(Tentative abstract) Isostaticity theory describes the continuous stress field in statically determinate systems, such as isostatic granular systems and open-cell cellular systems. In contrast to the paradigmatic elasticity theory, isostaticity theory decouples the stress and the strain fields. Its first-principles formulation, as well as explicit solutions, existed until now only in two-dimensions. The extension of the formalism to three-dimensions hinges on a derivation of three constitutive stress-structure relations to close the field equations, which proved to be quite elusive. This problem is resolved here, using insight from the two-dimensional analysis. The constitutive equations are derived in explicit form and they relate the local stress to a novel geometric tensor which characterises the geometry of the local microstructure. A discussion is presented towards the determination of the nature of the equations, i.e. hyperbolic versus elliptic, and whether the three-dimensional solutions give rise to force chains as the two-dimensional solutions do.
An observation is made that, under slow shear, granular systems come very close to the marginal rigidity state and are therefore nearly isostatic. This allows us to apply Isostaticity theory to examine the problem of jamming of sheared granular systems in two-dimensions. Two main results are reported: (i) force chains must develop in order to support the boundary loading just below the yield surface; (ii) in random isotropic media the force chains are roughly at $45\deg$ to the shearing direction.
This paper presents a new set of equations for the plastic yield rhgeology of granular systems. The equations are derived exactly for planar ordered systems and arguments are given in support of their validity to general granular systems both in two and in three dimensions. The equations are seamless with the equations of static stress transmission in planar assemblies so that when the strain rate vanishes the static stress field is automatically established. In contrast to most models of plasticity and yield of granular matter, these equations form a closed set, requiring no internal variables. We find that in $d$ dimensions ($d=2,3$) the equations describe $d^2+d+1$ degrees of freedom: the stress components, the strain rate components, the local rotational field and a scalar field that describes local sliding.
Isostaticity theory of stress transmission in disordered open-cell cellular solids is discussed and developed. A key concept is the static determinacy of the skeleton of such structures. The theory is described for two-dimensional systems, wherein the force and torque balance equations are closed by an explicit stress-structure constitutive relations. This makes it possible to determine stresses independently of stress-strain data, leading to field equations that differ significantly from conventional elasticity theory. As such they give rise to completely new micromechanics.
The general solution of the field equations are derived as a series expansion in the gradients of a local field that characterises the local microstructure. To lowest order the solutions are found explicitly. It is found that stresses are supported by non-uniform arches and force chains, to any order. These solutions are in marked contrast to elastic solutions.
This paper proposes a parameter-free model for nonequilibrium dynamics of disentanglement in polymer glasses. The model builds on a recently proposed theory for pullout of single chains from the bulk and applies to both internal deformations and failure at interfaces. It can be used to model failure of polymer/polymer, polymer/elastomer, and polymer/matrix interfaces and can also be relevant to the initial stages of cavitation in debonding of adhesives. By treating the dynamics, the model goes beyond existing theories that rely on thermodynamics alone and applies to systems that are far from equilibrium. The model requires knowledge of the distribution of disentanglement threshold forces, which is therefore derived here from a first-principle molecular argument. This distribution turns out to be dominated by large fluctuations, suggesting that it cannot be faithfully represented by one typical value. The model yields the stress as a function of time under a constant strain rate and provides a failure criterion. The deformation work per unit area is explicitly calculated. The results are given as a function of molecular properties and temperature. A numerical simulation is constructed for the dynamics and the results are in excellent agreement with the analytical calculations. Applications to various systems and phenomena are discussed.
We model ribbons as discrete line curves with an extra bivector degree of freedom. The additional degree of freedom can be represented by a classical Heisenberg spin vector coupled to the curve geometry. This makes it possible derive equations of motion for ribbons and analyse the generic dynamics of such objects. The resulting discrete Hamiltonian is analysed at the low energy approximation and two coupled equations of motion are derived. These equations are solved in various approximations, leading to a rich behaviour including formation of kinks and instanton-like conformations. The implications of the model to biological long molecules is discussed.
We derive a continuum approximation for the low-energy dynamics of classical Heisenberg spin chains with generally inhomogeneous coupling interactions between nearest neighbour spins, $J=J(x)$. We show that the known ferromagnetic and antiferromagnetic homogeneous cases can be obtained as particular cases of this formulation and report several new results: a) To lowest order in an energy parameter, $\epsilon$, we solve for chains with blocks of three spins, where within the block the couplings are (J,J,-J); b) We derive explicit equations for the spin configurations along the chain when $J(x)$ varies slowly along the chain; c) We discuss the statistics of the spins when $J(x)$ is fluctuating either randomly or with given correlations.
This paper addressses the kinetics and dynamics of a family of domain wall solutions (stripes) along classical antiferromagnetic Heisenberg spin chains at low energies. The equation of motion is derived and found to have long range position- and velocity-dependent two-body forces. A 'quiescent' regime is identified where the forces between walls are all repelling. Outside this regime some of the interactions are attractive, giving rise to wall collisions whereupon walls annihilate. The momentum of the system is found to be conserved in the quiescent regime and to suffer discontinuous jumps upon annihilation. The dynamics are elucidated by an exact solution for a double wall system and a numerical solution for a many-wall system. It is shown that on closed chains, the solutions of the equations gradually develop into periodic stripes and their stability is discussed.
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