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Journal Membership for August 2023: Attractors in harassed granular supplies

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Yida Zhang

Assistant Professor

Division of Civil, Environmental and Architectural Engineering, College of Colorado Boulder, Boulder, CO, 80303

Analysis lab: https://www.yidazhanggroup.com/

1. Introduction

An attractor is a set of states towards which a dynamic system tends to evolve, for all kinds of beginning situations of the system (Wikipedia). System values that get shut sufficient to the attractor values stay shut even when barely disturbed. A well-known instance is the Lorenz attractor: A system of abnormal differential equations initially proposed to explain the atmospheric convection (Lorenz, 1963) reveals chaotic behaviors, i.e., barely totally different preliminary situations result in essentially totally different options shortly after the beginning of the evolution. Nevertheless, all of them seemed to be attracted by however by no means attain sure factors within the part area, producing the well-known butterfly form (Fig. 1). The notion of an attractor gives an vital means to characterize chaotic, nonlinear, and complicated techniques.

 

Determine 1: An answer within the Lorenz attractor (Wikipedia)

Granular supplies are advanced. They include many easy subunits (i.e., grains) interacting with one another of their shut neighborhood by friction, rotation, interlocking, crushing, and abrasion. Emerged from these lower-scale interactions are the nonlinear and adaptative mechanical response of granular supplies: Its stress-strain relation is very nonlinear and historical past dependent; The material of granular assemblies (e.g., the orientations of contact, grain, and void) consistently evolves in adaptation to the utilized exterior stresses; In high-stress regime the place grain breakage can happen, the grain measurement and form can even self-organize to realize energetic and survival benefits. As a result of wealthy nonlinear interactions between grains, a predictive constitutive concept based mostly on the only real properties of the person particles has not been proposed but, though pc packages resolving the grain-scale interactions (e.g., discrete factor methodology) can efficiently mimic the macroscopic mechanical behaviors of granular supplies, i.e., the case of weak emergence (Bedau, 1997).

Identical to many different advanced techniques, granular supplies possess a number of distinct attractors that may be considered their “genes”. These attractors are extraordinarily helpful in constitutive theories to portrait the macroscopic mechanical response of granular supplies. On this article, I’ll try a quick assessment of some experimentally and numerically recognized attractors in granular techniques. The objective is to summarize the varied attractor-like ideas studied in soil mechanics, granular physics, and engineering mechanics communities, and hopefully to spark some discussions and inspirations among the many readers.

2. Vital State

The primary attractor-like idea discovered for sheared granular supplies is the Vital State Line (CSL) outlined within the void ratio e – imply stress p – deviatoric stress q area (Fig. 2). It states that each one granular supplies, when monotonically sheared to giant strains, will attain a stationary void ratio and stress ratio within the epq area. The vital state situation may be mathematically said by:

          q/p=M; e=e_c(p)          (1)

the place e is the void ratio; p the imply stress (efficient imply stress for saturated soils); q the deviatoric stress.

Determine 2: Vital state line within the e-p-q area (Wooden, 1990).

Casagrande (1936) was the primary to notice that ‘each cohesionless soil has a sure vital, through which state it could endure any quantity of deformation or precise movement with out quantity change.’ Wroth (1958) performed easy shear take a look at on 1-mm diameter metal balls and confirmed the emergence of a CSL within the epq area. The existence of CSL was then systematically recognized from experiments on reconstituted clays (Alan Bishop and Henkel, 1957; Roscoe et al., 1958). Subsequently, extra supporting knowledge have been collected on pure sand (Verdugo and Ishihara, 1996; Vesic and Clough, 1968) and rockfill (Marachi et al., 1972). The distinctiveness of CSL was additionally confirmed by discrete factor methodology (DEM) simulations (Huang et al., 2014; Nguyen et al., 2018).

