Interacting stress of multiple random elliptical holes under remote and arbitrary surface stresses

Abstract

Study of multiple elliptical hole problem is of great significance for optimization design and safety assessment in both geotechnical engineering and perforated materials concerned with elliptical holes. Currently, the existing kind of literature mainly focus on calculating the interacting stress of multiple elliptical holes under the remote stresses, less on that under the surface stresses. In this study, based on the derived fundamental solutions, the integral equations and superposition technique, a new analytical–numerical method is established to calculate the interacting stresses of an infinite elastic plane containing multiple elliptical holes with any size and orientation, under both remote stresses and arbitrarily distributed surface stresses applied on the elliptical holes. The new method is proved to be valid by existing methods (including the conformal mapping method, Laurent series method, complex function method, boundary element alternating method, and finite element method). Some computational examples of two elliptical holes (i.e., horizontal and oriented elliptical holes) under biaxial stresses at infinity (${\sigma }_x^{\infty }$=${\sigma }_y^{\infty }$) are given to analyze the influencing factors on the interacting tangential stresses at the elliptical boundaries. Research results show that the oriented elliptical hole usually has an amplifying effect on the horizontal one, while the horizontal elliptical hole has an amplifying or shielding effect on the oriented elliptical one, mainly depending on the orientation angle (α₂) of the oriented one. When the major axes of two elliptical holes are vertical to their center connecting-line, the plate with two elliptical holes has smaller stress concentration than that only with one elliptical hole, which is helpful for improving the strength of underground engineering and perforated materials. The proposed method has not only simple formulation (without singularity), high accuracy (due to the exact fundamental solution), and wider applicability (under both remote stresses and arbitrarily distributed surface stresses) than the common conformal mapping method, Laurent series method, complex function method and boundary element alternating method, but also the potential to be further developed for the isotropic/anisotropic problem of multiple arbitrarily shaped holes under the same complex stresses.

Keywords:

• Oval holes
• fundamental solutions
• complex variable
• superposition technique
• integral equations

Additional information

Funding

The authors gratefully acknowledge the financial supports by National Natural Science Foundation of China under Grant numbers 52078495 and 51874351 and Excellent Postdoctoral Innovative Talents Project of Hunan Province (grant no. 2020RC2001).