The Nottingham eprints service makes this work by researchers of the University of Nottingham available open access under the following conditions.

Transcription

1 Li, Xia (2016) Internal structure quantification for granular constitutie modeling. Journal of Engineering Mechanics. C /1-C /16. ISSN Access from the Uniersity of Nottingham reository: htt://erints.nottingham.ac.uk/38506/1/li% jem.df Coyright and reuse: The Nottingham eprints serice makes this work by researchers of the Uniersity of Nottingham aailable oen access under the following conditions. This article is made aailable under the Uniersity of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: htt://erints.nottingham.ac.uk/end_user_agreement.df A note on ersions: The ersion resented here may differ from the ublished ersion or from the ersion of record. If you wish to cite this item you are adised to consult the ublisher s ersion. Please see the reository url aboe for details on accessing the ublished ersion and note that access may require a subscrition. For more information, lease contact erints@nottingham.ac.uk

2 ON THE INTERNAL STRUCTURE QUANTIFICATION FOR GRANULAR CONSTITUTIVE MODELLING Xia Li a * a Deartment of Chemical and Enironmental Engineering, Faculty of Engineering, Uniersity Park, The Uniersity of Nottingham, Nottingham, NG7 2RD, UK ABSTRACT The imortance of internal structure on the stress-strain behaior of granular materials has been widely recognized. How to define the fabric tensor and to use it in constitutie modelling howeer remains an oen question. The definition of fabric tensor requires 1) identifying the key asects of structure information and 2) quantifying their imact on material strength and deformation. This aer addresses these issues by alying the homogenisation theory to interret the multi-scale data obtained from the discrete element simulations. Numerical exeriments hae been carried out to test granular materials with different article friction coefficients. More frictional articles tend to form less but larger oid cells, leading to a larger samle oid ratio. Uon shearing, they form more significant structure anisotroy and suort higher force anisotroy, resulting in higher friction angle. Material strength and deformation hae been exlored on the local scale with the article acking described by the oid cell system. Three grous of fabric tensor hae been coered in this aer. The first one is based on the contact ectors, which is the geometrical link between contact forces and material stress. And their relationshi with material strength has been quantified by the Stress-Force-Fabric relationshi. The second grou is based on as the statistics of indiidual oid cell characteristics. Material dilatancy has been interreted by tracing the oid cell statistics during shearing. The * Corresonding author. Address: Deartment of Chemical and Enironmental Engineering, Faculty of Engineering, Uniersity Park, Uniersity of Nottingham, Nottingham, NG7 2RD, United Kingdom, xia.li@nottingham.ac.uk. Tel: +44(0) Fax: +44(0)

3 last grou is based on the oid ectors, for their direct resence in the micro-structural strain definition, including those based on the oid ector robability density and mean oid ector. Correlations among arious fabric quantifications hae been exlored. The mean oid ector length and the mean oid cell area are arameters quantifying the internal structure size, and strongly correlated with each other. Anisotroy indices defined based on contact normal density, oid ector density, oid ector length and oid cell orientation are found effectie in characterizing loading-induced anisotroy. They are also closely correlated. In-deth inestigation on structural toology may hel establish the correlation among different fabric descritors and unify the fabric tensor definition. Deformation bands hae been obsered to continuously form, deelo and disaear oer a length scale of seeral tens of article diameters. Its relation to and imact on material deformation is an area of future inestigation. Keywords: Fabric quantification, Granular statistics, Homogenisation theory, Discrete Element Method (DEM) INTRODUCTION Different from metal, the comlexity in the stress-strain behaiour of granular materials is largely rooted in the acking formation and eolution uon shearing. It is widely acknowledged that the fabric tensor needs to be introduced into constitutie modelling to cature the main features of granular material behaiour. A number of fabric definitions hae been roosed (Satake 1982, Oda 1985, Li and Li 2009, Nguyen, Magoariec et al. 2009, Kruyt and Rothenburg 2014). Generally seaking, the aroriateness of fabric definition deends on its alication. Targeting at constitutie modelling, this aer interrets the material strength and deformation from the local scale in order to shed some light on the imortant and yet to answer questions, including 1) what is the most aroriate fabric definition used for modelling the material stress-strain behaiour and 2) how to effectiely incororate it to reflect the imact of internal 2

4 structure on the material stress-strain resonses. Among many interesting earlier discoeries, (Satake 1978) s grah-theoretical aroach is instrumental in establishing the corresondence between discrete and continuum reresentations and informing the adancement of homogenisation theory. (Satake 1983) relaced an assembly of grains with grahs and formulated the mathematical exressions of discrete granular mechanics. The imortance of oids has been recognized and emhasized by introducing dual articles to reresent oid saces. In line of Satake s ioneering work, (Bagi 1996) introduced the concets of two dual cell systems as the geometric reresentation of discrete assemblies, and building uon it, the duality of the stress and strain. (Li and Li 2009) extended the concet to three dimensional saces by modifying the Voronoi-Delaunay tessellation systems with consideration of whether the articles are in real contact or not. In two dimensional saces, their dual cell systems are equialent to Satake s dual grahs. Interestingly, the idea of describing the material internal structure with a tessellation system has also been deeloed, though searately, in the field of granular statistics by (Blumenfeld and Edwards 2006). Instead of using two dual systems, they reresent the granular structure with a set of grain olygons and oid olygons. With the internal structure described by the dual grahs or its analogues, the continuum scale stress tensor has been exressed in terms of article interactions and contact ectors which are geometrical quantities in the solid cell system connecting contact oints and article centres. This corresondence has been theoretically established on Newton s second law of motion (Christoffersen 1981, Rothenburg and Seladurai 1981, Bagi 1996, Kruyt and Rothenburg 1996, Li, Yu et al. 2009). In arallel, the continuum-scale strain tensor has been exressed in terms of article relatie dislacements and geometrical quantities in the oid cell systems based on the comatibility condition (Bagi 1996, Kruyt and Rothenburg 1996, Kuhn 1999, Li, Yu et al. 2009). The imortance of internal structure is self-eident with the resence of local 3

