We introduce and study a da Vinci Fluid, a fluid whose dissipation is dominated by solid friction. We analyse the flow rheology of a discrete model and then coarse-grain it to the continuum. We find that the model gives rise to behaviour that is characteristic of dense granular fluids. In particular, it leads to plug flow. We analyse the nucleation mechanism of plugs and their development. We find that plug boundaries generically expand and we calculate the growth rate of plug regions. In systems whose internal effective friction coefficient is relatively uniform we find that the linear size of plug regions grows as (time)$^{1/3}$. The suitability of the model to granular materials is discussed.

We discuss the micro-states of compressed granular matter in terms of two independent ensembles: one of volumes and another of boundary force moments. The former has been described in the literature and gives rise to the concept of compactivity - a scalar quantity that is the analogue of temperature in thermal systems. The latter ensemble gives rise to another analogue of the temperature - an angoricity tensor. We discuss averages under either of the ensembles and their relevance to experimental measurements. We also chart the transition from the micro-canoncial to a canonical description for granular materials and show that one consequence is that the well known exponential distribution of forces in granular systems subject to external forces is an immediate consequence of the canonical distribution, just as in the micro-canonical description E = H leads to exp(-H/kT). We then follow with a discussion of some unresolved issues. One is that the lack of ergodicity prevents convenient translation between time and ensemble averages and the problem is illustrated in the context of diffusion. Another issue is that it is unclear how to make use in the statistical formalism the emerging ability to predict stress fields exactly for given structures of granular systems.

We analyze a recently proposed continuous model for stress fields that develop in two-dimensional purely isostatic granular systems. We present a reformulation of the field equations, as a linear first order hyperbolic system, and show that it is very convenient both for analysis and for numerical computations. Our analysis allows us to predict quantitatively the formation and directions of stress paths and, from these, trajectories and magnitudes of force chains, given the structure in terms of a particular fabric tensor. We further predict quantitatively changes of stresses along the paths, as well as leakage and branching of stress from the main paths into the cones that they make in terms of the fabric tensor. Numerical computations in both Cartesian and cylindrical coordinates verify the analytic results and illustrate the rich behavior discovered. All the phenomena predicted by our solutions have been observed experimentally, suggesting that stresses in isostatic systems can form a base model for a more developed stress theory in granular materials.

A recently developed method is used for the analysis of structures of planar disordered granular assemblies. Within this method, the assemblies are partitioned into volume elements associated either with grains or with more basic elements called quadrons. Our first aim is to compare the relative usefulness of description by quadrons or by grains for entropic characterization. The second aim is to use the method to gain better understanding of the different roles of friction and grain shape and size distributions in determining the disordered structure. Our third aim is to quantify the statistics of basic volumes used for the entropic analysis. We report the following results. 1 Quadrons are more useful than grains as basic ‘‘quasiparticles’’ for the entropic formalism. 2 Both grain and quadron volume distributions show nontrivial peaks and shoulders. These can be understood only in the context of the quadrons in terms of particular conditional distributions. 3 Increasing friction increases the mean cell size, as expected, but does not affect the conditional distributions, which is explained on a fundamental level. We conclude that grain size and shape distributions determine the conditional distributions, while their relative weights are dominated by friction and by the pack formation process. This separates sharply the different roles that friction and grain shape distributions play. 4 The analysis of the quadron volumes shows that distributions, which are accepted to describe foamlike structures well, are too simplistic for general granular systems. 5 A range of quantitative results is obtained for the ‘‘density of states’’ of quadron and grain volumes and calculations of expectation values of structural properties are demonstrated. The structural characteristics of granular systems are compared with numerically generated foamlike Dirichlet-Voronoi constructions.

In this paper we characterize foams and tetrahedral structures in a unified way, by a simplified representation of both that conserves the system's topology. The paper presents a workflow for an automated characterization of the topology of the void space, using a partition of the void space into polyhedral cells connected by windows. This characterization serves as the basic input for the Edwards entropic formalism that deals with the statistical characterization of configurational disorder in granular aggregates and argued to work for foams. The Edwards formalism is introduced and simplified expectation-values are calculated.

The stress field equations for two-dimensional disordered isostatic granular materials are reformulated, giving new results beyond the commonly accepted force chains. Localized loads give rise to exactly determinable cones of influence, bounded by stress chains. Disorder couples same-source chains, attenuates stresses along chains, causes stress leakage from chains into the cone, and gives rise to branching. Chains from spatially separate sources do not interact. The formulation is convenient for computation and several numerical solutions are presented.

