Partial wave basis adapted to exterior boundary conditions of an elastic plate

Horia Hulubei National Institute For Physics And Nuclear Engineering, Romania ,Sergiu Cojocaru

doi:10.53081/mjps.2021.20-1.02

Horia Hulubei National Institute For Physics And Nuclear Engineering, Romania , Sergiu Cojocaru

Open Access

https://doi.org/10.53081/mjps.2021.20-1.02

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Abstract

An approach to describing normal elastic vibration modes in confined systems is presented. In a standard treatment of the problem, the displacement field is represented by a superposition of partial waves of a general form, e.g., plane waves. The unknown coefficients of superposition are then obtained from the equation of motion and the full set of boundary conditions. In the proposed approach, the functional form of partial waves is chosen in such a way as to satisfy the boundary conditions on exterior surfaces identically, i.e., even if the unknown quantities determined by the remaining constraints are found in an approximation, numerically or analytically. Some examples of solutions for composite elastic plates are discussed to illustrate the efficiency of the approach and its relevance for applications.

Highlights

We consider an infinite composite plate with layers of homogeneous materials labeled with index and briefly outline the general formalism
The standard approach is to use a universal basis set of incoming and outgoing plane waves, which does not depend on details of the structure. It is well suited for treating systems embedded in an infinite medium, and the problem is mapped into one of a linear algebra for the expansion coefficients by applying the complete set of boundary conditions
The set of independent material parameters is 8-dimensional: the longitudinal and transverse sound velocities in the two media, respective mass densities and thicknesses of the layers. These systems are quite common in a broad range of applications; Sergiu Cojocaru experimental or numerical exploration of the full parametric space is hopeless in view of understanding and engineering of their properties

Summary

Introduction

We consider an infinite composite plate with layers of homogeneous materials labeled with index (e.g., layer thickness ) and briefly outline the general formalism [e.g., 1–5]. The stress–strain relation in a given material is as follows: where is the Kronecker symbol,. Double index implies summation, are the two Lamé stiffness parameters and the strain tensor at a given point r = (x1=x, x2=y, x3=z) and time t is defined by respective space derivatives of the displacement field components. Are represented by a superposition of normal modes propagating in-plane with the wave vector q||. The relevant components of the stress tensor are as follows: Sergiu Cojocaru (2). Are the bulk longitudinal and transverse sound velocities in the respective materials of mass density. Quantities and defined as contain phase velocities of normal modes ( index is dropped)

Basis Set Functions

Clamped Plate

Summary