THE EFFECT OF PRE-STRESS ON THE VIBRATION AND STABILITY OF ELASTIC PLATES

Plane incremental vibrations superimposed on the pure homogeneous deformation of a rectangular block of incompressible isotropic elastic material are studied in detail. Frequency equations, which determine the frequencies of symmetric and antisymmetric modes of vibration in terms of the underlying deformation and stress and of the in-plane aspect ratio of the block, are obtained in respect of a general form of strain-energy function. Different cases, dependent on the properties of the strain-energy function and on the deformation and stress, are enumerated; results analogous to those found in the (compressible) linear theory are recovered for comparison. The case of zero frequency is of special interest since the frequency equations then become bifurcation equations. Each bifurcation equation determines a set of values of the deformation and stress at which, on a path from the undeformed stress-free configuration, stability of the underlying deformed configuration is lost and bifurcation into a (quasi-static) non-homogeneous incremental mode of deformation becomes possible. Results of Sawyers and Rivlin (1974) and Ogden (1984) emerge as special cases of those obtained here. For particular forms of strain-energy function calculations have been carried out to illustrate how the frequency depends on the deformation, stress and aspect ratio.