4,613 research outputs found
On the Microscopic Foundations of Elasticity
The modeling of the elastic properties of disordered or nanoscale solids
requires the foundations of the theory of elasticity to be revisited, as one
explores scales at which this theory may no longer hold. The only cases for
which microscopically based derivations of elasticity are documented are
(nearly) uniformly strained lattices. A microscopic approach to elasticity is
proposed. As a first step, microscopically exact expressions for the
displacement, strain and stress fields are derived. Conditions under which
linear elastic constitutive relations hold are studied theoretically and
numerically. It turns out that standard continuum elasticity is not
self-evident, and applies only above certain spatial scales, which depend on
details of the considered system and boundary conditions. Possible relevance to
granular materials is briefly discussed.Comment: 6 pages, 5 figures, LaTeX2e with svjour.cls and svepj.clo, submitted
to EPJ E, minor error corrected in v
Particle displacements in the elastic deformation of amorphous materials: local fluctuations vs. non-affine field
We study the local disorder in the deformation of amorphous materials by
decomposing the particle displacements into a continuous, inhomogeneous field
and the corresponding fluctuations. We compare these fields to the commonly
used non-affine displacements in an elastically deformed 2D Lennard-Jones
glass. Unlike the non-affine field, the fluctuations are very localized, and
exhibit a much smaller (and system size independent) correlation length, on the
order of a particle diameter, supporting the applicability of the notion of
local "defects" to such materials. We propose a scalar "noise" field to
characterize the fluctuations, as an additional field for extended continuum
models, e.g., to describe the localized irreversible events observed during
plastic deformation.Comment: Minor corrections to match the published versio
Stress response inside perturbed particle assemblies
The effect of structural disorder on the stress response inside three
dimensional particle assemblies is studied using computer simulations of
frictionless sphere packings. Upon applying a localised, perturbative force
within the packings, the resulting {\it Green's} function response is mapped
inside the different assemblies, thus providing an explicit view as to how the
imposed perturbation is transmitted through the packing. In weakly disordered
arrays, the resulting transmission of forces is of the double-peak variety, but
with peak widths scaling linearly with distance from the source of the
perturbation. This behaviour is consistent with an anisotropic elasticity
response profile. Increasing the disorder distorts the response function until
a single-peak response is obtained for fully disordered packings consistent
with an isotropic description.Comment: 8 pages, 7 figure captions To appear in Granular Matte
Towards a General Direct Product Testing Theorem
The Direct Product encoding of a string a in {0,1}^n on an underlying domain V subseteq ([n] choose k), is a function DP_V(a) which gets as input a set S in V and outputs a restricted to S. In the Direct Product Testing Problem, we are given a function F:V -> {0,1}^k, and our goal is to test whether F is close to a direct product encoding, i.e., whether there exists some a in {0,1}^n such that on most sets S, we have F(S)=DP_V(a)(S). A natural test is as follows: select a pair (S,S\u27)in V according to some underlying distribution over V x V, query F on this pair, and check for consistency on their intersection. Note that the above distribution may be viewed as a weighted graph over the vertex set V and is referred to as a test graph.
The testability of direct products was studied over various domains and test graphs: Dinur and Steurer (CCC \u2714) analyzed it when V equals the k-th slice of the Boolean hypercube and the test graph is a member of the Johnson graph family. Dinur and Kaufman (FOCS \u2717) analyzed it for the case where V is the set of faces of a Ramanujan complex, where in this case V=O_k(n). In this paper, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem?
Towards this goal we introduce the notion of coordinate expansion of a test graph. Roughly speaking a test graph is a coordinate expander if it has global and local expansion, and has certain nice intersection properties on sampling. We show that whenever the test graph has coordinate expansion then it admits a direct product testing theorem. Additionally, for every k and n we provide a direct product domain V subseteq (n choose k) of size n, called the Sliding Window domain for which we prove direct product testability
Force Chains, Microelasticity and Macroelasticity
It has been claimed that quasistatic granular materials, as well as nanoscale
materials, exhibit departures from elasticity even at small loadings. It is
demonstrated, using 2D and 3D models with interparticle harmonic interactions,
that such departures are expected at small scales [below O(100) particle
diameters], at which continuum elasticity is invalid, and vanish at large
scales. The models exhibit force chains on small scales, and force and stress
distributions which agree with experimental findings. Effects of anisotropy,
disorder and boundary conditions are discussed as well.Comment: 4 pages, 11 figures, RevTeX 4, revised and resubmitted to Phys. Rev.
Let
Scale separation in granular packings: stress plateaus and fluctuations
It is demonstrated, by numerical simulations of a 2D assembly of polydisperse
disks, that there exists a range (plateau) of coarse graining scales for which
the stress tensor field in a granular solid is nearly resolution independent,
thereby enabling an `objective' definition of this field. Expectedly, it is not
the mere size of the the system but the (related) magnitudes of the gradients
that determine the widths of the plateaus. Ensemble averaging (even over
`small' ensembles) extends the widths of the plateaus to sub-particle scales.
The fluctuations within the ensemble are studied as well. Both the response to
homogeneous forcing and to an external compressive localized load (and gravity)
are studied. Implications to small solid systems and constitutive relations are
briefly discussed.Comment: 4 pages, 4 figures, RevTeX 4, Minor corrections to match the
published versio
Shear-induced anisotropic decay of correlations in hard-sphere colloidal glasses
Spatial correlations of microscopic fluctuations are investigated via
real-space experiments and computer simulations of colloidal glasses under
steady shear. It is shown that while the distribution of one-particle
fluctuations is always isotropic regardless of the relative importance of shear
as compared to thermal fluctuations, their spatial correlations show a marked
sensitivity to the competition between shear-induced and thermally activated
relaxation. Correlations are isotropic in the thermally dominated regime, but
develop strong anisotropy as shear dominates the dynamics of microscopic
fluctuations. We discuss the relevance of this observation for a better
understanding of flow heterogeneity in sheared amorphous solids.Comment: 6 pages, 4 figure
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