Velocity Fluctuations in Dynamical Fracture: the Role of Microcracks

We address the velocity fluctuations of fastly moving cracks in stressed materials. One possible mechanism for such fluctuations is the interaction of the main crack with micro cracks (irrespective whether these are existing material defects or they form during the crack evolution). We analyze carefully the dynamics (in 2 space dimensions) of one macro and one micro crack, and demonstrate that their interaction results in a {\em large} and {\em rapid} velocity fluctuation, in qualitative correspondence with typical velocity fluctuations observed in experiments. In developing the theory of the dynamical interaction we invoke an approximation that affords a reduction in mathematical complexity to a simple set of ordinary differential equations for the positions of the cracks tips; we propose that this kind of approximation has a range of usefulness that exceeds the present context.Comment: 7 pages, 7 figure

Brief of Resondents

This Brief outlines the Respondents (State/Prosecution) side of legal support for why the Sheppard Case was conducted in a partial and Constitutional manner. The two questions they present include: DID THE PUBLICITY RELATING TO THE PETITIONER DEPRIVE HIM OF A FAIR TRIAL? and WAS THE PETITIONER DEPRIVED OF A PUBLIC TRIAL? Using precedent and examination of the Irvin v. Dowd, 366 U.S. 717, 723 (1961) case and the case of Rideau v. State of Louisiana, 373 U.S. 723, 83 S. Ct., 1417, 10 L. Ed. 2d, 663 (1963), (perhaps the two cases most favorable to petitioner\u27s position) reveal where the United States Supreme Court has drawn the boundaries for what constitutes a fair trial in the face of adverse publicity. These cases with such a pattern of facts reveals situations that caused the United States Supreme Court to reverse both convictions. Although slightly similar to Sheppard, various factors suggest that the deep-rooted prejudice that existed in the Rideau and Irvin cases did not exist in the Sheppard case–as noted by respondents. After an examination of the relevant factors surrounding the Sheppard jury and the publicity in the community, respondent finds that it can merely repeat the words of the United States District Court in United States v. Kahaner, 204 F. Supp. 921, 924 (1962): Publicity, in and of itself does not, foreclose a fair trial. The courts do not function in a vacuum and jurors are not required to be totally ignorant of what goes on about them. * * * In considering the mentioned complaints it is necessary to keep in mind the following legal principles: The mere fact of unfavorable publicity does not of itself raise a presumption of prejudice but prejudice must manifest itself so as to corrupt due process. Dennis v. U.S., 302 F 2d 5 (1962). Mere exposure to adverse publicity does not necessarily result in bias, prejudgment or other disqualification. U.S. v. Applegarth, 206 F. Supp. 686, 687 (1962). The mere fact that a juror has read newspaper accounts relative to a criminal charge is not in itself sufficient grounds for excusing a jury. Blumenfield v. U.S, 284 F. 2d 46, 51 (1960). Also, regarding the petitioner\u27s (Sheppard) assertion that the action of the trial court in setting aside the major portion of the courtroom for representatives of the news media was violative of his right to a public trial. Respondent contends that the trial court was justified in its actions. Equally, the respondent is aware that the petitioner is going to present statements to the effect that the trial judge expressed his belief in the guilt of the petitioner before the trial commenced. Accordingly, respondent invites the Court\u27s attention to the recent case of Hendrix v. Hand, 312 F. 2d (1962) wherein the Court of Appeals, in a habeas corpus proceeding held that the mere fact that a trial judge in a state criminal prosecution signed a statement in advance of trial relating to the judge\u27s belief that the defendant was guilty of the crime charged did not establish any infringement of defendant\u27s right to a fair trial

Does the continuum theory of dynamic fracture work?

We investigate the validity of the Linear Elastic Fracture Mechanics approach to dynamic fracture. We first test the predictions in a lattice simulation, using a formula of Eshelby for the time-dependent Stress Intensity Factor. Excellent agreement with the theory is found. We then use the same method to analyze the experiment of Sharon and Fineberg. The data here is not consistent with the theoretical expectation.Comment: 4 page

Extinction Rates for Fluctuation-Induced Metastabilities : A Real-Space WKB Approach

The extinction of a single species due to demographic stochasticity is analyzed. The discrete nature of the individual agents and the Poissonian noise related to the birth-death processes result in local extinction of a metastable population, as the system hits the absorbing state. The Fokker-Planck formulation of that problem fails to capture the statistics of large deviations from the metastable state, while approximations appropriate close to the absorbing state become, in general, invalid as the population becomes large. To connect these two regimes, a master equation based on a real space WKB method is presented, and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state. The details of the underlying microscopic process, smeared out in a mean field treatment, are shown to be crucial for an exact determination of the extinction exponent. This general scheme is shown to reproduce the known results in the field, to yield new corollaries and to fit quite precisely the numerical solutions. Moreover it allows for systematic improvement via a series expansion where the small parameter is the inverse of the number of individuals in the metastable state

