijklr

1kl ¼ Sm

ijklr0kl ð1Þ

where Scijkl and Sm

ijkl are the compliance tensors of the com-posite and matrix, respectively. By multiplication with the

T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134 127

elastic tensor for the isotropic matrix, Cmijkl, a relation

between the applied far-field stress and the effective stressapplied on to the matrix embedded agglomerate isobtained:

r0ij ¼ Cm

ijklScklmnr

1mn ð2Þ

For an aligned orthotropic composite and an isotropicmatrix subjected to a uniaxial tensile stress, this equationcan be simplified by neglecting the higher order termsand the relatively small off-axis stress components:

r0z ¼

Emð1� mÞEc

Lð1þ mÞð1� 2mÞr1z ð3Þ

where Em and m are the Young’s modulus and Poisson’sratio of the matrix, respectively. Ec

L is the Young’s modulusof the composite in the longitudinal direction. For our com-posite material, and other orthotropic materials with themain material axis aligned along the load direction, thetransverse stresses become negligible compared with thestress in the z direction. This is why a uniaxial far-fieldstress state is assumed also for the modelling case withan ellipsoid surrounded by an isotropic matrix only.

In the following, the stress perturbation aroundagglomerates is determined in two different ways; (i) thestress concentration factor from Eshelby theory, and (ii)the stress intensity factor from fracture mechanics. Themain difference by the two approaches is that the formeraccounts for the three-dimensional shape of the agglomer-ate, whereas the latter assumes that the defect is a two-dimension elliptical crack and thereby accounts for thelength scale of the defect. In the latter case, the three-dimensional ellipsoidal inclusion is collapsed into to two-dimensional elliptical crack in the direction of the mostminor of the semi-principal axes.

2.2. Stress concentration factor

The stress perturbation due to a general triaxial ellipsoi-dal cavity, such as the one represented in Fig. 2.a, wasderived by Sadowsky and Sternberg (1949) using ellipsoi-dal coordinates and a harmonic function. A more straight-forward method was then introduced by Eshelby (1957)who calculated the stress inside and outside the inhomo-geneity using an equivalent inclusion method. Theapproach presented here is based on the work of Ju andSun (1999, 2001), who presented an explicit solution forthe stress field in the matrix using Eshelby’s externaltensor.

The stress at a point p ¼ ðx1; x2; x3Þ in a Cartesian coor-dinate system, outside the ellipsoidal inhomogeneity cen-tred at ð0;0;0Þ, can be written as the sum of the far fieldstress in the infinite matrix, r0

ij; applied during mechanicaltesting and the stress perturbation due to the presence ofthe defect, r0ijðpÞ

rmij ðpÞ ¼ r0

ij þ r0ijðpÞ ð4Þ

Note that the stress field r0ij corresponds to the stress r0

z

expressed in the local coordinate system ðx1; x2; x3Þ definedby the principal directions the ellipsoid. The stress pertur-bation in the matrix is expressed as the product of the

exterior Eshelby tensor, GijklðpÞ; and the eigentstrain e0ij,

expressed in the coordinate system defined by the direc-tions of the axes of the ellipsoidal crack (Mura, 1987)

r0ijðpÞ ¼ CmijklGklmnðpÞe0

mn ð5Þ

An explicit expression of the tensor GijklðpÞ is given inGao and Ma (2010), and e0

ij is given by Ju and Sun (1999,2001). The full expressions of these two tensors can befound in Appendix A.

