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Local Analysis of the History Dependence in Tetrahedra Packings

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Local Analysis of the History Dependence in Tetrahedra Packings Local analysis of the history dependence in tetrahedra packings N. Nirmal Thyagu,1, 2 Max Neudecker,1 Simon Weis,3 Fabian M. Schaller,3, 4 and Matthias Schr¨oter1, 5, ∗ 1Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Goettingen, Germany 2Division of Physics, School of Advanced Sciences, VIT Chennai, Chennai - 600127, India 3Institut f¨urTheoretische Physik I, Friedrich-Alexander-Universit¨at,91058 Erlangen, Germany 4Institute of Stochastics, Karlsruhe Institute of Technology, 76049 Karlsruhe, Germany 5Institute for Multiscale Simulation, Friedrich-Alexander-Universit¨at,91052 Erlangen, Germany (Dated: October 22, 2018) The mechanical properties of a granular sample depend frequently on the way the packing was prepared. However, is not well understood which properties of the packing store this information. Here we present an X-ray tomography study of three pairs of tetrahedra packings prepared with three different tapping protocols. The packings in each pair differs in the number of mechanical constraints C imposed on the particles by their contacts, while their bulk volume fraction φglobal is approximately the same. We decompose C into the contributions of the three different contact types possible between tetrahedra { face-to-face (F2F), edge-to-face (E2F), and point contacts { which each fix a different amount of constraints. We then perform a local analysis of the contact distribution by grouping the particles together according to their individual volume fraction φlocal computed from a Voronoi tessellation. We find that in samples which have been tapped sufficiently long the number of F2F contacts becomes an universal function of φlocal. In contrast the number of E2F and point contacts varies with the applied tapping protocol. Moreover, we find that the anisotropy of the shape of the Voronoi cells depends on the tapping protocol. This behavior differs from spheres and ellipsoids and posses a significant constraint for any mean-field approach to tetrahedra packings. I. HISTORY DEPENDENCE IN GRANULAR particle. (The Set Voronoi cell of a given particle com- MATTER prises all points which are closer to the surface of the particle than to the surface of any other particle [13{15].) A number of experiments have shown that the mechan- Option a) has interesting theoretical consequences. ical properties of apparently identical granular samples This paper studies pairs of hard, frictional tetrahedra will differ depending on how they were prepared. Exam- packings which have approximately the same φglobal but ples include the volume response to shear [1], the force differ in their contact numbers Z. Such behavior is moment tensor in tapped packings [2{4], the increase of clearly beyond the reach of the Jamming paradigm which pressure with depth in a granular column [5{8], or the has been developed for soft, frictionless spheres [16, 17] pressure distribution [9] below a sandpile. Another type and where Z is considered to be a function of φglobal of history dependent behavior, which is sometimes re- alone. ferred to as memory effect, can be seen if a sample is The reason for this failure is not even primarily the compactified, e.g. by shearing or tapping with a driving shape of the particles: In the Jamming paradigm both 0 strength si, to a specific volume fraction φglobal. If the φglobal and Z are controlled simultaneously by the com- driving strength is then changed to some new value s0, pression of the soft particles. In contrast, granular parti- it is found that the subsequent evolution of φglobal differs cles change their contact number by changing their local for different values of si [10, 11]; even though all sam- geometry, not by compression. For tapped spheres and 0 ples now start at φglobal and are driven with the same ellipsoids [18], Z is controlled by the local volume frac- strength s0. A similar effect was found in granular gases tion φlocal which is computed for each individual particles [12]. by dividing its volume by the volume of its Voronoi cell In some sense the term history dependence simply ex- [19]. This behavior is in good agreement with statistical presses the fact that φglobal alone does not provide a com- mechanics approach [20{22], where Z is computed from plete description of the state of the packing. Given that a mean-field approximation of φ . arXiv:1501.04472v2 [cond-mat.soft] 1 Mar 2018 local the mechanical properties of the sample originate from Recently there has been some effort to evolve the Jam- forces transmitted between particles at their contacts it ming paradigm into a theory which describes history de- seems natural to extend the description by including in- pendence [23, 24]; if this approach can be expanded to formation on a) the number and spatial structure of the describe