Investigations of quasistatic vortex structures in 2D sand specimen under passive earth pressure conditions based on DEM and Helmholtz–Hodge vector field decomposition
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Abstract
The paper presents some twodimensional simulation results of granular vortexstructures in cohesionless initially dense sand during a quasistatic passive wall translation. 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. In order to detect vortexstructures, the Helmholtz–Hodge decomposition of a vector field from DEM calculations was used. This approach enabled us to distinguish both incompressibility and vorticity in the granular displacement field. In addition the predominant periods of vortices during horizontal wall movement were determined. The vortices were strongly connected to shear localization. They localized in locations where shear zones ultimately developed. In addition, the vortexstructures were calculated during plane strain compression.
Keywords
Earth pressure Granular material Discrete element method Vortexstructures Helmholtz–Hodge decomposition1 Introduction
Granular vortexstructures (swirling motion of several grains around its central point) were frequently observed in experiments on granular materials [1, 2, 3, 4] and in calculations using the discrete element method (DEM) [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The experiments were performed during Couette shearing [1], plane strain compression [2] and simple shear [3, 4]. The vortices were observed to form in association with the onset of peak stress. They appeared only occasionally, quickly dissipating [1, 2, 3, 4]. The granular vortexstructures were also obtained with the aid of the finite element method [17]. They became apparent in experiments [1, 2, 3, 4] and calculations (e.g. [5, 13, 14, 15, 16]) when the motion associated with uniform (affine) strain was subtracted from the actual granular deformation. They are reminiscent of turbulence in fluid dynamics [5], however the amount of the grain rotation is several ranges of magnitude smaller (\({\sim }0.01^{\circ }\)–\(0.1^{\circ }\)) than the fluid vortex rotation and granular flow is too slow to induce inertial forces characteristic for turbulences in fluid. According to Peters and Walizer [13] vortices represent an independent flow field following its own governing equations and satisfying its own (null) boundary conditions. Tordesillas et al. [11] showed that two classes of vortices emerging: primary ones concentrated in the shear zone and secondary ones forming next to the zone boundaries. A dominant mechanism responsible for the vortex formation was the breakage of force chains [11, 16]. The collapse of main force chains lead to a formation of larger voids and their buildup to a formation of smaller voids [11, 16]. Vortex dynamics were consistent with stickslip dynamics [16]. The vortices have been mainly observed in shear zones [11, 16, 18] which are the fundamental phenomenon in granular bodies. A precise mechanism behind the vortex evolution with its connection to shear localization in granular bodies still remains elusive. In numerical calculations, shear localization is usually identified in granular bodies by grain rotations (in DEM) or micropolar rotations or by an increase of void ratio (in FEM) [19].
The objective of this paper is to report the results of comprehensive 2D studies by DEM on vortexstructures in sand behind a rigid wall during its quasistatic passive translation by using the Helmholtz–Hodge decomposition (HHD) of a vector field [20, 21]. Attention was paid to the relationship between vortexstructures and shear localization with respect to the location and formation moment. The analyses were carried out with spheres with contact moments [16] to approximately capture the irregular grain shape. 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. A threedimensional discrete model YADE developed at University of Grenoble by F.V. Donze and his coworkers was applied [22, 23, 24, 25, 26, 27]. The discrete calculations were solely carried out with initially dense sand.