This chapter presents recent developments in entropic characterization of granular materials. The advantages of the formalism and its use are illustrated for calculation of structural characteristics, such as porosity fluctuations and the throat size distribution. I discuss the relations between the entropic formalism and stress transmission. It is argued that a new sub-ensemble of loading distributions is necessary, which introduces a tensor temperature-like quantity named angoricity.

It has been recognized that the concept of isostaticity holds the key to understanding stress transmission in cohesionless granular media. Here the field equations of isostaticity theory in two-dimensions are studied and solutions are derived.
The equations are first decoupled into integro-differential equations for the three independent stress components, highlighting the role of a particular position-dependent fabric tensor.
In disordered, but statistically isotropic, systems the fluctuations decay with length-scale and the decoupled equations can be expanded to first order and solved.
The zero-order solutions are obtained in closed form and give rise to force chains that propagate along straight characteristic lines. At this order solutions do not attenuate and chains cross one another without interference or scattering.
The analysis of the first order correction reveals emergence of weaker secondary force chains that branch off the zero-order chains and into a 'cone of influence' that they span.

The current controversy, whether hyperbolic equations can describe stresses in macroscopic materials, has spread recently from granular to cellular solids. Such equations are incompatible with elasticity theory, which underpins our understanding and modelling of solid mechanics in general. Hyperbolic equations, broadly termed isostaticity theory, give rise to arching and force chains that propagate with little attenuation, while conventional stresses decay away from a source.
We investigated the issue experimentally, using photoelastic cellular structures. Our primary observation is that localised boundary loads indeed propagate into cellular media along force chains, supporting a hyperbolic behaviour. This observation is particularly surprising in light of the fact that our samples are not necessarily statically determinate - a class of systems for which isostaticity theory was originally developed.
Our secondary observation is of another twist in the tale. While a hyperbolic theory would predict that force chains traverse the entire system, we find that they terminate before reaching the boundary. Hypothesising that pure hyperbolic behaviour can only be observed in skeletal structures, we attribute this to elastic correction introduced by finiteness of cell walls. To test this idea, we measure the length of force chains as cell wall are thinned at fixed topologies. We find that chains grow longer as structures approach the skeletal form, supporting our hypothesis. We therefore propose that: i) realistic irregular cellular solids should be regarded as two-phase composites of isostatic and elastic domains, necessitating a new 'stato-elastic' theory; ii) structural proximity to the skeletal state can be quantified by a new parameter \lambda which vanishes at this state and diverges for purely elastic materials; iii) near the skeletal state the dominant behaviour is hyperbolic with elliptic corrections that can be represented theoretically as a series expansion in powers of \lambda. Irregular cellular solids and foams are an important class of materials both due to their ubiquity in natural and man-made environments and due to the wide range of their technological applications. These results not only shed light on the ongoing controversy but also have significant implications for the modeling and understanding of the mechanics of these materials.

Granular materials can exist in infinitely many configurations, but under well defined external influences and conditions can exhibit perfectly reproducible behaviour, and therefore must be possible to describe by statistical mechanical laws. These must be quite different then the traditional, Hamiltonian, statistical mechanics since the dynamics involved in changing from one configuration to another is dominated by friction. The ergodic, self-sustained equilibration of of conventional statistical mechanics is then replaced by externally induced changes from one jammed configuration to another. Several questions arise, which we will attempt to answer here:
1. How does one specify a reproducible state of a granular system to which it can return after disturbance?
2. Stress due to external fields or boundary forces will propagate through the granular medium even if the grains are perfectly rigid and cannot be strained. This means that the stress is not related to the strain and stress-strain constitutive information is redundant. These are termed isostatic states. Even if the grains are not perfectly rigid the system can be isostatic. What are then the macroscopic constitutive equations required to determine the stress?
3. It appears that fluctuations of local properties are important to the stress equations. What exactly is the key quantity which fluctuates and how can the distribution of the fluctuations be found?
4. The three points above are theoretical in nature, but they must be supported experimentally. It is not difficult to generate spectacular but incomprehensible effects in granular materials, but there are experiments that really test basic fundamental concepts and these will be described.

