1 Introduction

This paper presents the latest advancement in the particle-scale modelling of clays, and demonstrates a realistic model capable of reproducing all the important mechanical features of behaviour arising from mainly electro-chemical interparticle forces. In this work, the Discrete Element Method (DEM) is used to simulate around 3200 kaolinite particles, which macroscopically demonstrate the correct mechanical behaviour of clay within the well-known critical state soil mechanics framework. It’s accepted that the “mechanical” behaviour of sand and clay are very similar, albeit at different stress levels and speeds of drainage, but the reason has remained a mystery because the particle interactions are very different.

The Discrete Element Method, as well as the closely-related Molecular Dynamics method, are popular tools for simulating and investigating the behaviour of particulate materials, and have been used to uncover many new insights and/or confirm existing theories for many coarse materials. However, it is only in recent years that researchers have begun turning their attention to modelling clays at the particle scale. This has been due to the uncertainty and difficulty in modelling clay platelet interactions, and the inherent inefficiency in modelling realistic clay particle shapes (thin platelets with large aspect ratios).

As such, there have been comparatively few studies using DEM to model and study clays (as opposed to sands for example), and many of the existing studies have been very simplified or exploratory in nature. The first to use DEM to simulate clay were Anandarajah and co-workers, initially in two dimensions using rods [1], and subsequently in three dimensions modelling kaolinite platelets as cuboids [2]. These seminal studies focussed on simulating one-dimensional compression of platelets, and sparked several later studies by others. For instance, Bayesteh and co-workers simulated the compression and swelling of clays using two-dimensional rods [3, 4]; and Ebrahimi et al. [5] studied the aggregation of montmorillonite platelets using disks. Similarly, Bandera et al. [6] used disks to simulate the isotropic compression of kaolinite platelets. Other researchers, including Guo and Yu [7], Sjoblom [8], Pagano et al. [9] and Aminpour and Sjoblom [10] all used rigid assemblies of spheres to model clay platelets and investigate a range of phenomena including sedimentation, compression, and shearing. As briefly summarised, a range of methods have been used to model realistic particle shapes (i.e. platelets). Additionally, most of these past studies have used different approaches to modelling the platelet interactions. This variety has provided an excellent basis for developing more advanced and/or realistic simulations of clay. In fact, very recently it has been shown that by accounting for the distinct surfaces on each platelet and implementing adhesion-dependent friction, it is possible to (qualitatively) reproduce further features of clay behaviour, such as a critical state line [11].

The work presented here extends this, by demonstrating a particle-scale model of kaolinite, which exhibits improved (more realistic) behaviour; demonstrating realistic compressibility, and significantly dilatant behaviour when sheared from an overconsolidated state. These advancements are made possible by implementing tangential and normal interactions between platelets based on the best available experimental evidence, and by careful consideration of the initial state of the platelets (inferred from SEM imaging). The aim of this paper is therefore to i) demonstrate how to achieve such realistic behaviour in any particle-scale model of clay; and ii) to use this advanced model to uncover new insights to what governs the macroscopic clay behaviour.

2 Numerical model

2.1 Background

The numerical model presented here is an advancement of a recently published model [11]. This model was a significant milestone in particle-scale modelling of clay: it outlined the essential steps and considerations necessary to reproduce qualitatively correct clay behaviour within the critical state soil mechanics framework. The present paper improves upon this model, is more realistic, and enables more quantitatively correct behaviour. Although all relevant details will be summarised, the main emphasis will be on the (significant) improvements.

To recap, it was shown [11] that to achieve irreversible (plastic) compression, and a realistic normal compression line, the friction and/or adhesion between the platelets needed to be weak enough to allow groups of aggregated or flocculated particles to break down. Conversely, however the friction or adhesion between platelets needed to be strong enough to provide resistance to shearing and enable steady states (avoiding liquefaction). To achieve this balance, it was necessary to compact an initial sample of platelets, to create aggregated ‘stacks’; and to use a combination of high and low friction coefficients. The friction coefficient used depended on whether the interacting surfaces were repulsive (low) or attractive (high). The high and low values of friction between adhesive and repulsive contacts had the effect of giving the soil shear strength, while still allowing aggregation of platelets. Unilaterally increasing the friction or adhesion between platelets prevented the formation of stacks and subsequent plastic compression.

