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The resonant modes of two-dimensional elastic elliptical objects are studiedfrom a modal formalism by emphasizing the role of the symmetries of theobjects. More precisely, as the symmetry is broken in the transition from thecircular disc to the elliptical one, the splitting up of resonances and levelcrossings are observed. From the mathematical point of view, this observationcan be explained by the broken invariance of the continuous symmetry group O(2)associated with the circular disc. The elliptical disc is however invariantunder the finite group C2v and the resonances are classified and associatedwith a given irreducible representation of this group. The main difficultyarises in the application of the group theory in elastodynamics where thevectorial formalism is used to express the physical quantities (elasticdisplacement and stress) involved in the boundary conditions. However, thismethod significantly simplifies the numerical treatment of the problem which isuncoupled over the four irreducible representations of C2v. This provides afull classification of the resonances. They are tagged and tracked as theeccentricity of the elliptical disc increases. Then, the splitting up ofresonances, which occurs in the transition from the circular disc to theelliptic one, is emphasized. The computation of displacement normal modes alsohighlights the mode splittings. A physical interpretation of resonances interms of geometrical paths is provided.

Elastodynamics and resonances in elliptical geometry