ORIGINAL PAPER

Mixtures of hard and soft grains: micromechanical behavior at large strains Guilhem Mollon1 Received: 22 February 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract In this paper, several simulations involving the compaction and the shearing of mixtures of hard and soft grains are performed in 2D plane-strain conditions. The multibody meshfree numerical tool developed for this purpose is first presented, and the focus is then put on the influence of the proportions of rigid and deformable grains in the mixture on the mechanical response at large strains. Dedicated postprocessing techniques reveal a wide range of behaviors, both in terms of macroscopic response and in micromechanical phenomena. Broadly speaking, the strength and the dilatancy of the mixture decrease when the proportion of soft grains is increased. There are, however, interesting exceptions to this trend at very high and very low contents of soft grains, which are analyzed in dedicated sections. This preliminary work paves the way to more comprehensive studies of this class of materials, which is still hardly understood but presents some potential in a wide range of applications. Keywords Granular materials · Soft and hard mixtures · Deformable grains · Meshfree methods · Discrete Element Modeling · Large strain behavior

1 Introduction Granular materials are extensively studied because they are ubiquitous in a large number of scientific and engineering fields such as geomechanics, geophysics, materials science, tribology, etc. Most of studies, both numerical and experimental, focus on granular materials composed of rigid grains since they are the most common in the concerned traditional industries, such as civil engineering. Typical numerical frameworks for such materials are continuous modeling (mostly based on Finite Element Modeling, FEM) for practical analysis of large systems and discrete modelling (mostly based on Discrete Element Modeling, DEM [1]) for a more detailed analysis of small systems. In a number of technical fields, however, the assumption of rigid grains does not hold any more, either because the material composing the grains is very soft (e.g. biological fluids [2], food industry [3], elasElectronic supplementary material The online version of this article (https://doi.org/10.1007/s10035-018-0812-3) contains supplementary material, which is available to authorized users.

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Guilhem Mollon [email protected] LaMCoS, INSA-Lyon, CNRS UMR5259, Université de Lyon, 69621 Villeurbanne, France

tomeric powders [4], etc.), because the applied load is very high (e.g. powder lubrication in tribology [5,6], ductile powder compaction [7,8], etc.), or because of thermal loading [9]. In such cases, practitioners are facing a general lack of knowledge in the scientific community, because the behavior of granular materials with soft grains has not been much investigated in the literature. This is especially true in terms of numerical simulation, because of the complexity that lies in the modeling of a large number of highly-deformable and interacting solids. For this reason, soft-grain systems are now gaining the attention of the community of granular science [10]. The more specific case of mixtures between rigid and soft grains is pivotal in a number of industrial fields such as soil reinforcement [11], forming processes, waste recycling [12], granular pastes, etc. In this paper, we first present a numerical tool which makes it possible to deal in a single simulation with rigid and highly-compliant grains. At the moment, this code is limited to plane-strain kinematics. We then employ this framework to run a set of simulations of shearing of mixtures composed of rigid and soft grains, with varying proportions of each kind. The bulk response of the mixture and the associated micromechanical behaviors are successively analyzed, and the last sections are dedicated to some interesting phenomena that occur at very low and very high content of soft grains.

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Fig. 1 Typical soft and rigid bodies in MELODY

Results show that a wide variety of behaviors can appear in this particular framework, and that the addition of soft grains in a granular medium may allow some control of the mixture mechanical response.

