Granular Matter (2010) 12:1–14DOI 10.1007/s10035-009-0155-1

Localised deformation patterning in 2D granular materialsrevealed by digital image correlation

Stephen A. Hall · David Muir Wood · Erdin Ibraim ·Gioacchino Viggiani

Received: 13 October 2008 / Published online: 17 November 2009© Springer-Verlag 2009

Abstract Tests have been performed on an analogue two-dimensional granular material in a special laboratory appa-ratus that allows the application of general stress or strainconditions. Digital image correlation of pairs of consecutivephotographs taken during the tests has enabled fields of dis-placement and hence strain to be determined. Thus directobservation of internal displacements and strains has beenpossible for a series of general strain increments with dif-ferent orientations of principal strain and different imposedangles of dilation. This analysis has successfully providedclear evidence of evolving internal structures of deformation.The observed evolving structures consist of bands of local-ised deformation and ‘cells’ of low deformation between thebands. The orientations of the identified localised featuresand cells are seen to depend on the applied strain path. Thecharacteristic features and dimensions of the bands have beenapproximately identified.

Keywords 2D analogue granular material · Rotationof principal axes · Digital image correlation · Strainlocalisation · Experimental analysis

1 Introduction

In soil mechanics we are used to treating granular materi-als as continua because this allows us to use concepts of

S. A. Hall (B) · G. ViggianiLaboratoire Sols Solides Structures Risques, CNRS/GrenobleUniversities, Grenoble, Francee-mail: stephen.hall@hmg.inpg.fr

D. Muir Wood · E. IbraimDepartment of Civil Engineering, University of Bristol,Bristol BS8 1TR, UK

stress and strain and standard numerical methods for anal-ysis. Soils are made up of individual particles, which arevery evident in sands and gravels, where the grain scaleis measured in millimetres or centimetres, but are less evi-dent in clays, in which the grains are very small (of micronor sub-micron molecular size). The challenge for model-ling is that the fundamental property of a continuum, that itshould appear to be continuous at whatever scale we chooseto observe it no matter how small, is clearly not supportedfor soils.

To be able to retain a continuum assumption we need to besure that we are working at a scale that is sufficiently largethat the particulate nature of the material is no longer evi-dent. Thus it is necessary to ensure that the size of our testspecimen or physical model is large with respect to any intrin-sic dimension of the soil. Typical particle size provides anobvious minimum to such dimension, but experiments haveshown that there are other larger dimensions that should alsobe taken into consideration. This paper presents some obser-vations of characteristic structuring and internal lengths inlaboratory tests on a two-dimensional particulate materialand analyses their nature.

This paper is concerned with the characterisation, usingdigital image correlation, of internal strains and strain local-isation patterning within a 2D laboratory model of a gran-ular material subjected to deviatoric loading along differ-ent strain paths. In the first part of the paper some dif-ferent possible scales of interest are discussed, followedby a presentation of the experimental procedure and theDigital Image Correlation (DIC) analysis. A number ofresults are then presented showing strain fields, whichexhibit patterns of localised deformation revealed by theDIC analysis. The implications of these results are thendiscussed.

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2 Intermediate scales

In standard soil mechanics testing, deformation is measuredat the macro scale, i.e., the scale of the specimen. How-ever, various techniques, including X-ray and photographicmethods, are available that allow better characterisation oflocal deformations throughout a specimen, and their evo-lution with time, at scales intermediate between the granu-lar and sample scales. Such methods can be classed as full-field methods (see [29]). Another important issue in under-standing the physics and mechanics of granular materials isthe way in which forces are transmitted through the grains.However, extracting stress information for granular mate-rials is not straightforward. This can be studied by photo-elasticity, which allows visualisation of stress distributionand transfer of load from a particle to its neighbour (e.g.,the pioneering work of Drescher [9,10] and the ongoingwork of Behringer’s group at Duke University, e.g., [12]and the many results on their webpage: http://www.phy.duke.edu/research/cm/behringer/). Additionally discrete elementmodelling can provide insight into granular behaviour (e.g.,the link between the displacement/strain and forces/stress),but should not be relied on as material truth since it remains amodelling approach and real experimental data are requiredto calibrate/validate the results. In this paper we focus on thekinematics, i.e., displacements and strain.