To calculate the stress tensor outside the ellipsoid in arobust way, the method used by Healy (Healy, 2009) wasadvocated: using Eq. (2), the stress tensor in the matrixdirectly outside the ellipsoidal inclusion was calculatedon the surface an ellipsoidal shell 0.01% larger than theinclusion. The stress concentration factor (SCF) is thengiven by the ratio between the largest first principal stressdirectly outside the ellipsoidal inclusion and the appliedfar-field stress, r0

ij (see Fig. 2b). Note that it can be shownthat the Eshelby exterior tensor, GijklðpÞ; is size indepen-dent, i.e. it only depends on the ratio between the lengthof the principal axes of the ellipsoid (a2=a1, a3=a1) (Healy,2009). The SCF does thus not capture any length scaleeffect, i.e. the stress concentration is the same for a macro-scopic agglomerate as for a microscopic one with the sameshape and orientation. The strength of the material may bepredicted by assuming that failure occurs when the criteria

rth ¼ rmax=SCF ð6Þ

is satisfied, where rmax is the strength of a sample withoutany agglomerate. In the present study all samples hadagglomerates of different geometries, and rmax was identi-fied by least square fitting of the experimental results.

2.3. Stress intensity factor

To account for size dependence, fracture mechanics canbe used. However, the inhomogeneity has now to be con-sidered as a flat two-dimensional crack. The assumptionused here is that a crack will propagate in the plane per-pendicular to the smallest of the semi-principal axes.Kassir and Sih (1966) have derived the stress intensity fac-tors (SIFs) for an elliptic slanted crack. Contrary to the SCF,the SIFs depend on the size of the agglomerate. The SIFs forthe fracture in opening and shearing mode are given by theexpressions (Sih and Cha, 1974)

K1 ¼r0sin2b

EðkÞ

ffiffiffiffiffia2

a1

ra2

1sin2aþ a22cos2a

� �1=4ð7Þ

K2 ¼�r0 sinbcosb

ða21sin2aþa2

2cos2aÞ1=4 1�ða2=a1Þ2h i

�ða2=a1Þ�1=2 a1 sinasinxðk2�mk02ÞEðkÞ�mk02KðkÞ

þ a2 cosacosxðk2�mÞEðkÞþk02KðkÞ

" #

ð8Þ

respectively, where K1 and K2 are the opening and shear

mode stress intensity factors, k ¼ 1� ða2=a1Þ2h i1=2

and

k0 ¼ 1� k2: The quantities E(k) and K(k) correspond to the

elliptical integrals of the first and second kind. The angles

128 T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134

b, x and a are defined in Fig. 3. The strength of the materialmay be predicted by assuming that the brittle failureoccurs when the mixed-mode fracture criteria

K1

K1c

� �m

þ K2

K2c

� �n

¼ 1 ð9Þ

is satisfied (Leonardo, 1979), where K1c and K2c are thefracture toughness values of the material in opening andshear mode, respectively. Here, K1c has been taken equalto 4 MPa�m1/2 according to experimentally determined val-ues (Park et al., 2004) whereas K2c has been approximatedusing a ratio K1c=K2c ¼ 0:8 (Melin et al., 2000). Further-more, the exponents m and n were taken to equal 1 and2, respectively, following the experimental work of Wu(1967).

3. Experimental procedures

3.1. Materials and manufacturing

The fibres used in this study are pulp fibres from Nor-way spruce (Picea Abies). They were extracted from woodby three commonly used processes: chemithermomechan-ical pulping (CTMP), thermomechanical pulping (TMP) andkraft pulping. A description of the fibre separation pro-cesses can be found in Chinga-Carrasco et al. (2009).

The matrix material is polylactic acid (PLA). PLA is ther-moplastic aliphatic polyester manufactured from starch-rich renewable resources such as maize, sugar beets orwheat. PLA has decent physical and mechanical properties,making it a candidate for substitution of certain petro-chemical thermoplastics (Wolf et al., 2005). Due to thegood adhesion to cellulose fibres, it is also suitable as

Fig. 1. Identification of the critical agglomerate: (a) a dog bone composite samodelling of the agglomerates using ellipsoids. The rupture plane is marked wiinterpretation of the references to colour in this figure legend, the reader is refe

matrix material in wood–fibre composites (Huda et al.,2006a; Huda et al., 2006b).