2 Numerical framework Dealing with deformable grains within a granular sample requires a continuum-based approach within the domain of each grain, and a contact-based discrete approach at the sample scale. Several attempts have been made to use FEM programs to represent the deformations of each grains [13– 16], but these approaches are somewhat limited by the fact that common FEM codes are not optimized to deal with a large number of bodies. Numerical ingredients such as optimized proximity and contact detection, which are central in DEM codes, are usually lacking in the FEM framework. One may also cite less common approaches such as the Non-Smooth Contact Dynamics (NSCD, [17]) or the Particle Finite Element Method (PFEM, [18,19]), since they are theoretically able to deal with both rigid and soft grains. A recent study proposed to use Material Point Method (MPM) to simulate the inner deformability of grains, with promising results [20]. Other attempts were made where the deformable grains were represented by shell elements (especially in the field of computational hemodynamics, [21–25]), but they remain limited to the case where the grains are similar to hollow spheres (biological fluids, etc.). Most of existing methods are susceptible to exhibit convergence problems in case of severely-distorted grains, because most of them rely on a mesh in some way. In two recent papers [26,27], a novel approach was proposed in order to retain the capabilities of both FEM and DEM in order to simulate samples of mixed rigid and highlycompliant grains. This framework is based on a multibody

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meshfree technique developed on purpose, which is currently limited to plane strains kinematics. Each deformable grain (Fig. 1) is discretized by a number of field nodes, which carry the degrees of freedom in displacement (just like in 2D FEM). The continuous displacement field is interpolated between these nodes using meshfree shape functions (Moving Least Squares), which are differentiable everywhere and typically encompass 10–25 nodes [28,29]. A weak formulation is then used to reformulate the continuous problem in a set of ordinary differential equations (ODE), including geometric non-linearities and inertial terms. Among the field nodes, those which are located on the border of the grain (“contact nodes”) are used for contact detection. A two-pass node-to-segment logic is applied in order to compute the contact forces (based on a penalization of the interpenetrations). Tangential forces based on prescribed contact laws such as friction or cohesion are computed locally (i.e. at each contact node, based on its own normal force) using a penalization approach as well. Normal and tangential forces are further distributed between the surrounding field nodes, in accordance with the values of their respective shape functions. The same contact algorithm is applied in a very natural way to rigid bodies (Fig. 1), which are then submitted to classical Newtonian dynamics as in DEM. An explicit adaptive solver is used to integrate in time the corresponding set of ODE. The whole numerical framework is implemented in C++ in an open-source code called Multibody ELement-free Open code for DYnamic simulations (MELODY), which is parallelized in an Open-MP framework and launched on the computational cluster of the LaMCoS. The code MELODY is distributed on http://guilhem.mollon.free.fr with its matlab preprocessor, and described in more details in [26,27].

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Fig. 2 Stabilized states of the six samples after frictionless compaction; a 0% of soft grains; b 20%; c 40%; d 60%; e 80%; f 100% (coloring convention identical to Fig. 1)

3 Simulations Six different simulations are performed, and each simulation comprises 400 circular grains (either rigid or compliant) with diameters uniformly sampled between 0.6 and 1 length unit (no intrinsic unit is used, since the simulations are dimensionless, but any appropriate unit system and time-space scale can be deduced by proper scaling). Every grain has a unit weight equal to 1, and the deformable grains are attributed a neoHookean hyperelastic model with a Young modulus equal to 1 and a Poisson coefficient equal to 0.495 (rendering the grains quasi-incompressible under the typical pressure levels of the study). The grains are positioned between two horizontal rigid walls (a lower wall and an upper wall), which possess a sinusoidal roughness√ with a length scale similar to that of the grains (equal to 3, in order to ensure a proper adhesion between the walls and the grains while avoiding a latticelike ordering in the vicinity of the wall). The lower wall is fixed in displacement, while the upper wall is used to apply the prescribed loads and displacements. The whole sample is periodic in the horizontal direction in order to enable large strains. In a first stage, the coefficient of friction between any

pair of contacting bodies is set to 0, and the upper wall is fixed in rotation and in horizontal displacement and submitted to a downwards pressure equal to 1. When the system has stabilized, all the friction coefficients are set to 0.5, and the upper wall is steadily accelerated in the horizontal direction in order to reach a constant velocity equal to 0.1 and to shear the compressed sampled. During this stage, the vertical pressure is kept constant and equal to 1 (pointing downwards). The chosen velocity (along with appropriate damping at the contacts between rigid grains and within the bulk of the soft ones) is chosen in order to avoid any inertial effect in the simulation. The quasi-static assumption is checked by comparing the total kinetic energy of the system with its potential energy (stored within contacts and related to the deformation of the soft grains), and making sure that the former is negligible when compared to the latter. The shearing is performed until a final shear strain of about 3–4, where a steady state is observed in all simulations in terms of stresses and kinematics. The inertial number [30] in the case of purely rigid grains is close to 5 × 10−3 , which classifies the situation as a very slightly dynamic dense flow. It should be noted that, when soft grains are involved, the use of the inertial number becomes questionable because it assumes

