Cut-off frequencies and correction factors of equivalent single layer theories

Shear and normal correction factors are used within equivalent single-layer first-order theories for homogeneous and layered structures, e.g. first-order shear deformation plate/shell theories, in order to improve the description of the transverse strains. Dynamic correction factors may be derived, following the original treatment for homogeneous plates (Mindlin, R.D., J. Appl. Phys. 1951), by imposing that certain wave propagation frequencies, e.g. the cut-off frequencies of the lowest cut-off frequency modes, match those of three-dimensional elasticity, which are typically obtained through computational procedures. In this paper we consider multilayered plates with principal material directions parallel to the geometrical axes and propose the use of a multiscale structural theory which allows the accurate closed-form derivation of the cut-off frequencies of the first thickness modes associated to plane-strain Rayleigh-Lamb waves. The theory captures the effects of the inhomogeneities on local fields and global behavior through homogenized equilibrium equations which depend on a limited number of variables independent of the number of layers. Explicit expressions are obtained for the lowest cut-off frequencies of multilayered wide plates with an arbitrary number of layers, with arbitrary layup and elastic constants, which are then used to define the dynamic correction factors of classical equivalent single-layer theories. The comparison with predictions obtained by matching exact elasticity solutions for highly-inhomogeneous bilayer media highlights the accuracy and potentials of the proposed approach.