Analytical solution for a vibrating rigid sphere with an elastic shell in an infinite linear elastic medium

💡 Research Summary

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The paper presents an exact analytical solution for a three‑dimensional dynamic elasticity problem in which a rigid sphere of radius a oscillates harmonically along the z‑axis while being surrounded by a concentric elastic shell (outer radius b) that is itself embedded in an infinite elastic medium. The core does not change its volume; its motion is prescribed as a displacement u₀ with amplitude U₀ and angular frequency ω. The authors start from the Navier equation for linear elasticity in the frequency domain, decompose the displacement field into a curl‑free (longitudinal) part u_L and a divergence‑free (transverse) part u_T, and show that each satisfies a Helmholtz equation with wave numbers k_L = ω/c_L and k_T = ω/c_T, where c_L and c_T are the longitudinal and shear wave speeds of the material.

Exploiting spherical symmetry, the displacement is expressed in terms of two radial scalar functions φ(r) (potential) and h(r) (rotational), leading to the compact form u = ∇