Relationship between vortex structures and shear localization in 3D granular specimens based on combined DEM and Helmholtz–Hodge decomposition
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Abstract
The paper presents threedimensional simulation results of granular vortex structures in cohesionless initially dense sand during quasistatic plane strain compression. The sand behaviour was simulated using the discrete element method (DEM). Sand grains were modelled by spheres with contact moments to approximately capture the irregular grain shape. The Helmholtz–Hodge decomposition of the displacement vector field obtained with DEM was used. The variational discrete multiscale vector field decomposition allowed for separating a vector field into the sum of three uniquely defined components: curl free, divergence free and harmonic. A direct correlation between vortex structures and shear localization was studied. The simulation results showed that vortex structures were closely connected to spontaneous shear localization. They localized early in locations wherein a shear zone ultimately developed. They were affected by the specimen depth.
Keywords
Plane strain compression Granular material Discrete element method Vortex structure Helmholtz–Hodge decomposition Shear localization1 Introduction
Localization of deformation in the form of narrow zones of intense shearing is a basic phenomenon in granular materials [1, 2, 3, 4, 5]. Localization under shear may occur in the interior domain in the form of a spontaneous shear zone as a single shear zone, a multiple or a regular pattern of zones. It may be also created at interfaces in the form of an induced single wall shear zone. Localized shear inside of materials is closely related to an unstable behaviour of the entire earth structure. In continuous and discontinuous numerical calculations and laboratory experiments, shear localization is usually identified in granular bodies by grain rotations or micropolar rotations or by an increase of void ratio in initially dense ones [4]. An understanding of the mechanism of the formation of shear zones is important since they act as a precursor of the ultimate soil failure. Thus it is of major importance to predict them very early for the safety of earth structures and soil behaviour optimization.
Recently Tordesillas et al. [6] and Kozicki and Tejchman [7, 8] have shown that shear localization may be predicted through socalled granular vortex structures defined as the roughly swirling (rotating) motion of several grains around a common central point. The collective particles rotated almost as rigid bodies, however the single particles inside did not rotate. The vortices were calculated at early stages in the prepeak deformation regime. A dominant mechanism responsible for the vortex formation was the breakage of force chains [6, 9]. The collapse of main force chains lead to a formation of larger voids and appearance of vortices and their buildup to a formation of smaller voids and disappearance of vortices. The vortex motion continuously appeared and disappeared during granular flow. This kinematic mode appeared to be prevalent in grains during granular flow. The calculations in [6, 7, 8, 9] were carried out under 2D conditions only. The 2D vortices were frequently observed in experiments on granular materials (Couette shear [10], plane strain compression [11] and simple shear [12, 13]) and in calculations using the discrete element method (DEM) [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. They became apparent in experiments and calculations (mainly in shear zones in the residual state) when the motion associated with uniform (affine) strain was subtracted from the actual granular deformation. They are reminiscent of turbulence in fluid dynamics [16], however the grain rotation is several ranges of magnitude smaller than the fluid vortex rotation. In addition, their life time is also shorter than of eddies during turbulent fluid flow. Moreover, granular flow is too slow to induce inertial forces characteristic for turbulences in fluid.
Motivation behind our present work is to calculate centres of 3D granular vortex structures and to find their relationship with shear localization in a deforming granular specimen (in contrast to the previous simplified 2D computations in [7, 8, 9]). The 3D vortex structures have not been calculated in granulates yet. When identifying granular vortex structures in a threedimensional kinematic field in dry sand during quasistatic plane strain compression, a novel approach was used based on the combined Helmholtz–Hodge decomposition (HHD) of a displacement vector field [24, 25] and the discrete element method (DEM) [26]. The plane strain compression test is one of the most important geotechnical laboratory tests to experimentally investigate both strength and shear localization in granular materials [4]. The analyses were carried out with spheres with contact moments [7] which were introduced to account for the effect of particle shape angularity, e.g. resistance to relative rotations due to particle interlocking. In order to accelerate the computation time, some simplifications were assumed in analyses: large spheres with contact moments, linear sphere distribution, linear normal contact model and no particle breakage [7]. The calculations were solely carried out with initially dense sand. A threedimensional discrete element model YADE developed at University of Grenoble by Donze and his coworkers was applied [27, 28, 29].
