Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended Finite Element Method

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Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended Finite Element Method

Danas Sutula

Prof. Stéphane Bordas Dr. Pierre Kerfriden Alexandre Barthelemy

24/07/2014

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Content

1. Problem statement 2. Crack growth

3. Discretization by XFEM 4. Implementation

5. Verification 6. Results

7. Summary

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• Consider a cracked linear-elastic isotropic solid subject to an external load whose quasistatic behavior can be described by the following total Lagrangian form:

• The solution for u(a) and a(t) are obtained by satisfying the stationarity of L(u,a) during the evolution of t , subject to Δ a

_i

≥ 0 :

Problem statement

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• The solution procedure at time t

^k

consists of

1. solving the variational form for u(a

^k

) _:

2. advancing the fracture fronts, such that Π(u,a

^k

) → Π(u,a

^k+1

) _follows the path of steepest descent while satisfying Griffith’s energy balance

Problem statement

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Crack growth

maximum hoop stress

• Post processing of solution to evaluate SIF

• Crack growth direction

• Crack growth criterion

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• Energy release rate w.r.t. crack increment direction, θ

_i

:

• The rates of energy release rate:

• Updated directions:

Crack growth

energy minimization

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• The discrete potential energy is given by:

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

_i

:

• The rates of the energy release rate:

Crack growth

energy minimization

, where:

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Discretization

XFEM

• Approximation function

singular tip enrichment discontinuous enrichment

standard part

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Implementation

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Implementation

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Verification

energy release rates

Π vs. θ Π vs. θ F

F -θ

Test case: square plate

with an edge crack with a

small kink loaded in

vertical tension

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Verification

energy release rates

G vs. θ G vs. θ Test case: square plate

with an edge crack with a small kink loaded in vertical tension

F

-θ

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Verification

energy release rates

(topological enr.)

dG/dθ vs. θ dG/dθ vs. θ Test case: square plate

with an edge crack with a small kink loaded in vertical tension

F

-θ

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Verification

energy release rates

(geometrical enr.)

dG/dθ vs. θ dG/dθ vs. θ Test case: square plate

with an edge crack with a small kink loaded in vertical tension

F

-θ

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Verification

energy min. VS. max-hoop

G

_min(Π)

/G

_hoop

vs. θ G

_min(Π)

/G

_hoop

vs. θ

F

F θ

Test case: square plate

with an inclined center

crack in vertical tension

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Verification

energy min. VS. max-hoop

F

Test case: square plate

with an inclined center

crack in vertical tension

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F

F p = F

Verification

energy min. VS. max-hoop

Test case: square plate

with a pressurized

inclined center crack in

vertical tension

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F

F p = F

Verification

energy min. VS. max-hoop

Test case: square plate

with a pressurized

inclined center crack in

vertical tension

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Verification

energy min. VS. max-hoop

F

Test case: square plate with a pressurized inclined center crack in vertical tension

G

_min(Π)

/G

_hoop

vs. θ G

_min(Π)

/G

_hoop

vs. θ

θ

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Results

10 crack problem

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Results

10 crack problem

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Results

10 crack problem

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Results

10 crack problem

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Results

10 crack problem

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Results

10 crack problem

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Results

10 crack problem

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Results

10 crack problem

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Results

10 crack problem

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Results

double cantilever problem

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Results

double cantilever problem

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Results

2 edge cracks; plate in vertical tension

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Results

2 edge cracks; plate in vertical tension

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Results

2 edge cracks; internal pressure loading

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Results

3 cracks; center crack pressure loading

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Results

Edge crack in a PMMA beam with 3 holes

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Results

Edge crack in a PMMA beam with 3 holes

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Results

Edge crack in a PMMA beam with 3 holes

(38)

Results

Edge crack in a PMMA beam with 3 holes

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Results

Edge crack in a PMMA beam with 3 holes

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Results

2 edge cracks and 2 holes (Khoeil et al. 2008)

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Results

2 edge cracks and 2 holes (Khoeil et al. 2008)

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Results

2 edge cracks and 2 holes (Khoeil et al. 2008)

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• A robust approach to determining multiple crack growth directions based on the principle of minimum energy

• Limitations undermining the max. hoop-stress criterion, e.g.

traction free cracks, absence of body loads etc., are overcome

• The energy minimization approach is characterized by mode-1 dominance at the end of each time-step (i.e. on convergence)

• The criteria lead to fracture paths that are, in the global sense, in very close agreement - virtue of self-similar crack growth

• Better accuracy and faster convergence of the fracture path can be obtained by taking a bi-section of the interval that is bounded by the respective criteria.

Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended Finite Element Method

Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended Finite Element Method

Danas Sutula

Prof. Stéphane Bordas Dr. Pierre Kerfriden Alexandre Barthelemy

24/07/2014

Content

1. Problem statement 2. Crack growth

3. Discretization by XFEM 4. Implementation

5. Verification 6. Results

7. Summary

• Consider a cracked linear-elastic isotropic solid subject to an external load whose quasistatic behavior can be described by the following total Lagrangian form:

• The solution for u(a) and a(t) are obtained by satisfying the stationarity of L(u,a) during the evolution of t , subject to Δ a

≥ 0 :

Problem statement

• The solution procedure at time t

consists of

1. solving the variational form for u(a

) :

2. advancing the fracture fronts, such that Π(u,a

) → Π(u,a

) follows the path of steepest descent while satisfying Griffith’s energy balance

Problem statement

Crack growth

maximum hoop stress

• Post processing of solution to evaluate SIF

• Crack growth direction

• Crack growth criterion

• Energy release rate w.r.t. crack increment direction, θ

:

• The rates of energy release rate:

• Updated directions:

Crack growth

energy minimization

• The discrete potential energy is given by:

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

:

• The rates of the energy release rate:

Crack growth

energy minimization

, where:

Discretization

XFEM

• Approximation function

singular tip enrichment discontinuous enrichment

standard part

Implementation

Implementation

Verification

energy release rates

Π vs. θ Π vs. θ F

F -θ

Test case: square plate

with an edge crack with a

small kink loaded in

vertical tension

Verification

energy release rates

G vs. θ G vs. θ Test case: square plate

with an edge crack with a small kink loaded in vertical tension

F

F

-θ

Verification

energy release rates

(topological enr.)

dG/dθ vs. θ dG/dθ vs. θ Test case: square plate

with an edge crack with a small kink loaded in vertical tension

F

F

-θ

Verification

energy release rates

(geometrical enr.)

dG/dθ vs. θ dG/dθ vs. θ Test case: square plate

with an edge crack with a small kink loaded in vertical tension

F

F

-θ

Verification

energy min. VS. max-hoop

) _:

) _follows the path of steepest descent while satisfying Griffith’s energy balance