Abstract
By mixing glass beads with a curable polymer we create a welldefined cohesive granular medium, held together by solidified, and hence elastic, capillary bridges. This material has a geometry similar to a wet packing of beads, but with an additional control over the elasticity of the bonds holding the particles together. We show that its mechanical response can be varied over several orders of magnitude by adjusting the size and stiffness of the bridges, and the size of the particles. We also investigate its mechanism of failure under unconfined uniaxial compression in combination with in situ xray microtomography. We show that a broad linearelastic regime ends at a limiting strain of about 8%, whatever the stiffness of the agglomerate, which corresponds to the beginning of shear failure. The possibility to finely tune the stiffness, size and shape of this simple material makes it an ideal model system for investigations on, for example, fracturing of porous rocks, seismology, or root growth in cohesive porous media.
Introduction
The mechanical and fracture properties of cohesive granular materials, where solid but deformable bonds ensure the rigidity of a granular packing, are relevant to a number of questions in powder aggregation^{1}, the strength of agglomerates^{2}, soil rheology^{3}, rock mechanics^{4} and geoengineering^{5}. As one example, sandstone is a porous rock widely used in construction, and is composed of grains of sand held together by a cement of, most often, calcite or silica. It is also the most relevant material for modelling underground aquifers or hydrocarbon reservoirs. Since the early work of Dvorkin and his collaborators on cemented aggregates^{6,7,8}, several approaches have been developed to overcome the intrinsic complexity of such heterogeneous media. Most progress has been made in the field of numerical modeling^{9,10,11}, while bottomup experimental approaches using simple systems and general constitutive models are still rare^{12,13,14,15}. Generally, one wants to be able to predict or control macroscopic properties of cohesive porous media, such as fluid permeability^{5}, fracture response^{16,17}, mechanical constitutive relations^{12,13,18,19}, or oil/water retention^{20}, and to evaluate how these properties depend on the composition of the material, such as grainscale heterogeneities^{20}, or its elasticity^{21,22}.
Here we describe a class of cohesive granular materials with tunable properties, and characterise the mechanical response of these materials to uniaxial compression. The materials are similar to wet granulates, like sand, but where the liquid in the capillary bridges between adjacent grains has been replaced by polydimethylsiloxane (PDMS), a curable elastomer, which is then crosslinked to produce “solid” capillary bridges (see Fig. 1a–c). Before curing, PDMS is a viscous liquid, and perfectly wets the glass beads. By mixing the two phases together (see Methods), a homogeneous material is easily made; the surface tension of the polymer phase naturally attracts it into liquid capillary bridges between particles, and distributes it evenly. The resulting composite is cohesive and malleable, and may be moulded into any desired shape. It can then be cured to harden the bridges by crosslinking the polymer, turning it from a pastelike material to a solid cohesive granular assembly.
For these materials, we focus our attention on conditions of soft bridges, and hard particles. In particular, in our system the Young’s modulus of the polymer bridges, E_{P} ~ 1 kPa−1 MPa, is several orders of magnitude smaller than the Young’s modulus of the beads, E_{b} ~ 60 GPa. Thus, we ensure that the mechanical properties of the composite materials are dominated by the deformation of the bridges, and not by the indentation of the beads into each other, such as would be the case with powders^{18,23}, for example. Our materials are, in fact, roughly in between the two limits of wet^{24,25} or charged^{26} granular materials, where bonds are weak and can reform after breaking, and weakly sintered^{27,28} or cemented^{6,7,8,16,29} materials, whose bonds are of comparable stiffness with that of the grains, and which are generally brittle.
We will show how one can control the Young’s modulus E of such cohesive granular materials by over two orders of magnitude, by changing the modulus E_{p} and volume fraction W of the polymer phase, and the diameter D of the particles in the granular phase. A simple model for the microcontacts, or bridges between particles, is also developed to explain how E should scale with these parameters. Finally, we report on the ultimate yielding of these materials, through shear failure under large strain, using in situ xray microtomography.
Results
The mechanical properties of cohesive granular materials depend on their microscopic structure and composition. Briefly, if the polymer bridges between particles are stiffer, or larger, we found the Young’s modulus of the bulk material to be higher, as might be expected. There is also some dependence of the material stiffness on the size of the particles used.
We prepared material samples with as broad a range of elastic properties as possible, by changing the bead size D, polymer fraction W (defined as the total volume of PDMS within a sample, divided by the sample volume), the Young’s modulus of the polymer E_{p}, and packing fraction ϕ_{b}. Details of the material preparation are given in the Methods section, at the end of this paper. Similar materials could also be made with other polymers (such as polyurethane or polybutadiene), extending the range of bulk elastic properties even further than those presented here. Additionally, the wetting properties of the beads and bridges could be adjusted by judicious choice of ingredients (e.g. polystyrene beads, surfacetreated beads, or more complex heterogeneities^{20,30} like Janus or patchy particles), as could the permeability of the final product.
To test the elastic response of our samples, we designed an unconfined uniaxial compression cell, with a geometry also suitable for insitu xray microtomography experiments. The test rig is sketched in Fig. 1d, and details of its construction, and the test procedures used, are given in the Methods section. We plot in Fig. 1e examples of stressstrain curves measured for various E_{p}, at a fixed bead diameter of 210 μm, and polymer content W = 2.3%. All curves show comparable behaviours, and these measurements are typical of all the results presented in this paper. There is a short nonlinear regime, extending to a strain of up to 2%. Here, this strain represents a displacement of ~100 μm, or one bead radius. This initial response corresponds to the progressive contact of the upper piston of the load cell with the sample, whose surface is uneven on this scale, rather than any inherent nonlinear elasticity. Once good contact is made between the piston and the sample, further compression shows a welldefined linear regime, from which a Young’s modulus E is extracted. Deviation from the linear behaviour is observed after a strain of about 8% on average, followed by plasticity and shear failure.
