Estimation of Stress Intensity Factor by Using a New Fast Multipole Dual-Boundary Element Method

Abstract

Cracks and defects are inevitable during the long-term use of structures. This study focuses on determining the stress intensity factors of multi-cracked structures by using a new fast multipole dual boundary element method. Numerical examples show that the results of the present method agree well with analytic solutions. When the crack distribution changes, the most unfavorable conditions also change. The shape of the defect has an effect on the stress intensity factors of nearby cracks. Among triangular, rectangular, hexagonal, and circular defects, when the area of the defect is identical, the triangular pore is more likely to induce crack propagation, while the circular pore is more secure. The above results can be used as a reference for structural design and optimization.

1. Introduction

Fracture analysis is crucial for assessing the safety and remaining life of structures [1,2,3]. The assessment parameters include stress intensity factor (SIF), J-integral, and strain energy release rate, among others. Notably, the SIF has emerged as a prevalent parameter due to its broad applicability.

Various methods are available for analyzing fracture problems, such as experimental measurements, analytical solutions, and numerical techniques. Experimental measurements [4,5,6,7,8] can obtain accurate and effective results but are time-consuming. Analytical formulas [9] are highly efficient and accurate but are suitable for some simple cases. Numerical techniques have the advantage of wide applicability; the most commonly used include but are not limited to, the meshless method [10], the finite element method (FEM) [11,12,13,14,15], and the boundary element method (BEM) [16,17,18]. These numerical techniques need to combine with some special treatments to obtain the SIF, such as the J-integral [19,20], the M-integral [21], the interaction integration [22], and the displacement and stress extrapolation methods [23]. The J-integral is path-independent, i.e., the computing results do not depend on the choice of integration path. The M-integral is also path-independent and has a definite relation to the J-integral. Compared with the J integral, the M-integral applies to a wider range but involves more complicated calculations. The interaction integration obtains the SIFs by establishing the auxiliary stress and displacement fields and can be used to calculate the SIFs in mixed mode. These integration methods all have high accuracy but are much more complicated than the displacement and stress extrapolation methods. The displacement extrapolation method has the advantages of wide applicability and simple implementation, making it especially suitable for multi-crack problems. Its accuracy can be further improved, which is one of the efforts of the present study.

In this study, a new fast multiple dual boundary element method (FMDBEM) combined with the displacement extrapolation method is developed to determine the SIFs of multi-cracked structures. Firstly, a novel variable-order asymptotic element (VAE) has been introduced to achieve highly accurate singular stress fields in the vicinity of the crack tip. Secondly, the crack-tip opening displacement (CTOD) and the least squares method are used to calculate the SIFs. Finally, the effects of defects on structural strength and the relationships between the SIF and the number of cracks are discussed.

2. Displacement Interpolation Method for SIFs

Considering a 2D cracked structure, the SIFs can be expressed as:

$$K_{\mathrm{I}}=\underset{r\Delta 0}{\lim }\sqrt{2\mathsf{\pi }}{\displaystyle \frac{\nu }{1+\kappa }}{\displaystyle \frac{∆u_n}{\sqrt{r}}},$$

(1)

$$K_{\mathrm{I}\mathrm{I}}=\underset{r\Delta 0}{\lim }\sqrt{2\mathsf{\pi }}{\displaystyle \frac{\nu }{1+\kappa }}{\displaystyle \frac{∆u_{\tau }}{\sqrt{r}}},$$

(2)

where $\kappa =3-4\nu$ for plane strain, $\kappa =(3-\nu )/(1+\nu )$ for plane stress, $\nu \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{’}\mathrm{s} \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}, ∆u_{\tau } \mathrm{a}\mathrm{n}\mathrm{d} ∆u_n$ are the displacement variation between the two crack surfaces along $\tau$ and n respectively, as seen in Figure 1.

Figure 1. Crack-tip opening displacement.