Constitutive modeling. The universality and robustness of CSL drove the institution of the Vital State Soil Mechanics (CSSM) concept (Schofield and Wroth, 1968; Wooden, 1990). CSSM has impressed quite a few soil constitutive fashions that make the most of CSL as their basic constructing block. Most of them are constructed following the structure of elastoplasticity (Gajo and Wooden, 1999; Jefferies, 1993; Manzari and Dafalias, 1997). Some others are formulated underneath hypoplasticity (Niemunis and Herle, 1997; von Wolffersdorff, 1996), thermodynamics with inner variables (TIV) (Collins and Muhunthan, 2003; Houlsby and Puzrin, 2007), or hydrodynamic concept (Jiang and Liu, 2015). 

3. Vital Material

There’s a rising consciousness {that a} extra full definition of vital state must make reference to the interior construction of granular supplies. In actual fact, it’s doable to indicate that the distinction between the precise void ratio and the vital state void ratio doesn’t absolutely decide the stress-strain evolution of the system (Theocharis et al., 2019). Many research discovered that sands ready to the identical preliminary void ratio reply in a different way underneath totally different shearing modes (Mooney et al., 1998; Vaid and Thomas, 1995; Yoshimine et al., 1998), implying the numerous impression of cloth anisotropy on the conduct of granular soils. Li and Dafalias (2012) proposed that, along with the void ratio and stress ratio, the second-rank cloth tensor should additionally attain its vital worth at vital state, i.e.:

          q/p=Me=e_c(p); F=F_c          (2)                                                                                                                    

the place F is the normalized deviatoric cloth tensor (i.e., tr(F)=0). This framework doesn’t constrain the definition of cloth tensors nor restrict the type of constitutive legal guidelines. Subsequently, it’s considered an extension of the unique CSSM and known as the Anisotropic Vital State Concept (ACST).

Many DEM and laboratory experiments have been adopted to check the speculation of a singular cloth attractor for granular soils. In most of those works, granular microstructure is represented by a traceless second-order cloth tensor outlined by both contact norms, void vectors, or the particle orientations. Fu and Dafalias (2011) and Wang et al. (2017) performed two-dimensional (2D) DEM simulations of elongated particle assemblies, confirming {that a} distinctive regular state is reached for every of those three cloth tensor definitions, no matter the preliminary void ratio and the orientation and depth of cloth anisotropy. Zhao and Guo (2013) performed a collection of true triaxial three-dimensional (3D) DEM exams and reported that the contact-based cloth tensor reaches a well-defined final envelop within the principal cloth area, the form of which is analogous however reciprocal to the vital stress envelop alongside the deviatoric airplane. They additional demonstrated that the primary joint invariant of the stress and cloth tensors at vital state Kc is unbiased of Lode angle. Nguyen et al. (2016, 2018) performed three-dimensional (3D) DEM triaxial exams and confirmed that the deviatoric part of contact-based cloth tensor at vital state is uniquely associated to p and e. As well as, they discovered that the coordination quantity CN at vital state is just a operate of p and unbiased of drainage situations and consolidation strategies. Kruyt and Rothenburg (2014) by 2D DEM research demonstrated that CN and contact-based cloth anisotropy A at vital state are capabilities of the interparticle friction and confining stress. When plotting these cloth states within the “cloth area” (i.e., A vs. CN plot), a vital cloth line may be recognized for the studied granular system. Zhu et al. (2016) and Deng et al. (2021) confirmed that the vital state void ratio and cloth descriptors achieved in a shear band at localized failure are an identical to these obtained by a diffusive failure sample. 

Lately, my group repeated the true triaxial DEM simulations of Zhao and Guo (2013) however prolonged the simulation to extraordinarily free sand underneath undrained shear and employed a brand new cloth tensor definition that preserves the hydrostatic part (as an alternative of the traceless tensor utilized in Eq. (2) and different works)(Wen and Zhang, 2022a). An attention-grabbing discovering is that even liquefied (or “unjammed” in granular physics neighborhood), apparently structureless granular assemblies exhibit a singular vital cloth after adequate shear. These vital cloth knowledge together with that of jammed granular soils makes a whole 3D vital cloth floor (Fig. 3), providing a well-defined, mathematically steady attractor for granular supplies within the principal cloth area. Constructing upon this preliminary discovery, we examined the vital cloth knowledge of quite a lot of granular packings which can be initially dense vs. free, isotropic vs. anisotropic, polydisperse vs. bi-disperse, with contact regulation being Hertz Mindlin vs. linear elastic, shear mode being easy shear vs. triaxial, monotonic vs. cyclic, boundary-driven shear vs. athermal quasistatic shear with Lees-Edwards boundary (Wen and Zhang, 2022a, b, 2023). All techniques exhibit comparable CFS at vital state, confirming the robustness of this attractor and its insensitivity to protocol/ system variations.

