The J and M integrals for a cylindrical cavity in a time-harmonic wave field

The invariant integrals are being widely used in the study of defects and fracture mechanics, mostly in elastostatics. However, the properties and the interpretation of these integrals in elastodynamics, especially in the case of time-harmonic excitation, remain unexplored. This contribution is focused on the derivation of the time average J integral for a cylindrical inhomogeneity and the M integral for a cylindrical cavity placed in a monochromatic plane elastic wave. It is shown in the context of antiplane linear elasticity that the J integral, or the material force acting on the inhomogeneity, resembles the radiation pressure force exerted on a dielectric cylinder by the normally incident electromagnetic wave. Based on the existing solution of this electrodynamic problem and the corresponding acoustic problem, the J integral is expressed as a function of the nondimensional wave number in the form of the partial wave expansion of the scattering theory. Employing the same classical method as with the J integral, the closed-form solution of the time average M integral for a traction-free cavity is also obtained as a function of the nondimensional wave number. The M integral, i.e., the expansion or scalar moment per unit length on an infinitely long circular cavity, is represented in terms of the scattering phase shifts as in the case of the J integral. Alternative expressions that are more convenient for numerical calculations for the cavity are also derived for both integrals. These calculations are carried out for the J and M integrals in a wide spectrum of frequencies. Asymptotic approximations of both integrals for low and high frequencies are presented. The long wavelength approximation, including the monopole and dipole contributions, has been provided for the J integral in the form of a simple analytical expression. The value of M integral in the vanishing frequency limit is also presented. In the opposite short wavelength limit, the corresponding asymptotic values are derived for both integrals. These solutions, which are valid for the empty cavity, are extended to the case of an inviscid fluid-filled cavity. The obtained results have a variety of engineering and geophysical implications. In particular, their applications to the flaw characterization of materials and further development of non-destructive evaluation techniques are briefly discussed in light of the frequency relationships derived for the J and M integrals.