To accurately obtain the SIFs, a set of data for ($r_i,{\tilde{K}}_{\mathrm{I}}^i$) and ($r_i,{\tilde{K}}_{\mathrm{I}\mathrm{I}}^i$) are first collected, as follows:

$${\tilde{K}}_{\mathrm{I}}^i=\sqrt{2\pi }{\displaystyle \frac{\nu }{1+\kappa }}{\displaystyle \frac{∆u_{ni}}{\sqrt{r_i}}},$$

(3)

$${\tilde{K}}_{\mathrm{I}\mathrm{I}}^i=\sqrt{2\pi }{\displaystyle \frac{\nu }{1+\kappa }}{\displaystyle \frac{∆u_{\tau i}}{\sqrt{r_i}}},$$

(4)

where $∆u_{ni}$ and $∆u_{\tau i}$ are the displacement along n and $\tau$ on the sample points, respectively. Then, the least squares method is used to fit $K_{\mathrm{I}}$, $K_{\mathrm{I}\mathrm{I}}$, as follows:

$$K_{\mathrm{I}}=\left(\sum _{i=1}^N{r}_i\sum _{i=1}^N{r}_i{\tilde{K}}_{\mathrm{I}}^i-N\sum _{i=1}^N{r}_i^2{\tilde{K}}_{\mathrm{I}}^i\right){\left[{\left(\sum _{i=1}^N{r}_i\right)}^2-N\sum _{i=1}^N{r}_i^2\right]}^{-1},$$

(5)

$$K_{\mathrm{I}\mathrm{I}}=\left(\sum _{i=1}^N{r}_i\sum _{i=1}^N{r}_i{\tilde{K}}_{\mathrm{I}\mathrm{I}}^i-N\sum _{i=1}^N{r}_i^2{\tilde{K}}_{\mathrm{I}\mathrm{I}}^i\right){\left[{\left(\sum _{i=1}^N{r}_i\right)}^2-N\sum _{i=1}^N{r}_i^2\right]}^{-1},$$

(6)

where N is the total number of sample points.

3. Determining Displacement and Stress Fields of Multi-Cracked Structures

For 2D cracked problems, the dual boundary integral equations are applied to the two surfaces of the cracks. The conventional boundary integral equation (CBIE) for displacements and the hyper-singular boundary integral equation (HBIE) [24] for tractions are written as follows:

$$c_{ij}\left(\mathit{y}\right)u_j\left(\mathit{y}\right)={\int }_{\Gamma }{U}_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right)t_j\left(\mathit{x}\right)\mathrm{d}\Gamma -{\int }_{\Gamma }{T}_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right)u_j\left(\mathit{x}\right)\mathrm{d}\Gamma ,\hspace{1em}\mathit{y}\in \Gamma ,\hspace{1em}\mathit{x}\in \Gamma ,$$

(7)

$$c_{ij}\left(\mathit{y}\right)t_j\left(\mathit{y}\right)={\int }_{\Gamma }{K}_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right)t_j\left(\mathit{x}\right)\mathrm{d}\Gamma -{\int }_{\Gamma }{H}_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right)u_j\left(\mathit{x}\right)\mathrm{d}\Gamma ,\hspace{1em}\mathit{y}\in {\Gamma }_c^{-},\hspace{1em}\mathit{x}\in \Gamma ,$$

(8)

where $\mathit{y}$ is the source point,$\mathit{x}$ is the field point, $t_j\left(\mathit{x}\right)$ and $u_j\left(\mathit{x}\right)$ are the traction and displacement at $\mathit{x}$, respectively, and $c_{ij}\left(\mathit{y}\right)$ is a coefficient. $U_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right)$, $T_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right)$, $K_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right),$ and $H_{ij}^{\ast }\left(\mathit{x},\mathit{y}\right)$ are the fundamental solutions. CBIE is applied to the boundary $\Gamma ={\Gamma }^{\ast }+{\Gamma }_c^{+}+{\Gamma }_c^{-}+{\Gamma }_b$ and CBIE + $\beta$HBIE = 0 is applied to the boundary ${\Gamma }_c^{-}$, where $\beta$ is a coupling coefficient, as seen in Figure 2.

Figure 2. The multi-cracked structure.

