Non-circular nano-inclusions with interface effects that achieve uniform internal strain fields in an elastic plane under anti-plane shear

Abstract

This paper verifies the existence of a single non-circular nano-inclusion with interface effect that achieves a uniform internal strain field in an elastic plane under uniform remote anti-plane shear loadings. The uniform strain field inside such a non-circular inclusion is prescribed via perturbations of the uniform strain field inside the analogous circular inclusion, and the unknown (non-circular) inclusion shape is characterized by a conformal mapping whose unknown coefficients are determined by a system of nonlinear equations. Numerical examples show various shapes of non-circular nano-inclusions with interface effects that achieve uniform internal strain fields. It is found that such a non-circular inclusion shape depends on the inclusion size and the specific uniform internal strain field. In particular, for given interface shear modulus and residual tension, it is shown that the minimum inclusion size required to guarantee the existence of such a non-circular inclusion usually increases as the shear modulus of the inclusion approaches that of the matrix. Moreover, the relationship between the shear traction jump across the interface and the curvature of the interface is discussed in detail for such non-circular inclusions.

A circular inhomogeneity incorporating surface/interface strain gradient elasticity

“…Recently, Dai and Gao and Dai et al showed that certain non-circular inhomogeneities (with conventional interface effects only, no gradient effects) could also achieve uniform internal stress field under uniform remote anti-plane shear loadings [26,27]. It is interesting to probe whether non-circular inhomogeneities with uniform internal stress fields still exist when interface gradient effects are taken into account.…”

Design of periodic harmonic holes with surface tension in plane deformations

“…to calculate the Jacobian matrix in the corresponding Newton-Raphson iteration. For detailed implementation of the Newton-Raphson iteration, one can refer to [13,14]. Once the finite coefficients a j (j = 1.N) are determined, it is easy to obtain the actual shape of the periodic harmonic holes in physical plane from the corresponding (truncated) mapping (8).…”

Harmonic holes with surface tension in an elastic plane under uniform remote loading

“…In particular, we use σ = e i θ ( 0 ≤ θ ≤ 2 π ) to denote the unit circle (where | ξ | = 1 ) in the ξ -plane that corresponds to the curve L in the z -plane. Using the mapping of equation (5), we can describe the term e − i α on the right-hand side of equation (4) by [14]…”

“…In such cases, the contribution of surface or interface effects [12, 13] (usually manifested as surface or interface tension or surface or interface elasticity) around internal nanosized holes or inclusions are known to play an important role in the determination of the corresponding stress distributions. The resulting inverse problems for the identification of optimal shapes in nanostructures or nanocomposites then become extremely challenging, perhaps explaining why so few investigations can be found in this area (see, for example, Dai and Gao [14]). In this paper, we make progress in this direction by incorporating the contribution of surface or interface effects and subsequently developing an efficient method for the construction of harmonic nanosize holes in an elastic plane subjected to uniform remote loading.…”