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Accurate loading analyses of curved cracks under mixed-mode conditions applying the J-integral

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Abstract

In linear elastic fracture mechanics the path-independent J-integral is a loading quantity equivalent to stress intensity factors (SIF) or the energy release rate. Concerning plane crack problems, \(J_k\) is a 2-dimensional vector with its components \(J_1\) and \(J_2\). These two parameters can be related to the mode-I and mode-II SIFs \(K_{\mathrm{I}}\) and \(K_{\mathrm{II}}\). To guarantee path-independence for curved crack geometries, an integration path along the crack faces must be considered. This paper deals with problems occurring at the numerical calculation of the J-integral in connection with the FE-method. Two new methods for accurately calculating values of \(J_2\) for arbitrary cracks are presented.

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Correspondence to Paul O. Judt.

Appendix

Appendix

Angular trigonometric functions of the n-th eigenfunction according to Eq. (3), (Kuna 2010):

$$\begin{aligned} M_{11}^{\left( n\right) }&= \frac{n}{2}\left\{ \left[ 2+\left( -1\right) ^n+\frac{n}{2}\right] \cos \left( \frac{n}{2}-1\right) \varphi \right. \nonumber \\&\left. -\left( \frac{n}{2}-1\right) \cos \left( \frac{n}{2}-3\right) \varphi \right\} \end{aligned}$$
(33a)
$$\begin{aligned} N_{11}^{\left( n\right) }&= \frac{n}{2}\left\{ \left[ -2+\left( -1\right) ^n-\frac{n}{2}\right] \sin \left( \frac{n}{2}-1\right) \varphi \right. \nonumber \\&\left. +\left( \frac{n}{2}-1\right) \sin \left( \frac{n}{2}-3\right) \varphi \right\} \end{aligned}$$
(33b)
$$\begin{aligned} M_{22}^{\left( n\right) }&= \frac{n}{2}\left\{ \left[ 2-\left( -1\right) ^n-\frac{n}{2}\right] \cos \left( \frac{n}{2}-1\right) \varphi \right. \nonumber \\&\left. +\left( \frac{n}{2}-1\right) \cos \left( \frac{n}{2}-3\right) \varphi \right\} \end{aligned}$$
(33c)
$$\begin{aligned} N_{22}^{\left( n\right) }&= \frac{n}{2}\left\{ \left[ -2-\left( -1\right) ^n+\frac{n}{2}\right] \sin \left( \frac{n}{2}-1\right) \varphi \right. \nonumber \\&\left. -\left( \frac{n}{2}-1\right) \sin \left( \frac{n}{2}-3\right) \varphi \right\} \end{aligned}$$
(33d)
$$\begin{aligned} M_{12}^{\left( n\right) }&= \frac{n}{2}\left\{ -\left[ \frac{n}{2}+\left( -1\right) ^n\right] \sin \left( \frac{n}{2}-1\right) \varphi \right. \nonumber \\&\left. +\left( \frac{n}{2}-1\right) \sin \left( \frac{n}{2}-3\right) \varphi \right\} \end{aligned}$$
(33e)
$$\begin{aligned} N_{12}^{\left( n\right) }&= \frac{n}{2}\left\{ -\left[ \frac{n}{2}-\left( -1\right) ^n\right] \cos \left( \frac{n}{2}-1\right) \varphi \right. \nonumber \\&\left. +\left( \frac{n}{2}-1\right) \cos \left( \frac{n}{2}-3\right) \varphi \right\} \end{aligned}$$
(33f)

Angular trigonometric functions of the n-th eigenfunction according to Eq. (3), (Kuna 2010):

$$\begin{aligned} F_1^{\left( n\right) }&\!=\!&\left[ \kappa \!+\!\left( -\!1\right) ^n\!+\!\frac{n}{2}\right] \cos \frac{n}{2} \varphi \!-\!\frac{n}{2}\cos \left( \frac{n}{2}\!-\!2\right) \varphi \nonumber \\\end{aligned}$$
(34a)
$$\begin{aligned} G_1^{\left( n\right) }&\!=\!&\left[ \!-\kappa +\left( \!-1\right) ^n\!-\! \frac{n}{2}\right] \sin \frac{n}{2}\varphi \!+\!\frac{n}{2} \sin \left( \frac{n}{2}\!-\!2\right) \varphi \nonumber \\\end{aligned}$$
(34b)
$$\begin{aligned} F_2^{\left( n\right) }&\!=\!&\left[ \kappa -\left( -\!1\right) ^n\!-\!\frac{n}{2}\right] \sin \frac{n}{2}\varphi \!+\!\frac{n}{2}\sin \left( \frac{n}{2}-2\right) \varphi \nonumber \\\end{aligned}$$
(34c)
$$\begin{aligned} G_2^{\left( n\right) }&= \left[ \kappa +\left( -1\right) ^n-\frac{n}{2}\right] \cos \frac{n}{2}\varphi +\frac{n}{2}\cos \left( \frac{n}{2}-2\right) \varphi \nonumber \\ \end{aligned}$$
(34d)

Complex potentials for the Griffith crack problem under mode-I and mode-II loading with the spatial coordinate \(z=x_1+i\,x_2\), (Kuna 2010):

$$\begin{aligned} \text{ mode-I: }\quad \Phi \left( z\right)&= -\frac{\sigma _{22}^\infty }{4}z+\frac{\sigma _{22}^\infty }{2}\left[ \sqrt{z^2-a^2}\right] \nonumber \\ \quad \Psi ^{\prime }\left( z\right)&= \frac{\sigma _{22}^\infty }{2}z-\frac{\sigma _{22}^\infty }{2}\frac{a^2}{\sqrt{z^2-a^2}}\nonumber \\ \text{ mode-II: }\quad \Phi \left( z\right)&= i\frac{\tau _{12}^\infty }{4}z-i\frac{\tau _{12}^\infty }{2}\left[ \sqrt{z^2-a^2}\right] \nonumber \\ \quad \Psi ^{\prime }\left( z\right)&= i\frac{\tau _{12}^\infty }{2}\frac{2z^2-a^2}{\sqrt{z^2-a^2}} \end{aligned}$$
(35)

Kolosov’s equations, (Kuna 2010):

$$\begin{aligned} \sigma _{11}+\sigma _{22}&=2\left[ \Phi ^{\prime }\left( z\right) +\overline{\Phi ^{\prime }\left( z\right) }\right] \nonumber \\ \sigma _{22}-\sigma _{11}+2i\tau _{12}&=2\left[ \bar{z}\,\Phi ^{\prime \prime }\left( z\right) +\Psi ^{\prime \prime }\left( z\right) \right] \nonumber \\ 2\mu \left( u_1+iu_2\right)&=\kappa \,\Phi \left( z\right) -z\,\overline{\Phi ^{\prime }\left( z\right) }-\overline{\Psi ^{\prime }\left( z\right) } \end{aligned}$$
(36)

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Judt, P.O., Ricoeur, A. Accurate loading analyses of curved cracks under mixed-mode conditions applying the J-integral. Int J Fract 182, 53–66 (2013). https://doi.org/10.1007/s10704-013-9857-9

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