numerical studies of shear banding in interface shear tests using a new strain...

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., (in press)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.589

Numerical studies of shear banding in interface shear testsusing a new strain calculation methodk

Jianfeng Wang}, Marte S. Gutierrez*,y,z and Joseph E. Dove}

Department of Civil and Environmental Engineering, 200 Patton Hall, Virginia Tech, Blacksburg,VA 24061, U.S.A.

SUMMARY

Strain localization is closely associated with the stress–strain behaviour of an interphase system subject toquasi-static direct interface shear, especially after peak stress state is reached. This behaviour is importantbecause it is closely related to deformations experienced by geotechnical composite structures. This paperpresents a study using two-dimensional discrete element method (DEM) simulations on the strainlocalization of an idealized interphase system composed of densely packed spherical particles in contactwith rough manufactured surfaces. The manufactured surface is made up of regular or irregular triangularasperities with varying slopes. A new simple method of strain calculation is used in this study to generatestrain field inside a simulated direct interface shear box. This method accounts for particle rotation andcaptures strain localization features at high resolution. Results show that strain localization begins with theonset of non-linear stress–strain behaviour. A distinct but discontinuous shear band emerges above therough surface just before the peak stress state, which becomes more expansive and coherent with post-peakstrain softening. It is found that the shear bands developed by surfaces with smaller roughness are muchthinner than those developed by surfaces with greater roughness. The maximum thickness of the intenseshear zone is observed to be about 8–10 median particle diameters. The shear band orientations, which aremainly dominated by the rough boundary surface, are parallel with the zero extension direction, which arehorizontally oriented. Published in 2007 by John Wiley & Sons, Ltd.

Received 2 April 2006; Revised 13 October 2006; Accepted 16 October 2006

KEY WORDS: interphase; strain localization; discrete element methods; shear band; surface roughness;numerical simulations

*Correspondence to: Marte S. Gutierrez, Via Department of Civil and Environmental Engineering, 200 Patton Hall,Virginia Tech, Blacksburg, VA 24061, U.S.A.yE-mail: [email protected] Professor.}Postdoctoral Research Associate.}Research Assistant Professor.kThis article is a U.S. Government work and is in the public domain in the U.S.A.

Contract/grant sponsor: National Science Foundation; contract/grant number: CMS-0200949

Published in 2007 by John Wiley & Sons, Ltd.

INTRODUCTION

Shear resistance develops when relative displacement occurs inside the interphase regionbetween a granular soil layer in contact with a rough, continuous manufactured or naturalmaterial surface. The interphase region consists of the rough surface with its asperities, and avariable thickness of granular material directly adjacent to the surface. The thickness of thegranular portion of the interphase is normally associated with the thickness of the zone ofintense shear deformation (shear band). The interface is a transitional zone between the surfaceand the granular soil inside the interphase (shear zone) where particle to surface interactionstake place. For a smooth surface, the shear zone has zero thickness and the two are the same.However, surface roughness causes strength mobilization within the interphase resulting involume change and higher shear resistance. During quasi-static shear of the interphase soil,strain localization is usually observed to be associated with the non-linear stress–strainbehaviour, resulting in a series of shear bands developing near the surface, especially after peakstate is reached. In this paper, the term ‘state’ is used to refer to the stress level that the soil isexperiencing. So, the peak state means the peak stress the soil undergoes. The mechanismunderlying the interphase strength and shear banding behaviour is critical to the design ofgeotechnical composite systems because the load–deformation behaviour of the interphaseregion controls the overall performance of the composite structure. It has been shown by Wang[1] that the evolution of fabric and contact force anisotropy at the boundary between the surfaceand granular soil controls interphase strength behaviour. However, the development of strainlocalization and shear banding occurring within the interphase region is still not fully known.

Previous-related research

Experimental and numerical investigations on the shear bands or rupture surfaces of clayey orgranular soils subject to various shearing conditions have contributed to our understanding ofshear band morphology. For example, Morgenstern and Tchalenko [2] studied the formation ofsmall-scale shear features within kaolin subject to direct shear tests. Their observations showeda series of structures and sub-structures were produced in a sequential manner, which coalesceto form the ultimate shear bands. A similar study of shear band patterns on dense sands subjectto direct shear was performed by Scarpelli and Wood [3] using a radiographic technique.Their study showed the important effects of kinematic constraints on the shear bandingbehaviour. Other valuable work on the shear banding behaviour of granular soils includesReferences [4–11].