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Fig. 3 Final state of the six samples after shearing; a 0% of soft grains; b 20%; c 40%; d 60%; e 80%; f 100%

that the totality of the macroscopic strain is accommodated by interparticle motions (the inner length scale is thus defined by the average particle diameter), which does not hold any more when grains can deform. The six simulations only differ in the proportion of soft grains among the rigid ones, which ranges from 0 to 100% by increments of 20%. Circular grains are first positioned on a regular hexagonal lattice, and their status (“rigid” or “soft”) is then assigned randomly prior to simulation. Captions of the six samples in a stabilized state after compaction and before shearing are provided in Fig. 2 (with the coloring convention defined in Fig. 1). In Fig. 3, the final stages of the six simulations are shown, with a color scale indicating the horizontal displacement of the matter. Color patterns seem to indicate that the flow regimes were quite different, and that the simulations with higher proportions of soft grains tend to form some sort of clusters of grains with coordinated motions. Fig. 4 shows that all the simulations exhibit a rather regular average velocity gradient between the lower and the upper wall. No long-lasting strain localization is observed under these loading conditions. A closer look however reveals that the

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regular laminar motion of the purely rigid grains (characterized by a constant velocity gradient and a negligible vertical velocity of the grains, Fig. 4a) is somewhat disturbed by the presence of the soft grains, leading to fluctuations of the velocity gradient and to the presence of noticeable vertical motions of individual grains. The plastic flow of these mixtures hence appears to be more “turbulent” than that of purely rigid samples (not in the sense that the flow is dominated by inertial effects, but in the sense of a significant disturbance of the laminar flow and of enhanced mixing). This is further confirmed in the Supplementary video 1, which clearly shows the different flow regimes.

4 Bulk mechanical response under shear During simulations, a number of macroscopic quantities are monitored. Fig. 5 shows the evolution of the shear stress as a function of the shear strain, for the six simulations. The curves are noisy because of the limited number of grains (which was chosen in order to limit the computational cost),

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Fig. 4 Flow patterns of the six samples at shear strains ranging between 3.07 and 3.80 (shown on 4 horizontal periods); a 0% of soft grains; b 20%; c 40%; d 60%; e 80%; f 100%

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Fig. 5 Evolution of the ratio shear stress/normal stress as a function of the shear strain, for different proportions of soft grains in the mixture

but the trends are easy to read. All six simulations first exhibit a quasi-linear regime, reach a stress peak, and then decrease until shear stress reaches a somewhat stable plateau (although rather large fluctuations are observed around this constant value). The qualitative behavior is similar for all simulations, but they all differ quantitatively in terms of the values of the peak and residual stresses, and of the strain level at which they are reached.

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The vertical displacement of the upper wall during shear is provided in Fig. 6a for the six simulations, as a function of its horizontal displacement. The same data are provided in Fig. 6b in terms of volumetric and shear strains. In all cases a dilatancy is observed, since the upper wall is submitted to an upward motion. This was to be expected, since the samples were compacted in a frictionless configuration but sheared with a coefficient of friction of 0.5, and were therefore strongly overconsolidated. All curves first show a quasi-linear increase, and in most cases the vertical position of the upper wall then seems to stabilize at a quite constant value (although, in some cases, this steady state is less clear and the vertical position of the wall still seems to follow an increasing trend). From the results of Figs. 5 and 6, some quantities of interest may be extracted. The peak friction angle is computed from the maximum value of the shear stress and the residual friction angle is computed by averaging the following stress plateau. The dilatancy angle is computed from the slope of the quasi-linear part of Fig. 6a. Those three angles are plotted in Fig. 7 as functions of the proportion of soft grains in the mixture. The general trends are monotonous, since all three angles generally decrease with the increase in the proportion of soft grains (from 24.3◦ to 19.1◦ for the peak friction angle, from 16.9◦ to 12.1◦ for the residual friction angle, and from 13.1◦ to 0.5◦ for the dilatancy angle). Broadly speaking, it is thus clear that adding soft grains in a classical granular material decreases its shear strength and renders it less dilatant. Three exceptions, however, deserve to be pointed out: (i) the sample with 100% of deformable grains has a peak