PLA and wood fibres were blended in a double-screwcompounding unit at SINTEF Materials and Chemistry, Nor-way. The mixture obtained was then moulded to the form ofdog-bone specimens. The mould was preheated in an ovenfor 10 min at 190 �C before being compressed at 50 �C and10 bar during 10 s. The dimensions of the dog bone shapedspecimens are 80 mm � 6 mm � 2 mm (Fig. 1), according tothe standard ASTM D638-IV. The weight fractions of fibresused in this study are 30% and 40%. For each weight fractionten samples were manufactured.

3.2. X-ray micro-computed tomography

XlCT is a non-destructive method that is used toobtain the three-dimensional structure of a sample.Two-dimensional X-ray projection images of the area ofinterest are taken in multiple directions. The X-ray pro-jections are used to computationally reconstruct thethree-dimensional microstructure of the sample, or moreprecisely, the spatial distribution of local X-ray attenua-tion coefficient. As the X-ray attenuation coefficient is dif-ferent for fibres and matrix, the resulting images can beused to distinguish between the two phases (Kak andSlaney, 1988).

In this work the imaging was done using a SkyScan1172 X-ray microtomograph employing X-ray energy of40 keV and power of 10 W. Each sample was scannedbefore and after mechanical testing. The mechanical testitself was performed in a tensile machine, outside thechamber of the XlCT. The voxel size was 4.25 lm. Thescanned area was chosen to contain the whole dog-boneexcept the wide clamping heads. In this way, it could be

mple, (b) identification of the agglomerates with XlCT, (c) geometricalth small blue stripes, (d) principal directions of the critical ellipsoid. (Forrred to the web version of this article.)

T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134 129

ensured that the mechanical failure of the sample wouldoccur in a location inside the image, such that thegeometry of the failure-initiating flaw could be analysed.

Fig. 3. Elliptical crack subjected to inclined loading.

3.3. Image analysis

The reconstructed images were affected by a cuppingartifact due to beam hardening which was removed usinga post processing filter. The signal to noise ratio or con-trast was low due to the similarity of X-ray absorptionin fibres and matrix. Direct thresholding would not bevery accurate and low pass filtering prior to thresholdingwould cause geometrical distortion (van Vliet, 1993). Amaximum a posteriori classifier (Boykov and Funka-Lea,2006; Greig et al., 1989), was used to overcome thoseproblems. It was assumed that the pixel valuescorresponding to the two phases could be described byGaussian distributions and that the clusters of interestwere mostly homogeneous, i.e. not porous. As a final step,the classification into matrix, void or fibre was manuallyverified by inspecting the segmented slices side by sidewith the XlCT images.

An ellipsoid was then fitted to the voxels of each of thesegmented agglomerates. The principal components ofthe points corresponding to the agglomerate were deter-mined, and used as ellipsoid semi-axis directions(Psychol, 1901). The lengths of the semi-axes were stan-dard deviations in the corresponding directions as givenby the principal component analysis. The fitting of the ellip-soids is illustrated in Fig. 1.

3.4. Mechanical testing

Prior to the test the specimens were kept at equilibriumconditions in an environmentally controlled room wherethe temperature was 23 �C and the relative humidity 50%for seven days. The tensile tests were performed in a labora-tory tensile test machine (Instron 5566). The machine wasequipped with a 10 kN load cell and a video extensometer.The specimens were tested at a cross-head speed of 1 mm/min.

Fig. 2. (a) Scheme of a general triaxial ellipsoidal cavity in a solid, and (b) the lasubjected to a stress, r0; aligned with the x3 axis.

4. Result and discussion

4.1. Interface between fibre agglomerate and matrix

Analysis of the XlCT images confirmed the assumptionthat agglomerates have poor bonding with the PLA matrix,after manufacturing and after mechanical testing. For indi-vidual fibres or small agglomerates, the bonding was morecomplete. For the larger agglomerates, a large number ofsmall voids can be observed at the agglomerate-matrixinterface. Occasionally, the void had coalesced into largerinterfacial openings. These interfacial voids appeared asblack zones in the XlCT images as shown in Fig. 4, sincethe X-ray absorption of the cavities was low.