5 geometrical quantities in these two discrete exressions. These theoretical deeloments in the homogenization theory hae also laid down the groundwork to systematically inestigate how the internal structure imacts on the stress-strain behaior from the local scale. In this study, numerical exeriments hae been carried out using the Discrete Element Method (DEM) (Cundall and Strack 1979) to roide the multi-scale data. A series of numerical simulations hae been carried out on granular assemblies with identical article geometries but different friction coefficients. The oid cell system has been constructed to describe article acking, and the continuum-scale material behaior is considered as the collectie resonse from all indiidual oid cells. Discussions hae been extended to the definition of fabric tensor, which seres as a necessary state ariable in constitutie modelling (Li and Dafalias 2012) NUMERICAL SIMULATIONS Numerical exeriments hae been carried out using the commercial ackage, Particle Flow Code (PFC2D), a two dimensional Discrete Element Method (DEM) software (Itasca Consulting Grou Inc. 1999). The boundary control algorithm introduced in (Li, Yu et al. 2013) has been used to imose the target loading ath. The articles are circular disks uniformly distributed in number within the range of (0.1mm, 0.3mm). The thickness of articles is set as 0.2 mm. The article interactions are of linear stiffness with a slider. The normal and tangential stiffnesses are set as N/m. A series of simulations hae been carried out with the article friction coefficient being 0.0, 0.1, 0.2, 0.5, 1.0 and 10.0 resectiely. The secimens are hexagonal excet for the case of 10.0, when the contact sliding is nearly rohibited, extremely large contact forces hae been obsered around the corner indicating local strong arching formation. The dodecagonal samle shae is hence used. The boundary roerties are set as the same as the article roerties. 4

6 96 97 Figure 1 Void ratio rior to shearing s article friction coefficient The samles are reared using the deosition method. Particles are generated in a rectangular region whose height is twice the width. The articles deosit ertically at graity 2 G 100m/s in the low daming enironment to form the initial acking, which is then 102 trimmed by the rescribed boundary and consolidated to 1000kPa for shearing. The scaled c graity is used to reduce comutational time. Such reared samles are exected to be initially anisotroic, although as shown later, of limited magnitude. For the series of numerical exeriments carried out in this study, the numbers of articles range from 3,443 to 3,938 deending on the article friction coefficient. The ratio between the samle size and the article diameter is around 60, and is belieed to be large enough to sere as reresentatie elements. Due to the difference in article friction coefficients, different initial structures are formed. Fig. 1 lots the oid ratio of the samles, an index of acking density, at their initial (re-shearing) states, which is obsered to increase with the increase in article friction coefficient. The acking with 10.0 has a similar oid ratio to the acking with 1.0. This information is not included in the figure for better illustration of the ariation when the friction coefficient aries between 0 and 1. In analogy to drained tests, samles are sheared in the ertical direction while the mean normal ressures are ket constant. The boundary control algorithm detailed in (Li, Yu et al. 2013) has been used to control the dislacements of boundary walls synchronously to imose the strain-controlled boundary, and to monitor the stress boundary using a sero-controlled mechanism. Local daming has been used to dissiate excess kinetic energy during shearing. Loading increments are only imosed when both the equilibrium criteria and the secimen boundary conditions are satisfactorily met. The material resonses are 5

7 shown in Fig. 2 by lotting the stress ratio q/ 2 and the olumetric strain against the deiatoric strain q, where 1 and 2 are the major and minor rincial stresses resectiely in two dimensional saces. The stress ratio is related to material frictional angle as 2sin Figure 2 Material resonses to shearing (a) Stress ratio b) Void ratio The deosition method is exected to roduce loose secimens. Most of the samles show strain hardening behaior howeer strain softening resonse has been obsered in samles with high article friction coefficients 1.0 and The friction angles are obsered to be low in general because circular articles hae been used in the simulations. Similar to the obserations in (Peyneau and Roux 2008), the samle made of frictionless articles ( 0.0 ) exhibits a low shear resistance and little olume change. It flows nearly as a fluid, with the samle friction angle as low as 4.6 o. A ery low and fluctuating olumetric strain u to 0.2% is obsered. The samle frictional angle increases gradually to 14 o when the article friction coefficient increases to 0.2. Howeer, further increase in article friction coefficient doesn t further increase the material shear resistance. This is consistent with the laboratory (Skinner 1969) and numerical (Thornton 2000, Antony and Sultan 2007, Huang, Hanley et al. 2014) obserations on 3D granular materials. The olume change exhibits more diersity. When the article friction coefficient increases from 0 to 0.2, the samle becomes more contractie with the olumetric strain with 0.2 going u to 1%. Howeer, when the article friction coefficient increases further to 0.5, the samle contracts slightly and then behaes dilatie. Further increase in article friction coefficient leads to more dilatie behaior with the olumetric strain with 10.0 as high as 2.8%. It is also obsered that although the ariation 6

