Numerical Solution of Integro-Differential Equations Modelling the Dynamic Behavior of a Nano-Cracked Viscoelastic Half-Plane

The scattering of time-harmonic waves by a finite, blunt nano-crack in a graded, viscoelastic bulk material with a free surface is considered in this work. Non-classical boundary conditions and a localized constitutive equation at the interface between crack and matrix, following the Gurtin-Murdoch surface elasticity theory are introduced. An efficient numerical technique is developed using integro-differential equations along the nano-crack line that is based on an analytically derived Green‘s function for the quadratically inhomogeneous half-plane. The dependence of the diffracted and scattered waves and of the local stress concentration fields on key problem parameters such as viscosity, inhomogeneity, surface elasticity, and interaction between the nano-crack and the free surface are all examined through an extensive parametric study.