Reconstruction of potential energy profiles from multiple rupture time distributions

We explore the mathematical and numerical aspects of reconstructing a potential energy profile of a molecular bond from its rupture time distribution. While reliable reconstruction of gross attributes, such as the height and the width of an energy barrier, can be easily extracted from a single first passage time (FPT) distribution, the reconstruction of finer structure is ill-conditioned. More careful analysis shows the existence of optimal bond potential amplitudes (represented by an effective Peclet number) and initial bond configurations that yield the most efficient numerical reconstruction of simple potentials. Furthermore, we show that reconstruction of more complex potentials containing multiple minima can be achieved by simultaneously using two or more measured FPT distributions, obtained under different physical conditions. For example, by changing the effective potential energy surface by known amounts, additional measured FPT distributions improve the reconstruction. We demonstrate the possibility of reconstructing potentials with multiple minima, motivate heuristic rules-of-thumb for optimizing the reconstruction, and discuss further applications and extensions.Comment: 20 pages, 9 figure

Autonomy and Singularity in Dynamic Fracture

The recently developed weakly nonlinear theory of dynamic fracture predicts $1/r$ corrections to the standard asymptotic linear elastic $1/\sqrt{r}$ displacement-gradients, where $r$ is measured from the tip of a tensile crack. We show that the $1/r$ singularity does not automatically conform with the notion of autonomy (autonomy means that any crack tip nonlinear solution is uniquely determined by the surrounding linear elastic $1/\sqrt{r}$ fields) and that it does not automatically satisfy the resultant Newton's equation in the crack parallel direction. We show that these two properties are interrelated and that by requiring that the resultant Newton's equation is satisfied, autonomy of the $1/r$ singular solution is retained. We further show that the resultant linear momentum carried by the $1/r$ singular fields vanishes identically. Our results, which reveal the physical and mathematical nature of the new solution, are in favorable agreement with recent near tip measurements.Comment: 4 pages, 2 figures, related papers: arXiv:0902.2121 and arXiv:0807.486

Equilibrium orbit analysis in a free-electron laser with a coaxial wiggler

An analysis of single-electron orbits in combined coaxial wiggler and axial guide magnetic fields is presented. Solutions of the equations of motion are developed in a form convenient for computing orbital velocity components and trajectories in the radially dependent wiggler. Simple analytical solutions are obtained in the radially-uniform-wiggler approximation and a formula for the derivative of the axial velocity $v_{\|}$ with respect to Lorentz factor $\gamma$ is derived. Results of numerical computations are presented and the characteristics of the equilibrium orbits are discussed. The third spatial harmonic of the coaxial wiggler field gives rise to group $III$ orbits which are characterized by a strong negative mass regime.Comment: 13 pages, 9 figures, to appear in phys. rev.

The Breakdown of Linear Elastic Fracture Mechanics near the Tip of a Rapid Crack

We present high resolution measurements of the displacement and strain fields near the tip of a dynamic (Mode I) crack. The experiments are performed on polyacrylamide gels, brittle elastomers whose fracture dynamics mirror those of typical brittle amorphous materials. Over a wide range of propagation velocities ($0.2-0.8c_s$), we compare linear elastic fracture mechanics (LEFM) to the measured near-tip fields. We find that, sufficiently near the tip, the measured stress intensity factor appears to be non-unique, the crack tip significantly deviates from its predicted parabolic form, and the strains ahead of the tip are more singular than the $r^{-1/2}$ divergence predicted by LEFM. These results show how LEFM breaks down as the crack tip is approached.Comment: 4 pages, 4 figures, first of a two-paper series (experiments); no change in content, minor textual revision

Unsteady Crack Motion and Branching in a Phase-Field Model of Brittle Fracture

Crack propagation is studied numerically using a continuum phase-field approach to mode III brittle fracture. The results shed light on the physics that controls the speed of accelerating cracks and the characteristic branching instability at a fraction of the wave speed.Comment: 11 pages, 4 figure

Some exact results for the velocity of cracks propagating in non-linear elastic models

We analyze a piece-wise linear elastic model for the propagation of a crack in a stripe geometry under mode III conditions, in the absence of dissipation. The model is continuous in the propagation direction and discrete in the perpendicular direction. The velocity of the crack is a function of the value of the applied strain. We find analytically the value of the propagation velocity close to the Griffith threshold, and close to the strain of uniform breakdown. Contrary to the case of perfectly harmonic behavior up to the fracture point, in the piece-wise linear elastic model the crack velocity is lower than the sound velocity, reaching this limiting value at the strain of uniform breakdown. We complement the analytical results with numerical simulations and find excellent agreement.Comment: 9 pages, 13 figure

Dynamic instabilities of fracture under biaxial strain using a phase field model

We present a phase field model of the propagation of fracture under plane strain. This model, based on simple physical considerations, is able to accurately reproduce the different behavior of cracks (the principle of local symmetry, the Griffith and Irwin criteria, and mode-I branching). In addition, we test our model against recent experimental findings showing the presence of oscillating cracks under bi-axial load. Our model again reproduces well observed supercritical Hopf bifurcation, and is therefore the first simulation which does so

Supersonic crack propagation in a class of lattice models of Mode III brittle fracture

We study a lattice model for mode III crack propagation in brittle materials in a stripe geometry at constant applied stretching. Stiffening of the material at large deformation produces supersonic crack propagation. For large stretching the propagation is guided by well developed soliton waves. For low stretching, the crack-tip velocity has a universal dependence on stretching that can be obtained using a simple geometrical argument.Comment: 4 pages, 3 figure

Evaluation of Noise Radiation Mechanisms in Turbulent Jets

Data from the direct numerical simulation (DNS) of a turbulent, compressible (Mach = 1.92) jet has been analyzed to investigate the process of sound generation. The overall goals are to understand how the different scales of turbulence contribute to the acoustic field, and to understand the role that linear instability waves play in the noise produced by supersonic turbulent jets. Lighthill’s acoustic analogy was used to predict the radiate sound from turbulent source terms computed from the DNS data. Preliminary computations (for the axisymmetric mode of the acoustic field) showgood agreement between the acoustic field determined from DNS and acoustic analogy. Further work is needed to refine the calculations and investigate the source terms. Work was also begun to test the validity of linear stability wave models of sound generation in supersonic jets. An adjoint-based method was developed to project the DNS data onto the most unstable linear stability mode at different streamwise positions. This will allow the evolution of the wave and its radiated acoustic field, determined by solving the linear equations, to be compared directly with the evolution of the near and far-field fluctuations in the DNS