In our previous paper [8] we calculated exactly the cores of 2D granular vortex structures in initially dense sand during a quasistatic 2D passive wall translation using the same approach. HHD proved to be an objective, universal and effective technique for identifying the centres of 2D vortex structures during granular flow that was directly based on single grain displacement increments from DEM (but not on displacement fluctuations). The method did not use any additional nonobjective parameters. It found the centres of all vortex structures. However it did not determine the size of vortex structures. A strong connection between the location of vortex structures and progressive shear localization was found in simulations. The vortex structures were the precursor of shear localization since they clearly concentrated in the area where shear zones ultimately later formed. Thus the ultimate shear zone pattern has already been detected in early loading stages. The vortex structures allowed to identify shear localization significantly earlier than, e.g. based on single grain rotations which were always the most reliable indicator of shear localization. They developed from the beginning of the deformation process. The vortex centres solely emerged in shear zones. They had a tendency to move along shear zones and their number varied (it was larger on average at the residual state). The righthanded vortices were dominant in the curved shear zone and lefthanded ones were dominant in the radial shear zone. In the residual state, local regions of dilatacy and contractancy alternately happened along shear zones with a dominance of local dilatancy. In addition, our preliminary analyses of plane strain compression in [8] detected 2D vortex structures very early at the location where a spontaneous shear zone ultimately occurred. Note that core regions of vortex structures may be detected using different methods (e.g. [6, 9, 31, 32, 33]).
The innovative points of the present paper are: (a) the calculation results of cores (rotational centres) of 3D granular vortex structures using an objective method from fluid dynamics (that has not yet been applied to granular materials), (b) the comparison between 3D and 2D vortex structures and (c) the investigations of the algorithm’s accuracy for finding the centres of 3D vortices. The vortex structures were directly related to the occurrence of spontaneous shear localization in the granular specimen. The numerical results of vortex structures and local volume changes along a granular shear zone were qualitatively compared with the experimental outcomes of plane strain compression on sand by Abedi et al. [11] and Chupin et al. [30].
The current paper is structured as follows. We first provide a brief summary of our explicit DEM model (Sect. 2). We next report on some DEM results of plane strain compression in sand (Sect. 3). The mathematical algorithm of HHD is described in Sect. 4. Then we demonstrate first 2D (Sect. 5) and later 3D results of HHD (Sect. 6) that demonstrate the efficacy of our approach to identify centres of vortex structures. Sect. 7 qualitatively compares the numerical results with the experimental ones. Finally, we draw conclusions as to the significance of our numerical results in the context of shear localization and failure in granular bodies (Sect. 8).
2 Threedimensional DEM model
where \(\vec {F}_n\) the normal force vector, \(\vec {F}_s\) the tangential force vector, \(K_{n}\)—the normal stiffness, \(K_{s}\)—the tangential stiffness, U—the penetration depth between discrete elements, \(\vec {N}\)—the unit normal vector at the contact point, \(\Delta \vec {X}_s\)—the incremental tangential displacement vector, \(E_{c}\)—the modulus of elasticity of the grain contact, \(\upsilon _{c}\)—Poisson’s ratio of the grain contact, \(R_{A}\) and \(R_{B}\)—the contacting grain radii, \(\mu \)—the interparticle friction angle, \(K_{r}\)—the rolling stiffness, \(\varDelta M\)—the contact moment increment, \(\Delta \vec {\omega }\)—the angular rotational increment vector, \(\beta \)—the dimensionless rolling stiffness coefficient, R—the equivalent grain radius, \(\eta \)  the dimensionless rolling coefficient that specifies the limit friction moment of the rolling motion, \(\alpha \)—damping parameter, \(\vec {F^{k}}\) and \(\vec {M^{k}}\) the \(k^{\mathrm{th}}\) components of the residual force and moment vector, \(\vec {v}^k\) and \(\vec {\omega }^k\)  the \(k^{th}\) components of the translational and rotational velocity.