Most of the previous studies on behaviour of the interphase between granular media andrough surfaces have focused on exploring and quantifying factors influencing peak and steadystate strength [12–17]. Uesugi and Kishida [12–14] introduced a widely used normalizedroughness parameter Rn to correlate with the peak friction angles of sand–steel surfaces.However, it has been concluded by Wang [1] that statistical geometric parameters alone cannotaccount for the actual contact conditions occurring at the boundary between the particles andsurface, and within the interphase. Therefore to date, only non-unique correlations betweeninterface strength and surface geometry have been achieved. Wang [1] has proposed a failurecriterion which is based on the contacts existing between the first layer of particles and thesurface. It accounts for the actual mechanism controlling particle and surface interaction and

J. WANG, M. S. GUTIERREZ AND J. E. DOVE

Published in 2007 by John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (in press)

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has proven to be successful in predicting interface strength for a range of particle distributionsand surface geometries.

However, very limited experimental work has been published on the shear deformation ofinterphase soils due to difficulty in collecting microscopic information interior to the sample.Uesugi et al. [18] observed the behaviour of sand particles in sand–steel friction in simple shearapparatus and described qualitatively their deformation characteristics due to interfaceroughness. Westgate and DeJong [19] used particle image velocimetry (PIV) to investigate theeffects of surface roughness, confining pressure and initial void ratio on sand–steel interfaceshear banding. This study illustrated the key role that surface roughness played on interphasedeformation.

The emergence of the discrete element method (DEM) [20, 21] and its numerical simulationprovide a powerful tool for studying particulate media and greatly facilitate the understandingof granular micromechanics. A rich body of microscopic information can be obtained fromDEM simulations and used to interpret the mechanical behaviour in a continuum-basedframework [1]. Using DEM simulations, Wang et al. [22, 23] conducted interface shear box testscomposed of densely packed spherical particles contacting rough manufactured surfaces andsuggested the use of sample height of 50 median particle diameters in order for full developmentof shear band. They also made a preliminary discussion on the effects of surface asperity slopeon interphase shear deformation and concluded that the relative grain to surface geometry wasthe controlling factor of the micromechanical behaviour. But these findings are insufficient toprovide a quantitative description of the effects of relative grain to surface geometry oninterphase shear banding behaviour.

In this paper, a new simple numerical-based strain calculation method, which is foundcapable of capturing characteristics of strain localization accurately inside the interphase region,is presented and applied to a numerical study to generate strain fields inside the interface shearbox. The numerical model of interface shear test is described, and a parametric study isperformed to study the influence of relative particle to surface geometry on the shear bandingstructures within the interphase region. Finally, the results of strain localization evolution,effects of surface geometry and shear band orientation are thoroughly discussed.

FORMULATION OF PROPOSED STRAIN CALCULATION METHOD

The spatial discretization approach is the most often used method to compute strains in adiscrete system [24–27]. This type of approach usually first discretizes the domain byconstructing a nodal graph and then interpolates between adjacent nodes to calculate thedisplacement and strain fields. For example, in evaluation of the strain field in an asphaltconcrete, Wang et al. [28] constructed a triangular graph by connecting particle centroids andcomputed displacement values using a linear interpolation function. Recently, Alshibli andAlramahi [29] used second-order polynomial functions to interpolate the local displacementsinside particle groups that consist of 10 particles each and share four particles betweenneighbouring groups to ensure the continuity of the strain field. However, O’Sullivan et al. [30]pointed out two problems of using the spatial discretization approach, which lead to inaccurateestimates of strain inside a granular media involving localization. These are: (1) negligence ofparticle rotation which plays an important role in the formation of shear band, and (2) use of

NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS


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a linear local interpolation function which results in erratic inter-element strain values within theshear band.

To avoid the above problems, O’Sullivan et al. [30] proposed a higher-order, non-local mesh-free method which employs a cubic spline interpolation function and takes into account particlerotation. The method was applied to 2D biaxial compression simulations. However, it wasfound by the authors that the higher-order mesh free interpolation functions overly smooththe displacement gradient within the shear band and thus failed to manifest the strainlocalization accurately inside the interface shear box. Given this fact, the authors propose hereina modified mesh-free method which does not use interpolation functions and calculatesthe displacement gradient directly based on movements of individual particles. The detailsof the method are described below. It is believed that the method is conceptually differentfrom [28–30].

