The J and M Integrals for a Cylindrical Cavity in a Time-Harmonic Wave Field

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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, have remained unexplored. Their study has a variety of engineering and geophysical implications, in particular, for the further development of non-destructive evaluation techniques. This contribution is focused on the derivation of the time average J integral for a cylindrical inhomogeneity and M integral for a cylindrical cavity placed in a monochromatic plane elastic wave of arbitrary wavelength. 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 for the J integral, the closed-form solution for 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 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. Rather different expressions for the cavity are also derived for both integrals, which can be used more conveniently for numerical calculations, and these calculations are carried out for 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 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 inviscid fluid-filled cavity. The obtained results can be used in the area of non-destructive evaluation for the flaw characterization by ultrasonic scattering methods. The derived frequency dependence of the J and M integral can be related to the measurable far-field scattering amplitudes. This relationship is relevant to the inverse-scattering approach, which can be applied to the characterization of materials in an attempt to infer geometrical characteristics of flow structures.