Numerical studies of shear banding in interface shear tests using a new strain calculation method k

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1 INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech., (in press) Published online in Wiley InterScience ( Numerical studies of shear banding in interface shear tests using a new strain calculation method k 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 to quasi-static direct interface shear, especially after peak stress state is reached. This behaviour is important because it is closely related to deformations experienced by geotechnical composite structures. This paper presents a study using two-dimensional discrete element method (DEM) simulations on the strain localization of an idealized interphase system composed of densely packed spherical particles in contact with rough manufactured surfaces. The manufactured surface is made up of regular or irregular triangular asperities with varying slopes. A new simple method of strain calculation is used in this study to generate strain field inside a simulated direct interface shear box. This method accounts for particle rotation and captures strain localization features at high resolution. Results show that strain localization begins with the onset of non-linear stress strain behaviour. A distinct but discontinuous shear band emerges above the rough surface just before the peak stress state, which becomes more expansive and coherent with post-peak strain softening. It is found that the shear bands developed by surfaces with smaller roughness are much thinner than those developed by surfaces with greater roughness. The maximum thickness of the intense shear zone is observed to be about 8 10 median particle diameters. The shear band orientations, which are mainly dominated by the rough boundary surface, are parallel with the zero extension direction, which are horizontally oriented. 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. y magutier@vt.edu z Associate Professor. } Postdoctoral Research Associate. } Research Assistant Professor. k This 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

2 J. WANG, M. S. GUTIERREZ AND J. E. DOVE INTRODUCTION Shear resistance develops when relative displacement occurs inside the interphase region between a granular soil layer in contact with a rough, continuous manufactured or natural material surface. The interphase region consists of the rough surface with its asperities, and a variable thickness of granular material directly adjacent to the surface. The thickness of the granular portion of the interphase is normally associated with the thickness of the zone of intense shear deformation (shear band). The interface is a transitional zone between the surface and the granular soil inside the interphase (shear zone) where particle to surface interactions take 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 in volume 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 strain behaviour, resulting in a series of shear bands developing near the surface, especially after peak state is reached. In this paper, the term state is used to refer to the stress level that the soil is experiencing. So, the peak state means the peak stress the soil undergoes. The mechanism underlying the interphase strength and shear banding behaviour is critical to the design of geotechnical composite systems because the load deformation behaviour of the interphase region 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 surface and granular soil controls interphase strength behaviour. However, the development of strain localization 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 or granular soils subject to various shearing conditions have contributed to our understanding of shear band morphology. For example, Morgenstern and Tchalenko [2] studied the formation of small-scale shear features within kaolin subject to direct shear tests. Their observations showed a series of structures and sub-structures were produced in a sequential manner, which coalesce to form the ultimate shear bands. A similar study of shear band patterns on dense sands subject to 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 banding behaviour. Other valuable work on the shear banding behaviour of granular soils includes References [4 11]. Most of the previous studies on behaviour of the interphase between granular media and rough surfaces have focused on exploring and quantifying factors influencing peak and steady state strength [12 17]. Uesugi and Kishida [12 14] introduced a widely used normalized roughness parameter R n to correlate with the peak friction angles of sand steel surfaces. However, it has been concluded by Wang [1] that statistical geometric parameters alone cannot account for the actual contact conditions occurring at the boundary between the particles and surface, and within the interphase. Therefore to date, only non-unique correlations between interface strength and surface geometry have been achieved. Wang [1] has proposed a failure criterion which is based on the contacts existing between the first layer of particles and the surface. It accounts for the actual mechanism controlling particle and surface interaction and