The current problem involves large strain localization inside the granular media, therefore theGreen-St. Venant large-strain tensor Eij is used

Eij ¼12ðui;j þ uj;i þ uk;iuk;jÞ ð1Þ

where ui;j is the displacement gradient tensor. It is based on the deformation measure related tothe reference configuration.

The mesh-free method used in this study employs a grid-type discretization over the referenceconfiguration. A schematic diagram of the approach is illustrated in Figure 1. A rectangular gridis generated that serves as the continuum reference space superimposed over the volume ofparticles. Then each grid point is assigned to an individual particle j which has the followingproperty:

dj

rj4

di

riði ¼ 1; 2; . . . ;Np; i=jÞ ð2Þ

Figure 1. Schematic diagram of the proposed meshfree method: (a) association of a grid point with certainparticle and (b) displacement of grid point and its associated particle.



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where ri is the radius of particle i; di is the distance between the grid point and the centroid ofparticle i; Np is the total number of particles within the volume (Figure 1(a)). If the ratio of thedistance between particle centroid and its associated grid point to the particle radius is the leastamong all the particles, then the grid point is considered as a point on the original or extendedrigid body of the particle. The displacement of the grid point is calculated by

ugx ¼ upx þ dðcosðy0 þ oÞ � cosðy0ÞÞ

ugy ¼ upy þ dðsinðy0 þ oÞ � sinðy0ÞÞ ð3Þ

where ugx; ugy and upx; u

py are the x and y displacement components of a grid point and a particle

centroid, respectively; d is the distance between the grid point and the particle centroid; y0 is theinitial phase angle of the position of a grid point relative to the particle centroid; and o is theaccumulated rotation of the particle (Figure 1(b)).

An important parameter involved in this method is the spacing between adjacent grid points,which has a direct control on the precision of the calculated strain values. The value of thisparameter is 0.64mm in the horizontal direction and 0.7mm in the vertical direction in thecurrent paper. This results in totally 41 rows and 201 columns of grid points over the entirevolume. As a general guidance, a grid spacing of a median particle diameter (D50Þ should sufficeto capture the shear localization at satisfactory resolution. This will be verified later bycomparing the results obtained using varying grid spacings.

The proposed method discretizes the reference space based on the simple algorithmrepresented by Equation (2) before any deformation occurs. Any random point (not necessarilythe grid point) inside the whole volume can be assigned to a particular particle using thisalgorithm. Then all the grid points that are assigned to the same single particle connect to form aregion that belongs to this particle. So if any point lies within this region, it will be considered asa point on the original or extended rigid body of this particle, and its displacement will becalculated using the principle of rigid body motion of the particle. This method takes intoaccount particle rotation and is found capable of capturing the strain features inside theinterface shear box more precisely than other methods.

NUMERICAL SIMULATIONS OF INTERFACE SHEAR TEST

Numerical model

The DEM model of direct interface shear device was developed by Wang [1] using PFC2D andvalidated against laboratory data [16]. The model is made up of a 128mm long, 28mm highshear box filled with a polydispersed mixture of spherical particles (maximum and minimumparticle diameters of 1.05 and 0.35mm, respectively) at a low initial porosity of 0.12. A sketch ofthe model is shown in Figure 2(a). The lower boundary is made up of a rough surface and two20mm long ‘dead zones’ placed at the ends of the box to avoid boundary effects. The roughsurface consisted of either regular or irregular triangular asperities, or of profiles of natural andmanufactured material surfaces made using a stylus profilometer. There is no particle-to-boundary friction within the dead zones. Shearing occurs by displacing the lower boundary ofthe box to the left at a constant velocity of 1mm per minute. Vertical and horizontal contactforces acting on all the asperities are summed to compute the normal and shear forces generatedalong the interface, respectively.



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The Hertz–Mindlin contact model was implemented in the simulations. A particle rollingresistance model [31] was also applied at both particle–particle contacts and particle–boundarycontacts. Physical constants used in the simulations include: particle density of 2650 kg=m3;shear modulus and Poisson’s ratio of the particles of 29GPa and 0.3, respectively; criticalnormal and shear viscous damping coefficient, both equal to 1.0; time step 5:0� 10�5 s; particle-to-surface asperity friction coefficient of 0.05; and interparticle and particle–boundary frictioncoefficient (except the dead zones) during shear equal to 0.5 and 0.9, respectively.