   

Determine 3: Left: vital cloth floor recognized by true triaxial shear on a polydisperse meeting (Wen and Zhang, 2022a). Knowledge factors within the determine signify the vital cloth knowledge of samples with totally different preliminary states. Proper: cloth evolution paths of initially unjammed, bi-disperse granular samples (Wen and Zhang, 2023). Relying on the preliminary density (indicated by the colour bar), some samples stay unjammed all through the shearing course of, whereas others develop sure shear stress and are thus thought-about to be shear jammed. All cloth paths converge to the CFS at regular state.

In distinction to DEM, in-situ experimental dedication of cloth evolution requires far more effort when it comes to take a look at designs and knowledge postprocessing. Oda (1972a, b) studied the cross sections of resin-impregnated sand specimens and located a powerful correlation between cloth anisotropy and stress ratio of triaxially loaded sand specimens. A extra frequent methodology of quantifying materials microstructure these days is the in-situ X-ray microtomography approach (X-μCT) (Andò et al., 2012; Desrues et al., 1996). On this method, miniature triaxial or oedometric exams have been performed on an X-μCT platform. X-ray scanning and mechanical loading have been alternated to acquire snapshots of the specimen’s inner construction all through the deformation course of. This X-ray picture dataset then requires large effort in put up processing to transform to bodily smart 3D fashions of the granular meeting, by which specimen’s cloth statistics may be extracted. Utilizing X-μCT, Imseeh et al. (2018) noticed that the contact-based cloth tensor certainly reaches a steady-state worth coinciding with the vital state situation in triaxial exams. Nevertheless, the evolution of cloth in the direction of such an attractor is discovered to be non-monotonic for some specimens. Wiebicke et al. (2020) and Zhao et al. (2021)  distinguished cloth evolution inside and outdoors of the shear band, and located that cloth anisotropy and coordination quantity method distinctive values at giant pressure (inside the band), no matter the preliminary cloth void ratio. In abstract, each DEM and experimental findings help the existence of an attractor within the cloth area for granular supplies. 

Constitutive modeling. Impressed by these findings, the neighborhood is now seeing a brand new spherical of developments on sand constitutive fashions acknowledging cloth evolution (Dafalias and Taiebat, 2016; Papadimitriou et al., 2019; Petalas et al., 2020; Tasiopoulou and Gerolymos, 2016; Wang et al., 2021; Zhang et al., 2020; Zhao and Gao, 2015). It’s value noting that the majority experimental and DEM research have centered on cloth descriptions based mostly on contact regular vectors. It’s cheap to imagine that different cloth descriptors comparable to these outlined on particle and void cell orientations additionally converge to regular state values at giant shear strains. Nevertheless, there’s a shortage of experimental knowledge to substantiate this hypothesis or quantifying their evolution sample in the direction of the vital state.  

4. Grain Dimension

The grain measurement distribution (GSD) of pure soil is consistently evolving resulting from crushing, agglomeration, mixing, and segregation (Foley, 2018; Johnson et al., 2012; Wang et al., 2002). Understanding the grain measurement dynamics throughout these processes is of nice engineering and scientific significances. In geotechnical engineering, grain measurement distribution supplies the primary indices characterizing and classifying sands. It has first-order impacts on the shear energy, deformability, hydraulic conductivity, and erodibility of granular soils (Kenney and Lau, 1985; Singh et al., 2021; Wooden and Maeda, 2008; Yang and Gu, 2013). In geoscience, grain measurement distribution encodes the historical past of earth’s or different planets’ floor (Anderson and Bunas, 1993; McKay et al., 1974). Every technique of crushing, agglomeration, mixing, and segregation drive a special mode of GSD evolution, and thus troublesome to establish common tendencies if they’re mentioned altogether. Herein, this text will concentrate on the grain measurement evolution dominated by grain crushing, often underneath excessive stress conditions that may be seen at many size scales (Fig. 4). 

