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Interacting damage models mapped onto Ising and percolation models.

Renaud Toussaint1, Steven R Pride

  • 1Department of Physics, University of Oslo, P. O. Box 1043 Blindern, 0316 Oslo, Norway. Renaud.Toussaint@fys.uio.no

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 21, 2005
PubMed
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This study introduces damage models on lattices, linking lattice damage to statistical mechanics. We show damage models with global load sharing map to percolation theory, while local load sharing maps to the Ising model.

Area of Science:

  • Statistical Mechanics
  • Materials Science
  • Condensed Matter Physics

Background:

  • Introduces a novel class of damage models on regular lattices with isotropic interactions.
  • Focuses on systems where stress perturbation from broken cells is weak compared to mean stress.

Purpose of the Study:

  • To develop and analyze damage models on lattices with quenched disorder and Griffith's principle.
  • To establish an isomorphism between these damage models and generalized Ising models.

Main Methods:

  • Direct integration over all realizations of quenched disorder to obtain damage configuration probability distributions.
  • Demonstration of isomorphism with generalized Ising models, percolation theory, and standard Ising models based on load-sharing rules.

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Main Results:

  • Damage models with global load sharing are isomorphic to percolation theory.
  • Damage models with local load sharing are isomorphic to the standard Ising model.
  • Classification of breakdown processes for power-law interactions and identification of universality classes.

Conclusions:

  • The probability distribution of configurations represents a maximum of Shannon's entropy under specific constraints.
  • Establishes a strong connection between quenched-disorder based damage models and standard statistical mechanics.
  • Provides insights into the universality class and autocorrelation length of breakdown transitions.