Parametric study

To systematically study the effects of surface geometry on the interphase shear bandingbehaviour, we conducted five groups of numerical simulations. In Groups 1, 2 and 3, regular

Figure 2. DEMmodel of direct interface shear box: (a) interphase composed of granular spheres in contactwith a rough surface and (b) particle columns pre-selected to identify interphase deformation.



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saw tooth asperities were used, with asperity height (RtÞ; asperity width (SwÞ and spacingbetween asperities (SrÞ varied in each group, respectively (Figure 3(a)). Table I provides theranges of variables and constant values used in these three groups of experiments. In Group 4,irregular saw tooth asperities are generated randomly and used as the lower boundary (Figure3(b)). Details of the analytical method and parameters used to generate the irregular surface aregiven in [1]. In Group 5, profiles of three coextruded geomembrane surfaces are used.

Sixteen evenly spaced particle columns are identified before shear. Each column is twice themedian particle diameter in width and extends for the full height of the specimen. A typicalprofile of the columns in a 28mm thick sample is shown in Figure 2(b). The displacements and

Figure 3. (a) Surface geometry used Groups 1, 2 and 3 and (b) Group 4. Note extreme roughness of surface.

Table I. Variables, constants of surface asperities and their values usedin the Experimental Groups 1, 2 and 3.

Group Variable Values of variables Rt=D50 Sr=D50 Sw=D50

1 Rt=D50 0.1, 0.2, 0.5, 1, 1.2, 1.5, 2, 3, 5 Varied 0 22 Sr=D50 0.2, 0.5, 1, 2, 3, 5, 8, 10 1 Varied 23 Sw=D50 0.2, 0.5, 1, 2, 3, 4, 5, 10 1 0 Varied



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velocities of particles in these columns are recorded during the shearing process and used for theanalysis of shear banding.

RESULTS AND DISCUSSIONS

In this section, results from numerical simulations of eight surfaces will be presented. The eightsurfaces, called surface A to surface H, are selected from all the five groups of numericalsimulations and give a representative collection of the interphase shear behaviour under theinfluence of various surface roughnesses. Descriptions of these surfaces are listed in Table II.

Macro and micromechanical interphase behaviour

A typical shear stress ratio t=s and volume change versus shear displacement relation fromsimulation A is shown in Figure 4. Due to the low initial porosity, distinct post-peak strainsoftening to steady state is observed (Figure 4(a)). Continuous dilation is also observed (Figure4(b)), with the highest dilation rate occurring around peak state and approaching zero at steadystate deformation. Similar curves were obtained from other simulations with varying surfacegeometries, but less dilation and mobilized peak stress ratio are typical for surfaces with lowerdegrees of relative particle to surface roughness.

The internal shear deformation of the sample is closely monitored by recording theaccumulated displacements at different stages of the test of the particles comprising the 16selected columns. Snapshots were taken at five instants as indicated on the stress–displacementcurve (Figure 4(a)): point A (pre-peak, 0.5mm), point B (pre-peak, 0.7mm), point C (peak,1mm), point D (post-peak, 2mm) and point E (steady state, 6.3mm).

Figure 5 shows the profiles of deformed columns inside the shear box at points C, D and E.During initial pre-peak shearing, the sample is deforming uniformly and pure shear distortion isthe dominating mode of deformation. With the onset of non-linear stress–displacementbehaviour, discontinuities in particle displacements are observed, leading to a narrow, localizedshear deformation zone at about 10 median particle diameters above the surface asperities. Theextent of strain localization is small prior to peak state therefore it cannot be readily observedfrom deformed columns (Figure 5(a)). After peak state, rolling and sliding of particles over theasperities leads to the rapid reduction of shear stress and expansion of the plastic zone of strainlocalization. Curved profiles of deformed columns above the asperities can be clearly seen at thepost-peak state (Figure 5(b)). At the steady state, a fully developed shear band is observed,which has a higher porosity than the rest part of the sample due to dilation, as shown inFigure 5(c).

Table II. Eight surfaces whose simulation results are presented.

