Rafi Blumenfeld: Research interests and plans

(This file is constantly changing. It has been updated only schematically since I moved to the UK, but I intend to get down to that very soon. )

I. A statistical theory for patterns of evolving interfaces: References
  1. R. Blumenfeld and A. Aharony "Breakdown of multifractal behaviour in diffusion limited aggregates" Phys. Rev. Lett. 62, 2977-2980 (1989).
  2. R. C. Ball and R. Blumenfeld "Exact results on exponential screening in two-dimensional diffusion limited aggregation" Phys. Rev. A, R828-R831 (1991).
  3. R. C. Ball, P. W. H. Barker and R. Blumenfeld "Sidebranch selection in fractal growth" EuroPhys. Lett. 16, 47-52 (1991).
  4. R. Blumenfeld, "Two dimensional Laplacian growth can be mapped onto Hamiltonian dynamics" Phys. Lett. A 186, 317-322 (1994).
  5. R. Blumenfeld, "Formulating a first-principles statistical theory of growing surfaces in two-dimensional Laplacian fields" Phys. Rev. E 50, 2952 (1994).
  6. R. Blumenfeld and R. C. Ball, "Two dimensional Laplacian growth as a system of creating and annihilating particles" Phys. Rev. E 51, 3434 (1995).
  7. R. Blumenfeld, "Pattern formation in Laplacian growth: Theory" Center for Nonlinear Studies Newsletter 112, April 1995.
  8. Y. Jiang and R. Blumenfeld, "First-principles estimates of the fractal dimension and the multifractal spectrum of diffusion-Limited-aggregates" in preparation.

II. Models for crack propagation

  1. Slow cracking:

    There are two approaches that I am taking in this context: The first uses a formalism that is close to the one constructed for the above Laplacian growth processes. This formalism applies to two-dimensional stress fields too, albeit with some modifications. That formalism needs to be adapted to two-dimensional stress fields and this is currently being carried out. I have made some preliminary progress in this direction but still have not reached the stage that I have in the Laplacian growth, mostly due to lack of time. In my opinion, an enthusiastic student here will be able to turn this into an exciting thesis.

    The second approach consists of replacing the complicated boundary of the fracture by a regular envelope that can be solved for in the quasi-static limit. This approach has been initiated by R. C. Ball and I and allows to carry out a stability analysis which shows that an initially isotropic system pattern becomes anisotropic. Namely, a marginally stable angle emerges between globally emanating arms which is just below 120 degrees. The stability analysis can be modified to include a Gibbs-Thomson term (similar to surface tension) but this calculation has not been carried out yet. An interesting phenomenon that I have found with R. C. Ball is that the stress along the simple envelope of the fracture develops logarithmic oscillations. Although these oscillations stem from subdominant terms in the quasi-static stress field, they are enhanced by the dynamic instability during the propagation and therefore are likely to overshadow the dominant static terms. These oscillations result in interesting scale- ivariant sidebranching patterns as observed in simulations and measurements. To observe this phenomenon we constructed a general morphological correlator (see below) that is sensitive to logarithmic oscillations. Such oscillations have been observed recently in other systems, for which this correlator should be possible to be useful.

  2. Fast cracking:

    Mesoscale modeling and analysis: During fast propagation the system is far from equilibrium and quasi-static calculations break down close to the propagating front. I investigate a new minimal model in the mesoscale that I developed recently for fracture in amorphous brittle media. This model incorporates the effects of the changes in material properties and the appearance of microcracks in the process zone. Due to enhanced scattering of the shear waves the stress relaxes very slowly in the process zone and the crack can overtake the shear waves. The theory that I propose turns out to fit together several pieces of a long standing puzzle in the field, including observations of: i) Limiting velocities that are well below the predicted Rayleigh wave speed; ii) Oscillatory instabilities in the propagation speed; and iii) the anomalously slow stress relaxation in the process zone. I am currently working on the rich statistics that emerges from this picture and the connection to the structure of the fracture surfaces observed in many measurements. I hope to check this model against measurements on PMMA with experimentalist collaborators.

    Atomistic analysis: Together with Brad Holian from Los Alamos and Peter Gumbsch from Max Planck Inst. in Sttutgart, I have been studying a minimal model (again) that describes the highly nonlinear behaviour on the atomic scale. The results are surprisingly accurate for such a simple model and give a very clear insight into lattice trapping, the low propagation rate in single crystals and the nonlinear excitations that build up in front of the propagating crack. Moreover, this model opens the door to run atomistic simulations of fracture for selected small regions in an otherwise continuously simulated system. This is a Hybrid numerical model, where atomic localised simulators feed information through boundary conditions to continuum simulators.

  3. Relations between dynamics and fracture roughness:

    I have started to study the relations between the fracture dynamics with microcracking in its front and the observed so-called self-affinity of the roughness of crack surfaces. This issue is significant in Materials Science due to the huge effort to reliably correlate between fracture roughness and material mechanical properties; a crucial step in designing better materials. I have recently coauthored a comprehensive review on fracture roughness from this perspective [Progress in Materials Science 38, 425 (1994)]. With the aid of a newly devised method for quantifying scale-invariant morphologies, I am currently investigating the dependence of the characteristics of the fracture surface on the dynamics and on material parameters.

    I have long maintained that the above three issues are connected and consequently I organised a workshop on "Fractal Analysis and Modelling of Materials"

    whose main theme was the integration of the dynamics, materials parameters and roughness characteristics.

  1. R. C. Ball and R. Blumenfeld "Universal scaling of the stress field at the vicinity of a wedge crack in two dimensions and oscillatory self-similar corrections to scaling" Phys. Rev. Lett. 65, 1784-1787 (1990).
  2. R. Blumenfeld and R. C. Ball "Onset of scale-invariant pattern in growth processes: The cracking problem" Physica A 177, 407-415 (1991).
  3. R. Blumenfeld and R. C. Ball "Onset of scaling behaviour in 2D slow cracking" Modern Phys. Lett. B 5, 1567-1573 (1991).
  4. V. Milman, R. Blumenfeld, N. A. Stelmashenko and R. C. Ball, "Experimental measurements of the roughness of brittle cracks" Phys. Rev. Lett. 71, 204 (1993).
  5. V. Milman, N. A. Stelmashenko and R. Blumenfeld, "Fracture surfaces: A critical review and a morphological analysis of scanning tunneling microscopy measurements", Progress in Materials Science 38, 425-474 (1994).
  6. R. Blumenfeld, "Nonequilibrium brittle fracture propagation: Steady state, velocity oscillations and intermittency" Phys. Rev. Lett. 76, 3703 (1996).
  7. B. L. Holian, R. Blumenfeld and P. Gumbsch, "An Einstein ice-skater model for brittle fracture propagation" Phys. Rev. Lett. 78, 78 (1997).
  8. R. Blumenfeld, in Fractal Analysis and Modelling of Materials: New Directions, Eds. R. Blumenfeld and A. R. Bishop (World Scientific, ca. 1999).

IIIa. Fast data transmission and communication through noisy conduits: