Energy Minimizing Multi-Crack Growth in Linear Elastic Fracture Using - - PowerPoint PPT Presentation

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Energy Minimizing Multi-Crack Growth in Linear Elastic Fracture Using - - PowerPoint PPT Presentation

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Energy Minimizing Multi-Crack Growth in Linear Elastic Fracture Using The Extended Finite Element Method Danas Sutula Prof. Stphane Bordas Dr. Pierre Kerfriden 01/04/2016 Content 1. Motivation 2. Problem statement 3. Crack growth 4.

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Energy Minimizing Multi-Crack Growth in Linear Elastic Fracture Using The Extended Finite Element Method

Danas Sutula

  • Prof. Stéphane Bordas
  • Dr. Pierre Kerfriden

01/04/2016

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Content

1. Motivation 2. Problem statement 3. Crack growth 4. Discretization by XFEM 5. Implementation 6. Verification 7. Results 8. Summary

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM 2

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Problem statement

  • 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 Δai≥ 0:

3 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Problem statement

  • The solution procedure at time tk consists of

1. solving the variational form for u(ak): 2. advancing the fracture fronts, such that Π(u,ak) → Π(u,ak+1) follows the path of steepest descent while satisfying Griffith’s energy balance

4

  • Danas Sutula

Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Crack growth

maximum hoop stress

  • Post processing of solution to evaluate SIF [Yau, 1980]
  • Crack growth direction [Erdogan & Shi, 1963]
  • Growth criterion [Irwin, 1957; Hayashi & Nemat-Nasser, 1981]

5 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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  • Energy release rate w.r.t. crack increment direction, θi:
  • The rates of energy release rates:
  • Updated directions (using Newton):

Crack growth

energy minimization

6 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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  • The discrete potential energy is given by:
  • Energy release rate w.r.t. crack increment direction θi :
  • The rates of the energy release rate:

Crack growth

energy minimization

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, where: , where:

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Discretization

XFEM

  • Approximation function [Belytschko et al., 2001]

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singular tip enrichment discontinuous enrichment standard part

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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9 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

Implementation

how to compute 𝜀K ?

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10 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

Implementation

how to compute 𝜀K ?

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Verification

rotational energy release rates

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Π vs. θ F F

  • θ

Test case: square plate with an edge crack with a small kink loaded in vertical tension

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Verification

rotational energy release rates

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G vs. θ Test case: square plate with an edge crack with a small kink loaded in vertical tension F F

  • θ

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Verification

rotational energy release rates

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(topological enr.) dG/dθ vs. θ Test case: square plate with an edge crack with a small kink loaded in vertical tension F F

  • θ

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Verification

rotational energy release rates

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(geometrical enr.) dG/dθ vs. θ Test case: square plate with an edge crack with a small kink loaded in vertical tension F F

  • θ

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Verification

energy min. VS. max-hoop

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F F Test case: square plate with an inclined center crack in vertical tension

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Verification

energy min. VS. max-hoop

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Gmin(Π)/Ghoop vs. θ

F F θ Test case: square plate with an inclined center crack in vertical tension

Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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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; 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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39 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

Results

Edge crack in a PMMA beam with 3 holes

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40 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

Results

Edge crack in a PMMA beam with 3 holes

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41 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

Results

Edge crack in a PMMA beam with 3 holes

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42 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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)

43 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Results

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

24 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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Results

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

45 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM

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  • A robust approach to determining multiple crack growth

based on the principle of minimum energy within XFEM;

  • Limitations undermining the max. hoop-stress criterion are
  • vercome, e.g. assumptions about geometry and loading;
  • The energy minimization approach is characterized by mode-I

field dominance at the crack tip (post-increment);

  • Both criteria lead to fracture paths solutions that are in close

agreement (strong correlation with local symmetry, i.e. KII=0);

  • Better accuracy and faster convergence of fracture path

solutions can be obtained by taking a bi-section of the interval that is bounded by the respective criteria.

Summary

46 Danas Sutula Energy minimizing multi-crack growth in linear elastic fracture using XFEM