13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- FE-percolation model of Butcher [2], the particle field is decomposed into “percolation elements” based upon the resolution of the finite-element mesh to better capture local stress and strain gradients within the microstructure. Void nucleation and coalescence occur within each percolation element with a global search for void coalescence occurring between elements at the end of each time step. The general percolation modeling strategy is outlined in Figure 2 and discussed in the subsequent sections. The interested reader is referred to [2] for additional details. Finite-element model of a notched tensile test Finite-elements are mapped onto a measured particle field Micromechanical models are applied to each particle to characterize local void initiation and evolution leading to fracture Σ1 Σ1 σ1 σ1 Figure 1: Multi-scale percolation model applied to a finite-element simulation of a tensile test. Call Subroutine for the Constitutive Model Update Stress Tensor and Plastic Strain Determine the Macroscopic Stress Tensor and Plastic Strain in the Element Test for Element Failure Delete Element Yes No Call User Defined Subroutine for the Percolation Elements Obtain Nodal Displacements of the Elements from the FE Code Finite-Element Simulation at Time t Call the Percolation Subroutine and Update Global Porosity Compute the Strain Increment of the Element from the Nodal Displacements Divide the Nodal Displacements and Strain Increment into Sub-increments Reached Number of Sub-increments? No Update Centroids of All Objects in the Element using the Nodal Displacement Increment Evaluate the Deformation gradient, Spin Tensor and Plastic Spin Tensor for All Objects and Update the Object Orientation Determine the Stress State within the Particles and Evaluate the Nucleation Model Nucleation? Create Penny-Shaped Void Yes Test for Coalescence for ALL Void and Crack Neighbours Search for Neighbour Elements, Particles, Voids and Cracks using Characteristic Lengths Evaluate Growth and Shape Evolution of the Voids and Cracks Coalescence or Overlap? Merge Objects to Create a New Crack Update the Total Porosity in the Material and Other Volume Averaged Statistics Call the Percolation Subroutine and Update Global Porosity Return Averaged Properties to the Constitutive Model to Account for Material Softening and Failure of the Element Yes Figure 2: Flow-chart of the percolation modeling process 2. Constitutive model The percolation material model was written as a user-defined subroutine for LS-DYNA [6] to integrate the stress state, analyze the microstructure for void evolution and return the stress tensor
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