Uniqueness and stability in the continuum theory of granular materials

Meccanica ◽  
1976 ◽  
Vol 11 (1) ◽  
pp. 18-22
Author(s):  
Franco Angotti
Keyword(s):  
Granular Materials ◽  
Continuum Theory ◽  
The Continuum

Free Energy of Granular Materials in Static Equilibrium

1979 ◽  
Vol 46 (4) ◽  
pp. 944-945 ◽  
Author(s):  
M. Shahinpoor ◽  
G. Ahmadi
Keyword(s):  
Free Energy ◽  
Granular Materials ◽  
Continuum Theory ◽  
Gravitational Effect ◽  
Experimental Results ◽  
Static Equilibrium ◽  
The Continuum

We employ the continuum theory of granular materials due to Goodman and Cowin and some experimental results due to P. G. Nutting to arrive at a functional from for the free energy of granular materials in static equilibrium. The results obtained indicate the dominance of gravitational effect, modify and enlarge the results previously obtained by J. T. Jenkins.

A unified formulation for plasticity models of granular and other materials

1995 ◽  
Vol 450 (1938) ◽  
pp. 37-49 ◽  
Keyword(s):  
Granular Materials ◽  
Potential Model ◽  
Continuum Theory ◽  
Potential Models ◽  
Special Cases ◽  
Plastic Potential ◽  
Physically Based ◽  
Relationship Of ◽  
The Relationship ◽  
The Continuum

A general physically based rule for the kinematics of planar deformation of granular materials is presented and a pair of kinematic equations governing the velocity field is obtained. The method results in a unified derivation of the equations for the double sliding, free rotating model, the double shearing model and the plastic potential model for granular materials. The formulation of the double sliding, free rotating model for the kinematics of granular materials presented here is consistent with the continuum theory of constitutive equations and demonstrates the relationship of this model both to the double shearing model and to the plastic potential models. Expressions for the rate of working of the stresses are presented. A constitutive equation, expressed in terms of planar tensor quantities, is also presented that has the three models as special cases.

Confirming the continuum theory of dynamic brittle fracture for fast cracks

Nature ◽  
10.1038/16891 ◽  
1999 ◽  
Vol 397 (6717) ◽  
pp. 333-335 ◽  
Author(s):  
Eran Sharon ◽  
Jay Fineberg
Keyword(s):  
Brittle Fracture ◽  
Continuum Theory ◽  
Dynamic Brittle Fracture ◽  
The Continuum

scholarly journals Matrix integrals & finite holography

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Dionysios Anninos ◽  
Beatrix Mühlmann
Keyword(s):  
Critical Exponents ◽  
Continuum Theory ◽  
Point Of View ◽  
Leading Order ◽  
Minimal Models ◽  
Point Evaluation ◽  
Brst Cohomology ◽  
Matrix Integrals ◽  
The Matrix ◽  
The Continuum

Abstract We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

On the Motion of Granular Materials Due to Shear

2000 ◽  
Author(s):  
Mehrdad Massoudi ◽  
Tran X. Phuoc
Keyword(s):  
Finite Difference ◽  
Granular Materials ◽  
Continuum Model ◽  
Theory Model ◽  
Governing Equations ◽  
Flat Plates ◽  
Material Parameters ◽  
Finite Difference Technique ◽  
Linear Differential ◽  
The Continuum

Abstract In this paper we study the flow of granular materials between two horisontal flat plates where the top plate is moving with a constant speed. The constitutive relation used for the stress is based on the continuum model proposed by Rajagopal and Massoudi (1990), where the material parameters are derived using the kinetic theory model proposed by Boyle and Massoudi (1990). The governing equations are non-dimensionalized and the resulting system of non-linear differential equations is solved numerically using finite difference technique.