X-ray radiography has in the past been used to record thepositions of lead markers placed in a soil (e.g., [23]). Anal-ysis of the changing position of such markers can give someinformation about the strain field in the soil, but the gridfor the location of the markers is usually somewhat coarse.Photographic techniques, such as sterophotogrammetry, havebeen used to highlight the development of strain localisationin plane-strain tests on soils (e.g., [7]). X-ray radiographyor tomography can also be used to assess density changesin soil samples (e.g., [2,5,6,22,34]). Recent developmentshave also demonstrated the use of X-ray micro-tomographyto obtain quite detailed information about particles and voids([1,11,14,20,30]), especially with respect to shear bands andother localised phenomena.

Localised density changes observed with X-ray radiogra-phy or tomography are clearly related to localised volumetricstrain (concentrated dilation). Furthermore, since it is knownthat dense sand dilates as it is sheared, it has often beenassumed that zones of dilation observed in X-ray radiographsor tomograms are also zones of localisation of shear defor-mation (rupture surfaces, shear bands), which they may bebut this is not a necessary equivalence (Fig. 1a). Whatever theorigin of the observed density changes, the first length scalethat is clearly much larger than the diameter of a single par-ticle of sand relates to the thickness of the dilation (or shear)bands: a number of experimental observations for a variety ofsands consistently suggest a value of the order of 10–20 d50

Fig. 1 a Dilation bands in dense sand over a displacing trapdoor ([25]);b,c dilation bands and zones of dilation around rotating blade in densesand (b radiograph from Cambridge University archive; c interpretationof b)

(e.g., [21,22,34]). Furthermore, in the presence of imposedrotation of principal axes especially, patterns of repeated dila-tion bands have been observed (Fig. 1b,c) which have a cell-size controlling the repeat of the pattern. Observations fromseveral different test configurations suggest that the repeatlength is of the order of 30 d50 ([32–34]). This defines a sec-ond length scale. The description of a sample of a granularmaterial as a continuum requires a scale of observation suffi-ciently large that the material appears to be homogeneous. Ifcells of localised deformation develop within a sample, thenprobably at least 10 such cells need to be included to have achance of a plausible continuum. Therefore a typical sampledimension would need to be of the order of 300 d50.

3 Experimental procedure

The tests described here were performed in the 1γ 2ε biaxialshear apparatus in Grenoble ([17–19]) (Fig. 2). This appara-tus was developed to apply general stress or strain conditionsto a two dimensional analogue granular material. The appara-tus consists of a containment parallelogram (OABC), whichencloses the analogue material. Point O of the parallelogramis fixed; sides AB and OC are able to rotate; sides AO andBC always remain horizontal; and all four sides are able toextend or contract. The control of the apparatus is basicallykinematic (the kinematics are defined by the measurement ofthe lengths of the sides Li (t) and the corner angle γ (t)). Themeasurement of the forces along x and y directions at threecorners (B, C and O), and the equilibrium of the sides BCand OC, allow the estimation of the normal and tangentialforces on the sides. Assuming a homogeneous state of stress,the normal and tangential stresses are further obtained, fromwhich the stresses referred to orthogonal x :y axes can be

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Fig. 2 a 1γ 2ε apparatus (after [19]) and b example of the sample usedin this study

Fig. 3 Schneebeli material: distribution of sizes of rods

deduced. Details of the measurement systems and the con-trol of the test parameters are given by [19].

Tests were performed using a so-called Schneebeli mate-rial [24]: a 2D analogue granular material formed of a com-bination of three different diameters (1.5, 3 and 3.5 mm) of60 mm long PVC rods. The grading of this granular materialis shown in Fig. 3. Taking d50 ≈ 3 mm, the ratio of sampledimension to particle size is 150–200, which would seem tosatisfy some of the demands of the intermediate scales thathave been noted.

The PVC rods were placed and spread in the apparatusin small quantities from the bottom to the top of the par-

Fig. 4 a Mohr’s circle of stress indicating the Coloumb directions(lines of maximum stress obliquity); b Mohr’s circle of strain incre-ment indicating the Roscoe directions (zero extension lines) (after [4]).Note that in these tests the principal directions, 1 and 2, do not necessar-ily coincide with the horizontal and vertical directions, x and y, nor dothe principal axes of strain increment necessarily coincide with thoseof stress

allelogram. Once the top of the apparatus was reached, thesides of the parallelogram were brought into contact with therods. The initial dimensions and density of the specimen werealways the same: Lxo = 593 mm; L yo = 482 mm; densityρ ≈ 1.1 mg/m3.