The aim of this paper is to formulate a new framework for characterisation of porous materials and for obtaining first-principles structure-permeability relations. The formaslism is based on a representation of the porous medium in skeletal form and involves several steps. First, the skeleton is characterised by a fabric tensor that describes the local pore structure. Second, the fabric tensor is used to construct a configurational entropy of the porous medium, based on Edwards' compactivity concept. Both the fabric tensor and the entropic analysis are initially illustrated in two-dimensions and are then extended to three-dimensional systems. Third, the local pore-scale permeability is expressed in terms of the structure of the skeleton and the degrees of freedom that comprise the phase space of configurational states. Fourth, the pore-scale permeability is calculated as an expectation value over the partition function.
We propose that the same procedure can be used to find the conductivity of the porous medium when it is filled with brine water. This makes it possible to derive a theoretical relation between these two transport properties.

A new model is proposed for force transmission through the cytoskeleton. A general discussion is first presented on the physical principles that underlie the modelling of this phenomenon. Some fundamental problems of conventional models - continuous and discrete - are examined. It is argued that mediation of focused forces is essential for good control over intracellular mechanical signals. The difficulties of conventional continuous models to describe such mediation are traced to a fundamental assumption rather than to them being continuous. Relevant advantages and disadvantages of continuous and discrete modelling are discussed. It is concluded that favouring discrete models is based on two misconceptions which are clarified. The model proposed here is based on the idea that focused propagation of mechanical stimuli in frameworks over large distances (compared to the mesh size) can only occur when considerable regions of the CSK are isostatic. The concept of isostaticity is explained and a recently developed isostaticity theory is briefly reviewed. The model enjoys several advantages: it leads to good control over force mediation; it explains nonuniform stresses and action at a distance; being continuos it makes it possible to model long-scale force propagation; it enables prediction of individual force paths. To be isostatic or nearly so, cytoskeleton networks must possess specific structural characteristics and these are quantified explicitly. Finally, several implications of the new model are discussed. In particular, it is suggested that the mechanism for static force transmission may give insight into the dynamics of reorganisation of the CSK. Many of the results are amenable to experimental measurements, providing a testing ground for the proposed picture.

Auxeticity is the result of internal structural degrees of freedom that get in the way of affine deformations. This paper proposes a new understanding of strains in disordered auxetic materials. A class of iso-auxetic structures is identified, which are auxetic structures that are also isostatic, and these are distinguished from conventional elasto-auxetic materials. It is then argued that the mechanisms that give rise to auxeticity are the same in both classes of materials and the implications of this observation on the equations that govern the strain are explored. Next, the compatibility conditions of Saint Venant are demonstrated to be irrelevant for the determination of stresses in iso-auxetic materials, which are governed by balance conditions alone. This leads to the conclusion that elasticity theory is not essential for the general description of auxetic behaviour. One consequence of this is that characterization in terms of negative Poisson's ratio may be of limited utility. A new equation is then proposed for the dependence of the strain on local rotational and expansive fields. Central to the characterization of the geometry of the structure, to the iso-auxetic stress field equations, and to the strain-rotation relation is a specific fabric tensor. This tensor is defined here explicitly for two-dimensional systems, however disordered. It is argued that, while the proposed dependence of the strain on the local rotational and expansive fields is common to all auxetic materials, iso-auxetic and elasto-auxetic materials may exhibit significantly different macroscopic behaviours.

A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies.

Granular materials can exist in infinitely many configurations, but under well defined external influences and conditions can exhibit perfectly reproducable behaviour, and therefore must be possible to describe by statistical mechanical laws. These must be quite different then the traditional, Hamiltonian, statistical mechanics since the dynamics involved in changing from one configuration to another is dominated by friction. Thus the ergodic, self-sustained equilibration of of conventional statistical mechanics is replaced by externally induced changes from one jammed configuration to another. Several questions arise, which we will attempt to answer here:

\item{1.} How does one specify a reproducible state of a granular system to which it can return after disturbance, by repeating the history of its formation?

\item{2.} Stress due to external fields or boundary forces will propagate through the granular medium even if the grains are perfectly rigid and cannot be strained. This means that the stress is not related to the strain and stress-strain constitutive information is redundant. What are then the constitutive equations required to determine the stress?

\item{3.} It appears that fluctuations are important to the stress equations. What exactly is the key quantity which fluctuates and how can the distribution of the fluctuations be found?