No forces were transmitted when grains were separated. The elastic contact constants were specified from the experimental data of a triaxial compression sand test and could be related to the modulus of elasticity of grain material E and its Poisson ratio v [34, 35]. In Eq. 5, the angular rotational increment vectors do not depend on the spherical grain radii in contrast to equations that take the different radii into account [36, 37]. The rolling stiffness \(K_{r}\) in Eq. 5 is related to the tangential stiffness \(K_{s}\) in Eq. 3 by the formula in [38]. Because the proposed DEM is a fully dynamic formulation, a local nonviscous damping scheme was applied [39] in order to dissipate excessive kinetic energy in a discrete system and facilitate convergence towards quasistatic equilibrium (Eq. 8). The effect of damping was insignificant in quasistatic calculations [34, 35]. Although a nonlinear contact law is more realistic, a linear contact law provides similar results with the significantly reduced computation time [35] and therefore was used in the present simulations.
The following five main local material parameters are necessary in our DEM simulations: \(E_{c}\) (modulus of elasticity of the grain contact), \(v_{c}\) (Poisson’s ratio of the grain contact), \(\mu \) (interparticle friction angle), \(\beta \) (rolling stiffness coefficient) and \(\eta \) (limit rolling coefficient). In addition, a particle radius R, particle mass density \(\rho \) and numerical damping parameter \(\alpha \) are required. The DEM material parameters: \(E_{c}\), \(v_{c}, \mu ,\beta ,\eta \) and \(\alpha \) were calibrated using the corresponding homogeneous axisymmetric triaxial laboratory test results on Karlsruhe sand with the different initial void ratio and lateral pressure by Wu [40]. The procedure for determining the material parameters in DEM was described in detail by Kozicki et al. [34, 35]. The index properties of Karlsruhe sand are: mean grain diameter \(d_{{ 50}}=0.50\) mm, grain size between 0.08 mm and 1.8 mm, uniformity coefficient \(U_{c}=2\), maximum specific weight \(\gamma _{d}^{max}=17.4\hbox { kN/m}^{3}\), minimum void ratio \(e_{min}=0.53\), minimum specific weight \(\gamma _{d}^{min}=14.6~\hbox {kN/m}^{3}\) and maximum void ratio \(e_{max}=0.84\). The sand grains are classified as subrounded/subangular. The following material constants were found in DEM by fitting numerical outcomes with experimental ones during homogeneous triaxial compression: \(E_{c}=0.3\) GPa, \(v_{c} =0.3\), \(\mu =18^{{\circ }}\), \(\beta =0.7\), \(\eta =0.4~\rho =2.55\hbox { g/cm}^{3}\) and \(a=0.08\). Note that the constants \(E_{c}\) and \(v_{c}\) do not correspond to the elastic constants of grains [34, 35]
3 DEM results of plane strain compression
Figure 3 demonstrates one typical evolution of the mobilized internal friction angle (calculated with principal stresses from Mohr’s equation) versus the vertical normal strain \(\varepsilon _{1}=u_{1}/h\) and volumetric strain \(\varepsilon _{v}\) versus \(\varepsilon _{{ 1}}\) for the granular specimen [7]. Figures 4 and 5 show the distribution of sphere rotations \(\omega \) and void ratio e at four different values of axial strain in the vertical midsection slice with the area of \(4\times 14\hbox { cm}^{2 }\) and thickness of \(5\times d_{{ 50}}\) (1.25 cm with \(d_{{ 50}}=2.5\) mm) cut out from the granular specimen [7] (Fig. 2c). Both the quantities were calculated from the volumetric cell \(V_{c}=5d_{{ 50}}\times 5d_{{ 50}}\times 5d_{{ 50}}\) moved by \(d_{{ 50}}\) in two directions within the slice in order to create a 2D grid of the averaged values in the cell. The cell size, which was smaller than the shear zone thickness \(t_{s}\), was chosen with preliminary calculations. The averaging cell larger than \(V_{c}\) caused the results too diffusive and with the smaller cell volume \(V_{c}\), the results started to too strongly fluctuate.