Variation with polymer stiffness
Without changing the structure of the cohesive granular material, its elastic response can be adjusted by varying the stiffness of its polymer bridges. PDMS is a crosslinkable polymer whose Young’s modulus can range over several orders of magnitude by changing the ratio of oligomer (base) to crosslinker, prior to curing. By varying the mass ratio of base to crosslinker from 66:1 to 10:1, we measured Young’s moduli of bulk PDMS samples from E_{P} ~ 1 kPa to 1.5 MPa, respectively. As shown in Fig. 2a, these values are in good agreement with results reported elsewhere^{31,32,33}, despite differences in curing times and temperatures. Outside of this range of mass ratios the stiffness of the PDMS does not vary significantly, as it is already almost completely crosslinked in the harder limit, while for the softest case the PDMS does not fully solidify, and remains tacky to the touch. Like many other elastomers, PDMS is essentially incompressible, with a Poisson ratio close to 0.5.
Material samples were made by mixing different preparations of PDMS with 210 μm diameter glass beads. For these experiments the polymer content was fixed at W = 2.3%. This choice ensures that welldefined capillary bridges connect neighbouring beads, without merging into clusters; as will be described in the next section, we are in the pendular regime^{34}. The bead volume fraction ϕ_{b} was measured in these samples to be, on average, 0.574, with a standard deviation (representing the reproducibility between samples) of ±0.01. In general, ϕ_{b} did not vary strongly with E_{P}. However, as Fig. 2b shows, there may be a slight decrease in packing fraction in the hardest samples.
For these conditions, as summarised in Fig. 2c, we found that the Young’s modulus E of the cohesive granular material was about an order of magnitude higher than that of the PDMS in its bridges, but very much lower than that of the glass beads. In other words the elasticity of the composite material increases with, and is controlled by, that of the PDMS, despite the fact that the polymer represents only a small minority phase. By varying E_{P} from 1 kPa to 1.5 MPa, we could tune E from 90 kPa to 6 MPa, respectively. Finally we note that at the lowest concentration of crosslinker the PDMS remains somewhat liquidlike even after curing; it still made a rigid cohesive sample, but the properties of this material may be controlled more by capillary forces, rather than the elasticity of the bridges.
Variation with polymer content
Next, we studied how the size and shape of the polymer bridges affects the elasticity of the cohesive granular material, while keeping all other properties of the polymer and beads constant. For comparison, it is wellknown that the mechanical properties of wet granular materials are linked to their internal liquid morphology^{24,34,35}. In particular, the strength of such materials increases rapidly with liquid content in the pendular regime, i.e. as long as the fluid remains in individual capillary rings, like those shown in Fig. 1a. Beyond some threshold, additional fluid does not create stronger bonds, but instead causes the rings to coalesce into larger structures, like the trimer shown in Fig. 1c. Above this socalled pendularfunicular transition, the strength of something like a sandcastle is surprisingly independent of its liquid content^{24}. Here we demonstrate similar results for the scaling of the cohesive granular material’s Young’s modulus, E, with polymer content W.
For this set of experiments we prepared a range of samples with different polymer contents, from W = 0.3% to 8.2%, for fixed bead size D = 210 μm and polymer modulus E_{P} = 250 kPa. The range of W studied allows us to probe the pendular regime, starting from the first observable capillary bridges at low W, but also extends to the funicular regime, as we will shortly demonstrate. As shown in Fig. 3a, we observed two types of response, depending on the volume fraction of polymer. For W between 0.3 and 2.7%, E depends strongly on W, and varies by more than an order of magnitude over this range. Above W = 2.7%, E continues to increase with added polymer, but much more slowly than below the threshold value.
By looking at pieces of these samples through a microscope we found that for volume fractions higher than W = 0.7% the polymer was homogeneously distributed throughout the sample, forming welldefined bridges between adjacent beads. The samples were then ground up in a mortar, with a pestle, and a single layer of detached beads was spread on a glass slide, for each sample. Images of these beads were taken by digital microscopy, the radii r of the polymer bridges remaining adhered to the particles were measured manually using ImageJ^{36}, and the bridge areas were calculated, assuming circular crosssections. At least 31 different bridges were analysed from each sample. As shown in Fig. 3b, the mean crosssectional area of the bridges, A, increases linearly with W, up to several percent of polymer content. Trimers were first observed at W* = 2.7%, and this critical value is shown on Fig. 3a to correspond well to the change in elastic response of the cohesive granular media. There is a similar kink at W* in the dependence of the packing fraction of the beads, ϕ_{b}, with the amount of polymer (see Fig. 3c). Finally, trimers occur when three adjacent bridges touch each other, and this situation is expected when the bridge volume V_{c} ≃ 0.058R^{3}, with R the radius of the beads, if one assumes that all the liquid goes into the bridges^{24}. For our 210 μm samples this predicts a critical bridge area of A_{c} = 6.7 × 10^{3} μm^{2}. As Fig. 3b shows, in this case W* also corresponds to the point where the upper tail of the distribution of A crosses A_{c}, but where the average A is still only around 4 × 10^{3} μm^{2}. For higher polymer content, the largest bridges will coalesce, increasing the fraction of clusters in the samples. For example, one can see on Fig. 3b that the resulting trimming of the bridge size distribution leads to a significant deviation from the linear relation between A and W for W = 4.6%. Although we were not able to quantify the absolute fraction of trimers, or larger clusters, these results qualitatively agree with the behaviour of wet granular packings, which suggests a ratio of trimers to bridges of about 5%, when W = 3.5%^{24,34}.
