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Crack propagation on highly heterogeneous composite materials
Miguel Patrício
Motivation
Macroscopic view:
- a (foot)ball (round object)
Microscopic view:
- thick round-ish skin
- fissures and cracks
- collection of molecules
- simple (?) problem
- not so accurate
-complicated problem
- accurate
Motivation
Macroscopic view:
- a (foot)ball (round object)
Microscopic view:
- thick round-ish skin
- fissures and cracks
- collection of molecules
Best of both
worlds???
Model crack propagation
Macroscopic view Microscopic view
Matrix
Inclusions
Problem formulation
“Determine how (and whether) a given crack will propagate.”
- Where to start?
Problem formulation
“Determine how (and whether) a given crack will propagate.”
- What makes the problem complicated?
Simplify
“Determine how (and whether) a given crack will propagate.”
- Microstructure
- Crack propagation (how)
???
Assume: Static crack
Starting point
- Static crack is part of the geometry
“Determine whether a given crack will propagate in a homogenised medium.”
What homogenised medium?
Microstructure to macrostructure
?
MacrostructureMicrostructure
Microstructure to macrostructure
Homogenisation
MacrostructureMicrostructure
Homogenisation
MacrostructureMicrostructure
Assume: There exists a RVE
Mathematical homogenisation
MacrostructureMicrostructure
Assume: There exists a RVE Periodical distribution
Mathematical homogenisation
Microstructure
Linear elastic materials:Hook’s lawElasticity tensors
Mathematical homogenisation
Microstructure
averagingprocedure
( and )
Example
-0.5 0.5
0.5 Young’s modulus
Poisson’s ratio
Young’s modulus
Poisson’s ratio
Homogenised solution
Example
Exact solution
Horizontal component of the displacements
Mathematical homogenisation
- “Sort of” averaging procedure
- Loss of accuracy
- Alternatives do exist (heterogeneous multiscale method, multiscale finite elements…)
- Periodic structures (but not only)
- Simplifies problem greatly
Crack propagation (homogeneous case)
Assume: pre-existent static crack homogeneous material
Crack propagation (homogeneous case)
Question: will the crack propagate?
Crack propagation (homogeneous case)
Question: will the crack propagate?
(one possible)Answer: look at the SIFs
Crack tip
How to compute the SIFs?
Crack propagation (homogeneous case)
Question: will the crack propagate?
Why look at the SIFs?- Solve elasticity problem (FEM)
- Determine the stresses
- Crack will propagate when
- Direction of crack propagation
- Compute
+ how?
Step by step- Pull the plate
- Compute displacements and stresses
- Check propagation criterion
compute SIFs
- If the criterion is met, compute the direction of propagation
Increment crack (update geometry)
-What length of crack increment?
Example
- FEM discretisation (ABAQUS)
- Crack modelled as a closed line
- Open crack (after loading):
Example
Crack propagation
- Homogeneous media
- how and whether the crack will propagate
- Pre-existing crack
- Incrementation approach
- What about heterogeneous media?
Crack propagation
- What about heterogeneous media?
Idea: employ homogenisation and apply same procedure
Bad
Local effects
Local effects
Crack tip in material A Crack tip in material B
Crack tip in
homogenised material
Crack propagation
- What about heterogeneous media?
Idea: employ homogenisation and apply same procedure
Bad
Because the local structure may not be neglected when the SIFs are computed
FEM will not work!!!
Crack propagation (composite material)
Assume: pre-existent static crack composite material
Domain decomposition
Assume: pre-existent static crack composite material
- Partition computational domain
- Instead of one heavy problem, solve many light problems
- Allows for a complex problem to be divided into several subproblems
Domain decomposition
- Schwarz procedure dates back to the XIX century
- Parallelization may be implemented
- Deal with different problems where different phenomena exists
- May overlap or not
Homogenisable
Hybrid Approach
Homogenisable
Schwarz
(overlapping)
Homogenisation
Crack
Hybrid approach for the SIF
Layered materialCrack
Hybrid approach for the SIF
Crack
- Employ homogenisation far away from the crack
- Use Schwarz overlapping scheme
- Uses homogenisation where possible; resolves heterogeneous problem where necessary
Hybrid approach
- Combines homogenisation and domain decomposition
- More than one micro region may be considered
- Accuracy depends on the accuracy of homogenisation or, on other words, on how much the material is homogenisable in the macro region
- Why not domain decomposition?
- Why not homogenisation?
- Domain decomposition divides problem in subproblems
Summary
- Homogenisation yields macroscopic equations
- Fracture propagation can be implemented by incrementing the crack
- Hybrid approach combines these two techniques
Main open question
Assume: pre-existent static crack composite layered material
- What happens when the crack hits
the interface between the layers?
A few references
Model crack propagation
Macroscopic view Microscopic view
Linear elastic homogeneous plate
Plate composed by linear elastic homogeneous constituents
Matrix
Inclusions
Layeredmaterial
?Othermicro-structure
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