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

Plane incremental vibrations superimposed on the pure homogeneous defo rmation of a rectangular block of incompressible isotropic elastic mat erial 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 r atio 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 enumerat ed; results analogous to those found in the (compressible) linear theo ry are recovered for comparison. The case of zero frequency is of spec ial interest since the frequency equations then become bifurcation equ ations. Each bifurcation equation determines a set of values of the de formation and stress at which, on a path from the undeformed stress-fr ee configuration, stability of the underlying deformed configuration i s lost and bifurcation into a (quasi-static) non-homogeneous increment al mode of deformation becomes possible. Results of Sawyers and Rivlin (1974) and Ogden (1984) emerge as special cases of those obtained her e. For particular forms of strain-energy function calculations have be en carried out to illustrate how the frequency depends on the deformat ion, stress and aspect ratio.