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MSc Civil Eng. Adrian Egger, ETH Zürich

Prof. Dr. Savvas Triantafyllou, University of Nottingham

Prof. Dr. Eleni Chatzi, ETH Zurich

Revisiting Fracture via a

Scaled Boundary Multiscale Approach

16/05/2018Revisiting Fracture via a Scaled Boundary Multiscale Approach 1

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1. Motivation

2. Salient Features of MsSBFEM

3. Calculation of Stress Intensity Factors

4. Numerical Examples

1. Unit Cell with embedded slant crack subject to tension

2. Plate with multiple cracks in tension

5. Conclusions

6. Questions

Contents

16/05/2018 2Revisiting Fracture via a Scaled Boundary Multiscale Approach

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Scaled Boundary Finite Element Method (SBFEM) Computationally efficient

Semi-analytical

SIFs can be determined accurately and with ease

Computational challenges for large domains remain

Extended multiscale finite element method (EMsFEM)

Coarse mesh: solve governing equations of the problem

Fine mesh: account for fracture phenomena

Motivation

16/05/2018 3

1. Motivation

2. MsSBFEM theory

3. SIF calculation

4. Num. Examples

5. Conclusion

6. Questions

Revisiting Fracture via a Scaled Boundary Multiscale Approach

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MsSBFEM Theory

16/05/2018 4

1. Motivation

2. MsSBFEM theory

i. SBFEM

ii. MsFEM

3. SIF calculation

4. Num. Examples

5. Conclusion

6. Questions

Revisiting Fracture via a Scaled Boundary Multiscale Approach

Coordinates:

Displacements:

𝑥 𝜉, 𝜂 = 𝑥𝑂 + 𝜉𝑥(𝜂)

= 𝑥𝑂 + 𝜉 𝑵 𝑛 {𝒙}

𝑢 𝜉, 𝜂 = 𝑵𝑢(𝜂) {𝒖(𝝃)}

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General solution as power series

Having performed the eigen-decomposition

Equating displacement modes and force modes

on the boundary 𝑢 𝜉 = 1 :

MsSBFEM Theory

16/05/2018Revisiting Fracture via a Scaled Boundary Multiscale Approach 5

1. Motivation

2. MsSBFEM theory

i. SBFEM

ii. MsFEM

3. SIF calculation

4. Num. Examples

5. Conclusion

6. Questions

𝑢 𝜉 = 𝜙𝑖 𝜉− 𝜆𝑖 [𝑐𝑖]

𝜙𝑖 = eigenvector

𝜆𝑖 = eigenvalue

𝑐𝑖 = integration constant

Displacements:

Forces:

𝑢 𝜉 = [𝜙−𝑢]𝜉− 𝜆− [𝑐−]

𝑞 𝜉 = [𝜙−𝑞]𝜉− 𝜆− [𝑐−]

𝑲𝑏𝑜𝑢𝑛𝑑𝑒𝑑 = [𝜙−𝑞][𝜙−𝑢]−1Stiffness Matrix:

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Difference FEM and MsFEM

Basis functions (G) map the response between fine (micro)

and coarse (macro) mesh

MsSBFEM Theory

16/05/2018Revisiting Fracture via a Scaled Boundary Multiscale Approach 6

1. Motivation

2. MsSBFEM theory

i. SBFEM

ii. MsFEM

3. SIF calculation

4. Num. Examples

5. Conclusion

6. Questions

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MsSBFEM TheoryMsFEM construction of basis functions

16/05/2018Revisiting Fracture via a Scaled Boundary Multiscale Approach 7

1. Motivation

2. MsSBFEM theory

i. SBFEM

ii. MsFEM

3. SIF calculation

4. Num. Examples

5. Conclusion

6. Questions

Linear Periodic

+ • Simple implementation • Local periodicity included

• Softer behaviour

- • Too stiff

• Too restrained

• No local variation

• Requires adjacent pairs

• Computational effort

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SBFEM expression for the stresses

Inspection of modal representation yields

Singularity for -1 < λ < 0

By matching expressions with

the exact solution:

Calculating Stress Intensity Factors

16/05/2018Revisiting Fracture via a Scaled Boundary Multiscale Approach 8

1. Motivation

2. MsSBFEM theory

3. SIF calculation

4. Num. Examples

5. Conclusion

6. Questions

𝜎 𝜉, 𝜂 = [𝑫] 𝑩1 𝜂 𝑢 𝜉 ,𝜉 +1𝜉𝑩2 𝜂 {𝑢(𝜉)}

𝜎𝑠 𝜉, 𝜂 = 𝜞𝑖 𝜂 𝜉− 𝜆𝑠 − 𝑰 {𝒄𝑠}

𝜞𝑖 = 𝑫 −𝜆𝑖 𝑩1 + 𝑩2 [𝝓𝑖]

where:

𝐾𝐼𝐾𝐼𝐼= 2𝜋𝐿0

𝑖=𝐼,𝐼𝐼

𝑐𝑖Γ𝑦𝑦(𝜂 = 𝜂𝐴)𝑖

𝑖=𝐼,𝐼𝐼

𝑐𝑖Γ𝑥𝑦(𝜂 = 𝜂𝐴)𝑖

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Numerical Examples:Unit Cell with embedded slant crack in tension

16/05/2018

1. Motivation

2. MsSBFEM theory

3. SIF calculation

4. Num. Examples

i. Unit cell intro

ii. Unit cell results

iii. Plate intro

iv. Plate results

5. Conclusion

6. Questions

Property Value

E-modulus 200 [N/mm2]

Poisson ratio 0.3

Side length L 1 [mm]

Crack angle 30°

Crack length a variable

Tension force 0.1 [N/mm]

Quantities of interest:

• SIF K1 and K2

at right crack tip

Revisiting Fracture via a Scaled Boundary Multiscale Approach

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Numerical Examples:Unit Cell with embedded slant crack in tension

16/05/2018

1. Motivation

2. MsSBFEM theory

3. SIF calculation

4. Num. Examples

i. Unit cell intro

ii. Unit cell results

iii. Plate intro

iv. Plate results

5. Conclusion

6. Questions

< 1 1-2 2-5 5-10 > 10

Revisiting Fracture via a Scaled Boundary Multiscale Approach

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Numerical Examples:Plate with multiple cracks in tension

16/05/2018

1. Motivation

2. MsSBFEM theory

3. SIF calculation

4. Num. Examples

i. Unit cell intro

ii. Unit cell results

iii.Plate intro

iv. Plate results

5. Conclusion

6. Questions

Property Value

E-modulus 200 [N/mm2]

Poisson ratio 0.3

Side length 5 [mm]

Crack angle 30°

Crack length variable

Tension force 0.1 [N/mm]

Quantities of Interest:

• SIFs K1 and K2

at right crack tip

of crack 1-5Q4 Q8 Q12 Q16

1

2

3

4

5

Revisiting Fracture via a Scaled Boundary Multiscale Approach

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Numerical Examples:Plate with multiple cracks in tension

16/05/2018

1. Motivation

2. MsSBFEM theory

3. SIF calculation

4. Num. Examples

i. Unit cell intro

ii. Unit cell results

iii. Plate intro

iv.Plate results

5. Conclusion

6. Questions

Revisiting Fracture via a Scaled Boundary Multiscale Approach

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MsSBFEM

Cracks successfully incorporated

at the microscale

Testing of linear and periodic

Q4, Q8, Q12 and Q16 elements

Q4 and Q8 elements not

recommended when cracks present

No benefit from using quadratic BC

Q12 and Q16 elements deliver best

performance

Accurate for ratios of a/L ≤ 0.7

For larger ratios, boundary effects

difficult to capture with just 12 or 16

coarse nodes and current choice of

micro basis functions

Conclusion and Outlook

16/05/2018 13

Micro basis functions

Linear BC better represent the

physical behaviour of the crack

Periodic BC better account for the

uncracked behaviour of the UC

Propose hybrid BCs as seen below

Revisiting Fracture via a Scaled Boundary Multiscale Approach

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This research was performed under the auspices of the

Swiss National Science Foundation (SNSF), Grant #

200021 153379, A Multiscale Hysteretic XFEM Scheme

for the Analysis of Composite Structures

Further, we would like to extend our gratitude to Dr.

Konstantinos Agathos for the insightfull discussions.

Grants and Acknowledgements

16/05/2018 14Revisiting Fracture via a Scaled Boundary Multiscale Approach

Thank you for your attention!

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BACKUP

16/05/2018The Scaled Boundary Finite Element Method for the Efficient Modelling of Linear Elastic Fracture 16

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SBFEM derivation I

5/16/2018 17Adrian Egger | FEM II | HS 2015

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SBFEM derivation II Geometry transformation

Jacobian

Differential unit volumen

5/16/2018 18Adrian Egger | FEM II | HS 2015

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SBFEM derivation III The linear differential operator L may thus be written as:

5/16/2018 19Adrian Egger | FEM II | HS 2015

with

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SBFEM derivation IV Assuming an analytical solution in radial direction:

And therefore the strains and stresses become:

5/16/2018 20Adrian Egger | FEM II | HS 2015

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SBFEM derivation V Setting up the virtual work formulation:

5/16/2018 21Adrian Egger | FEM II | HS 2015

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SBFEM derivation VI

5/16/2018 22Adrian Egger | FEM II | HS 2015

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SBFEM derivation VII

5/16/2018 23Adrian Egger | FEM II | HS 2015

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SBFEM derivation VIII Introducing some substitutions

Leads to some significant simplifications

5/16/2018 24Adrian Egger | FEM II | HS 2015

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SBFEM derivation IX

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SBFEM derivation X

5/16/2018 26Adrian Egger | FEM II | HS 2015