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About the animations
  1. Microbranching experiments were performed by Sharon and Fineberg (1996). Initial stress is applied by clamping the top and bottom of a PMMA sheet, and a sharp crack is created by a razor blade. They observed that fracture surfaces are rough and more microbranching occurs when a crack velocity is greater than a critical value. Such physical phenomena are investigated by using the PPR potential-based cohesive zone model in conjunction with the extrinsic cohesive zone model. To address mesh orientation dependence in 4k structured meshes, and to avoid undesirable crack patterns, nodal perturbation and edge-swap operators are introduced.

    Related Publications:
    E. Sharon, and J. Fineberg. "Microbranching instability and the dynamic fracture of brittle materials." Physical Review B Vol. 54, No. 10, pp. 7128–7139, 1996.

    Z. Zhang, G. H. Paulino, and W. Celes. "Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials." International Journal for Numerical Methods in Engineering. Vol. 72, No. (8), pp. 893–923, 2007.

    K. Park. Potential-based fracture mechanics using cohesive zone and virtual internal bond modeling. PhD Thesis, University of Illinois at Urbana-Champaign, 2009.

  2. For a mixed-mode dynamic fracture problem, Kalthoff and Winkler (1987) tested a doubly notched specimen to investigate the failure mode transition with respect to the loading rates. They observed that relatively lower loading rate resulted in brittle failure with a crack propagation angle of about 70º, while higher loading rate generated a shear band ahead of the initial notch with a negative angle of about −10º. Brittle fracture behavior is simulated by employing the extrinsic cohesive zone model. The crack tip region is adaptively refined to capture nonlinear crack tip behavior while a far field from a crack tip is adaptively coarsened on a basis of coarsening criterion. Animation 2 shows the adaptive mesh refinement and coarsening.

    Related Publications:
    J. F. Kalthoff, ans S. Winkler. "Failure mode transition at high rates of shear loading." International Conference on Impact Loading and Dynamic Behavior of Materials 1, pp. 185–195, 1987.

    K. Park. Potential-based fracture mechanics using cohesive zone and virtual internal bond modeling. PhD Thesis, University of Illinois at Urbana-Champaign, 2009.

  3. Strain energy associated with problem described above (2).

  4. Topology optimization of a 2D cantilever beam subject to tip loading (applied at lower right corner) using adaptive mesh refinement and coarsening

  5. Topology optimization of a 3D cantilever beam subject to tip loading (applied at the mid-depth) using adaptive mesh refinement and coarsening

  6. Deformed shape of cantilever beam with moment applied to free end and load factor versus free end displacement 

  7. Deformed square frame under tensile loading and load factor versus displacements. Example from: Kjell Mattiasson.  "Numerical results from Large Deflection Beam and Frame Problems Analysed by Means of Elliptic Integrals."  International Journal for Numerical Methods in Engineering, Vol.17, Nos.1, pp.145-155, 1980.

  8. Deformed square frame under compression loading and load factor versus displacements. Example from: Kjell Mattiasson.  "Numerical results from Large Deflection Beam and Frame Problems Analysed by Means of Elliptic Integrals."  International Journal for Numerical Methods in Engineering, Vol.17, Nos.1, pp.145-155, 1980.

  9. Deformed shape of the Lee Frame and load factor versus displacements.

  10. Load factor versus displacement of a 12 Bar 3D Truss