In the current work, it will be shown that by starting with an aggregated sample of stacked platelets—which is demonstrated to be appropriate for clay—excellent macroscopic behaviour is achieved, without being overly sensitive to the values of platelet adhesion and friction. This allows more reasonable values of both adhesion and friction to be used (estimated from experimental data), as well as their influence on the macroscopic behaviour to be assessed. This approach provides a numerical clay that behaves (as desired) more mechanically like a sand, consistent with the well-established CSSM framework and models such as Cam clay, Granta Gravel [12].

2.2 Platelets

This study focusses on simulating kaolin clay, which consists of platelets of the mineral kaolinite. There is a wealth of experimental data available for this material, and so has been the subject of many numerical simulations. Kaolinite platelets are thin and flat, and often have a pseudo-hexagonal shape. Platelet diameters typically lie in the range 0.1–4.0 µm, and common aspect ratios are between 5–100 [13].

Due to their size, interactions between platelets are determined by the surface chemistry [14] rather than by mechanical forces. Conventionally (especially in geotechnical engineering), it was assumed that the two sides (‘faces’) of kaolinite platelets were identical, both possessing a negative surface charge. This view can be attributed to the difficulty in distinguishing between the two face surfaces. In recent years, through the use of Atomic Force Microscopy (AFM), it has been established that the two faces of kaolinite platelets are chemically different, and in most conditions have opposite surface charges [15,16,17]. The mineral kaolinite itself consists of alternating silica tetrahedral and alumina octahedral sheets. A single repeating ‘unit’ consists of a tetrahedral silica sheet bonded via shared oxygen atoms to an alumina octahedral sheet. Each repeating unit is around 0.72 nm thick [13], so a typical platelet contains many. Due to these inherent alternating sheets, it can be assumed that each platelet has one external silica tetrahedral face, and one external alumina octahedral face. In typical conditions (pH < 8), the silica face possess a negative surface charge, and the alumina face possesses a positive surface charge.

In this work, kaolinite platelets are simulated using rigid arrays of spheres (Fig. 1). To model the two different faces, the platelets are constructed from two layers of spheres, as shown. Using multiple spheres to model each platelet means many pairwise interactions are computed between any two neighbouring platelets. This method makes it relatively easy to implement categorically different interactions between different surfaces of the platelets, and automatically accounts for arbitrary orientations. Here, a range of platelet diameters is used, however all platelets are constructed from 0.04 µm diameter spheres, and have a thickness of 0.055 µm. This approach can be summarised as a convenient means to achieve (i) the correct particle shape, (ii) the desired interactions between different platelet surfaces.

Fig. 1
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Platelet modelled using spheres in DEM

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2.3 Initial particle structure

For most conditions of interest (pH < 8), the silica face has a negative surface charge, and the alumina face has a positive surface charge. Further details of the interactions between the surfaces is given in the next section; for now it is only necessary to state that like-charged faces demonstrate repulsion, and opposite-charged surfaces demonstrate long-range attraction.

The unique feature of this study (in contrast to previous models), is that the platelets are initially created in aggregated ‘stacks’ (Fig. 2). These stacks are formed by creating platelets in close proximity, which are held together by the attraction between oppositely charged faces. Earlier work on clays, including previous work by the authors, has often followed the same procedure used when modelling coarse-grained materials: i.e. particles are generated separately, in a loose state with no forces existing between them, and are then allowed to settle under gravity or are compacted as necessary. This has proved to be a sufficient approach for bulky particles, which have mechanical interactions and a relatively narrow range of stable porosities. However, for adhesive and platy particles, this method creates a very porous, open structure, with the material resembling a gel. To subsequently obtain a state resembling a dense clay slurry (porosity ≈ 0.6–0.7) requires subsequent compaction (compressive loading). This compaction must be great enough to overcome the particle adhesion, meaning that large attractive forces between the platelets will require an unrealistic compaction effort, and/or the sample will maintain an unrealistic porosity when subjected to compression. In the authors’ previous work, this constrained the strength of adhesion that could be modelled between platelets, and prevented a proper investigation of its role.

