CMDS-12

We generalize the Ericksen-Leslie continuum model of liquid crystals to allow for dynamically evolving line defect distributions. Defects in orientational and positional order are represented through the incompatibility of the director and deformation ‘gradient’ fields. Due to their geometric basis as rigorous spatial densities, conservation laws for the defect fields are immediate. This dynamics is shown to satisfy the constraints, in this case quite restrictive, imposed by material frame indifference. The phenomenon of permeation appears as a natural consequence of our kinematic approach. The approach also leads to a natural unification of the Ericksen-Leslie director vector field based model and the De Gennes orientational tensor order-parameter field. As a consequence, it provides a dynamics for the kinematically powerful tensor field in terms of the physically-motivated and experimentally well-characterized director model. The resulting tensor dynamical equation allows us to deal with disclinations in a direct and precise manner.

Joint work with Kaushik Dayal

Fluctuation relations establish rigorous identities for the nonequilibrium averages of observables. Starting from a general transport master equation with time-dependent rates, we employ the stochastic path integral approach to study statistical fluctuations around such averages. We show how under nonequilibrium conditions, rare configurations of the discrete particles underlying a transport process imply massive fluctuations. These flucutations encode vital information on the microscopic mechanisms driving a system out of equilbrium. We illustrate our results on the paradigmatic example of a mesoscopic RC circuit.

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The talk will review both statics and dynamics of electrical breakdown in composites emphasizing the interplay of disorder and geometry.

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Hydraulic jump is an ancient problem with a not very satisfactory quantitative solution.However,an observation by Schuetzhold and Unruh in the last decade opened up a new dimension in the study of this ubiquitous phenomenon.They pointed out an analogy with gravitational objects with a boundary which allows only one way passage.In this talk we will discuss different aspects of the jump in one and two dimensions and talk about some rather intriguing experiments done in the last few years.

Buckling of a straight elastic column subject to compressive end thrust occurs at a critical load for which the straight configuration of the column becomes unstable and simultaneously ceases to be the unique solution of the elastic problem (so that instability and bifurcation are concomitant phenomena). Buckling is known from ancient times: it has been experimentally investigated in a systematic way by Pieter van Musschenbrok (1692-1761) and mathematically solved by Leonhard Euler (1707-1783), who derived the differential equation governing the behaviour of a thin elastic rod suffering a large bending, the so-called `elastica' (see Love, 1927).

Through centuries, engineers have experimented and calculated complex structures, such as frames, plates and cylinders, manifesting instabilities and bifurcations of various forms (Timoshenko and Gere, 1961). Until now structures exhibiting bifurcation and instability under tensile load of fixed direction and point of application (in other words `dead') have never been found, so that the word `buckling' is commonly associated by engineers to compressive loads.

We show that structures buckling in tension exist and we substantiate this statement with both theoretical and experimental proofs.

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We consider lattices from bistable links (rods). Each rod can be in equilibrium at two different lengths, it possesses a non-unimodal force-elongation dependence. The assembly mimics materials under phase transition and destructable structure. A lattice with bistable links has multiple equlibria, some of them (still states) are not accompanied with inner stresses. We describe a set of such states and transitions between them. We also discuss instabilities and uncertainties of the state of bistable structures, waves of transition in the structures, and an adequate "dynamic homogenization."

A special attention is paid to bistable links structures with irreversible transition that model a damageable elements and are called "waiting links". These structures are designed to withstand an impact: they delocalize an impact stress, spread and dissipate an impact energy. They delocalize damage: due to inner instabilities, it stops a damage development where it has started, let it happence in a different link, and then again. Waiting links structures absorb many times more impact energy than a non-structured material. They also transport, dissipate, and radiate the impact energy by originating transitional waves of partial damage, which propagate similarly to the falling dominos train. We investigate a controllable dynamics of transition, calculate the speed of transition wave, introduce a damage tensor and describe its evolution. A numerical examples show protective structures that distribute partial damage over a large area and quickly transform the impact energy into high-frequency dissipative modes.

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We experimentally studied the melting and freezing behaviors of colloidal crystals composed of diameter tunable microgel spheres by bright-field and confocal video microscopies. The melting behaviors of three-dimensional (3D), two-dimensional (2D) and multilayer thin films of both single crystals and polycrystals were systematically studied with single-particle dynamics. Thick films (>4 layers) melt heterogeneously, while thin films (<5 layers) melt homogeneously even in polycrystals. A novel heterogeneous melting at dislocation is discovered in 5- to 12-layer films. The equilibrium phase behaviors are different in three thickness regimes: thick films have a liquid-solid coexistence regime which decreases with the film thickness and vanishes at 4 layers, thin films melt into the liquid phase in one step, while monolayers melt in two steps with an intermediate hexatic phase. Superheated crystals and homogeneous melting in 3D were directly visualized and studied by locally heating single crystals with a focused beam of light. In the freezing studies, we experimentally tested four empirical 2D freezing criteria in a thermal system for the first time and suggested four new freezing criteria. The critical nucleus size and the line tension in 2D nucleation have also been measured for the first time.

