Stress Fields Around Nanosized Holes Or Inclusions Of Arbitrary Shape

Stress analysis of inhomogeneities(including holes and inclusions) is of great importance in solid mechanics. Surface/interface tension imposed on the hole surface or the matrix-inclusion interface,which is usually ignored for macro-scale materials, will play an important role for micro-scale and nano-scale materials due to the high ratio of surface to bulk. Therefore, this paper focuses on the hole/inclusion problem with surface/interface tension effect taken into account.Elastic field around a hole/inclusion with surface/interface tension is examined based on the complex variable method. The shape of the hole/inclusion is characterized by a conformal mapping which maps the exterior of the hole/inclusion onto the exterior of the unit circle in the image plane.With the aid of the conformal mapping, the complex potentials of the matrix and inclusion are expressed in a series form with unknown coefficients to be determined by Fourier expansion method.Detailed numerical results are shown for ellipse, triangle, square, rectangle and so on. A basic conclusion is that the stress field around the hole/inclusion increases remarkably with decreasing hole/inclusion size and changes rapidly nearby the point of maximum curvature. In addition, several conclusions can be drawn as below: for holes, it is shown that maximum hoop stress usually appears not at the point of maximum curvature. Conversely, it is found from most of the numerical examples that a local minimum hoop stress generally appears at the point of maximum curvature. Moreover, it is found that this local minimum hoop stress for holes of certain shapes turns out to be the global minimum hoop stress. For inclusions, with increasing hardness of the inclusion, it is found that the hoop and normal stresses of the inclusion at the interface increase, while those of the matrix at the interface decrease. Particularly, it is found that the shear stresses at the interface for the case of the matrix and inclusion with identical shear modulus are much larger than those for the case of the matrix and inclusion with different shear moduli.