1,815 research outputs found
Rate dependent shear bands in a shear transformation zone model of amorphous solids
We use Shear Transformation Zone (STZ) theory to develop a deformation map
for amorphous solids as a function of the imposed shear rate and initial
material preparation. The STZ formulation incorporates recent simulation
results [Haxton and Liu, PRL 99 195701 (2007)] showing that the steady state
effective temperature is rate dependent. The resulting model predicts a wide
range of deformation behavior as a function of the initial conditions,
including homogeneous deformation, broad shear bands, extremely thin shear
bands, and the onset of material failure. In particular, the STZ model predicts
homogeneous deformation for shorter quench times and lower strain rates, and
inhomogeneous deformation for longer quench times and higher strain rates. The
location of the transition between homogeneous and inhomogeneous flow on the
deformation map is determined in part by the steady state effective
temperature, which is likely material dependent. This model also suggests that
material failure occurs due to a runaway feedback between shear heating and the
local disorder, and provides an explanation for the thickness of shear bands
near the onset of material failure. We find that this model, which resolves
dynamics within a sheared material interface, predicts that the stress weakens
with strain much more rapidly than a similar model which uses a single state
variable to specify internal dynamics on the interface.Comment: 10 pages, 13 figures, corrected typos, added section on rate
strengthening vs. rate weakening material
Strain localization in a shear transformation zone model for amorphous solids
We model a sheared disordered solid using the theory of Shear Transformation
Zones (STZs). In this mean-field continuum model the density of zones is
governed by an effective temperature that approaches a steady state value as
energy is dissipated. We compare the STZ model to simulations by Shi, et
al.(Phys. Rev. Lett. 98 185505 2007), finding that the model generates
solutions that fit the data,exhibit strain localization, and capture important
features of the localization process. We show that perturbations to the
effective temperature grow due to an instability in the transient dynamics, but
unstable systems do not always develop shear bands. Nonlinear energy
dissipation processes interact with perturbation growth to determine whether a
material exhibits strain localization. By estimating the effects of these
interactions, we derive a criterion that determines which materials exhibit
shear bands based on the initial conditions alone. We also show that the shear
band width is not set by an inherent diffusion length scale but instead by a
dynamical scale that depends on the imposed strain rate.Comment: 8 figures, references added, typos correcte
Spontaneous thermal runaway as an ultimate failure mechanism of materials
The first theoretical estimate of the shear strength of a perfect crystal was
given by Frenkel [Z. Phys. 37, 572 (1926)]. He assumed that as slip occurred,
two rigid atomic rows in the crystal would move over each other along a slip
plane. Based on this simple model, Frenkel derived the ultimate shear strength
to be about one tenth of the shear modulus. Here we present a theoretical study
showing that catastrophic material failure may occur below Frenkel's ultimate
limit as a result of thermal runaway. We demonstrate that the condition for
thermal runaway to occur is controlled by only two dimensionless variables and,
based on the thermal runaway failure mechanism, we calculate the maximum shear
strength of viscoelastic materials. Moreover, during the thermal
runaway process, the magnitude of strain and temperature progressively localize
in space producing a narrow region of highly deformed material, i.e. a shear
band. We then demonstrate the relevance of this new concept for material
failure known to occur at scales ranging from nanometers to kilometers.Comment: 4 pages, 3 figures. Eq. (6) and Fig. 2a corrected; added references;
improved quality of figure
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Further biembeddings of twofold triple systems
We construct face two-colourable triangulations of the graph 2Kn in an orientable surface; equivalently biembeddings of two twofold triple systems of order n, for all n ξ 16 or 28 (mod 48). The biembeddings come from index 1 current graphs lifted under a group ℤn/4 × 4
Distributive and anti-distributive Mendelsohn triple systems
We prove that the existence spectrum of Mendelsohn triple systems whose
associated quasigroups satisfy distributivity corresponds to the Loeschian
numbers, and provide some enumeration results. We do this by considering a
description of the quasigroups in terms of commutative Moufang loops.
In addition we provide constructions of Mendelsohn quasigroups that fail
distributivity for as many combinations of elements as possible.
These systems are analogues of Hall triple systems and anti-mitre Steiner
triple systems respectively
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