Dislocation Threading Through an Epitaxial Film: an Analysis Based on the Peierls-Nabarro Concept

1993 ◽  
Vol 308 ◽  
Author(s):  
G. E. Beltz ◽  
L. B. Freund
Keyword(s):  
Critical Thickness ◽  
Epitaxial Film ◽  
Continuum Theory ◽  
Dislocation Core ◽  
Critical Layer Thickness ◽  
Cutoff Parameter ◽  
Mismatch Strain ◽  
Realistic Description ◽  
Cutoff Radius ◽  
The Continuum

ABSTRACTThe Peierls-Nabarro theory of crystal dislocations is applied to estimate the critical thickness of a strained layer bonded to a substrate for a given mismatch strain. Previous analyses were based on the continuum theory of elastic dislocations, and hence depended on the artificial core cutoff parameter r0. The Peierls-Nabarro theory makes use of an interplanar shear law, which leads to a more realistic description of the stresses and displacements in the vicinity of a dislocation core, thus eliminating the need for the core cutoff parameter. The dependence of the critical layer thickness on the mismatch strain in films with a diamond cubic lattice is found to be similar to that predicted by the continuum elastic dislocation theory, provided that a core cutoff radius equal to about one-tenth the Burgers displacement is used.

scholarly journals Contributions to the theory of the specific heat of crystals I—Lattice theory and continuum theory

1935 ◽  
Vol 148 (864) ◽  
pp. 365-383 ◽  
Keyword(s):  
Specific Heat ◽  
Empirical Formula ◽  
Lattice Theory ◽  
Continuum Theory ◽  
Quantum Statistics ◽  
Physical Basis ◽  
General Explanation ◽  
The Continuum

The variation of the specific heat of crystals with temperature received a satisfactory general explanation as soon as quantum statistics were applied to the motion of the particles of which a crystal is composed. The first formula on such a basis, proposed by Einstein, gave fairly satisfactory results, but showed large discrepancies at the lowest temperatures; an empirical formula due to Nernst and Lindemann gave better agreement but lacked a satisfactory physical basis. Two consistent theories were advanced practically simultaneously by Debye and Born and v. Kármán. Debye's theory is based on the ingenious idea of replacing a crystal by a continuum as far as the distribution of the vibrations is concerned, cutting off the spectrum at a suitable point. Because of its inherent simplicity and because it can be applied to non-crystals as well as to crystals, the continuum theory has taken precedence and now is practically the only one which received attention.

scholarly journals Edge modes of gravity. Part II. Corner metric and Lorentz charges

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Laurent Freidel ◽  
Marc Geiller ◽  
Daniele Pranzetti
Keyword(s):  
Phase Space ◽  
Lorentz Invariance ◽  
Discrete Series ◽  
Continuum Theory ◽  
Symmetry Algebra ◽  
Area Spectrum ◽  
Area Element ◽  
Bf Theory ◽  
Tetrad Gravity ◽  
The Continuum

Abstract In this second paper of the series we continue to spell out a new program for quantum gravity, grounded in the notion of corner symmetry algebra and its representations. Here we focus on tetrad gravity and its corner symplectic potential. We start by performing a detailed decomposition of the various geometrical quantities appearing in BF theory and tetrad gravity. This provides a new decomposition of the symplectic potential of BF theory and the simplicity constraints. We then show that the dynamical variables of the tetrad gravity corner phase space are the internal normal to the spacetime foliation, which is conjugated to the boost generator, and the corner coframe field. This allows us to derive several key results. First, we construct the corner Lorentz charges. In addition to sphere diffeomorphisms, common to all formulations of gravity, these charges add a local $$ \mathfrak{sl} $$ sl (2, ℂ) component to the corner symmetry algebra of tetrad gravity. Second, we also reveal that the corner metric satisfies a local $$ \mathfrak{sl} $$ sl (2, ℝ) algebra, whose Casimir corresponds to the corner area element. Due to the space-like nature of the corner metric, this Casimir belongs to the unitary discrete series, and its spectrum is therefore quantized. This result, which reconciles discreteness of the area spectrum with Lorentz invariance, is proven in the continuum and without resorting to a bulk connection. Third, we show that the corner phase space explains why the simplicity constraints become non-commutative on the corner. This fact requires a reconciliation between the bulk and corner symplectic structures, already in the classical continuum theory. Understanding this leads inevitably to the introduction of edge modes.

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