The samples were all subjected to three consecutive phasesof loading:

1. isotropic loading up to σx = σy = 50 kPa;2. biaxial test with lateral stress σx = 50 kPa (constant) and

σy increased up to 80 kPa;3. strain controlled test following a linear strain path in the

(εx , εy , γxy) space.

Note that in this paper the usual sign convention of soilmechanics for stress and strain (compression positive) isadopted throughout.

The tests are characterised by the fixed orientation of theprincipal axes of strain increment, ξ , in the third phase ofeach test, and by the imposed (positive or negative) angleof dilation, ψ (see Fig. 4). A full description of the test pro-gramme and stress–strain responses obtained is given by [15](see Fig. 5 for summary stress–strain plots; note that in thisinstance we refer to the macroscopic strain, i.e., at the scaleof the sample, whereas later, from the DIC analysis we willpresent maps of the internal, local strains).

4 Image analysis

Modern techniques of digital photography and digital imagecorrelation can provide detailed information concerning thefields of displacement and hence strain in a test specimen.Such techniques require only that there is a visible surface(or a volume, if using X-ray tomography images) with suffi-cient character or texture so that successive images may be

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reliably compared, thus dispensing with the need for the dis-crete markers traditionally used in radiographic studies (asin Fig. 1a). Many optical techniques exist to assess displace-ment and strain fields over the surface of a test specimen,of these digital image correlation is the most accessible (see[29], for a geomechanics-based overview).

Digital image correlation techniques have been usedincreasingly over the last 20 years or so in studies of themechanics of a diverse range of materials (an historical ref-erence is provided by [26]; and an application to geomateri-als by [31], see also [29]). DIC is essentially a mathematicaltool for assessing the spatial transformation (translations anddistortions) of images. Various developments of the DIC con-cept have been presented not only within solid mechanics,but also in fluid mechanics (where the procedure is oftenreferred to as particle image velocimetry (PIV)) and in other,more diverse fields such as animation or film special effectsand medical imaging.

The DIC analysis presented in this paper was carried outusing the code PhotoWarp developed at Laboratoire 3S-R asa two dimensional version of a previous code for 3D (volu-metric) DIC used for time-lapse analysis of hydrocarbon res-ervoirs under production, based on 3D seismic images (see[13]). PhotoWarp follows the same basic steps as most DICprocedures for strain analysis: (1) definition of nodes distrib-uted over the first image; (2) definition of a region centredon each node (the correlation window); (3) calculation of acorrelation coefficient for all 2D displacements of the cor-relation window within an area (the search window) aroundthe target node in the second image; (4) definition of the dis-crete displacement (integer number of pixels), given by thedisplacement with the best correlation; (5) sub-pixel refine-ment (because the displacements are rarely integer numbersof pixels); (6) calculation of the strains based on the deriveddisplacements and a continuum assumption.

Each DIC implementation has its variations and additionsto the general methodology outlined above. For example,there are different approaches for the sub-pixel refinementafter the initial discrete estimate of the nodal displacementshas been made (based on the displacement in the searchwindow of the correlation window that returned the maxi-mum cross-correlation value). The specific procedure usedin PhotoWarp involves a functional description of the local2D correlation field (describing the variation in correlationover the 9 positions within a 3 × 3 (±1) pixel, search rangeabout the best integer shift value) and a Newton–Raphsoniteration to find the maximum of this local cross-correlationsurface.

Besides the image resolution (which was approximately0.19 mm/pixel for each 13.5 Mpixel image), the key param-eters in this application of DIC are the distance between thecalculation nodes (here 10 pixels in x and y), and the corre-lation window size (here a square of side 41 pixels, which is