8 in stress ratio occurs mainly in the first 10% deiatoric strain, the change in olumetric strain continues until much larger strain leels FABRIC QUANTIFICATION PERTINENT TO MATERIAL SHEAR RESISTANE The external loading is transmitted throughout the secimen ia the force-bearing structure. Fig. 3 lots the force chains at the initial states. The heterogeneity in article interaction is clear from the figure. It is obsered that strong forces aear eriodically oer eery few article diameters. Since the chosen samle size is much larger than the dimension exhibited in force heterogeneity, the samles are considered as reresentatie elements for stress analyses. Comaring Fig.3(a) & (b), samles of higher article friction coefficients exhibit a eriodicity oer a slightly larger length scale Figure 3 Contact force distribution rior to shearing (a) g 0.0 and (b) g 1.0. (The thickness of the black lines is roortional to the magnitude of contact forces) The Stress-Force-Fabric Relationshi Granular materials are known for its ability to self-organize their internal structure. Anisotroy deelos as a result of shearing and makes an imortant contribution to material shear resistance. This section addresses the fabric quantification ertinent to the shear resistance of granular material in aid of the Stress-Force-Fabric relationshi, which was originally roosed by (Rothenburg and Bathurst 1989). It was established based on the micro-structural definition of stress tensor, linking the continuum scale stress tensor with contact forces c f i and contact ectors c i as: c c i f j V cv (1) 7

9 in which V stands for the olume of interest. Note that a contact oint is identified only when there is non-zero interaction between two entities. At an internal contact oint between two articles, there is always a air of action and reaction forces corresonding to two contact ectors ointing from the contact oint to each article centre. They are counted as two contacts. Howeer, an external contact oint between article and boundary wall is only counted once. (Li and Yu 2013) emloyed the theory of directional statistics (Kanatani 1984) to inestigate the statistics of article-scale information, characterised the directional deendence of article-scale information with direction tensors and formulated the Stress-Force-Fabric relationshi in the tensorial form. The notations used in (Li and Yu 2013, Li and Yu 2014) are followed in this aer. Examination of the article-scale statistics suorts the following simlifications: 1) There is a slight and isotroic statistical deendence between contact forces and contact ectors which can be aroximated by i f j n = i n f j where n is a scalar around for all the simulations. In this exression, * n denotes the alue of ariable * in direction n, and * n denotes the aerage alue of all terms of * sharing the same direction n ; 2) The contact ector length is isotroic; 3) The contact normal robability density can be sufficiently accurately aroximated by u to the 2 nd rank olynomial series of unit directional ector n ; 4) The mean contact force f n can be sufficiently accurately aroximated by u to the 3 rd rank olynomial series of unit directional ector n. Eq. (1) can be conerted into integration oer direction by grouing the terms with the same contact normal directions together. Combined with the aboe obserations, the simlified Stress-Force-Fabric relationshi can be written as: 8

10 192 N f 1 c 0 f 0 1 h G ji D G 2V 2 (2) where is the article coordination number, N is the number of articles, 0 is the directional aerage of mean contact ector and f 0 is the directional aerage of mean contact force, h is a scalar accounting for the contribution from the joint roducts which increases slightly from 0 to around 0.01 during shearing. In two dimensional saces, the direction tensor 197 for contact normal density is D c c ccos sin d, where sin cos c c c c d denotes the magnitude 198 c of directional ariation and 2 indicates the referred rincial direction of contact normal 199 density. G f f f cos sin B sin cos f f f is the 2 nd rank tensor characterizing the directional deendence of contact forces, where f B denotes the magnitudes of directional ariation, indicates its referable rincial direction. It is worth ointing that f f G coers the contributions from both the normal contact force comonents and the tangential contact force comonents. c G is defined similar to f G but characterises the statistics of contact ectors. Aroximation using Eq. (2) has been found to gie exact matches of the continuum-scale stress, and roides a alid oint to interret material strength from the article scale Fabric quantification The micro-structural stress definition gien in Eq. (1) shows that the article-scale geometrical information linked to the material stress is contact ectors. And the SFF relationshi gien as Eq. (2) roides the analytical relationshi quantifying the correlation between the contact ectors and material stress state. Considering the different nature in the normal and tangential force-dislacement relationshi, the terms in Eq. (1) has been groued based on their contact normal directions, and the deiatoric tensor c D in Eq. (2) reflects the 9