These observations all show how the stiffness of an aggregate sample is related to the sizes and shapes of its bridges. As long as the bridges remain isolated, then E ~ A, as in Fig. 3d. For the lowest W the polymer does not spread smoothly through the sample, and the result is a softer sample than otherwise expected. If W > W*, then any further addition of polymer will end up mainly filling the pore space between groups of particles, rather than strengthening the elastic bonds that hold the sample together. This suggests that the slow increase in E observed for W > W* could then result from the growth of bridges from the smaller end of their size distribution, rather than from pore filling and the coalescence of bridges into more complex structures.
Variation with volume fraction and bead diameter
As we have seen, the volume fraction of the beads, ϕ_{b}, in fully cured samples may change along with other parameters such as E_{p} and W. This may affect the density of bead contacts, or bridges, in different samples. It is therefore necessary to test for any dependance of E on ϕ_{b}. To do so we used a coupling between ϕ_{b} and the diameter of the beads D in finite size samples to disturb the homogeneity of the packing and vary the average value of ϕ_{b}. In particular, we found that confinement in a mould leads to an ordering of the outer layer of beads, as in other granular media^{37,38}. This creates a different packing of particles near the walls, than in the bulk. By changing the size of such samples, relative to the size of their constituent beads, we could thus adjust their total average volume fraction.
In a first series of experiments we kept the geometry of the samples fixed to a cylinder of 4.65 mm × 4.9 mm (base diameter × height), while increasing the bead size from D = 55 μm to D = 775 μm. We also fixed W = 2.3% and E_{P} = 1 MPa. As shown in Fig. 4a, the mean packing fraction in these samples varied from ϕ_{b} = 0.59 to 0.53, a range at least as broad as those already encountered in Figs 2b and 3c. Next, by using moulds with dimensions much larger than the bead size (i.e. at least 25 beads across, see Methods), we sought to minimise boundary effects, and keep ϕ_{b} constant. As demonstrated in Fig. 4a, the volume fractions of these larger samples were all consistent with an average value of ϕ_{b} = 0.584 ± 0.01, regardless of particle diameter. These two sets of experiments, summarised in Fig. 4b, allowed us to independently test the effects of the beads’ packing density and size on the elasticity of the cohesive material. We found no significant dependance of E on ϕ_{b}, but instead observed an unexpected variation in E, of nearly one order of magnitude, across the range of bead sizes tested.
This dependence of material elasticity on bead size is somewhat surprising, as linear elasticity is inherently scalefree. In other words, if all lengths in the system are scaled up by a constant factor, we would expect the elastic response to remain unchanged. That this is not, in fact, true here suggests that there is an additional lengthscale in the system, such as a minimum bead separation. One lengthscale that can be excluded from consideration here is the capillary length. For the samples discussed above we also measured the average bridge and bead radii. As shown in Fig. 4c, there is no deviation from the linear relation between d and D expected for capillary bridges^{39,40}, showing that gravity can be safely neglected here, even for millimetric beads. Any significant effect of drainage of the liquid polymer in the granular pile prior to curing was also rejected by measuring the size of the bridges in a cylindrical sample of 16 mm × 38 mm (base diameter × height) made with the biggest (2 mm) beads. After curing the sample was cut into three pieces of 1cm height and the bridge size distribution was measured for each piece, showing almost no differences with respect to the position in the sample, as shown in Fig. 4c.
To summarise, we have shown that perturbations of the average volume fraction of the beads appear to have little observable influence on the mechanical properties of the material. However, we have observed a strong dependance of E on the bead size, which implies that there is some additional lengthscale that is relevant to the physics here. Although the origins of this effect remain unclear, we have ruled out a possible influence of drainage.
Behaviour at larger compressive strains
So far we have limited our study to the linear regime of the stressstrain curves, from which the Young’s modulus of a sample can be extracted. Now, we briefly investigate the behaviour of the cohesive granular material for larger compressive strains, focusing on the end of the linear regime. Although deviation from a linear elastic response is not necessarily equivalent to yielding of the sample, we shall presently show how it corresponds to the onset of shear failure in the compressed sample. We therefore take it to be a good approximation of the yielding point.
For each of the compression experiments discussed above we determined the limit strain ε_{L} by the first point of the stressstrain curve that deviates significantly from a linear fit, as demonstrated in Fig. 5a. The results are shown in Fig. 5b. Interestingly, we found that all samples, whether they varied in bead size, bridge size, or bridge composition, were consistent with a constant ε_{L} = 8% (positive onetailed ttest with a pvalue of 0.49). This suggests that the plasticity of the material is controlled by a limiting strain, rather than a limiting stress.
To explore further the internal deformations of the sample, in this limit of large strains, we performed an in situ uniaxial compression experiment on one of our samples using microcomputed xray tomography (microCT); details of this test are given in the Methods section. The use of precision monodisperse beads here allowed us to reconstruct the positions of nearly all the beads in the sample after a series of compressive steps. Four xray scans were made at different strains, as indicated in Fig. 5c, and the position vectors were found for each bead i after each compression step n. The relative displacements of the beads, were calculated as , where the angular brackets represent averaging over all beads (i.e. we subtract the samplewide average displacement).
