We numerically and theoretically study the macroscopic properties of dense, sheared granular materials. In this process we first consider an invariance in Newton's equations, explain how it leads to Bagnold's scaling, and discuss how it relates to the dynamics of granular temperature. Next we implement numerical simulations of granular materials in two different geometries--simple shear and flow down an incline--and show that measurements can be extrapolated from one geometry to the other. Then we observe nonaffine rearrangements of clusters of grains in response to shear strain and show that fundamental observations, which served as a basis for the shear transformation zone (STZ) theory of amorphous solids [M. L. Falk and J. S. Langer, Phys. Rev. E. 57, 7192 (1998); M.R.S. Bull 25, 40 (2000)], can be reproduced in granular materials. Finally we present constitutive equations for granular materials as proposed by Lemaître [Phys. Rev. Lett. 89, 064303 (2002)], based on the dynamics of granular temperature and STZ theory, and show that they match remarkably well with our numerical data from both geometries.

We numerically study the jamming transition in particulate systems with attraction by investigating their mechanical response at zero temperature (T=0). We find three regimes of mechanical behavior separated by two critical transitions--connectivity and rigidity percolation. The transitions belong to different universality classes than their lattice counterparts, due to force balance constraints. We also find that these transitions are unchanged at low temperatures and resemble gelation transitions in experiments on colloidal and silica gels.

We examine two basic assumptions of kinetic theory-binary collisions and molecular chaos-using numerical simulations of sheared granular materials. We investigate a wide range of densities and restitution coefficients and demonstrate that kinetic theory breaks down at large density and small restitution coefficients. In the regimes where kinetic theory fails, there is an associated emergence of clusters of spatially correlated grains.For granular materials, kinetic theory has been the primary strategy used to systematically derive hydrodynamics equations, starting from elementary assumptions about grain-grain interactions [1]. This has led to much interest in applying predictions from the theory to realistic granular flows (for reviews, see [2]), and recent work continues in this direction [3,4]. However, kinetic theory strictly applies only to dilute gases, and the extent that it applies to the dense regime remains unclear.In this Letter we perform tests of the fundamental assumptions of kinetic theory, using the Contact Dynamics (CD) algorithm. We find that kinetic theory is severely limited by the assumption that only binary interactions occur between grains. Instead, an effective many body interaction arises that is a direct consequence of persistent contacts in the dense regime. In Fig. 1 we characterize the failure of the binary collision assumption, using spatial force correlations to approximate the average number of grains N c that form a cluster in contact. As we see, the cluster size increases when going to low restitution coefficient and high density, which should limit the relevance of kinetic theory. This Letter provides quantitative estimates of this breakdown.Most kinetic theory research starts with the Boltzmann equation, which is derived from the BBGKY hierarchy, and then finds its solutions [1,4,5]. However, certain assumptions are necessary to derive the Boltzmann equation. We begin by discussing two of these assumptions.Consider a system of N grains and the evolution equation for the N -body probability distribution function (pdf) f (N ) (r,p), where (r,p) denotes the set of all positions and momenta for the system (with the notatioñ r = { r i }). This equation is simply a statement of conservation of probability and reads

Unfolded proteins may contain native or non-native residual structure, which has important implications for the thermodynamics and kinetics of folding as well as for misfolding and aggregation diseases. However, it has been universally accepted that residual structure should not affect the global size scaling of the denatured chain, which obeys the statistics of random coil polymers. Here we use a single-molecule optical technique, fluorescence correlation spectroscopy, to probe the denatured state of set of repeat proteins containing an increasing number of identical domains, from two to twenty. The availability of this set allows us to obtain the scaling law for the unfolded state of these proteins, which turns out to be unusually compact, strongly deviating from random-coil statistics. The origin of this unexpected behavior is traced to the presence of extensive non-native polyproline II helical structure, which we localize to specific segments of the polypeptide chain. We show that the experimentally observed effects of PPII on the size scaling of the denatured state can be well-described by simple polymer models. Our findings suggest an hitherto unforeseen potential of non-native structure to induce significant compaction of denatured proteins, affecting significantly folding pathways and kinetics.

We study the response of dry granular materials to external stress using experiment, simulation, and theory. We derive a Ginzburg-Landau functional that enforces mechanical stability and positivity of contact forces. In this framework, the elastic moduli depend only on the applied stress. A combination of this feature and the positivity constraint leads to stress correlations whose shape and magnitude are extremely sensitive to the nature of the applied stress. The predictions from the theory describe the stress correlations for both simulations and experiments semiquantitatively. DOI: 10.1103/PhysRevE.80.060303 PACS number͑s͒: 45.70.Ϫn, 64.60.Ϫi, 83.80.Fg A striking feature of dry granular materials and other athermal systems is that they form force chain networks in response to applied stress, with large forces distributed inhomogeneously into linear chainlike structures ͓1,2͔. A number of experimental studies have visualized and quantified these networks in granular systems using carbon paper ͓3͔ and photoelastic techniques ͓4,5͔. These studies demonstrated that geometrical and mechanical properties of force chain networks are acutely sensitive to preparation procedures, especially near the jamming transition ͓6͔. For example, in isotropically compressed systems, force networks are ramified with only short-ranged spatial correlations of the stress. In contrast, in sheared systems, aligned force chains give rise to longer-ranged stress correlations in the compressive direction.Developing theoretical descriptions for the mechanical properties of granular media is challenging for several important reasons ͓7-10͔: ͑1͒ since tensile stresses are absent in dry granular materials they only remain intact via applied stress, making the limiting zero-stress isostatic state-where the number of degrees of freedom matches the number of constraints ͓11-13͔-problematic; ͑2͒ forces at the microscopic level are indeterminate due to friction and disorder; ͑3͒ granular materials are athermal, so that conventional energy-based statistical approaches are not appropriate; and ͑4͒ near isostaticity we expect fluctuations to be important, both within a single realization of a system and from realization to realization. New methods are needed to bridge the gap between force networks at small length scales and continuum elastoplastic theory at large scales, and to capture the highly sensitive, fluctuating behavior of granular systems near jamming.In this Rapid Communication, we construct a model for stress fluctuations based on grain-scale force and torque balance, positivity of contact forces, and entropy maximization. We then calculate stress correlations and predict differences for systems under isotropic compression versus pure shear. We also perform complementary numerical simulations and experimental studies of jammed granular systems in twodimensions ͑2D͒ subject to isotropic compression and pure shear. The stress correlation functions from theory, simulation, and experiment are in qualitative and in some cases quantitative...