Volume Integral Equation Method (VIEM)

A number of analytical techniques are available for the stress analysis of inclusion problems when the geometries of inclusions are simple (e.g., cylindrical, spherical or ellipsoidal) and when they are well separated [9, 41, 52]. However, these approaches cannot be applied to more general problems where the inclusions are anisotropic and arbitrary in shape, particularly when their concentration is high. Thus, stress analysis of heterogeneous solids or analysis of elastic wave scattering problems in heterogeneous solids often requires the use of numerical techniques based on either the finite element method (FEM) or the boundary integral equation method (BIEM). However, these methods become problematic when dealing with elastostatic problems or elastic wave scattering problems in unbounded media containing anisotropic and/or heterogeneous inclusions of arbitrary shapes. It has been demonstrated that the volume integral equation method (VIEM) can overcome such difficulties in solving a large class of inclusion problems [6,10,20,21,28–30]. One advantage of the VIEM over the BIEM is that it does not require the use of Green’s functions for anisotropic inclusions. Since the elastodynamic Green’s functions for anisotropic media are extremely difficult to calculate, the VIEM offers a clear advantage over the BIEM. In addition, the VIEM is not sensitive to the geometry or concentration of the inclusions. Moreover, in contrast to the finite element method, where the full domain needs to be discretized, the VIEM requires discretization of the inclusions only.

Analysis of wave scattering in pipes with non-axisymmetric and inclined angle defects using finite element modeling

In this paper, elastic wave scattering in hollow pipes with non-axisymmetric and inclined angle defects is studied using finite element (FE) simulations. A comb array transducer is employed in the FE code to excite the pipe in its first longitudinal mode using a 10-cycle sine modulated excitation signal at 120[Formula: see text]kHz central frequency. Defects with variations in geometrical shapes such as depths, axial and circumferential lengths, and inclined angles are investigated to provide detailed analysis of wave propagation patterns and mode conversions in a 12-in diameter pipe. The influence of each geometrical parameter and also possible newborn modes is studied both in time and wavenumber-frequency domain via circumferential order identification approach and dispersion curves. Results show that the depth of a non-axisymmetric circumferential defect has the minimum influence on the propagating mode while crack’s width can influence the measured longitudinal mode in a sinusoidal pattern which is a function of excitation signal’s wavelength. Further, the propagating mode can exhibit higher contribution of either axisymmetric or non-axisymmetric modes based on the reflection patterns, depending on its angle and axial length.

Static elastic cloaking, low-frequency elastic wave transparency and neutral inclusions

New connections between static elastic cloaking, low-frequency elastic wave scattering and neutral inclusions (NIs) are established in the context of two-dimensional elasticity. A cylindrical core surrounded by a cylindrical shell is embedded in a uniform elastic matrix. Given the core and matrix properties, we answer the questions of how to select the shell material such that (i) it acts as a static elastic cloak, and (ii) it eliminates low-frequency scattering of incident elastic waves. It is shown that static cloaking (i) requires an anisotropic shell, whereas scattering reduction (ii) can be satisfied more simply with isotropic materials. Implicit solutions for the shell material are obtained by considering the core–shell composite cylinder as a neutral elastic inclusion. Two types of NI are distinguished, weak and strong with the former equivalent to low-frequency transparency and the classical Christensen and Lo generalized self-consistent result for in-plane shear from 1979. Our introduction of the strong NI is an important extension of this result in that we show that standard anisotropic shells can act as perfect static cloaks, contrasting previous work that has employed ‘unphysical’ materials. The relationships between low-frequency transparency, static cloaking and NIs provide the material designer with options for achieving elastic cloaking in the quasi-static limit.