The experimental curves were satisfactorily reproduced in our DEM simulations of initially dense sand [7] in spite of the fact that the real grain shape, mean grain size and grain size distribution of Karlsruhe sand were not taken into account and the assembling process generated a higher coordination number than in experiments [41]. The calculated shear zone (location, width and inclination) and maps of void ratio (Fig. 5) were also realistic with respect to the experiments. The DEM results of resultant grain rotations (Fig. 4) were qualitatively in agreement with other tests on granular bodies including shear localization wherein single grain rotations were measured in artificial granular materials [4].
4 Helmholtz–Hodge decomposition (HHD)
4.1 Method of calculations

\(\varGamma \) is the domain where the vector field \(\vec {\xi }\) is defined – the total volume of all tetrahedrons (or triangle areas) where the Delaunay triangulation was performed,

u is the discrete scalar potential at the node \('i'\, u\left( {\vec {r}}\right) = \sum _{i} \phi _i (\vec {r})u_i\),

\(\vec {v}\) is the discrete vector field at the node \('i'\, \vec {v}(\vec {r})=\sum _i \phi _i (\vec {r}){\vec {v_l}}\),

\(\phi _i \left( {\vec {r}} \right) \) is the piecewiselinear basis function (shape function) valued 1 at \({\vec {r_l}} \) (the ith node) and valued 0 at all other nodes,

\(\vec {r}\) is the spatial coordinate in \(\varGamma \) using the Cartesian coordinate system \(\vec {r}=(x,y,z)\).
4.2 Boundary conditions
5 Numerical 2D results using HHD/DEM approach
Figure 7 shows the evolution of the displacement vector field \(\vec {\xi }\) during deformation \(\varepsilon _{1}\) (the sphere displacement increment directions are marked by the white arrows). The scale attached denotes the sphere displacement vector length during \(IT=1000\) iterations in [mm/iteration], ranging from 0 up to 0.1 mm/iteration. Based on the incremental displacement vector length and vector direction changes, the displacement field inside the specimen (midregion) started to be nonuniform from the beginning of deformation. An inclined internal shear zone was already well visible along the specimen width for \(\varepsilon _{1}=2\%\) (Fig. 7b). Beyond the internal shear zone the material was practically rigid. Based on the displacement vectors, one can recognize some righthanded vortex structures whose size is limited to the width of the shear zone.
The evolution of the vector field curl \({\vec {\nabla }} \times \vec {v}\) (divergencefree component related to vorticity) during deformation \(\varepsilon _{1}\) is presented in Fig. 8. The scale denotes the component of the vector potential \(\vec {v}\) perpendicular to the specimen in [\(\hbox {mm}^{2}\)/iteration]), changing from \(\,0.50 \hbox { mm}^{2}\)/iter up to \(0.05~\hbox {mm}^{2}\)/iter. The green circles describe the local minima of the scalar field \(\Vert v\Vert \) (equivalent to centres of righthanded vortices) and red circles the local maxima of the scalar field \(\Vert v\Vert \) (equivalent to centres of lefthanded vortices). These local extrema of the scalar field \(\Vert v\Vert \) were defined in such a way that the values of \(\Vert v\Vert \) in all neighbouring nodes in the mesh created by the Delaunay triangulation were smaller/larger than \(\Vert v\Vert \) at the node in question. The diameter of each circle indicates the relative magnitude of extremum points.
The vortex structures emerged from the beginning of the specimen deformation. Initially their centres were seemingly randomly located. However, by \(\varepsilon _{1}\ge 1\%\), they were quickly aligned along the line of the ultimate location of the spontaneous shear zone (Fig. 8a–c). In the range of \(\varepsilon _{1}=\) 3–4% they were more concentrated around the shear zone midpoint (Fig. 8d, e) before a final shear direction was chosen. Thus the final shear zone turned out to be encoded in the grain kinematics far before the stress peak (\(\varepsilon _{1}=5\%\)). This outcome is in accordance with our earlier calculation results for plane strain compression based on displacement fluctuations [7] and calculation results based on bottlenecks in force transmission through the contact network [49]. The righthanded vortices (green circles) dominated during progressive shear deformation. The lefthanded vortices (red circles) were evidently in minority. The distance of vortexcentres and their intensity along the shear zone was varying in time. Some single vortices also occurred (very intermittently) beyond the shear zone. Note that the size of vortex structures could not be directly deduced from HHD since the vortex structures corresponded to points only that were associated with their rotational centres. The size of vortices may be detected with the aid of other methods, based on the vector displacement field of single particles [6, 50, 51, 52]].