Fig. 2
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Initial Stack of platelets. This stack acts as a unit, and is replicated throughout the initial sample

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Creating the initial sample using ‘pre-formed’ stacks of platelets circumvents this issue. Stacks of platelets are inherently more bulky, and when allowed to settle freely (or are lightly compacted), they do not form such a loose, open structure. This allows realistic, slurry-like porosities (or voids ratios) to be obtained, without an unrealistic compaction effort and irrespective of the adhesive forces between platelets. The use of stacks therefore allows larger (and potentially more realistic) adhesive forces to be modelled, and their role to be properly investigated. In addition to these practical advantages, creating an initial sample with pre-existing stacks of platelets is also consistent with images of real clay slurries and suspensions; and so can be considered as a sensible and realistic approach. For example, Fig. 3 shows SEM images obtained during this study, from a clay slurry prepared at a pH of around 5, with voids ratios of around 2. These images show numerous bulky, aggregated stacks of platelets; intuitively this might not be expected in a slurry, but it is observed. In some cases, for example Fig. 3c, it remains unclear whether the book-like stacks are made of multiple platelets adhered together, or if they are simply larger mineral particles that have undergone weathering, exposing internal layers. However, what is clear is that there is minimal evidence of a loose structure formed by individual platelets. Further images of kaolin slurries can be found in the literature [16, 18, 19], as well as images of suspensions with a much smaller clay content [20,21,22], all of which corroborate this view. We acknowledge here that in this paper, by starting with ‘breakable clay stacks’ we are facilitating a compaction mechanism similar to sand but the interparticle forces at play are very different indeed.

Fig. 3
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SEM images of kaolin slurry, prepared at pH ≈ 5, e ≈ 2.3

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2.4 Interactions

Any discrete particle model requires the interactions between particles to be defined, and for clays this involves a significant degree of uncertainty. It has not been possible to obtain any direct measurements of the forces between individual clay platelets; and in fact it remains impossible to measure the forces between two flat surfaces [surface force measurements are obtained using at least one curved surface;, 14]. In most previous studies of kaolinite, platelet interactions were often based on the classical DLVO theory. The DLVO theory computes the force between like-charged particles based on the interaction of their diffuse layers and van der Waals forces. The DLVO theory has many uses, and is essential in the interpretation/calibration of surface force experiments; however, it cannot account for any attraction between surfaces, and away from ideal conditions, often fails to accurately describe real particle behaviour [13, 23]. Recent work [11] has shown that it is possible instead to estimate the platelet surface interactions using experimental Atomic Force Microscopy (AFM) data. AFM typically involves measuring the force between a negatively charged silica tip and a flat surface, and has been used to probe individual kaolinite platelets. It is through this work that the multifaceted nature of kaolinite platelets has been revealed [17]. Typical AFM data is shown in Fig. 4, reproduced from Kumar et al. [24], which shows force-distance measurements between a negative probe and the negative silica and positive alumina faces of kaolinite platelets. These two curves show characteristic repulsive and attractive interactions, as one would expect between like- and opposite-charge surfaces. It is straightforward to implement monotonically repulsive long-range interactions in DEM—if the stiffness at close-range is great enough to prevent any physical overlap, the only parameter that needs consideration is the range of interaction. Modelling attractive interactions between platelets involves more uncertainty, as this requires the magnitude of attraction forces to be defined (i.e. the ‘pull off’ force for platelets of given size; the minimum in Fig. 4).

Fig. 4
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AFM data showing force-distance interactions between silica tip and two opposite faces of kaolinite platelet. Data from Kumar et al. [24]

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Force-distance measurements between negatively-charged silicon tips and the positively-charged alumina-octahedral faces of individual kaolinite platelets are available in Gupta and Miller [25], Liu et al. [26] and Kumar et al. [24]. Similar measurements, between a silica nano-sphere and an alumina substrate can be found in Veeramasuneni et al. [27]. These studies [24,25,26] reported surface charge densities of the silica tips that were very similar to those measured on the silica face of the kaolinite particles. Thus, if one assumes that the probing material (silica) is representative of the silica face of kaolinite platelets, it is possible to estimate the force-distance relationship between equivalent flat surfaces by using the Derjaguin approximation [14]. This simple procedure [see [11, 28] takes into account the radius of curvature of the interacting surfaces, and using the available AFM data, suggests pull-off forces per unit area of between 100–1000 nN/µm2 between the silica and alumina faces of kaolinite platelets. In previous work [11], a relatively low value of 100 nN/µm2 was used—due to the fact that larger pull-off forces resulted in an unrealistically strong flocculated material—requiring a very large compaction effort. In this work, due to the improved method of creating the sample, greater adhesion forces can be explored. As such, a range of values are used: 100, 500, 1000 nN/µm2; which allows an evaluation of the effects of adhesion.

