General solution for inhomogeneous line inclusion with non-uniform eigenstrain
- 27 Downloads
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.
KeywordsInhomogeneous 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).
- 1.Aguilar, S., Tabares, R., Serna, C.: Microstructural transformations of dissimilar austenite–ferrite stainless steels welded joints. J. Mater. Phys. Chem. 1, 65–68 (2013)Google Scholar
- 4.Ballarini, R.: An integral-equation approach for rigid line inhomogeneity problems. Int. J. Fract. 33, R23–R26 (1987)Google Scholar
- 8.Christensen, R.M.: Mechanics of Composite Materials. Wiley, New York (1979)Google Scholar
- 9.Claussen, N., Ruehle, M., Heuer, A. H. (eds.): Advances in ceramics, Vol. 12, p. 352. Science and technology of zirconia II. The American Ceramic Society, Columbus, OH (1984)Google Scholar
- 18.Herbig, M., Kuzmina, M., Haase, C., Marceau, R.K.W., Gutierrez-Urrutia, I., Haley, D., Molodov, D.A., Choi, P., Raabe, D.: Grain boundary segregation in Fe–Mn–C twinning-induced plasticity steels studied by correlative electron backscatter diffraction and atom probe tomography. Acta Mater. 83, 37–47 (2015)CrossRefGoogle Scholar
- 19.Hickel, T., Nazarov, R., McEniry, E.J., Leyson, G., Grabowski, B., Neugebauer, J.: Ab Initio based understanding of the segregation and diffusion mechanisms of hydrogen in steels. J. Met. 66, 1399–1405 (2014)Google Scholar
- 21.Hull, D., Bacon, D.J.: Introduction to Dislocations, 5th edn. Elsevier, Kidlington (2011)Google Scholar
- 23.Inglis, C.E.: Stresses in a plate due to the presence of cracks and sharp corners. Inst. Naval Archit. Lond. 55, 219–230 (1913)Google Scholar
- 39.Muskhelishvili, N.I.: Some Problems of Mathematical Theory of Elasticity (English translation from the third Russian edition). Noordhoff Ltd., Groningen (1953)Google Scholar