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

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

Content 1. Motivation 2. Problem statement 3. Crack growth 4. Discretization by XFEM 5. Implementation 6. Verification 7. Results 8. Summary Energy minimizing multi-crack growth in linear Danas Sutula 2 elastic fracture using XFEM

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 Δ a i ≥ 0 : Energy minimizing multi-crack growth in linear Danas Sutula 3 elastic fracture using XFEM

• Problem statement • The solution procedure at time t k consists of solving the variational form for u(a k ) : 1. advancing the fracture fronts, such that Π( u,a k ) → Π( u,a k+1 ) follows 2. the path of steepest descent while satisfying Griffith’s energy balance Energy minimizing multi-crack growth in linear Danas Sutula 4 elastic fracture using XFEM

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

Crack growth energy minimization • Energy release rate w.r.t. crack increment direction, θ i : • The rates of energy release rates: • Updated directions (using Newton): Energy minimizing multi-crack growth in linear Danas Sutula 6 elastic fracture using XFEM

Crack growth energy minimization • The discrete potential energy is given by: • Energy release rate w.r.t. crack increment direction θ i : , where: • The rates of the energy release rate: , where: Energy minimizing multi-crack growth in linear Danas Sutula 7 elastic fracture using XFEM

Discretization XFEM • Approximation function [Belytschko et al., 2001] discontinuous enrichment standard part singular tip enrichment Energy minimizing multi-crack growth in linear Danas Sutula 8 elastic fracture using XFEM

Implementation how to compute 𝜀 K ? Energy minimizing multi-crack growth in linear Danas Sutula 9 elastic fracture using XFEM

Implementation how to compute 𝜀 K ? Energy minimizing multi-crack growth in linear Danas Sutula 10 elastic fracture using XFEM

Verification rotational energy release rates Test case: square plate with an edge crack with a small kink loaded in vertical tension F - θ F Π vs. θ Energy minimizing multi-crack growth in linear Danas Sutula 11 elastic fracture using XFEM

Verification rotational energy release rates Test case: square plate with an edge crack with a small kink loaded in vertical tension F - θ F G vs. θ Energy minimizing multi-crack growth in linear Danas Sutula 12 elastic fracture using XFEM

Verification rotational energy release rates Test case: square plate with an edge crack with a small kink loaded in vertical tension F (topological enr.) - θ F d G/ d θ vs. θ Energy minimizing multi-crack growth in linear Danas Sutula 13 elastic fracture using XFEM

Verification rotational energy release rates Test case: square plate with an edge crack with a small kink loaded in vertical tension F (geometrical enr.) - θ F d G/ d θ vs. θ Energy minimizing multi-crack growth in linear Danas Sutula 14 elastic fracture using XFEM

Verification energy min. VS. max-hoop Test case: square plate with an inclined center crack in vertical tension F F Energy minimizing multi-crack growth in linear Danas Sutula 15 elastic fracture using XFEM

Verification energy min. VS. max-hoop Test case: square plate with an inclined center crack in vertical tension F θ F G min(Π) /G hoop vs. θ Energy minimizing multi-crack growth in linear Danas Sutula 16 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 20 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 21 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 22 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 23 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 24 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 25 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 26 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 27 elastic fracture using XFEM

Results 10 crack problem Energy minimizing multi-crack growth in linear Danas Sutula 29 elastic fracture using XFEM

Results double cantilever problem Energy minimizing multi-crack growth in linear Danas Sutula 30 elastic fracture using XFEM

Results double cantilever problem Energy minimizing multi-crack growth in linear Danas Sutula 34 elastic fracture using XFEM

Results 2 edge cracks; internal pressure loading Energy minimizing multi-crack growth in linear Danas Sutula 36 elastic fracture using XFEM

Results 3 cracks; center crack pressure loading Energy minimizing multi-crack growth in linear Danas Sutula 37 elastic fracture using XFEM

Results Edge crack in a PMMA beam with 3 holes Energy minimizing multi-crack growth in linear Danas Sutula 38 elastic fracture using XFEM

Results Edge crack in a PMMA beam with 3 holes Energy minimizing multi-crack growth in linear Danas Sutula 39 elastic fracture using XFEM

Results Edge crack in a PMMA beam with 3 holes Energy minimizing multi-crack growth in linear Danas Sutula 40 elastic fracture using XFEM

Results Edge crack in a PMMA beam with 3 holes Energy minimizing multi-crack growth in linear Danas Sutula 41 elastic fracture using XFEM

Results Edge crack in a PMMA beam with 3 holes Energy minimizing multi-crack growth in linear Danas Sutula 42 elastic fracture using XFEM

Results 2 edge cracks and 2 holes (Khoeil et al. 2008) Energy minimizing multi-crack growth in linear Danas Sutula 43 elastic fracture using XFEM

Results 2 edge cracks and 2 holes (Khoeil et al. 2008) Energy minimizing multi-crack growth in linear Danas Sutula 24 elastic fracture using XFEM

Results 2 edge cracks and 2 holes (Khoeil et al. 2008) Energy minimizing multi-crack growth in linear Danas Sutula 45 elastic fracture using XFEM

Summary • 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 overcome, 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. K II =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. Energy minimizing multi-crack growth in linear Danas Sutula 46 elastic fracture using XFEM