Fig. 6 Dilation of the sample for different proportions of soft grains in the mixture; a evolution of the vertical displacement of the upper wall as a function its horizontal displacement; b volumetric strain as a function of the shear strain

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Fig. 7 Peak friction angle, residual friction angle, and dilatancy angle, as functions of the proportion of soft grains

friction angle larger than those with proportions of 80% and even 60%, (ii) the sample with 20% of deformable grains has a residual friction angle larger than that of the sample with only rigid grains, and (iii) the sample with only rigid grains shows a final dilation (final increase of the vertical position of the upper wall) lower than for the samples with 20 and 40% of soft grains. These anomalies will be analyzed in details in Sects. 6 and 7.

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Figure 8a shows the value of the shear strain at which the stress peak occurs, as a function of the proportion of soft grains. It shows that the progressive addition of soft grains in a granular mixtures reduces its overall stiffness and significantly delays the appearance of the peak stress. For the sample containing only soft grains, the peak occurs at a shear strain close to 1.5, which is almost 70 times more than for the sample containing only rigid grains. This is confirmed by Fig. 8b, which shows the secant shear modulus of the six packings (computed at a level of stress equal to half that of the stress peak). This graph shows that, when the proportion of soft grains is equal to 0 and 20%, this modulus has a rather constant and very high value, meaning that adding 20% of soft grains to the rigid mixture does not seem to strongly disturb the way strong force chains of rigid grains percolate from one wall to the other. We observe, however, a significant drop of this modulus for larger values of the proportion of soft grains (by a factor 22 between the packings with 20 and 40% of soft grains, for example). When the proportion of soft grains reaches 100%, the shear modulus of the packing is slightly lower than that of the material composing the grains. This is because, at that level of shearing, most of the macroscopic shear is accommodated by elastic deformation of the grains, and the solid fraction is close to 1 (see next paragraph). The evolution of the solid fraction of the six samples during shear is plotted in Fig. 9. In agreement with Fig. 6, all six samples show a decrease of the solid fraction in the first part of the shearing and then reach a plateau. The values of the initial and residual values of the solid fraction are summarized

Fig. 8 Deformability of the packing as a function of the proportion of soft grains; a level of macroscopic shear strain at which the stress peak occurs; b secant shear modulus

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Fig. 9 Solid fraction as a function of the shear strain, for different proportions of soft grains in the mixture

Fig. 11 Local snapshots of the von Mises equivalent stress fields in the residual state; a 20% of soft grains; b 80% of soft grains; c 100% of soft grains

Fig. 10 Solid fractions in the compacted and residual states, as functions of the proportion of soft grains

in Fig. 10, as functions of the proportion of soft grains in the mixture. It appears that adding soft grains to a granular sample strongly reduces its porosity, both in the dense state and in the critical sheared state. The completely soft sample compacted in a frictionless configuration leads to a solid fraction very close to 100%, meaning that the porosity has virtually disappeared. Interestingly, the samples with similar amounts of rigid and deformable grains (for proportions of 20–60% of soft grains) are those for which the difference between

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the initial and the residual solid fractions are the largest. It would seem that such mixtures have a larger ability to create porosity during shear than the samples composed purely of rigid or soft grains, for which the residual solid fraction is closer to the initial one.

