Stress analysis for an orthotropic elastic half plane with an oblique edge crack and stress intensity factors

Abstract

The main purposes of the present paper are as follows: (1) to analytically derive the general solution (stress functions) for an orthotropic elastic half plane with a crack or a notch; (2) to derive the Riemann–Hilbert equation as the analytical method; (3) to solve the present problem using two methods, one is a Cauchy integral method and other is a Riemann–Hilbert method; and (4) to derive the general expressions of the stress intensity factor (SIF) for a crack problem. The stress functions obtained by the Riemann–Hilbert equation are compared with those obtained by Cauchy integral method. Then, it is confirmed that the same stress functions can be obtained. The stress functions are expressed by any elastic constants. Therefore, Mode I and II SIFs can be calculated for any elastic constants. Some examples are shown. It is stated from the results of the examples that the negative Mode I SIFs exist for certain oblique edge crack angles and elastic constants. Because the derived stress functions are expressed using a mapping function, other geometrical shapes can be analyzed by changing the mapping function.

Appendix 1

The method to calculate stress intensity factors (SIFs) of Modes I and II is stated. The crack in the physical plane shown in Fig. 4a is first considered. The small distance “r” on the extended crack line at the crack tip is considered. The crack on the x-axis must be considered. The crack in the physical plane is shown in Fig. 4b. The crack in the analytical plane is shown in Fig. 5b.

The coordinate $\sigma_{10}$ at the crack tip C₁ on the unit circle on the $z_{1}\hbox{-}$plane is

$$\sigma_{10} = {{(1 - 2q_{1} + i)} \mathord{\left/ {\vphantom {{(1 - 2q_{1} + i)} {(1 - 2q_{1} - i)}}} \right. \kern-0pt} {(1 - 2q_{1} - i)}}$$

(66)

which is derived from the first derivative

$$\omega^{\prime}_{1} (\sigma_{10} ) = 0.$$

(67)

The crack surfaces are rotated on the x₁-axis (see Fig. 5), and the following equation using Eqs. (45), (53) and (66) is derived:

$$k_{1} = \mathop {\lim }\limits_{\begin{subarray}{l} r \to 0 \\ \theta = 0 \end{subarray} } \sqrt {2r} \sigma_{y} = \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \sqrt {2\left\{ {\omega_{1} (\zeta_{1} ) - \omega_{1} (\sigma_{10} )} \right\}e^{{iq_{1} \pi }} } 2\text{Re} \left[ {\frac{{\varPhi^{\prime}_{0} (\zeta_{1} )}}{{\omega^{\prime}_{1} (\zeta_{1} )}} + \frac{{\varPsi^{\prime}_{0} (\zeta_{2} )}}{{\omega^{\prime}_{2} (\zeta_{2} )}}} \right].$$

(68)

The first term in Re[] of the above equation is considered:

$$\begin{aligned} & \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \sqrt {2\left\{ {\omega_{1} (\zeta_{1} ) - \omega_{1} (\sigma_{10} )} \right\}} \frac{{\varPhi_{0}^{{\prime }} (\zeta_{1} )}}{{\omega_{1}^{{\prime }} (\zeta_{1} )}} = \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \left[ {2\frac{{\omega_{1} (\zeta_{1} ) - \omega_{1} (\sigma_{10} )}}{{\left\{ {\omega_{1}^{{\prime }} (\zeta_{1} )} \right\}^{2} }}} \right]^{1/2} \varPhi_{0}^{{\prime }} (\zeta_{1} ) \\ & \quad = \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \left[ {2\frac{{\omega_{1}^{{\prime }} (\zeta_{1} )}}{{2\omega_{1}^{{\prime }} (\zeta_{1} )\omega_{1}^{{\prime \prime }} (\zeta_{1} )}}} \right]^{1/2} \varPhi_{0}^{{\prime }} (\zeta_{1} ) = \left[ {\frac{1}{{\omega_{1}^{{\prime \prime }} (\sigma_{10} )}}} \right]^{1/2} \varPhi_{0}^{{\prime }} (\sigma_{10} ) = \frac{{\varPhi_{0}^{{\prime }} (\sigma_{10} )}}{{\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} }}, \\ \end{aligned}$$

(69)

and the second term (see Eq. (52))