11 anisotroy in contact normal density. The anisotroy in contact ector is a secondary factor which can be characterized in terms of c G. These two asects can be combined and quantified in terms of one fabric tensor. This section summarises their definitions and calculations based on directional statistical theories Fabric quantification for contact normal density Contact normal based fabric tensor is one of the most widely used index in characterizing the loading induced anisotroy (Oda, Nemat-Nasser et al. 1985), and aears in Eq. (2) as D c c ccos sin d, which is called the fabric tensor of the third kind (Kanatani 1984). sin cos c c c It describes the ariation of contact normal density oer direction. An equialent definition is the fabric tensor of the second kind distribution can be aroximated as: c F (Kanatani 1984). With them, the contact normal density 1 1 E ( n ) F n n 1 D n n (3) c c c i j i j E0 E0 where E0 d 2 in the two dimensional saces. c c c D and c F are interchangeable as F D (4) They can be determined from the fabric tensor of the first kind, also referred to the moment tensor c N in (Kanatani 1984, Li and Yu 2013) as where the moment tensor can be calculated as: c c 1 F 4N 4 and c c 1 D 4N 2, where 1 n, 2 n, and total number of contacts. c 1 M i j 1 i j N n n n n (5) M N n being the unit ectors reresenting contact normals. M is the 10

12 Fabric quantification for contact ector anisotroies The anisotroy in mean contact ector could be an additional contributor to material stress ratio as listed in the Stress-Force-Fabric relationshi, Eq. (2), for non-sherical articles (Li and Yu 2014), although its anisotroy magnitude is often found to be secondary comared with that 237 of contact normal density. The mean contact ector j n can be aroximated as c c n 0, or equialently in terms of the fabric tensor 0 1 c H G n G n j j ji i is the directional aerage of mean contact ector., where Fabric quantification combining contact normal and contact ector anisotroies A combined account for the contribution of material fabric to stress state may include both contact normal density and contact ector anisotroy, and be defined on the contact ector based moment tensor as: c 1 M c L in j ( ) 1 i n j E i n jd M n n (6) c c c c c Substituting Eq. (3) into Eq.(6) leads to L 0 G D Dim G jm 1 1 in 2D saces. 246 Note c D and c G are deiatoric tensors. Neglecting the joint roducts of higher rank terms for simlicity and denoting the normalized deiator tensor as Stress-Force-Fabric relationshi can be rewritten as: c c 2L c c C c G D 2, the L kk where L c ii 0. strength. N c f 0 f 0 C G ji (7) 2V c C roides an exlicit account of the imact of internal structure on material The micromechanical interretation of material shear resistance In this study, disk-shaed articles are used. The mean contact ector has been found 11

13 254 nearly isotroic so that c c G =0 and C D 2. For all the simulations, the rincial fabric c directions are the same as the loading direction, and the material anisotroy can be characterized in terms of the degrees of contact normal anisotroy c d, which is lotted in Fig. 4(a). Een for frictionless articles, shearing results in structure anisotroy, although of limited magnitude. More significant fabric anisotroy deelos in more frictional articles. Uon shearing, the contact normal anisotroy increases mostly monotonically, although in more frictional samles, its rate of increases is obsered to be higher and reaches a stronger anisotroy at the critical state. When the friction coefficient increases further beyond 0.5, the eolutions of contact normal anisotroy are obsered to no longer change. This is similar to the obseration made in (Huang, Hanley et al. 2014) based on 3D DEM simulations Figure 4 The micro-mechanical contributors to material strength (a) Contact normal anisotroy c d, and (b) Contact force anisotroy f B Information on contact force anisotroy f B is lotted in Fig. 4(b). While article friction coefficient increases, both the contact normal anisotroy and the contact force anisotroy increase. The contact force anisotroy howeer exhibits a eak before aroaching the critical state, coincident with the occurrences of eak stress ratio followed by strain softening. It is interesting to oint out that no matter what the article friction coefficient is, the anisotroy in contact force is of similar magnitude with contact normal anisotroy, which is better shown in Fig. 5 by lotting the two anisotroies against each other. The reference line indicates when the two anisotroic degrees are equal to each other. The strong correlation between the contact normal anisotroy and the contact force anisotroy is eident with most data oints falling near the reference line. Shearing motiates contact force anisotroy slightly faster and higher than 12

14 the deeloed contact normal anisotroy. For samles made of ery rough articles, contact force anisotroy was obsered to be higher than the contact normal anisotroy at the early stage of shearing. When aroaching the critical state, the two anisotroies become equal Figure 5 Correlation between the fabric and contact force anisotroy In a summary, SFF relationshi suorts the effectieness of c D and c C as the fabric tensor definition to study the material stress and hence strength. The force anisotroy is found strongly associated with the obsered fabric anisotroy, in articular at the critical state. Hence, material shear strength can be determined from the fabric anisotroy should there be an established fabric-force correlation VOID CELL STATISTICS AND MATERIAL DILATANY In this section, the relationshi between material dilatancy and the eolution of oid cell statistics will be exlored by iewing a granular assembly as a collection of oid cells. The oid cell system is formed by connecting contact oints and article centres. Particles without contribution to the global force transmission, including those with few than two contact oints, are excluded during the oid cell construction. The number of constitutie articles in oid cells should be no less than 3. Fig. 6 roides an examle by resenting the oid cell system with 0.5. The color scheme is associated with the oid cell area. The oid cells between boundary articles and walls hae been identified in order to tessellate the whole sace enclosed by the secimen boundaries Figure 6 The oid cell system at re-shearing stage ( 0.5 )

