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 circumstances 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 extraordinary differential equations initially proposed to explain the atmospheric convection (Lorenz, 1963) displays chaotic behaviors, i.e., barely totally different preliminary circumstances 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 section house, producing the well-known butterfly form (Fig. 1). The notion of an attractor presents an vital means to characterize chaotic, nonlinear, and complicated methods.
Determine 1: An answer within the Lorenz attractor (Wikipedia)
Granular supplies are complicated. They include many easy subunits (i.e., grains) interacting with one another of their shut neighborhood by way of 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 dimension and form also can self-organize to realize energetic and survival benefits. As a result of wealthy nonlinear interactions between grains, a predictive constitutive idea primarily based on the only properties of the person particles has not been proposed but, though laptop applications resolving the grain-scale interactions (e.g., discrete ingredient methodology) can efficiently mimic the macroscopic mechanical behaviors of granular supplies, i.e., the case of weak emergence (Bedau, 1997).
Identical to many different complicated methods, granular supplies possess a number of distinct attractors that may be thought to be 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 methods. The objective is to summarize the assorted 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. Essential State
The primary attractor-like idea discovered for sheared granular supplies is the Essential State Line (CSL) outlined within the void ratio e – imply stress p – deviatoric stress q house (Fig. 2). It states that every one granular supplies, when monotonically sheared to massive strains, will attain a stationary void ratio and stress ratio within the e–p–q house. The essential state situation could be mathematically acknowledged 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: Essential state line within the e-p-q house (Wooden, 1990).
Casagrande (1936) was the primary to notice that ‘each cohesionless soil has a sure essential, through which state it might endure any quantity of deformation or precise circulate with out quantity change.’ Wroth (1958) carried out easy shear check on 1-mm diameter metal balls and confirmed the emergence of a CSL within the e–p–q house. 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 information 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 ingredient methodology (DEM) simulations (Huang et al., 2014; Nguyen et al., 2018).
Constitutive modeling. The universality and robustness of CSL drove the institution of the Essential State Soil Mechanics (CSSM) idea (Schofield and Wroth, 1968; Wooden, 1990). CSSM has impressed quite a few soil constitutive fashions that make the most of CSL as their elementary 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 inside variables (TIV) (Collins and Muhunthan, 2003; Houlsby and Puzrin, 2007), or hydrodynamic idea (Jiang and Liu, 2015).
3. Essential Material
There’s a rising consciousness {that a} extra full definition of essential state must make reference to the interior construction of granular supplies. In truth, it’s potential to point out that the distinction between the precise void ratio and the essential state void ratio doesn’t totally 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 influence of material anisotropy on the conduct of granular soils. Li and Dafalias (2012) proposed that, along with the void ratio and stress ratio, the second-rank material tensor should additionally attain its essential worth at essential state, i.e.:
q/p=M; e=e_c(p); F=F_c (2)
the place F is the normalized deviatoric material tensor (i.e., tr(F)=0). This framework doesn’t constrain the definition of material tensors nor restrict the type of constitutive legal guidelines. Subsequently, it’s thought to be an extension of the unique CSSM and known as the Anisotropic Essential State Concept (ACST).
Many DEM and laboratory experiments have been adopted to check the speculation of a novel material attractor for granular soils. In most of those works, granular microstructure is represented by a traceless second-order material tensor outlined by way of both contact norms, void vectors, or the particle orientations. Fu and Dafalias (2011) and Wang et al. (2017) carried out two-dimensional (2D) DEM simulations of elongated particle assemblies, confirming {that a} distinctive regular state is reached for every of those three material tensor definitions, no matter the preliminary void ratio and the orientation and depth of material anisotropy. Zhao and Guo (2013) carried out a sequence of true triaxial three-dimensional (3D) DEM checks and reported that the contact-based material tensor reaches a well-defined final envelop within the principal material house, the form of which is analogous however reciprocal to the essential stress envelop alongside the deviatoric aircraft. They additional demonstrated that the primary joint invariant of the stress and material tensors at essential state Kc is impartial of Lode angle. Nguyen et al. (2016, 2018) carried out three-dimensional (3D) DEM triaxial checks and confirmed that the deviatoric element of contact-based material tensor at essential state is uniquely associated to p and e. As well as, they discovered that the coordination quantity CN at essential state is barely a perform of p and impartial of drainage circumstances and consolidation strategies. Kruyt and Rothenburg (2014) by way of 2D DEM research demonstrated that CN and contact-based material anisotropy A at essential state are capabilities of the interparticle friction and confining strain. When plotting these material states within the “material house” (i.e., A vs. CN plot), a essential material line could be recognized for the studied granular system. Zhu et al. (2016) and Deng et al. (2021) confirmed that the essential state void ratio and material descriptors achieved in a shear band at localized failure are equivalent to these obtained by way of a diffusive failure sample.