Fig. 5 Loading paths for each test in the normalised planes; s is themean stress (σx + σy)/2, t is the radius of the Mohr’s circle of stress,i.e., the maximum shear stress, (τ 2

xy +β2)1/2, β is the deviatoric stress,(σx − σy)/2, and so is the consolidation mean stress, with x and y thehorizontal and vertical cartesian co-ordinates

reduced to 11 pixels for the sub-pixel derivation). This meansthat the correlation windows overlap slightly for adjacentnodes, which provides some inherent smoothing, but this issmall and should not compromise the subsequent strain cal-culation. These parameters were optimised from extensivetesting [8]. The parameter testing also allowed the stabilityof the results to be confirmed and to ensure that any pat-terns observed were not just numerical artifacts of the DICalgorithms adopted. For example, a possible artificial causeof patterning is inaccuracy in the sub-pixel refinement. Itis well-known that without sub-pixel resolution a steppeddisplacement field is achieved, as only integer values of dis-placement can be resolved, this subsequently produces linesof high strain corresponding with the jumps between integerdisplacement values (see Fig. 6 left column). The resultantstrain field is thus patterned according to the jumps in dis-placements. This pattern is clearly different from the pat-terning revealed by the DIC-derived strain images wheresub-pixel analysis is used (as demonstrated in Fig. 6 rightcolumn). Furthermore, consistency between results obtainedwith different combinations of parameters (different searchand correlation window sizes) provides confidence in thevalidity of the observations.

5 Results

Results of the DIC for pairs of photographs taken during sev-eral of the tests (see Table 1) are shown in Figs. 7, 8, 9, 10,11, 12 and 13. Note that these represent a small selectionfrom a much larger number of tests (see [15]). The selectioncovers a range of (fixed) orientations of the principal axes ofstrain increment and the imposed angle of dilation. The vol-ume change was dilative in tests A-T21(1.5d), A-T30(5.3d),A-T37(4.5d) and A-T9(15.4d) and compressive in testsA-T0(3.8c), A-T21(2.9c) and A-T42(9.6c).

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Fig. 6 Test A-T0(3.8c): DICresults for test A-T0(3.8c):without (first column) and with(second column) sub-pixelrefinement

Table 1 Details of tests used for digital image correlation

Test γ1 (%) γ2 (%) ξ◦ ψ◦ α◦ φ◦ R1◦ R2◦ C1◦ C2◦

A-T0(3.8c) 0 1.1 0 −3.9 0 19 −48 48 −35 35A-T21(2.9c) 0.2 1.2 22 −3.0 18 23 −24 69 −15 51

A-T42(9.6c) 0.5 2.4 42 −9.4 36 17 −7 92 0 72

A-T21(1.5d) 1.7 3.7 21 1.5 22 24 −23 66 −11 55

A-T30(5.3d) 7.0 9.0 29 5.2 33 25 −13 71 1 66

A-T37(4.5d) 7.0 9.0 37 4.7 40 26 −6 80 9 72

A-T9(15.4d) 0.7 2.5 10 14.7 9 27 −28 48 −17 46

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Fig. 7 Test A-T0(3.8c):volumetric and shear strains(a and b respectively) and clinear features identified fromthe strain images darker (purple)lines were identified in the shearstrain image and lighter (green)lines are from the volumetricstrain image—the upper plotassembles all the identifiedlines). The dominant directionsof the identified features areshown on the ‘orientation plot’with the “Roscoe” and“Coulomb” orientations,superposed (see text for details)

Fig. 8 Test A-T21(2.9c):volumetric and shear strains(a and b, respectively) and clinear features identified fromthe strain images (as Fig. 7)

Figures 7, 8, 9, 10, 11, 12 and 13 show in each case col-oured maps of magnitude of volumetric strains between thetwo photographs in the upper figure and maximum shearstrains in the lower picture. There are many more or less clearlinear (or sub-linear) features in each of the pairs of images.Many of these linear features have been identified separately

in each image of the pair and the combined identificationshave been assembled on the upper picture (which shows thevolumetric strains). Automatic identification of these featuresand their orientations has proved to be more challenging thanit might appear at first view, therefore it has been performedentirely by eye. This is obviously a somewhat subjective

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Fig. 9 Test A-T42(9.6c):volumetric and shear strains(a and b, respectively) and clinear features identified fromthe strain images (as Fig. 7)

Fig. 10 Test A-T21(1.5d):volumetric and shear strains(a and b, respectively) and clinear features identified fromthe strain images (as Fig. 7)

process, but adequate for the simple objective of obtainingan indication of the length, spacing and orientation of thelocalised features that are certainly present. Some reason forconfidence in this approach is provided by the observed broadagreement between the patterns deduced independently fromthe shear strains and the volumetric strains. On the other hand

it is evident that some features are more marked on one plotrather than the other; there is often a more striking concen-tration of shear strain than of volumetric strain.