Determine 4: Grain breakage throughout scales.

Quantitative descriptions of the diploma of grain crushing and monitoring grain measurement evolution was primarily tried in geotechnical engineering and geomorphology communiteis. Hardin (1985) hypothesized that each one particles in a pattern of soil may very well be crushed to fines (particles with measurement lower than 0.074 mm) underneath sufficiently excessive stresses. This was, nevertheless, not help by the remark that fault gouge supplies which have skilled excessive compression and shearing exhibit a self-similar GSD characterised by a fractal dimension of two.6 (Sammis et al., 1987), corroborating the fragmentation concept of Turcotte (1986):

          N(x>d)=Advert^(-α)          (3)

the place N is the variety of particles with diameter bigger than d; A is a continuing of proportionality; α is the fractal dimension. The existence of a GSD attractor described by Eq. (3) with a common fractal dimension (2.5~2.7 for 3D, 1~1.3 for 2D) for severely harassed granular supplies was later supported by a growth of proof from laboratory investigations (Coop et al., 2004; McDowell and Bolton, 1998; Nakata et al., 2001a), DEM simulations (Ben-Nun and Einav, 2010; McDowell and de Bono, 2013), and geological observations (Billi, 2005; Marone and Scholz, 1989). See Fig. 5 (left) for an instance. It’s value highlighting that the final word fractal GSD seemed to be insensitive to minor alterantions of the preliminary properties of the packing nor the totally different modes of fragmentation (Ben-Nun and Einav, 2010), hinting that the collective breakage of grain meeting is a strong self-organized course of (Bak, 2013). In actual fact, Fig.5 (left) resembles the rank vs. frequency traits of many different advanced techniques that exibits self-organization, comparable to metropolis measurement (Zipf’s regulation), earthquake (Gutenberg-Richter regulation), and many others. See Fig. 5 (proper) for instance. 

 

Determine 5: Left: GSD of gouge and breccia supplies collected from the Mattinata fault (Billi, 2005). Proper: Variety of cities through which the inhabitants exceeds a given measurement or, equivalently, the relative rating of cities vs. their inhabitants round 12 months 1920 (Bak, 2013; Zipf, 2016).   

The sample of GSD evolution may be additionally considered mechanistically because of dynamic competitors between two micromechanisms: (1) smaller particles can stand up to greater deviatoric stress and thus are tougher to interrupt (Kendall, 1978; McDowell, 2001; Nakata et al., 2001b); (2) throughout collective breakage, giant particles get surrounded and supported by smaller particles (i.e., the so-called cushioning impact) and turn into much less more likely to break (Ben-Nun et al., 2010; Tsoungui et al., 1999). This explains the existence of an final GSD and packing configuration the place the chance of crushing of any of the particles within the system are equal and asymptotically approaches zero as confining stress will increase.

It’s value to say a number of latest observations which will problem the universality of the final word GSD. Particularly, gap-graded soils appear to “keep in mind” their preliminary GSD even after being loaded to very giant stress ranges (Zhang and Baudet, 2013; Zhang et al., 2017) (Fig. 6 left). Whether or not such a result’s a pure consequence of self-organized crushing, or there are some unknown mechanisms that interrupt such self-organization for gap-graded soils, is at the moment not clear. It has additionally been identified that, though the final word GSDs generated from extreme shearing and high-pressure compression are each fractal, they seem to have totally different fractal dimensions even for a similar sand (Miao and Airey, 2013). Gradings ensuing from shear seem like total finer than these from compression (Fig. 6 proper), which can be defined by the upper mobility of grains throughout shearing. Extra knowledge and investigations are wanted to additional perceive these deviations. 

 

Determine 6: Left: Hole-graded soils don’t exhibit a mono-fractal GSD after crushing (Zhang et al., 2017). Proper: samples harassed underneath oedometric compression and ring shear exams develop totally different final GSDs (Miao and Airey, 2013).

