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.

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

Numerical formulation of an edge crack in a half-plane

Fig. 19

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.