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-stressed incompress ible elastic layer near its cut-off frequencies is derived. Leading-order s olutions for displacement and pressure are obtained in terms of the long wa ve amplitude by direct asymptotic integration. A governing equation, togeth er with corrections for displacement and pressure, is derived from the seco nd-order problem. A novel feature of this (two-dimensional) hyperbolic gove rning equation is that, for certain pre-stressed states, time and one of th e two (inplane) spatial variables can change roles. Although whenever this phenomenon occurs the equation still remains hyperbolic, it is clearly not wave-like. The second-order solution is completed by deriving a refined gov erning equation from the third-order problem. Asymptotic consistency, in th e sense that the dispersion relation associated with the two-dimensional mo del concurs with the appropriate order expansion of the three-dimensional r elation at each order, is verified. The model has particular application to stationary thickness vibration of, or transient response to high frequency shock loading in, thin walled bodies.