Surface A B C D E F G H

Group No. 1 3 1 2 4 4 5 3Description Rt=D50 ¼ 1 Sw=D50 ¼ 10 Rt=D50 ¼ 5 Sr=D50 ¼ 5 See

Figure 8See

Figure 8See

Figure 8Sw=D50 ¼ 3



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Model validation using laboratory test results

Validation of the numerical model has been made in [1] by comparing global stress–displacement response and particle movement to existing laboratory test data [16, 19]. It wasfound that the boundary effects play a significant role on the interphase shear behaviour. Thecurrent size of the numerical box is chosen such that the length is about 180 median particlediameters (D50Þ; which meets the minimum laboratory requirement of 70 D50 for shear box sizeused in the conventional soil testing. A 20mm long frictionless ‘dead zone’ is placed at both endsof the box to avoid the lateral boundary effects, as mentioned above. In addition, the effects ofmodel height on the behaviour of interphase shear band has been quantified in detail in [23] andit was found that a maximum shear band with thickness of 8–10 D50 can be formed if the modelheight reaches 40–50 D50: The numerical model used a height of 40 D50: Particle to boundaryfriction, which was found to have major impacts on the initial stiffness behaviour and particlemovement, has been assigned a high value (0.9) to prevent the sliding of the granular samplewith respect to the boundaries. The readers are referred to [1] for more details of the model.

A comparison between the simulation results and laboratory data by Westgate and DeJong[19] is shown in Figure 6. The simulation data are for particle displacements of column 9 fromsurface B and surface H recorded at 7mm shear displacement. The laboratory data wereobtained from two types of sand–steel interfaces, namely sandblasted steel surface and epoxied

Figure 4. Macroscopic interface behaviour of Simulation A: (a) stress–displacement relationshipand (b) volume change–displacement relationship.



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sand–steel surface, but were the average positions and displacements for each row of the PIVpatch mesh (21 rows by 28 columns) recorded at the same shear displacement. The laboratorysoil specimen has a dimension of 101.6mm (length) �63:4 mm (width) �25:4 mm (height). Thematerial used in the experiments is a uniform Ottawa sand with D50 ¼ 0:74 mm: Two

Figure 5. Deformed particle columns inside the shear box from Simulation A: (a) at peak, 1mmdisplacement; (b) at post-peak, 2mm displacement; and (c) at steady state, 6.3mm displacement.



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parameters, the average roughness (RaÞ and average slope (DaÞ were used to quantify theroughness of the steel surfaces. Specifically, Ra is defined as the average deviation of profileheight ZðxÞ from the mean line, i.e. Ra ¼ ð1=LÞ

R L0jZðxÞj dx; where L is the total length of the

profile; Da is defined as the arithmetic average of the rate of profile height change over the wholelength, i.e. Da ¼ ð1=LÞ

R L0 jdZ=dxj dx:

In Figure 6, profiles of particle displacements from the simulations agree very well with thelaboratory data, showing an excellent match in the lower shear band. At the top of the box, slipof the whole sample is observed in the laboratory test but zero slip occurs in the simulations.This is because the particle to boundary friction is lower in the laboratory test than in thesimulation. A similar slip at the top wall has also been observed in our simulations if a lowerparticle to boundary friction coefficient is adopted.

The surfaces used in the simulations are not reproductions of the steel surfaces used byWestgate and DeJong in their laboratory tests. Difference of roughness quantified by Ra and Da

exists between the two groups of surfaces, and will lead to different degrees of shear localizationinside the interphase region. In addition, other differences between the simulations andlaboratory tests including granular material, initial density, interparticle friction and particle tosurface friction, etc. will also contribute to the dissimilar interphase shear behaviour and makethe piecewise comparison of the results between the simulations and laboratory tests pointless.However, it is important to realize that the comparison shown in Figure 6 is made as avalidation of our numerical model in terms of producing physically sound and realisticinterphase shear behaviour. The overall agreement of the displacement profiles observed inFigure 6 suggests that our numerical model is able to effectively eliminate the boundary effectsand correctly simulate the shear band created by a rough surface.

Evolution of strain localization

The accuracy of the proposed method on capturing the localization features of the granularmedia is first validated by comparing the displacement profile of grid points calculated usingEquations (2) and (3) to the displacements of particle columns recorded during the simulations.An example is shown in Figure 7 which compares the calculated and recorded displacements in

Figure 6. Comparison of particle horizontal displacements from simulations with laboratory data.