In our previous paper we calculated 2D vortex and antivortex structures during a quasistatic plane strain compression test solved by the DEM [28]. Twodimensional vortex and antivortex structures were determined by a method based on orientation angles of displacement fluctuation vectors of neighbouring single spheres [28]. The proposed method used for the detection of 2D granular vortex/antivortexstructures was very effective. The method detected all vortex and antivortexstructures regardless of the displacement vector length. The 2D vortex and antivortex structures solely appeared in the main inclined shear zone. Their occurrence did not depend on the specimen depth. The antivortices turned out be the best precursor of the location of shear zones since they appeared from the beginning of the deformation process, i.e. significantly earlier than e.g. based on the average cumulative grain rotation. Other results showed that the increase of the grainnonregularity decreased the predominant period of lefthanded vortices and antivortices by 25–40%. This method had however 3 disadvantages: (1) it operated on displacement fluctuation vectors (not directly on displacement increment vectors), (2) it was solely designed for 2D granular flow and (3) it depended on several parameters such as: region size of the average background translation, averaging region size of local displacement increments, searching circle radius and grid move distance [18]. The method used in this paper has the following advantages over the previous one: (1) it operates directly on displacement increment vectors, (2) it is designed for identifying vortices in both two and threedimensional kinematic fields, (3) it does not need any additional parameters for calculations and (4) it can be used for DEM and FEM calculations. The method does not determine the size of vortexstructures. There exist also the methods which take the vortex size into account [29]. However this size depends on the displacement vector lengths, displacement vector rotations, ratio between the height and width of vectors assumed for describing the vortex shape and number of grains comprising the vortex. In the paper [16] in order to mathematically describe 2D vortices, the displacement fluctuation vector of each grain in the neighbourhood of each central grain was decomposed into 2 vectors: the normal and tangential to the movement direction of each central grain. Only the tangential displacement fluctuations were assumed to be responsible for a vortex, i.e. if the neighbouring grains had solely the tangential displacement fluctuation component, the central grain was assumed to be located in a vortex midpoint. In order to avoid the prescribing pure shearing along a shear zone as a vortex, a certain limitation was imposed, i.e. the condition that the sum of the tangential displacement fluctuations had to be at least twice as big as the sum of the normal displacement fluctuations (to be classified as a part of the vortex). It was assumed in Peters and Walizer [13] that a 2D vortex was connected with the rigid body rotation (spin) in the displacement fluctuation field. During 2D calculations described in Tordesillas et al. [11], vortex cores were identified by partitioning a vector displacement field into triangles, each analyzed with respect to a directionspanning property (the vectors at the triangle vertices were located within a certain range of angles, i.e. each vector pointed to a unique direction range). A vortex was then identified for each vortex core with the vortex boundary set by the last member vector rotating in the vortex direction. The full extent of each vortex was established by a systematic check of the direction of neighbouring vectors (in the second ring, third ring, etc.) at increasing radial distance from the vortex core until a vector was found that rotated in an opposite direction to the corresponding vortex.
2 Threedimensional DEM model
3 DEM results of passive earth pressure model tests
The DEM calculation results were described in detail in Nitka et al. [16]. The simulations were performed for a 2D sand body of \(l_{w}=0.40\,\hbox { m}\) length and \(h_{w}=0.20\,\hbox {m}\) height in order to compare with experiments with Karlsruhe sand \((d_{50}=0.5\,\hbox { mm})\) [37, 38]. The vertical retaining wall and the bottom of the granular specimen were assumed to be stiff and very rough, i.e. there was no relative displacement along vertical and bottom surface [19]. Since the experiments were idealized as a 2D boundary value problem and the effect of the specimen depth in the out of plane direction turned out to be almost negligible during direct shearing in DEM calculations [15] in order to significantly accelerate simulations, the computations were performed with the specimen depth equal to the grain size (i.e. one layer of spheres was simulated along the depth only).
The spheres with \({d_{50}=1.0~\hbox {mm}}\), characterized by a linear grain size distribution, were assumed (grain size range 0.5–1.5 mm, 62,600 spheres). Along the height of 200 mm, about 200 spheres were used. The initial void ratio of sand, obtained by generating random spheres above a box and then allowing them to fall down by gravity, was \(e_{o}=0.62\). The loading speed was slow enough to ensure that the tests were conducted under quasistatic conditions. The calculated mean inertial number (which quantifies the significance of dynamic effects) in the midlength of the curved shear zone for the maximum horizontal earth pressure force was in the analyses \(I = \frac{{\dot{\gamma } {d_{50}}}}{{\sqrt{\frac{P}{{\rho }}} }} = ~5\cdot {10^{ 4}}\, (\dot{\gamma }=0.75 1/\hbox {s}\)—the shear rate, \(P=5~ \hbox {kPa}\)—the mean pressure and \(\rho =2.55 \hbox { g/cm}^{3})\). The inertial number obviously changed along the specimen height and in time during the entire granular flow. The value of \(I\le 10^{3}\) corresponds usually to a quasistatic regime [39].
