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

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.

Automated summary

The main purposes of this paper are to analytically derive the general solution for an orthotropic elastic half plane with a crack or a notch, derive the Riemann-Hilbert equation as the analytical method, solve the problem using two methods, and derive general expressions of the stress intensity factor for a crack problem. It is confirmed that the stress functions obtained by the Riemann-Hilbert equation and Cauchy integral method are the same, and can be expressed by any elastic constants for calculating Mode I and II SIFs.