3 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS has proven to be successful in predicting interface strength for a range of particle distributions and surface geometries. However, very limited experimental work has been published on the shear deformation of interphase 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 shear apparatus and described qualitatively their deformation characteristics due to interface roughness. Westgate and DeJong [19] used particle image velocimetry (PIV) to investigate the effects of surface roughness, confining pressure and initial void ratio on sand steel interface shear banding. This study illustrated the key role that surface roughness played on interphase deformation. The emergence of the discrete element method (DEM) [20, 21] and its numerical simulation provide a powerful tool for studying particulate media and greatly facilitate the understanding of granular micromechanics. A rich body of microscopic information can be obtained from DEM simulations and used to interpret the mechanical behaviour in a continuum-based framework [1]. Using DEM simulations, Wang et al. [22, 23] conducted interface shear box tests composed of densely packed spherical particles contacting rough manufactured surfaces and suggested the use of sample height of 50 median particle diameters in order for full development of shear band. They also made a preliminary discussion on the effects of surface asperity slope on interphase shear deformation and concluded that the relative grain to surface geometry was the controlling factor of the micromechanical behaviour. But these findings are insufficient to provide a quantitative description of the effects of relative grain to surface geometry on interphase shear banding behaviour. In this paper, a new simple numerical-based strain calculation method, which is found capable 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 shear box. The numerical model of interface shear test is described, and a parametric study is performed to study the influence of relative particle to surface geometry on the shear banding structures 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 a discrete system [24 27]. This type of approach usually first discretizes the domain by constructing a nodal graph and then interpolates between adjacent nodes to calculate the displacement and strain fields. For example, in evaluation of the strain field in an asphalt concrete, Wang et al. [28] constructed a triangular graph by connecting particle centroids and computed displacement values using a linear interpolation function. Recently, Alshibli and Alramahi [29] used second-order polynomial functions to interpolate the local displacements inside particle groups that consist of 10 particles each and share four particles between neighbouring 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 inaccurate estimates of strain inside a granular media involving localization. These are: (1) negligence of particle rotation which plays an important role in the formation of shear band, and (2) use of

4 J. WANG, M. S. GUTIERREZ AND J. E. DOVE a linear local interpolation function which results in erratic inter-element strain values within the shear band. To avoid the above problems, O Sullivan et al. [30] proposed a higher-order, non-local meshfree method which employs a cubic spline interpolation function and takes into account particle rotation. The method was applied to 2D biaxial compression simulations. However, it was found by the authors that the higher-order mesh free interpolation functions overly smooth the displacement gradient within the shear band and thus failed to manifest the strain localization accurately inside the interface shear box. Given this fact, the authors propose herein a modified mesh-free method which does not use interpolation functions and calculates the displacement gradient directly based on movements of individual particles. The details of the method are described below. It is believed that the method is conceptually different from [28 30]. The current problem involves large strain localization inside the granular media, therefore the Green-St. Venant large-strain tensor E ij is used E ij ¼ 1 2 ðu i;j þ u j;i þ u k;i u k;j Þ ð1þ where u i;j is the displacement gradient tensor. It is based on the deformation measure related to the reference configuration. The mesh-free method used in this study employs a grid-type discretization over the reference configuration. A schematic diagram of the approach is illustrated in Figure 1. A rectangular grid is generated that serves as the continuum reference space superimposed over the volume of particles. Then each grid point is assigned to an individual particle j which has the following property: d j r j 4 d i r i ði ¼ 1; 2;...; N p ; i=jþ ð2þ Figure 1. Schematic diagram of the proposed meshfree method: (a) association of a grid point with certain particle and (b) displacement of grid point and its associated particle.