The relative displacements within the sample are visualised in Fig. 5d. There we show how the displacement field evolves toward a nonlinear response during the compression test. Analysis of the first 2% of the strain (between points 1 and 2) confirms that this initial phase corresponds to the progressive contact of the rough surface of the sample with the piston. Variations of the displacement field in the linear regime between points 2 and 3 remain smooth, as expected for a linear elastic response, other than a few isolated features (slip events, local rearrangements) near the edges of the load cell. However, it is interesting to note that the displacement field is not entirely uniform and already shows the onset of a gradient that will evolve towards a shear band in the plastic regime. For higher strains, between points 3 and 4, we cross the limiting strain ε_{L}. Now we can clearly see the intermediate stages of shear failure, demonstrated by the shear band, or slip plane, that has developed across the sample. This type of yielding behaviour is expected for homogeneous and isotropic solids failing under compression, when the deviatoric stress (or strain) reaches the yield stress (or strain). Several amorphous materials, such as metallic glasses^{41}, granular materials^{42}, or foams^{43}, also form shear bands when failing. However, the exact process by which these local plastic rearrangements lead to final persistent shear planes at failure remains unclear and is still a topic of active research^{44,45}.
Discussion
We have shown how to make stiff, porous granular materials, and characterised their mechanical properties. As demonstrated above, the elastic response of this class of cohesive material can be tuned by changing the stiffness and size of its polymer bonds, as well as the diameter of its beads. A full micromechanical model, which would link macroscopic responses such as elasticity, fracture and plasticity to microscopic modes of deformations, is beyond the scope of this paper. Nevertheless, we will see here how simple models of elastic bridges, combined with finite element simulations (performed using Comsol Multiphysics), can be used to derive simple scaling relations that can explain most of the observed findings.
In these materials there is a large difference between the Young’s moduli of the glass beads and of the polymer bonds: E_{p}/E_{b} < 10^{−4}. Since the modulus of the composite material also scales as E ~ E_{p}, this suggests that we can neglect the deformation of the beads when considering elastic responses. Indeed, the stress transmitted by actual contact of the beads should obey the Hertz theory of contacts^{46}, and lead to a strongly nonlinear constitutive relation^{47}. This is not what we see.
A linear elastic response can also be observed when granular materials are cemented with a hard cement, when . This is the case for glass or PMMA cemented by epoxy^{8} or frozen tetradecane^{29}, for example. However, in such systems force propagation still occurs by Hertzian contacts between the beads, in parallel with deformation of the cement^{8,29}. This leads to a stiffness of the packing that is comparable to that of the grains: E ~ E_{b}. Again, this is not what we see: here E/E_{b} < 10^{−3}.
Therefore, we will instead consider a model where the elastic response of the aggregate is only due to elastic deformations of the polymer bridges. Interestingly, the effective Young’s modulus of such a model can still be an order of magnitude higher than that of the bridge material, a result similar to the data shown in Fig. 2c. This can be understood by first considering the simple example of the uniaxial compression of a cylinder of radius a and height h with bonded surfaces (i.e. there is no slip between the sample and the pistons, during compression). Williams and Gamonpilas^{48} derived an analytic relation between the “apparent” measured Young’s modulus of this cylinder, E_{a}, and the true Young’s modulus of the material composing it, E_{p}:
where ν is the Poisson ratio of the material and S = a/h the aspect ratio of the cylinder. We show the results of Eq. 1 in Fig. 6a and compare them with a finite element simulation of a bonded cylinder under uniaxial compression. Here E_{a} can be significantly higher than E_{p}, especially for large aspect ratios and for materials with a Poisson ratio close to 0.5, such as is the case for elastomers. For perfectly incompressible materials (ν = 0.5), Eq. 1 reduces to E_{a}/E_{p} = 1 + S^{2}/2, and for large aspect ratios, S ≫ 1, it further shows a scaling of the apparent stiffness with the crosssectional area of the cylinder. Although a cylinder is only a rough approximation of a solid capillary bridge, this example shows how bonding between the beads and the bridges, along with incompressibility of the polymer bridges and their large aspect ratios, can lead to a Young’s modulus of the aggregate that is higher than that of the polymer. Also, the model predicts a linear relation between E_{a} and E_{p}, and a scaling of E_{a} with the crosssectional areas of the polymer bridges, in fair agreement with the measurements reported in Figs 2c and 3d, respectively.
To go towards a more realistic description of the mechanics of the polymer bridges, we consider now a system made up of two beads of radius R, initially separated by a surfacetosurface distance l_{0} and connected by an elastic bridge consisting of a cylinder of radius r, height H and Poisson ratio ν, which has been truncated by the spherical caps of the intruding beads (see Fig. 6b). As in the example discussed above, the bridge is firmly bonded to the beads, so that its contact areas remain fixed. We performed a series of simulations in this configuration by displacing the centres of both beads towards each other, and measuring the total force required to hold them there (see Methods). The effective strain, ε*, was defined as the relative change of the centretocentre distance between the beads, and the effective stress, σ*, as the restoring force divided by the bead crosssectional area. In the limit of small strains the relationship between ε* and σ* is linear and does not depend on the sign of the deformation (i.e. compression vs. tension). We thus define an effective Young’s modulus as E* = σ*/ε*, for small strains. The results of these simulations are shown in Fig. 6b for different gaps l_{0} and bridge sizes r, and taking nearly incompressible bridges with ν = 0.49. As in the case of the bonded cylinder, we see that here E* can be significantly higher than E_{p}, in particular for beads close to contact (e.g. for l_{0}/R = 0.02). Specifically, like the data shown in Fig. 2c, the bridgeparticle assembly is up to about an order of magnitude stiffer than the material in the bridge itself.
