M A M

Institute of Mechanics

& Advanced Materials I

3D fatigue fracture modeling by isogeometric boundary element methods

Xuan Peng, Elena Atroshchenko, Pierre Kerfriden, Stephane Bordas

Cardiff University, UK

April 2016

ACME 2016

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Outline

•Motivation

•IGABEM formulation for crack modeling

•Surface breaking cracks and trimmed NURBS

•Embedded cracks

•Conclusion

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Motivation

http://met-tech.com/

fatigue cracking of nozzle sleeve

Fatigue Fracture Failure of Structure

•Initiation: micro defects

•Loading : cyclic stress state (temperature, corrosion)

Numerical methods for crack growth

•Volume methods:

FEM, XFEM/GFEM, Meshfree

•Boundary methods: BEM

Bordas & Moran, 2006

XFEM+LEVEL SET

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Motivation

Challenges in volume-based methods

Efficiency & Accuracy

XFEM

IGABEM

crack

crack

calculation

stress analysis

mesh Remeshing (FEM)

Local mesh refinement

IGA

Kelvin fundamental solution

Navier equation:

Kelvin solution: assuming a unit concentrated force applied on a point in the infinite domain , we seek and for any point

for 3D problem, the expressions are:

Let source point “s” approach to the boundary

Boundary integral equation: direct method

Betti’s theorem

State (1) : auxiliary state, using Kelvin solution

State (2): real state, neglecting body force

Take derivative w.r.t. “s”

-could be used to

crack problem, dual equations

singular matrix introduced by

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Boundary integral equations (BIEs) and crack modeling

•Collocation: Greville Abscissae

•Displacement BIE

•Traction BIE

•NURBS approximation

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Singular integration

•Singularity subtraction technique (SST)

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Singular integration

•Conformal mapping for SST

•Rong et al 2014, EAWBE

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Evaluation of stress intensity factors (SIFs)

•Virtual crack closure integrals (VCCI)

•M integral

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Penny-shaped crack under remote tension (embedded crack)

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Penny crack under remote tension (embedded crack)

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Penny crack under remote tension (embedded crack)

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Penny-shaped crack under remote tension

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NURBS-represented crack growth algorithm

•Fatigue fracture: Paris law

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Numerical example of Mode-I penny crack growth (first 10 steps)

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Numerical example of inclined elliptical crack growth (first 10 steps)

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Modeling techniques for surface breaking cracks

•Surface discontinuity is introduced

•Crack trimming curve

•Trimmed NURBS technique

•Phantom node method

For surface breaking cracks, discontinuity is introduced along the body boundary. Crack initiation and propagation is difficult due to highly smoothness in basis functions

Crack can be considered as trimming curve. Then the problem is converted into trimmed NURBS modeling

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Integration and collocation for trimmed NURBS

•Beer et al 2015, CMAME

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Convergence study for a cube with cylindrical cutout

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Example of surface breaking cracks: edge crack under uniform tension

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Conclusions & Future work

•T-spline for local refinement

•Acceleration algorithm

•Dual BIEs are used for NURBS-represented fracture modeling

•Improved numerical singular integration scheme

•Approaches for SIFs evaluation

•Fatigue crack growth algorithm

•IGABEM for trimmed NURBS and surface crack modeling

•Improve the integration and collocation schemes for trimmed NURBS

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Many thanks for YOUR attention

The financial support by FP7-ITN under grant No. 289361 "Integrating Numerical Simulation and Geometric Design Technology” is gratefully acknowledged