5 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS where r i is the radius of particle i; d i is the distance between the grid point and the centroid of particle i; N p is the total number of particles within the volume (Figure 1(a)). If the ratio of the distance between particle centroid and its associated grid point to the particle radius is the least among all the particles, then the grid point is considered as a point on the original or extended rigid body of the particle. The displacement of the grid point is calculated by u g x ¼ up x þ dðcosðy 0 þ oþ cosðy 0 ÞÞ u g y ¼ up y þ dðsinðy 0 þ oþ sinðy 0 ÞÞ ð3þ where u g x ; ug y and up x ; up y 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; y 0 is the initial phase angle of the position of a grid point relative to the particle centroid; and o is the accumulated 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 this parameter is 0.64 mm in the horizontal direction and 0.7 mm in the vertical direction in the current paper. This results in totally 41 rows and 201 columns of grid points over the entire volume. As a general guidance, a grid spacing of a median particle diameter (D 50 Þ should suffice to capture the shear localization at satisfactory resolution. This will be verified later by comparing the results obtained using varying grid spacings. The proposed method discretizes the reference space based on the simple algorithm represented by Equation (2) before any deformation occurs. Any random point (not necessarily the grid point) inside the whole volume can be assigned to a particular particle using this algorithm. Then all the grid points that are assigned to the same single particle connect to form a region that belongs to this particle. So if any point lies within this region, it will be considered as a point on the original or extended rigid body of this particle, and its displacement will be calculated using the principle of rigid body motion of the particle. This method takes into account particle rotation and is found capable of capturing the strain features inside the interface 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 and validated against laboratory data [16]. The model is made up of a 128 mm long, 28 mm high shear box filled with a polydispersed mixture of spherical particles (maximum and minimum particle diameters of 1.05 and 0.35 mm, respectively) at a low initial porosity of A sketch of the model is shown in Figure 2(a). The lower boundary is made up of a rough surface and two 20 mm long dead zones placed at the ends of the box to avoid boundary effects. The rough surface consisted of either regular or irregular triangular asperities, or of profiles of natural and manufactured material surfaces made using a stylus profilometer. There is no particle-toboundary friction within the dead zones. Shearing occurs by displacing the lower boundary of the box to the left at a constant velocity of 1 mm per minute. Vertical and horizontal contact forces acting on all the asperities are summed to compute the normal and shear forces generated along the interface, respectively.

6 J. WANG, M. S. GUTIERREZ AND J. E. DOVE Figure 2. DEM model of direct interface shear box: (a) interphase composed of granular spheres in contact with a rough surface and (b) particle columns pre-selected to identify interphase deformation. The Hertz Mindlin contact model was implemented in the simulations. A particle rolling resistance model [31] was also applied at both particle particle contacts and particle boundary contacts. Physical constants used in the simulations include: particle density of 2650 kg=m 3 ; shear modulus and Poisson s ratio of the particles of 29 GPa and 0.3, respectively; critical normal and shear viscous damping coefficient, both equal to 1.0; time step 5: s; particleto-surface asperity friction coefficient of 0.05; and interparticle and particle boundary friction coefficient (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 banding behaviour, we conducted five groups of numerical simulations. In Groups 1, 2 and 3, regular

7 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS 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 used in the Experimental Groups 1, 2 and 3. Group Variable Values of variables R t =D 50 S r =D 50 S w =D 50 1 R t =D , 0.2, 0.5, 1, 1.2, 1.5, 2, 3, 5 Varied S r =D , 0.5, 1, 2, 3, 5, 8, 10 1 Varied 2 3 S w =D , 0.5, 1, 2, 3, 4, 5, Varied saw tooth asperities were used, with asperity height (R t Þ; asperity width (S w Þ and spacing between asperities (S r Þ varied in each group, respectively (Figure 3(a)). Table I provides the ranges 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 (Figure 3(b)). Details of the analytical method and parameters used to generate the irregular surface are given 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 the median particle diameter in width and extends for the full height of the specimen. A typical profile of the columns in a 28 mm thick sample is shown in Figure 2(b). The displacements and

