Archive of Applied Mechanics

, Volume 88, Issue 4, pp 613–627 | Cite as

Complex variable formulation for a rigid line inclusion interacting with a generalized singularity

  • Lifeng Ma
  • Biao Wang
  • Alexander M. Korsunsky


Analytical solution for a rigid line inclusion embedded in an infinite-extended plane under non-uniform loading is still a challenging problem in inclusion mechanics, which has both theoretical and applied significance in material engineering. In this paper, by directly employing Kolosov–Muskhelishvili stress potentials, a rigid line inclusion interacting with a generalized singularity is addressed in the framework of plane deformation, with the help of the superposition principle. It should be pointed out that the generalized singularity in this study can represent a point force, an edge dislocation, a point moment, a point nucleus of strain, and even remote uniform load, etc. The solutions can be used as kernel functions for integral equation formulations of rigid line inclusion–substrate system models using the Green’s function method. With this framework, stress field and stress intensity factors at the line inclusion ends are analyzed. The application of the solutions is demonstrated with two simple examples: (i) the rigid line inclusion under remote loading is studied, and it is strictly confirmed with rigorous proof that the remote shear load will not arouse stress concentration; (ii) a rigid line inclusion interacting with a dislocation is investigated, and the full solution is given. These examples also partially validate the general solution derived this study.


Rigid line inclusion Generalized singularity Green’s function method Plane deformation 



This work is partially supported by National Natural Science Foundation of China (Grant No. 41630634).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.S&V Lab, Department of Engineering MechanicsXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Engineering ScienceUniversity of OxfordOxfordUK

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