After mechanical testing, the presence of agglomeratesat the fracture interface has been investigated. The XlCTscans consistently confirmed the presence of large agglom-erates at the fracture surface. An example with extremely

rgest principal stress calculated imminently outside an ellipsoidal defect

Fig. 4. XlCT slices showing voids in the agglomerate-matrix interface for (a) kraft 30% material, and (b) kraft 40% specimen. Unlike scanning electronmicrographs of polished cross-sections, the XlCT images shows the interior microstructure and is generally not hampered by damage and artifacts fromspecimen machining and preparation. The length scale applies to both cross-section images.

Fig. 5. Visualisations of a XlCT image of the fracture surface of a sample (30% kraft fibres). The arrows are pointing toward the critical agglomerate, whichinitiated final failure.

Table 1Validation of the analytical solution of the stress concentration by comparison with the results from the finite element analysis.

Inclusion (a1, a2, a3) SCF: analytical solution SCF: FE simulation Difference (%)

Sphere (1,1,1) 2.05 2.05 �0.2Prolate Straight (1,1,5) 1.13 1.12 1.0Oblate Straight(1,5,5) 7.14 7.32 �2.5Prolate, tilted 60� 2.45 2.58 �5.0Oblate, tilted 60� 3.42 3.36 1.7

130 T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134

large fibre agglomerate is presented in Fig. 5: the upperpart of the fracture surface is convex whereas the lowerpart is concave. From this unevenness of either fracturesurface, there are many fibres protruding, which revealsthe presence the fibre agglomerate. Fracture has likely ini-tiated from the exposed interface between the agglomer-ate and the surrounding matrix.

4.2. Stress field and model validation

The analytical solution for the stress concentrationpresented in the modelling section above and in

Appendix A has been validated by comparing theanalytical results with the ones obtained using a com-mercial finite element (FE) software, ANSYS 11.Symmetry planes were used when applicable to reducethe computational time. Around 50000 tetrahedral ele-ments were created for each simulation. The relativedifference between the SCF predictions using the ana-lytical model and the one obtained through the FE sim-ulation did not exceed 5% in any of cases, as compiledin Table 1. An example of this comparison is illustratedin Fig. 6.

Fig. 6. Stress concentration around a sphere subjected to traction determined by (a) FE simulation visualized in a matrix octant, and (b) by the analyticalsolution, shown on the surface of the inclusion. In both cases the SCF is equal to 2.05 for the chosen material properties.

Table 2Average measured longitudinal Young’s modulus and co-fitted strength of avirtual composite having no agglomerates (Eq. (6)).

wf ð%Þ EcLðGPaÞ rmaxðMPaÞ

30 5.9 9040 6.5 96

T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134 131

4.3. Strength correlation

The material properties used to predict the strength aresummarized in Table 2. Em and m were set to 2.3 GPa and0.35, respectively according to literature data (Almgrenet al., 2008; Almgren et al., 2010). The Young’s modulus,Ec

L can be estimated using for instance the laminate anal-ogy (Joffre et al., 2013) but since tensile tests were carriedout within this study, the experimental values were used.The results obtain by using the SIF and the SCF approachesare presented in Fig. 7. Additional information about thevalue of size and orientation of the largest ellipsoid forthe different specimens is provided in Appendix B.

The results obtained by using the SCF approach are pre-sented in Fig. 7(a). The predicted strength using the SCFtheory does not correlate well with the experimentalresults. The coefficient of determination is merelyR2 = 0.22. The results obtained by using the SIF approach