8 J. WANG, M. S. GUTIERREZ AND J. E. DOVE velocities of particles in these columns are recorded during the shearing process and used for the analysis of shear banding. RESULTS AND DISCUSSIONS In this section, results from numerical simulations of eight surfaces will be presented. The eight surfaces, called surface A to surface H, are selected from all the five groups of numerical simulations and give a representative collection of the interphase shear behaviour under the influence 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 from simulation A is shown in Figure 4. Due to the low initial porosity, distinct post-peak strain softening to steady state is observed (Figure 4(a)). Continuous dilation is also observed (Figure 4(b)), with the highest dilation rate occurring around peak state and approaching zero at steady state deformation. Similar curves were obtained from other simulations with varying surface geometries, but less dilation and mobilized peak stress ratio are typical for surfaces with lower degrees of relative particle to surface roughness. The internal shear deformation of the sample is closely monitored by recording the accumulated displacements at different stages of the test of the particles comprising the 16 selected columns. Snapshots were taken at five instants as indicated on the stress displacement curve (Figure 4(a)): point A (pre-peak, 0.5 mm), point B (pre-peak, 0.7 mm), point C (peak, 1 mm), point D (post-peak, 2 mm) and point E (steady state, 6.3 mm). 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 is the dominating mode of deformation. With the onset of non-linear stress displacement behaviour, discontinuities in particle displacements are observed, leading to a narrow, localized shear deformation zone at about 10 median particle diameters above the surface asperities. The extent of strain localization is small prior to peak state therefore it cannot be readily observed from deformed columns (Figure 5(a)). After peak state, rolling and sliding of particles over the asperities leads to the rapid reduction of shear stress and expansion of the plastic zone of strain localization. Curved profiles of deformed columns above the asperities can be clearly seen at the post-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 in Figure 5(c). Table II. Eight surfaces whose simulation results are presented. Surface A B C D E F G H Group No Description R t =D 50 ¼ 1 S w =D 50 ¼ 10 R t =D 50 ¼ 5 S r =D 50 ¼ 5 See See See S w =D 50 ¼ 3 Figure 8 Figure 8 Figure 8

9 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS Figure 4. Macroscopic interface behaviour of Simulation A: (a) stress displacement relationship and (b) volume change displacement relationship. 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 was found that the boundary effects play a significant role on the interphase shear behaviour. The current size of the numerical box is chosen such that the length is about 180 median particle diameters (D 50 Þ; which meets the minimum laboratory requirement of 70 D 50 for shear box size used in the conventional soil testing. A 20 mm long frictionless dead zone is placed at both ends of the box to avoid the lateral boundary effects, as mentioned above. In addition, the effects of model height on the behaviour of interphase shear band has been quantified in detail in [23] and it was found that a maximum shear band with thickness of 8 10 D 50 can be formed if the model height reaches D 50 : The numerical model used a height of 40 D 50 : Particle to boundary friction, which was found to have major impacts on the initial stiffness behaviour and particle movement, has been assigned a high value (0.9) to prevent the sliding of the granular sample with 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 from surface B and surface H recorded at 7 mm shear displacement. The laboratory data were obtained from two types of sand steel interfaces, namely sandblasted steel surface and epoxied

10 J. WANG, M. S. GUTIERREZ AND J. E. DOVE Figure 5. Deformed particle columns inside the shear box from Simulation A: (a) at peak, 1 mm displacement; (b) at post-peak, 2 mm displacement; and (c) at steady state, 6.3 mm displacement. sand steel surface, but were the average positions and displacements for each row of the PIV patch mesh (21 rows by 28 columns) recorded at the same shear displacement. The laboratory soil specimen has a dimension of mm (length) 63:4 mm (width) 25:4 mm (height). The material used in the experiments is a uniform Ottawa sand with D 50 ¼ 0:74 mm: Two

11 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS Figure 6. Comparison of particle horizontal displacements from simulations with laboratory data. parameters, the average roughness (R a Þ and average slope (D a Þ were used to quantify the roughness of the steel surfaces. Specifically, R a is defined as the average deviation of profile height ZðxÞ from the mean line, i.e. R a ¼ð1=LÞ R L 0 jzðxþj dx; where L is the total length of the profile; D a is defined as the arithmetic average of the rate of profile height change over the whole length, i.e. D a ¼ð1=LÞ R L 0 jdz=dxj dx: In Figure 6, profiles of particle displacements from the simulations agree very well with the laboratory data, showing an excellent match in the lower shear band. At the top of the box, slip of 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 the simulation. A similar slip at the top wall has also been observed in our simulations if a lower particle to boundary friction coefficient is adopted. The surfaces used in the simulations are not reproductions of the steel surfaces used by Westgate and DeJong in their laboratory tests. Difference of roughness quantified by R a and D a exists between the two groups of surfaces, and will lead to different degrees of shear localization inside the interphase region. In addition, other differences between the simulations and laboratory tests including granular material, initial density, interparticle friction and particle to surface friction, etc. will also contribute to the dissimilar interphase shear behaviour and make the 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 a validation of our numerical model in terms of producing physically sound and realistic interphase shear behaviour. The overall agreement of the displacement profiles observed in Figure 6 suggests that our numerical model is able to effectively eliminate the boundary effects and 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 granular media is first validated by comparing the displacement profile of grid points calculated using Equations (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