Not too long ago, my group repeated the true triaxial DEM simulations of Zhao and Guo (2013) however prolonged the simulation to extraordinarily unfastened sand underneath undrained shear and employed a brand new material tensor definition that preserves the hydrostatic element (as an alternative of the traceless tensor utilized in Eq. (2) and different works)(Wen and Zhang, 2022a). An fascinating discovering is that even liquefied (or “unjammed” in granular physics neighborhood), apparently structureless granular assemblies exhibit a novel essential material after enough shear. These essential material information together with that of jammed granular soils makes an entire 3D essential material floor (Fig. 3), providing a well-defined, mathematically steady attractor for granular supplies within the principal material house. Constructing upon this preliminary discovery, we examined the essential material information of a wide range of granular packings which are initially dense vs. unfastened, isotropic vs. anisotropic, polydisperse vs. bi-disperse, with contact legislation 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 methods exhibit comparable CFS at essential state, confirming the robustness of this attractor and its insensitivity to protocol/ system variations.
Determine 3: Left: essential material floor recognized by way of true triaxial shear on a polydisperse meeting (Wen and Zhang, 2022a). Knowledge factors within the determine signify the essential material information of samples with totally different preliminary states. Proper: material 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 material paths converge to the CFS at regular state.
In distinction to DEM, in-situ experimental willpower of material evolution requires rather more effort when it comes to check designs and information postprocessing. Oda (1972a, b) studied the cross sections of resin-impregnated sand specimens and located a robust correlation between material 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 checks have been carried out on an X-μCT platform. X-ray scanning and mechanical loading have been alternated to acquire snapshots of the specimen’s inside construction all through the deformation course of. This X-ray picture dataset then requires great effort in publish processing to transform to bodily smart 3D fashions of the granular meeting, by way of which specimen’s material statistics could be extracted. Utilizing X-μCT, Imseeh et al. (2018) noticed that the contact-based material tensor certainly reaches a steady-state worth coinciding with the essential state situation in triaxial checks. Nevertheless, the evolution of material 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 material evolution inside and outdoors of the shear band, and located that material anisotropy and coordination quantity method distinctive values at massive pressure (throughout the band), no matter the preliminary material void ratio. In abstract, each DEM and experimental findings help the existence of an attractor within the material house 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 material 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 price noting that the majority experimental and DEM research have centered on material descriptions primarily based on contact regular vectors. It’s affordable to consider that different material descriptors corresponding to these outlined on particle and void cell orientations additionally converge to regular state values at massive shear strains. Nevertheless, there’s a shortage of experimental information to substantiate this hypothesis or quantifying their evolution sample in the direction of the essential state.
4. Grain Measurement
The grain dimension distribution (GSD) of pure soil is consistently evolving as a result of crushing, agglomeration, mixing, and segregation (Foley, 2018; Johnson et al., 2012; Wang et al., 2002). Understanding the grain dimension dynamics throughout these processes is of nice engineering and scientific significances. In geotechnical engineering, grain dimension distribution gives the primary indices characterizing and classifying sands. It has first-order impacts on the shear power, 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 dimension distribution encodes the historical past of earth’s or different planets’ floor (Anderson and Bunas, 1993; McKay et al., 1974). Every means of crushing, agglomeration, mixing, and segregation drive a special mode of GSD evolution, and thus tough to determine common traits if they’re mentioned altogether. Herein, this text will deal with the grain dimension 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 dimension evolution was primarily tried in geotechnical engineering and geomorphology communiteis. Hardin (1985) hypothesized that every one particles in a pattern of soil might be crushed to fines (particles with dimension 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 idea 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 burdened granular supplies was later supported by a increase 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 price highlighting that the last 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 sturdy self-organized course of (Bak, 2013). In truth, Fig.5 (left) resembles the rank vs. frequency traits of many different complicated methods that exibits self-organization, corresponding to metropolis dimension (Zipf’s legislation), earthquake (Gutenberg-Richter legislation), 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 dimension or, equivalently, the relative rating of cities vs. their inhabitants round 12 months 1920 (Bak, 2013; Zipf, 2016).
