Modelling and analysis of the dynamic behavior of inhomogeneous continuum containing a circular inclusion

Based on the complex function theory, an universal approach of solving the dynamic stress concentration around a circular inclusion in two-dimensional (2D) inhomogeneous medium is presented. The inhomogeneous Helmholtz equation with variable coefficient is converted to the standard Helmholtz equation by using the general conformal transformation technique analytically. An inhomogeneity with the density varying as a smooth function of two spatial coordinates and the constant elastic modulus is established to verify the accuracy of analytical results. As a typical example, an exponential variation of the density is introduced for analyzing the dynamic stress concentration factor (DSCF) around the inclusion. Numerical results show the efficiency of the method and the effects of the medium inhomogeneity, reference wave number of the background medium, the reference wave number and shear modulus ratios between the background medium and inclusion.