5 Micromechanical analysis In this section, a closer look is taken to local behavior within the sheared samples. Detailed snapshots of Fig. 11 show that the deformable grains are submitted to very large deformations (but at a quasi-constant volume), which is explained by two facts: (i) they are very soft compared to the vertical applied pressure, and (ii) the friction coefficient between the grains leads to a good transmission of shear stresses at con-

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Fig. 12 Increment in the norm of the Cauchy–Green strain between two close times in the residual state (macroscopic strain increment norm on the same period is 0.01); a 0% of soft grains; b 20%; c 40%; d 60%; e 80%; f 100%

tacts, and hence to a large magnitude of the shear strain of the individual grains. Deformable grains trapped between rigid grains are also submitted to extreme deformations because of stress concentrations. This is especially the case at low contents of soft grains, since each soft grain is likely to be isolated and to be only in contact with rigid grains. This heuristic is confirmed by the von Mises stress magnitudes shown in Fig. 11, which shows that isolated soft grains are submitted to maximum stresses. In such cases, one may expect a quicker damaging of the soft grains under shear. In the case of a small amount of rigid grains in a sample of soft grains, Fig. 11 also indicates that stress concentrations are likely to occur in the neighborhood of rigid grains. An animated view of the stress fields is provided in Supplementary video 2. In order to get a deeper understanding of the flow regimes of the different mixtures, a specific postprocessing technique is employed. At a certain time, a Delaunay triangulation is performed on a cloud of points composed (i) of all the field nodes belonging to deformable grains and (ii) of all the con-

tact nodes and the centers of inertia of the rigid grains. Hence, this triangulation encompasses the totality of the sheared domain between the rigid walls, and includes both the solid matter and the porosity. For each triangle of this partition, it is then possible to compute a strain tensor based on the increment of motion of its three nodes between two successive saved states (these states should be close enough in order to avoid inversion of the Delaunay triangles, but large enough to allow some time-averaging of the local shear rates). The increment in the norm of the Green–Lagrange strain tensor is then used as a measure of the local strain rate, and makes it possible to determine if the macroscopic shear is locally accommodated by deformation of the grains or by interparticle motion. Snapshots (in the residual state) and animated views (Supplementary video 3) are provided for the six simulations in Fig. 12, and give precious insights into the different flow regimes. Figure 12a shows that, in the case of a sample composed of purely rigid grains, the flow regime obviously consists in

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Fig. 13 Evolution of the coordination number as a function of the shear strain, for different proportions of soft grains in the mixture

Fig. 14 Coordination numbers in the initial compacted state and in the residual state, as functions of the proportion of soft grains

interparticle motion, and is spread in a rather diffuse way on the sample. The flow in this case is essentially laminar. The addition of 20% of soft grains modifies this regime, by introducing two new features: a contribution of bulk shearing in the soft grains (yellow patches in Fig. 12b), and the appearance of hard clusters of typically 8–10 rigid grains with coordinated motions (uniform blue patches in Fig. 12b). Such hard clusters of grains are expected to be stronger than the surrounding soft grains, because their deformation