15 Void cell characterisation and oid cell based fabric tensor Viewing a granular material as an assembly of oid cells, the material fabric tensor can be defined as the statistical aerage of indiidual oid cell characteristics. The loo tensors used in (Nguyen, Magoariec et al. 2009, Kruyt and Rothenburg 2014) are such examles. Howeer, there is no unique way in doing so. Here, the indiidual oid cell is characterized based on the area moment of inertia, and the oid cell based fabric tensor is roosed as their statistical aerage as one examle of its kinds Characterisation of indiidual oid cells Void cells may hae different and irregular shaes. A single dimension is inadequate to describe the geometry of indiidual oid cells. Factors of rimary interest are the size of the oid cell, its shae and the orientation. The area moment of inertia I rr A i jda, where r i is the ector from the location of the area element da to the area centre of oid cell, contains all the necessary information and can be otentially used. Based on the area moment of inertia I, the tensor Z is used to describe the local cell geometry: 316 Z 4 I (8) A Its rincile direction gies information on the oid cell orientation. Z J Figure 7 det Z s. oid cell area In the case of an ellise of semi-major axis of length a and semi-minor axis of lengthb, a b. Note that the area of the ellise is ab J det Z det Z Z, where Z denotes the Jacobian determinant of tensor Z. This suggests that 14

16 det Z may sere as an effectie estimation of oid cell areas. Fig. 7 lots det Z against the area of oid cells for all the oid cells shown in Fig. 6. The red line in the figure 327 lots the reference line y x. Desite their irregular shae, the data hae been found lying closely to, with most data slightly aboe, the reference line. The shae of an ellise can be described by the index a b a b. For a circle, the index is equal to 0 and for an ellise with infinite asect ratio, it is 1. In terms of the tensor defined in Eq.(8), the equialent exression is the oid cell anisotroy index Z1 Z 1 2 Z 1 1 Z2, where 1 of the fabric tensor Z and Z 2 are the major and minor rincial alues Z. Fig. 8 resents information on the shae of oid cells by lotting the 334 robability density function d P d, where x x P reresents the robability of oid cells whose shae factor is no larger than x, and dp reresents the robability of oid cells whose shae factor falls within x x d 2 Z Z 1 Z Z 1 x d 2. Fig. 8(a) lots the robability density function at the initial state while Fig. 8(b) lots the robability density function after 20% deiatoric strain. It is obsered that most oid cells are anisotroic with the highest robability around 0.2. For larger friction coefficients, the area fraction occuied by more anisotroic oid cells becomes slightly larger while that by less anisotroic oid cells becomes slightly smaller Figure 8 Probability Density Function d P x d (a) Deiatoric strain 0% (b) Deiatoric 344 strain 20% The fabric tensor for indiidual oid cell fabric as A 1 2, the minor rincial fabric as S is hence defined such that the major rincial A 1 2 and the rincial directions 15

17 are the same as those of is Z1 Z 2. Z. Note that the ratio between the major and minor rincial fabrics Anisotroy in oid cell orientation The orientation of oid cells can be reresented by a unit ector. Similar to contact normal density, the oid cell orientations can be characterised by the direction tensor with the form 353 D S S S cos sin d sin cos S S S (9) and calculated from its moment tensor, where S S d is the anisotroy index and the rincial direction. The anisotroy index has been lotted in Fig. 9. The rincial direction has been all around 90 o. The figure suggests that material anisotroy has deeloed as a result of more oid cells orienting towards the loading direction, similar to the obseration reorted in (Nguyen, Magoariec et al. 2012). Figure 9 Anisotroy in oid cell orientations Void cell based fabric quantification The continuum-scale fabric tensor is defined as the aerage of oid cell fabric tensors as: 365 F S 1 S (10) N A The fabric tensors of indiidual oid cells hae been calculated from the oid cell geometries obtained from DEM simulations, and used to calculate the macro fabric tensor defined in Eq. 368 (10). The first inariant Fii S S A is the aerage oid cell area. The deiatoric art of F is an area-weighted measure of oid cell shaes. The anisotroy index of oid cell-based fabric F S S S S tensor, Eq.(10), is defined as d 2F 1 F 2 F 1 F 2 S S, where F 1 and F 2 are the 16

18 rincial alues of the fabric tensor 10 shows the eolution of the anisotroy index S F. The rincial direction is obsered around 90 o. Fig. S d during shearing, whose attern is obsered in great similarity as that of contact normal density in Fig. 4(a) and that of oid cell orientation in Fig. 9, suggesting a strong correlation among these fabric indices, which will be exlored later in this aer. Material dilatancy and oid cell statistics Figure 10 Anisotroy index of Dilatancy is the change in samle olume or oid ratio during shearing. For 2D granular assemblies, the total area of assembly can be exressed as: S F A sam is equal to the summation of all oid cell areas and 383 Asam A N A (11) N where A denotes the area of the -th oid cell, N the total number of oid cells, and A 385 the aerage oid cell area. The total article (solid) area As A N A, where N 1 A denotes the area of the -th article, N is the total number of articles and A is the aerage article area, a constant throughout the test. The oid ratio of the granular assembly can hence be formulated as: Asam N A e 1 1 (12) A N A s The total number of contacts can be found by summing u the coordination numbers of all articles, which howeer may be slightly different from that summing oer all the oid cells since in the oid cell system each article-wall contact is counted twice. Should the samle size 393 be large enough, the difference is small and negligible, M N N, where the oid cell 17

