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Crack propagation in anisotropic brittle materials: from a phase-field model to a shape optimization approach

Tim Suchan, Chaitanya Kandekar, Wolfgang E. Weber, Kathrin Welker

Abstract

The phase-field method is based on the energy minimization principle which is a geometric method for modeling diffusive cracks that are popularly implemented with irreversibility based on Griffith's criterion. This method requires a length-scale parameter that smooths the sharp discontinuity, which influences the diffuse band and results in mesh-sensitive fracture propagation results. Recently, a novel approach based on the optimization on Riemannian shape spaces has been proposed, where the crack path is realized by techniques from shape optimization. This approach requires the shape derivative, which is derived in a continuous sense and used for a gradient-based algorithm to minimize the energy of the system. Due to the continuous derivation of the shape derivative, this approach yields mesh-independent results. In this paper, the novel approach based on shape optimization is presented, followed by an assessment of the predicted crack path in anisotropic brittle material using numerical calculations from a phase-field model.

Crack propagation in anisotropic brittle materials: from a phase-field model to a shape optimization approach

Abstract

The phase-field method is based on the energy minimization principle which is a geometric method for modeling diffusive cracks that are popularly implemented with irreversibility based on Griffith's criterion. This method requires a length-scale parameter that smooths the sharp discontinuity, which influences the diffuse band and results in mesh-sensitive fracture propagation results. Recently, a novel approach based on the optimization on Riemannian shape spaces has been proposed, where the crack path is realized by techniques from shape optimization. This approach requires the shape derivative, which is derived in a continuous sense and used for a gradient-based algorithm to minimize the energy of the system. Due to the continuous derivation of the shape derivative, this approach yields mesh-independent results. In this paper, the novel approach based on shape optimization is presented, followed by an assessment of the predicted crack path in anisotropic brittle material using numerical calculations from a phase-field model.
Paper Structure (12 sections, 30 equations, 9 figures, 2 algorithms)

This paper contains 12 sections, 30 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modelling
  3. Fracture model
  4. Adaption of the fracture model to the phase-field setting
  5. Adaption of the fracture model to the shape optimization setting
  6. Numerical implementation
  7. Implementation of the phase-field approach
  8. Implementation of the shape optimization approach
  9. Numerical examples
  10. Boundary Value Problem 1: V-notch -- Isotropy
  11. Boundary Value Problem 2: Single-edge notch test -- Anisotropy
  12. Summary and conclusion

Figures (9)

  • Figure 1: Replacement of an infinitesimally thin fracture for the single-edge notch test (left) by an open curve as part of the boundary of $\mathcal{B}$ (right).
  • Figure 2: Mesh for V-notch: Nodes - 6900, Elements - 13434
  • Figure 3: Fracture evolution results: phase-field (left) and shape optimization (right).
  • Figure 4: Mesh for SENT: Nodes - 34863, Elements - 69014
  • Figure 5: Fracture paths for different angles $\theta$ of the anisotropic material.
  • ...and 4 more figures