Analytical solution for the stress field of hierarchical defects: multiscale framework and applications

Hierarchical defects are defined as adjacent defects at different length scales. Involved are the two scales where the stress field distribution is interrelated. Based on the complex variable method and conformal mapping, a multiscale framework for solving the problems of hierarchical defects is formulated. The separated representations of mapping function, the governing equations of potentials, and the stress field are subsequently obtained. The proposed multiscale framework can be used to solve a variety of simplified engineering problems. The case in point is the analytical solution of a macroscopic elliptic hole with a microscopic circular edge defect. The results indicate that the microscopic defect aggregates the stress concentration on the macroscopic defect and likely leads to global propagation and rupture. Multiple micro-defects have interactive effects on the distribution of the stress field. The level of stress concentration may be reduced by the coalescence of micro-defects. This work provides a unified method to analytically investigate the influence of edge micro-defects within the scope of multiscale hierarchy. The formulated multiscale approach can also be potentially applied to materials with hierarchical defects, such as additive manufacturing and bio-inspired materials.

Interaction between a generalized screw dislocation in the matrix and an inhomogeneity containing an elliptic hole in piezoelectric–piezomagnetic composite materials

The paper deals with the interaction of a generalized screw dislocation and an elliptic inhomogeneity containing a confocal elliptic hole in a magneto-electro-elastic composite material. Exact solutions are derived for the case where the generalized screw dislocation is located in the matrix under a remote anti-plane shear stress field, an in-plane electric field, and a magnetic field. Based on the complex variable method, the complex potentials of both the matrix and the inhomogeneity are obtained in series, and analytic expressions for the generalized stress and strain field, the image force, the generalized stress intensity factor of the blunt crack tip, and the energy release rate are derived explicitly. The presented solutions include some previous solutions, such as pure elastic, piezoelectric, piezomagnetic, and circular inclusions. Typical numerical examples are presented and the influences of the dislocation position, the volume of inhomogeneity, and the elliptic hole on these physical quantities are discussed. The results show that the magneto-electro-elastic coupling effect has a great influence on the image force and the equilibrium position of dislocation, especially when the dislocation approaches the interface; the coupling effect makes the image force on the screw dislocation follow different variation laws in piezoelectric–piezomagnetic composite materials compared with elastic materials.