These simulations allow us to test the range of the analytic model that treats an elastic bridge as a simple bonded cylinder. To do so, we replace the bridge between the beads by a full (i.e. untruncated) cylinder of radius r and height H (see Methods and Fig. 6b). We see in Fig. 6b that this approximation is valid when the bridge has a shape close to a cylinder (H ~ l_{0}, i.e. for small r/R or large l_{0}/R) but that it fails to describe the cases of large bridges and beads close to contact. Over a larger range of parameter space the simulations instead match the relation E*/E_{p} ∝ (r/R)^{2}, which is in good agreement with our measurements of E ∝ r^{2} in the pendular regime. Figure 3d showed that this result holds experimentally up to . It is known that in wet granular assemblies, capillary bridges are formed between beads in contact or very close to contact^{40}. However, liquid bridges can also span a finite distance, so that the average number of capillary bridges per bead is at least ~10–15% higher than the number of dry contacts for a given packing fraction^{40,49}. The wide range of validity for the relation E ∝ r^{2} in our experimental results could then indicate that the elastic behaviour of the aggregate is mainly governed by these extended bridges.
We have seen that simple contactmechanics models can enlighten the underlying physics of our system and provide good quantitative comparisons to our experimental findings. They even suggest the importance of small gaps between beads, when bridges are made, to the macroscopic response of the material. An additional characteristic length scales, such as this gap width, is necessary to explain the observed dependence of E on the bead size (Fig. 4b). However, it also shows that other smallscale features could affect the scaling of the elastic response of cohesive granular materials. To explore models of these materials further would require a detailed micromechanical understanding of other modes of deformation, besides tension or compression, such as shear, torsion or bending. More realistic numerical simulations of the whole packing, similar to those describing the behaviour of aggregates cemented with stiff cements^{16,50,51}, could also clarify, for example, the role of intergranular friction and bead roughness. Furthermore, the link between the microscopic details (i.e. at the bridge or bondscale) to the macroscopic elastic properties of the sample could be investigated using latticesolid models, which simulate large ensembles of discrete particles interacting via springs, and which are widely used to model the mechanical properties of rocks^{52,53,54}. However, one of the main differences between existing models and our system is the peculiar dependance of the bridge stiffness on the initial bead separation, which originates principally from the incompressibility of the polymer bonds.
In several places we have noted that our materials can behave differently to the more wellknown cemented aggregates. One useful feature of our cohesive aggregates worth highlighting is the large strains that they can accommodate. The value of the limiting strain is ε_{L} ~ 8%, compared to typical values of 1–2% of yield strain observed for aggregates cemented with hard cement^{22}. Elastomers such as PDMS remain elastic for very large deformations (e.g. PDMS can be reversibly stretched by more than 40% of its length^{55}), perhaps explaining this difference. The fact that yielding is observed for a constant strain, largely independent of sample stiffness and of the microscopic parameters W, E_{p} and D is remarkable. This behaviour indicates that yielding of the aggregate could be a purely geometric effect linked to local rearrangement of the beads, in a similar manner to the yielding of amorphous systems such as colloidal suspensions^{56} and dense emulsions^{57}, for example. Further understanding of the yielding properties of the material via modelling would require a micromechanical model taking into account plastic deformations at the bridge scale, for example partial debonding or fracture at the contacts^{58}, as observed in failure of weakly cemented granular materials^{29}.
In summary, we have investigated the mechanical properties of a cohesive granular material obtained by mixing glass beads with a curable elastomer, and shown how its elasticity can be finely tuned by controlling the stiffness of the polymer phase and its volume fraction within the sample. We have seen how scaling relations for the Young’s modulus of the aggregate can be obtained by using simple models, which in turn show good agreement with our experimental findings. We have also found that the stiffness of the aggregate depends on the bead size, while remaining largely insensitive to variations in the volume fraction of the beads. Finally, we have shown that this material exhibits a linear elastic response, even for large strains, and a strain at yield that is independent of its strength and microscopic details. The high tunability of the properties of this model cohesive granular material makes it a good candidate for investigations on problems involving cohesive porous media, such as the hydraulic fracture of porous rocks, seismology, biofouling, or root growth in porous matrices.
Methods
Glass beads
All beads were made of sodalime glass with a Young’s modulus between 60 GPa and 70 GPa (manufacturer’s info). Prior to use, beads were first cleaned for two hours in a strong surfactant solution (20%, by weight, Hellmanex III, Hellma Analytics, diluted with water), rinsed, further cleaned with a solution of 1M NaOH, and rinsed again several times with water, until reaching neutral pH. Deionized (Millipore) water was used for all cleaning steps. Finally, the beads were dried in an oven at 90 °C overnight.
The densities of each type of bead were measured with a precision of ±5 kg/m^{3} using a pycnometer, and the results are summarised in Table 1. The 210 μm beads were used for most tests, while the monodisperse 200.9 μm beads were reserved for the tomography experiments. The remaining beads were used to test the effects of particle size on material properties. The particle size distributions of the beads were measured by optical microscopy using a minimum of 100 beads per bead type. The mean diameters agreed with manufacturer specifications to better than 5%, and are shown in Table 1. The polydispersity, also shown, corresponds to the standard deviation of the distribution of bead diameters, divided by the mean. To limit the effects of packing effects near the walls of the container, during casting and curing, the diameter and height of the samples could be varied along with bead size. The various specimen sizes are also listed in Table 1.
