Defects and boundary layers in non-Euclidean plates

We investigate the behavior of non-Euclidean plates with constant negative Gaussian curvature using the F\"oppl-von K\'arm\'an reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers -- deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic profile that have additional boundary layers near their lines of inflection. These additional boundary layers are a new phenomenon in thin elastic sheets and are necessary to regularize jump discontinuities in the azimuthal curvature across lines of inflection. We rigorously derive scaling laws for the width of these boundary layers as a function of the thickness of the sheet

Multiscale characterization of damage tolerance in barium titanate thin films

Barium titanate is a brittle, lead free ferroelectric and piezoelectric ceramic used in patterned and thin film forms in micro- and nano-scale electronic devices. Both during deposition and eventually during service, this material system develops stresses due to different loads acting on the system, which can lead to its failure due to cracking in the films and/or interface delamination. In situ microcantilever bending based fracture experiments and tensile tests based on shear lag tests in combination with digital image correlation were used to understand the cracking behavior of barium titanate films when deposited on flexible substrates. For the first time, the fracture behavior of these nanocrystalline barium titanate films has been quantified in terms of fracture toughness, fracture strength, and interface shear stresses for different film thicknesses. Critical defect size is estimated using the above information as a function of film thickness. It is found that damage tolerance in terms of fracture strength depends on film thickness. Furthermore, compared to a bulk single crystal, barium titanate fracture resistance of the nanocrystalline thin films is reduced. Both effects need to be considered in engineering design of reliable devices employing micro- and nano-scale barium titanate thin film structures

Steady Stokes flow with long-range correlations, fractal Fourier spectrum, and anomalous transport

We consider viscous two-dimensional steady flows of incompressible fluids past doubly periodic arrays of solid obstacles. In a class of such flows, the autocorrelations for the Lagrangian observables decay in accordance with the power law, and the Fourier spectrum is neither discrete nor absolutely continuous. We demonstrate that spreading of the droplet of tracers in such flows is anomalously fast. Since the flow is equivalent to the integrable Hamiltonian system with 1 degree of freedom, this provides an example of integrable dynamics with long-range correlations, fractal power spectrum, and anomalous transport properties.Comment: 4 pages, 4 figures, published in Physical Review Letter

Universal Scaling Properties in Large Assemblies of Simple Dynamical Units Driven by Long-Wave Random Forcing

Large assemblies of nonlinear dynamical units driven by a long-wave fluctuating external field are found to generate strong turbulence with scaling properties. This type of turbulence is so robust that it persists over a finite parameter range with parameter-dependent exponents of singularity, and is insensitive to the specific nature of the dynamical units involved. Whether or not the units are coupled with their neighborhood is also unimportant. It is discovered numerically that the derivative of the field exhibits strong spatial intermittency with multifractal structure.Comment: 10 pages, 7 figures, submitted to PR

Asymptotic power law of moments in a random multiplicative process with weak additive noise

It is well known that a random multiplicative process with weak additive noise generates a power-law probability distribution. It has recently been recognized that this process exhibits another type of power law: the moment of the stochastic variable scales as a function of the additive noise strength. We clarify the mechanism for this power-law behavior of moments by treating a simple Langevin-type model both approximately and exactly, and argue this mechanism is universal. We also discuss the relevance of our findings to noisy on-off intermittency and to singular spatio-temporal chaos recently observed in systems of non-locally coupled elements.Comment: 11 pages, 9 figures, submitted to Phys. Rev.

Stigma as a fundamental hindrance to the United States opioid overdose crisis response.

Alexander Tsai and co-authors discuss the role of stigma in responses to the US opioid crisis