Computational homogenization of brittle fracture

• Applicant: Professor Dr. Matti Schneider
• Subject Area: Mechanics
• Term: from 2019 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 426323259

• Final Report Year: 2024

Final Report Abstract

To design components according to requirements, it is crucial to characterize the mechanical behavior of the underlying material until fracture or failure. Simulation tools play a decisive role in reducing the considerable experimental effort, in particular for microstructured materials with direction-dependent mechanical properties. The aim of the DFG-funded project was to determine the material parameters of brittle materials with the help of microstructure simulations. In particular, materials of industrial relevance, which typically have a rather complex microstructure as a consequence of the manufacturing process, should be treatable. Based on specific mathematical homogenization results, the effective crack energy, which represents a lower bound on the crack resistance of the heterogeneous material, needs to be determined using efficient numerical methods. More precisely, the effective crack energy may be computed as the area of the minimal surface, weighted by the phasewise crack resistance, which intersects the microstructure with a given mean normal direction. In order to calculate this quantity efficiently, a duality result from convex analysis is helpful, which determines the effective crack energy as the maximum flow through the microstructure in the specified normal direction, where the flow field must be limited point by point by the phasewise crack resistance. In the course of the project, a series of efficient solvers based on the fast Fourier transform (FFT) were developed for the problem at hand, which are able to handle large three-dimensional microstructures with periodic boundary conditions. A special finite volume discretization is used as well as a solver based on the Alternating Direction Method of Multipliers with adaptive step size control. The developed methodology is general enough to treat anisotropic crack resistance on the microscale as well as interfaces with reduced crack resistance. If the possible normal directions are scanned, an effective crack energy area can be determined, which is used in simulations on the component scale.

Publications

• An FFT‐based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture. International Journal for Numerical Methods in Engineering, 121(7), 1367-1387.
  Schneider, Matti
  
• Fast implicit solvers for phase-field fracture problems on heterogeneous microstructures. Computer Methods in Applied Mechanics and Engineering, 363, 112793.
  Ernesti, Felix; Schneider, Matti & Böhlke, Thomas
  
• A fast Fourier transform based method for computing the effective crack energy of a heterogeneous material on a combinatorially consistent grid. International Journal for Numerical Methods in Engineering, 122(21), 6283-6307.
  Ernesti, Felix & Schneider, Matti
  
• Computing the effective crack energy of heterogeneous and anisotropic microstructures via anisotropic minimal surfaces. Computational Mechanics, 69(1), 45-57.
  Ernesti, Felix & Schneider, Matti
  
• Computing the effective crack energy of microstructures via quadratic cone solvers. PAMM, 21(1).
  Ernesti, Felix; Schneider, Matti & Böhlke, Thomas
  
• Characterizing digital microstructures by the Minkowski‐based quadratic normal tensor. Mathematical Methods in the Applied Sciences, 46(1), 961-985.
  Ernesti, Felix; Schneider, Matti; Winter, Steffen; Hug, Daniel; Last, Günter & Böhlke, Thomas
  
• Investigations on the influence of the boundary conditions when computing the effective crack energy of random heterogeneous materials using fast marching methods. Computational Mechanics, 71(2), 277-293.
  Ernesti, Felix; Lendvai, Jonas & Schneider, Matti
  
• A computational multi-scale approach for brittle materials, Karlsruher Institut für Technologie (KIT), Diss.
  Ernesti, F.
  
• Accounting for weak interfaces in computing the effective crack energy of heterogeneous materials using the composite voxel technique. Archive of Applied Mechanics, 93(10), 3983-4008.
  Ernesti, Felix & Schneider, Matti