XFEM modeling for curved fracture in the anisotropic fracture toughness medium

Abstract

The materials with anisotropic fracture toughness are familiar in nature, e.g., anisotropic rocks, woods, and crystals. The deflecting crack propagation behaviors are often observed in these materials due to the anisotropic fracture toughness property. In this paper, the extended finite element method (XFEM) is developed for modeling the crack extending behavior in anisotropic fracture toughness medium. First, anisotropic fracture toughness profiles are introduced and embedded into XFEM, and the crack deflecting direction is predicted based on maximum energy release rate criterion. To capture the details of the twisting crack path as accurate as possible in XFEM, a mesh independent piecewise linear crack model is developed numerically. Then several numerical examples in studying the curved crack path in a material with the anisotropic fracture toughness property are given. With the techniques of XFEM embedded with anisotropic fracture toughness, the crack path in such anisotropic materials could be predicted and designed.

Appendix A: The SIFs after crack deflection

Amestoy and Leblond [31] demonstrated that, the SIFs of fracture deflecting in direction $\theta$ with a small enough length, $\{\tilde{K}_I(\theta ), ~\tilde{K}_{II}(\theta )\}$, would linearly depend on the SIFs before deflecting, $\{K_I,K_{II}\}$. Their results could be shown, with $\theta =m\pi$, as:

$$\begin{aligned} \begin{aligned}&\tilde{K}_I(m\pi ) =F_{11}(m)K_I+F_{12}(m)K_{II},\\&\tilde{K}_{II}(m\pi ) =F_{21}(m)K_I+F_{22}(m)K_{II}, \end{aligned} \end{aligned}$$

(A.1)

and the coefficients in Eq. (A.1) expanded up to 20th order series are obtained as:

$$\begin{aligned}&\begin{aligned} F_{11}(m)&={} 1-\frac{3 m^2 \pi ^2}{8}+\left( \pi ^2-\frac{5 \pi ^4}{128}\right) m^4 \\&\quad \, + \left( \frac{\pi ^2}{9}-\frac{11 \pi ^4}{72}+\frac{119 \pi ^6}{15360}\right) m^6\\&\quad \, {}+5.0779 m^8-2.88312 m^{10}\\&\quad \, -0.0925 m^{12}+2.996 m^{14}\\&\quad \, {}-4.059 m^{16}+1.63 m^{18}+4.1 m^{20}, \end{aligned} \end{aligned}$$

(A.2a)

$$\begin{aligned}&\begin{aligned} F_{12}(m)=&{}-\frac{3 m \pi }{2}+\left( \frac{10 \pi }{3}+\frac{\pi ^3}{16}\right) m^3 \\&+ \left( -2 \pi -\frac{133 \pi ^3}{180}+\frac{59 \pi ^5}{1280}\right) m^5\\&{}+12.3139 m^7-7.32433 m^9\\&+1.5793 m^{11}+4.0216 m^{13}\\&-6.915 m^{15}+4.21 m^{17}+4.56 m^{19}, \end{aligned} \end{aligned}$$

(A.2b)

$$\begin{aligned}&\begin{aligned} F_{21}(m)={}&\frac{m \pi }{2}-\left( \frac{4 \pi }{3}+\frac{\pi ^3}{48}\right) m^3 \\&+ \left( -\frac{2 \pi }{3}+\frac{13 \pi ^3}{30}-\frac{59 \pi ^5}{3840}\right) m^5\\&{}-6.17602 m^7+4.44112 m^9\\&-1.534 m^{11}-2.07 m^{13}\\&{}+4.684 m^{15}-3.95 m^{17}-1.32 m^{19}, \end{aligned} \end{aligned}$$

(A.2c)

$$\begin{aligned}&\begin{aligned} F_{22}(m)={}&1- \left( 4+\frac{3 \pi ^2}{8}\right) m^2+\left( \frac{8}{3}+\frac{29 \pi ^2}{18}-\frac{5 \pi ^4}{128}\right) m^4\\&{} +\left( -\frac{32}{15}-\frac{4 \pi ^2}{9}-\frac{1159 \pi ^4}{7200}+\frac{119 \pi ^6}{15360}\right) m^6\\&+10.5825 m^8\\&{}-4.78511 m^{10}-1.8804 m^{12}\\&+7.28 m^{14}-7.591 m^{16}\\&{}+0.25 m^{18}+26.6681 m^{20}. \end{aligned} \end{aligned}$$

(A.2d)

It is noted that $F_{11}(m)$ and $F_{22}(m)$ are even functions; $F_{12}(m)$ and $F_{21}(m)$ are odd functions.

Besides Amestoy and Leblond, He and Hutchinson [37, 38] showed a table for how the coefficients $\{F_{11}, F_{12}, F_{21},$$F_{22}\}$ depend on m, which could be used with Eq. (A.1). Nuismer [39] also gave simple relations on $\tilde{K}_I(\theta ;K_I,K_{II})$ and $\tilde{K}_{II}(\theta ;K_I,K_{II})$, which are similar to Eq. (A.1):

$$\begin{aligned} \begin{aligned}&\tilde{K}_I(\theta ) = \frac{1}{2}\cos {\frac{\theta }{2}}\left[ K_I (1+\cos \theta )-3K_{II}\sin {\theta }\right] ,\\&\tilde{K}_{II}(\theta ) = \frac{1}{2}\cos {\frac{\theta }{2}}\left[ K_I \sin \theta +K_{II}(3\cos {\theta }-1)\right] . \end{aligned} \end{aligned}$$

(A.3)

A comparison among the results of Amestoy and Leblond [31], He and Hutchinson [37, 38], and Nuismer [39] is also given in the appendix of Gao et al. [7]. It is noted that the result of Amestoy and Leblond is both accurate enough and easy to use, therefore is also adopted in this paper.