To simplify the expressions of the fundamental solutions, complex notation is used. The integrals on a typical boundary element in Equations (7) and (8) can be expressed as follows:

$$\begin{gathered} {\mathcal{D}}^t\left(z_s\right)={\mathcal{D}}_1^t\left(\mathit{y}\right)+\mathrm{i}{\mathcal{D}}_2^t\left(\mathit{y}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{2\mu (1+\kappa )}}{\int }_{\Gamma }[\kappa G\left(z_s,z\right)t\left(z\right)-(z_s-z)\overline{G^{\prime }\left(z_s,z\right)t\left(z\right)}+\kappa \overline{G\left(z_s,z\right)}t\left(z\right)]d\Gamma , \end{gathered}$$

(9)

$$\begin{gathered} {\mathcal{D}}^u\left(z_s\right)={\mathcal{D}}_1^u\left(\mathit{y}\right)+\mathrm{i}{\mathcal{D}}_2^u\left(\mathit{y}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-{\displaystyle \frac{1}{\left(1+\kappa \right)}}{\int }_{\Gamma }\left[\kappa G^{\prime }\left(z_s,z\right)n\right(z)u\left(z\right)-(z_s-z)\overline{G^{\prime \prime }\left(z_s,z\right)n\left(z\right)u\left(z\right)} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\overline{G^{\prime }\left(z_s,z\right)}(n\left(z\right)\overline{u\left(z\right)}+\overline{n\left(z\right)})u\left(z\right)]d\Gamma , \end{gathered}$$

(10)

$$\begin{gathered} {\mathcal{F}}^t\left(z_s\right)={\mathcal{F}}_1^t\left(\mathit{y}\right)+\mathrm{i}{\mathcal{F}}_2^t\left(\mathit{y}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{\left(1+\kappa \right)}}{\int }_{\Gamma }\left[\right(G^{\prime }\left(z_s,z\right)t\left(z\right)-\overline{G^{\prime }\left(z_s,z\right)t\left(z\right)}\left)n\right(z_s) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left.\left({\kappa G}^{\prime }\left(z_s,z\right)t\left(z\right)-\left(z_s-z\right)\overline{G^{\prime }\left(z_s,z\right)t\left(z\right)}\right)\overline{n\left(z_s\right)}\right]d\Gamma , \end{gathered}$$

(11)

$$\begin{gathered} {\mathcal{F}}^u\left(z_s\right)={\mathcal{F}}_1^u\left(\mathit{y}\right)+\mathrm{i}{\mathcal{F}}_2^u\left(\mathit{y}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=-{\displaystyle \frac{2\mu }{\left(1+\kappa \right)}}{\int }_{\Gamma }\left\{\right(G^{\prime \prime }\left(z_s,z\right)n\left(z\right)u\left(z\right)+\overline{G^{\prime \prime }\left(z_s,z\right)n\left(z\right)u\left(z\right)}\left)n\right(z_s) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+[\overline{G^{\prime \prime }\left(z_s,z\right)}\left(n\left(z\right)\overline{u\left(z\right)}+\overline{n\left(z\right)}u\left(z\right)\right)+(z_s-z\left)\overline{G^{\prime \prime \prime }\left(z_s,z\right)}\overline{n\left(z\right)}\overline{u\left(z\right)}\right]\overline{n\left(z_s\right)}\}d\Gamma , \end{gathered}$$

(12)

where $z = x_1 +\mathrm{i}{x}_2, z_s = y_1 + \mathrm{i}{y}_2, t\left(z\right) = t_1 + \mathrm{i}{t}_2$, $u\left(z\right) = u_1 + \mathrm{i}{u}_2$, $n\left(z\right) = n_1 + \mathrm{i}{n}_2$, ${\mathrm{i} = \sqrt{-1}, \left(\right)}^{\prime } = \partial \left(\right)/\partial z_s$, $\overline{\left(\right)}$ indicates the complex conjugate.

$$\begin{gathered} \begin{array}{l}G\left(z_s,z\right)=-{\displaystyle \frac{1}{2\mathsf{\pi }}}\ln \left(z-z_s\right)\\ \begin{array}{c}{G}^{\prime }\left(z_s,z\right)={\displaystyle \frac{\partial G\left(z,z_s\right)}{\partial z_s}}={\displaystyle \frac{1}{2\mathsf{\pi }}}{\displaystyle \frac{1}{z-z_s}} \\ G^{\prime \prime }\left(z_s,z\right)={\displaystyle \frac{\partial G^{\prime }\left(z,z_s\right)}{\partial z_s}}={\displaystyle \frac{1}{2\mathsf{\pi }}}{\displaystyle \frac{1}{{\left(z-z_s\right)}^2}}\end{array}\end{array} \\ {G}^{\prime \prime \prime }\left(z_s,z\right)={\displaystyle \frac{\partial G^{\prime \prime }\left(z,z_s\right)}{\partial z_s}}={\displaystyle \frac{1}{2\mathsf{\pi }}}{\displaystyle \frac{2}{{\left(z-z_s\right)}^3}} \end{gathered}$$

(13)

The conventional elements are difficult to accurately describe with regard to the singular stresses near the V-notch or crack tip region for the existence of singularity. Therefore, a novel VAE has been developed to capture the high gradient variation of singular stress fields around the V-notch or crack tip region, and discontinuous linear elements (DLEs) are arranged on all other boundaries, as shown in Figure 3.