Figure 5 shows the interactions modelled between the kaolinite surfaces when using a pull-off force of 500 nN/µm2. This plot shows both the attractive interaction between silica and alumina kaolinite faces as well as the purely repulsive interaction implemented between identical kaolinite surfaces. Repulsive interactions are defined to exhibit the similar close-range stiffness and range as the attractive interactions. For simplicity, both silica-silica and alumina-alumina interactions are considered to be identical. All interactions defined in the model have the same range of interaction, comparable to the experimental data shown in Fig. 4. Much like the force-distance measurements obtained using AFM, these interactions computed in the simulations implicitly include all colloidal forces between the surfaces (e.g. van der Waals, ion correlation forces, hydration forces etc.), and assume an acidic environment (pH 5–6) with a moderate electrolyte content (0.01–0.1 mol/L), corresponding to both common soil conditions and to lab-prepared samples of kaolin without any adjustment [19]. The edges of kaolinite platelets are reported to have a negative surface charge [16, 17], with a lower density compared to that of the silica face (meaning the interaction forces per unit area are smaller for the edges). For this reason, and based on previous work by the authors which have shown the influence of the edge surface charge to be negligible [29], no separate consideration is given to the edges in this work; the edge interactions happen naturally as a result of the silica-alumina layers.

Fig. 5
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Platelet surface interactions used in the DEM model

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Tangential platelet interactions are defined using a simple linear (mechanical) tangential stiffness with a coefficient of friction. Like the normal interactions, choosing suitable tangential interactions involves a great deal of uncertainty. No direct measurements of forces have been made between the surfaces of kaolinite platelets (it has not been possible to shear one single kaolin platelet past another), and at this scale it is known that frictional forces tend to increases with the ‘true’ contact area rather than with the normal load [30]. In the preceding work [11], it was found that using a large coefficient of friction of 1.0 between adhesive contacts, and a low value of 0.01 between repulsive contacts was necessary to reproduce qualitatively correct shearing behaviour. These arbitrary values were chosen based on the fact that adhesion is known to have a significant effect on the measured friction forces between smooth materials [14, 30, 31]. For example, it has been shown that large sheets of mica (which is smooth and chemically similar to the silica face of kaolinite), can demonstrate apparent friction coefficients > 1.0 when sliding in dry adhesive contact, and a very low value when in purely repulsive contact [32, 33]. In this work, due to the improved method of creating the sample (with pre-existing stacks), more moderate friction coefficients can be used, without compromising the macroscopic behaviour. For this reason, a coefficient of friction of 0.5 is used for adhesive contacts, and a value of 0.05 is used for purely repulsive contacts; these values were chosen with the aim of being more realistic, and were estimated based on available data. Friction coefficients of between 0.25–0.50 have been reported for sliding between adhesive mica or silica surfaces in various liquids [33,34,35]; and values of 0.02–0.05 for sliding between repulsive mica surfaces in aqueous solutions [32]. However these values remain a simplification—aside from the uncertainty over accurate friction coefficients between kaolinite surfaces, it is known that Amonton’s law does not always apply at this scale for adhesive contacts, and furthermore the friction forces would be expected to vary depending on the alignment of the crystal lattices [36].