Figure 2A shows the evolution of the resultant normalized horizontal earth pressure force (earth pressure coefficient) \(K_{p}=2E_{h} /\left( \gamma {h}_{w}^{2}d_{50}\right) \) versus the normalized horizontal wall displacement \(u_{h}/h_{w} (h_{w}=0.2~\hbox {m}, E_{h}\)—the horizontal force acting on the wall, \(\gamma =16.75~\hbox { kN/m}^{2}\)—the initial unit weight) for \(d_{50}=1~\hbox { mm}\) from DEM simulations. The normalized horizontal earth pressure force evolved typically for initially dense granulates in biaxial compression, triaxial compression and direct shearing. The specimen exhibited the initial strain hardening up to the peak \(\left( u_{h}/h_{w}=0.038\right) \), followed by some softening before the common asymptote was reached (critical state). The horizontal force fluctuated after the peak that was attributed to the buildup and collapse of force chains—the main carrier of stresses transferred within the granular assembly [16]. The earth pressure coefficient was \(K_{p}^{{ max}}=30\) for \(d_{50}=1~\hbox { mm}\). It can be thus anticipated that for \(d_{50}=0.5~\hbox { mm}\) (real sand), \(K_{p}^{{ max}}\) should be about 25–27. The value of \(K_{p}^{{ max}}=30\) for \(d_{50}=1~\hbox { mm}\) was slightly smaller than \(K_{p}^{{ max}}=31\) obtained by FEM \(\left( d_{50}=0.5~\hbox {mm}\right) \) [38] and was closer to the engineering earth pressure coefficients [40].
The distribution of single sphere rotations \(\omega \) during wall translation is presented in Fig. 2B which are usually the best indicator for shear localization in DEM (red denotes the sphere rotation \(\omega >+30^{\circ }\) and blue \(\omega <30^{\circ }\), dark grey is related to the sphere rotation in the range \(5^{\circ }\le \omega \le 30^{\circ }\), light grey in the range \(30^{\circ }\le \omega \le 5^{\circ }\), medium grey in the range \(5^{\circ }\le \omega \le 5^{\circ }\), positive sign means clockwise rotation). Such a colour convention made shear zones clearly observable (only particles within shear zones significantly rotated). There existed a clear grain separation between a clockwise (red) and an anticlockwise (blue) rotation—the majority of ‘red grains’ was located within the dominant curve shear zone while the majority of ‘blue grains’ was placed within the straight radial shear zone (however there existed also a small amount of blue grains within the ‘red shear zone’ and vice versa). Based on grain rotations, a curved shear zone started to develop along the specimen bottom for the normalized wall translation of \(u_{h}/h_{w}=0.02\) (Fig. 2Bb). It was fully developed for \(u_{h}/h_{w}=0.06\). Its thickness was \(t_{s}=20~\hbox { mm } (20 \times d_{50})\). The radial shear zone started later to form for \(u_{h}/h_{w}=0.04\) (Fig. 2Bd) and for \(u_{h}/h_{w}=0.06\) connected the curved shear zone. Its thickness was \(t_{s}=10~\hbox { mm } (10\times d_{50})\). There was a satisfactory agreement between DEM simulation results and real experimental outcomes [41, 42] and FE results [19, 38].
4 Helmholtz–Hodge decomposition
A variational calculus approach was used [44] which allowed for finding the vector fields \(\vec {\nabla }u\) and \(\vec {\nabla } \times \vec {\nu }\) by examining the difference between the unknown vector field and provided field \(\vec {\xi }\) (see Eqs. 11, 12). By requesting that this difference is minimum (the minimum was found by requesting that the derivatives of the functionals were equal to zero, Eqs. 13, 14), the vector fields \(\vec {\nabla }u\) and \(\vec {\nabla }\times \vec {\nu }\) were explicitly determined. The explicit calculation for \(\vec {\nabla }u\) was given in Eqs. 15 and 16 and for \(\vec {\nabla }\times \vec {\nu }\) in Eq. 17.
