1,211 research outputs found
Shear-induced rigidity of frictional particles: Analysis of emergent order in stress space
Solids are distinguished from fluids by their ability to resist shear. In
traditional solids, the resistance to shear is associated with the emergence of
broken translational symmetry as exhibited by a non-uniform density pattern,
which results from either minimizing the energy cost or maximizing the entropy
or both. In this work, we focus on a class of systems, where this paradigm is
challenged. We show that shear-driven jamming in dry granular materials is a
collective process controlled solely by the constraints of mechanical
equilibrium. We argue that these constraints lead to a broken translational
symmetry in a dual space that encodes the statistics of contact forces and the
topology of the contact network. The shear-jamming transition is marked by the
appearance of this broken symmetry. We extend our earlier work, by comparing
and contrasting real space measures of rheology with those obtained from the
dual space. We investigate the structure and behavior of the dual space as the
system evolves through the rigidity transition in two different shear
protocols. We analyze the robustness of the shear-jamming scenario with respect
to protocol and packing fraction, and demonstrate that it is possible to define
a protocol-independent order parameter in this dual space, which signals the
onset of rigidity.Comment: 14 pages, 17 figure
Semitoric integrable systems on symplectic 4-manifolds
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a
pair of real-valued smooth functions J, H on M for which J generates a
Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall
introduce new global symplectic invariants for these systems; some of these
invariants encode topological or geometric aspects, while others encode
analytical information about the singularities and how they stand with respect
to the system. Our goal is to prove that a semitoric system is completely
determined by the invariants we introduce
Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations
Inspired by Katz-Mazur theorem on crystalline cohomology and by
Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon
of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the
Hodge bundle over any Teichm\"uller curve. We also discuss the connections
between the two polygons and the integral of eigenvalues of the curvature of
the Hodge bundle by using Atiyah-Bott, Forni and M\"oller's works. We obtain
several applications to Teichm\"uller dynamics conditional to the conjecture.Comment: 37 pages. We rewrite this paper without changing the mathematics
content. arXiv admin note: text overlap with arXiv:1112.5872, arXiv:1204.1707
by other author
Rigidity transitions in zero-temperature polygons
We study geometrical clues of a rigidity transition due to the emergence of a
system-spanning state of self stress in under-constrained systems of individual
polygons and spring networks constructed from such polygons. When a polygon
with harmonic bond edges and an area spring constraint is subject to an
expansive strain, we observe that convexity of the polygon is a necessary
condition for such a self stress. We prove that the cyclic configuration of the
polygon is a sufficient condition for the self stress. This correspondence of
geometry and rigidity is akin to the straightening of a one dimensional chain
of springs to rigidify it. We predict the onset of the rigidity transition
using a purely geometrical method. We also estimate the transition strain for a
given initial configuration by approximating irregular polygons as regular
polygons. These findings help determine the rigidity of an area-preserving
polygon just by looking at it. Since two-dimensional spring networks can be
considered as a network of polygons, we look for similar geometric features in
under-constrained spring networks under isotropic expansive strain. In
particular, we observe that all polygons attain convexity at the rigidity
transition such that the fraction of convex, but not cyclic, polygons predicts
the onset of the rigidity transition. Interestingly, acyclic polygons in the
network correlate with larger tensions, thus, forming effective force chains.Comment: 12 pages, 10 figure
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