Figure 3. A novel variable-order asymptotic element.

The displacements and tractions on the VAE are assumed as follows:

$$\left\{\begin{array}{c}{u}_i={\displaystyle \sum _{k=1}^N}{\left({\displaystyle \frac{1+\vartheta \xi }{2}}\right)}^{d_k}{\tilde{u}}_i^{\left(k\right)}-{\displaystyle \frac{1+\vartheta \xi }{2}}{\displaystyle \sum _{k=1}^N}{\tilde{u}}_i^{\left(k\right)}+{\displaystyle \frac{1+\vartheta \xi }{2}}{u}_i^{N+1}\\ t_i={\displaystyle \sum _{k=1}^N}{\left({\displaystyle \frac{1+\vartheta \xi }{2}}\right)}^{s_k}{\tilde{t}}_i^{\left(k\right)}-{\displaystyle \frac{1+\vartheta \xi }{2}}{\displaystyle \sum _{k=1}^N}{\tilde{t}}_i^{\left(k\right)}+{\displaystyle \frac{1+\vartheta \xi }{2}}{t}_i^{N+1}\end{array}\right., i=\mathrm{1,2},$$

(14)

where $s_k$ and $d_k$ are the stress and displacement exponents [25], respectively, and ${\tilde{t}}_i^{\left(k\right)}$ and ${\tilde{u}}_i^{\left(k\right)}$ are the traction and displacement coefficients, respectively. If the V-notch/crack tip is at $\xi$ = −1.0, $\vartheta$ = 1.0. If the V-notch/crack tip is at $\xi$ = 1.0, $\vartheta$ = −1.0. $t_i^{N+1}$ and $u_i^{N+1}$ are the traction and displacement at the other end of the VAE away from the tip, respectively. This feature allows the VAE to be directly connected to conventional elements without any connective elements. The new VAE can be suited for arbitrary angles with corresponding $d_k$ and $s_k$.

In the fast multipole boundary element method for 2-D elastic problems, the boundary integrals are grouped into two categories through a quadtree structure: the far-field integrals on the elements that are far from the source point and the near-field integrals on the elements that are close to the source point. For the former, the fast multipole expansions are used for rapid evaluation. For the latter, direct evaluations are employed in the conventional boundary element method. More details can be found in refs. [26,27].

4. Numerical Results

4.1. A Rectangular Plate with an Internal Kinked Crack

A rectangular plate with an internal kinked crack under tension is considered, as illustrated in Figure 4. The slant angle of the crack is β and the length is 2c. The plate has a width of w, a height of 2h, $\sigma =10$, E = 21, and Poisson’s ratio ν = 0.35.

Figure 4. A rectangular plate with an internal kinked crack.

For the convenience of discussion, the normalized stress intensity factors $F_{\mathrm{I}}$, $F_{\mathrm{I}\mathrm{I}}$, and error are defined as follows:

$$F_{\mathrm{I}}={\displaystyle \frac{K_{\mathrm{I}}}{\sigma \sqrt{\pi c}}}$$

(15)

$$F_{\mathrm{I}\mathrm{I}}={\displaystyle \frac{K_{\mathrm{I}\mathrm{I}}}{\sigma \sqrt{\pi c}}}$$

(16)

$$\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}={\displaystyle \frac{\mathrm{a}\mathrm{b}\mathrm{s}(\mathrm{F}\mathrm{M}\mathrm{D}\mathrm{B}\mathrm{E}\mathrm{M}-\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c})}{\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c}}}\times 100\%$$

(17)

The FMDBEM-VAE denotes that the VAE is used at the crack tip, while the FMDBEM-DLE denotes that the conventional discontinuous linear element (DLE) is used at the crack tip. The normalized stress intensity factors $F_{\mathrm{I}}$ and $F_{\mathrm{I}\mathrm{I}}$obtained by the FMDBEM-VAE and FMDBEM-DLE are listed in Table 1 and Table 2, respectively. The total number of elements used by both the FMDBEM-VAE and FMDBEM-DLE are 66.

Table 1. Normalized stress intensity factors $F_{\mathrm{I}}$.

b/a	Analytic [9]	FMDBEM-VAE		FMDBEM-DLE
		Solution	Error	Solution	Error
0.05	0.7058	0.7140	0.0116	0.6570	0.0691
0.2	0.6532	0.6583	0.0078	0.6055	0.0730
0.6	0.6161	0.6198	0.0060	0.5906	0.0414
1	0.6054	0.6047	0.0012	0.5925	0.0213

Table 2. Normalized stress intensity factors $F_{\mathrm{I}\mathrm{I}}$.

b/a	Analytic [9]	FMDBEM-VAE		FMDBEM-DLE
		Solution	Error	Solution	Error
0.05	0.4184	0.4145	0.0093	0.3807	0.0901
0.2	0.4888	0.4842	0.0094	0.4442	0.0912
0.6	0.5491	0.5532	0.0075	0.5096	0.0719
1	0.5698	0.5653	0.0079	0.5447	0.0441

From Table 1 and Table 2, it can be observed that F_I and F_II obtained by the FMDBEM-VAE are in good agreement with the analytical solutions. In contrast, there is a notable discrepancy between the analytical solution and the results obtained by the FMDBEM-DLE. The conventional DLE struggles to capture the high gradient variation of the singular stress fields, while the VAE are capable of precisely describing the singularity.

Figure 5 and Figure 6 show F_I and F_II under different angles $\beta$, respectively. With the increase in $\beta$, F_I decreases, while F_II increases first, and then decreases. The mutation of F_II is nearly at $\beta ={60}^{°}$. Compared with the analytical solutions, the FMDBEM-VAE provides more accurate results than the FMDBEM-DLE.

Figure 5. The normalized stress intensity factor F_I under different $\beta$.

Figure 6. The normalized stress intensity factor F_II under different $\beta$.

Next, the convergence of the FMDBEM-VAE is discussed, as seen in Figure 7. From Figure 7, we can see that the error of the FMDBEM-VAE approaches a fixed value under different b/a, which shows the FMDBEM-VAE has good convergence properties.

Figure 7. The convergence of the FMDBEM-VAE.

4.2. A Rectangular Plate with Multiple Parallel Inclined Cracks

A rectangular plate with multi-parallel inclined cracks under tension is considered, as illustrated in Figure 8. The slant angle of the crack $\beta ={45}^{°}$ and the length 2a = 6 mm. The plate has a width of w = 50 mm and a height of l = 100 mm, the distance between the cracks b = 10 mm, ${\sigma }_y=10 \mathrm{M}\mathrm{P}\mathrm{a}$, E = 21 $\mathrm{G}\mathrm{P}\mathrm{a}$, ν = 0.35, ${\sigma }_x=B{\sigma }_y$, B = 1.0, 2.0, and n is the number of cracks.

Figure 8. Geometry of multi-parallel inclined cracks.

From Figure 9, it can be observed that when B is equal to 0.5, the F_I of n = 1 is larger than the ones of n = 3 and 5 when β < ${45}^{°}$, while the F_I of n = 1 is smaller than the ones of n = 3 and 5 when β$\ge {45}^{°}$. When B is equal to 1.0 or 2.0, the F_I of n = 1 is larger than the ones of n = 3 and 5 when β$\le {30}^{°}$, while the F_I of n = 1 is smaller than the ones of n = 3 and 5 when β > ${45}^{°}$. This is because when β is small, the cracks are primarily subjected to the vertical load (${\sigma }_y$), and their arrangement allows them to collectively support the vertical force. As a result, the greater the number of cracks, the lower the F_I.

Figure 9. The normalized stress intensity factor of multi-parallel inclined cracked structure.

With the increase in β, the cracks are primarily subjected to the horizontal load (${\sigma }_x=B{\sigma }_y$), but their arrangement does not allow them to cooperate in supporting the horizontal force and even weakens the structure. Thus, the greater the number of cracks, the higher the F_I.