Nonlinear lattice model of viscoelastic Mode III fracture

We study the effect of general nonlinear force laws in viscoelastic lattice models of fracture, focusing on the existence and stability of steady-state Mode III cracks. We show that the hysteretic behavior at small driving is very sensitive to the smoothness of the force law. At large driving, we find a Hopf bifurcation to a straight crack whose velocity is periodic in time. The frequency of the unstable bifurcating mode depends on the smoothness of the potential, but is very close to an exact period-doubling instability. Slightly above the onset of the instability, the system settles into a exactly period-doubled state, presumably connected to the aforementioned bifurcation structure. We explicitly solve for this new state and map out its velocity-driving relation

Propagating mode-I fracture in amorphous materials using the continuous random network (CRN) model

We study propagating mode-I fracture in two dimensional amorphous materials using atomistic simulations. We used the continuous random network (CRN) model of an amorphous material, creating samples using a two dimensional analogue of the WWW (Wooten, Winer & Weaire) Monte-Carlo algorithm. For modeling fracture, molecular-dynamics simulations were run on the resulting samples. The results of our simulations reproduce the main experimental features. In addition to achieving a steady-state crack under a constant driving displacement (which had not yet been achieved by other atomistic models for amorphous materials), the runs show micro-branching, which increases with driving, transitioning to macro-branching for the largest drivings. Beside the qualitative visual similarity of the simulated cracks to experiment, the simulation also succeeds in explaining the experimentally observed oscillations of the crack velocity

Front Propagation up a Reaction Rate Gradient

We expand on a previous study of fronts in finite particle number reaction-diffusion systems in the presence of a reaction rate gradient in the direction of the front motion. We study the system via reaction-diffusion equations, using the expedient of a cutoff in the reaction rate below some critical density to capture the essential role of fl uctuations in the system. For large density, the velocity is large, which allows for an approximate analytic treatment. We derive an analytic approximation for the front velocity depe ndence on bulk particle density, showing that the velocity indeed diverge s in the infinite density limit. The form in which diffusion is impleme nted, namely nearest-neighbor hopping on a lattice, is seen to have an essential impact on the nature of the divergence

Arrested Cracks in Nonlinear Lattice Models of Brittle Fracture

We generalize lattice models of brittle fracture to arbitrary nonlinear force laws and study the existence of arrested semi-infinite cracks. Unlike what is seen in the discontinuous case studied to date, the range in driving displacement for which these arrested cracks exist is very small. Also, our results indicate that small changes in the vicinity of the crack tip can have an extremely large effect on arrested cracks. Finally, we briefly discuss the possible relevance of our findings to recent experiments.Comment: submitted to PRE, Rapid Communication

Answer and Return of Writ

Respondent says that the record in this case affirmatively shows that the petitioner (Sheppard) was awarded a full and fair hearing in the state courts, resulting in reliable findings of fact, and that the state courts applied correct constitutional standards in disposing of the various claims of the petitioner. Respondent denies that the trial court erred in refusing to grant petitioner (Sheppard) a new trial on the ground of newly discovered evidence. Respondent denies that any relevant material or substantial evidence was suppressed by the prosecution or that any unjust tactics were used by the prosecuting authorities in the trial of this case. Respondent denies that the petitioner (Sheppard) was prevented from having a fair and impartial trial by the action of the trial judge as a result of the actions of the bailiffs in permitting the jurors to make telephone calls. Respondent denies that the trial judge coerced the jury into reaching a verdict, and avers that the fact that the jury deliberated for a period of five days merely shows the carefulness and consideration that the jury gave the mass of testimony and over 200 exhibits in the case. Respondent denies each and every other allegation in the petition not herein admitted to be true. Affirmatively, respondent alleges that petitioner was convicted in a court that had jurisdiction of his person and of the crime involved, that petitioner was not deprived of any of his constitutional rights, and that the facts upon which petitioner (Sheppard) relies, even if true, constitute mere error in the trial court, which is not cognizable in an action of habeas corpus. For the foregoing reasons, respondent prays that the petition herein be dismissed

Answer and Return of Writ

Respondent says that the record in this case affirmatively shows that the petitioner (Sheppard) was awarded a full and fair hearing in the state courts, resulting in reliable findings of fact, and that the state courts applied correct constitutional standards in disposing of the various claims of the petitioner. Respondent denies that the trial court erred in refusing to grant petitioner (Sheppard) a new trial on the ground of newly discovered evidence. Respondent denies that any relevant material or substantial evidence was suppressed by the prosecution or that any unjust tactics were used by the prosecuting authorities in the trial of this case. Respondent denies that the petitioner (Sheppard) was prevented from having a fair and impartial trial by the action of the trial judge as a result of the actions of the bailiffs in permitting the jurors to make telephone calls. Respondent denies that the trial judge coerced the jury into reaching a verdict, and avers that the fact that the jury deliberated for a period of five days merely shows the carefulness and consideration that the jury gave the mass of testimony and over 200 exhibits in the case. Respondent denies each and every other allegation in the petition not herein admitted to be true. Affirmatively, respondent alleges that petitioner was convicted in a court that had jurisdiction of his person and of the crime involved, that petitioner was not deprived of any of his constitutional rights, and that the facts upon which petitioner (Sheppard) relies, even if true, constitute mere error in the trial court, which is not cognizable in an action of habeas corpus. For the foregoing reasons, respondent prays that the petition herein be dismissed