Asymptotically Correct Analytical Model for Flexural Response of a Two-Layer Strip with Contrast Elastic Constants

Abstract

The paper is concerned with the asymptotically consistent models for a two-layer beam-strip with high-contrast elastic properties. In the general case, the strip is acted upon by dynamic body forces varying in the thickness direction, and by variable surface forces. The two dimensionless parameters, the thickness-towavelength ratio and the ratio of moduli for soft and stiff layers, are taken as small parameters with the first one being assumed as the main small parameter. A solution of the 2D boundary-value problem is sought in the form of an asymptotic series in the main small parameter. Step-by-step integration of the 2D equations in the thickness direction results in a sequence of relations for stresses and displacements in the form of polynomials in the transverse coordinates with coefficients depending on time and the axial coordinate. The novel asymptotically consistent 1D differential equations, corresponding to the Bernoulli-Euler and Timoshenko-Reissner models, governing vibration (or static deformation) of a beam-strip are derived. As an example, free vibrations of a simply-supported two-layer beam consisting of the soft and stiff layers are analysed relying on both models at different correlation between the geometrical and physical parameters of layers.