12 J. WANG, M. S. GUTIERREZ AND J. E. DOVE Figure 7. Comparison of the calculated displacement profile of grid point columns to the displacements of particle columns recorded during Simulation A: (a) at post-peak, 2 mm displacement and (b) at steady state, 6.3 mm displacement. Simulation A at 2 mm post-peak state (point D) and 6.3 mm steady state (point E). The grid point columns 20, 50, 100 and 150 were selected with initial horizontal positions corresponding to the particle columns 2, 5, 9 and 13, respectively. It is seen that the calculated displacements fit the recorded data very well at both shearing stages, indicating the proposed method is excellent in capturing the displacement field characteristic of a granular media. The evolution of strain localization in Simulation A can be better visualized by the shear strain contours computed at point A to point E shown in Plate 1. The shear strain field is calculated using the proposed strain calculation method given in Equations (1) (3), and is shown overlain on the original configuration of the sample. A series of rupture bands emerge in a sequential way to form the final shear band. The patterns of rupture bands are similar to those

13 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS observed by Morgenstern and Tchalenko [2] and Scarpelli and Wood [3]. Here, we adopt the same notation for rupture bands, S k ; as used by Scarpelli and Wood [3] to describe the evolution of strain localization inside the interface shear box. The subscript k refers to the order of appearance of the rupture bands in time. In this paper, we only report primary and fully developed structures of rupture bands which result from boundary movement. Local and secondary structures, which are smaller and are not fully developed, are not included in the analysis. Strain localization occurs very early with the appearance of rupture bands S 1 at both the ends of the rough surface at about 0.5 mm displacement (Plate 1(a) and (b)). But the band at the right end (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 in the passive region. Another primary S 2 rupture band, immediately following the S 1 rupture bands, forms horizontally along the rough surface. Due to the kinematic constraint of the boundary movement, this rupture band is forced to be aligned with the direction of zero extension line (Plate 1(b)). Subsequently, a series of secondary bands that originate from the primary S 1 and S 2 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 other with continuous shearing, finally resulting in a mound-shaped region of shear localization at peak state (Plate 1(c)). After peak state, the rupture bands expand rapidly with post-peak strain softening, but as can be seen, the S 1 band at the right end and S 2 band are the most intensely developed bands (Plate 1(d)). The mound region corresponds to the localized zone under the influence of interface shear and is a visual representation of the interphase region. The peak of the mound is located approximately at the middle of the box, with a height of about 25 median particle diameters above the asperities. But the most intensely sheared zone at the bottom of the mound is only 8 10 median particle diameters thick. In the steady state with large shear displacement, a thin and long shear zone of intense strain localization close to the surface emerges (Plate 1(e)). Due to the lower resolution necessary in this plot, the large mound-shaped localization zone shown in earlier stages cannot be fully visualized. The dependence of the accuracy of the computed strain field on the size of discretization is demonstrated by repeating the computation using another two grid spacings, which are 1.28 mm in the horizontal and 1.4 mm in the vertical, and 0.32 mm in the horizontal and 0.35 mm in the vertical. The results of shear strain contours of Simulation A at 2 mm shear displacement are compared for different grid spacings in Plate 2. It can be seen that some features of strain localization are lost and a much lower resolution of shear strain field results when a coarser mesh grid is used, as shown in Plate 2(a), where the grid spacing is nearly twice D 50 : When the grid spacing is reduced to about 0.5 D 50 ; as in Plate 2(c), a shear strain field with even higher resolution 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 Plate 2(b), with no additional significant details revealed by spending much higher computational efforts. Therefore, it is concluded that a grid spacing of a median particle diameter would be sufficient to capture the strain localization at a satisfactory resolution. Shear strain contours at point A to point E in the stress displacement curve calculated using the method proposed by O Sullivan et al. [30] are shown in Plate 3 for comparison. Because a high-order mesh free interpolation function was adopted, the characteristics of the rupture bands 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