Constitutive modeling. The identification of an attractor within the GSD airplane has facilitated the event of many continuum fashions for crushable granular supplies. The main concept on this regard is the Continuum Breakage Mechanics (CBM) (Einav, 2007a, b). The recognized GSD attractor is used to outline a brand new inner state variable referred to as breakage that varies from 0 to 1 within the technique of crushing (Fig. 7). Such enrichment permits the coupling of the vitality, microstructure, and stress-strain response of the granular meeting. The thermomechanical formulation of CBM bears some similarity with continuum injury mechanics (CDM) for brittle solids and can also be rooted within the concept of linear elastic fracture mechanics (Einav, 2007c). Inside the framework of CBM, quite a lot of mechanical and bodily properties of crushable granular supplies at the moment are probably modelled in quantitative method. To call a couple of, these fashions cowl subjects on environment-dependent fragmentation of rock aggregates (Shen and Buscarnera, 2022a; Zhang and Buscarnera, 2018), breakage-induced creep and rest (Alaei et al., 2021; Zhang and Buscarnera, 2017), high-strain charge comminution and penetration (Kuwik et al., 2022), grain-size dependent yielding (Zhang et al., 2016), water retention and permeability evolution (Esna Ashari et al., 2018; Gao et al., 2016; Singh et al., 2021), pressure localization in granular rocks (Collins-Craft et al., 2020; Das et al., 2013; Nguyen and Einav, 2010; Tengattini et al., 2014), anisotropy and fabric-dependent breakage (Marinelli and Buscarnera, 2019; Shen and Buscarnera, 2022b). CBM fashions that acknowledges or predicts different attractors such because the vital state line in epq area have been additionally proposed (Tengattini et al., 2016). 

Determine 7: Left: Definition of breakage variable B. Proper: predicted GSD vs. experimental knowledge. (Einav, 2007a)

5. Grain Form

Grain form coevolves with grain measurement throughout fragmentation. Growing proof has pointed in the direction of the existence of an attractor for the grain form, in complementary to the GSD attractor, for crushed granular supplies. Admittedly, analysis on grain form evolution in fragmentation/ comminution continues to be at early stage in comparison with GSD evolution research, and plenty of conclusions will not be particular and require additional validations. The scenario can also be sophisticated by the big variety of grain form descriptors proposed up to now (Anusree and Latha, 2023). This part will try to summarize a couple of research on this regard and briefly introduce their implications to constitutive modelling of granular supplies.

Within the broad context of fragmentation dynamics, Domokos et al. (2015) discovered that rock fragments, whether or not generated by slowly evolving weathering or from fast breakup induced by explosion and hammering, reveals a self-similar form distribution quantified by surface-to-volume ratio (Fig. 8). The dominating elongation ratio and the flatness ratio of fragmented particles exhibit dependency on grain measurement, i.e., bigger particles are usually extra rounded. The existence of such form attractor is elegantly defined by a discrete stochastic mannequin of fragmentation (Domokos et al., 2020). Different grain form attractors created by totally different mechanical processes and geological settings from river pebbles (Novák-Szabó et al., 2018) to boulders on asteroids (Michikami et al., 2010) have additionally been recognized within the literature. 

Determine 8: Likelihood distribution of the form parameter for fragments generated by weathering and hammering (Domokos et al., 2015)