19 coordination number denotes the aerage number of constitutie articles in oid cells. It should be no less than 3 in two dimensional granulate systems. The material oid ratio can hence be rewritten as: A e 1 (13) A The olume change tendency, i.e., the dilatancy of granular material, can be quantified as the change in the samle oid ratio uon shearing, and studied by tracing the eolution of oid cell 400 statistics, in articular A A and during shearing number Fig. 11(a) lots the article coordination number and the oid cell coordination for re-sheared samles with different article friction coefficients. Fig. 11(b) 403 roides information of A A and at arious friction coefficients. The data of are close to those of 1.0, and not shown in the figures. Note that the stability condition of two dimensional infinite granulate system imoses the requirement of the minimal coordination number being 3. The coordination numbers slightly smaller than 3 hae been obsered in this study is artially because non-load bearing articles (rattlers) are resent in the system, but not excluded in article coordination number. It is also because of the boundary effect. At each boundary-article contact oint, there are two force comonents contributing to the system stability. They are counted twice in oid cell construction, but only once when calculating the article coordination number. For the same reasons, the relationshi between the article coordination number and the oid cell coordination number is found to slightly deiate from the Euler s relation for laner grahs 2 2 (Satake 1985) Figure 11 The internal structure at initial states (a) Coordination number; (b) Void cell characteristics 18

20 The figures show clearly that the article friction coefficient has a significant effect on oid cell characteristics. For frictionless articles, the article coordination number is only slightly larger than that of oid cells. The aerage oid cell area and the aerage article area are close. When the articles become frictional, the article coordination number reduces while the oid cell coordination number increases. More frictional articles tend to form fewer but larger oid cells. It is obsered that with increasing friction coefficients, the number of oid cells dros, accomanied with an increase in oid cell area. As a result, the aerage oid cell area almost doubles when the article friction changes from =0 to =10. The increase in oid cell area exceeds the reduction in oid cell number, resulting in larger oid ratios obsered at higher friction coefficients. The eolutions of the samle oid ratio e and the oid cell characteristics, including 429 A A, the article coordination number and the oid cell coordination number, hae 430 been lotted in Fig. 12. Eq. (13) reeals that the change in the oid ratio e is resulted from the 431 cometition between A A and. As seen in Fig. 12, when samles are sheared, the 432 increase in oid cell coordination number is obsered and accomanied by an increase in the 433 mean oid cell area. When the increase in A A exceeds that in, the samle dilates with an increase in oid ratio. Otherwise, the samle contracts with a reduced oid ratio. With zero and low article frictions, the article and oid cell coordination numbers remain almost constant during shearing. Howeer, for highly frictional articles, shearing causes significant reduction in article coordination number and increase in oid cell coordination number at the early stage of shearing, but this effect is oertaken by the increase 439 in A A. Samles show significant dilatie resonses. These changes during shearing are 440 associated with the deeloment of oid cell anisotroies resented in Figs. 8, 9 &

21 442 Figure 12 Eolution of oid cell statistics to shearing (a) Void ratio e, (b) A A, (c) 443 Particle coordination number and (d) Void cell coordination number The oid cell coordination number Frictional articles tend to form larger oid cells with higher coordination number. Grouing the oid cells according to their coordination number, the total samle area can be exressed as: 449 (14) A H A N h A sam ali ali i3 ali i3 ali 450 where Hal i is the number of oid cells whose coordination number is i, h H N ali ali 451 reresents its robability and A al i the aerage area of such oid cells. The samle oid hence becomes: i3 A sam N e 1 1 h A A (15) ali A N ali s where N stands for the total number of oid cells. Fig. 13 gies the robability and the aerage area of oid cells with different coordination numbers at the initial and sheared states. It shows clearly that there is a close correlation between the aerage oid cell area and the coordination number. The correlation can be roughly aroximated by the olynomial function of ower 2, and is found indeendent of article friction coefficients. Particles with higher friction coefficients are more likely to form oid cells with more constitutie articles, hence the robability of oid cells with a larger coordination 461 number is higher. Shearing alters the correlation between A A and the cell coordination 462 number slightly. Data at 20% deiatoric strain are shown in Fig. 13(b). At the same 20