\item{4.} The three points above are theoretical in nature, but they must be supported experimentally. It is not difficult to generate spectacular but incomprehensible effects in granular materials, but there are experiments that really test basic fundamental concepts and these will be described.

The isostaticity theory for stress transmission in macroscopic planar particulate assemblies is extended here to non-rigid particles. It is shown that, provided that the mean coordination number in $d$ dimensions is $d+1$, macroscopic systems can be mapped onto equivalent assemblies of perectly rigid particles that support the same stress field. The error in the stress field that the compliance introduces for finite systems is shown to decay with size as a power law. This leads to the conclusion that the isostatic state is not limited to infinitely rigid particles both in two and in three dimensions, and paves the way to an application of isostaticity theory to more general systems.

Progress is reported on several questions that bedevil understanding of granular systems: (i) are the stress equations elliptic, parabolic or hyperbolic? (ii) how can the often-observed force chains be predicted from a first-principles continuous theory? (iii) How do we relate insight from isostatic systems to general packings? Explicit equations are derived for the stress components in two dimensions including the dependence on the local structure. The equations are shown to be hyperbolic and their general solutions, as well as the Green function, are found. It is shown that the solutions give rise to force chains and the explicit dependence of the force chains trajectories and magnitudes on the local geometry is predicted. Direct experimental tests of the predictions are proposed. Finally, a framework is proposed to relate the analysis to non-isostatic and more realistic granular assemblies.

A recent theory for stress transmission in isostatic planar granular assemblies and trivalent cellular solids predicts a constitutive equation that couples the stress field to the local microstructure [R. C. Ball and R. Blumenfeld, Phys. Rev. Lett. 88, 115505 (2002)]. The theory has been difficult to apply to macroscopic systems because the constitutive equation becomes trivial when coarse-grained by a simple area-average. This problem is resolved here for arbitrary planar topologies. The solution is based on the observation that a staggered order makes it possible to write the equation in terms of a reduced geometric tensor that can be upscaled. The method proposed here makes it possible to apply this idea to realistic systems whose staggered order is generally 'frustrated'. The method consists of a systematic renormalization, which removes the frustration and enables the use of the reduced geometric tensor. A calculation of the stress due to a defect in a honeycomb cellular system is presented as an example.

The marginally rigid state is a candidate paradigm for what makes granular material a state of matter distinct from both liquid and solid. The coordination number is identified as a discriminating characteristic, and for rough-surfaced particles we show that the low values predicted are indeed approached in simple two-dimensional experiments. We show calculations of the stress transmission, suggesting that this is governed by local linear equations of constraint between the stress components. These constraints can in turn be related to the generalized forces conjugate to the motion of grains rolling over each other. The lack of a spatially coherent way of imposing a sign convention on these motions is a problem for scaling up the equations, which leads us to attempt a renormalization-group calculation. Finally, we discuss how perturbations propagate through such systems, suggesting a distinction between the behaviour of rough and smooth grains.

This paper proposes a new volume function for calculation of the entropy of planar granular assemblies. This function is extracted from the antisymmetric part of a new geometric tensor and is rigorously additive when summed over grains. It leads to the identification of a conveniently small phase space. The utility of the volume function is demonstrated on several case studies, for which we calculate explicitly the mean volume and the volume fluctuations.

Stress transmission in planar open-cell cellular solids is analysed using a recent theory developed for marginally rigid granular assemblies. This is made possible by constructing a one-to-one mapping between the two systems. General trivalent networks are mapped onto assemblies of rough grains, while networks where Plateau rules are observed, are mapped onto assemblies of smooth grains. The constitutive part of the stress transmission equations couples the stress directly to the local rotational disorder of the cellular structure via a new fabric tensor. An intriguing consequence of the analysis is that the stress field can be determined in terms of the microstructure alone independent of stress-strain information. This redefines the problem of structure-property relationship in these materials and poses questions on the relations between this formalism and elasticity theory. The deviation of the stress transmission equations from those of conventional solids has been interpreted in the context of granular assemblies as a new state of solid matter and the relevance of this interpretation to the state of matter of cellular solids is discussed.

The equation of motion of twists on classical antiferromagnetic Heisenberg spin chains are derived. It is shown that twists interact via position- and velocity-dependent long-range two-body forces. A quiescent regime is identified wherein the system conserves momentum. With increasing kinetic energy the system exits this regime and momentum conservation is violated due to walls annihilation. A bitwist system is shown to be integrable and its exact solution is analysed. Many-twist systems are discussed and novel periodic twist lattice solutions are found on closed chains. The stability of these solutions is discussed.