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requires dilatancy. Hence, they are likely to trigger more localized strains in their surroundings because they can less easily accommodate shear by themselves. This observation is further confirmed in the case of 40% of deformable grains (Fig. 12c), where most of the shear is accommodated by the localized deformation of the soft grains and hard clusters are more numerous. When the proportion of soft grains reaches 60%, the hard clusters are still present but tend to become rarer because of the lower proportion of rigid grains. Figure 12d also shows a change in the behavior of the deformable grains, with the appearance of localized shear events (yellow segments in Fig. 12d). This is related to the fact that the deformable regions are now much larger than in the previous cases, and that intergranular slip can occur more easily between fortuitously aligned soft grains. When the proportion of soft grains reaches 80% (Fig. 12e), no more rigid clusters are present in the flow, and most of the macroscopic shear is accommodated by localized slip between patches of soft grains. A closer look at Fig. 12e and at the Supplementary video 3 seems to indicate that these shear lines often nucleate around the few remaining rigid grains, which act as stress concentrators. Quite often, the slip lines are even joining two rigid grains, creating a network of slip lines separating coordinated clusters of soft grains. Finally, when the sample is composed of soft grains only, the shear is accommodated by much longer and broader slip zones, which can develop more freely thanks to the absence of rigid grains (Fig. 12f). These slip zones delimitate large areas of undisturbed grains, and lead to a flow regime based on the relative motions of soft aggregates. This situation is hence very far from the laminar regime of purely rigid grains. Besides, Supplementary video 3 shows that these localized slip zones appear in “bursts”, when the internal energy stored by the elastic deformation of a cluster of soft gains is suddenly released by the slip at the interface between this cluster and its neighbor. Hence, the flow is dominated by events of local reorganization and of restitution of the stored elastic energy that trigger localized slip. The coordination number (average number of contacting grains for a given grain) is plotted as a function of the shear strain for the six simulations in Fig. 13. The general trend is that the progressive addition of soft grains in the mixture increases the coordination number, both in the initial compacted state and in the residual sheared state. For all the simulations, there is a decrease of the coordination number as the shear develops. These results are summarized in Fig. 14, which provide some quantitative data: when going from the completely rigid to the completely soft samples, the initial coordination number increases from 4.0 to 5.7, and the residual coordination number increases from 3.3 to 5.3. Curves of Fig. 14 show a similar behavior to that observed in Fig. 10 for the solid fraction: the samples with similar proportions of soft and rigid grains are those for which the initial compacted

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Fig. 15 Typical force networks in the residual state; a 0% of soft grains; b 20%; c 40%; d 60%; e 80%; f 100%

state is the most affected by the shear. This leads to a more important decrease of the coordination number during shear than in the samples composed purely of rigid or soft grains. Typical forces networks in the residual state are plotted in Fig. 15, with a color and width scale quantifying the contact forces (segments link the centers of mass of contacting grains, either rigid or compliant). In the case of a soft grain in contact with either a rigid grain or another soft grain, the total contact force is integrated on the whole contact area. Figure 15 shows that the addition of soft grains in a granular sample strongly modifies the force network, by progressively inhibiting the presence of strong force chains. The difference between the purely rigid case and the case with 20% of soft grains already shows that this small amount of deformable matter is sufficient to disrupt the force network, by slightly restraining the percolation of chains of purely rigid grains from one wall to the other, and by promoting the appearance of a weak network. This is in good agreement with the increase in the coordination number between the two cases. It should be noted, however, that this observation only holds for the residual state, but that Fig. 8b shows that it is not valid at very low strain levels. In the extreme case of purely soft grains, strong chains have almost disappeared and the force network is diffuse and homogeneous.

6 Peak behavior at low content of rigid grains As stated in Sect. 4 and Fig. 7, the peak friction angle of the sample with 100% of soft grains is larger than those of samples with 60 or 80% of soft grains. This result is surprising, because it contradicts the general trend that the strength increases with the proportion of rigid particles. An observation of the stress curves of Fig. 5 reveals that the sample without rigid grains is macroscopically softer than the others (i.e. the slope of its quasi-linear part is lower), but that the shear stress curve keeps growing up to much larger strains. The larger peak stress in this case is clearly related to the delay in the appearance of the peak. Another interesting macroscopic result is obtained by observing Fig. 13, which shows that the three samples with 60, 80 and 100% of soft grains do start at the same coordination number of about 5.7 at the initial compacted state. However, as shear develops, this coordination number quickly drops for the samples with 60 and 80% of soft grains, but stays constant until a strain of about 1 for the purely soft sample. It would thus appear that this sample has the ability to retain its initial microstructure up to much larger strains.