The HHD on irregular grids has been already used e.g. in graphics [43]. In order to create a grid, the centre of each sphere was a node in a Delaunay triangulation [45] and the ith node had the coordinate \({\vec {r}}_{i}\). Then the discrete piecewiseconstant vector field \(\vec {\xi }\left( {\vec {r}_i }\right) =\mathop {\sum } \nolimits _k \psi _k ( \vec {r}) \vec {\xi }_{k}\) was created by assigning the constant vector value \(\vec {{\xi }_k}\) to each kth tetrahedron \((\psi _{k}\) is the piecewiseconstant basis function equal to 1 inside the kth tetrahedron and 0 otherwise). This value was calculated as the average of sphere displacement increments \(\vec {d_n}\) which constituted each tetrahedron \(\vec {{\xi }}_k =1/4\mathop {\sum }\nolimits _{n=1}^{n=4} \vec {d_n }\) in the 3D case or each triangle \(\vec {\xi }_k=1/3\mathop {\sum } \nolimits _{n=1}^{n=3} \vec {d_n }\) in the 2D case. Since u and \(\vec {\nu }\) are the piecewise linear functions described using a piecewiselinear basis shape function \(\phi _i ( {\vec {r}})\), their derivatives \(\nabla \) will be piecewiseconstant, hence the solution for the piecewiseconstant \(\vec {\xi }(\vec {r})\) discrete vector field is exact [43].

\({\varGamma }\) 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’ \(u( {\vec {r}})= \mathop {\sum }\nolimits _{i} \phi _{i} (\vec {r}) u_{i}\),

\(\vec {\nu }\) is the discrete vector field at the node ‘i’ \(\vec {\nu }(\vec {r})=\mathop {\sum }\nolimits _{i} \phi _{i}(\vec {r})\vec {\nu }_{i}\),

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

\(\vec {r}\) is the spatial coordinate in \(\varGamma \) using the Cartesian coordinate system \(\vec {r}=(x,y,z)\).
\(\left {T_k } \right \)—the tetrahedron volume (triangle area),
\(\vec {(\nabla }{\phi }_i)_k \)—the vector orthogonal to the tetrahedron face f (triangle edge f for 2D cases) opposite to the ith node in the kth tetrahedron (triangle), pointing towards the ith node with the magnitude of \(\frac{{ area}\left( f \right) }{3\left {T_k } \right }\left( {\hbox {or }\frac{\hbox {length}\left( f \right) }{2\left {T_k } \right }\,\hbox {for 2D cases}} \right) \),
N(i)—the set of all tetrahedrons (triangles) containing the ith node.
Equations 18 and 19 describe the ith row of 2 sparse matrices and were numerically solved for the unknowns \(u_{i }\) and \(\vec {\nu }_{i}\) using the Eigen library with a biconjugate gradient stabilized solver [45]. The third component of Eq. 10 the harmonic vector field (which contains a nonintegrable field component) is determined as \(\vec {h}=\vec {\xi }\vec {\nabla }u \vec {\nabla } \times \vec {\nu }\). The sphere displacement increments were calculated during 10,000 iterations \(\left( \Delta u_{h}/h_{w}=0.002\right) \) [16].
5 Numerical results
5.1 Passive wall translation
The calculation results are described in Figs. 3, 4, 5 and 6. Figure 3 shows the evolution of the vector displacement increment field \(\vec {\xi }\) during the normalized horizontal wall translation \(u_{h}/h_{w}\) (sphere displacement increment directions are marked by white arrows). The scale attached denotes the sphere displacement increment vector length during 10,000 iterations in [mm/iteration] which changes between 0 and 1 mm. Based on the displacement increment vector length and vector direction changes, a curved shear zone between the wall bottom and free upper boundary already started in the first calculation step. Later it moved to the right to reach its ultimate position due to wall friction along the bottom that was close to the maximum resultant normalized horizontal earth pressure force of Fig. 2A \((u_{h}/h_{h}=0.04)\). Behind the curved shear zone the material was totally rigid. A radial shear zone was approximately created for \(u_{h}/h_{w}=0.065\).
The evolution of the scalar field gradient \(\vec {\nabla }u\) (curlfree component related to compressibility) during normalized wall translation \(u_{h}/h_{w}\) is described in Fig. 4. The scale attached denotes the scalar potential u in \([\hbox {mm}^{2}/\hbox {iteration}] \) (sign (−)—dilatancy, sign (\(+\))—contractancy), changing between \(4\) and \(2~\hbox { mm}^{2}/\hbox {iter}\). The green circles describe the sources (local minima of the scalar potential u  local dilatancy minima) and the red circles denote the sinks (local maxima of the scalar potential u  local contractancy maxima). The local extrema of the scalar field u were defined by requesting that the values of u in all neighbouring nodes in the mesh created by the Delaunay triangulation were smaller/larger than \(u_{i}\) at the node in question.
