On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and discontinuous shocks

We study the generalization of the dispersionless Kadomtsev - Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2+1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel IST, and it has been recently shown to be a prototype model equation in the description of the two dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single valued discontinuous shocks. Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if $m(n-1)\le 2$. At last, the analytic aspects of such a wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a discontinuous shock. These results, contained in the 2012 master thesis of one of the authors (FS), generalize those obtained by one of the authors (PMS) and S.V.Manakov for the dKP equation in n+1 dimensions with quadratic nonlinearity, and are obtained following the same strategy.Comment: 31 pages, 11 figure

Slow crack growth in polycarbonate films

We study experimentally the slow growth of a single crack in polycarbonate films submitted to uniaxial and constant imposed stress. The specificity of fracture in polycarbonate films is the appearance of flame shaped macroscopic process zones at the tips of the crack. Supported by an experimental study of the mechanical properties of polycarbonate films, an analysis of the stress dependence of the mean ratio between the process zone and crack lengths, during the crack growth, show a quantitative agreement with the Dugdale-Barenblatt model of the plastic process zone. We find that the fracture growth curves obey strong scaling properties that lead to a well defined growth master curve

Attractive and repulsive cracks in a heterogeneous material

We study experimentally the paths of an assembly of cracks growing in interaction in a heterogeneous two-dimensional elastic brittle material submitted to uniaxial stress. For a given initial crack assembly geometry, we observe two types of crack path. The first one corresponds to a repulsion followed by an attraction on one end of the crack and a tip to tip attraction on the other end. The second one corresponds to a pure attraction. Only one of the crack path type is observed in a given sample. Thus, selection between the two types appears as a statistical collective process.Comment: soumis \`a JSTA

Parametric Analysis on the Static and Modal Response of Folded Metamaterials

Metamaterials have been studied and analyzed in the past three decades because of their outstanding properties. Generally speaking, a metamaterial is a material that exhibits a mechanical behavior that does not depend only on the bulk material but also on the geometrical configuration in which it lies. This aspect leads to the possibility of tuning and engineering the structural response. One of the most interesting properties is the auxetic behavior of metamaterial. An auxetic material shows a global negative Poisson’s ratio. Shock absorption, acoustic dissipation, and shape morphing are some of the most popular employment for auxetic materials. In this article, we focus on the response of folded material under static and dynamic load conditions. Folded materials consist of folding a sheet under specific geometrical constraints. One of the most famous is the Miura-ori pattern, which comes from the origamifolding technique. The geometrical parameters, such as folding angles and edge lengths, play a fundamental role in achieving the desired auxetic behavior. These geometrical parameters define a unit cell that can be stacked into a periodic structure. This article proposes an experimental parametric study of the thickness impact on the auxetic behavior while edge dimensions and folding angles are fixed. The geometrical complexity of the pattern forced us to use additive manufacturing for the specimen fabrication. In particular, we choose Fused Filament Fabrication (FFF) using polymers like ABS and PLA. Digital Image Correlation (DIC) is used for monitoring the displacement and strain fields onto the Miura-ori surface under tensile load. Finally, Time Averaged Speckle Interferometry is employed for evaluating the modal response by using a quasi-full out-of-plane sensitivity setup

Discrepancy between sub-critical and fast rupture roughness: a cumulant analysis

We study the roughness of a crack interface in a sheet of paper. We distinguish between slow (sub-critical) and fast crack growth regimes. We show that the fracture roughness is different in the two regimes using a new method based on a multifractal formalism recently developed in the turbulence literature. Deviations from monofractality also appear to be different in both regimes

Roughness of tensile crack fronts in heterogenous materials

The dynamics of planar crack fronts in heterogeneous media is studied using a recently proposed stochastic equation of motion that takes into account nonlinear effects. The analysis is carried for a moving front in the quasi-static regime using the Self Consistent Expansion. A continuous dynamical phase transition between a flat phase and a dynamically rough phase, with a roughness exponent $\zeta=1/2$, is found. The rough phase becomes possible due to the destabilization of the linear modes by the nonlinear terms. Taking into account the irreversibility of the crack propagation, we infer that the roughness exponent found in experiments might become history-dependent, and so our result gives a lower bound for $\zeta$.Comment: 7 page

Chronography of the Milky Way's Halo System with Field Blue Horizontal-Branch Stars

In a pioneering effort, Preston et al. reported that the colors of blue horizontal-branch (BHB) stars in the halo of the Galaxy shift with distance, from regions near the Galactic center to about 12 kpc away, and interpreted this as a correlated variation in the ages of halo stars, from older to younger, spanning a range of a few Gyrs. We have applied this approach to a sample of some 4700 spectroscopically confirmed BHB stars selected from the Sloan Digital Sky Survey to produce the first "chronographic map" of the halo of the Galaxy. We demonstrate that the mean de-reddened g$-$r color, , increases outward in the Galaxy from $-$0.22 to $-$0.08 (over a color window spanning [$-$0.3:0.0]) from regions close to the Galactic center to ~40 kpc, independent of the metallicity of the stars. Models of the expected shift in the color of the field BHB stars based on modern stellar evolutionary codes confirm that this color gradient can be associated with an age difference of roughly 2-2.5 Gyrs, with the oldest stars concentrated in the central ~15 kpc of the Galaxy. Within this central region, the age difference spans a mean color range of about 0.05 mag (~0.8 Gyrs). Furthermore, we show that chronographic maps can be used to identify individual substructures, such as the Sagittarius Stream, and overdensities in the direction of Virgo and Monoceros, based on the observed contrast in their mean BHB colors with respect to the foreground/background field population.Comment: 6 pages, 4 figures, ApJ letter

Discriminating between competing models for the allosteric regulation of oncogenic phosphatase SHP2 by characterizing its active state

The Src-homology 2 domain containing phosphatase 2 (SHP2) plays a critical role in crucial signaling pathways and is involved in oncogenesis and in developmental disorders. Its structure includes two SH2 domains (N-SH2 and C-SH2), and a protein tyrosine phosphatase (PTP) domain. Under basal conditions, SHP2 is auto-inhibited, with the N-SH2 domain blocking the PTP active site. Activation involves a rearrangement of the domains that makes the catalytic site accessible, coupled to the association between the SH2 domains and cognate proteins containing phosphotyrosines. Several aspects of this transition are debated and competing mechanistic models have been proposed. A crystallographic structure of SHP2 in an active state has been reported (PDB code 6crf), but several lines of evidence suggests that it is not fully representative of the conformations populated in solution. To clarify the structural rearrangements involved in SHP2 activation, enhanced sampling simulations of the autoinhibited and active states have been performed, for wild type SHP2 and its pathogenic E76K variant. Our results demonstrate that the crystallographic conformation of the active state is unstable in solution, and multiple interdomain arrangements are populated, thus allowing association to bisphosphorylated sequences. Contrary to a recent proposal, activation is coupled to the conformational changes of the N-SH2 binding site, which is significantly more accessible in the active sate, rather than to the structure of the central β-sheet of the domain. In this coupling, a previously undescribed role for the N-SH2 BG loop emerged

Average crack-front velocity during subcritical fracture propagation in a heterogeneous medium

We study the average velocity of crack fronts during stable interfacial fracture experiments in a heterogeneous quasibrittle material under constant loading rates and during long relaxation tests. The transparency of the material (polymethylmethacrylate) allows continuous tracking of the front position and relation of its evolution to the energy release rate. Despite significant velocity fluctuations at local scales, we show that a model of independent thermally activated sites successfully reproduces the large-scale behavior of the crack front for several loading conditions