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Simulation A at 2mm post-peak state (point D) and 6.3mm steady state (point E). The gridpoint columns 20, 50, 100 and 150 were selected with initial horizontal positions correspondingto the particle columns 2, 5, 9 and 13, respectively. It is seen that the calculated displacements fitthe recorded data very well at both shearing stages, indicating the proposed method is excellentin capturing the displacement field characteristic of a granular media.

The evolution of strain localization in Simulation A can be better visualized by the shearstrain contours computed at point A to point E shown in Plate 1. The shear strain field iscalculated using the proposed strain calculation method given in Equations (1)–(3), and isshown overlain on the original configuration of the sample. A series of rupture bands emerge ina sequential way to form the final shear band. The patterns of rupture bands are similar to those

Figure 7. Comparison of the calculated displacement profile of grid point columns to the displacements ofparticle columns recorded during Simulation A: (a) at post-peak, 2mm displacement and (b) at steady

state, 6.3mm displacement.



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observed by Morgenstern and Tchalenko [2] and Scarpelli and Wood [3]. Here, we adopt thesame notation for rupture bands, Sk; as used by Scarpelli and Wood [3] to describe the evolutionof strain localization inside the interface shear box. The subscript k refers to the order ofappearance of the rupture bands in time. In this paper, we only report primary and fullydeveloped structures of rupture bands which result from boundary movement. Local andsecondary structures, which are smaller and are not fully developed, are not included in theanalysis.

Strain localization occurs very early with the appearance of rupture bands S1 at both the endsof the rough surface at about 0.5mm displacement (Plate 1(a) and (b)). But the band at the rightend (passive zone) is oriented at a greater slope from the horizontal than the one at the left end(active zone). The difference is due to the greater mobilized strength and dilation occurring inthe passive region. Another primary S2 rupture band, immediately following the S1 rupturebands, forms horizontally along the rough surface. Due to the kinematic constraint of theboundary movement, this rupture band is forced to be aligned with the direction of zeroextension line (Plate 1(b)). Subsequently, a series of secondary bands that originate from theprimary S1 and S2 bands develop inside the interphase due to local effects of surface roughness.

All the primary bands and their secondary structures expand and coalesce with each otherwith continuous shearing, finally resulting in a mound-shaped region of shear localization atpeak state (Plate 1(c)). After peak state, the rupture bands expand rapidly with post-peak strainsoftening, but as can be seen, the S1 band at the right end and S2 band are the most intenselydeveloped bands (Plate 1(d)). The mound region corresponds to the localized zone under theinfluence of interface shear and is a visual representation of the interphase region. The peak ofthe mound is located approximately at the middle of the box, with a height of about 25 medianparticle diameters above the asperities. But the most intensely sheared zone at the bottom of themound is only 8–10 median particle diameters thick. In the steady state with large sheardisplacement, a thin and long shear zone of intense strain localization close to the surfaceemerges (Plate 1(e)). Due to the lower resolution necessary in this plot, the large mound-shapedlocalization zone shown in earlier stages cannot be fully visualized.

The dependence of the accuracy of the computed strain field on the size of discretization isdemonstrated by repeating the computation using another two grid spacings, which are 1.28mmin the horizontal and 1.4mm in the vertical, and 0.32mm in the horizontal and 0.35mm in thevertical. The results of shear strain contours of Simulation A at 2mm shear displacement arecompared for different grid spacings in Plate 2. It can be seen that some features of strainlocalization are lost and a much lower resolution of shear strain field results when a coarsermesh grid is used, as shown in Plate 2(a), where the grid spacing is nearly twice D50: When thegrid spacing is reduced to about 0.5 D50; as in Plate 2(c), a shear strain field with even higherresolution is obtained, with minute details of strain localization more clearly depicted. However,all the major important features of primary and secondary bands are as well captured in Plate2(b), with no additional significant details revealed by spending much higher computationalefforts. Therefore, it is concluded that a grid spacing of a median particle diameter would besufficient to capture the strain localization at a satisfactory resolution.

Shear strain contours at point A to point E in the stress–displacement curve calculated usingthe method proposed by O’Sullivan et al. [30] are shown in Plate 3 for comparison. Because ahigh-order mesh free interpolation function was adopted, the characteristics of the rupturebands inside the shear zone, especially at small strain pre-peak and peak stages (Plate 3(a)and (b)), are not visible. The overall shape of the shear zone is also marked, with significant



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noises taking place in the upper part of the box. However, at large shear deformation, the shearband is in good agreement with Plate 1(e). Therefore, it is apparent from this comparison thatthe current method is capable of detecting pre-peak and peak rupture bands much moreaccurately than O’Sullivan’s method.

