A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

A two-dimensional model for extensional motion of a pre-stressedincompressible elastic layer near its cut-off frequencies isderived. Leading-order solutions for displacement and pressureare obtained in terms of the long wave amplitude by direct asymptoticintegration. A governing equation, together with correctionsfor displacement and pressure, is derived from the second-orderproblem. A novel feature of this (two-dimensional) hyperbolicgoverning equation is that, for certain pre-stressed states,time and one of the two (in-plane) spatial variables can changeroles. Although whenever this phenomenon occurs the equationstill remains hyperbolic, it is clearly not wave-like. The second-ordersolution is completed by deriving a refined governing equationfrom the third-order problem. Asymptotic consistency, in thesense that the dispersion relation associated with the two-dimensionalmodel concurs with the appropriate order expansion of the three-dimensionalrelation at each order, is verified. The model has particularapplication to stationary thickness vibration of, or transientresponse to high frequency shock loading in, thin walled bodies.