The area contraction and expansion for a nano-void under four different kinds of loading

摘要

This paper provides an explanation for the authors' recent conclusions as why the M-integral could be negative for a nano-void (hole) in elastic materials under tensile, bi-axes tensile, and bi-axes tensile-compress loadings (Hui T, Chen YH, in ASME J Appl Mech 77:021019-1-9, 2009; 024505-1-5, 2009). Attention is focused on whether the area of the nano-void under each of four different kinds of loading is contractive when compared to the original geometrical area without loading. The four kinds of loading are as follows: (1) simple tensile loading; (2) bi-axes tensile loading; (3) bi-axes tensile-compression loading; and (4) purely shear loading. It is concluded that, unlike those for a macro/micro hole, the area of the nano-void under relatively lower amplitude of the loadings (1) and (2) is always smaller than the original area of the same void before loading. This reduction in the nano-void area is induced from the surface effect including the surface tension and the surface Lame constants along the nano-void rim. Thus, these two kinds of loading can either decrease or increase the nano-void area depending on the amplitude of the loading. Under a relatively lower tensile loading, the area contraction occurs but not always corresponding to the negative value of the M-integral, whereas under a relatively higher tensile loading, the area expansion occurs but not always corresponding to the positive value of the A/-integral. This again verifies that the positive value of the M-integral does not always correspond to the energy release due to the nano-hole expansion, rather, the area contraction might yield the energy release either. Under bi-axes tensile compression or purely shear loading, this feature disappears.