Sample preparation
PDMS was prepared by thoroughly mixing Sylgard 184 (Dow Corning) base and curing agent in the desired weight ratio (from 10:1 to 66:1) and then degassing the resulting mixture under vacuum. The density of the degassed PDMS was measured to be 1010 kg/m^{3}, and did not vary within instrument precision over the range of mixing ratios used. Samples were made by mixing a set amount of PDMS uniformly into glass beads (typically ~10 mL of beads) in a mortar with a pestle, taking care that the PDMS was imbibed directly into the bead pack, rather than adhering to the pestle or mortar wall. A subset of the mixture was then cast into a desired mould, where it was gently compressed into shape. To ensure sample homogeneity, the moulded samples were vibrated at a fixed acceleration of 2 g for two minutes at each of the following frequencies: 20 Hz, 100 Hz, 200 Hz, and 500 Hz. This provided an acceleration that is close to, but below, the fluidisation onset of wet granulates^{49}. After resting for one hour, to ensure equilibration of the capillary bridges, samples were then baked at 75 °C for 14 hours. The volume fraction of the beads, ϕ_{b}, in any sample was obtained by measuring the mass (m_{s}) and volume (V_{s}) of that sample, the density of the beads ρ_{b}, and the masses of PDMS (m_{p}) and beads (m_{b}) mixed in the mortar prior to curing, and using the relation ϕ_{b} = ρ_{s}m_{b}/ρ_{b}(m_{p} + m_{b}), where ρ_{s} = m_{s}/V_{s}.
Mechanical tests
Unconfined uniaxial compression tests were performed using two different custommade testing machines. The results from the two instruments showed no noticeable differences, for tests on samples prepared under identical conditions. The Young’s modulus of PDMS, E_{P}, was determined from cylindrical samples of dimensions (base diameter × height) of 15.00 mm × 8.00 mm, while the Young’s modulus of the cured aggregate samples, E, was measured on cylinders of dimensions 4.65 mm × 4.90 mm, unless stated otherwise. The cylinder heights could vary by ~10% between samples, but were measured individually for each test, and the measured values were used in calculating the sample strain. A droplet of sunflower oil was deposited on the surface of the pistons before each experiment, to prevent sticking of the sample during compression. The results do not depend on the sample size or aspect ratio, as one can see on Fig. 2c., where we show results of three measurements at different E_{p} for cylinders of diameter 15.00 mm × height 7.70 mm, at W = 2.3%, D = 210 μm and ϕ_{b} = 0.56–0.57.
During a mechanical test compression was controlled via a stepper motor of submicron resolution, while the restoring force was monitored using either an analytical balance (Denver Instruments) or a compact highprecision load cell (model 31E, Honeywell), with an effective force resolution of 5 mN. Compression consisted of discrete steps of 7.6 μm. After each step the load cell was continuously monitored, until the restoring force reported by it had reached an equilibrium value. Only then was the next compression step made. This led to an average compression rate of about 5 μm/min. Tests made with significantly longer or shorter intervals between compression steps showed no significant variation in the resulting stressstrain curves. Each value of E given here is the result of at least three independent measurements, with standard deviations shown as error bars.
Xray microtomography
The sample used for in situ xray microtomography was prepared with monodisperse beads of diameter 200.9 ± 1.9 μm (see Table 1) using a volume fraction W = 3.7 % of PDMS prepared with a 10:1 weight ratio of base to curing agent (E_{P} = 1.5 MPa). The Young’s modulus of the sample was measured as E = 7.9 MPa and the bead volume fraction was ϕ_{b} = 0.59 ± 0.01. A series of four xray computed microtomography scans (GE Nanotom) was performed at different fixed strains (see Fig. 5), using the setup sketched in Fig. 1d. The sample was slowly compressed by 120 μm between each scan, using the same methods described above for mechanical testing. Tomograms were acquired with a tungsten target and an acceleration voltage of 120 keV. Each scan consisted of a set of 2000 projections with a resolution of 1132 × 1132 pixels and a voxel size of 5 μm. Beads were detected within the reconstructed volume using Matlab. In particular, after thresholding using Otsu’s algorithm^{59} and 3D erosion of the binary images to separate neighbouring beads, particle positions were determined by finding the centroids of each individual connected volume. By using beads that were monodisperse, both in size and shape, we could detect all beads within the sample and measure their centrepositions to within about one voxel’s precision. The displacements in bead positions between subsequent scans were generally small, as compared to the bead diameter. We thus tracked the motion of individual beads through the compression by assuming that the two closest bead positions, from any pair of successive scans, correspond to the same particle.
Comsol modelling
Comsol Multiphysics 4.0 was used to build a 3D finiteelement model of an elastic bridge. The bridge was generated by subtracting two spheres of radius R and surfacetosurface separation l_{0} from a cylinder of Young’s modulus E_{p}, Poisson’s ratio ν = 0.49 and radius r (see Fig. 6b). A finite translation δ was imposed on one of the spherebridge interfaces while the other one was kept fixed, using noslip boundary conditions for both. The total reaction force F_{n} exerted on the displaced interface was measured, giving an effective Young’s modulus E* = F_{n}(2R + L_{0})/(δπR^{2}).
