Crack propagation on highly heterogeneous composite materials Miguel Patrício.

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


- What makes the problem complicated?

Simplify


- 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


Homogenisation


Assume: There exists a RVE

Mathematical homogenisation


Assume: There exists a RVE Periodical distribution


Microstructure

Linear elastic materials:Hook’s lawElasticity tensors


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


- “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


Question: will the crack propagate?



(one possible)Answer: look at the SIFs

Crack tip

How to compute the SIFs?



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


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


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

Crack propagation on highly heterogeneous composite materials Miguel Patrício.

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