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

• Published in: IMA journal of applied mathematics 2001-08, Vol.66 (4), p.357-385
• Main Authors: Pichugin, Aleksey V., Rogerson, Graham A.
• Format: Article
• Language: English
• Subjects: asymptotic analysis; Classical and quantum physics: mechanics and fields; Classical mechanics of continuous media: general mathematical aspects; elasticity; Elasticity, plasticity: general mathematical aspects; Exact sciences and technology; Physics; pre; stress; waves
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/66.4.357

A two‐dimensional model for extensional motion of a pre‐stressed incompressible elastic layer near its cut‐off frequencies is derived. Leading‐order solutions for displacement and pressure are obtained in terms of the long wave amplitude by direct asymptotic integration. A governing equation, together with corrections for displacement and pressure, is derived from the second‐order problem. A novel feature of this (two‐dimensional) hyperbolic governing equation is that, for certain pre‐stressed states, time and one of the two (in‐plane) 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 governing equation from the third‐order problem. Asymptotic consistency, in the sense that the dispersion relation associated with the two‐dimensional model concurs with the appropriate order expansion of the three‐dimensional relation 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.