The interactions shown in Fig. 5 are the net platelet-to-platelet interactions. These interactions are therefore the sum of many individual sphere-to-sphere interactions. In the DEM simulations, interactions are computed between individual spheres, and so to achieve these desired platelet-to-platelet interactions, the sphere-to-sphere interactions must be calibrated. This straightforward process is done by adjusting the sphere-to-sphere interaction laws whilst measuring the net force-distance function between two platelet surfaces [37]. To capture the shapes of the curves in Fig. 5, custom interaction laws are used. These custom laws have the form: \(F\sim {r}^{-\alpha }-{r}^{-\beta }\) for the attractive case, similar to the well-known Lennard Jones equation; and \(F\sim {r}^{-\alpha }+{r}^{-\beta }\) for repulsive interactions; where F is the normal force, r is the separation, and α, β are constants (α > β). The parameters α, β as well as the constant of proportionality are adjusted such that the desired platelet interactions (Fig. 5) are obtained. However, any custom interaction law can be used for this purpose, and the final parameters after calibration will depend on the density of spheres in each platelet. Further details, and a guide to calibrating any desired platelet interaction can be found in de Bono and McDowell [37]. It is worth pointing out that in previous studies which have used spheres to model kaolinite platelets [8], it was implied that the net platelet interactions are simply the sphere-to-sphere interactions scaled by the number of interacting spheres. However, the surface separation between two interacting spheres will almost certainly not equal the surface separation between the two platelets—a problem which is exacerbated when using a lattice arrangement of spheres. It is essential that the sphere-to-sphere interactions are calibrated, and furthermore, to ensure repeatability it is important that the spheres are given random internal displacements within each platelet to avoid any interlocking effect [37]. A summary of all defined interactions is given in Table 1. To minimise the number of input variables in this initial set of simulations, the boundaries are assumed to have a negative surface charge, and are considered identical to the silica faces of the kaolinite particles. This means that any platelet surface interacting with a boundary experiences the same interaction per unit area as it would with a silica surface of a platelet; although no friction forces act with the boundaries to reduce any potential boundary effects.

Table 1 Summary details of interactions between platelet surfaces
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2.5 Simulation procedure

To explore the behaviour of the numerical clay specimens within the critical state framework, numerical samples are compressed and then sheared in triaxial conditions. As explained above, for all simulations here, the platelets are initially created in aggregated stacks. For simplicity using this new approach, a single stack template is used (shown in Fig. 2), and replicated throughout the sample. Each of these stacks contains a single 1.5 µm diameter platelet, two 1.0 µm diameter platelets, and four platelets with diameters of 0.75 µm.

Three initial samples are used, which are identical except for the varying adhesion (the pull-off force between opposite charged faces). Following creation of the platelets/stacks, the three samples are one-dimensionally compacted, then isotropically compressed, and finally a series of triaxial shear tests are simulated from a range of states.

Figure 6a shows a freshly created sample, consisting of 3234 platelets, in 462 stacks. Each stack is created separately in a sparse arrangement, with a random orientation. The stacks are then allowed to settle freely under gravity, with the stacks and platelets clearly showing a degree of alignment amongst themselves due to the attraction and repulsion between the different faces. Even in the simulation with the weakest pull-off force (adhesion), nearly all stacks remain intact during this stage. Figure 6b shows the same sample as in (a), but after a vertical stress of 10 kPa is applied.

Fig. 6
figure 6

Image showing settling stacks of platelets (a); and the same sample compacted under a vertical stress of 10 kPa (b)

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After vertical compaction (applying a vertical stress of 100 kPa and then unloading), the three samples (with varying adhesion) all exhibit voids ratios of around 2–2.5, corresponding to laboratory prepared samples following the same preparation. The samples are then subjected to isotropic compression and/or shearing, as detailed below.

2.6 Effects of adhesion (pull-off force)

2.6.1 Normal compression

The isotropic normal compression behaviour for the three samples (with pull-off forces of 100, 500 and 1000 nN/µm2) is shown in Fig. 7a. It appears that the strength of adhesion (the pull-off force per area of contact), the only varied parameter, does not noticeably affect the compressibility. It appears to shift the compression curve upwards in e–log p space, without affecting the rate of compression. This can be attributed to the greater adhesion maintaining a looser structure during the initial one-dimensional compaction. However, the stronger samples (greater pull-off forces), appear to also yield later (at higher stresses). This is more evident in Fig. 7b, which shows the number of ‘macro particles’ against applied mean stress. The term ‘macro’ is used to refer to both single platelets and stacks inclusively (a single platelet may be considered as the smallest stack). Using this definition, all samples are created with the same number of macro particles; breakage or splitting of stacks will increase the number of particles, whilst platelets joining together will decrease the number of macro particles. Figure 7b shows that in each simulation, macroscopic yielding and plastic compression occurs simultaneously with a rapid increase in the apparent number of particles. This is analogous to the yielding of coarse-grained materials, in which yielding and plastic compression is caused by particle breakage (the strength of platelet adhesion seems analogous to particle crushing strengths). Given that platelets joining together reduces the number of macro particles, what this shows is that the normal compression of clay is due to the “fracture” of aggregated stacks. The reason why the normal compression of clay and sand are similar is now apparent.