14 J. WANG, M. S. GUTIERREZ AND J. E. DOVE noises taking place in the upper part of the box. However, at large shear deformation, the shear band is in good agreement with Plate 1(e). Therefore, it is apparent from this comparison that the current method is capable of detecting pre-peak and peak rupture bands much more accurately 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 resistance and the associated strain localization behaviour. Plates 4 and 5 show the shear strain distributions at peak state and steady state, respectively, from Simulation B to Simulation G with varying surface geometries. Profiles of surface geometry of all the six surfaces are shown in Figure 8. It can be seen from Plates 4(a) (c) that the shear localization zones developed by regular saw tooth surfaces at peak state are generally more continuous and expansive than those by irregular surfaces (Plates 4(d) (f)). This is attributed to the relatively large spatial variation of surface roughness of the irregular surface. Higher mobilized friction resistance will take place at the sections with greater roughness and correspondingly, more intense shear deformation will occur at the same place. Generally speaking, the surfaces which can sustain higher stress ratios will result in greater shear deformation and thicker shear band. The structures of local rupture bands 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 the irregular surface, with the size of the mound corresponding to the roughness of that particular section of the surface. However, within each local mound-shaped zone, the structure of rupture bands 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 shape but apparently the two shear bands developed by surface B and E (Plates 5(a) and (d)) are much more diffused than the others. This is because the peak stress ratios measured on surface B and E are and 0.3, respectively much lower than the strength ratio (0.63) of the granular material. For surfaces with roughness high enough to fully mobilize the material strength, such as surface A (Plate 1(e)) and surface C (Plate 5(b)), the shear band thickness reaches its maximum value of about 8 median particle diameters. As mentioned above, the excessively large displacement gradient close to the rough surface at steady state has dwarfed the relatively small displacement discontinuity in the higher region away from the surface when shear localization first appears around peak state. To better examine the scope of shear localization, displacement profiles of column 9 recorded at post-peak state (2 mm shear displacement), which is located at the middle of the box, are shown in Figure 9. As can be seen from Figure 9, surfaces A and C resist greater shear deformation than other surfaces because they sustain the highest stress ratios (fully mobilizing the material strength). A few particles are trapped inside their asperity valleys and travel nearly the same amount of distance as the surfaces. It is found that surfaces which have large enough roughness to fully mobilize the granular material strength will develop the thickest shear band. The maximum thickness of the shear band due to post-peak localization is about 8 10 median particle diameters from the surface asperities. Surfaces with lower degrees of roughness will sustain a lower stress ratio and accordingly, develop a less intense and thick shear band. This is clearly demonstrated by the displacement profiles of surfaces B and E, whose slope of the upper linear portion and displacement gradient inside the lower shear zone are the smallest. Surfaces

15 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS 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. D, F and G fall between these two extremes, indicating their intermediate degrees of roughness to resist shear deformation. Interestingly, the irregular surface F has a similar shear band pattern to surfaces A and C but the slope of the upper linear portion falls at the lower end, i.e. being the same as surfaces

16 J. WANG, M. S. GUTIERREZ AND J. E. DOVE Figure 9. Horizontal displacements of particles in Column 9 at 2 mm displacement in Simulation A to G. B and E. Surface F sustains a peak stress ratio of 0.51, which is almost the upper bound for an irregular surface even if it possesses a greater roughness. This is mainly attributed to the lower initial stiffness of the irregular surface as compared to the regular saw tooth surface with comparable roughness. Lower shear stress will build up on the irregular surface than on the regular surface due to a same amount of shear displacement during the initial shearing before shear localization occurs, resulting in a smaller amount of pure shear rotation provided that the shear modulus of a given granular material remains constant. The pure shear rotation of the granular material stops after shear localization starts, and is replaced by the plastic flow inside the shear band above the surface. The peak stress ratio, however, is related to the extent of the shear band and is ultimately governed by the relative particle to surface interaction [1]. The large variation of the asperity slopes along the length of the irregular surface will usually make the contact forces acting on the surface counteract with each other, leading to a lower macroscopic stress 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 of the shear bands, especially the primary S 2 band, which is almost horizontally oriented above the rough surface for all the cases, as shown in Plate 5. This a natural result due to the horizontal movement of the rough surface which forces the primary S 2 band to be aligned with the zero extension direction. The orientations of other shear bands, however, are not clearly defined in the interface shear box due to the complicated effects of surface roughness and its interaction with the interphase material. These effects on the shear banding behaviour are best demonstrated by the shear localization zone shown in Plates 4 and 5. It should be clarified here that the profound effects of the rough boundary on the shear banding behaviour are exactly what we mean to investigate in this paper. This is somewhat different from the conventional studies on the shear band of granular soils in which mechanics of the granular soil itself is the primary target and boundary constraint is minimized.

