Harmonic solutions of micropolar elastodynamics

The equations of micropolar elastodynamics are considered for an unbounded continuum subjected to a body force and a body couple. These act harmonically with the same real frequency , but with individual arbitrary spatial distributions. Over a harmonic state, the displacement and microrotation are related to two radiation conditioned harmonic vectors, each acquiring three eigenvalue contributions, assuming a noncritical -frequency. Altogether, four distinct eigenvalues are admissible. If ²<2²₀, ₀ being a frequency parameter of the continuum, two of these are real while two are purely imaginary. But if ²<2²₀, then all admissible eigenvalues are real. Each eigenvalue contribution resolves into a series of Hankel and Bessel functions coupled to Hankel type transforms of: (i) spherical integrals which, in turn, can be expanded via spherical harmonics for the 3-dimensional problem, (ii) circular integrals for the 2-dimensional problem. Axisymmetric and spherically symmetric results are deduced in 3-dimensions. Asymptotic solutions are also established; they disclose long-range formation of radially attenuated spherical (or circular) waves propagating with, generally, anisotropic amplitudes but, invariably, isotropic eikonals.If, in the absence of a body couple, a body force acts radially in 3-dimensions with a spherically symmetric strength, then the elastic displacement behaves likewise while the microrotation vanishes identically. Another application is made to a 2-dimensional problem for a 1 × 3 source system of body force plus body couple without longitudinal variation but with magnitudes symmetric about a longitudinal axis.As approaches a certain critical frequency , dependent solely on the continuum, at least two eigenvalues approach the same value. The phenomenon is explored for a continuum consistent with ²<2²₀ and under the hypothesis ²<2²₀. All admissible eigenvalues are then real throughout an -neighbourhood of . Here, two associated eigenvalue contributions behave singularly. Nevertheless, their essential singularities cancel out within the relevant combination. Examination of a far-field suggests that critical frequency attainment sets off a slow instability in the 2-dimensional configuration. In the 3-dimensional configuration, however, it preserves stability and eliminates radial attenuation; an exact solution is formulated for this case.