We also performed simulations of full cylinders to compare with analytical results obtained by Williams and Gamonpilas^{48} (see Fig. 6a), and to test modelling of a bridge with such a bonded cylinder of height (see Fig. 6b). Adapting Eq. 1 to this geometry and our definition of E* leads to:
Additional Information
How to cite this article: Hemmerle, A. et al. A cohesive granular material with tunable elasticity. Sci. Rep. 6, 35650; doi: 10.1038/srep35650 (2016).
References
Iveson, S., Litster, J. & Ennis, B. Fundamental studies of granule consolidation part 1: Effects of binder content and binder viscosity. Powder Technol. 88, 15–20 (1996).
Kendall, K. & Stainton, C. Adhesion and aggregation of fine particles. Powder Technol. 121, 223–229 (2001).
Mitchell, J. & Soga, K. Fundamentals of soil behavior (Wiley New York, 1976).
Goodman, R. Introduction to rock mechanics (Wiley New York, 1989).
Turcotte, D., Moores, E. & Rundle, J. Super fracking. Phys. Today 67, 34 (2014).
Dvorkin, J., Mavko, G. & Nur, A. The effect of cementation on the elastic properties of granular material. Mech. Mater. 12, 207–217 (1991).
Dvorkin, J., Nur, A. & Yin, H. Effective properties of cemented granular materials. Mech. Mater. 18, 351–366 (1994).
Dvorkin, J., Berryman, J. & Nur, A. Elastic moduli of cemented sphere packs. Mech. Mater. 31, 461–469 (1999).
Jing, L. A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering. Int. J. Rock. Mech. Min. 40, 283–353 (2003).
Potyondy, D. & Cundall, P. A bondedparticle model for rock. Int. J. Rock. Mech. Min. 41, 1329–1364 (2004).
Jiang, M., Zhang, W., Sun, Y. & Utili, S. An investigation on loose cemented granular materials via DEM analyses. Granul. Matter 15, 65–84 (2013).
Delenne, J., El Youssoufi, M., Cherblanc, F. & Bénet, J. Mechanical behaviour and failure of cohesive granular materials. Int. J. Numer. Anal. Met. 28, 1577–1594 (2004).
Delenne, J., Soulié, F., El Youssoufi, M. & Radjaї, F. From liquid to solid bonding in cohesive granular media. Mech. Mater. 43, 529–537 (2011).
Papamichos, E. Constitutive laws for geomaterials. Oil Gas Sci. Technol. 54, 759–771 (1999).
Rieser, J., Arratia, P., Yodh, A., Gollub, J. & Durian, D. Tunable capillaryinduced attraction between vertical cylinders. Langmuir 31, 2421–2429 (2015).
Affes, R., Delenne, J., Monerie, Y., Radjaї, F. & Topin, V. Tensile strength and fracture of cemented granular aggregates. Eur. Phys. J. E 35, 1–15 (2012).
Birchall, J., Howard, A. & Kendall, K. Flexural strength and porosity of cements. Nature 289, 388–390 (1981).
Kendall, K., Alford, N. & Birchall, J. Elasticity of particle assemblies as a measure of the surface energy of solids. P. Roy. Soc. Lond. A Mat. 412, 269–283 (1987).
Goehring, L. Drying and cracking mechanisms in a starch slurry. Phys. Rev. E 80, 036116 (2009).
Murison, J. et al. Wetting heterogeneities in porous media control flow dissipation. Phys. Rev. Applied 2, 034002 (2014).
Holtzman, R., Szulezuwski, M. & Juanes, R. Capillary fracturing in granular media. Phys. Rev. Lett. 108, 264504 (2012).
Delenne, J., Topin, V. & Radjaї, F. Failure of cemented granular materials under simple compression: experiments and numerical simulations. Acta Mech. 205, 9–21 (2009).
Johnson, K., Kendall, K. & Roberts, A. Surface energy and the contact of elastic solids. P. Roy. Soc. Lond. A Mat. 324, 301–313 (1971).
Scheel, M. et al. Morphological clues to wet granular pile stability. Nat. Mater. 7, 189–193 (2008).
Li, J. et al. Similarity of wet granular packing to gels. Nat. Commun. 5 (2014).
Chen, S., Li, S., Liu, W. & Makse, H. Effect of longrange repulsive coulomb interactions on packing structure of adhesive particles. Soft Matter 12, 1836–1846 (2016).
Rice, R., McKinney, K., Wu, C., Freiman, S. & Donough, W. Fracture energy of Si3N4. J. Mater. Sci. 20, 1392–1406 (1985).
Arató, P., Besenyei, E., Kele, A. & Wéber, F. Mechanical properties in the initial stage of sintering. J. Mater. Sci. 30, 1863–1871 (1995).
Langlois, V. & Jia, X. Acoustic probing of elastic behavior and damage in weakly cemented granular media. Phys. Rev. E 89, 023206 (2014).
Walther, A. & Müller, A. Janus particles. Soft Matter 4, 663–668 (2008).
Brown, X., Ookawa, K. & Wong, J. Evaluation of polydimethylsiloxane scaffolds with physiologicallyrelevant elastic moduli: interplay of substrate mechanics and surface chemistry effects on vascular smooth muscle cell response. Biomaterials 26, 3123–3129 (2005).
Ochsner, M. et al. Microwell arrays for 3D shape control and high resolution analysis of single cells. Lab Chip 7, 1074–1077 (2007).
Balaban, N. Q. et al. Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates. Nat. Cell. Biol. 3, 466–472 (2001).
Herminghaus, S. Wet Granular Matter, a Truly Complex Fluid (World Scientific, 2013).