Initially global contractancy and later global dilatancy occurred in the granular specimen. The global dilatancy was the largest after the stress peak (Fig. 4d) and later diminished. The local contractancy maxima and local dilatancy minima started to develop in two main shear zones before the stress peak (Fig. 4c). In the residual state \((u_{h}/h_{w}\ge 0.075\), Fig. 2A), local regions of dilatancy and contractancy alternately happened along both shear zones (with the prevalence of dilatancy). This outcome is in accordance with the distribution of local void ratio in a shear zones in calculations by DEM [16]. Note that local dilatant regions may be connected to the collapse of main force chains and creation of vortices and local contractant regions may be connected to the buildup of main force chains and disappearance of vortices [16].
Figure 5 presents the evolution of the vector field curl \(\vec {\nabla } \times \vec {\nu }\) (divergencefree component related to vorticity) during normalized wall translation \(u_{h}/h_{w}\). The scale denotes the component of the vector potential \(\vec {\nu }\) perpendicular to the specimen in \([\hbox {mm}^{2}/\hbox {iteration}])\), changing from \(8 \hbox { mm}^{2}/\hbox {iter}\) up to \(8~\hbox {mm}^{2}/\hbox {iter}\). The green circles describe the local minima of the scalar field \(\Vert v\Vert \) (righthanded vortices) and red circles the local maxima of the scalar field \(\Vert v\Vert \) (left handed vortices). The local extrema of the scalar field \(\Vert v \Vert \) were interpreted as vortices and were calculated in the same way as the local extrema of the scalar field u. The vortexstructures appeared from the wall translation beginning. They were immediately concentrated in regions of the shear zones’ occurrence. Thus the ultimate shear zone pattern turned out to be encoded in the grain kinematics from the deformation beginning. This outcome is in accordance with our earlier calculation results for plane strain compression based on displacement fluctuations [18] using the method described in Gould et al. [28] and calculation results based on bottlenecks in force transmission through the contact network [50]. Righthanded and lefthanded vortices alternately occurred. The righthanded vortices dominated in the curved shear zone and lefthanded vortices dominated in the radial shear zone. Their distance along the shear zones was different.
Finally Fig. 6 shows the evolution of the harmonic increment vector field \(\vec {h}\) during normalized wall translation \( u_{h}/h_{w}\). The scale denotes the increment vector length during 10’000 iterations in [mm/iteration] changing from 0 mm up to 0.8 mm. The evolution \(\vec {h}\) was practically the same independently of \(u_{h}/h_{w}\) since the wall continuously moved at the same displacement increment.
5.2 Plane strain compression
Our numerical outcomes with respect to vortexstructures were next checked during quasistatic plane strain compression. The details of DEM 3D calculations are given in Kozicki and Tejchman [18]. The granular specimen used in DEM had the same size as in the experiments by Vardoulakis [51], namely: the width \(b=4~\hbox {cm}\), height \(h=14~\hbox {cm}\) and depth \(l=8 \hbox { cm}\) (outofplane direction) (Fig. 12a). The linear grain distribution curve was assumed; the grain diameter range was between 1.25 and 3.75 mm with \(d_{50}=2.5~\hbox {mm}\). About 56,000 spheres were used with the same material constants. The initial void ratio was \(e_{o}=0.53\). The flexible vertical walls were assumed to model the membrane surrounding the specimen in experiments (Fig. 12a). Both the front and rear specimen sides \(4\hbox { cm}^{2} \times 14 \hbox { cm}^{2}\) were blocked in a perpendicular direction to the specimen to enforce plane strain conditions. The bottom surface \(4~\hbox {cm}^{2} \times 8~\hbox {cm}^{2}\) was fixed in a vertical direction and the top surface \(4~\hbox {cm}^{2} \times 8~\hbox {cm}^{2}\) was subjected to a constant vertical displacement \(u_{\nu }\). Along the top, bottom and membrane granular surfaces, the interparticle friction angle was \(\mu =0\). During the loading process, the constant confining pressure of \(\sigma _{c}=200 \hbox { kPa}\) was applied through the flexible membrane. The evolution of the mobilized internal friction angle (calculated with principal stresses from the Mohr’s equation) versus the vertical normal strain and displacement vectors of spheres in the specimen are shown in Fig. 12b, c. Similarly as in real experiments [51], the initially dense specimen showed an asymptotic behaviour; it exhibited initially small elasticity, hardening (connected first to contractancy and then dilatancy), reached a peak strength at about of \(\varepsilon _{1}=5\%\), gradually softened and dilated reaching a residual state at the large vertical strain of 25–30% (Fig. 