The sample of GSD evolution could be additionally seen mechanistically because of dynamic competitors between two micromechanisms: (1) smaller particles can face up to greater deviatoric stress and thus are harder to interrupt (Kendall, 1978; McDowell, 2001; Nakata et al., 2001b); (2) throughout collective breakage, massive 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 price to say a number of current observations that will problem the universality of the last word GSD. Particularly, gap-graded soils appear to “keep in mind” their preliminary GSD even after being loaded to very massive 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 presently not clear. It has additionally been identified that, though the last 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 look like general finer than these from compression (Fig. 6 proper), which can be defined by the upper mobility of grains throughout shearing. Extra information 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 burdened underneath oedometric compression and ring shear checks develop totally different final GSDs (Miao and Airey, 2013).
Constitutive modeling. The identification of an attractor within the GSD aircraft has facilitated the event of many continuum fashions for crushable granular supplies. The main idea on this regard is the Continuum Breakage Mechanics (CBM) (Einav, 2007a, b). The recognized GSD attractor is used to outline a brand new inside state variable referred to as breakage that varies from 0 to 1 within the means 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 harm mechanics (CDM) for brittle solids and can also be rooted within the idea of linear elastic fracture mechanics (Einav, 2007c). Inside the framework of CBM, a wide range 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 price 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 essential state line in e–p–q house have been additionally proposed (Tengattini et al., 2016).
Determine 7: Left: Definition of breakage variable B. Proper: predicted GSD vs. experimental information. (Einav, 2007a)
5. Grain Form
Grain form coevolves with grain dimension throughout fragmentation. Rising 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 remains to be at early stage in comparison with GSD evolution research, and lots of conclusions is probably not particular and require additional validations. The scenario can also be sophisticated by the massive 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 way of slowly evolving weathering or from fast breakup induced by explosion and hammering, displays 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 dimension, i.e., bigger particles are usually extra rounded. The existence of such form attractor is elegantly defined by way of 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 way of weathering and hammering (Domokos et al., 2015)
The universality of grain form could be additionally noticed from granular assemblies collectively crushed underneath high-pressure compression and shear. Website positioning et al. (2020) carried out 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 enough to induce pervasive breakage (See Fig. 9). Miao and Airey (2013) studied the last word states of a carbonate sand underneath totally different stress circumstances. Each GSD and grain form evolves in the direction of a gentle state after enough loading. Nevertheless, samples subjected to steady shearing (by way of a hoop shear gadget) produces an final grain form that has a barely greater facet ratio than that obtained from high-pressure oedometric compression checks. Ueda et al. (2013) carried out 2D DEM simulation of oedometric compression checks on granular assemblies with preliminary shapes starting from good circle to elongated hexagon. They noticed that every one samples arrived at a secure facet ratio after crushed to close final state, regardless of the presence of drastically totally different crushing modes corresponding to cleavage destruction, bending fracture, and edge abrasion (See Fig. 9). By way 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 could be approximated by a traditional or Weibull distribution, accompanied with the discount of median grain sizes. Ma et al. (2019) carried out a mixed finite and discrete ingredient methodology (FDEM) research of the identical course of and reported comparable conclusions for different form descriptors corresponding 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 dimension on the onset of yielding, whereas it modifications to the abrasion of native asperities with additional improve of stress post-yielding. That is per 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 perform of shear pressure is noticed. Lastly, it’s price to say the idea of form most popular orientation (SPO) of survivor grains in fault gouge (Cladouhos, 1999). The truth that SPO could be recognized and used to deduce the earlier kinematic historical past of the fault hints that the evolution of grain dimension, grain form, and material 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 understood to influence the mechanical properties of sand together with stiffness, power, 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 just lately by Buscarnera and Einav (2021). They confirmed that such form attractors could be integrated in CBM through further shape-related inside state variables (ISVs). The form ISVs are launched within the saved elastic vitality of the granular meeting in a approach 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 acquire of the system, which have to be balanced with the vitality dissipation and the exterior work enter to the fabric ingredient. By proposing a coevolution legislation between grain form and dimension, the evolution of grain form in the direction of an final attractor is predicted all through the course of loading. The speculation has a one-descriptor formulation primarily based on facet ratio and a two-descriptor model utilizing the elongation ratio and flatness ratio (i.e., the Zingg aircraft) as form ISVs. The expected form evolution path agrees effectively with experimental and DEM information from sands with totally different preliminary grain morphologies (Fig. 9). It’s anticipated that, as rising experimental information 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 information (Left) (Ueda et al., 2013) and experimental information (proper) (Website positioning et al., 2020)
6. Concluding Remarks
Attractors disclose vital data for in any other case intangible and complicated methods. 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. Definitely, rather more must be carried out to raised quantify these attractors and perceive how totally different preliminary states evolve in the direction of their regular states for burdened granular supplies. It’s envisioned that new theories or analysis methodology for complicated methods and self-organization might deliver new insights into granular mechanics analysis, if correctly mixed with the extra conventional continuum mechanics method.
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