also frictional and hard particles it would be contacts, and b) the distribution of contact forces. More- very valuable. over we might want to include, for reasons shown below, Option b) is directly connected to the fact that fric- c) properties of the Set Voronoi cells surrounding each tional packings at finite pressures are hyperstatic [25{29] i.e. there exist many possible force configurations which fulfill a set of given boundary conditions [30]. Measur- ing the distribution of contact forces experimentally is ∗ [email protected] a hard problem; it has presently only been achieved in 2 systems of compressive particles [31, 32] or photoelastic discs [33]. Experiments with the latter method find a a) clear dependence on the preparation protocol [34] . Option c) might initially not look too promising: at least for spheres neither the distribution of the volumes of the Voronoi cells [35] nor the distribution of their shapes [36] shows any signature of the preparation con- ditions. However, for packings of tetrahedra we find that b) the preparation history does indeed influence the shape and size distribution of the Voronoi volumes. c) II. PACKINGS OF TETRAHEDRA An interesting difference between spheres and tetrahe- dra is that the latter have not only one but four different types of contacts: face-to-face (F2F) contacts, which are mechanically equivalent to 3 individual point contacts, edge-to-face (E2F) contacts (equivalent to two point con- tacts) and the vertex-to-face (V2F) and edge-to-edge (E2E) contacts which we group here as point contact. This distinction is important because the different con- tact types fix each a different number of degrees of free- dom. The total number of constraints of a particle C is P therefore i Ci where i goes over the three contact types FIG. 1. Tetrahedra packing. a) Polypropylene particle made (F2F, E2F, point). Each Ci equals miZi where Zi is the by injection moulding, the edge length is 7mm. b) Set Voronoi cell around a tetrahedron with a local volume frac- respective contact number and mi the number of con- straints per particle fixed by this type of contact. The dif- tion of 0.52. The Set Voronoi tessellation is computed by ferent values of m can be understood for frictional con- spawning 514 point on the surface of each tetrahedron and i them merging the Voronoi cells of these points. c) Rendering tacts as follows: All contacts impose 3 translational con- of an interior section of a tetrahedra packing after all particles straints, E2F contacts add 2 rotational constraints (thus, have been identified from an X-ray tomogram. Particles are a total of 5 constraints), F2F contacts prohibit 3 differ- colored according to the number of constraints fixed by their ent rotations (thus, a total of 6 constraints). As these neighbors. The global volume fraction is 0.53, the average constraints are shared between two tetrahedra, we ob- constraint number is 15.5. tain the constraint multipliers mF 2F = 3:0, mE2F = 2:5, and mV 2F = mE2E = 1:5. Each tetrahedron has 6 degrees of freedom (3 trans- lations and 3 rotations), therefore a mechanical stable been a quest for the densest tetrahedra packing [53{55], packing requires C to be at least 6, the so-called isostatic which is presently believed to be at φglobal ≈ 0:8563 [56]. contact number. However, typical packings of frictional Experiments using necessarily frictional tetrahedra ex- tetrahedra have C values in the range 12 to 18 [37] and plore a much lower range of φglobal [57] than the nu- are therefore strongly hyperstatic [38]. Contrary claims merical studies of frictionless tetrahedra described above. of isostaticity [39] are based on the erroneous use of fric- They have been characterized with respect to their me- tionless constraint multipliers mi while analyzing exper- chanical properties under uniaxial compression [58, 59] imental i.e. frictional packings. (Due to the absence of and shown to have history dependent values of C and Z tangential forces the frictionless mi are always smaller for identical values of φglobal [37]. than their frictional counterparts.) Because of the importance of the constraint number for the mechanical properties, we will discuss in the re- mainder of this paper C rather than Z. Packings of frictionless tetrahedra have recently been III. EXPERIMENTAL SETUP an active field of research because of the existence of quasi-crystals [40], hyperuniformity [41], the effect of al- tered particle shape on packing density [42{45], the dis- Experiments were performed with the same polypropy- tribution of contact types [46{48], and the behavior of lene tetrahedra as in reference [37]. The particles, figure binary systems of tetrahedra and octahedra [49]. The- 1a shows an example, were made by mould casting, their oretical approaches include a mean field ansatz [21, 22] edge length is 7 mm and their coefficient of static friction and a local motif analysis [50{52]. Moreover there has equals 0.8 [60].
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