Effects of surface geometry on strain localization behaviour

When the surface geometry changes, it will influence the mobilized interface friction resistanceand the associated strain localization behaviour. Plates 4 and 5 show the shear straindistributions at peak state and steady state, respectively, from Simulation B to Simulation Gwith varying surface geometries. Profiles of surface geometry of all the six surfaces are shown inFigure 8.

It can be seen from Plates 4(a)–(c) that the shear localization zones developed by regular sawtooth surfaces at peak state are generally more continuous and expansive than those by irregularsurfaces (Plates 4(d)–(f)). This is attributed to the relatively large spatial variation of surfaceroughness of the irregular surface. Higher mobilized friction resistance will take place at thesections with greater roughness and correspondingly, more intense shear deformation will occurat the same place. Generally speaking, the surfaces which can sustain higher stress ratios willresult in greater shear deformation and thicker shear band. The structures of local rupturebands developed by irregular surfaces are also more complicated than those by regular surfaces.Local mound-shaped localization zones are more likely to form at different sections of theirregular surface, with the size of the ‘mound’ corresponding to the roughness of that particularsection of the surface. However, within each local mound-shaped zone, the structure of rupturebands is similar to that of a regular surface, as described above.

In Plate 5, the six fully developed shear bands at steady state possess the same general shapebut apparently the two shear bands developed by surface B and E (Plates 5(a) and (d)) are muchmore diffused than the others. This is because the peak stress ratios measured on surface B andE are 0.215 and 0.3, respectively much lower than the strength ratio (0.63) of the granularmaterial. For surfaces with roughness high enough to fully mobilize the material strength, suchas surface A (Plate 1(e)) and surface C (Plate 5(b)), the shear band thickness reaches itsmaximum value of about 8 median particle diameters.

As mentioned above, the excessively large displacement gradient close to the rough surface atsteady state has dwarfed the relatively small displacement discontinuity in the higher regionaway from the surface when shear localization first appears around peak state. To betterexamine the scope of shear localization, displacement profiles of column 9 recorded at post-peakstate (2mm shear displacement), which is located at the middle of the box, are shown inFigure 9. As can be seen from Figure 9, surfaces A and C resist greater shear deformation thanother surfaces because they sustain the highest stress ratios (fully mobilizing the materialstrength). A few particles are trapped inside their asperity valleys and travel nearly the sameamount of distance as the surfaces. It is found that surfaces which have large enough roughnessto fully mobilize the granular material strength will develop the thickest shear band. Themaximum thickness of the shear band due to post-peak localization is about 8–10 medianparticle diameters from the surface asperities. Surfaces with lower degrees of roughness willsustain a lower stress ratio and accordingly, develop a less intense and thick shear band. This isclearly demonstrated by the displacement profiles of surfaces B and E, whose slope of the upperlinear portion and displacement gradient inside the lower shear zone are the smallest. Surfaces



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D, F and G fall between these two extremes, indicating their intermediate degrees of roughnessto resist shear deformation.

Interestingly, the irregular surface F has a similar shear band pattern to surfaces A and C butthe slope of the upper linear portion falls at the lower end, i.e. being the same as surfaces

Figure 8. Profiles of surface geometry in Simulation B to G: (a) Simulation B; (b) Simulation C;(c) Simulation D; (d) Simulation E; (e) Simulation F; and (f) Simulation G.



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B and E. Surface F sustains a peak stress ratio of 0.51, which is almost the upper bound for anirregular surface even if it possesses a greater roughness. This is mainly attributed to the lowerinitial stiffness of the irregular surface as compared to the regular saw tooth surface withcomparable roughness. Lower shear stress will build up on the irregular surface than on theregular surface due to a same amount of shear displacement during the initial shearing beforeshear localization occurs, resulting in a smaller amount of pure shear rotation provided that theshear modulus of a given granular material remains constant. The pure shear rotation of thegranular material stops after shear localization starts, and is replaced by the plastic flow insidethe shear band above the surface. The peak stress ratio, however, is related to the extent of theshear band and is ultimately governed by the relative particle to surface interaction [1]. The largevariation of the asperity slopes along the length of the irregular surface will usually make thecontact forces acting on the surface counteract with each other, leading to a lower macroscopicstress ratio than a regular surface with similar roughness.