22 463 coordination number, A A is smaller at the sheared states than that in the initial state, 464 indicating the deendence of aerage oid cell area on oid cell anisotroy Figure 13 Void cell statistics at different coordination number ( =0.5) (a) Deiatoric strain 0%; (b) Deiatoric strain 20% VOID VECTOR BASED FABRIC QUANTIFICATION AND MATERIAL STRAIN Using the oid cell system, the strain of a granular assembly can be considered as the olume weighted aerage of oid cell strains. The micro-structural strain definition exresses the continuum-scale material strain in terms of article relatie dislacements and oid ectors (Bagi 1996, Kruyt and Rothenburg 1996, Kuhn 1999, Li, Yu et al. 2009), and insired the definition of oid ector fabric tensors The micro-structural strain tensor Following the sign conention defined in (Li, Yu et al. 2009), the comressie strain is ositie. nx ( ) denotes the normal direction on the boundary surface at oint x, ositie when ointing inwards. In two dimensional saces, the dislacement gradient tensor aeraged oer the samle area A could be ealuated as: 1 1 e u A dl A d j, i A A u n (16) where u ji, denotes the dislacement gradient and L is the boundary of the area of interest A. The line integral on the right hand side follows the counter-clockwise integration aths oer the boundary of the area A. With reresents the two dimensional ermutation tensor B , nidl dx j. Eq. (16) becomes: 21

23 e u dx x du (17) jk jk i k k i A A B B With the material internal structure reresented by the oid cell system, Eq. (17) can be discretized into: 488 e x u u (18) jk jk k i k i A A A L A L 489 where i is the ector starting from the contact oint to the oid cell centre, referred to as the 490 oid ector. Eq. (18) is a double summation. The inner summation * L runs oer the boundary L of oid cell and * is a summation oer all the oid cells within the samle area A. For A granular materials, no matter how the samle is diided into sub-domains; the weighted sum of local dislacement gradient tensors is always the same (Bagi 1993). Denoting 494 e jk k i A L u (19) as the local dislacement gradient tensor defined on the oid cell, the samle dislacement gradient tensor can be written as the area-weighted aerage oer all the oid cells: e 1 A e A A (20) It is erified that such estimated samle dislacement gradient is in good agreement with the alue obtained from samle boundary Void ector based fabric quantification The micro-structural strain definition gien in Eq. (18) shows that the key geometrical information bridging-u the continuum scale strain and the article-scale relatie dislacements is oid ector, which connects the contact oint to the oid cell centre. This insired the oid ector based fabric tensor definitions (Li and Li 2009). The mathematical treatment has been detailed in (Li and Yu 2011) and alied to analyze the contact ectors in the reious session. 22

24 Fabric quantification based on oid ector robability density To describe the directional deendence of oid ectors, it is of interest to know in each direction 1) their robability density and 2) their reresentatie (or mean) alue. The directional robability density of oid ectors can be quantified in terms of a second rank deiatoric tensor 510 D cos sin d sin cos (21) following the similar rocedure to rocess information on contact normal and oid cell orientations Fabric quantification based on oid ector length As a descrition of oid cell shae in aerage, the directional deendence of mean oid ector has been characterized in terms of the second rank deiatoric tensor G as cos sin B sin cos so that the mean oid ector in direction n can be aroximated 1 cos 0 2 B n (22) where in two dimensional saces, the unit direction ector is equialently exressed as n cos,sin. Based on the mean oid ector length, (Li and Li 2009) roosed the oid ector based fabric tensor as: H 0 G (23) 523 The oid ector based moment tensor The oid ector based moment tensor can be considered as a combined account of the anisotroies in oid ector density and mean oid ector length. It has been used in (Fu and Dafalias 2015) in structural characterization. The moment tensor can be found as 23

25 M i j 1 i j L n n. Similar to reious discussions on contact ectors, L can be M determined from D and H. In two dimensional saces, 1 1 L 0 G D Dim G 1 jm Internal structure size during shearing As shearing continues, anisotroy in oid ectors deelos and is quantified with the two 532 anisotroy indices d, B. Both anisotroies are obsered to be significant. For all the simulations in this study, both anisotroies align in the loading direction. And similarity is obsered between their eolutions and those in contact normal density and oid cell orientation. The directional aerage of oid ector length 0 is regarded as a measure of the oid cell size, and lotted in Fig. 14. It is shown that samles with larger article friction coefficients hae a larger oid ector length, corresonding to larger oid cells Figure 14 Directional aerage of oid ector length CORRELATION BETWEEN DIFFERENT FABRIC QUANTIFICATIONS So far, a number of fabric quantifications hae been listed in this aer and defined as the statistical characterisatics of contacts, oid cells and oid ectors, resectiely. They are chosen because of their releance to material strength and deformation, and formulated based on the directional statistical theory (Kanatani 1984, Li and Yu 2011). The deeloment of constitutie model howeer requires minimizing the number of ariable and arameters. It is hence imortant to exlore the correlations among arious fabric quantifications (Fu and Dafalias 2015). The similarities obsered in their eolution attern is encouraging. In this session, the 24

