Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
||
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
||
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
||
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
||
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)}
||
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:
||
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
||
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
||
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
||
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
||
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
||
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
||
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
||
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
||
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!
||
BACKUP
16/05/2018The Scaled Boundary Finite Element Method for the Efficient Modelling of Linear Elastic Fracture 16
||
SBFEM derivation I
5/16/2018 17Adrian Egger | FEM II | HS 2015
||
SBFEM derivation II Geometry transformation
Jacobian
Differential unit volumen
5/16/2018 18Adrian Egger | FEM II | HS 2015
||
SBFEM derivation III The linear differential operator L may thus be written as:
5/16/2018 19Adrian Egger | FEM II | HS 2015
with
||
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
||
SBFEM derivation V Setting up the virtual work formulation:
5/16/2018 21Adrian Egger | FEM II | HS 2015
||
SBFEM derivation VI
5/16/2018 22Adrian Egger | FEM II | HS 2015
||
SBFEM derivation VII
5/16/2018 23Adrian Egger | FEM II | HS 2015
||
SBFEM derivation VIII Introducing some substitutions
Leads to some significant simplifications
5/16/2018 24Adrian Egger | FEM II | HS 2015
||
SBFEM derivation IX
5/16/2018 25Adrian Egger | FEM II | HS 2015
||
SBFEM derivation X
5/16/2018 26Adrian Egger | FEM II | HS 2015