1. Institute of Mechanics and Advanced Materials An Adaptive Multiscale Method for Modelling of Fracture in Polycrystalline Materials Ahmad Akbari R., Pierre Kerfriden*, Stéphane Bordas Institute of Mechanics and Advanced Materials, School of Engineering, Cardiff University, UK

2. Institute of Mechanics and Advanced Materials 1-Introduction: Fracture a multiscale phenomena  Multiscale methods: Hierarchical vs. concurrent multiscale methods  homogenization: formulation of averaging theorem, criteria, and coupling formula  Concurrent multiscale method: formulations 2-Adaptive multiscale method:  mesh adaptivity based on GOEE 3-Results  Polycrystalline microstructure  L-shape  notched beam 4- Conclusion

3. Institute of Mechanics and Advanced Materials Coupling of macroscopic and microscopic levels The volume averaging theorem is postulated for: 1) Strain tensor: 2) Virtual work (Hill-Mandel condition): 3) Stress tensor: Computational Homogenization Definition of an RVE

4. Institute of Mechanics and Advanced Materials Hierarchical multiscale method: FE Scheme 2 In softening regime:  Lack of scale separation  At the macroscale is mesh dependent The macroscopic constitutive law is not required Non-linear material behaviour can be simulated Microscale behaviour of material is monitored at each load step Advantages and abilities: Drawbacks:

5. Institute of Mechanics and Advanced Materials Concurrent Multiscale method Decomposing the problem into two coarse mesh and fine mesh sub-domains. Least square method is used to define the non-conforming meshes relation: where And is an extractor for the fine mesh nodes on the interface,.

6. Institute of Mechanics and Advanced Materials Concurrent Multiscale method Lagrange multipliers technique is used to enforce the prefect continuous connection between the sub-domains: At the stationary point we have: Lagrangian: Where is the potential energy of the system, and are the Lagrange multipliers. A local arc-length method is employed to control crack propagation speed: Where c is the extractor of the maximum variation of displacement jump at the fine scale, and is a limit for the maximum variation of the displacement jump.

7. Institute of Mechanics and Advanced Materials Adaptive mesh refinement Recovery-based goal-oriented error estimator Dual problem: Quantity of interest is a function of maximum damage at the microscopic RVE sample for each element. Where is the unit vector corresponding to the softest orientation of the macroscopic tangent stiffness tensor which is obtained by analysing the acoustic tensor. Acoustic tensor: Index notation Voigt notation

8. Institute of Mechanics and Advanced Materials 123 FE 2 FE + Concurrent 2 Mesh refinement Hybrid method Adaptive mesh refinement

9. Institute of Mechanics and Advanced Materials Results Material microstructure Constitutive model for grains: where are the stiffness, the stress, and the strain tensors in the principal material coordinate system, respectively. The constitutive equation in the global coordinate system can be developed by using transformation matrix, : The potential failure of the interface between adjacent grains is described by a cohesive model in the local coordinate

10. Institute of Mechanics and Advanced Materials Results Example 1: Example 2:

11. Institute of Mechanics and Advanced Materials Conclusion A hybrid multiscale method was developed for modeling of fracture in polycrystalline materials: A local arc-length was used to control crack speed at the process zones. A goal-oriented error estimation was employed to have optimal mesh at each time step. The robustness of the method was shown by two examples.