Fig. 7. Strength prediction using (a) the stress concentration factor (b)

are presented in Fig. 7(b). In contrast to the SCF predic-tions, there is a clear correlation between the strength val-ues obtained from experimental testing and from thelength-scale dependent SIF model. The coefficient of deter-mination is R2 = 0.67, which shows an apparent correlationdespite the large scatter in strength. The measuredstrength of the 9 samples having a weight fraction of 30%ranged from 55 to 70 MPa, whereas the strength of the 7samples with a weight fraction of 40% ranged from 61 to74 MPa. Existing models used for the prediction of thestrength of short fibre composite, (Cox, 1952; Fu andLauke, 1996; Fukuda and Chou, 1981, 1982; Zhu et al.,1994), do not account for the variability of strength. Witha criterion not only based on the overall properties of theconstituents, but also including the dimensions of the crit-ical defect, it is possible to reduce the scatter and improvethe predictive precision. The most straight forward way toaccount for the size of a critical effect is to adopt a fracturemechanics approach, as has been advocated in the pre-sented study. Since the defects are generally inclined, itwas necessary to quantify fracture toughness for the differ-ent modes of fracture, and use a mixed-mode fracturecriterion.

The bottle-neck has hitherto been to measure thegeometry of the critical defect. With the ongoing develop-ment of X-ray computed tomography, this hurdle is

the stress intensity factors and a mixed-mode fracture criterion.

132 T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134

becoming increasingly conquerable, which opens doors todevelopment of more accurate and mechanism-based fail-ure criteria. The simple linear-elastic fracture mechanicsmodel used here to relate the strength to the size of thecritical defect could be refined to reduce scatter andimprove predictability further. The first assumption hereis that the agglomerates are effectively acting as voids,i.e. they are completely debonded from the matrix, asobserved experimentally at high applied stresses closeto failure. The interaction between adjacent fibre agglom-erates is also neglected for composites with a low volumefraction of fibres. Some shielding effect can be expected,though, by the presence of other smaller agglomeratesin the vicinity of the critical defect. Furthermore, theagglomerates are not perfectly ellipsoidal, but havethree-dimensional irregular features. In average, however,the agglomerates can be regarded as flat ellipsoids, not farfrom two-dimensional ellipsoidal cracks. The suppressionof the minor radius is motivated by the fact that we candirectly investigate the effect of agglomerate size onstrength by a fracture mechanics model.

5. Conclusions

Failure criteria for strength of short fibre compositesare generally inspired by success of various rules of mix-tures for homogenization of elastic properties. This exper-imental study shows that failure of wood-fibre reinforcedPLA composite is initiated from the most severe defect inthe material, typically a large transversely oriented fibreagglomerate. From X-ray computed tomography, it waspossible to measure the dimensions and orientation ofthe critical defect. The defect was modelled by an ellip-soid, but it was found that the stress concentrationcaused by the ellipsoidal inclusion did not correlate verywell with the measured tensile strength. This led to thehypothesis that not only the shape of the critical defect,but also the length scale of the defect affects the strength.A significantly better correlation with the measuredstrength was obtained by use of a mixed-mode fracturecriterion. It was thus demonstrated with three-dimen-sional image analysis that the strength of a heterogeneouscomposite material is controlled by the geometry and sizeof its most severe defect, and not only by averagemechanical properties of the composite constituents.Further work is, however, necessary to formulate animproved strength model for engineering purposes.X-ray computed tomography techniques also need to berationalized to identify and quantify critical interiordefects.

Acknowledgements

This work was financially supported by WoodWisdom-Net, project WoodFibre3D. Former colleagues at Wallen-berg Wood Science Centre and Department of SolidMechanics at KTH are gratefully acknowledged for theirhelp in tensile testing and numerical analysis.

Appendix A. Stress concentration factor

For a point p ¼ ðx1; x2; x3Þ outside the ellipsoidal inclu-sion centred at O ¼ ð0;0;0Þ. The Eshelby tensor, used inEq. (1) can be expressed as (Gao and Ma, 2010)