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Fig. 16 Pre- and post-peak behaviors of the two samples with 80 and 100% of soft grains

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Fig. 17 Examples of force chains in the mixture with 20% of soft grains

This interpretation is confirmed by the fields of von Mises stress and of strain increments provided in Fig. 16 for the two samples with 80 and 100% of soft grains. These fields are plotted at 4 different macroscopic shear strain levels: during the quasi-linear part of both samples (strain of 0.50), at the peak of the 80% soft sample (strain of 0.93), at the peak of the 100% soft sample (strain of 1.45), and in the residual state of both samples (strain of 3.00). At a strain of 0.50, both samples show limited magnitudes in stress, but the sample with a few rigid grains is clearly submitted to a few stress concentrations, and some localized shear strains in small areas. These events occur in the neighborhood of rigid grains, while the macroscopic stress peak has not occurred yet. In the sample with only soft grains, the stress and strain increment fields are much more homogeneous. This observation still holds at a macroscopic strain of 0.93, at which the purely soft sample shows its first signs of very localized strain and of stress peaks. Meanwhile, the sample with 80% soft grains has reached its stress peak, and its stress and strain increment fields are very heterogeneous because of the onset of failure. Specifically, important localized strains appear in quite broad areas, generally close to the rigid grains and sometimes connecting them. At a macroscopic strain of 1.45, the purely soft sample has also reached its peak, and the localized events coalesce in broader areas of large strain rate, indicating failure of the sample. The stress field is maximum, but is still rather homogeneous. Finally, at a macroscopic strain of 3.00, both samples are in their residual state, and exhibit the typical strain rate patterns described in the previous section and in Fig. 12. Meanwhile, the stress fields are

now very heterogeneous, with patches of low and high stress levels, but do not exhibit any percolation of large stress networks that would be analogous to force chains in purely rigid samples. This is in good agreement with the force networks analyzed in the previous section and in Fig. 15. From this analysis, it comes that the larger peak strength of the purely soft sample is related to the absence of rigid grains in the mixture. In the cases of 60 and 80% soft grains, they act as stress concentrators and therefore accelerate the coalescence of very localized slip events and lead the stress peak to occur earlier, which limits its magnitude. When the rigid grains become more numerous (i.e. for proportions of soft grains lower than 60%), this effect disappears because of the presence of rigid clusters which increase the strength of the mixture, as described in the previous section.

7 Residual behavior at low content of soft grains Numerical results provided in Fig. 7 seem to indicate that the addition of a limited quantity of soft grains in a granular material increases the residual friction angle (from 16.9◦ to 18.3◦ in the present case). Such a property is interesting since it could make it possible to increase artificially the strength of some natural or synthetic granular materials, by mixing them with a limited quantity of soft grains. However, the micromechanical explanation of this phenomenon is not straightforward. A closer comparison of the samples with 0

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and 20% of soft grains in all the previously presented results provides some possible leads, which are detailed below. Figure 5 shows that the presence of 20% of soft grains does not decrease significantly the magnitude of the peak stress, but that it delays the post-peak stress drop in a very significant way when compared to the purely rigid case. It might be expected that the large scale failure of a mixture with 20% of soft grains would thus be more progressive and less sudden than without them. Figure 6a shows that the sample with 20% of soft grains leads to the largest vertical displacement of the upper wall. It is thus the one for which the residual solid fraction is the largest when compared to that of the dense state. This observation is probably a partial explanation of the previous one: the post-peak stress drop is delayed because the sample needs a larger amount of shear strain to dilate up to its residual state. Figure 8b reveals that the shear modulus of the two packings are almost identical, showing that the strong force chains of rigid grains that drive the mixture behavior at low strains is not disturbed by the presence of 20% of soft grains. Figures 11a and 14 show respectively that the isolated soft grains are submitted to extreme deformations when stuck between only rigid grains, and that their presence increases strongly the residual coordination number of the mixture (from 3.3 to 4.0 in the present case). Hence, it would seem that the soft grains improve the connectivity of the mixture, but also create some intricacy in the contact network because of their typical non-convex shapes. Figure 12b shows the presence of the hard clusters in the mixture with 20% soft grains, while such meso-structures do not exist in the purely rigid case. As stated above, these clusters seem to act as hard inclusions in the medium, since they are harder to shear than the surrounding mixture. They may have a role in the observed increase of the residual strength. It has been proposed [10] that soft grains were able to stabilize the force chains. This intuition is partially confirmed by a close inspection of the force networks of the mixture containing 20% of soft grains (Fig. 17). In several cases (although not systematically), force chains are indeed composed of rigid grains surrounded by soft grains. A tentative assumption is that their deformability increases their contact area with rigid grains and renders their contacts more “ductile”, preventing sudden failure of the force chains. This is confirmed by the curves of Fig. 5, which show that the shear stress fluctuates more in the case of purely rigid grains than when 20% of soft grains are added. Hence, it may be speculated that a small proportion of soft grains increases the residual strength by stabilizing the force chains without disturbing too much their percolation from one wall to the other. This benefit is lost when the proportion of soft grains is too large, indicating that there may be an optimal concentration of each kind of grain which may allow to maximize the

