Global energy minimization for multiple fracture growth

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Global energy minimization for multiple fracture growth

Danas Sutula

Prof. Stephane Bordas & Dr. Pierre Kerfriden

(10/12/13)

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Discretization: XFEM

Extended Finite Element Method (XFEM)

• Introduced by Ted Belytschko et al. (1999) for elastic problems

T. Belytschko, Northwestern Univeristy

Fracture process (>300 cracks)

• vertical tensile loading

Fracture of “XFEM” using XFEM

• pressure driven crack propagation

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Discretization: XFEM

Approximation function :

singular tip enrichment discontinuous

enrichment standard part

Enriched nodes - “Heaviside”

- ”crack tip”

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Crack propagation: traditional approach

Evaluation of stress intensity factors (SIF)

• The interaction integral (Yau 1980)

Crack growth criterion for mixed mode fracture

• Direction that maximises the energy release (Nuismer 1975)

Crack growth direction

• orthogonal to maximum hoop stress

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(1) – from current solution (2) – known auxiliary solution

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Crack propagation: optimization of directions

• Energy release rate w.r.t crack increment direction:

• The rates of the energy release rate are given by:

• where, in a discrete setting, the potential energy is:

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• The discrete potential energy:

• The discrete energy release rate:

• The rates of the energy release rate

(where: )

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𝟎

(where: )

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𝟎

(where: )

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𝟎

(where: )

expensive

remote interaction

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Implementation

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Implementation

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𝟎

(where: )

Implementation

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Updated directions:

Implementation

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• Square plate in pure mode-I loading with varying increment angle

Potential energy vs. crack increment angle -75^o

0^o

Verification: energy release rates

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mesh= 50 x 50, 𝑟_𝑡𝑖𝑝 = 2.5ℎ_𝑒, ∆𝑎_𝑖𝑛𝑐 = 0.1

Energy release rate vs. crack increment angle Rate of energy release vs. increment angle

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• Geometrical tip enrichment ( )

• Topological enrichment ( )

mesh = 200 x 200, 𝑟_𝑡𝑖𝑝 = 10.0ℎ_𝑒 mesh = 100 x 100, 𝑟_𝑡𝑖𝑝 = 5.0ℎ_𝑒

mesh = 50 x 50, 𝑟_𝑡𝑖𝑝 = 2.5ℎ_𝑒 𝑟𝑒_{𝐻𝑠 𝐿2} = 0.130

𝑟𝑒_{𝐻𝑠 𝐿2} = 0.076 𝑟𝑒_{𝐻𝑠 𝐿2} = 0.044

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• Geometrical tip enrichment ( )

• Topological enrichment ( )

mesh = 200 x 200, 𝑟_𝑡𝑖𝑝 = 10.0ℎ_𝑒 mesh = 100 x 100, 𝑟_𝑡𝑖𝑝 = 5.0ℎ_𝑒

mesh = 50 x 50, 𝑟_𝑡𝑖𝑝 = 2.5ℎ_𝑒 𝑟𝑒_{𝐻𝑠 𝐿2} = 0.130

𝑟𝑒_{𝐻𝑠 𝐿2} = 0.076 𝑟𝑒_{𝐻𝑠 𝐿2} = 0.044

mesh = 100 x 100, ∆𝑎_𝑖𝑛𝑐 = 0.050

mesh = 50 x 50, ∆𝑎_𝑖𝑛𝑐 = 0.100 𝑟𝑒_{𝐻𝑠 𝐿2} = 0.130 𝑟𝑒_{𝐻𝑠 𝐿2} = 0.134 𝑟𝑒_{𝐻𝑠 𝐿2} = 0.139 mesh = 200 x 200, ∆𝑎_𝑖𝑛𝑐 = 0.025

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Results: Double cantilever problem

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F

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F

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Results: Two Edge Cracks

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Results: Two cracks protruding from holes

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Results: 5 randomly distributed cracks

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What’s next

• Energy release rates w.r.t crack increment length:

,

 Robust crack growth criterion for multiple fractures

 Global energy minimization w.r.t. both ,