GijlmðpÞ ¼ Sð1Þil dijdlm þ Sð2Þij ðdildjm þ dimdjlÞ þ Sð3Þi xlxmdijxlxm

þ Sð4Þl xixjdlmxlxj þ Sð5Þi ðxjxmdilxjxm þxjxldimxjxlÞ

þ Sð6Þj ðxjxmdjlxixm þxixmdjmxixlÞ þ Sð7Þijlmxixjxlxm

ðA1Þwhere

Sð1Þil ðkÞ¼ 18pð1�mÞ ½2mIiðkÞ� IlðkÞþa2

i IilðkÞ�

Sð2Þij ðkÞ¼ 18pð1�mÞ ð1�mÞ½IiðkÞþ IjðkÞ�� IjðkÞþa2

i IilðkÞ� �

Sð3Þi ðkÞ¼a1a2a3

DZk

2ð1�mÞxi

Sð4Þl ðkÞ¼a1a2a3

DZ1

2ð1�mÞ ðkxl�2mÞ

Sð5ÞI ðkÞ¼a1a2a3

DZ1

2ð1�mÞ ðkxi�1þmÞ

Sð6ÞJ ðkÞ¼a1a2a3

DZ1

2ð1�mÞ ðkxj�1þmÞ

Sð7ÞijlmðkÞ¼a1 a2 a3

DZ21

2ð1�mÞxixjxlxm 2�2kðxiþxjþxlþxmÞ�khþ 4kgZ

h iðA2Þ

and

xi ¼1

a2i þ k

ðA3Þ

where m is the Poisson ratio. The constant k is determinedby a new imaginary ellipsoid for each point p such as

x21

a21 þ k

þ x22

a22 þ k

þ x23

a23 þ k

¼ 1 ðA4Þ

Since Eq. (A4) does not have any simple analyticalsolution, k has been evaluated numerically. Note thatsince the point p is outside the inclusion k is strictly posi-tive. The parameters D, Z, h and g are determined by therelations

D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

1 þ kÞða22 þ kÞða2

3 þ kÞq

; Z ¼ x2m

ða2m þ kÞ2

; h

¼ xm and g ¼ x2m

ða2m þ kÞ3

: ðA5Þ

The functions IiðkÞ and IijðkÞ in Eq. (A2) are defined bythe integrals

IiðkÞ ¼ 2pa1a2a3

Z 1

k

dt

ða2iþtÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1þtð Þ a22þtð Þ a2

3þtð Þp

IijðkÞ ¼ 2pa1a2a3

Z 1

k

dtða2

iþtÞða2

jþtÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2

1þtÞða22þtÞða2

3þtÞp

ðA6Þ

In the case where the inclusion is a void, the eigenstraine0 in Eq. (1) can be expressed as

e0ij ¼ � ~Sijkl � Iijkl

� ��1e1kl ðA7Þ

T. Joffre et al. / Mechanics of Materials 75 (2014) 125–134 133

where Iijkl is the identity tensor, e1ij corresponds to the

strain due to the applied far-field stress r1ij ¼ Cmijkle1ij

� �,

and ~S is Eshelby interior tensor defined as (Eshelby, 1957)

~Sijkl ¼1

4ð1� vÞ Sð1Þik ð0Þdijdkl þ Sð2Þij ð0Þðdikdjl þ dildjkÞh i

ðA8Þ

where Sð1Þik and Sð2Þij can be derived from Eq. (A2) by setting kequal to 0.

Appendix B. Geometry of the largest agglomerates forthe tensile-tested specimens

Table B1Weight fraction of fibres, ultimate tensile strength, princi-pal semi-axis lengths and main inclination angle of thecritical defect for each sample.

wf ½%�

rmax½MPa� a1 [lm] a2 [lm] a3 [lm] b [�]

30

62.3 357 183 96 71 62.7 177 119 44 71 68.1 116 73 28 75 56.1 707 187 124 76 59.5 234 100 58 71 66.9 169 119 46 44 63.4 209 119 47 59 64.7 125 114 27 56 66.0 163 119 42 80

40

67.9 218 117 61 65 69.4 344 161 94 85 73.1 473 246 88 88 60.8 147 126 47 60 66.2 257 201 72 66 69.8 132 109 30 69 74.3 158 131 51 74

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