$$\begin{aligned} & \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \sqrt {2\left\{ {\omega_{1} (\zeta_{1} ) - \omega_{1} (\sigma_{10} )} \right\}} \frac{{\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}}}{{\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\omega_{1}^{{\prime }} (\zeta_{1} ) + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\overline{{\omega_{1}^{{\prime }} (\zeta_{1} )}} \frac{{\bar{\zeta }_{1} }}{{\zeta_{1} }}}} \\ & \quad = \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \left[ {2\frac{{\omega_{1} (\zeta_{1} ) - \omega_{1} (\sigma_{10} )}}{{\left\{ {\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\omega_{1}^{{\prime }} (\zeta_{1} ) + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\overline{{\omega_{1}^{{\prime }} (\zeta_{1} )}} \frac{{\bar{\zeta }_{1} }}{{\zeta_{1} }}} \right\}^{2} }}} \right]^{1/2} \left[ {\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}} \right]_{{\zeta_{1} = \sigma_{10} }} \\ & \quad = \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \left[ {2\frac{{\omega_{1}^{{\prime }} (\zeta_{1} )}}{{2\left\{ {\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\omega_{1}^{{\prime }} (\zeta_{1} ) + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\overline{{\omega^{\prime}_{1} (\zeta_{1} )}} \frac{{\bar{\zeta }_{1} }}{{\zeta_{1} }}} \right\}\left\{ {\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\omega_{1}^{{\prime \prime }} (\zeta_{1} ) + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\frac{{\overline{{\partial \omega_{1}^{{\prime }} (\zeta_{1} )}} }}{{\partial \bar{\zeta }_{1} }}\frac{{\partial \bar{\zeta }_{1} }}{{\partial \zeta_{1} }}\frac{{\partial \bar{\zeta }_{1} }}{{\partial \zeta_{1} }}} \right\}}}} \right]^{1/2} \left[ {\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}} \right]_{{\zeta_{1} = \sigma_{10} }} \\ & \quad = \left[ {\frac{{\omega_{1}^{{\prime \prime }} (\zeta_{1} )}}{{\left\{ {\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\omega_{1}^{{\prime \prime }} (\zeta_{1} ) + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\frac{{\overline{{\partial \omega_{1}^{{\prime }} (\zeta_{1} )}} }}{{\partial \bar{\zeta }_{1} }}\frac{{\partial \bar{\zeta }_{1} }}{{\partial \zeta_{1} }}\frac{{\partial \bar{\zeta }_{1} }}{{\partial \zeta_{1} }}} \right\}^{2} }}} \right]_{{\zeta_{1} = \sigma_{10} }}^{1/2} \left[ {\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}} \right]_{{\zeta_{1} = \sigma_{10} }} \\ & \quad = \frac{{\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} \left[ {\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}} \right]_{{\zeta_{1} = \sigma_{10} }} }}{{\left\{ {\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\omega_{1}^{{\prime \prime }} (\zeta_{1} ) + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\overline{{\omega_{1}^{{\prime \prime }} (\zeta_{1} )}} \frac{{\partial \bar{\zeta }_{1} }}{{\partial \zeta_{1} }}\frac{{\partial \bar{\zeta }_{1} }}{{\partial \zeta_{1} }}} \right\}_{{\zeta_{1} = \sigma_{10} }} }} \\ & \quad = \frac{{\left[ {\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}} \right]_{{\zeta_{1} = \sigma_{10} }} }}{{\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\frac{{\overline{{\omega_{1}^{{\prime \prime }} (\sigma_{10} )}} }}{{\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} }}\left( { - \frac{{\left| {\sigma_{10} } \right|^{2} }}{{\sigma_{10}^{2} }}} \right)^{2} }}. \\ \end{aligned}$$

(70)

Therefore,

$$k_{1} = 2\sqrt {e^{{iq_{1} \pi }} } \text{Re} \left[ {\frac{{\varPhi^{\prime}_{0} (\sigma_{10} )}}{{\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} }} + \frac{{\left[ {\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}} \right]_{{\zeta_{1} = \sigma_{10} }} }}{{\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\frac{{\overline{{\omega_{1}^{{\prime \prime }} (\sigma_{10} )}} }}{{\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} }}\left( { - \frac{{\left| {\sigma_{10} } \right|^{2} }}{{\sigma_{10}^{2} }}} \right)^{2} }}} \right].$$

(71)

Similarly, using $\tau_{xy}$ in Eq. (63), the equation for k₂ is derived:

$$\begin{aligned} k_{2} & = \mathop {\lim }\limits_{\begin{subarray}{l} r \to 0 \\ \theta = 0 \end{subarray} } \sqrt {2r} \tau_{xy} = \mathop {\lim }\limits_{\begin{subarray}{l} \zeta_{1} \to \sigma_{10} \\ \theta = 0 \end{subarray} } \sqrt {2\left\{ {\omega_{1} (\zeta_{1} ) - \omega_{1} (\sigma_{10} )} \right\}e^{{iq_{1} \pi }} } ( - 2)\text{Re} \left[ {s_{1} \frac{{\varPhi^{\prime}_{0} (\zeta_{1} )}}{{\omega^{\prime}_{1} (\zeta_{1} )}} + s_{2} \frac{{\varPsi^{\prime}_{0} (\zeta_{2} )}}{{\omega^{\prime}_{2} (\zeta_{2} )}}} \right] \\ & = - 2\sqrt {e^{{iq_{1} \pi }} } \text{Re} \left[ {s_{1} \frac{{\varPhi_{0}^{{\prime }} (\sigma_{10} )}}{{\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} }} + s_{2} \frac{{\left[ {\frac{{\partial \varPsi_{0} (\zeta_{2} )}}{{\partial \zeta_{1} }}} \right]_{{\zeta_{1} = \sigma_{10} }} }}{{\frac{{s_{2} - \bar{s}_{1} }}{{s_{1} - \bar{s}_{1} }}\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} + \frac{{s_{1} - s_{2} }}{{s_{1} - \bar{s}_{1} }}\frac{{\overline{{\omega_{1}^{{\prime \prime }} (\sigma_{10} )}} }}{{\sqrt {\omega_{1}^{{\prime \prime }} (\sigma_{10} )} }}\left( { - \frac{{\left| {\sigma_{10} } \right|^{2} }}{{\sigma_{10}^{2} }}} \right)^{2} }}} \right]. \\ \end{aligned}$$

(72)

The values k₁ and k₂ are calculated by Eqs. (71) and (72).