The evolution of the scalar field gradient \({\vec {\nabla }}u\) (curlfree component related to compressibility) during deformation \(\varepsilon _{1}\) is described in Fig. 9. The green circles describe the sources (local minima of the scalar potential u  centres of local dilatancy regions) and the red circles denote the sinks (local maxima of the scalar potential u—centres of local contractancy regions). The magnitude of sources and sinks was again expressed by the different diameter of green and red circles. The scale attached denotes the scalar potential u in [\(\hbox {mm}^{2}\)/iteration] (sign ()  sources in the scalar potential field u, sign (+) – sinks in the scalar potential field u), varying between \(0.1\hbox { mm}^{2}/\hbox {iter}\) and \(0.25\hbox { mm}^{2}/\hbox {iter}\).
The local dilatancy (sources) and local contractancy extremum points (sinks) started to develop in the shear zone region for \(\varepsilon _{1}>1\%\) (thus clearly before the stress peak \(\varepsilon _{1}=1\%\), Fig. 3b). During progressive shear deformation, the sources and sinks alternately happened inside of the shear zone area. The magnitude of sources and sinks was the smallest in the residual state (Fig. 9i). This outcome is in accordance with the alternating distribution of local void ratio in shear zones during DEM calculations [9]. Note that local dilatant regions are usually connected to the collapse of main force chains and creation of vortices, and local contractant regions are linked with the buildup of main force chains and disappearance of vortices [9].
Finally Fig. 10 shows the evolution of the harmonic increment vector field \(\vec {h}\) during deformation \(\varepsilon _{1}\). The scale denotes the vector length in [mm/iteration] changing from 0 mm up to 0.06 mm. The evolution \(\vec {h}\) was practically the same independently of \(\varepsilon _{1}\) since the top boundary continuously moved at the same displacement increment. Some irregularities in the harmonic field appeared due to large vectors (see Fig. 6) since the vector field of sphere displacements was not smoothed. The irregularities were the most pronounced in the shear zone (Fig. 10f–i). The 2D results depicted in Figs. 7, 8, 9 and 10 were very similar as those during passive wall translation [8].
6 Numerical 3D results using combined HHD/DEM approach
Figures 12, 13, 14 and 15 present the centres of vortex structures at 4 different specimen vertical crosssections (middepth, front side, rear side and 2/3 depth of the granular specimen) with increasing vertical normal strain \(\varepsilon _{1}\). The circles with the different diameters in Figs. 12, 13, 14 and 15 denote the spots where the vertical crosssectional slices intersected the 3D tubes of Fig. 11. The circle diameters do not again represent the spatial size of vortices (that are not defined in the present paper) but the magnitude of the vector \({\vert }\vec {v}{\vert }\), related to the rotational velocity around the vortex axis. The threshold value of \(\delta =\) 2–5% was tested. The larger values of \(\delta \) resulted in too many tubes (contributing to an obscure image).
The 3D results are certainly more exact that the 2D results, since a significantly larger number of spheres was captured in 3D computations (56,000 spheres against 8000 spheres in the 2D slice), that caused a smoother vector field and smaller calculation errors. Slightly less vortex structures occurred in the shear zone and beyond the shear zone in 3D conditions than in 2D ones (compare Fig. 8 with 12, 13, 14 and 15). However, the 2D vortex calculations were performed for the slice of the thickness of \(5\times d_{\mathrm{50}}\) by ignoring the zcoordinate of each sphere. Since all sphere coordinates were projected onto the plane, it might happen therefore that two spheres, which were at the distance of \(5 \times d_{\mathrm{50}}\) on the opposite sides of the slice, could lie very next to each other. Since their real 3D distance was large, they might move in significantly different planeprojected directions. When two nodes, close to each other in the 2D \(\xi \)vector field, moved in significantly different directions, a disturbance in the vector field occurred that contributed to the larger number of vortices.