Fig. 7
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Normal compression behaviour in terms of voids ratio and stress (a); number of particles versus applied stress (b)

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Shown in Fig. 7a is also the compressibility, where the slope of the normal compression lines is around Cc = 0.5, and the slope of the unload-reload line is Cs = 0.2. These compare favourably with experimental data for kaolin, with typical values of Cc for kaolin between 0.2–0.4 [38,39,40]. The unload-reload loop shown in one test also demonstrates realistic hysteresis [41, 42].

2.7 Shearing

Triaxial shearing behaviour for the three samples is shown in Fig. 8. These constant volume tests were performed from normally consolidated states of 100 kPa mean stress. All materials demonstrate strain hardening, and overall contractile behaviour (a decrease in mean effective stress). The material with the weakest adhesion (pull-off force of 100 nN/µm2) demonstrates slightly unique behaviour: a rapid loss in mean stress, followed by a partial recovery in deviatoric stress. The two stronger materials do not exhibit such a rapid decrease in mean stress, and their stress paths compare well to experimental data [43, 44]. These two stress paths are reminiscent of data for ‘dispersed’ kaolin samples [45], which typically have a slightly more upright, dilatant stress path compared to more ‘flocculated’ samples. In experiments, the principal difference between ‘dispersed’ (alkaline) and ‘flocculated’ (acidic) samples are their initial density, so one may assume that the more dilative response arises due to the higher density of ‘dispersed’ samples. In this case, the initial density is essentially constant across the three simulations, and the more dilative behaviour appears to arise from the increased strength of adhesion.

Fig. 8
figure 8

Constant volume (undrained) triaxial shear behaviour. Deviatoric stress versus strain (a); and stress paths (b)

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All materials demonstrate final stress ratios of around 1.2–1.4, again, comparing excellently with experimental data [38, 46]. The results indicate that the weakest material (pull-off force of 100 nN/µm2) may exhibit a slightly lower ultimate value (~ 1.2) compared to the stronger two materials. Overall, the shearing behaviour shown here compares more favourably to experimental data than any previous numerical simulations. Previous work, using samples without pre-existing stacks of platelets, consistently show excessive contraction [10], liquefaction, and very low ultimate shear strengths [11].

Some insight to what occurs at the particle level can now be gained by analysing the platelet associations and interactions in the three tests. As can be seen in Fig. 7b, at the start of shearing (p = 100 kPa), each material has a different number of macro particles. It is worth noting here that a platelet is considered to be ‘stacked’ to another platelet if they are in adhesive contact, parallel (± 2°) and the surface separation is less than or equal to 2 nm (Fig. 5).

The material with the weakest adhesion (pull-off force of 100 nN/µm2) has undergone the most degradation of stacks, and so possesses the most macro particles; the strongest material (1000 nN/µm2) has the fewest. Interestingly, in each of the three triaxial tests, the number of stacks remain approximately constant—however these aggregated stacks lose and gain platelets during the course of shearing. Figure 9a shows the number of single (non-stacked) platelets during the tests, and Fig. 9b shows the average macro particle size. The weakest material is unique in that as soon as shearing begins there is a decrease in free single platelets; this coincides with the rapid drop in mean stress. As shearing progresses, there is a subsequent growth in the number of single, non-stacked platelets. The stronger materials both show a slow and small increase in single platelets. From this data, as well as from quantifying the number and type of interactions, it is clear that in the case of weaker adhesion, platelets can more easily dislodge and rearrange themselves within the sample. It appears that as soon as the sample is subjected to shear deformation, the many single platelets are able to move to more favourable positions easily, which likely contributes to the drop in mean stress. The ‘phase transformation point’ as it would be called in a sand, corresponding to dp/dq = 0 in Fig. 8b for the weakest sample, corresponds to the onset of plastic dilation: the volume is constant and beyond this point p is increasing (elastic volume reduction) and plastic volume increase occurs. This point appears to coincide with when the number of free platelets reaches a minimum.