Each test was controlled to have a straight strain path sothat, for each test, the inclination, ξ , of the principal axes ofthe strain increment and the value of the angle of dilation ψ

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Fig. 11 Test A-T30(5.3d):volumetric and shear strains(a and b, respectively) and clinear features identified fromthe strain images (as Fig. 7)

Fig. 12 Test A-T37(4.5d):volumetric and shear strains(a and b, respectively) and clinear features identified fromthe strain images (as Fig. 7)

were kept constant (Fig. 5). Therefore, it is possible to definequite clearly the orientation of the macroscopic major prin-cipal (compressive) strain, irrespective of the magnitude ofthe strain increment between the two pictures analysed.

The notion that regions of shear localisation in radio-graphic images such as those in Fig. 1 are associated with

directions of zero extension in the sand is generally attributedto Roscoe [22] (see also [16]). The zero extension directionsare oriented at an angle π/4 −ψ/2 each side of the principalstrain increment direction, where ψ is the angle of dilation(Fig. 4b). However, in soil mechanics there is another pos-sible interpretation (traditionally associated to the name of

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Fig. 13 Test A-T9(15.4d):volumetric and shear strains(a and b, respectively) and clinear features identified fromthe strain images (as Fig. 7)

Table 2 Orientations of observed features

Test R1◦ R2◦ C1◦ C2◦ F1◦ F2◦ F3◦ F4◦ F5◦

A-T0(3.8c) −48 48 −35 35 −37 (C1) −18 42 (R2) 60A-T21(2.9c) −24 69 −15 51 −78 −58 −48 30 63 (R2)

A-T42(9.6c) −7 92 0 72 −70 −16 22 60 (C2?) 85 (R2)

A-T21(1.5d) −23 66 −11 55 −58 27 45 (C2?) 66 (R2)

A-T30(5.3d) −13 71 1 66 −58 −5 (C1?) 30

A-T37(4.5d) −6 80 9 72 −65 −7 (R1) 14 (C1) 72 (C2)

A-T9(15.4d) −28 48 −17 46 −63 −41 33

Coulomb) where shear bands are rather thought of as linesof maximum stress obliquity (i.e., the shear to normal stressratio). In this case, shear band orientation will be at an angleπ/4−φ/2 each side of the principal stress direction, where φis the mobilized friction angle at failure (Fig. 4a). These twoorientations are traditionally referred to as the Roscoe andCoulomb angles, respectively. Generally received wisdomsuggests that the Roscoe directions will tend to be observedwhen the configuration is somewhat constrained kinemati-cally. Where kinematic restraint is less marked, shear local-isation is more likely to occur along directions controlledpurely by the stress state, i.e., the Coulomb directions. Thereexist experimental observations and theoretical argumentssuggesting an alternative shear band orientation that dependson both the friction and dilatancy angles, i.e., the so-calledArthur/Vardoulakis direction, which roughly bisects the Cou-

lomb and Roscoe directions (assuming that principal strainincrements and principal stresses are coaxial) [3,27,28].

The dominant directions of the features observed in thestrain images are indicated by black lines on the orientationplot between the strain image pairs. These ‘dominant direc-tions’ are again chosen by eye to give an indication of thefavoured orientations; these seem in general to fall into cleargroups, give or take a few degrees. In each of the figures theRoscoe directions have been calculated and shown in red onthe orientation plot, with a central line indicating the prin-cipal direction of strain increment. The Coulomb directionshave been indicated in each orientation plot as blue lines, witha central line indicating the principal direction of stress.

Table 1 gives some detailed information about the testsand the strain range (γ1 to γ2) for each pair of photographs.The angles ξ , ψ , α, φ are the direction of major principal

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Fig. 14 Test I65-T0(6.9d): a incremental internal shear strains forimposed strain increments noted and b the linear features identifiedfrom a; after the first picture, lines from the previous picture are darker(red) and new lines are lighter (green)

Fig. 15 Test I65-T0(6.9d): a stress ratio t/s and shear strain ε1 − ε2;b mean stress s and shear strain ε1 − ε2

strain increment, the angle of dilation, the direction of majorprincipal stress and the angle of friction. The table also liststhe Roscoe angles (R1, R2) and the Coulomb angles (C1,C2). Whereas the global strain axes and dilation angle areconstant during each test, the principal axes of stress in gen-eral vary along each test path—the observation of which wasone of the objectives of the test programme (see [15])—andthe mobilised friction also develops. The Coulomb directionsindicated relate to the stress state corresponding to the secondphotograph (apart from test A-T9(15.4d) for which the finalstresses were spurious). The Coulomb angles have also beencomputed for the initial photograph of each pair (initial prin-cipal stress directions and initial mobilised friction): thesevalues are not shown, but the difference is not large relativeto the spread of directions of features in the strain pictures.

