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Limits with Oscillating Discontinuities

An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...
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Magnetically Induced Rotating Rayleigh-Taylor Instability
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Global bifurcation criterion for oscillatory crack path instability.

Van-Bac Pham1, Hans-Achim Bahr, Ute Bahr

  • 1Institut für Festkörpermechanik, Technische Universität Dresden, D-01062 Dresden, Germany. van_Bac.Pham@tu-dresden.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 23, 2008
PubMed
Summary

A new global bifurcation criterion predicts crack propagation patterns, matching experimental results for oscillating cracks in glass. This method ensures pure mode I stress fields for accurate instability predictions.

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Area of Science:

  • Materials Science
  • Solid Mechanics
  • Fracture Mechanics

Background:

  • Quasistatic crack propagation can transition from straight to oscillatory behavior.
  • Understanding this transition is crucial for predicting material failure.
  • Previous local criteria have shown limitations in explaining oscillatory crack paths.

Purpose of the Study:

  • Derive a global bifurcation criterion for the transition to oscillatory crack propagation.
  • Simplify the criterion for practical application to sine-shaped crack paths.
  • Validate the criterion against experimental observations of crack instability.

Main Methods:

  • Derivation of a global bifurcation criterion based on pure mode I stress fields (K_{II}=0) along the entire crack path.
  • Simplification of the criterion by applying K_{II}=0 only at specific phases for sine-shaped crack paths.
  • Thermoelastic analysis of a hot glass slab cooling in water to obtain temperature fields.
  • Comparison of critical wavelengths from the derived criterion with experimental data.

Main Results:

  • The global bifurcation criterion accurately predicts the transition from straight to oscillatory crack growth.
  • A simplified version of the criterion is sufficient for sine-shaped crack paths.
  • Calculated critical wavelengths for oscillating crack growth show excellent agreement with experimental results.
  • Local bifurcation criteria yield incorrect predictions for oscillatory crack path instability.

Conclusions:

  • The derived global bifurcation criterion provides a robust method for analyzing crack propagation instability.
  • The simplified criterion offers a computationally efficient approach for predicting oscillatory crack behavior.
  • This research validates the importance of considering global stress field conditions in fracture mechanics.