Spherical nano-inhomogeneity with the Steigmann–Ogden interface model under general uniform far-field stress loading

Abstract

An explicit solution, considering the interface bending resistance as described by the Steigmann–Ogden interface model, is derived for the problem of a spherical nano-inhomogeneity (nanoscale void/inclusion) embedded in an infinite linear-elastic matrix under a general uniform far-field-stress (including tensile and shear stresses). The Papkovich-Neuber (P-N) general solutions, which are expressed in terms of spherical harmonics, are used to derive the analytical solution. A superposition technique is used to overcome the mathematical complexity brought on by the assumed interfacial residual stress in the Steigmann-Ogden interface model. Numerical examples show that the stress field, considering the interface bending resistance as with the Steigmann–Ogden interface model, differs significantly from that considering only the interface stretching resistance as with the Gurtin–Murdoch interface model. In addition to the size-dependency, another interesting phenomenon is observed: some stress components are invariant to interface bending stiffness parameters along a certain circle in the inclusion/matrix. Moreover, a characteristic line for the interface bending stiffness parameters is presented, near which the stress concentration becomes quite severe. Finally, the derived analytical solution with the Steigmann–Ogden interface model is provided in the supplemental MATLAB code, which can be easily executed, and used as a benchmark for semi-analytical solutions and numerical solutions in future studies.