Lattices near mechanical
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Frames consisting of nodes connected
pairwise by rigid rods or central-force springs,
possibly with preferred relative angles
controlled by bending forces, are useful models
for systems as diverse as architectural
structures, crystalline and amorphous solids,
sphere packings and granular matter, networks of
semi-flexible polymers, proteins, origami, and
an increasing number of lab-constructed
micron-scale metamaterials. The rigidity
of these networks depends on the average
coordination number z of the nodes: If z
is small enough, the frames have internal
zero-frequency modes, and they are “floppy”; if
z is large enough, they have no internal zero
modes and they are rigid. The critical
point separating these two regimes occurs at a
rigidity threshold that for central forces in
d-dimensions occurs at or near coordination
number zc = 2d. At and near the rigidity
threshold, elastic frames exhibit unique and
interestingproperties, including extreme
sensitivity to boundary conditions, power-law
scaling of elastic moduli with (z- zc), and
diverging length and time
scales.
This talk will explore elastic and mechanical
properties and mode structures of model periodic
lattices, such as the square and kagome lattices
with central-force springs, that are just on
verge of mechanical instability. It will discuss
the origin and nature of zero modes and
elasticity of these structures under both
periodic (PBC) and free boundary conditions
(FBC), and it will derive general conditions [1]
(a) under which the zero modes under the two
boundary conditions are essentially identical
and (b) under which phonon modes are gapped with
no zero modes in the periodic spectrum but
include zero-frequency surface Rayleigh waves in
the free spectrum. In the former situation,
lattices are generally in a type of critical
state that admits states of self-stress in
which there can be
tension in bars with zero force on any node, and
distortions away from that state give rise to
surface modes under free boundary conditions
whose degree of penetration into the bulk
diverges at the critical state. The gapped
states have a topological characterization,
similar to that of topological insulators, that
define the nature of zero-modes at the boundary
between systems with different topology.
References:
[1] K. Sun, A. Souslov, X. M. Mao, and T.C.
Lubensky, PNAS 109, 12369-12374 (2012).
[2] C.L. Kane and T.C. Lubensky, Nature Physics
10, 39-45(2014)