References:
[1] Y. Peng, Z.-R. Wang, A. Alsayed, A. Yodh, and Y. Han, Phys. Rev. Lett. 104, 205703 (2010)
[2] Z.-R. Wang, A. Alsayed, A. Yodh, and Y. Han, J. Chem. Phys. 132, 154501 (2010)
[3] Y. Han, N. Y Ha, A. Alsayed, and A. Yodh, Phys. Rev. E 77, 041406 (2008)

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Frictional properties of dense granular matter under pressure are closely associated with the friction on faults, because faults contain powdered rock that is ground up by the fault motion of the past. We shall discuss the velocity dependence of steady-state kinetic friction coefficient, which determines instability/stability of sliding friction. Focusing on the cooperative motion of grains, we derived a constitutive law describing the nature of granular friction in a wide range of shear rate. The result is compared with simulation and experiment. In addition, crackling noise in sheared granular matter is analyzed in terms of aftershocks and compared with experiments and earthquakes on natural faults.

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Spatiotemporal correlations of the spring-block (Burridge-Knopoff) model of earthquakes are studied by means of numerical computer simulations in both one and two dimensions. As a constitutive relation, a simple velocity-weakening friction law or the rate-and-state friction law are employed. Particular attention is paied to the nucleation process prior to mainshock and its relation to the underlying constitutive law. Statistical properties of earthqukaes such as the size distribution often exhibit "characteristic" behaviors, while "near-critical" behavior is observed under certain conditions. Continuum limit of the model is also discussed.

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We find a method to control morphology of desiccation crack patterns by using memory effect of paste. Pastes made of powder and water remember the direction of external mechanical fields, such as vibration and flow. These memories in pastes are sustained as microscopically anisotropic network structures of powder particles. When the pastes are dried, memories in pastes are visualized as macroscopically anisotropic crack patterns. Thus, by imprint memories into pastes before drying, the morphology of desiccation crack patterns can be controlled to be anisotropic ones, such as lamellar, radial, ring, spiral, and so on.

References:
[1] A. Nakahara and Y. Matsuo, J. Phys. Soc. Jpn. 74 (2005) 1362.
[2] A. Nakahara and Y. Matsuo, Phys. Rev. E, 74 (2006) 045102(R).
[3] Y. Matsuo and A. Nakahara, in preparation.

We show how to construct localized elliptic cell problems for homogenization with non-separated scales, high-contrast and arbitrary deterministic coefficients. Randomness, scale separation, mixing or "epsilon-sequences" are not required because the proposed method solely relies on the compactness of the solution space. The support of cell problems can be localized to arbitrarily small subsets of the whole domain and explicit approximation error estimates are obtained as a function of the size of those subsets. We show how the proposed method extends to the wave equation, elastodynamics and molecular dynamics. Various parts of this talk are joint work with L. Zhang, L. Berlyand, M. Federov, M. Desbrun, L. Kharevych and P. Mullen.

We discuss different methodology for studying fracture propagation at different length scale: 1) Atomistic modelling at nano-scale (using LAMMPS) 2) Pore scale modelling (using PFC) 3) Lattice beam modelling at mesoscopic scale 4) Mean-filed modelling using FBM and RFM 5) Reservoir scale modelling (Using DFN and DEM).

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The flocking model due to Vicsek and co-workers, in which point particles with fixed speed align with their neighbors, in the presence of noise, displays a nonequilibrium phase transition as noise is decreased or mean density increased, from a disordered to an ordered state where all particles move coherently. We present a one-loop self-consistent treatment of the role of density fluctuations, in the Toner-Tu continuum field theory corresponding to the Vicsek model. We show that the interaction of the order parameter with density fluctuations renders the transition discontinuous, in agreement with recent large-scale computer simulations.