The maximum number of righthanded vortices was 2–15 before the peak stress (\(\varepsilon _{1}\le 5\%\)) and about 40–50 after the peak stress (\(\varepsilon _{1}\ge 5\%\)) (Fig. 16a). Their number decreased from the beginning of deformation up to the stress peak (\(\varepsilon _{1}=5\%\)), increased up to \(\varepsilon _{1}=10\%\) and then remained the same up to \(\varepsilon _{1}=30\%\). The maximum number of episodic lefthanded vortices was about 20 (Fig. 16b).
The calculation results of Sects. 5 and 6 clearly indicate that the vortexmotion kinematics should be explicitly considered in granular materials within continuum mechanics modelling, since it is a preferable motion type during granular flow (granular material tends to organize its flow into a collection of vortices, similarly to fluids).
7 Comparison with experiments
In plane strain compression laboratory tests on a sand specimen by Abedi et al. [11] and Chupin et al [30] (\(b=4\) cm, \(h=14\) cm, \(l=8\) cm and \(d_{\mathrm{50}}=0.84\) mm), the digital image correlation (DIC) technique was used to detect vortex structures and volumetric strain nonuniformity in the spontaneous inclined shear zone in the residual state. DIC is a noncontact experimental technique to measure surface displacements on a deforming solid [54]. In order to determine the vortices in the form of a rotational motion of several grains around its central point, the displacement fluctuation vectors of small clusters of grains were calculated from the mean displacement vector field. A systematic vortex formation and vortex disappearance was observed in the inclined shear zone after the stressstrain peak that propagated from the left side up to the right side. The vortices were shortlived. The maximum 5 righthanded vortices and lefthanded vortices temporarily occurred in the shear zone. They had a periodic character. In addition, a pattern of nearly periodic spatial variation of local volume changes (including dilatant and contractant regions) along the length of the shear zone was also observed in the critical state. Our calculations showed that vortex structures and local volume changes were observed to occur in the shear zone throughout the entire deformation regime (note that [11] and [30] did not report experimental observations before the peak). The computed number of vortex structures was also significantly higher than this in experiments. This difference between the calculation outcomes and experimental results is due to: a) a different method used to detect vortices, b) a smaller mean grain diameter \(d_{\mathrm{50}}\) in the experiment \(d_{\mathrm{50}}=0.84\) mm (in DEM: \(d_{\mathrm{50}}=2.5\) mm) and c) a lack of the real sand grain shape in DEM. Our numerical detection method (based on single grain displacements) is an objective method that detects the centres of all vortex structures independently of their size. The experimental detection’s method was however based on displacement fluctuations of small clusters of grains (not single grains) and was strongly limited by the accuracy of DIC. Moreover DIC is sensitive to the assumed subset size and image length resolution [54].
8 Conclusions

The occurrence of vortex structures was closely related to shear localization. The vortices proved to be an early precursor of shear localization since their centres concentrated in the region where the shear zone ultimately formed. They developed throughout deformation. They mainly emerged in the main inclined internal shear zone and were strongly nonuniform in a spatial arrangement. Vortices rotating in a clockwise direction mainly occurred due to the shearing direction. Lefthanded vortices were rarely observed. Thus, the vortex structures allowed identification of shear localization earlier than, for example, based on single grain rotations or an increase of void ratio.

The number of 3D vortices spatially and temporarily changed along the specimen depth.

The centres of local regions of dilatancy and contractancy alternately happened in the shear zone with a dominance of local dilatant regions.

An early prediction possibility of shear localization through the formation of vortex structures creates an interesting perspective for a detection of impending failure in granular bodies within continuum mechanics (inherently connected with shear localization).
Notes
Acknowledgements
The authors would like to acknowledge the support by the Grant 2011/03/B/ST8/05865 “Experimental and theoretical investigations of microstructural phenomena inside of shear localization in granular materials” financed by the Polish National Science Centre (NCN).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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