26 oid cell based fabric tensor different fabric quantifications. S F has been used as a reference to discuss the correlaton among Among all the fabric tensors, two of them contains informaton reflecting oid cell size. They are the fabric tensor based on oid ector length fabric tensor H, Eq. (23) and the oid cell based S F, Eq. (10). The directional aeraged oid ector length 0 in V H and the 554 mean oid cell area Fii S S A in F are lotted against each other in Fig. 15, showing a strong correlation in between. It confirms that 0 can be considered as an effectie descritor of material internal structure size. The correlation is indeendnt of article friction coefficient Figure 15 Correlations between internal structure size descritors All the fabric tensors contains material anisotroy information. The anisotroy deeloed 562 in contact ector length c G ji is not elaborated here because its effect is secondary. The 563 anisotroy index F S d in the oid cell based fabric quantification F, Eq. (10) is shown correlated with other anisotroy indices, including in oid cell orientation, Eq. (9), c d in contact normal density, Eq. (3), d in the oid ector orientation, Eq. (21) and S d B in the mean oid ector length, Eq. (23) in Fig. 16. The strong correlation among these anisotroy confirms the obserations made in (Li, Yu et al. 2009, Fu and Dafalias 2015). The anisotroy indices associated with oid ectors are exected to be closely related that in oid cells, as confirmed in Fig. 16(c) & (d). In-deth inestigation into structural toology may hel to establish the correlation analytically and to unify the fabric tensor definitions Figure 16 Correlations between the oid cell-based anisotroy and other anisotroy indices (a) Contact normal robability density; (b) Void cell orientation; (c) Void ector robability 25

27 574 density and (d) Mean oid ector length DISCUSSION ON STRAIN HETEROGENITY Obseration of deformation attern Strain heterogeneity is another imortant feature of granular materials. The deformation descritor in Eq. (19) is defined for each indiidual oid cell and offers a iew of satial distribution of material deformation. Take the configuration when the oid cell system is constructed as the reference undeformed configuration. The relatie dislacements occurring during the subsequent 0.5% deiatoric strain increments are extracted from the DEM simulations and used to calculate the dislacement gradient tensor of each oid cell as er Eq. (19). Fig. 17 shows the local dislacement gradients of each oid cell when the samle was sheared from 15% to 15.5% deiatoric strain. The four comonents of non-affine dislacement gradient tensor, defined as the deiation of the local strain from the samle aerage dislacement gradient tensor, for the samle with g 0.5 are lotted in the searate sub- figures. It is obsered that there are localized banding structures where the strain is much more significant than the remaining of areas. This is similar to the obseration made in (Kuhn 1999) that sli deformation was most intense within thin obliquely micro bands. Different from the eriodic boundaries used in (Kuhn 1999), the samle boundaries are rigid walls which imose uniform dislacement gradient field. These banding structures do not ersist during shearing. Subsequent loading continuously destroys the existing banding structures and romotes the formation of new bands in other locations. It is interesting to note that although certain banding features are commonly obsered in the four lots; the atterns for the two shear strain comonents are obsered to be different from those for the two normal strain comonents. 26

28 w Furthermore, bands of comonent e tend to roagate in the ertical direction while the 12 w attern shown by comonent e extends in the horizontal direction Figure 17 Patterns of non-affined deformation gradient obsered from deiatoric strain q 15% to q 15.5% ( g 0.5) (a) (b) 11 (c) 12 and (d) The distance between deformation bands is in the order of tens of article diameters. It is seeral times larger than the internal scale in force chain heterogeneity. Shearing brings about continuous formation, deeloment and dissolution of deformation bands, causing synchronized swing in the material shear stresses as seen in Fig. 2(a). The deeloments of the force chain heterogeneity and the deformation bands are belieed to be critical to the deformation and failure of granular systems. It is an area of future research. Considering the heterogeneity in material deformation, the samle size may need to be further enlarged to sere as a reresentatie element Probability distributions The samle deformation gradient tensor gien in Eq. (20) can be interreted as an integral oer all the ossible local deformation gradient alues as 615 e W e de (24) e 616 in which W e A 1 e e e e e lim A e 0 e 2, 2 is the area fraction density function. It is the 617 area fraction of oid cells whose dislacement gradient comonent e falls within the range e e e 2, e e 2 normalized by the deformation increment e. Eq. (24) deals with the four comonents of dislacement gradient tensor searately. The Einstein summation oer 27

29 620 the reeated subscrits doesn t aly here Figure 18 Area fraction density of the four dislacement gradient comonents ( g 0.5, from q 15% to 15.5% ) (a) normal comonents and (b) shear comonents q Fig. 18 lots the area fraction density function for the four comonents of dislacement gradient tensor. The data are again taken from the samle with g 0.5 when sheared from 15% to 15.5% as shown in Fig. 17. For all the simulations in this study, the highest q q area fraction occurs at zero or near zero deformation. The area fraction decreases quickly as the magnitude of strain comonent increases. Howeer, it is worth noting that there exists a large area fraction where local deformation is much more rominent than the continuum scale aerage 0.5%. Although the samles are loaded in the biaxial mode, significant shear strains are obsered, indicating rigid body rotation or deformation deiated away from the ertical direction are imortant deformation mechanisms in local oid cells. The continuum-scale deformation is of small magnitudes because there are significant ortions of ositie as well as negatie strain comonents which comensate each other. Particle friction coefficient has a significant influence on deformation distribution. Samles of smooth articles show more disersed but more significant oid cell deformations. Fig. 19 resents the robability distribution of oid cell deformations by lotting the area fraction of ositie and negatie normal strains and the aerages of ositie and negatie shear strain comonents resectiely. The shae of function W for the two shear comonents is e symmetric with resect to x 0, corresonding to the obseration that the area fractions for the ositie and negatie shear comonents are around 50%, although not lotted here