17 NUMERICAL STUDIES OF SHEAR BANDING IN INTERFACE SHEAR TESTS CONCLUSIONS In this paper, a numerical study of shear banding behaviour in the interface shear test composed of 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 generate strain field inside the interface shear box. This method avoids using any interpolation function which is conventionally adopted in any other numerical method and takes into account particle rotation which plays an important role in the shear band formation. The calculated strain field is found to clearly outline the overall shape of shear localization zone and capture the characteristics of rupture bands more accurately than that calculated using the method proposed 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 deformation into a narrow zone above the surface asperities around peak state. For a regular surface, a series of rupture bands emerge in a sequential way to form the major and subordinate structures of the final shear zone at the post-peak stage. The irregular surfaces tend to develop irregular zones of shear localization due to spatial distribution of surface roughness along its surface length, but the 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 stress ratios and develop thicker shear bands. The surfaces which can fully mobilize the material strength will develop the most intense shear band with maximum thickness of about 8 10 median particle diameters. Results of shear band orientation show that the effect of lower boundary movement is dominant, making the major shear band aligned with the horizontal zero extension direction. ACKNOWLEDGEMENTS This material is based upon work supported by the National Science Foundation under Grant No. CMS Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. REFERENCES 1. Wang JF. Micromechanics of granular media: a fundamental study of interphase systems. Ph.D. Dissertation, Virginia tech, Blacksburg, Virginia, Morgenstern NR, Tchalenko JS. Microscopic structure in kaolin subjected to direct shear. Ge otechnique 1967; 17: Scarpelli G, Wood DM. Experimental observations of shear band patterns in direct shear tests. IUTAM Conference on Deformation and Failure of Granular Materials, Delft, 1982; Roscoe KH, Schofield AN, Wroth CP. On the yielding of soils. Ge otechnique 1958; 8(1): Arthur JRF, Dunstan T. Rupture layers in granular media. IUTAM Conference on Deformation and Failure of Granular Materials, Delft, 1982; Oda M, Nemat-Nasser S, Konishi J. Stress-induced anisotropy in granular masses. Soils and Foundations 1985; 25(3): Vermeer PA. The orientation of shear bands in biaxial tests. Ge otechnique 1990; 40(2): Bardet JP, Proubet J. Shear-band analysis in idealized granular material. Journal of Engineering Mechanics (ASCE) 1992; 118(2):