Shear band orientation

It was found that the kinematic movement of the rough boundary dominates the formation ofthe shear bands, especially the primary S2 band, which is almost horizontally oriented above therough surface for all the cases, as shown in Plate 5. This a natural result due to the horizontalmovement of the rough surface which forces the primary S2 band to be aligned with the zeroextension direction. The orientations of other shear bands, however, are not clearly defined inthe interface shear box due to the complicated effects of surface roughness and its interactionwith the interphase material. These effects on the shear banding behaviour are bestdemonstrated by the shear localization zone shown in Plates 4 and 5. It should be clarifiedhere that the profound effects of the rough boundary on the shear banding behaviour are exactlywhat we mean to investigate in this paper. This is somewhat different from the conventionalstudies on the shear band of granular soils in which mechanics of the granular soil itself is theprimary target and boundary constraint is minimized.

Figure 9. Horizontal displacements of particles in Column 9 at 2mm displacement in Simulation A to G.



DOI: 10.1002/nag

CONCLUSIONS

In this paper, a numerical study of shear banding behaviour in the interface shear test composedof assemblage of densely packed spherical particles in contact with rough surfaces is presented.A new simple numerical-based method of strain calculation is used in this study to generatestrain field inside the interface shear box. This method avoids using any interpolation functionwhich is conventionally adopted in any other numerical method and takes into account particlerotation which plays an important role in the shear band formation. The calculated strain field isfound to clearly outline the overall shape of shear localization zone and capture thecharacteristics of rupture bands more accurately than that calculated using the methodproposed by O’Sullivan et al. [30], especially at small strain peak or pre-peak stages.

Results show that strain localization starts early with the occurrence of non-linear stress–strain behaviour. Discontinuities of particle displacements lead to localized shear deformationinto a narrow zone above the surface asperities around peak state. For a regular surface, a seriesof rupture bands emerge in a sequential way to form the major and subordinate structures of thefinal shear zone at the post-peak stage. The irregular surfaces tend to develop irregular zones ofshear localization due to spatial distribution of surface roughness along its surface length, butthe structure of rupture bands within each local shear zone is similar to that of a regular surface.It has been found that the surfaces with greater roughness will generally sustain higher stressratios and develop thicker shear bands. The surfaces which can fully mobilize the materialstrength will develop the most intense shear band with maximum thickness of about 8–10median particle diameters. Results of shear band orientation show that the effect of lowerboundary movement is dominant, making the major shear band aligned with the horizontal zeroextension direction.

ACKNOWLEDGEMENTS

This material is based upon work supported by the National Science Foundation under Grant No. CMS-0200949. Any opinions, findings, and conclusions or recommendations expressed in this material are thoseof the authors and do not necessarily reflect the views of the National Science Foundation.

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Plate 1. Evolution of strain localization and rupture bands in Simulation A: (a) at pre-peak, 0.5mmdisplacement; (b) at pre-peak, 0.7mm displacement; (c) at peak, 1mm displacement; (d) at post-peak, 2mmdisplacement; and (e) at steady state, 6.3mm displacement. Positive and negative values indicate shear

strain in counter-clockwise and clockwise directions, respectively (same in Plate 2 to Plate 5).


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Plate 2. Effects of size of discretization on the computed shear strain field. Grid spacing of: (a) 1.28mm inthe horizontal and 1.4mm in the vertical; (b) 0.64mm in the horizontal and 0.7mm in the vertical; and (c)

0.32mm in the horizontal and 0.35mm in the vertical.


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Plate 3. Evolution of strain localization in Simulation A calculated using O’Sullivan’s method: (a) at pre-peak, 0.5mm displacement; (b) at pre-peak, 0.7mm displacement; (c) at peak, 1mm displacement; (d) at

post-peak, 2mm displacement; and (e) at steady state, 6.3mm displacement.


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Plate 4. Shear strain distribution and rupture bands at peak state in Simulation B to G: (a) Simulation B;(b) Simulation C; (c) Simulation D; (d) Simulation E; (e) Simulation F; and (f) Simulation G.


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Plate 5. Shear strain distribution and rupture bands at steady state in Simulation B to G: (a) Simulation B;(b) Simulation C; (c) Simulation D; (d) Simulation E; (e) Simulation F; and (f) Simulation G.


DOI: 10.1002/nag

numerical studies of shear banding in interface shear tests using a new strain...

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