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resulting strength. These assumptions require much deeper analysis and understanding, which are left for further studies.

8 Conclusions and perspectives This study presented several simulations of compaction and shearing of mixtures of rigid and soft grains, and revealed a broad range of behaviors and of phenomena in this rather unstudied class of materials. The main findings are the following: – The general trend is that the progressive addition of soft grains in a granular sample (going from the purely rigid to the purely soft samples) decreases its peak and its residual strengths, and delays the appearance of the stress peaks. It also tends to decrease the porosity of the mixture and to increase its average coordination number. – Plastic flow regimes are strongly affected by the proportion of soft grains. While the purely rigid sample exhibits a quasi-laminar flow with a constant velocity gradient and limited vertical motions of the grains, adding soft grains triggers localized shearing in soft regions and appearance of clusters of rigid grains which act as hard inclusions. When the mixture is mostly composed of soft grains (80– 100%), shear is accommodated by broad but temporary shear bands which separate undisturbed aggregates of soft grains. Initial laminar flow is strongly disrupted and a noticeable mixing occurs. – The presence of a small amount of rigid grains in a soft granular sample modifies the micromechanical behavior. It seems that each rigid grain concentrates stresses and facilitates the initiation of local slips in its neighborhood. Hence, a few rigid grains are sufficient to reduce the peak stress of the mixture by triggering an earlier failure than in the purely deformable case, for which the strain and stress fields before the peak are much more homogeneous. – The presence of a small amount of soft grains in a rigid sample seems to improve its residual strength properties because they increase the connectivity of the granular assembly, and possibly because they improve the stability of the force chains without disrupting too much their percolation from one wall to the other. There is however a certain amount of soft grains (between 20 and 40% in the present study) beyond which the shear stiffness of the packing significantly drops, dramatically changing the behavior at low strains. Several assumptions and observations made in this paper will require a deeper analysis, especially because the samples considered in the study were very limited is size (mostly for

Mixtures of hard and soft grains: micromechanical behavior at large strains

computational reasons). The analysis was also restricted to 2D plane-strain kinematics because the numerical tool allowing a 3D simulation of this kind of systems does not exist yet, although there does not seem to be any major obstacle to its future implementation (except maybe the associated computational cost). Besides, only a small portion of the parametric space was investigated. Among the important parameters that were not addressed, the amount of the normal pressure applied to a given material (or conversely the stiffness of the soft grains of a mixture submitted to a given confining pressure) deserves more attention. Interesting effects may also be foreseen if complex shapes [31–33] and varying sizes are considered for both kinds of grains, and if other types of loadings (e.g. oedometric tests, cyclic shearing, hopper flow [34] or kinetic flows [35]) are investigated. Acknowledgements Valuable and useful comments by the two anonymous reviewers are gratefully acknowledged by the author.

Compliance with ethical standards Conflict of interest The author acknowledges that this study contains original material, as a result of a purely academic study without any kind of private funding or conflict of interest. Its publication has been approved tacitly by the responsible authorities at the institute where the work has been carried out.

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