Fig. 9
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Number of single platelets in sample versus strain (a); average macro particle size (b)

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2.8 Micro mechanical analysis

2.8.1 Normal compression

Focussing on the intermediate simulation—which models a pull-off force of 500 nN/µm2—several further particle-scale mechanisms can be analysed. The primary mechanism for plastic compression is deformation of the stacks. After vertical compaction (and before normal compression), the material forms a stable, ‘jammed’ structure. The only way for platelets to rearrange into a denser packing is therefore by deformation of the bulky stacks, and/or disintegration. Figure 10 shows the distribution of stack sizes before and after compression. At the start of compression (10 kPa), the sample is mainly composed of stacks comprising seven platelets, due to how the sample was created. After isotropic compression to 1000 kPa, one can see a distribution of sizes has emerged, with stack size inversely correlating with frequency. Tentative evidence suggests that if a much larger sample was used (if computationally possible), it is likely a fractal distribution of sizes would emerge.

Fig. 10
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Size distribution of macro particles before and after compression

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Figure 11 shows cross-sectional images of the sample before and after compression. In this figure, single platelets are coloured white and all stacks are shaded; different shading is used to differentiate between stacks. At 10 kPa, in Fig. 11a, large stacks are visible in which the platelets appear perfectly parallel, and there are very few single platelets. In Fig. 11b, at 1000 kPa there are many more single platelets, and the stacks are not as large. In the lower right of these figures, it can be seen that some stacks remain perfectly intact, and appear only to experience a vertical displacement. However, the largest stacks in (a) clearly disintegrate into smaller stacks, with the platelets becoming non-aligned.

Fig. 11
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Internal cross section on vertical plane, before (a) and after (b) isotropic compression. White platelets are single, unstacked platelets. Shaded platelets are those forming larger stacks, shading is used to differentiate different stacks

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By considering which constituent spheres within each platelets are interacting with another platelet, it is possible to divide the total surface area of the platelets into available surface area, and unavailable area (in contact). The available surface area of the material during the course of normal compression is shown in Fig. 12. The disintegration or deformation of any stack must expose new surface area from between the platelets. The plot shows that before yielding, there is no change; then an increase in surface area as the stacks degrade and new area is exposed; and then a subsequent decrease as the overall number of interactions increases. It is interesting to note that the available surface area of the material shows no significant overall change during compression, despite the apparent degradation of stacks. This can be queried by analysing the area in contact–referring to all surface(s) of platelets which are interacting with any other platelet or boundary. This area can be categorised as either internal to a stack or external. Figure 12 shows that this new surface area exposed through the degradation of stacks is taken up by the formation of new contacts. These contacts are between non-parallel platelets, and maybe attractive or repulsive. So although bulky stacks are degrading, platelets are becoming more disordered and more closely packed.

Fig. 12
figure 12

Plots showing available particle surface area (top) and different types of contact area (bottom)

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It is also possible to categorise the physical type of interactions between platelets as either face-to-face, edge-to-face or edge-to-edge [11]. In these simulations, there is an overall trend of an increase in edge-to-face interactions during compression. An interesting finding from the authors’ previous work was that a characteristic measure of separation between edge-to-face contacts appeared to correlate with the macroscopic porosity, and this also seems to be the case here, as shown in Fig. 13a. The separation of any edge-to-face contact can be measured as the shortest distance between the centre of mass of one platelet and the surface of the other, as shown in the figure. The median separation, across all edge-to-face interactions, appears to correlate linearly with the voids ratio. This is consistent with the notion speculated from experimental observations using SEM and mercury intrusion porisometry, that the reduction in voids ratio is due to a collapse of edge-to-face contacts.

Fig. 13
figure 13

Voids ratio as a function of edge-to-face separation (a); and frequency distribution of edge-to-face separations (b)

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The reason behind this correlation may be speculated that face-to-face interactions, by their definition, are parallel platelets with a very small separation and no significant space between them. Edge-to-edge contacts, which are few in number, similarly don’t enclose any volume. Edge-to-face contacts are unique in that they may partially enclose an internal volume, of variable size, and therefore it seems reasonable that these types of platelet contacts may tell us something about the overall porosity. Counting each edge-to-face contact in the sample, Fig. 13b shows the distribution of separations. This plot shows data before and after compression, and corresponds very well to pore size distributions estimated using mercury intrusion porosimetry [18, 47]. The modal separation or ‘pore size’ stays the same before and after compression, but the frequency decreases and there are more smaller pores after compression.