In total, there are interactions between multiple cracks. When the number and the location of cracks change, the most unfavorable conditions of the structures also change.

4.3. A Rectangular Plate with Different Types of Defects

A rectangular plate with different types of defects under tension is considered, as illustrated in Figure 10. The length of the crack is 2 mm. The width and height of the plate is 40 mm × 40 mm, σ = 10 $\mathrm{M}\mathrm{P}\mathrm{a}$, Young’s modulus E = 21 $\mathrm{G}\mathrm{P}\mathrm{a}$, Poisson’s ratio ν = 0.35, fracture toughness is 100 N/mm^1.5, and ${∆a}_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.2 \mathrm{m}\mathrm{m}$. Different defects are distributed between cracks, and the area of the defects is equal.

Figure 10. Geometry of the rectangular plate with different types of defects.

The VAE is adaptable to V-notches with varying opening angles, necessitating only minor modifications to the stress exponents. The stress and displacement exponents differ for V-notches with various opening angles [25], as shown in Table 3.

Table 3. Variable-order asymptotic terms.

		Crack	Triangular Defect (60°)	Square Defect (90°)	Hexagonal Defect (120°)
sk	1	0	0	0	0
	2	−0.5	−0.4878	−0.4556	−0.3843
	3	0.5	0.4710	0.6293	0.8335
dk	1	0	0	0	0
	2	0.5	0.5122	0.5444	0.6157
	3	1.5	1.4710	1.6293	1.8335

Stress intensity factors play a fundamental role in crack growth processes. For each increment of the crack propagation, the stress intensity factors are evaluated. The effective stress intensity factor $K_{\mathrm{e}\mathrm{q}}$ [28] of each crack tip is obtained by

$$K_{\mathrm{e}\mathrm{q}}=K_{\mathrm{I}}{\mathrm{c}\mathrm{o}\mathrm{s}}^3\left({\displaystyle \frac{{\theta }_c}{2}}\right)-3K_{\mathrm{I}\mathrm{I}}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\left({\displaystyle \frac{{\theta }_c}{2}}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{{\theta }_c}{2}}\right).$$

(18)

where the crack initiation angle ${\theta }_c$ [28] can be determined based on the maximum circumferential stress criterion.

$$K_{\mathrm{I}}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_c+K_{\mathrm{I}\mathrm{I}}\left(3\cos {\theta }_c-1\right)=0,$$

(19)

$${\theta }_c=2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{K_{\mathrm{I}}-\sqrt{K_{\mathrm{I}}^2+8K_{\mathrm{I}\mathrm{I}}^2}}{4K_{\mathrm{I}\mathrm{I}}}}\right).$$

(20)

If the $K_{\mathrm{e}\mathrm{q}}$ exceeds the material toughness $K_{\mathrm{I}\mathrm{C}}$, the crack will propagate forward along the direction of ${\theta }_c$, resulting in the formation of a new multi-cracked structure. It should be noted that crack propagation does not occur when $K_{\mathrm{e}\mathrm{q}}<K_{\mathrm{I}\mathrm{C}}$, and the simulation procedures of multiple-crack propagation can be seen in ref. [27] in detail.

Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 show the K_I at node A~F under different propagation steps and different defects. The K_I gradually increases with the increase in the crack propagation steps, indicating that the structure is becoming weaker. With different defects, K_I exhibits the following phenomenon: triangle > rectangle > hexagon > circle. When the area of the defect is identical, the triangular defect is more likely to induce crack propagation, while the circular defect is more secure. The K_I for the hexagonal and circular cases are almost the same, so a hexagonal defect can be replaced by a circular defect for the analysis of complex engineering problems. Additionally, the crack propagation paths after 10 incremental steps are shown in Figure 11.

Table 4. The K_I of node A under different propagation steps (N/mm^1.5).

Step	Triangle	Rectangle	Hexagon	Circle
1	17.7397	17.5459	17.4917	17.4645
2	19.5476	19.3336	19.2722	19.2421
3	21.2185	20.9838	20.9143	20.8848
4	22.7875	22.5368	22.4578	22.4253
5	24.2907	24.0194	23.9309	23.8967
6	25.7445	25.4524	25.3532	25.3174
7	27.1575	26.8446	26.7391	26.7070
8	28.5666	28.2349	28.1188	28.0815
9	29.9818	29.6222	29.4965	29.4535
10	31.4113	31.0254	30.8881	30.8380

Table 5. The K_I of node B under different propagation steps (N/mm^1.5).

