Example for Presentation

1
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

• 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
Computational Homogenization
4
2
Hierarchical multiscale method FE Scheme
Drawbacks
Advantages and abilities
The macroscopic constitutive law is not
required Non-linear material behaviour can be
simulated Microscale behaviour of material is
monitored at each load step

• In softening regime
• Lack of scale separation
• At the macroscale is mesh dependent

5
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

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Concurrent Multiscale method

• Lagrange multipliers technique is used to
  enforce the prefect continuous connection between
  the sub-domains

At the stationary point we have

• 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.
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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

Voigt notation
Index notation
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Adaptive mesh refinement
Hybrid method
Mesh refinement
1
2
3
2
2
FE
FE Concurrent
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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
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Results
Example 1
Example 2
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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.