2.9 Normally consolidated and overconsolidated shearing

Continuing with the intermediate material with a pull-off force of 500 nN/µm2, further shear tests are now shown from both normally consolidated and overconsolidated states. In addition to the normally consolidated triaxial test from p = 100 kPa shown above, a further such test from p = 400 kPa, and two overconsolidated tests from p = 100 kPa with OCRs of 5 and 10 were performed. The deviatoric responses are shown in Fig. 14, and Fig. 15 shows the stress paths in terms of e, q, and p. Figure 15a shows the emergence of a critical state line, parallel with the normal compression line and linear in e–log p space, and shows clearly dilatant behaviour for the overconsolidated samples. The critical state line is also visible in Fig. 15b, and reveals a critical state constant M (the ultimate stress ratio) of around 1.5, agreeing excellently with typical experimental values [40, 46]. Although the deviatoric (and mean) stresses appear to be still increasing at the end of the tests, the stress ratio M has stabilised and contractile and dilatant trends are clear; and it is only computational constraints that prevented continuing simulations (and indeed performing more). The two normally consolidated tests show very similar-shaped stress paths, as one would expect for real clays [43]. The heavily overconsolidated stress paths shows clear dilatant behaviour, with the stress path rising almost vertically in q-p space before then showing an increase in mean stress at an approximately constant stress ratio. This behaviour is well known for heavily overconsolidated clays: the stress path usually is seen to be approximately at constant p and constant volume (with the implication of approximate isotropic elasticity) until yielding on the Hvorslev surface before the stress state heads to a critical state. It is worth pointing out that such dilatant behaviour has not before been captured in any particle-scale model of clay, and that these results therefore present the state-of-the-art in capturing realistic clay behaviour. The only occasion in which a critical state line has been achieved in a numerical model, only very mildly dilatant behaviour was observed [11].

Fig. 14
figure 14

Deviatoric stress responses from normal and overconsolidated triaxial tests

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Figure 16 shows vertical cross sections of a sample before-and-after triaxial shearing. The presence of stacks and localised alignment of platelets is visible in both. Although not immediately visible, after shearing the platelets are more uniformly aligned horizontally. This is confirmed in Fig. 17, which plots the platelet orientations with respect to the horizontal plane before and after compression, and then after shearing. This plot reveals an overarching trend, with platelets tending to orientate themselves between 20°–30° to the horizontal. The greatest change appears to occur during shearing, and corresponds closely with interpretations of SEM images [48].

Fig. 15
figure 15

Stress paths in voids ratio-stress space (a); and in terms of q-p (b)

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Fig. 16
figure 16

Images before (a) and after (b) triaxial shearing.

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Fig. 17
figure 17

Frequency plot showing platelet orientations with respect to the horizontal plane

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3 Conclusions

Using the discrete element method, this paper demonstrates the very latest and most realistic particle-scale model of clay. The importance of how the sample is created is highlighted, and it is clearly shown that this has a significant impact on how the model behaves. It appears conclusive that creating platelets in aggregated stacks leads to greatly improved macroscopic behaviour, closely resembling real clays and explaining why, for the first time, the mechanical plastic behaviour of clay is similar to sand, albeit with very different particle interaction laws. By creating pre-formed stacks of platelets, it becomes possible to explore and investigate the influence of adhesion (the pull-off force) between platelets, and this was shown to affect the compressive yield strength but not the compressibility. The attraction between single platelets is not something that can be measured directly in real clays, however real platelet interactions can be adjusted by controlling the environment—for example by adjusting the fluid pH or electrolyte content. As such, the findings here offer use in explaining observed behaviour.

Creating the sample using stacks (consistent with images obtained from clay slurries), allowed more realistic interactions to be modelled, and ultimately led to achieving quantitatively correct behaviour. The behaviour demonstrated here—the shear strength, compressibility, and dilation—all agree well with experimental data for kaolin, and are the most realistic behaviour achieved to date in a particle-scale model. In particular, it has been shown that normal compression of clay appears to be the ‘fracture’ of aggregated stacks and the evolution of a distribution of pore sizes (edge to face spaces) and that a critical state line and Hvorslev-type yielding for heavily overconsolidated clays can be observed.