Step	Triangle	Rectangle	Hexagon	Circle
1	17.9160	17.6824	17.6155	17.5846
2	19.7887	19.5194	19.4410	19.4053
3	21.5343	21.2277	21.1370	21.0962
4	23.1922	22.8432	22.7392	22.6943
5	24.7936	24.3991	24.2766	24.2270
6	26.3610	25.9131	25.7737	25.7175
7	27.8817	27.3732	27.2264	27.1865
8	29.4479	28.8697	28.6950	28.6498
9	31.0435	30.3820	30.1730	30.1235
10	32.6891	31.9264	31.6816	31.6253

Table 6. The K_I of node C under different propagation steps (N/mm^1.5).

Step	Triangle	Rectangle	Hexagon	Circle
1	18.7003	18.3206	18.2467	18.1949
2	20.6381	20.2082	20.1221	20.0640
3	22.4441	21.9622	21.8623	21.7982
4	24.1582	23.6207	23.5051	23.4350
5	25.8110	25.2126	25.0793	25.0025
6	27.4277	26.7619	26.6078	26.5241
7	29.0309	28.2888	28.1101	28.0191
8	30.6422	29.8121	29.6048	29.5054
9	32.2827	31.3503	31.1083	30.9998
10	33.9742	32.9216	32.6376	32.5191

Table 7. The K_I of node D under different propagation steps (N/mm^1.5).

Step	Triangle	Rectangle	Hexagon	Circle
1	18.0266	17.6907	17.6432	17.6044
2	19.7882	19.4133	19.3624	19.3183
3	21.3859	20.9745	20.9193	20.8679
4	22.8637	22.4116	22.3479	22.2969
5	24.2400	23.7620	23.6803	23.6305
6	25.5477	25.0452	24.9473	24.8898
7	26.8000	26.2955	26.1748	26.1025
8	28.0427	27.4792	27.3606	27.2843
9	29.2744	28.6610	28.5350	28.4506
10	30.5122	29.8464	29.7130	29.6258

Table 8. The K_I of node E under different propagation steps (N/mm^1.5).

Step	Triangle	Rectangle	Hexagon	Circle
1	18.1501	17.8010	17.7386	17.6994
2	19.9566	19.5652	19.4910	19.4475
3	21.6080	21.1735	21.0860	21.0378
4	23.1444	22.6653	22.5622	22.5100
5	24.5953	24.0712	23.9493	23.8918
6	25.9884	25.4148	25.2730	25.2090
7	27.3484	26.7220	26.5545	26.4843
8	28.6958	28.0086	27.8131	27.7371
9	30.0538	29.2954	29.0691	28.9863
10	31.4451	30.6061	30.3426	30.2489

Table 9. The K_I of node F under different propagation steps (N/mm^1.5).

Step	Triangle	Rectangle	Hexagon	Circle
1	19.0531	18.5251	18.4664	18.4160
2	20.9559	20.3651	20.2979	20.2396
3	22.6986	22.0472	21.9693	21.9044
4	24.3244	23.6113	23.5210	23.4503
5	25.8655	25.0884	24.9837	24.9069
6	27.3493	26.5041	26.3824	26.2992
7	28.7994	27.8809	27.7385	27.6493
8	30.2398	29.2413	29.0729	28.9777
9	31.6926	30.6048	30.4056	30.3039
10	33.1819	31.9914	31.7560	31.6460

Figure 11. The crack propagation path after 10 increment steps.

5. Conclusions

The stress intensity factors of 2D elastic plates with multiple cracks are calculated in the present study. Some interesting conclusions can be drawn, as follow:

• The conventional discontinuous linear element struggles to capture the high gradient variation of the singular stress fields near the crack tip, while the novel variable-order asymptotic element can describe the singularity accurately.
• For the structure with multi-parallel inclined cracks, there are interactions between multiple cracks. When the crack distribution changes, the most unfavorable conditions of the structures also change.
• For cracks near different defects, the triangular defect is more likely to induce crack propagation, while the circular defect is more secure when the area of the defect is identical.

The present method can deal with the multiple cracks in a single domain accurately, more complicated cases such as random cracks and interface cracks in multi-material joints need to be considered carefully; this is our future work.