12b). During deformation a distinct internal inclined shear zone occurred inside the sand specimen which was marked by shear strain, larger grain rotation and volume increase. The thickness of the inclined interior shear zone \(t_{s}\) was on average in the residual state for \(\varepsilon _{1}=30\%\) about \(t_{s}=25~\hbox { mm}~(10\times d_{50})\) based on strain deformation in the specimen. The calculated shear zone inclination to the bottom was \(60^{\circ }\) at \(\varepsilon _{1}=10\%\) and \(67^{\circ }\) at \(\varepsilon _{1}=30\%\). Based on the cumulative grain rotation, the internal inclined shear zone might be noticed for \(\varepsilon _{1}\approx 3\%\) [18]. Due to a rather small number of spheres along the specimen width (about 16), the effect of boundary conditions during calculations of vortexstructures was weakened by introducing virtual particles outside boundaries [47] (Fig. 12c). Artificial nodes were added in the Delaunay’s triangular mesh at the distance of up to 50 mm around the specimen (with the grid internode distance of 1 mm) (Fig. 12c). The vector \(\vec {\xi }\) in these artificial nodes was calculated using the Gaussian averaging for true specimen nodes with the averaging radius of \(80~\hbox {mm } (2\times b)\). Some wild vectors in Fig. 12c appeared if single grains suddenly undergo large displacements.
Figure 13 presents the evolution of the vector field curl \(\vec {\nabla } \times \vec {\nu }\) (divergencefree component related to vorticity) during the normalized top displacement \(u_{\nu }/h\) in the vertical midsection slice with the area of \(4\hbox { cm}^{2} \times 14 \hbox { cm}^{2}\) and thickness of \(5\times d_{50} (1.25~\hbox { cm})\) (Fig. 12a) cut out from the granular specimen [18]. The vortexstructures appeared from the deformation beginning. They were immediately concentrated in the region of the mean shear zone occurrence. Righthanded vortices were dominant in the shear zone.
In order to fully validate the proposed method for detecting vortexstructures, several various boundary value problems with shear localization will be investigated as. e.g. during simple and direct shearing and wall shearing [38]. The grain properties, shear rates and grain size distribution ranges will be changed. Moreover, 3D vortexstructures will be calculated during plane strain compression [18] using HHD. The effect of the different definition of the rolling velocity [33] will be also investigated. In parallel, similar calculations will be carried out within micropolar hypoplasticity [38].
6 Conclusions

The HHD allowed for separating a vector field into the sum of three uniquely defined components: curl free, divergence free and harmonic. It proved to be an objective, universal and effective technique for identifying all vortexstructures during granular flow which was directly based on single grain displacement increments (but not on displacement fluctuations). The method did not use any additional nonobjective parameters. A large number of spheres was however required to avoid the effect of boundary conditions assumed. The size of vortexstructures could not be deduced since they corresponded to points only which were associated with the centre of shear zones. The method may be used for the detection of 3D vortices.

A strong connection between the location of vortexstructures and progressive shear localization was found out. The vortexstructures were the precursor of shear localization since they clearly concentrated in the area where a curved and radial shear zone ultimately later formed. Thus the ultimate shear zone pattern was detected in early loading stages. The vortexstructures allowed to identify shear localization significantly earlier than e.g. based on single grain rotations which were always a reliable indicator of shear localization. They developed from the deformation process beginning. They solely emerged in main shear zones. They had a tendency to move along shear zones. Their number varied and 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 curved shear zone, the predominant period of righthanded vortices was 4% of u / h during the entire wall movement. In the radial shear zone, the predominant period of lefthanded vortices was also 4% of u / h.

In the residual state, local regions of dilatacy and contractancy alternately happened along globally dilatant shear zones with a dominance of local dilatancy.

An early prediction possibility of shear localization through vortexstructures may open new perspectives for a detection of impending failure in granular bodies (inherently connected with shear localization) within continuum mechanics.
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.
Compliance with ethical standards
Conflict of interest
We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
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