18 J. WANG, M. S. GUTIERREZ AND J. E. DOVE 9. Alshibli KA, Sture S. Shear band formulation in plane strain experiments of sand. Journal of Geotechnical and Geoenvironmental Engineering (ASCE) 2000; 126(6): Masson S, Martinez J. Micromechanical analysis of the shear band behaviour of a granular material. Journal of Engineering Mechanics 2001; 127(10): Rechenmacher AL. Grain-scale process governing shear band initiation and evolution in sands. Journal of the Mechanics and Physics of Solids 2006; 54: Uesugi M, Kishida H. Influential factors of friction between steel and dry sands. Soils and Foundations 1986; 26(2): Uesugi M, Kishida H. Frictional resistance at yield between dry sand and mild steel. Soils and Foundations 1986; 26(4): Kishida H, Uesugi M. Tests of interface between sand and steel in the simple shear apparatus. Ge otechnique 1987; 37(1): Dove JE, Harping JC. Geometric and spatial parameters for analysis of geomembrane/soil interface behaviour. Proceeding of Geosynthetics, Boston, MA, vol. 1. Industrial Fabrics Association International: Roseville, MN. 1999; Dove JE, Jarrett JB. Behavior of dilative sand interface in a geotribology framework. Journal of Geotechnical and Geoenvironmental Engineering 2002; 128(1): Paikowsky SG, Xi F. Photoelastic quantitative study of the behaviour of discrete materials with application to the problem of interfacial friction. Research Report, Air Force Office of Scientific Research, Research Grant F , Uesugi M, Kishida H, Tsubakihara Y. Behavior of sand particles in sand steel friction. Soils and Foundations 1988; 28(1): Westgate ZJ, DeJong JT. Evolution of sand structure interface response during monotonic shear using particle image velocimetry. Proceedings of Geocongress ASCE: Atlanta, GA. 20. Cundall PA. A computer model for simulating progressive large scale movements in blocky rocky systems. Proceedings of the Symposium of the International Society of Rock Mechanics, Nancy, France, vol. 2(8), 1971; Cundall PA, Strack ODL. A discrete numerical model for granular assemblies. Ge otechnique 1979; 29(1): Wang JF, Gutierrez MS, Dove JE. Strain localization in an idealized interphase system. Proceedings of 2005 Joint ASME/ASCE/SES Conference on Mechanics and Materials, Baton Rouge, Louisiana, Wang JF, Dove JE, Gutierrez MS, Corton JD. Shear deformation in the interphase region. In Powders and Grains 2005, Garcia-Rojo R, Herrmann HJ, McNamara S (eds). A.A. Balkema: Rotterdam, 2005; Thomas PA. Discontinuous deformation analysis of particulate media. Ph.D. Thesis, Department of Civil Engineering, University of California, Berkeley, Cambou B, Chaze M, Dedecker F. Change of scale in granular materials. European Journal of Mechanics, Vol. A (Solids) 2000; 19: Dedecker F, Chaze M, Dubujet P, Cambou B. Specific features of strain in granular materials. Mechanics of Cohesive-Frictional Materials 2000; 5: Bagi K, Bojtar I. Different microstructural strain tensors for granular materials. In Proceedings of the 4th International Conference on Analysis of Discontinuous Deformation, Bicanic N (ed.). University of Glasgow: U.K., 2001; Wang LB, Frost JD, Lai JS. Noninvasive measurement of permanent strain field from rutting in Asphalt concrete. Transportation Research Board #1687, 1999; Alshibli KA, Alramahi B. Microscopic evaluation of strain distribution in granular materials during shear. Journal of Geotechnical and Geoenvironmental Engineering (ASCE) 2006; 132(1): O Sullivan C, Bray JD, Li SF. A new approach for calculating strain for particulate media. International Journal for Numerical and Analytical Methods in Geomechanics 2003; 27: Wang JF, Gutierrez MS, Dove JE. Effect of particle rolling resistance on interface shear behaviour. Proceedings 17th ASCE Engineering Mechanics Conference, Newark, Delaware, 2004;

19 Plate 1. Evolution of strain localization and rupture bands in Simulation A: (a) at pre-peak, 0.5 mm displacement; (b) at pre-peak, 0.7 mm displacement; (c) at peak, 1 mm displacement; (d) at post-peak, 2 mm displacement; and (e) at steady state, 6.3 mm displacement. Positive and negative values indicate shear strain in counter-clockwise and clockwise directions, respectively (same in Plate 2 to Plate 5).

20 Plate 2. Effects of size of discretization on the computed shear strain field. Grid spacing of: (a) 1.28 mm in the horizontal and 1.4 mm in the vertical; (b) 0.64 mm in the horizontal and 0.7 mm in the vertical; and (c) 0.32 mm in the horizontal and 0.35 mm in the vertical.

21 Plate 3. Evolution of strain localization in Simulation A calculated using O Sullivan s method: (a) at prepeak, 0.5 mm displacement; (b) at pre-peak, 0.7 mm displacement; (c) at peak, 1 mm displacement; (d) at post-peak, 2 mm displacement; and (e) at steady state, 6.3 mm displacement.

22 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.

23 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.