The orientations of observed features and the estimatedRoscoe and Coulomb directions are listed in Table 2. Thecorrelation taken as a whole is not conclusive. Although theredo appear to be a number of close matches with one of theRoscoe directions, there are many more features which donot appear to have any link with either Roscoe or Coulombdirections.

The features that have been marked on the strain plots inFigs. 7, 8, 9, 10, 11, 12 and 13 represent the accumulationof information contained in the granular material over theentire increment of strain and stress between the two pho-tographs. In any single test the strain concentrations appear

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Fig. 16 Test I65-T0(6.9d):cumulative internal shear strainsfor the accumulated strainsnoted

in different locations and at different orientations at differ-ent stages during the test. Figure 14 collects observationsmade in test I65-T0(6.9d) over successive strain increments(this test differed from those discussed previously in thatthe stress state before the strain-controlled linear-strain pathloading was isotropic, σx = σy = 65 kPa). The stress-strainresponse and variation of mean stress are shown in Fig. 15:this is a test with no imposed rotation of principal axes andwith an imposed dilation angle of about 6.9◦.

Figure 14a shows the incremental shear strain fields andFig. 14b shows lines indicating—again somewhat subjec-tively—the locations and orientations of features that areobserved. After the first picture, lines from the previous pic-ture are coloured red and new lines are coloured green. Recallthat these are all incremental and not cumulative images, each

covering a strain range of around 0.5–1.0%. In each image,although new features appear, the old ones do not disap-pear—in other words they remain at least somewhat active.The kinematics of the overall internal deformation mecha-nism are thus quite complex. Figure 16 shows the cumulativestrain maps for the same test and strain increments. Interest-ingly, whilst the incremental strain maps indicate that roughlythe same localisation features exist from one step to the next(although with new features emerging), such features are notfixed in space. As a result, by the end of the test the accumu-lated shear strain is essentially distributed across the sample.In other words, strain is incrementally localised in space onlyfor a given time step and moves between steps.

Treating the features that have been identified as shearbands of some sort, one would expect that localisation of

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shearing would be associated with volumetric strain, and thisis generally borne out by the previous figures for the severaltests. However, as shear deformation proceeds, we expectthat the angle of dilation will change and that the kinematicenvironment that encouraged the original localisation willno longer seem so desirable. Localisation with some slightlydifferent orientation will then occur and shearing proceedson a combination of the two localisations—just as is apparentin the macroscopic features shown in Fig. 1a.

In the spirit of attempting to extract information aboutintermediate scales from the observations of strain localisa-tions in each increment, an estimate of thicknesses and spac-ings of the identified features has been made from the resultsshown in Fig. 14. First, the band thickness is estimated asindicated in Fig. 17. It is very evident at this scale that thefeatures are certainly not continuous: there are little bursts ofshear strain separated by regions of inactivity. The solid blackcircles are drawn to indicate the approximate thickness of theshear strain features: there is variation but within a narrowrange. Typical thicknesses are around 10.8 mm which corre-sponds to about 3.5d50. It should not surprise us that this sizeis much lower than that usually found for sands. We are test-ing a strictly two-dimensional granular material composedof circular grains with a very narrow range of particle diam-eters. In principle, shearing between nondeforming regionscan, in such a material, take place within a thickness of justtwo particle diameters.