The transmission of stress through a marginally stable granular pile in two dimensions is exactly formulated in terms of a vector field of loop forces, and thence in terms of a single scalar potential. This leads to a local constitutive equation coupling the stress tensor to fluctuations in the local geometry. For a disordered pile of rough grains this means the stress tensor components are coupled in a frustrated manner. In piles of rough grains with long range staggered order, frustration is avoided and a simple linear theory follows. We show that piles of smooth grains can be mapped onto a pile of unfrustrated rough grains, indicating that the problems of rough and smooth grains may be fundamentally distinct.

This paper examines the effect of cooling on disentanglement forces in polymers and the implications for both single chain pullout and polymer dynamics. I derive the explicit dependence of the distribution of these forces on temperature, which is found to exhibit a rich behaviour. Most significantly, it is shown to be dominated by large fluctuations up to a certain temperature $T_0$ that can be determined from molecular parameters. The effects of these fluctuations on chain friction are analysed and they are argued to undermine the traditional melt-based models that rely on a typical chain friction coefficient. A direct implication for first principles calculation of viscosity is discussed. This quantifies the limit of validity of such descriptions, such as Rouse dynamics and the Tube model, and pave the way to model polymer dynamics around the glass transition temperature.

This paper addressses the kinetics and dynamics of a family of domain wall solutions 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 the colliding 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 illustrated by an exact solution for a double wall system and a numerical solution for a many-wall system. On circular chains the equations support stable periodic stripes that can rotate as a rigid body. It is found that the stripes are more stable the faster they rotate. The periodic structure can be destabilised by perturbing the walls' angular velocity in which case there is a transition to another periodic structure, possibly via a cascade of annihilation events.

The Belavin-Polyakov equation t u = t × t s with t 2 = 1 has been shown recently to describe the evolution of a wide class of space curves, and has several physical applications. Here, we obtain a hierarchy of exact multi-twist solutions for this nonlinear system. As an illustration, we apply our results to continuum magnetic models. When u and s denote temporal and spatial variables respectively, these twists describe very low-energy domain walls travelling along an antiferromagnetic spin chain. When they denote independent spatial variables, the solutions represent twists in the static configuration (texture) of a two-dimensional ferromagnet.

Recent experiments on the pullout of single polymer chains have revealed a complex behavior of the force fluctuations. This paper analyzes the pullout process theoretically and numerically and shows that these fluctuations can be made to shed light on disentanglement dynamics. To facilitate the analysis, I first derive the probability density function of the threshold force needed to disentangle one entanglement point. This function is found to be dominated by large fluctuations, which bears directly on the observed statistics. The average and variance of the force are calculated, and a numerical investigation of the dynamics is carried out to check the results. Finally, applications to deformations in several macroscopic systems are discussed.

I discuss thin magnetic layers in the context of a two-dimensional ferromagnetic Heisenberg spin system. In the low energy regime it is shown that the system follows a Belavin-Polyakov type of equation. It is argued that unlike in traditional systems where these equations occur only under uniform boundary conditions, the boundary conditions in this case are less restrictive, allowing for a new family of solutions. These solutions consist of magnetic domains of spins that are oriented at relative opposite directions. The boundaries between regions are sharp on the continuous scale but within a domain wall the magnetization changes orientation continuously from one ground state to another. All the magnetic energy in the system is shown to concentrate along the domain walls. It is therefore argued that most favorable for the wails is to rearrange in a hierarchical or fractal fashion because such an arrangement lowers the overall energy density. It is suggested that this hierarchical structure of magnetic domain boundaries should be observable by magnetic force microscopy. Recent results suggest that such configurations may also dominate the structure of domain walls in magnetostrictive materials and magnetic nanotubes.

Recent years witnessed an explosion of research activity in single-wall nanotubes (SWNT). Their phenomenal strength is accompanied by a relative softness to radial deformations, which we propose to exploit for actively controlling the shape of SWNT-based composite materials. We show that if the SWNT are supplemented with magneto-elastic interactions then magnetic domain walls can form on the curved surface. The magneto-elastic coupling leads to surface wrinkles along the domain walls. We suggest that, by applying an external magnetic field, one can control the number and location of the wrinkles, thus magneto-actively controlling the material properties.