The universality of grain form may be additionally noticed from granular assemblies collectively crushed underneath high-pressure compression and shear. Search engine marketing et al. (2020) performed oedometric compression together with in-situ X-μCT on two quartz sands with totally different grain morphologies. They discovered that steady compression can mitigate morphological variations, particularly when the stress is adequate to induce pervasive breakage (See Fig. 9). Miao and Airey (2013) studied the final word states of a carbonate sand underneath totally different stress situations. Each GSD and grain form evolves in the direction of a gentle state after adequate loading. Nevertheless, samples subjected to steady shearing (by a hoop shear gadget) produces an final grain form that has a barely greater facet ratio than that obtained from high-pressure oedometric compression exams. Ueda et al. (2013) carried out 2D DEM simulation of oedometric compression exams on granular assemblies with preliminary shapes starting from good circle to elongated hexagon. They noticed that each one samples arrived at a steady facet ratio after crushed to close final state, regardless of the presence of drastically totally different crushing modes comparable to cleavage destruction, bending fracture, and edge abrasion (See Fig. 9). By means of a novel hybrid peridynamics (PD) and non-smooth contact dynamics (NSCD) simulation of oedometric crushing course of, Zhu and Zhao (2021) noticed that the distributions of a number of form components (elongation, flatness, facet ratio) method to a gentle profile which may be approximated by a standard or Weibull distribution, accompanied with the discount of median grain sizes. Ma et al. (2019) performed a mixed finite and discrete factor methodology (FDEM) research of the identical course of and reported comparable conclusions for different form descriptors comparable to surface-volume ratio, sphericity, and convexity.  Their simulation signifies that the dominate type of grain breakage is the splitting of particles into a number of fragments of comparable measurement on the onset of yielding, whereas it adjustments to the abrasion of native asperities with additional enhance of stress post-yielding. That is according to the DEM research of easy shear mimicking the situation of faut gouges  Mair and Abe (2011), the place a faster decay of grain splitting than grain abrasion as a operate of shear pressure is noticed. Lastly, it’s value to say the idea of form most well-liked orientation (SPO) of survivor grains in fault gouge (Cladouhos, 1999). The truth that SPO may be recognized and used to deduce the earlier kinematic historical past of the fault hints that the evolution of grain measurement, grain form, and cloth are intimately coupled for granular supplies underneath extreme shear. Evaluation of fault gouge supplies counsel that the grain form is very depending on the mineral (e.g., quartz, feldspar) for bigger grains, however the variations diminish for smaller grains (Heilbronner and Keulen, 2006).  

Constitutive modeling. In soil mechanics, grain form is thought to impression the mechanical properties of sand together with stiffness, energy, and packing density (Alshibli and Cil, 2018; Cho et al., 2006; Liu and Yang, 2018). The identification of a possible grain form attractor offers one other motivation to include grain form descriptors in soil constitutive fashions. Nevertheless, as of now, fashions that incorporating grain form and its dynamic evolution throughout breakage are nonetheless uncommon. To my data, the one try is made lately by Buscarnera and Einav (2021). They confirmed that such form attractors may be integrated in CBM through extra shape-related inner state variables (ISVs). The form ISVs are launched within the saved elastic vitality of the granular meeting in a method just like the breakage variable. This dependency is motivated by assuming a linear scaling between the elastic pressure vitality and the floor space of the particles. Perturbation of the grain form thus causes vitality launch or achieve of the system, which should be balanced with the vitality dissipation and the exterior work enter to the fabric factor. By proposing a coevolution regulation between grain form and measurement, the evolution of grain form in the direction of an final attractor is predicted all through the course of loading. The idea has a one-descriptor formulation based mostly on facet ratio and a two-descriptor model utilizing the elongation ratio and flatness ratio (i.e., the Zingg airplane) as form ISVs. The expected form evolution path agrees nicely with experimental and DEM knowledge from sands with totally different preliminary grain morphologies (Fig. 9). It’s anticipated that, as rising experimental knowledge on form attractors emerge, extra theoretical developments will try to include grain form within the continuum description of granular supplies.

Determine 9: Grain form evolution predicted by the CBM mannequin of Buscarnera and Einav (2021) in comparison with DEM knowledge (Left) (Ueda et al., 2013) and experimental knowledge (proper) (Search engine marketing et al., 2020)

6. Concluding Remarks

Attractors disclose vital info for in any other case intangible and complicated techniques. They supply the backbones for phenomenological macroscopic fashions to explain the system properties with out resorting to the detailed interplay and properties of the constituting subunits. Actually, far more must be achieved to raised quantify these attractors and perceive how totally different preliminary states evolve in the direction of their regular states for harassed granular supplies. It’s envisioned that new theories or analysis methodology for advanced techniques and self-organization may convey new insights into granular mechanics analysis, if correctly mixed with the extra conventional continuum mechanics method.

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