Dashed lines in Fig. 17 show the location of a pair of neigh-bouring shear localisation features: the open circle indicatesthe spacing to be about 31.3 mm, which is equivalent to about10.4d50 or 3 times the thickness of the localisations. A similar

Fig. 17 Example of feature identification carried out for the images inFig 14(b) and summarised in Fig 18. Dashed lines show the locationof a pair of neighbouring shear localisation features, the open circleindicates the measured band spacing and the filled circles the measuredlocalisation thicknesses

procedure has been used to collect information about spac-ings of the localisations in Fig. 14 b. In each of the incremen-tal pictures open circles have been drawn where either pairsof more or less parallel new features have emerged or where anew feature appears to be more or less parallel to an existingfeature. The spread of spacings is shown as a histogram inFig. 18a, which indicates an average value of about 174 pix-els (≡32.8 mm), corresponding to about 11d50 or about 3.1times the observed thickness of the strain localisation fea-tures. The histogram in Fig. 18b collects these spacings byincrement. For most increments the spacings are rather dis-tributed over a range from 10–45 mm and it is only in incre-ments 3 and 5, corresponding to strain increments between1.61–2.12 and 2.85–3.66%, that there seems to be any clearerindication of a concentration of spacings around the 31.3 mmeventual average. These two increments both correspond tostrain softening episodes (see Fig. 15).

Noting that the localisations in the cumulative strain fieldsappear gradually to occupy more and more of the sample

Fig. 18 Test I65-T0(6.9d): a cumulative histogram of spacings oflocalisation features seen in incremental pictures (Fig 14); b histogramof spacings of localisation features seen in individual incremental pic-tures

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(Fig. 16), the application of a similar procedure to find thespacings of all the features that have accumulated by the endof a test shows a dominant spacing of about 20 mm and thusabout 6.5d50, only about twice the thickness of typical fea-tures. However, these spacings are not necessarily betweenfeatures that are related at the time of their formation.

6 Discussion

Whilst the features of localised deformation described in theprevious section are not an artifact of the DIC, questions maystill be posed about their origin.

Are these features a product of some detail of the apparatusor loading system? The boundaries are to some extent seg-mental with individual ‘barrettes’ visible in Fig. 2. The comb-like nature of these loading devices is intended to smooth outthe load transfer. If the internal features had this source thenone would expect that the locations and spacing would be insome way constant—and this does not appear to be observed.

Are the features a consequence of the rather narrow grad-ing of circular two-dimensional particles? This is harder torefute. Even though two of the rod sizes are rather close, thepresence of the smaller size should discourage ‘crystallisa-tion’ of groups of particles. However, the two-dimensionalityof the material is inevitable.

The analysis of dimensions and orientations here islargely, and deliberately, approximate, but the results sug-gest that patterning is indeed seen in these two dimensionalexperiments (for a range of imposed strain paths—both com-pressive and dilative). This deserves further investigation,focussing on more thorough statistical analysis of the geom-etry of the features that are seen.

Finally, it is noted that the displacements observed over thetime intervals, or more importantly strain intervals, betweeneach pair of photographs are relatively large and of the sameorder of magnitude as the grain size. The question remains asto whether within a given strain increment what is observedis the result of a single localisation event or the superposi-tion of multiple such events. With the current data it is notpossible to respond to this question. In future work manymore images (100’s rather than about 10) will be acquiredfor each test, which will enable analysis of finer incrementsof strain as well as more detailed analysis of the life-timeof the individual localised features and the patterns that theyform.

7 Conclusion

Against a background of various experimental and analyticalobservations of the development of features within samplesand model tests on granular materials it is suggested that there

are a number of routes by which material controlled scale-lengths greater than the size of individual particles enter intothe response of the granular system. These ‘routes’ includethickness of shear bands or other forms of localisation and‘cell size’ of patterns of localisation. This theme has beencontinued in this work with direct observation of internal dis-placements and strains, using photographs and DIC, withintwo-dimensional samples of a Schneebeli rod material testedunder a series of general strain increments with different ori-entations of principal strain increments and different imposedangles of dilation.

The DIC analyses of these tests on an analogue gran-ular material have successfully provided clear evidence ofevolving internal structures of deformation. These structuresconsist of bands of localised deformation and ‘cells’ of lowdeformation between the bands. The orientations of the iden-tified localised features and cells are seen to depend on theapplied strain path. Characteristic spacings of the bands (cell-sizes) have been qualitatively identified. However, questionsremain as to the interpretation of the patterns and spacings.

Acknowledgments The experimental work was performed in Greno-ble with the support of a British Council Alliance grant. The assistanceof Jack Lanier in carrying out the tests and F. Bonnel in setting up thecamera for digital photography is gratefully acknowledged. Further-more we thank the editor, Stefan Luding, plus the two referees for theirhelpful feedback.

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