We study the dynamics of reversible breakdown in a driven Random Resistor cum Tunneling-bond Network (RRTN). For a pedagogical review on the RRTN, see [1]. To model any generalised failure/breakdown, we use the paradigm of dielectric breakdown of a classical insulator between a pair of next nearest neighbour metallic bonds (above a microscopic threshold voltage due to 'semi-classical tunneling'). To wit, this threshold field may stand for the minimum field for the onset of classical fluid motion against the capi- llary (surface tension) forces in a disordered porous media or that for the onset of classical mechanical motion on a rigid surface due to the fric- tional forces, etc. In the electrical paradigm, the first-passage time of electrical charge through the whole system (i.e., onset time of the macro- scopic dielectric breakdown) is the general fracture/breakdown-time. Using the local equation of continuity at each node to enforce the global conti- nuity, we find an important and practically useful result that in large disordered systems, the breakdown-time bears a constant ratio to the sample dependent macroscopic relaxation time of the whole system [2] (i.e., the breakdown time is predictable on an average). Further, our earlier results show that the early dynamics is scale-free with two power-law regimes [1], as observed in many complex systems of Nature with statistically correlated randomness (including earthquakes). Eventually, the dynamics becomes exponential, i.e., acquires a time-scale, as it approaches a steady state, which is very robust against arbitrarily chosen initial field distributions. This strong memory attribute of the steady state, in spite of its intrinsic disorder, should be very useful for cognitive processes, learning, fault-tolerant coding, etc.[3]. We do also look at some aspects of 'information entropy' (athermal) in our model taking care of the correlated randomness of the t-bonds. Early results show some interesting trends in finite size systems. [1] A.K. Sen, "Nonlinear response, semi-classical percolation and break- down in the RRTN model" in "Quantum and Semi-classical Percolation and Breakdown in Disordered Solids," Lecture Notes in Physics v.762, Eds. A.K. Sen et.al., pp.21-82 (Springer, Berlin, 2009). [2] A.K. Sen, "Dynamics of breakdown in the RRTN model for nonlinear systems," in Proc. 11th Intl. Conf. on "Continuum Models and Discrete Systems" (CMDS11), Eds. D. Jeulin and S. Forest, pp.251-256 (Ecole des Mines de Paris, Paris, 2008). [3] A.K. Sen, "Strong memory and recognition in the RRTN model" in Proc. 4-th "Intl. Conf. on Natural Computn" (ICNC2008), Eds. M. Guo, et.al., v.3, p.339-343 (CPS, IEEE Computer Society, Los Alamitos, 2008).

We show, using molecular-dynamics simulations, that a two-dimensional, unstressed, Lennard-Jones solid exhibits droplet ï¬uctuations characterized by nonafï¬ne deviations from local crystallinity. The fraction of particles in these droplets increases as the mean density of the solid decreases and approaches 20% of the total number in the vicinity of the ï¬uid-solid phase boundary. We monitor the geometry, local equation of state, density correla- tions, and Van Hove functions of these droplets. We provide evidence that these nonafï¬ne heterogeneities should be interpreted as being droplet ï¬uctuations from nearby metastable minima. The local excess pressure of the droplets plotted against the local number density shows a van der Waal loop with distinct branches corresponding to ï¬uidlike compact and string-like glassy droplets. The distinction between ï¬uidlike and glassy droplets disappears above a well-deï¬ned temperature. When, on the other hand, an external stress is imposed, only string-like droplets remain which percolate for a value of stress much lower than the yield point. The percolating droplets of large non-affine regions appear to be precursors of shear bands along which the solid begins to flow when stress finally exceeds the yield threshold. We identify the percolation of non-affine droplets with the onset of an-elasticity in the ideal solid.

When studying the statics of a truss made of a discrete network of nodes joined by linear elastic springs, the kernel of the quadratic potential energy (space of floppy modes) plays an essential role. In particular when studying the asymptotics of the truss as the number of nodes becomes large : in the continuous homogenized description of the system one can recover the floppy modes. The fact that one does not commonly consider systems with finite dimension kernels other than the space of rigid motions is the reason why only displacements or rotations, or their dual counterparts, forces or torques, are transmitted by usual materials. We show how to design a truss in order to get an extra floppy mode which corresponds to a constant dilatation of the medium. Thus, forcing the dilatation in some part of the domain will tend to fix it on the whole domain. The mechanical interactions which are present in this continuum and responsible of this "transmission of dilatation" are not classical. We discuss and illustrate them using the discrete system of springs. We then give some examples of different possible floppy modes and the associated continuous models.

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It is well established that including spatial structure and stochastic noise in models for predator-prey interactions invalidates the classical deterministic Lotka-Volterra picture of neutral population cycles. In contrast, stochastic models yield long-lived, but ultimately decaying erratic population oscillations, which can be understood through a resonant amplification mechanism for density fluctuations. Monte Carlo simulations of spatial stochastic predator-prey systems yield striking complex spatio-temporal structures. These spreading activity fronts induce persistent correlations between predators and prey species. We employ field-theoretic methods based on the Doi-Peliti representation of the master equation for stochastic particle interaction models to address fluctuation-induced renormalizations of the oscillation frequency and damping.

References:
- M. Mobilia, I.T. Georgiev, and U.C.T., J. Stat. Phys. 128, 447 (2007) [q-bio.PE/0512039]
- U.C.T., in preparation (2010)

In Homogenization Theory,one homogenizes microstructures and this yields a set of macro coefficients at homogenization scale.If we decrease the scale,one finds that these coefficients do not provide adequate approximation to the heterogeneous medium.Using Bloch wave analysis, we introduce another set of macro coefficients which is relevant besides the first one.We present a comparative study of their properties.For example, somewhat surprisingly,the signs of these two sets are opposite to each other.We also compare their behaviour on certain optimal structures.