Pietsch, W., Hoffman, E. & Rumpf, H. Tensile strength of moist agglomerates. Ind. Eng. Chem. Prod. Res. Dev. 8, 58–62 (1969).
Schneider, C., Rasband, W. & Eliceiri, K. NIH image to ImageJ: 25 years of image analysis. Nat. Meth. 9, 671–675 (2012).
Desmond, K. & Weeks, E. Random close packing of disks and spheres in confined geometries. Phys. Rev. E 80, 051305 (2009).
Jerkins, M. et al. Onset of mechanical stability in random packings of frictional spheres. Phys. Rev. Lett. 101, 018301 (2008).
Willett, C. D., Adams, M. J., Johnson, S. A. & Seville, J. P. K. Capillary bridges between two spherical bodies. Langmuir 16, 9396–9405 (2000).
Herminghaus, S. Dynamics of wet granular matter. Adv. Phys. 54, 221–261 (2005).
Johnson, W. Bulk glassforming metallic alloys: Science and technology. MRS Bull. 24, 42–56 (1999).
Li, W., Rieser, J., Liu, A., Durian, D. & Li, J. Deformationdriven diffusion and plastic flow in amorphous granular pillars. Phys. Rev. E 91, 062212 (2015).
Kabla, A., Scheibert, J. & Debregeas, G. Quasistatic rheology of foams. part 2. continuous shear flow. J. Fluid Mech. 587, 45–72 (2007).
Maloney, C. & Lematre, A. Amorphous systems in athermal, quasistatic shear. Phys. Rev. E 74, 016118 (2006).
Le Bouil, A., Amon, A., McNamara, S. & Crassous, J. Emergence of cooperativity in plasticity of soft glassy materials. Phys. Rev. Lett. 112, 246001 (2014).
Hertz, H. On the contact of elastic solids. J. reine angew. Math. 92, 156–171 (1881).
Makse, H., Gland, N., Johnson, D. & Schwartz, L. Granular packings: Nonlinear elasticity, sound propagation, and collective relaxation dynamics. Phys. Rev. E 70, 061302 (2004).
Williams, J. & Gamonpilas, C. Using the simple compression test to determine Young’s modulus, Poisson’s ratio and the Coulomb friction coefficient. Int. J. Solids Struct. 45, 4448–4459 (2008).
Fournier, Z. et al. Mechanical properties of wet granular materials. J. Phys.Condens. Mat. 17, S477 (2005).
Holtzman, R. Micromechanical model of weaklycemented sediments. Int. J. Numer. Anal. Met. 36, 944–958 (2012).
Carmona, H. A., Wittel, F. K., Kun, F. & Herrmann, H. J. Fragmentation processes in impact of spheres. Phys. Rev. E 77, 051302 (2008).
Place, D. & Mora, P. The lattice solid model to simulate the physics of rocks and earthquakes: Incorporation of friction. J. Comput. Phys. 150, 332–372 (1999).
Wang, Y., Abe, S., Latham, S. & Mora, P. Implementation of particlescale rotation in the 3d lattice solid model. Pure Appl. Geophys. 163, 1769–1785 (2006).
Zhao, G.F., Fang, J. & Zhao, J. A 3d distinct lattice spring model for elasticity and dynamic failure. Int. J. Numer. Anal. Met. 35, 859–885 (2010).
Johnston, I., McCluskey, D., Tan, C. & Tracey, M. Mechanical characterization of bulk Sylgard 184 for microfluidics and microengineering. J. Micromech. Microeng. 24, 035017 (2014).
Boulogne, F. et al. Structural anisotropy of directionally dried colloids. EPL 105, 38005 (2014).
Hébraud, P., Lequeux, F., Munch, J. P. & Pine, D. J. Yielding and rearrangements in disordered emulsions. Phys. Rev. Lett. 78, 4657–4660 (1997).
Léopoldès, J., Conrad, G. & Jia, X. Onset of sliding in amorphous films triggered by highfrequency oscillatory shear. Phys. Rev. Lett. 110, 248301 (2013).
Otsu, N. A threshold selection method from graylevel histograms. IEEE T. Syst. Man. Cyb. 9, 62–66 (1979).
Acknowledgements
The authors would like to thank W. Keiderling and M. Richter for technical assistance and help designing the compression tests, and R. Holtzman, C. MacMinn, and S. Herminghaus for fruitful discussions.
Author information
Affiliations
Contributions
A.H., M.S. and L.G. conceived the experiments. A.H. conducted the experiments and analysed the results. A.H. and L. G. wrote the manuscript, while all authors discussed the results and reviewed the manuscript.
Ethics declarations
Competing interests
A.H., M.S. and L.G. are inventors on a patent application regarding the cohesive granular material (PCT/EP2016/067079).
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Hemmerle, A., Schröter, M. & Goehring, L. A cohesive granular material with tunable elasticity. Sci Rep 6, 35650 (2016). https://doi.org/10.1038/srep35650
Received:
Accepted:
Published:
Further reading

Cold compression of ceramic spraydried granules: Role of the spatial distribution of the binder
Ceramics International (2020)

Mechanical Behavior of Cemented Granular Aggregates under Uniaxial Compression
Journal of Materials in Civil Engineering (2019)

Lightweight Porous Glass Composite Materials Based on Capillary Suspensions
Materials (2019)

Effect of cement matrix on mechanical properties of cemented granular materials
Powder Technology (2019)

Toughening Nanoparticle Films via Polymer Infiltration and Confinement
ACS Applied Materials & Interfaces (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.