The $M$-Integral Analysis for a Nano-Inclusion in Plane Elastic Materials Under Uni-Axial or Bi-Axial Loadings

This paper deals with the $M$-integral analysis for a nano-inclusion in plane elastic materials under uni-axial or bi-axial loadings. Based on previous works (Gurtin and Murdoch, 1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57, pp. 291–323; Mogilevskaya, et al., 2008, “Multiple Interacting Circular Nano-Inhomogeneities With Surface/Interface Effects,” J. Mech. Phys. Solids, 56, pp. 2298–2327), the surface effect induced from the surface tension and the surface Lamé constants is taken into account, and an analytical solution is obtained. Four kinds of inclusions including soft inclusion, hard inclusion, void, and rigid inclusions are considered. The variable tendencies of the $M$-integral for each of four nano-inclusions against the loading or against the inclusion radius are plotted and discussed in detail. It is found that in nanoscale the surface parameters for the hard inclusion or rigid inclusion have a little or little influence on the $M$-integral, and the values of the $M$-integral are always negative as they would be in macroscale, whereas the surface parameters for the soft inclusion or void yield significant influence on the $M$-integral and the values of the $M$-integral could be either positive or negative depending on the loading levels and the surface parameters. Of great interest is that there is a neutral loading point for the soft inclusion or void, at which the $M$-integral transforms from a negative value to a positive value, and that the bi-axial loading yields similar variable tendencies of the $M$-integral as those under the uni-axial tension loading. Moreover, the bi-axial tension loading increases the neutral loading point, whereas the bi-axial tension-compression loading decreases it. Particularly, the magnitude of the negative $M$-integral representing the energy absorbing of the soft inclusion or void increases very sharply as the radius of the soft inclusion or void decreases from 5 nm to 1 nm.