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Eshelby formalism for nano-inhomogeneities. (English) Zbl 1370.74068
Summary: The Eshelby formalism for inclusion/inhomogeneity problems is extended to the nano-scale at which surface/interface effects become important. The interior and exterior Eshelby tensors for a spherical inhomogeneous inclusion with the interface stress effect subjected to an arbitrary uniform eigenstrain embedded in an infinite alien matrix, and the stress concentration tensors for a spherical inhomogeneity subjected to an arbitrary remote uniform stress field are obtained. Unlike their counterparts at the macro-scale, the Eshelby and stress concentration tensors are, in general, not uniform inside the inhomogeneity but are position-dependent. They have the property of radial transverse isotropy. It is also shown that the size-dependence of the Eshelby tensors and the stress concentration tensors follow very simple scaling laws. Finally, the Eshelby formula to calculate the strain energy in the presence of the interface effect is given.

MSC:
74G70 Stress concentrations, singularities in solid mechanics
74B05 Classical linear elasticity
74E05 Inhomogeneity in solid mechanics
74M25 Micromechanics of solids
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