11,845 research outputs found
Elastic collapse in disordered isostatic networks
Isostatic networks are minimally rigid and therefore have, generically,
nonzero elastic moduli. Regular isostatic networks have finite moduli in the
limit of large sizes. However, numerical simulations show that all elastic
moduli of geometrically disordered isostatic networks go to zero with system
size. This holds true for positional as well as for topological disorder. In
most cases, elastic moduli decrease as inverse power-laws of system size. On
directed isostatic networks, however, of which the square and cubic lattices
are particular cases, the decrease of the moduli is exponential with size. For
these, the observed elastic weakening can be quantitatively described in terms
of the multiplicative growth of stresses with system size, giving rise to bulk
and shear moduli of order exp{-bL}. The case of sphere packings, which only
accept compressive contact forces, is considered separately. It is argued that
these have a finite bulk modulus because of specific correlations in contact
disorder, introduced by the constraint of compressivity. We discuss why their
shear modulus, nevertheless, is again zero for large sizes. A quantitative
model is proposed that describes the numerically measured shear modulus, both
as a function of the loading angle and system size. In all cases, if a density
p>0 of overconstraints is present, as when a packing is deformed by
compression, or when a glass is outside its isostatic composition window, all
asymptotic moduli become finite. For square networks with periodic boundary
conditions, these are of order sqrt{p}. For directed networks, elastic moduli
are of order exp{-c/p}, indicating the existence of an "isostatic length scale"
of order 1/p.Comment: 6 pages, 6 figues, to appear in Europhysics Letter
Effective medium theory of elastic waves in random networks of rods
We formulate an effective medium (mean field) theory of a material consisting
of randomly distributed nodes connected by straight slender rods, hinged at the
nodes. Defining novel wavelength-dependent effective elastic moduli, we
calculate both the static moduli and the dispersion relations of ultrasonic
longitudinal and transverse elastic waves. At finite wave vector the waves
are dispersive, with phase and group velocities decreasing with increasing wave
vector. These results are directly applicable to networks with empty pore
space. They also describe the solid matrix in two-component (Biot) theories of
fluid-filled porous media. We suggest the possibility of low density materials
with higher ratios of stiffness and strength to density than those of foams,
aerogels or trabecular bone.Comment: 14 pp., 3 fig
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