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Energy release rate approximation for edge cracks using higher-order topological derivatives

Abstract

Topological derivatives provide the variation of a functional when an infinitesimal hole is introduced into the domain. In Silva et al. (J Mech Phys Solids 59(5):925–939, 2011), the authors developed a first-order approximation of the energy release rate associated with a small crack at any boundary location and at any orientation using a first-order topological derivative. This approach offers significant computational advantages over other methods because (i) it requires only a single analysis while other methods require an analysis for each crack size-location-direction combination, and (ii) it is performed on the non-cracked domain, removing the need to create very fine meshes in the vicinity of the crack. In the present study, higher-precision approximations of the energy release rate are developed using higher-order topological derivatives which allow the analyst to accurately treat longer cracks and determine the crack lengths for which the first-order approximation is accurate.

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Notes

  1. 1.

    In the equations that follow, the \(\hat{{\mathbf {x}}}\) and \(\alpha \) dependencies of the domain \(\Omega _\varepsilon \) and boundaries \(\partial \Omega _\varepsilon \) and \(\gamma _\varepsilon \) are dropped for conciseness.

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Acknowledgements

The authors gratefully acknowledge the support of the National Science Foundation (Award CMI-1200086). The authors also appreciate the rewarding comments from the reviewers, the invaluable conversations with Tom Curtin of BEASY USA - Computational Mechanics Inc, and the excellent work of Mariana Silva Sohn at the University of Illinois at Urbana-Champaign. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

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Correspondence to Kazem Alidoost.

A weight functions

A weight functions

In Sect. 3.3, the weight functions for an edge crack in a half-plane were used to relate the stress intensity factors to the load applied to the crack surface \(\gamma _\varepsilon \). These weight functions were calculated numerically using the boundary element method on a \(2w \times 2h\) domain. It was determined that dimensions of \(h,w\ge 100\varepsilon \) were required for the numerical approximation to sufficiently represent a half-plane (Fig. 18).

Fig. 18
figure18

Numerical formulation of an edge crack in a half-plane

Fig. 19
figure19

Prescribed tractions \(\tilde{{\mathbf {t}}}_{k}\) imposed in order to solve a \(h_{i1}^{(0)}(\alpha )\), \(h_{i1}^{(1)}(\alpha )\), and \(h_{i1}^{(2)}(\alpha )\) and b \(h_{i2}^{(0)}(\alpha )\), \(h_{i2}^{(1)}(\alpha )\), and \(h_{i2}^{(2)}(\alpha )\)

The prescribed traction \(\tilde{{\mathbf {t}}}\) on the crack face can be expressed as a polynomial in terms of the scaled coordinate \(\xi \),

$$\begin{aligned} \tilde{{\mathbf {t}}}(\xi ) = \tilde{{\mathbf {t}}}_{0}+\xi \tilde{{\mathbf {t}}}_{1}+\xi ^2\tilde{{\mathbf {t}}}_{2}+\cdots . \end{aligned}$$
(53)

The stress intensity factors are then given in terms of the prescribed traction as

$$\begin{aligned} \begin{aligned} \begin{pmatrix} {K}_{I}(\alpha ) \\ \\ {K}_{II}(\alpha ) \\ \end{pmatrix}\ \ = \,&\sqrt{\pi \varepsilon } \begin{pmatrix} h_{11}^{(0)}(\alpha ) &{} h_{12}^{(0)}(\alpha ) \\ \\ h_{21}^{(0)}(\alpha ) &{} h_{22}^{(0)}(\alpha ) \\ \end{pmatrix} \begin{pmatrix} T^{(0)}_{\theta \theta } \\ \\ T^{(0)}_{r\theta } \\ \end{pmatrix}\ \ \ \\&+ \,\sqrt{\pi \varepsilon } \begin{pmatrix} h_{11}^{(1)}(\alpha ) &{} h_{12}^{(1)}(\alpha ) \\ \\ h_{21}^{(1)}(\alpha ) &{} h_{22}^{(1)}(\alpha ) \\ \end{pmatrix} \begin{pmatrix} T^{(1)}_{\theta \theta } \\ \\ T^{(1)}_{r\theta } \\ \end{pmatrix}\ \ \ \\&+ \sqrt{\pi \varepsilon } \begin{pmatrix} h_{11}^{(2)}(\alpha ) &{} h_{12}^{(2)}(\alpha ) \\ \\ h_{21}^{(2)}(\alpha ) &{} h_{22}^{(2)}(\alpha ) \\ \end{pmatrix} \begin{pmatrix} T^{(2)}_{\theta \theta } \\ \\ T^{(2)}_{r\theta } \\ \end{pmatrix}\ \ \ \!\!\!+\! \cdots , \end{aligned} \end{aligned}$$
(54)

where \(\tilde{{\mathbf {t}}}_{k} = {\mathbf {T}}^{(k)}{\mathbf {e}}_\theta \).

In this work, the weight functions \(h_{ij}^{(k)}(\alpha )\) are calculated by solving problems with prescribed tractions on the crack face of \(\tilde{{\mathbf {t}}}_k=\xi ^{k}{\mathbf {e}}_\theta \) and \(\tilde{{\mathbf {t}}}_k=\xi ^{k}{\mathbf {e}}_r\). The former is used to calculate \(h_{i1}^{(k)}(\alpha )\) while the latter is used to calculate \(h_{i2}^{(k)}(\alpha )\). For example, when \(\tilde{{\mathbf {t}}}_k=\xi ^{k}{\mathbf {e}}_\theta \) then \(T^{(j)}_{r\theta }=0\) and Eq. (54) becomes

$$\begin{aligned} \begin{aligned} \begin{pmatrix} {K}_{I}(\alpha ) \\ \\ {K}_{II}(\alpha ) \\ \end{pmatrix}&= \, \sqrt{\pi \varepsilon } \begin{pmatrix} h_{11}^{(0)}(\alpha ) \\ \\ h_{21}^{(0)}(\alpha ) \\ \\ \end{pmatrix} T^{(0)}_{\theta \theta } \\&\quad + \, \sqrt{\pi \varepsilon } \begin{pmatrix} h_{11}^{(1)}(\alpha ) \\ \\ h_{21}^{(1)}(\alpha ) \\ \end{pmatrix} T^{(1)}_{\theta \theta } \\&\quad + \, \sqrt{\pi \varepsilon } \begin{pmatrix} h_{11}^{(2)}(\alpha ) \\ \\ h_{21}^{(2)}(\alpha ) \\ \end{pmatrix} T^{(2)}_{\theta \theta } +\ \cdots . \end{aligned} \end{aligned}$$
(55)

The \(h_{ij}^{(k)}(\alpha )\) were calculated for \(0 \le \alpha \le 70^{\circ }\) in increments of \(1^{\circ }\). The prescribed tractions used to solve for \(h_{ij}^{(0)}(\alpha )\), \(h_{ij}^{(1)}(\alpha )\), and \(h_{ij}^{(2)}(\alpha )\) are shown in Fig. 19.

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Alidoost, K., Geubelle, P.H. & Tortorelli, D.A. Energy release rate approximation for edge cracks using higher-order topological derivatives. Int J Fract 210, 187–205 (2018). https://doi.org/10.1007/s10704-018-0271-1

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Keywords

  • Topological derivative
  • Energy release rate
  • Edge cracks
  • Asymptotic analysis