New formulation of graph-theoretical approach in the mechanics of granular materials

Satake, Masatada

doi:10.1016/0167-6636(93)90028-p

Cited by 41 publications

(35 citation statements)

References 3 publications

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“…Such sets of constraint equations have been recognized in the literature [55,53,56] and were recently used to determine forces from the positions of jammed packings of soft disks [64]. Clearly each of these loop constraints is independent of the others, since each void polygon has distinct edges.…”

Section: Coordination and Isostaticitymentioning

confidence: 99%

Disordered contact networks in jammed packings of frictionless disks

Ramola¹,

Chakraborty²

2016

J. Stat. Mech.

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Abstract. We analyse properties of contact networks formed in packings of soft frictionless disks near the unjamming transition. We construct polygonal tilings and triangulations of the contact network that partitions space into convex regions which are either covered or uncovered. This allows us to characterize the local spatial structure of the packing near the transition using well-defined geometric objects. We construct bounds on the number of polygons and triangulation vectors that appear in such packings. We study these networks using simulations of bidispersed disks interacting via a one-sided linear spring potential. We find that several underlying geometric distributions are reproducible and display self averaging properties. We find that the total covered area is a reliable real space parameter that can serve as a substitute for the packing fraction. We find that the unjamming transition occurs at a fraction of covered area A * G = 0.446(1). We determine scaling exponents of the excess covered area as the energy of the system approaches zero E G → 0 + , and the coordination number zg approaches its isostatic value ∆Z = zg − zg iso → 0 + . We find ∆A G ∼ ∆E G 0.28(1) and ∆A G ∼ ∆Z 1.00(1) , representing new structural critical exponents. We use the distribution functions of local areas to study the underlying geometric disorder in the packings. We find that a finite fraction of order Ψ * O = 0.369(1) persists as the transition is approached.

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Section: Coordination and Isostaticitymentioning

confidence: 99%

Disordered contact networks in jammed packings of frictionless disks

Ramola¹,

Chakraborty²

2016

J. Stat. Mech.

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“…Also consider materials where their microscale subelements geometries could be tessellated. For example, Figure 1 shows two optional tessellations for granular medium, discussed with other options by Satake [1978]. Due to the random physical properties of the subelements and their random geometrical configuration within the tessellation yielding occurs gradually within the RVE and this is often ignored.…”

Section: Micro Meso Probabilistic Approachmentioning

confidence: 99%

A thermomechanical framework of plasticity based on probabilistic micromechanics

Einav

Collins

2008

JOMMS

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A conventional thermomechanical elasto-plastic constitutive modelling framework is proposed but which allows an effective physical interpretation of the micromechanical internal variables and inherent parameters in a representative volume element (RVE). The statistical distribution of the elastic-plastic parameters within the RVE is described using a simple statistical method of probabilistic rearrangement (MPR). This method facilitates deriving analytical approximating formulas to the stress-strain response of the RVE under unidirectional monotonic loading conditions. The applicability of the MPR is validated numerically by comparing the analytical formulas against numerical experiments of a probabilistic boundary value problem initially under monotonic conditions. The assumptions associated with the MPR are embedded within the simple micro thermomechanical constitutive framework and further evaluated numerically by applying the methodology for the case of cyclic loads. For the limited experimental program the constitutive modelling framework seems to give a rather effective estimation of the full boundary value problem. The results for the cyclic case demonstrate how hysteresis behaviour of materials could be modelled without incorporating kinematic hardening parameters.

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“…Thus, for the Satake graph G, the substitutions f → f , and u → u in (15-16) yields the quasi-static equilibrium of forces and the compatibility of relative velocities (or displacements) [31,38]:…”

Section: Graph Theory For Extrinsic Modesmentioning

confidence: 99%

“…In a schema dating back to the early works of Satake [31,37,38] 5 , we let particle centroids define the the nodes or vertices j ∈ [1, N ] defined by an appropriate Delaunay triangulation [1,19,20,37]. Note that this triangulation should be generally based on (minimal) separation between particle surfaces rather than particle centroids.…”

Section: Graph Theory For Extrinsic Modesmentioning

confidence: 99%

See 1 more Smart Citation

From Granular Matter to Generalized Continuum

Goddard

2008

Mathematical Models of Granular Matter

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Summary. Following a cursory review and synthesis of multipolar continua, the rudiments of graph theory, and granular mechanics, a graph-theoretic, energy-based homogenization is proposed for the systematic derivation of multipolar stress and kinematics in granular media. This provides a weakly non-local hierarchy of multipolar field equations for quasi-static mechanics based on polynomial representations of the kinematics of the type employed in past works. As an improvement on those works, a method is proposed for avoiding "overfitting" of fluctuations based on the so-called "Generalized Additive Method" of statistics. Among other results, it is shown that the standard formula for Cauchy stress in granular media may break down owing to multipolar effects, and that granular rotations in the typical granular medium should not lead to Cosserat effects, as the lowest-order departure from the simple-continuum model.

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New formulation of graph-theoretical approach in the mechanics of granular materials

Cited by 41 publications

References 3 publications

Disordered contact networks in jammed packings of frictionless disks

Disordered contact networks in jammed packings of frictionless disks

A thermomechanical framework of plasticity based on probabilistic micromechanics

From Granular Matter to Generalized Continuum

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