General solution for inhomogeneous line inclusion with non-uniform eigenstrain

  • Lifeng MaEmail author
  • Yike Qiu
  • Yumei Zhang
  • Guang Li


The inhomogeneous line inclusion problem has various backgrounds in practical application such as graphene sheet-reinforced composites, and hydrogen embrittlement, grain boundary segregation in metallic materials. Due to the long-standing mathematical difficulty, there is no explicit analytical solution obtained except for the thin ellipsoidal inhomogeneity and rigid line inhomogeneity. In this paper, to find the deformation state due to the presence of such kind of elastic inhomogeneities, the inhomogeneous line inclusion problem is tackled in the framework of plane deformation. Firstly, the fundamental solution for a point-wise residual strain is presented and its deformation strain field is derived. By using Green’s function method, the homogeneous line inclusion problem with non-uniform eigenstrain is formulated and an Eshelby tensor-like line inclusion tensor is derived. From the line inclusion concept, the classical edge dislocation is revisited. Also, by virtue of this model, some elementary line homogenous inclusion problems are explored. Secondly, based on the homogeneous line inclusion solution, the inhomogeneous line inclusion problem is formulated using the equivalent eigenstrain principle, and its general solution is derived. Then, an inhomogeneous edge dislocation model is proposed and its analytical solution is presented. Furthermore, to demonstrate the application of the proposed inhomogeneous line inclusion model, a typical thin inclusion under remote load is studied. This study provides a general solution for inhomogeneous thin inclusion problems. The models and their solutions introduced here will also find application in the mechanics of composites analysis, heterogeneous material modeling, etc.


Inhomogeneous line inclusion Line inclusion tensor The equivalent eigenstrain principle Inhomogeneous edge dislocation Non-uniform eigenstrain distribution 



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


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.S&V Lab, Department of Engineering MechanicsXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Materials Science and Engineering, State Key Laboratory of Advanced Processing and Recycling of Non-ferrous MetalsLanzhou University of TechnologyLanzhouChina

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